The economics of spaceship Earth

Rob Hart

c© Draft date October 26, 2016

rob.hart@slu.se

Contents

Chapter 1. Introduction 11.1. Economic growth on spaceship Earth 11.2. Questions regarding economic growth on spaceship Earth 51.3. Economic methodology 61.4. This book 7

Part 1. The driving forces of economic growth 11

Chapter 2. Growth fundamentals 132.1. Production, GDP, and growth 132.2. What drives growth? Reasoning from first principles 142.3. Growth drivers in the real economy 16

Chapter 3. Labour, capital, and neoclassical growth: The Solow and Ramsey models 173.1. The Solow model with constant technology 173.2. Solow with exogenous technological progress 213.3. The Ramsey model 21

Chapter 4. Endogenous growth theory 234.1. Ideas and long-run growth 234.2. A simple model of endogenous growth 234.3. The effect of endogenizing growth 27

Part 2. Natural resources and neoclassical growth 29

Chapter 5. Neoclassical growth and nonrenewable resources 315.1. Land 315.2. An abundant resource, costly to extract 325.3. A limited resource, costless to extract (Hotelling/DHSS) 33

Chapter 6. Pre-industrial human development 416.1. Population dynamics 416.2. A simple model of Malthusian growth 42

Chapter 7. Resource stocks 457.1. The elephant in Hotelling’s room 457.2. An extraction model with inhomogeneous stocks 487.3. Concluding discussion on resource stocks in the very long run 56

Chapter 8. Renewable natural resources and pollution as inputs 578.1. Renewables 578.2. Pollution 59

Part 3. Demand for natural resources and energy 61

Chapter 9. Directed technological change and resource efficiency 639.1. Resource efficiency in the production function 639.2. A simple model of directed technological change 659.3. Evidence regarding the DTC model 699.4. Links between knowledge stocks 71

Chapter 10. Structural change and energy demand 7310.1. Introduction to structural change 7310.2. Preliminary analysis 75

iii

10.3. Structural change without income effects 7810.4. A model of endogenously expanding variety 8410.5. Evidence for income effects in energy demand 88

Part 4. Substitution between resource inputs, and pollution 91

Chapter 11. Substitution between alternative resource inputs 9311.1. A simple model with alternative resource inputs 9311.2. Technological change 9611.3. An alternative to lock-in: The Fundamentalist economy 98

Chapter 12. The EKC and substitution between alternative inputs 10112.1. Introduction 10112.2. The model 10412.3. The elements of the model 10712.4. Case studies of pollution reduction 11212.5. Conclusions 115

Chapter 13. Production, pollution and land 117

Part 5. Policy for sustainable development 119

Chapter 14. Is unsustainability sustainable? 12114.1. Lessons from historical adaptation 12114.2. Financial and other crises 12114.3. Uncertainty and future crises 121

Chapter 15. Growthandenvironment 12315.1. Taxation contra Command and Control 12315.2. Precaution 12315.3. Lessons from observations of structural change 123

Chapter 16. The no-growth paradigm 12516.1. Consumerism 12516.2. ‘Green’ consumerism 12516.3. Conspicuous consumption, labour, and leisure 125

Chapter 17. Mathematical appendix 12917.1. Growth rates 12917.2. Continuous and discrete time 13017.3. A continuum of firms 131

Bibliography 133

CHAPTER 1

Introduction

1.1. Economic growth on spaceship Earth

The metaphor of Earth as a spaceship—made famous by Kenneth Boulding (1966)—helps tobring home the obvious truth that we live together on a planetwith finite resources. Furthermore,inputs from beyond the Earth—such as the flow of energy from the sun—are limited, and thereare enormous obstacles to resource extraction from other planets, let alone their colonization. Thenecessity of living together on the finite surface of the Earth suggests the need for cooperationand fairness, as emphasized by Orwell in ‘The road to Wigan Pier’.1 On the other hand, Bouldingfocuses on the necessity of the careful use of the finite natural resources available to us, and theavoidance of fouling our own nest with pollution. In this book we follow Boulding by focusing onresource use and pollution rather than cooperation and fairness. Boulding describes the transitionin the human imagination from the idea of the frontier economy in which scarcity is only everlocal, to the idea of global limits. We aim to understand boththe ‘frontier’ and ‘spaceship’ phasesof development as part of a single long-run process; even when we have the mindset of the frontier,we are already on the spaceship.

This book thus concerns the development of the global—spaceship—economy in the verylong run. How can we make sense of the changes in the human economy that we have seen ofthe last couple of centuries, and even the last 20000 years? In the light of our explanation of thepast, what future scenarios are possible, or indeed likely,regarding long-run global productionof goods and services, given the finite nature of the Earth, its natural resources, and the inflowof energy from the sun? Furthermore, what policies are called for in order to achieve desirablelong-run outcomes with regard to (sustainable) long-run production and the quality of the livingenvironment? Over the last 100 years and more the world has witnessed economic growth whichis not only uniquely rapid, but also astonishingly steady, by which I mean that the increase inproduction of goods and services per capita in the richest countries has occurred at a remarkablyconstant rate. Can this increase be maintained, and if it is maintained, will that be at the expenseof the quality of the Earth’s environment or other species?

To get some perspective on these questions we need data, observations. We begin by pre-senting data regarding long-run growth in GDP per capita, and also growth in population. Wedo this for two periods: pre-industrial (up to 1700) and industrial (after 1800). Regarding thepre-industrial period, Figure 1.1 shows that production per capita was remarkably constant fromprehistoric times through to around 1700.2 Global population, on the other hand, increased veryslowly up to around 5000 BCE, from which point it grew at a rather constant exponential rate(note the logarithmic scale) up to 1700 CE. Because we have used a (natural) log scale, the slopesof the line tell us growth rates directly: thus we can read off that the growth rate of populationbetween 5000 BC and 1700 was approximately 4.5/6700, i.e. less than 0.1 percent per year. (Formore on the mathematics behind growth rates and the use of thelogarithmic scale see the Mathe-matical appendix, Chapter 17.1.) The explanation of these two trends is straightforward and wasfirst put forward by Thomas Malthus (1798); we return to it in Chapter 6.

After 1800 things look very, very, different, as we see in Figure 1.2. During the 19th centuryboth global product and population grow at around 0.5 percent per year, whereas in the 20thcentury they grow faster: almost 2 percent per year for average global product per capita, and 1.5percent per year for population. Note that the growth rate intotal global product is the sum of thegrowth rates in population and global product per capita, and was therefore slightly over 3 percentper year during the 20th century.

1‘The world is a raft sailing through space with, potentially, plenty of provisions for everybody; the idea that we mustall co-operate and see to it that everyone does his fair shareof the work and gets his fair share of the provisions seems so bla-tantly obvious that one would say that no one could possibly fail to accept it unless he had some corrupt motive for clingingto the present system.’ Orwell (1958), p.203. Available athttps://archive.org/details/roadtowiganpie00orwe.

2Note however that the data on which the figure is based could becalled into question.

1

2 1.1. Economic growth on spaceship Earth

−20000 −10000 0 1700

0

1

2

3

4

5

year

Average yearly production per capita

Global population

Lo

gsc

ale,

no

rmal

ized

Figure 1.1. Global product and population, historical (data from Brad DeLong).Both variables are normalized to start at zero. Population grows by a factor ofapproximately exp[5], i.e. about 150. Average yearly production per capita isclose to 100 USD throughout.

1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000

0

0.5

1

1.5

2

2.5

Average global product per capita, normalized

Global population, normalized

Lo

gsc

ale

Figure 1.2. Global product and population, modern (data from Maddison(2010) and US Census Bureau).

We now turn to resource consumption rates, and prices, as illustrated in Figure 1.3.3 In Figure1.3(a) we show data for an aggregate of the 17 most important metals measured by factor expen-diture (not including uranium). The results are striking: over a period of more than 100 years,growth in global consumption of metals almost exactly tracks growth in global product, whereasthe real price is almost exactly constant. The result is thatthe share of metals in global productis also constant, since expenditures (the product of price and quantity) track global product. InFigure 1.3(b) we see the same result for global primary energy supply through combustion. Again,quantity tracks global product while the weighted energy price is almost constant. After a slight

3Note: Global product data from Maddison (2010). Metals: Al, Cr, Cu, Au, Fe, Pb, Mg, Mn, Hg, Ni, Pt, Rare earths,Ag, Sn, W, Zn. All metals data from Kelly and Matos (2012). Energy: Coal, oil, natural gas, and biofuel. Fossil quantitydata from Boden et al. (2012). Oil price data from BP (2012). Coal and gas price data from Fouquet (2011); note thatthese data are only for average prices in England; we make the(heroic) assumption that weighted average global pricesare similar. Biofuel quantity data from Maddison (2003). Biofuel price data from Fouquet (2011); again, we assume thatthe data are representative for global prices, and we extrapolate from the end of Fouquet’s series to the present assumingconstant prices. The older price data is gathered from historical records and is not constructed based on assumptions aboutelasticities of demand or similar. Sensitivity analysis shows that the assumptions are not critical in driving the results.

Chapter 1. Introduction 3

decline in the early 19th century, the factor share of energyremains almost constant, although inthe short run it is tightly linked to price changes.4

1900 1920 1940 1960 1980 2000−1

−0.5

0

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Global productExpenditurePriceQuantity

1850 1900 1950 2000−1

0

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Global productExpenditurePriceQuantity

Nat

ura

llo

gsc

ale

Figure 1.3. Long-run growth in total consumption and prices, compared togrowth in total global product, for (a) Metals, and (b) Primary energy fromcombustion.

Finally we look at data from two specific pollutants, sulphuremissions in the United King-dom, and global CFCs: Figure 1.4. Sulphur has regional effects, therefore we look at emissionsfrom a specific country; CFCs have global effects, hence we look at global emissions. In bothcases the figures show very rapidly rising emissions up to thepoint at which regulatory action istaken against the pollutant. For clarity we show the date on which the first major regulation wasintroduced; further (stricter) regulations followed thereafter. The natural interpretation of the datais that the increasing trend for emissions was rapidly turned around through regulation.

The data in Figures 1.1–1.3 suggest two questions regardingthe long-run development of thehuman population and its economy. Firstly, how and when willthe exponential growth in popula-tion be stopped? Secondly, how and when will the exponentialgrowth in resource and energy usebe stopped? These questions arise since, on a finite planet, it is clear that such exponential growthcannot continue indefinitely. (Note that there is no physical necessity for economic growth perse to slow down or stop in the long run, since global product isultimately measured in monetary,not physical, units; there is no logically necessary connection between (a) the value of goods andservices produced per year and (b) any physical quantity.) Figure 1.4 leads to many more ques-tions: what causes the rise and then fall of polluting emissions in these cases; is this a generalpattern, and if so are we heading for a cleaner future or will new pollutants constantly appear?More generally, when will the disruption of ecosystems through pollution and overexploitationstop? Will it stop at all?

Regarding the first question, about population growth, opinions differ but the broad outlinesof the answer are clear: population growth will slow down andstop over the next few decades, aslong as economic security, health care, education, and women’s rights continue to spread acrossall people of the world; we thus pay relatively little attention to this question in the book. Theanalysis we do present is to be found in Chapter 6 on Malthus, population growth, and economicgrowth.

Regarding the second question, it is one of the key questionstackled in this book. It bearsrepeating: how and when will the exponential growth in resource and energy use be stopped?To get a handle on the time frame within which the observed growth must slow down or stopwe look first at resource extraction, and then at fossil energy. Regarding resource extraction,consider the following calculation. The current physical extraction rate of minerals is of the orderof 1010 tonnes per year globally (this can be calculated using data from Kelly and Matos, 2012).

4Note that in Figure 1.3 we show log-normalized data to emphasize relative growth rates, but what is thelevelof thefactor share of resources compared to labour and capital? Based on the above data sources, the factor share of resources issignificant but not large. For instance, the factor share of crude oil in the global economy in 2008 was 3.6 percent, whereasthe factor share of the 17 major metals was just 0.7 percent. Note also that what we present as factor costs do not all go tothe factor in a general equilibrium sense: payments to capital and labour make up a large part of the costs of mineral andenergy resources.

4 1.1. Economic growth on spaceship Earth

1850 1900 1950 2000−2

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1950 1960 1970 1980 1990 2000−4.5

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GDP

Glob.prod.

Sulphur

Clean air act

CFC

Montreal protocol

Nat

ura

llo

gsc

ale

Figure 1.4. UK Sulphur emissions compared to total UK GDP, and global CFCproduction (CFC11+CFC12) compared to total global product. Sulphur: bothnormalized to zero in 1956, the date of introduction of the first of a long se-ries of regulations restricting emissions. CFCs: both normalized to zero in1987, the date of signing of the Montreal protocol. Data: Maddison (2010)(GDP), Stern (2005) (Sulphur), AFEAS (CFCs). AFEAS data downloaded fromhttp://www.afeas.org/data.php, 9 Nov. 2014. Two anomalous points in the sul-phur data have been altered.

Assume that this rate continues to grow at the same rate as it has done over the past 100 years,i.e. approximately 3 percent per year. Then extraction would be multiplied by a factor 20 eachcentury, and in 700 years we would be mining and using minerals roughly equal to the entireearth’s crustevery year, based on a figure of 2× 1019 tonnes for the Earth’s crust.5 Regardingfossil energy, Rogner (1997) estimates the total resource base of (extractable) fossil fuels at 5000Gtoe, that is 5×1012 tons of oil equivalent. The current 2012 consumption rate offossil fuels is1.09×1010 toe, according to the BP Statistical Review of World Energy.So given a 3 percent peryear rate of increase in consumption, total stocks would be exhausted in just 90 years from now.At which point the global climate would of course be in meltdown. If we instead assume that therate of consumption remains constant at today’s level then stocks last for almost 500 years.

The mention of climate suggests a third question: when will the disruption of ecosystemsthrough pollution and overexploitation stop? Will it stop at all? It is not obvious that this phe-nomenon—ecosystem disruption—must stop. Consider CFCs. The large-scale emission of thesechemicals into the environment was a disastrous mistake with terrible consequences, and it hasnow all-but ceased. Yet the economy continues to grow, and welfare continues to increase. Thereis no guarantee that such mistakes will not be made again; indeed, it is almost certain that suchmistakeswill be made again. Similarly, the ‘mistake’ of CO2 emissions leading to climate changewill probably not bring down the global economy either, although it may lead to the loss of a signif-icant percentage of extant species globally. In a nutshell,might ‘unsustainability’ be sustainable?The control of pollution is also an important theme in the book.

So, what (if anything) will stop galloping resource extraction and environmental damage?Here we list some possibilities which will be discussed at various point in the book. Note that thesepossibilities are not logical alternatives, rather they are interlinked. For instance, environmentalpolicy may work by stimulating the development and adoptionof clean or efficient technology.

• Resource scarcity;• Resource efficiency;• Clean technology;• Satiation;• Enlightenment;• Deindustrialization;• Green consumerism;• Environmental policy;• Economic collapse.

5Extraction data from Kelly and Matos (2012).

Chapter 1. Introduction 5

1.2. Questions regarding economic growth on spaceship Earth

In the previous section we raised a number of questions regarding the economics of spaceshipEarth. For instance, how and when will the exponential growth in resource and energy use bestopped? And when will the disruption of ecosystems throughpollution and over-exploitationstop? Will it stop at all? However, there are many other relevant questions that could be raisedbased on the overview of the current situation that we presented. In this section we very brieflyconsider different types of question, and link these questions to different analytical or scientificapproaches. The aim is to clarify the approach taken in this book, and also to give some context,to show that many other approaches may also give valuable lessons regarding economic growthand sustainability. This is particularly important given that the questions posed in this book arerather tightly defined or narrow, as is the methodology used to answer them.

Consider first the following three questions.

(1) (a) What is desirable?(b) What is feasible?(c) What is optimal?

The first question might be asked by a philosopher. What characterizes a good life? Whatcharacterizes a good society? What characterizes a good society taking into account not only thepeople (and animals, plants, etc.) of today, but also those of the future? There are many differentideas about a good society. A couple of famous ones are those of utilitarianism and Rawls. Ac-cording to utilitarianism we should strive to maximize utility, which is typically considered as thesum of individual utilities. Utility may be a function of various factors; in economics we typicallyfocus inconsumptionas the key to utility. However, all utilitarian measures aretypically conse-quentialist, that is they measure the rightness of actions and more general moral rules according totheir outcomes, with the best actions leading to the greatest sum of utility. A fascinating and deepanalysis of issues regarding morality and choices affecting the future is Derek Parfit’sReasonsand persons.6 He subjects standard utilitarian reasoning to a searching examination from whichit emerges severely battered. If for instance we should maximize happiness, is it better to have100 billion people scratching out a living but enjoying a tiny bit of happiness each, or 5 billiongenuinely happy people? And what responsibilities do we have to future generations since ourspecific choices lead to their creation? (If we had done things differently, different people wouldhave been born.)

According to Rawls (1971), good moral rules (or indeed rulesfor how society is run) arerules which would be chosen by the citizens of that society ifthey had to make their choices frombehind a ‘veil of ignorance’, a veil which prevented each player from knowing anything about theirown position in that society (i.e. whether they would be richor poor, or have high or low status).Rawls claims that—when faced with a choice from behind the veil—we would choose moraland social principles based on liberty, equality of opportunity, and help for the least advantaged.In an intergenerational context—where the veil prevents usfrom knowing into which generationwe would be born—these principles translate into equality of opportunity across generations (sofuture generations should not be denied opportunities thatwe have, and vice versa), and also helpfor the least advantaged generations, so we should work hardto help the poorest generations ratherthan the richest. If we assume that global growth will carry on indefinitely this implies that weshould look after ourselves, at least in a material sense, since future generations will be richer!

The second question from our list—what is feasible—might beasked by an engineer. Oncea feasible solution to a problem has been found, the engineermight be satisfied. Or, if there is alimit on the budget, the problem might be to find a feasible solution within budget. This brings usto the third question, what is optimal. This is of course the classic question asked in economics,where we seek not just general rules of morality, but also specific ways of organizing society(including production and trade of goods and services) in order to find the ‘optimal’ allocation ofresources.

How do we know whether an allocation is optimal? Clearly we need a criterion, and thecriterion typically used in economics is utilitarian: we want to maximize the sum of utility or(more generally) expected utility. But how to measure utility? Economists typically assume thatit is an increasing function of consumption. Furthermore, we assume that the utility of consuminga given quantity goes down the further into the future that consumption takes places; exponential

6Parfit (1984).

6 1.3. Economic methodology

discounting. Therefore we have

max∑

h

Uh,

where Uh =∑

t

uh(ct)βt.

HereUh is the net present value of the utility of householdh, uh(ct) is the instantaneous flow ofutility at time t, andβ is the discount factor (which is less than 1). The reduction of utility toconsumption is of course highly controversial outside economics, but most economists take it forgranted. Furthermore, we also tend to take for granted the assumption that all households at agiven time are identical (or can be treated as such), which istypically made in order to increasethe tractability of macroeconomic models.7

A utility function can help us to rank alternative options, but how do we know what optionsare available? In order to know this, we must understand how the economy works, and howit can be controlled or managed. To build up such an understanding, we use highly simplifiedmodels which describe the agents in the economy (such as the government or regulator, firms, andhouseholds), their endowments (what they own), and the technology (what possibilities there areto produce outputs using available inputs). Given such a model we can tackle questions such asthe following.

(2) (a) What would happen given laissez-faire?(b) What would happen given business-as-usual (b.a.u.)?(c) What would be the effect of regulations such as high taxes on fossil fuels, or strong

support for research into renewable energy?(d) What would asocial plannerdo?(e) What should aregulatordo?

The first three questions relate to what happens in the economy under different circumstances.Question 2(a): what will happen in the absence of regulationby government? That is, what willhappen if the government allows all agents to do what they like without imposing rules or taxes,except the rule of law (implying for instance the protectionof private property or the right ofindividuals to own things). Question 2(b) is about the future in case the government continues toapply the policies which it currently applies, no more and noless, and the 2(c) is about the effectof specific policies.

Questions 2(d) and (e) are more subtle, since to answer them we need to be able to predictthe future and then choose between alternative options. Question 2(d) presupposes the existenceof an agent who has the power to fully control the allocation of resources in the economy withoutthe need for regulatory instruments; all other agents simply do as the social planner instructsthem. The planner is (fortunately) benevolent, i.e. she wants the best for everyone, she wants tomaximize the sum of utilityU. What would such a social planner do, how would she allocateresources? Would she, for instance, instantly cut fossil-fuel extraction? Or would she investmassively in research into alternative energy sources? If we can work out what our imaginarybenevolent dictator would do, this gives us a yardstick against which to measure the results of ourefforts to manage the economy through regulation (emissions taxes, technology standards, etc.),i.e. a yardstick which helps us to answer the question 2(e). How close can we get to the allocationin the planned economy?

1.3. Economic methodology

Having discussed the types of questions normally tackled ineconomics, we now briefly dis-cuss neoclassical economic methodology in general, and macroeconomic methodology in particu-lar. In neoclassical economics we build precisely defined model economies in which it is possibleto calculate outcomes exactly, both given laissez faire (i.e. no regulatory intervention), and giveninterventions such as the application of taxes or command-and-control regulations. We then usethese models to draw conclusions about how real economies work and are likely to develop orreact to regulatory interventions. In macroeconomics our focus is on the economy as a wholerather than a specific set of firms or markets.

7Note that we could (and sometimes do) use a Rawlsian criterion in which the aim is to maximize the utility of theleast well off, themaximincriterion:

maxminu1,u2, . . . ,un.

This states that we should maximize the smallest member of the set of utilities, i.e. we should concentrate our efforts onensuring that the least well-off individual increases their utility.

Chapter 1. Introduction 7

The precisely defined models of neoclassical economics consist of sets of equations. Themodels are generally very drastic simplifications of real economies: it is common for instance toassume that only one type of good ever gets produced in a modeleconomy, and furthermore thatthere is only one type of labour. Some of the single good is consumed, while a proportion is keptback by firms in the form of capital to help in further production. It is then natural to ask whetherwe can ever use such models to learn anything of importance about real economies?

If a highly simplified model is to teach us about the real economy, it seems reasonable tosuppose that it should be through a similar mechanism to thatby which a parable teaches us aboutlife. A good model helps us to organize our analysis of the economy, and allows us an insightinto how the real, complex economy works, and how it is likelyto react to changed circumstances(such as shortages of raw materials, or the introduction of atax on fossil fuels).

The point can be made more explicitly through a caricature ofhownot to do macroeconomics.Consider the aggregate data shown in Figure 1.3. The data areconsistent with a model in whichlong-run expenditure is a constant fraction of global product. Furthermore, the following aggre-gate production function is also consistent with this result:

Y= (AlL)1−β(ArR)β,(1.1)

whereAl is labour productivity,L is labour,Ar is resource productivity,R is the quantity of re-source input (pricewr ), andβ is a parameter less than 1. It is straightforward to show thatgivenperfect markets (so price=marginal revenue product)wrR/Y= β, i.e. expenditure on the resourceis a constant fraction of total product. Having noticed thisproperty of the function, thewrongthing to do is to draw the conclusion that the production function (1.1) is an appropriate descrip-tion of production in the economy, and to use this to (for instance) predict the effect of policyinterventions. A prediction which would follow from this isthat there is no point in boostingenergy-efficiencyAr to try to reduceR, because rises inAr cause effective energy inputsArR tobecome cheaper, causing producers to use more energy, negating the effect.

Why should we not draw the conclusion that the production function (1.1) is an appropriatedescription of production in the economy, and use this to predict the effect of policy interventions?There are many different ways to answer this question. Perhaps the simplest is that we haveno evidence that (1.1) is thecorrect description of the economy, or even close to being correct.All we know is that it can match one subset of aggregate data. There are likely to be manyother functions or models which also match the data, but which may give completely differentpredictions concerning the effect of boostingAr .

In order to have any confidence in our model we need much more evidence regarding itssuitability as a description of the economy. In particular as economists we want our model toinclude amicroeconomic mechanismor microfoundations: this is a mechanism through which thebehaviour of individual agents—behaviour which can be explained as a result of the incentivesfaced by these agents—leads to the observed aggregate quantities and trends. Furthermore, weneed to see evidence that this microeconomic mechanism is infact relevant to the case in hand. Fora famous statement of the case for microfoundations—or perhaps the caseagainstpolicy modelswithout microfoundations—see Lucas (1976). According to Lucas, we cannot expect to predictthe effects of a policy experiment based only on patterns in aggregate historical observations,because the policy experiment will change the rules of the game and therefore the old patternsmay no longer apply. We must instead understand the rules of the game and how it is played—in economic terms we must know agents’ preferences, the technologies available to them, whatresources they are endowed with, etc.—in order to understand and predict the effect of a policyintervention. This book is largely concerned with this endeavour, i.e. understanding the rules ofthe game and how it is played in order to understand and predict the effects of policy interventions.

1.4. This book

This book is based on economic analysis. However, this does not mean that we accept theeconomist’s habit of equating utility and income. Instead,we focus primarily on questions 2(b)and (c) above—what will happen under various scenarios—thus conveniently obviating the needto explore the more difficult questions 1(a), (b), and (c). It turns out that predicting the future isquite hard enough as it is.

We will also consider 2(e), policy, but based on goals which are exogenously determined(such as reducing CO2 emissions) rather than calculated within the model. For instance, whenconsidering climate policy we simply assume that society has decided (in its wisdom) on a goalof drastically reducing CO2 emissions, and investigate how best that goal can be achieved. Thus,

8 1.4. This book

again, we avoid difficult questions about optimal overall choices, and focus on more limited ques-tions.

Our avoidance of questions such as 1(a), (b), and (c) impliesin practice that we avoid somehuge and important debates within the fields of economics andsustainability. One such debate isthat about the appropriatediscount rateto use when assessing public policy, a debate made famousby the report of Sir Nicholas Stern (2006) for the UK Treasuryon the economics of climate change.More generally, in the main part of the book we ignore the issue of intergenerational equity andjustice completely. However, in the concluding part (Part 5) we broaden the discussion somewhat.

We almost completely ignore problems linked to the fact thatthere is no one global gov-ernment, but rather a collection of states which are very different from one another: they havedifferent economies, different values, and also different endowments of natural resources such asfossil fuels. This leaves individual states that wish to pursue a cleaner, greener global developmentpath with a much more complex problem than that which would face a global government; theymust either seek international agreement, or—if they act unilaterally—they must weigh up boththe direct effects of their actions on the global environment, and also theindirect effects via theeffect onothercountries actions. These effects may be in the opposite direction to that desired, aform of rebound effect.

In Part 1,The driving forces of economic growthwe try to understand what drives the growthprocess; is it for instance driven by increasing resource exploitation, or changing preferences? Wefind that economic growth is driven by technological change,while phenomena such as increasingresource exploitation are consequences of this growth rather than causes. In the last chapter ofthis part we investigate what drives technological change,testing the idea that it is driven bydeliberate investment by households or firms in innovation,i.e. the discovery and development ofnew technology. An understanding of these processes will beimportant later on when we considerhow economic policy can change the trajectory of the economy.

In Part 2,Natural resources and neoclassical growth, we set up and solve some very simplemodels in which natural resource inputs are essential to theproduction of a single final good. Weare interested in finding out what effect scarcity of nonrenewable resources has on the rate of eco-nomic growth, and also the consumption rates and prices of the resource stocks. We continue towork within the single-sector neoclassical framework. That is, we assume an economy with a sin-gle product, which may either be consumed or retained withinfirms as capital, to help with futureproduction. Furthermore, we assume that the final-good production function is Cobb–Douglas, asin the baseline neoclassical model.

In Part 3,Demand for natural resources and energy, we turn from a focus on the supply ofnonrenewable resources to demand. We begin by testing whether the forces of directed techno-logical change are the cause of the increase in resource consumption rates observed in the globaleconomy; that is, has the cheapness of resources led to a failure to develop resource-efficient tech-nologies? And then, turning to prediction of the future, we ask whether increasing resource priceswould be likely to trigger a massive increase in resource efficiency, thus helping to reduce theneed for resources. We conclude that the answer is likely to be ‘no’, on both counts. The failureof the DTC model to account for the historical data puts the spotlight on the alternative, substitu-tion on the consumption side. Has resource and energy consumption risen so rapidly because ofsubstitution towards resource- and energy-intensive consumption categories, a form of structuralchange? If we accept the clear evidence from the energy sector that the energy-efficiency of indi-vidual products has indeed increased dramatically, the answer to the above question must be ‘Yes’,since these two alternatives exhaust all the possibilitiesfor explaining why aggregate efficiencyhas failed to increase. Is this backed up by the evidence, andif so, what are the implications?

In Part 4,Substitution between resource inputs, and pollution, we turn from the analysis ofoverall demand for nonrenewable resources and energy, to the analysis of demand for alternative—substitutable—resource inputs. Such inputs may be alternatives such as iron and aluminium, orcoal and gas. But they could also be coal and wind, or coal and solar. Clearly, such substitutionhas major implications for the analysis of polluting emissions, and after an initial (more general)analysis, we turn our focus to the prediction and regulationof polluting emissions. We show thatrising income drives substitution from dirty inputs, such as coal, to cleaner inputs such as gas orrenewables. This helps us to understand the observed tendency of polluting emissions to first riseover time, and then decline: the rises are driven by the increase in overall resource demand, drivenby growth; and the declines are driven by switches to cleanerinputs.

In Part 5,Policy for sustainable development, we take a step back and consider the impli-cations of the analysis presented earlier in the book, and also put it into a broader context. Weexamine the role of green consumerism, and also consider arguments for government policies

Chapter 1. Introduction 9

to encourage shorter working hours and thus less productionand consumption in the economy.Section under construction!

Part 1

The driving forces of economic growth

The history of the human economy over the last 20000 years is one of growth, first growthin population (and hence also the total value of production)and later on growth in the value ofproduction per capita as well. Economic growth is thus central to the analysis of this book, andin this (first) part of the book we investigate the driving forces of this growth. Is for instanceeconomic growth driven by increasing resource exploitation, or changing preferences? We findthat economic growth is driven by technological change, while phenomena such as increasingresource exploitation are consequences of this growth rather than causes. In the last chapter ofthis part we investigate what drives technological change,testing the idea that it is driven bydeliberate investment by households or firms in innovation,i.e. the discovery and development ofnew technology. An understanding of these processes will beimportant later on when we considerhow economic policy can change the trajectory of the economy.

CHAPTER 2

Growth fundamentals

In this chapter we define what we mean by economic growth, and narrow down the list ofcandidates with which to explain long-run growth. We show that in the long run increases inthe productivity of labour are essential to drive economic growth; such productivity increases areinextricably linked to the concept of technological change. A further consequence of increasinglabour productivity may be increasing use of non-renewablenatural resources, and also increasinguse of renewable natural resources, and increasing polluting emissions.

2.1. Production, GDP, and growth

Economic growth is growth in GDP, gross domestic product, i.e. the value of all the goodsand services produced within an economy during a given year.Consider an economy with onlyone final product. Then GDP is simply the quantity of that goodproduced multiplied by its price.RealGDP is quantity× price in constant dollars. Growth in real GDP (in percent peryear) is thensimply the growth rate of production of the good.1

In an economy with many products the problem of measuring GDPgrowth is more complex.NominalGDP is simply quantity× price for each product, summed over all products, and growthin nominal GDP is the growth rate of this sum. But what aboutreal GDP? If prices are all constantand the range of goods available is also constant then there is no problem: nominal and real GDP(and their growth rates) are the same. But if the prices of different goods change at different rates,and completely new goods also appear, then the problem is much more complex. We do not godeeply into this problem here: suffice it to say that growth accountants try to measure the changein consumer’s (and firms’) willingness-to-pay for the set ofgoods produced, in constant dollars.If consumers are willing to pay 3 percent more (in constant dollars) for the ‘basket’ of goodsproduced in the economy in 2014 than for the goods produced in2013 then GDP growth musthave been 3 percent.

In economic models we describe production using productionfunctions, which describe thequantity of a good (or potentially several goods) produced as a function of the quantities of theinputs used. We now investigate the nature of production functions, focusing on one case anddiscussing it by comparison to descriptions from chemistryand physics. The function on whichwe focus is for the production of iron from iron ore. Assume wehave an economy with householdswho supply labourL, a stock of iron (III) oxide Fe2O3, and a stock of coal in the form of purecarbon,C. The workersL take the iron ore and coal and—using furnaces—reduce the oreto pureiron, with a by-product of carbon dioxide. A simple chemicaldescription of this process is

2Fe2O3+3C−−−→ 4Fe+3CO2.

Note that—as in all meaningful equations—there is a form of balance or equality between thetwo sides. In this case, the number of atoms of each type (iron, oxygen, and carbon) is identicalon each side, and the equation shows how these atoms are rearranged through a chemical reaction.

Another way of describing the process is to track the energy released during the chemicalreaction. Assume that 2 moles of iron (III) oxide and three moles of carbon react (as above)within a reaction chamber at constant temperature and pressure, where the chamber is part of anisolated system. Then we know that

∆G = ∆H−T∆Sint,

where∆G is the change in Gibbs free energy,∆H is the enthalpy change of reaction,T is thetemperature, and∆Sint is the entropy change within the reaction chamber. This equation is mathe-matical (rather than chemical) hence each side of the equation must be equal to the same number,and furthermore these numbers must be in the same units. In this case, the units are units of energy,so we choose the standard SI unit ofjoules. SinceT has units of kelvin (the temperature scaleused in physics and chemistry),S must have units ofjoules per kelvin; otherwise, the units wouldnot be equal!

1Note that we always work with real, not nominal, quantities in this book, unless otherwise stated.

13

14 2.2. What drives growth? Reasoning from first principles

The above equation is an accounting relationship, trackingthe flow of energy in the reaction.An economic production function describing the same process is an accounting relationship track-ing costly inputs and outputs. Assume that a firm employs labour to run the process, and thatvast piles of iron ore and coal are freely available; furthermore, they are so large that they arethought to be inexhaustible. Assume also that facilities such as furnaces are also abundant andfreely available. So the only costly input for the firm is labour. How then do we write the firm’sproduction function? Very generally, we can write

Yi = F(l i),

whereYi is the quantity in tons of the final product (iron) made by the firm per year,F is a function,andl i is the number of workers employed (on an annual basis). Much more specifically we couldwrite

Yi = Al l i .

Now we have assumed that there areconstant returnsin labour, so if the firm doubles its labourinputs, the quantity of final productYi produced will double. Note that we can assign units toAl ,since the units on each side of the equation must balance: theunits ofAl must betons worker−1

year−1. ThusAl is a measure of the firm’s productivity.Note that the firm is also using iron ore, coal, and capital in the production process, but we

do not include them in the production function because they are assumed to be free and abundant.Furthermore, the firm also produces carbon dioxide (see the chemical equation) but again sincethis has no value we leave it out of the production function. However, as soon as one of theother inputs or outputs becomes valuable (or as soon as we realize that it is valuable) then we caninclude it in the production function. For instance, assumethat the pile of coal starts to run out,and firms start competing on a market to buy coal inputs. Then we should include coal in theproduction function. If we assume that the firm has no flexibility whatsoever about its productionprocess, and that a fixed quantity of coal is required for eachton or iron produced, then we shoulduse a Leontief production function:

Yi =min(Al l i ,Ar r i).

This reads as follows: productionYi is equal to the smallest of the following two quantities;effective labour inputsAl l i and effective coal inputsAr r i . The units ofr i are tons of coal per year,hence the units ofAr must be tons of iron per ton of coal. Alternatively, the firm may find that itcan trade off more labour for less coal (because with more labour inputs the workers can ensurethat the coal is used more effectively) in which case some more complex functionF is called forwhere

Yi = F(Al l i ,Ar r i).

In a similar way we can add capital and carbon dioxide to the production function, with eachaddition making the function more complex, and therefore these additions are only undertakenwhen they are relevant, i.e. when they add to the explanatorypower of the models. In economicanalysis focusing on issues other than natural resources and environment it is common to leaveout resource inputs and polluting outputs from the analysis, since although they are not free theydo not typically make up a large proportion of firm’s costs. This does not make these analyseswrong, or in contradiction with the laws of physics; it simply means that they are focused on thequestion at hand. Of course, in this book the questions are all about resource inputs and pollutingoutputs, hence it is essential to include these quantities in our production functions.

2.2. What drives growth? Reasoning from first principles

We saw in the introduction that average global growth has been remarkably constant andsustained over a period of more than 200 years. What has driven this process, and if such growthis to continue into the future, what will drive it? We now turnto these questions. First we consideran economy where there is only one scarce input—labour—thenwe add capital and physicalresource flows.

2.2.1. Labour. We begin with a model in which production involves the application of lim-ited labour—together with abundant machines and raw materials—to make a single final product.The final product is measured byY, unitswidgets year−1. Labour is homogeneous (all workers arethe same) so we can measure its quantity in a single dimension, L, unitsworkers.2 The production

2One worker is then one person who works full-time throughoutthe year, or (for instance) two people who eachwork half-time throughout the year.

Chapter 2. Growth fundamentals 15

function is then as follows:

Y= ALL,

whereAL is labour productivity and has unitswidgets year−1 worker−1. It is essential to includeAL in the function, otherwise the units on either side of the equation cannot be the same. Theproperties of this function are intuitively reasonable. For instance, if we double the input oflabour L while holding its productivityAL constant then production doubles. Similarly, if wedouble labour productivity and hold labour inputs constantthen production also doubles.

Now consider growth. Assume first thatL (labour) is the same as population. Then we canof course increase total productionY if population increases. However, production per capitaY/Lwould not then change; in the literatureY/L is frequently denotedy (as opposed toY for totalproduction). Another way to generate growth would be if everyone worked longer hours, or if agreater proportion of the population joined the labour force. However, the long-run possibilitiesusing this method are clearly very limited, given the high level of labour-force participation wealready see, and the limited hours in the day. The conclusionfrom this discussion is therefore thatthe only way to raise production per capita in this economy inthe long run is to raiseAL, labourproductivity.3

2.2.2. Labour and capital. Now assume that machines (or more generallycapital) are notabundant, i.e. they are no longer free but rather they are costly and their use must be accountedfor. Might an increase in the quantity of capital be the driving force for growth? Since there isstill only one product, widgets, then capitalK is simply the number of widgets kept back withinfirms to help in the production process: it therefore has units of widgets. Our general productionfunction becomes

Y= F(ALL,AKK),

whereK has unitswidgets, and AK has unitswidgets year−1widget−1. So widgets—the finalproduct—may either be consumed or kept back within firms, where they can be used as tools ormachines. Workers need these machines in order to produce, and the more machines each workerhas access to, the more she can produce. Therefore the function F must be increasing in its twoarguments, effective inputs of capital and labour.

Consider now doubling both capital and labour while holdingtheir respective productivitiesconstant. Then we have the same number of workers per machine, hence production per workermust also be the same, and total productionY must double. This implies that there are constantreturns to scale inK andL together, implying in turn that there must be decreasing returns to scalein either ofK or L separately: for instance, doublingK while holdingL constant will not lead toY doubling. Finally, the more capital we have (for a fixed number of workers) the less the benefitof adding even more capital should be: formally,F′′K < 0. To understand this, assume that the‘machines’ are actually hammers: once workers have one hammer each there is little or no benefitto saving up additional hammers.

These arguments tell us straight away that simply accumulating capital cannot give long-rungrowth. Consider again an economy in which workers use hammers to make final goods. Ifthere is initially less than one hammer each then accumulation of capital (hammers) can boostproduction. But when every worker has a hammer then accumulation of additional capital willscarcely boost production at all, and when there is a mountain of hammers for each worker thenadditional hammers will definitely not boost production.

Since neither increases in labour nor capital can drive long-run growth in this model, it mustfollow that technological progress—growth inAL andAK —lies behind long-run growth inY. Infact we can go further, if we rewrite the production functionas follows:

y= f (AL,AKK/L).

The functionf should have the same properties asF, implying thatAL (labour productivity) mustrise to deliver long-run growth, whereasAK (capital productivity) does not necessarily need torise, since capital per worker may rise instead.

3We typically ignore the distinction between labour and population, implying that we assume that the ratio of laboursupply to population is constant. However, in Chapter 16 we study labour supply explicitly.

16 2.3. Growth drivers in the real economy

2.2.3. Labour, capital, and resources.Assume now that we also need to account for flowsof natural resourcesR (which might for instance have units tons/year if we have a single homo-geneous resource). We then also need resource productivityAR, units widgets ton−1, and theproduction function (per capita) is

y= f (AL,AKK/L,ARR/L).

Again, f must have the property that there are diminishing returns toARR/L, so long-run growthcannot be driven purely by increases inR (nor by increases inAR); all three of the ‘augmented’inputs—AL, AKK/L, andARR/L—must grow together to give long-run growth.4

Summing up, a precondition for long-run growth is increasing labour-productivityAL. Onthe other hand, the productivity of capitalAK does not necessarily need to increase, as it may bepossible instead to increase the quantity of capitalK.5 Furthermore, if stocks of nonrenewableresources are very large, then growth may be sustainable over a very long period (hundreds oreven thousands of years) without increases inAR; we can simply increase the rate of extractionRinstead. Nevertheless, in thevery long runAR must increase, at least if we are restricted to ‘space-ship Earth’ as our source of material inputs, since resourceflows must be bounded by physicallimits.

2.3. Growth drivers in the real economy

In the above analysis we assumed a single unchanging product, widgets, and a single typeof labour. This is an approach frequently taken in macroeconomic models in general, and inparticular in the neoclassical growth model which is the subject of the next chapter. Do theconclusions about the centrality of technological change boosting labour productivity carry overto more realistic economies with many different products, and many different types of worker? Itturns out that they do, with some minor qualifications.

Consider first an economy with a single product, but a varietyof labour types, where someworkers are more productive than others. Clearly then we canincrease GDP if the number ofhighly productive workers increases while the number of less productive workers decreases. How-ever, this is effectively an increase in labour productivity, even though the productivity of the mostproductive workers in the economy is unchanged. Furthermore, growth driven by this process willeventually peter out once the entire labour force is of the productive type; to sustain growth, newtechnologies must be introduced which allow further increases in labour productivity.

When there are multiple products the fundamental conclusion from above still applies: inorder to increase production, increasing labour-productivity AL is essential. However, many otherquestions are raised. For instance, given a range of products growth may be achieved throughincreases in the quality, not just the quantity, of what is produced. Thus there may be differenttypes of technological progress:processinnovations which allow more of the same product to bemade from given input, andproductinnovations which allow the production of new goods. Thenew goods may be more valuable per se, or they may just bedifferent. If there is a love of varietythen an increase in the variety of products with a fixed total quantity boosts the value of productionand hence GDP.6 Furthermore, a greater variety of products may translate into a greater varietyof capital goods, and also potentiallyintermediate goods, i.e. products that are not bought byconsumers, but are bought by production firms and used as inputs forfinal goods. When a greatervariety of capital and intermediate goods is available, final-good producers may be able to raisetheir productivity.

Another possibility is that goods may beintensivein different inputs. That is, different goodsmay require inputs in different proportions. Thus artificial light requires a lot of energy and littlelabour to produce, whereas education services require a lotof labour and little energy. So artificiallight is energy-intensive, whereas education islabour-intensive. Given this observation is shouldbe clear that shifts in the patterns of production and consumption—between goods of differentquality, and between goods intensive in different inputs—may have important implications fordemand for resource and energy inputs, and also polluting emissions and environmental quality.These implications are lost when we restrict ourselves to single-sector models, i.e. models withjust a single product. We return to this type of analysis in Chapter 10.

4The quantity of anaugmentedinput is the quantity of the physical input multiplied by thelevel of the relevantinput-augmenting (or factor-augmenting) technology or productivity.

5In fact in real economies the distinction between the quantity of quality (or productivity) of capital is meaningless,and we can only talk about effective capital inputsAKK.

6Note however that this type of growth may be almost impossible for statistical agencies to detect let alone measure.

CHAPTER 3

Labour, capital, and neoclassical growth: The Solow andRamsey models

In this chapter we turn to the basic neoclassical growth model without including natural re-sources in the analysis. The key differences compared to the simple analysis of the previouschapter are that we specify the production function, and we specify the process through whichcapital is accumulated. The key conclusion confirms that which we drew above: capital accumu-lation cannot drive growth, increases in labour productivity drive growth. We focus on the modelof Solow (1956), in which household behaviour is highly simplified. We also briefly mentionthe more sophisticated Ramsey–Cass–Koopman’s model whichputs household choices into anintertemporal optimization framework.1

3.1. The Solow model with constant technology

In this section we set up the most basic Solow model indiscrete time, rather than continuoustime. This choice is essentially arbitrary; we choose discrete time because it is slightly easier toexplain aspects of the intuition behind the model in such a context. For more on discrete andcontinuous time see Chapter 17, Section 17.2.

3.1.1. The production function. Solow focuses on labour and capital; implicitly, he as-sumes that resources are so cheap that they can be treated as being available to the firm for free,and therefore do not need to be included in the (economic) production function. He assumes thatthere is just one product,Y, which is thus used for both consumption and capital. Furthermore,labourL is supplied inelastically, so if we know the population thenwe knowL.

So in general we can writeY= F(K,L). What determines how much of the productY we canmake from given inputs of labour and capital? By definition itis theproductivityof our inputs. Ifproduction is measured in widgets/day, capital is measured in widgets, and labour is measured inworkers, then in general there must be two productivity indices—we denote themAK andAL —with unitswidgets day−1 widget−1 andwidgets day−1 worker−1 respectively. It must be possibleto write the production function in this form since the left and right hand sides of any meaningfulequation must have the same units. So now we have

Y= F(AKK,ALL).

These productivities may in general change over time as (forinstance) our knowledge and skillsimprove.

Now we wish to constrain the properties of the function. Firstly, we assume that having morecapital and labour available is never a bad thing for production, in other words

F′K ,F′L ≥ 0.

Second, we assume that the more we increase one input—while holding the other constant—thesmaller is the marginal increase in production, i.e.2

F′′K ,F′′L ≤ 0.

Third, we assume that each input is essential, hence there isno production whenK or L arezero, and furthermore the marginal productsF′K andF′L approach zero when the quantity of therespective input approaches infinity.

Fourth, and finally, we assumeconstant returns to scalein the physical inputsK andL: thatis, if we doubleK and doubleL but hold the productivitiesAK andAL constant, output doubles.In a sense this assumption is already implied by the way we have defined the productivity indices;given this definition, the new assumption is thatchanging the quantities K and L alone does notaffect the productivity indices.

1Also called the Ramsey model. See Ramsey (1928), Cass (1965), Koopmans (1963).2Note that we also allow for the possibility that the marginalproductivity of an input may be constant in the quantity

of that input, at least over some interval.

17

18 3.1. The Solow model with constant technology

Figure 3.1. A Solovian economy in which hammers are the final good.

This production function tells us immediately that accumulation of capital will boost produc-tion, but only up to a point; as the amount of capital in the economy increases, returns to furtherincreases in capital diminish (F′′K < 0). Think of the economy with workers and tools. Increasingthe number of tools available may boost production if tools are short, but once there are plenty oftools available then no further benefits will be felt.

3.1.2. Technology.Solow thought of technology as knowledge, a non-rival and non-excludablegood which can therefore be used simultaneously throughoutthe global economy. In the baselinemodel we therefore assume thatAL andAK are the same in all economies. Furthermore, since wehave no mechanism which causes them to grow, we assume in our baseline model that they areconstant. In the next section—based on the failure of the baseline model—we allowAL to growexogenously, i.e. for reasons not explained in the model.

3.1.3. Savings and capital according to Solow.Solow makes crucial assumptions aboutsavingsandcapital. Regarding saving, he assumes that a fixed proportions of final-good produc-tion is saved as capital, while the remainder is consumed. Furthermore, he assumes that the stockof capital depreciates at a constant rateδ. That is, in discrete time, if we have 10 units of capitalin periodt then (in the absence of saving) there will only be (1− δ)×10 units left in periodt+1.Note that this depreciation is entirely independent of whether the capital goods are actually usedin production or not. The equation governing the evolution of the capital stock can therefore bewritten

Kt+1 = (1− δ)Kt+ sYt.

Recalling the production function, we can write

Kt+1 = (1− δ)Kt+ sF(Kt,Lt).

To get an intuitive grip on the model economy, consider Figure 3.1. What are the values ofK, L,Y, ands? Assume that the economy is in a steady state, i.e. nothing changes over time. What mustbe the value ofδ?

3.1.4. Firm optimization. In the model we assume that there are very many small firms inperfect competition. Mathematically we can think of a unit mass of firms, the result of which isthat the economy behavesas if there is just one firm, but that the single firm is neverthelessaprice-taker. Saying that the firm is a price taker is the same as saying that the price it pays forinputs is equal to the marginal product of those inputs, and the price it takes for its product isequal to marginal cost. This one firm is known as therepresentativefirm. For more on this seethe mathematical appendix, Chapter 17.3.

Chapter 3. Labour, capital, and neoclassical growth 19

The representative firm hires capital and labour on the market. The firm’s problem is asfollows:

maxK≥0,L≥0

F(K,L)−wK K −wL L.(3.1)

The prices paid by the firm—wK for renting capital andwL for labour—are equal to the marginalproducts of the inputs, that iswK = F′K , andwL = F′L. Note that total costs are thenKF′K +LF′L. Ifprofits are to be zero these must be equal to total income whichis justY, since we normalize theprice of the final good to 1.

3.1.5. Cobb–Douglas production function.The standard functional form for the produc-tion function in the Solow model is the Cobb–Douglas. Furthermore, technological progress isassumed to increase the productivity of labour, not capital. That is,

Y= (ALL)1−αKα,(3.2)

whereα is between 0 and 1. Compare this to the simple economy above inwhich Y = ALL.Now, because both labour and capital are needed for production, a doubling in labour productivityincreases production, but does not give a doubling in production. You should verify that thefunction has all of the properties described above.

A special property of the Cobb–Douglas function is that the elasticity of substitution betweenthe inputs is 1. Assume that the relative prices of the inputschange, and that we have a repre-sentative firm which is a price taker. Then the elasticity of substitution is the percentage changein relative input quantities chosen by a firm divided by the relative price change, times−1. Tocalculate it, note first that quantities are simplyK andL. Relative prices are given by the relativemarginal products, and∂Y/∂L = (1−α)Y/L, whereas∂Y/∂K = αY/K. Hence

wK K

wL L=

α

1−α;

KL=

wL

wK

α

1−α;

∂K/L∂wK /wL

= −K/L

wK /wL

.

This confirms the unit elasticity of substitution, and highlights another feature of the productionfunction: the relative returns to the factors are fixed. Capital takes a proportionα of total returns,and labour takes a proportion 1− α. To get the intuition, assume that the quantity of capitalavailable increases; one result of this increase is that theprice of capital decreases relative to thewage (the price of labour). Given Cobb–Douglas, the decrease in price exactly compensates for theincrease in quantity such that returns to capital remain unchanged relative to returns to labour. Thisis an attractive feature of the model, since in reality we observe a remarkably constant divisionof returns between labour (70 percent) and capital (30 percent), across different economies anddifferent times.

3.1.6. Dynamics.Consider the Solowian economy of Figure 3.1 in which the productionfunction is Cobb–Douglas andAL is fixed. Assume thatα = 1/3 andδ = 0.1. What isAL? Whatis s? Is the economy in long-run equilibrium?

In Figure 3.1 there are 10 workers, each with a hammer. ThusL = 10 andK = 10. ProductionY is 5 hammers per period, soY= 5. So

Y= A1−αL ·101−α ·10α.

So we know thatA2/3L = 0.5. Furthermore, we see that 20 percent of production is saved, so

s= 0.2. Finally, the economy is in long-run equilibrium because the quantity of capital remainsconstant over time: savings (which are equal to investment)are 1 hammer per period, whichexactly compensates for depreciation (10 percent of 10 hammers per period).

Imagine instead the same economy, but that in the initial period there is only one hammersaved as capital. Work out production and saving, and depreciation, and hence the number ofhammers available in the next period. Describe how the economy develops over time. Yourdescription should match Figure 3.2.

Three things are notable from the figure. Firstly, even at thedesperately low initial level ofcapital, initial production is at almost 50 percent of its long-run level. Secondly, the transition tothe long-run equilibrium is rather rapid. Each dot represents one year, so we see that in just 15years the initially very capital-poor economy is close to 90percent of the production of the richeconomy. Finally, and most fundamentally, there is zero growth in the long run.

20 3.1. The Solow model with constant technology

0 1 2 3 4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

capital, hammers

prod

uctio

n an

d in

vest

men

t, ha

mm

ers

Investment

Production

Necessary investments

Figure 3.2. The transition path in the Solowian economy of Figure 3.1, startingwith one hammer.

Mathematically, we can easily work out the long-run steady state of the model. It is whensY= δK, i.e. when investment matches depreciation. Using the production function we have

s(ALL)1−αKα = δK,

hence K1−α = s(ALL)1−α/δ

and K = ALL(s/δ)1/(1−α),

Y/L = AL(s/δ)α/(1−α).

So GDP per capita is a linear function of labour productivity, and weakly increasing in the savingsrates.

Now assume instead that capital is available for hire on a global market at pricewK per year,and that we have many separate economies. If we index the economies byi then we have (foreconomyi)

Yi = (ALLi)1−αKα

i .(3.3)

Economyi has exogenously given levels of labourLi ; all economies have access to the sameknowledge and therefore have the same technologyAL. What is economyi’s level of capitalKi

and productionYi in long-run equilibrium, i.e. when the quantity of capitalKi is optimal?WhenKi is optimal then marginal returns to capital must be equal to the price, i.e.

αYi/Ki = wK .

Thus we see straight away that the ratio of GDP to capital is constant across countries! Further-more, after substituting in the expression forYi and rearranging we have

Ki = ALLiw−1/(1−α)K

.

This tells us the optimal quantity of capital. Substitute itback into the expression forY to yield

Yi/Li = ALw−α/(1−α)K

.(3.4)

So—again—GDP per capita rises linearly with labour productivity AL. Now it declines slowlyin the rental price of capital, rather than increasing ins, the savings rate.

3.1.7. Conclusions on Solow with constant technology.The above results show that theSolow model with constant technology is incapable of explaining any of the key empirical obser-vations about long-run growth and differences in GDP between countries. GDP does not grow inthe long run in the model, and differences in GDP between countries should be modest, even ifsavings rates differ drastically between the countries.

Chapter 3. Labour, capital, and neoclassical growth 21

The model is nevertheless useful, because it shows us whatcannotdrive long-run growth.Capital accumulation! This is a remarkable conclusion given that the long-run aggregate produc-tion function appears to have the form

Y/L = K/L.

So even though long-run production per capita is linearlycorrelatedwith the long-run capital percapita, simply accumulating capital will not give growth!

3.2. Solow with exogenous technological progress

Given that capital accumulation alone cannot drive growth,we turn back to labour productiv-ity AL. Even though we cannot explainwhy it grows, it is clear that it must grow in the long run—at least in the model economy—to explain the growth process.Note that in this section we switchto continuous time which is convenient when there is positive long-run growth.

So let’s assume thatAL grows exogenously at a constant rategAL . For good measure letLgrow too, at raten. Now assume that there exists abalanced growth path(b.g.p.) along whichall variables grow at constant rates (note that these rates may differ from one another, and may bezero or negative). What are the characteristics of such a path, if it exists?

We have

Y= (ALL)1−αKα

AL/AL = gAL

L/L = n

K = sY− δK.

Now, since we know that we are on a b.g.p. we know thatK/K must be constant, hencesY/K − δmust also be constant, in turn implying thatY/K must be constant. Returning to the productionfunction we therefore know that

(ALL/K)1−α

must be constant on a b.g.p., implying that

K/K = AL/AL+ L/L.

In words, the capital stock grows as the same rate as augmented labour inputs, keeping the ratioof effective labour to capital constant. But, from the productionfunction, we know that

Y/L = AL[K/(ALL)]α,

implying that on such a b.g.p.Y/L grows at the same rate asAL! That is, per-capita GDP grows atthe same rate as labour productivity.

Is such a b.g.p. stable, and (if so) how quickly does the economy approach such a b.g.p. if ithas been knocked off course by a shock? It is stable, for the same reasons as the economy withouttechnological progress: when there is ‘too little’ capitalsavings outstrip depreciation and capital(per effective unit of labour) accumulates rapidly; when there is too much capital depreciation getsthe upper hand and capital (per . . . ) declines. Furthermore,simulations show that this process israther rapid.

Together these facts tell us very clearly that differences in the quantities of the factors ofproduction (in particular capital) available in different countries cannot explain the very large andvery persistent differences in the productivity of labour (or GDP per capita) between countries.Instead the difference must lie in the different ability to make productive use of the factors (labour,capital, resources) available.

This analysis is supported by empirical evidence; for instance, we know that economies re-cover rapidly from a sudden loss of physical capital. This iswell illustrated by the case of Germanrecovery after WW2, as shown in Figure 3.3. We see that after just 15 years the economy has al-most regained its old trend line, despite the very drastic loss of capital and hence also productivityin 1945. This matches well to a model economy with standard parameters, starting on its balancedgrowth path.

3.3. The Ramsey model

The Solow model is good for explaining the medium-run evolution of the capital stock, andphenomena which are tightly linked to that. For instance, ifa country’s capital stock is largelydestroyed due to a calamity such as war or an earthquake, the Solow model predicts that the capital

22 3.3. The Ramsey model

1920 1930 1940 1950 1960 1970 19800

0.2

0.4

0.6

0.8

1

GD

P /

GD

P 1

988

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

GD

P /

GD

P 6

8

Figure 3.3. Testing the Solow model. Upper panel: German real GDP, com-pared to an estimated constant-growth trend (growth rate 2.3 percent per year).Data: Maddison (2010). Lower panel: Model simulation assuming the samegrowth trend in labour productivity and parametersα = 0.3, δ = 0.1, s= 0.2,with a massive shock to capital (the stock is divided by a factor of 30 at the endof year 24).

stock and GDP will recover rather rapidly; the evidence bears this out (consider for instance thegrowth of West Germany’s economy after the end of WW2).

A better model for doing the above—and many other things—is the Ramsey model, whichtakes the Solow model and endogenizes the saving rates; that is, in the Ramsey model householdschoose the balance between consumption and saving in order to maximize their utility, rather thansimply by following a rule of thumb.3 The result is that (for instance) a catastrophic loss of capitalmay lead to a rise in the saving rate because of the greater effect of saving on growth, but acountervailing decline in the saving rate due to the desire to smooth consumption over time. Whatactually happens depends on the balance between these effects.

We do not go into the solution of the Ramsey model, since this is beyond the scope of thisbook. However, you should know that the key to endogenizing the savings rate is the householdutility function. Recall from the Introduction that we typically write a household utility maximiza-tion problem as follows:

max∑

h

Uh,

where Uh =∑

t

uh(ct)βt.

HereUh is the net present value of the utility of householdh, uh(ct) is the instantaneous flow ofutility at time t, andβ is the discount factor (which is less than 1). But what is the form of theinstantaneous utility functionuh(ct)? The standard utility function used in the Ramsey model isthe CIES, as follows:

u=c1−σ−1

1−σ.

As long asσ > 0 this implies that marginal utility is declining in consumption, which means thathouseholds have a preference for consumption smoothing. Furthermore, sinceβ < 1 they alsohave a preference for consuming now rather than later. In an economy in which technologicalprogress drives consumption growth, both of these factors militate against investment and towardsconsumption today.

Note that in an open economy we should also allow for international capital flows, which willunambiguously favour rapid recovery from a negative shock to capital.

3Also called the Ramsey–Cass–Koopmans model. See Ramsey (1928), Cass (1965), Koopmans (1963).

CHAPTER 4

Endogenous growth theory

This chapter has three main aims. Firstly, to introduce the modelling of endogenous techno-logical change (and hence endogenous growth) in general. Secondly, to introduce the particularmodel that we will then develop and use through this part of the book. Thirdly, to show thatendogenizing technological change as such makes little or no difference to the conclusions wedrew earlier from the neoclassical model; among other things this shows that we can safely ignorecapital in our models in many circumstances.

4.1. Ideas and long-run growth

Much of the ‘new’ literature on growth has been preoccupied with the incentives agents haveto perform research or generate new knowledge or ideas. A keyinsight is that ideas possess a veryspecial property—when compared to other goods such as pizzaor cars—a property which mayreduce incentives to produce ideas. This is that they arenon-rival in consumption, meaning thatone agent’s use of the good does not hinder another person from using it, either consecutively oreven simultaneously. This is clearly not true of cars, and even less so of pizza.

Return to the Solow model, and assume that someone founds a research firm and (after muchwork) finds a better way to make hammers such thatA increases. What happens: (a) if there arepatent laws; (b) if there are not?

If there are patent laws, the owner of the new knowledge will have an advantage in production,and can take over production in the entire economy and becomefabulously rich. On the other hand,if there are no patent laws then the discoverer of the new knowledge will gain no benefit from itwhatsoever, since all firms will use the knowledge and the owner’s firm will still make zero profits.In total the owner’s firm will make a loss, because the firm willbe unable to cover the costs of theresearch. Neither of these extreme results seems to make much sense. What is the way forward?

In general, we can say that a more sophisticated model is clearly required, a model in whichthere is a variety of products with their own production functions, and where researchers canprotect their intellectual property in some way, for instance through taking out patents. Agentsthen have an incentive to perform research (the incentive being the expected flow of profit in thefuture). Furthermore, the resultant discoveries may form the basis for further advances by laterresearchers, thus potentially leading to a sustainable growth process.

In the literature there are two main traditions concerning how to model growth. In bothresearch traditions anintermediate sectoris introduced, where there are patents and market power,and the intermediate goods produced in this sector are inputs—together with labour—into thefinal-good sector where there is perfect competition. Thesetraditions are theRomertradition (seefor instance Romer (1990)), and theAghion–Howittor Schumpeteriantradition (see for instanceAghion and Howitt (1992)). However, in this book we use a different basic model in which thereare no intermediate goods, just a range of final goods which are imperfect substitutes for oneanother. This model is simpler, and in some respects it has more explanatory power as well. Forthe purposes of this chapter it is the best choice, partly dueto its simplicity, and partly due to thefact that we later build on it in our model of directed technological change (Chapter 9).

4.2. A simple model of endogenous growth

A key part of all models of endogenous growth is the relaxation of the restriction in Solowianmodels that knowledge is completely non-excludable. The non-rivality of knowledge is indis-putable; any number of agents can use the same piece of knowledge at the same time. However,knowledge is definitely not always non-excludable: it is often possible for an inventor to keep newknowledge secret; alternatively, it may be possible to prevent others from using new knowledgein production through systems of intellectual property rights such as patents.

If one firm has a patent and another does not then it is clearly important to distinguish betweenthe firms, and it turns out that in general we need to distinguish between analysis at the firm leveland at the aggregate level in endogenous growth models. In this book individual firms are typically

23

24 4.2. A simple model of endogenous growth

indexed using a subscripti, and the quantities of inputs used by such firms, and outputs produced,are denoted using lower case (small) letters. So for instance aggregate labour isL and labourused by firmi is l i . Furthermore, we typically assume that there exists aunit continuumof firms.1

This is a highly convenient way of handling the idea that there are many firms such that eachfirm is ‘small’ and the demand of each firm for inputs has no effect on the input price. Given acontinuum of firms then each one is ‘infinitely small’ and therefore a price taker with regard toinputs. Furthermore, if firms are symmetric (using the same quantities of inputs to produce thesame quantities of outputs) then we have for instance

L =∫ 1

0l idi = l,

wherel is labour demanded by the representative firm. So aggregate quantities are equal to firm-level quantities. For more on this see the mathematical appendix, Chapter 17.3.

4.2.1. The model environment.Now we turn to the model. Assume that a unit continuumof infinitely lived households, where the representative household hasLt members, maximizes theutility U of the household, which is given by

U =∞∑

t=0

βtCt.

So time is discrete, and utility is linear in consumption, discounted by a factorβ per period.The mass of products that can be made is variable, with each product made by a single firm.

Denote the mass of products made byN, and assume for now thatN = 1. Index the products byi,and focus on producti, quantityyi . The production function for producti is as follows:

yi = (Ali l i)1−αkαi .(4.1)

Herel i is the number of workers hired by the firm at pricewL , andki is the amount of capital hiredby the firm at pricewK . The price of labour is determined endogenously (within theeconomy),and the price of capital depends on depreciation and the interest rate, as we see below.

In periodt−1 a firm which plans to make goodi in periodt must decide how much to investin development of the production technology which in turn determinesAli , labour productivity.The production function forAlit is as follows:

Alit = ALt−1(ζzit−1)φ,

whereδA is depreciation,ALt−1 is general knowledge in the previous period,ζ is a productivityparameter,zit−1 is the quantity of research inputs bought by the firm, andφ is a positive parameter.So the more research inputs are bought by the firm, the more productive it will be. The price ofresearch inputs iswZt−1.

Capital is foregone consumption, and depreciates completely from one period to the next. So

Kt = Yt−1−Ct−1,

whereCt is consumption. So total capital in periodt is equal to the total production kept backfrom the previous period. The price of capital is simply 1/β, which is of course greater than one:to buy capital in periodt the firm must compensate the seller for delaying their consumption byone period, and thus pay the price 1/β per unit of capital instead of 1, where 1 is the price of finalgoods.

General knowledgeA is made up of the knowledge of each of the individual firms. In generalwe haveALt = F(ALt), whereALt is the set of all knowledge levelsAlit . However, for simplicitywe specify the functionF from the start:

ALt =

[∫ 1

0Aψlit di

]1/ψ

.

The parameterψ is greater than or equal to one. Ifψ = 1 then we simply add up firm knowledgelevels to get general knowledge. Ifψ > 1 this implies that if one firm has more knowledge than theothers then that firm makes an especially large contributionto general knowledge. On the otherhand, if firm knowledge levels are all the same (which they arein the model equilibrium) then

ALt = Alit ,

whereAlit is the knowledge level of any particular firmi.

1In the literature this may also be denoted a unit mass of firms,or a continuum of firms measure one.

Chapter 4. Endogenous growth theory 25

Finally, aggregate productionY is a function of the individual production levelsyi which areall equal toy in symmetric equilibrium:

Yt =

[∫ 1

0yηitdi

]1/η

= yt.

The parameterη is between 0 and 1, implying that the goods are not perfect substitutes, and eachproducer has a degree of market power. The price of the aggregate good is normalized to 1.

4.2.2. A single firm’s problem. Based on this information we can set up the optimizationproblem facing a firm which has decided to produce goodi in periodt. (Note that in Nash equi-librium no two firms will produce the same product in the same period.) The Lagrangian for theproblem is as follows:

Lit = pyityit −wlt l it −wktkit −wzt−1zit−1/β−λit

[

Alit −ALt−1(ζzit )φ]

.

SoL is equal to revenue minus costs, minus the Lagrange multiplier λ times the restriction onlabour productivity.

To solve the problem we need to know how the price of goodi, pi , is determined. For nowwe assume that goodi is produced by a single monopolistic firm with productivityAli . Then wehave

pi = (y/yi)1−η

hence MR=∂pi

∂yiyi + pi = ηpi ,

where MR is marginal revenue. So when firmi raises its production of goodi by one unit, itsrevenue goes up not bypi (the price of that unit at the start) but byηpi whereη is less than 1.Hence firms havemarket powerand can raise the price they receive for their goods by restrictingproduction.

Now take the first-order condition orf.o.c.on labour (using the above result) to yield

wL l i = l i∂piyi

∂yi

∂yi

∂l i= η(1−α)y1−ηyηi .(4.2)

Analogously we have (from the f.o.c.s for capital and productivity Ali )

wK ki = ηαy1−ηyηi(4.3)

and λiAli = η(1−α)y1−ηyηi .(4.4)

Finally, take the f.o.c. in research labour to show that

wzt−1zit−1/β = φλit Alit ,

and substitute in the f.o.c. for labour productivity to obtain

wzt−1zit−1/β = ηφ(1−α)y1−ηt yηit .(4.5)

Now use these results to derive an expression for the profits made by firmi, i.e. revenue minuscosts:

πi = pityit − (wlt l it +wktkit +wzt−1zit−1/β)

=

1−η[

1+φ(1−α)]

y1−ηt yηit .

This tells us that if 1−η[

1+φ(1−α)]

> 0 then firmi will make positive profits, hence there shouldbe an incentive for further firms to enter, raising the total mass of firms above 1. On the other hand,if 1−η

[

1+φ(1−α)]

< 0 then firm profits are negative, and firms should exit thus pushing down thetotal mass of firms. To make sense of this we simply need to assume thatN = 1 is an endogenousoutcome becauseη is an increasing function ofN; when more firms enter, competition betweenfirms intensifies, the elasticity of substitution between products increases (approaching 1), andprofits go down. So whenN is low, η is low, 1− η

[

1+φ(1−α)]

> 0, profits are positive andNincreases. WhenN is high the reverse is the case.

26 4.2. A simple model of endogenous growth

4.2.3. Aggregate allocation.We are now in a position to work out the aggregate allocationin the economy, assuming exogenous supply of labourL. From equation 4.1 and givenki = K andAli = AL we have

Y= (ALL)1−αKα.

Furthermore, the f.o.c.s in labour and capital imply that

wK K

wL L=

α

1−α.

So the relative shares of capital and labour are fixed atα/(1−α), just as in the neoclassical model.What is the quantity of capital,K? This is straightforward to calculate since (equation 4.3)in

symmetric equilibrium

wkK = ηαY,

and we know thatwk = 1/β. Hence

K = ηαβ(AlL)1−αKα

= (ηαβ)1/(1−α)AlL,

and Y= AlL(ηαβ)α/(1−α).(4.6)

But what isAl? We know thatAl = Al−(ζz−)φ, where the subscript minus-sign indicates the previ-ous period. And from (4.2) and (4.5) we know that

wz−z−wlL

= βφ,

hence z− = βφL(wl/wz−).

Substitute this in to the growth equation for knowledge to obtain

Al/Al− = [βφζL(wl/wz−)]φ.

This result effectively gives us the (endogenous) rate of research investment, and hence alsogrowth.2

To close the model and find the growth rate ofAL we need to know more about the supplyof the research inputZ. One option is to fix the quantity of research labour exogenously, but thengrowth is hardly endogenous. Another would be to assume thatit comes from the overall poolof labour, so for instance we might haveL = LY+ LZ, whereLY is labour devoted to productionandLZ is labour devoted to research, and both a paid the same wage. This option is slightly morecomplex, and we compromise by assuming that research labouris supplied elastically at pricewL , implying that individuals are prepared to work overtime asresearchers from the normal wage.Given the results above we have

wL L+wK K = ηY,

and hence

(wl−/β)Z− = (1−η)Y.

But we also know that in the aggregate

wl− = η(1−α)Y−/L−,

soZ−L−=β(1−η)η(1−α)

YY−.

If we setφ = 0.1 and other parameters to standard values (capital share 1/3, growth rate 2 percentper year, interest rate 5 percent per year), and set the period length to 10 years, then we findthat research labour is around 7 percent of production labour. Note that if we raised the wage toresearch labour then the quantity would diminish in proportion. The rather high rate of researchreflects the assumption that all sectors are characterized by research, technological progress, andcreative destruction. In a more sophisticated model we could mix stagnant and growing sectors,leading to lower overall research investment.3

2Note that there is a little more work to do since we should workoutwl/wz−.3Note the possibility that goodi could be produced by competitive firms using old technology.The production

function would then be

yi = (Alit−1lit )1−αkαit .

Chapter 4. Endogenous growth theory 27

4.3. The effect of endogenizing growth

The big difference from the neoclassical model is that we now have an endogenous mecha-nism through whichAL rises. Furthermore, the growth rate increases in proportion toLφ, where wesetφ = 0.1. So if the population doubles, we expect the growth rate to be multiplied by 20.1 = 1.07.In the next chapter we will see whether such a result makes anysense in a very long run context.Here we look at other effects that arise from the endogenization of the growth rate.

4.3.1. Aggregate production.The aggregate production function in the above model is, aspreviously noted,

Y= (ALL)1−αKα.

This is exactly as in the neoclassical model! And if we added the need for a natural resourceR in the production function, the endogenization of the growth process still has no effect on thelong-run aggregate production function.

The above conclusion does not only apply to our particular endogenous growth model, itapplies to those of Romer and Aghion–Howitt too. See for instance Chapter 16 of Aghion andHowitt’s recent book,4 and compare their analysis to ours, in Chapter 5. Both derivea conditionfor positive long-run growth. In the neoclassical model thecondition is (1−α− β)gA > βθ: thegrowth rate ofAL multiplied by the labour share must be greater than the resource share multipliedby the rate of decline in resource extraction. In the Aghion–Howitt model the condition is thatthe growth rate ofAL must be greater than the resource share multiplied by the rate of declinein resource extraction: the slight difference arises from two factors which are not directly linkedto endogenous growth (firstly, there is no capital in their model, and secondly there are constantreturns to labour and the intermediate input, and increasing returns to labour, the intermediate,and the resource flow.)

What then do we need to add to these models in order to start getting a real understanding oflinks between growth and environment? Below we will see thatwe need to add at least two char-acteristics to the models: different types of technological progress (augmenting different inputs);and different types of consumption (intensive in different inputs). In the remainder of this part ofthe book we focus on the former.

4.3.2. Capital. Another aspect of the neoclassical model scarcely affected by the endoge-nization of growth is the role of capital: capital accumulation plays a small role in long-rungrowth in the neoclassical model, and it also plays a small role in long-run growth in endogenousgrowth models.

Since these firms are competitive then we know from the first-order conditions that

wkki

wl li=

α

1−α,

and sincewk = 1/β we can solve forki in terms ofwl andli , and substitute into the production function to obtain

yi = A1−αlit−1lit

(

wlαβ

1−α

)α

.

Production costs arewl li +wkki , so profits are as follows:

πi = pi A1−αlit−1lit

(

wlαβ

1−α

)α

−1

1−αwl li .

This equation gives us a condition for the competitive firms using old technology to be unable to compete with the monop-olist using new technology: profits of the competitive firms must be zero at most when the monopolist sets her preferredprice, pi = 1. When this is the case we call the innovations made each period drastic, otherwise they arenon-drastic. Thecondition for drastic innovations is

0≥ A1−αlit−1wα

lt

(

αβ

1−α

)α

−1

1−αwlt ,

hence wlt ≥ Alit−1(1−α)(αβ)α/(1−α).

In symmetric equilibrium we know that

wl L = η(1−α)Y,

where Y= Al L(ηαβ)α/(1−α).

Insert this into the condition to obtain

Al/Alit−1 ≥ η−1/(1−α).

When this condition is not fulfilled thenpit is determined by the competitive firms using old technology,but productionwill be from the monopolistic firms using new technology, as long as they do not make negative profits at this price. Wefocus on the drastic case.

4The economics of growth, Aghion and Howitt. (2009).

28 4.3. The effect of endogenizing growth

The role of capital is small in the sense that capital accumulation does not drive the growthprocess, capital accumulation is a consequence of technological progress, which is the fundamen-tal driver of growth. Capital accumulation is thus a link in the chain connecting investment in newtechnology to growth in production. Capital can therefore safely be ignored in many situations.To see this, return to the model and in particular equation 4.6,

Y= Al L(ηαβ)α/(1−α).

Incorporating the constant intoAL we have

Y= A′LL

in the long run. This shows that there are likely to be many circumstances in which we cansafely ignore capital and treat production as if it were based purely on labour inputs. We take thisapproach in most of the remainder of this book.

Part 2

Natural resources and neoclassical growth

In this part of the book we set up and solve some very simple models in which natural resourceinputs are essential to the production of a single final good.We are interested in finding outwhat effect scarcity of nonrenewable resources has on the rate of economic growth, and also theconsumption rates and prices of the resource stocks. We continue to work within the single-sectorneoclassical framework. That is, we assume an economy with asingle product, which may eitherbe consumed or retained within firms as capital, to help with future production. Furthermore, weassume that the final-good production function is Cobb–Douglas, as in the baseline neoclassicalmodel.

We begin, in Chapter 5, by returning to the neoclassical model of Chapter 3, and addinga homogeneous nonrenewable resource to the production function. We consider three types ofresource: firstly, one available in a fixed quantity, i.e. land; secondly, one with a very large homo-geneous stock, costly to extract (consider iron ore, or coal); and thirdly we consider a Hotellingresource which is free to extract but available in a strictlylimited quantity. In Chapter 6 we applythis framework to explain the broad outlines of economic andpopulation development over the last20000 years or more. To do so we build and analyse a model economy in which a fixed quantityof land (the surface of the Earth) is an essential input to production, and both labour productivityand population evolve endogenously over time. In Chapter 7 we return to resource stocks andconsider the more realistic but also much more complex case of a large inhomogeneous resourcestock. Finally, in Chapter 8 we consider very briefly the situation when renewable resources areessential to production, and when polluting emissions diminish or damage productive capacity (orutility).

We find that the neoclassical growth model can be set up in a waythat is consistent with muchof the aggregate data that we presented in Figures 1.1–1.4, but that it provides insufficient detailfor policy analysis. In particular we show that to say more about optimal policy we need modelswith multiple products, and to have some understanding about the nature of consumer preferencesover those products. Furthermore, we need an understandingabout how technological change canaffect the relative productivities of alternative inputs.

CHAPTER 5

Neoclassical growth and nonrenewable resources

We now return to the neoclassical growth model of Chapter 3, and add a nonrenewable re-source to the production function. We begin with the simple case ofland; it is simple because thequantity of land is fixed. We then turn to the case in which the flow of resourcesR comes from anon-renewable stock which is very large and homogeneous, inwhich case increasing demand—driven by the increasing productivity of labour—drives increases in resource extraction while notaffecting the resource price. Finally, we consider the case made famous by Hotelling (1931), inwhich a homogeneous resource is available in a fixed total quantity, and costs nothing to extract.In this case, if the resource is essential to production then—if production is to be sustainable—itmust be extracted and consumed at a decreasing rate over time, at least in the long run.

5.1. Land

We begin this chapter with a neoclassical growth model with zero population growth, and addthe need for land in the production function. The quantity ofland is simply fixed atR, and thereis exogenous labour-augmenting technological progress.

Y= (ALL)1−α−βKαRβ

AL/AL = gAL

K = sY− δK.

To characterize the development of the economy, assume thatthere exists a balanced growth pathon which all variables grow at constant rates. This implies that K/K is constant, which implies(from the capital-accumulation equation) thatsY/K − δ is constant, implying in turn that the ratioof productionY to capitalK is constant. This implies thatY/Y= K/K, and hence (differentiatingthe production function with respect to time and substituting)

Y/Y= (1−α−β)gAL +αY/Y.

Rearrange to obtain

gY = gy =1−α−β

1−αgAL .

This confirms the reasonableness of the original assumption, i.e. that there exists a balancedgrowth path on which all variables grow at constant rates.

So given labour-augmenting technological progress, whenβ > 0 (i.e. land is a non-negligiblefactor of production) the growth rate of production is slowed. The underlying reason why land inthe production function slows growth whereas capital does not is that land is non-reproducible (bycontrast to capital).

The price or landwR can be obtained by considering the representative producer’s profit-maximization problem,

maxπ = pY(ALL)1−α−βKαRβ− (wL L+wK K+wRR).

Normalize the price of the final good,pY, to one. Then take the first-order condition inR to showthatwR = βY/R, and hence that

wR/wR = Y/Y.

So the price of land tracks growth in GDP.Finally, if we setβ = 0 (implying that land is not important) then we obtaingY = gy = gAL , just

as we did in the basic neoclassical model with the productionfunctionY= (ALL)1−αKα. Note thatwe also obtained the same result in the even simpler model in whichY= ALL.

How relevant is this model to reality? Clearly, the total supply of land is fixed. The onlytestable prediction of the model (apart from results that more-or-less follow by assumption) is thatthe price of land should grow at the overall growth rate. Long-run data on land prices is hard tocome by, but the prediction seems to be broadly correct.

31

32 5.2. An abundant resource, costly to extract

5.2. An abundant resource, costly to extract

Now we consider a mineral resource input instead of land. We assume that there is a verylarge homogeneous stock of this mineral (consider for instance iron ore). The input is costly toextract, since extraction requires the use of labour, capital, and the resource input, inputs whichcould otherwise have been used to produce the final good. There is exogenous labour-augmentingtechnological progress. Denoting final-good production byYY we have

YY = (ALLY)1−α−βKαYRβY;

AL/AL = gAL ;

K = sYY− δK;

R= φ(ALLR)1−α−βKαRRβR.

Hereφ is a parameter determining the relative productivity of theinputs in extraction comparedto final-good production, and total labourL is the sum ofLY andLR, and similarly for capital andthe resource. Because of the symmetry between the two production functions (forY andR), andthe fact that we have constant returns to scale, it follows the inputs (L, K, andR) will be usedin equal proportions in each case, it follows that we can reformulate the model with all inputsbeing devoted to production of the final good, some of which isconsumed, some of which goes tocapital, and some of which goes to resource extraction. Denoting total final-good production byY(thus returning to standard notation), and denoting final goods devoted to extraction asX we have

Y= (ALL)1−α−βKαRβ;

AL/AL = gAL;

K = s(Y−X)− δK;

R= φX.

Finally, note that total extraction costs are simplyX, since the price of the final good is normalizedto 1. And since we assume perfect markets, the resource pricewR is equal to unit extraction costsX/R, hence

wR= 1/φ.

Now, as previously, assume a b.g.p. and then characterize it. On a b.g.p.K/K is constant,which implies (from the capital-accumulation equation) that s(Y−X)/K− δ is constant, implyingin turn thatY, X, andK all grow at equal rates, and hence (from the extraction function) R alsogrows at this rate. Differentiate the production function with respect to time, andsubstitute forK/K andR/R to yield

Y/Y= (1−α−β)gAL+αY/Y+βY/Y.

Rearrange to yield

Y/Y= gAL .

So the need for the resource imposes no brake at all on the growth rate of production, sinceresource use tracks production (by contrast to land, which is fixed).

Note that the predictions of this model do a pretty good job ofmatching the data in Figure1.3, especially the left panel (metals): the resource priceis constant, and resource extraction tracksoverall growth. The broad outline of the story told by the model is simple and almost certainlycorrect: on the demand side, technological progress drivesincreasing demand for the resource; onthe supply side, it implies that when a fixed proportion of labour and capital is devoted to resourceextraction, the flow of extraction increases exponentiallywhile price remains constant.

The success of this simple model in accounting for historical data is striking, but so is itsunsuitability for predicting the future. The model predicts that resource extraction will continueto increase exponentially, indefinitely. As mentioned in the introduction, the current physicalextraction rate of minerals is of the order of 1010 tonnes per year globally. Assume that this ratecontinues to grow at the same rate as it has done over the past 100 years, i.e. approximately 3percent per year. Then extraction would be multiplied by a factor 20 each century, and in 700years we would be mining and using minerals roughly equal to the entire earth’s crustevery year,based on a figure of 2×1019 tonnes for the Earth’s crust. To predict the future we clearly need amodel with limited, inhomogeneous resource stocks. The rest of this part of the book is devotedto such models.

Chapter 5. Neoclassical growth and nonrenewable resources 33

5.3. A limited resource, costless to extract (Hotelling/DHSS)

In the previous section we had a resource that was costly to extract but available in unlimitedquantity. Now we go the other way and assume a homogeneous resource available in a known, fi-nite quantity. This takes us into the heart of the literaturefrom the 20th century, with the Hotellingrule from 1931 and the DHSS model from 1974.

5.3.1. General set-up.Recall that the Solow model concerns how production of a singlefinal good may increase over time through: (i) investment of the final good as capital to be usedin the production process in the future; and (ii) exogenous technological change. What if we add(limited, free) resources as inputs to production? We then have the Dasgupta–Heal–Solow–Stiglitzmodel, developed in the 1970s.1

Y= KαRβ(ALL)1−α−β

AL/AL = gAL

L/L = n

K = Y−C− δK

S ≥∫ ∞

0Rtdt.

Note that we have added a constant rate of population growth,n, and we have not fixed thesaving rate, instead we simply say that investment is the difference between productionY andconsumptionC. The flow of resource inputs isR, and the sum (integral) of resource inputs overinfinite time cannot exceed the initial stock of the resource(‘in the ground’) which we denoteS.Note that we can keep producing final goodsY even asR approaches zero; so if the final good ishammers, and the resource input is iron, we can make an unlimited number of hammers from afixed stock of iron (without recycling). The model is thus setup from the start to favour feasibilityof sustainable growth. Note that with a different choice of production function—such as Leon-tief—we can rule out the above property directly; given Leontief (and no resource-augmentingtechnological progress), you needx grams of iron per hammer, full stop.

5.3.2. Exogenous technological progress.What if we have the general set-up above, withgAL > 0, a fixed saving rates, andn= δ = 0? That is,

Y= KαRβ(ALL)1−α−β,

AL/AL = gAL ,

K = sY,

S ≥∫ ∞

0Rtdt.

Under these circumstances it is easy to show that we can maintain constant consumption if we setits initial level low enough.

(1) First assume balanced growth, i.e. all growth rates are constant.(2) Second, note that for any initial level of resource useR0 we can always find a rate of

decay in resource consumptionθ such that the resource is asymptotically exhausted:

S =∫ ∞

0R0e−θtdt = [−(R0/θ)e−θt]∞0 = R0/θ.

The lower isR0/S, the lower is the rate of declineθ.(3) Third, note that ifY grows at some constant rategY, while a constant proportions is

invested, andK/K is constant, thenY/K must also be constant. ThusY and K mustgrow at the same rate,gY = gK . This follows sinceK/K = sY/K.

(4) Fourth, differentiate the production function w.r.t. time and usegY = gK to find an ex-pression forgY in balanced growth:

gY =

(

1−β

1−α

)

gAL −βθ

1−α.

So by choosing initial resource consumption, and henceθ, we can choose the long-run growth rategY up to a maximum ofgAL[1−β/(1−α)].

1The original papers are Dasgupta and Heal (1974), Solow (1974), Stiglitz (1974).

34 5.3. A limited resource, costless to extract (Hotelling/DHSS)

Recall that when the flow of resource input was fixed (the case with land), we had

gY =

(

1−β

1−α

)

gAL .

This is identical to the expression above whenθ approaches zero, as we would expect, since whenθ = 0 the resource flow is constant. Put differently, the need for resource flows to decline puts apenalty on to the growth rate.

5.3.3. Hotelling. What will the solution be in a market economy? To answer this questionwe need to know the resource price. The resource is free to extract, but because it is available infinite quantity it isscarceand hence it will have a non-zero price on the market due to scarcityrent (also known as Hotelling rent, resource rent, etc.).

Imagine you own a stock of some resource, which you keep in a large pressurized under-ground tank. To extract the resource, you simply turn the tap. You have no alternative uses for thetank once the resource is exhausted, and there are no environmental considerations.

In order to decide when to extract the resource you consider the price path. Is the resourcepricewR increasing over time? If so, then all else equal you will keepthe resource and sell it later.On the other hand, if you extract and sell the resource then you can put the money in the bank, andit will grow at ratewm, wherewm is the interest rate (we express it this way to emphasize thattheinterest rate is the rental price of money). So your rule is toextract if the resource price is rising ata rate slower thanwm, and not extract if the price is rising at a rate faster thanwm. If wR/wR= wm

then you are indifferent.Now imagine you are one of a vast number of such resource holders, so the market for selling

the resource is competitive (as are all other markets in the economy). How do the others plantheir extraction? Presumably, they think the same way as youdo. Assume that the current price ishigh, and thus all resource owners expect slowly-rising prices in the future; then everyone wantsto sell. Now assume instead that the current price is low, andthus all resource owners expectrapidly-rising prices; then no-one wants to sell. In the former case—when everyone wants to sell—the price must drop in a discrete step down until it reaches apoint at which it is no longer thecase that everyone wants to sell. In the latter case it must rise in a discrete step up until it reachesa point at which it is no longer the case that no-one wants to sell. The equilibrium price is thesame in both cases; it is the point from which the price is expected to rise at exactly the rate ofinterest. At this point resource owners are indifferent between holding on to their resources andselling their resources: we have an equilibrium.

We have thus intuited the equilibrium price path for the resource:

wR

wR= wm,

wherewR is the price of the resource andwm is the price of (borrowing) money, i.e. the interestrate.

Now return to the overall model. Given perfect markets we know that the Hotelling rule mustbe obeyed, i.e.

wR/wR= wm.

But the resource price is just the marginal product of the resource for the representative producer,i.e.

wR = βY/R.

Chapter 5. Neoclassical growth and nonrenewable resources 35

Differentiate this expression and use the Hotelling result, andour result regarding the b.g.p., toshow that on a b.g.p. we have

wm= wR/wR= (gAL + θ)[1−β/(1−α)].

If the interest ratewm is simply equal to the pure rate of time preferenceρ—based on the simplestpossible model of preferences over time,U =

∫ ∞

0 cte−ρtdt—then we have

θ =ρ

1−β/(1−α)−gAL ,

so the optimal rate of decline of resource use is increasing in the degree of impatience of agents(ρ) and in the weight of resources in the production function (β), but decreasing in the rate oftechnological progressgAL . We can now put this back into the equation forgY to yield

gY = gAL −ρβ/(1−α)

1−β/(1−α),

and hence θ = ρ−gY.

So the growth penalty due to the need to gradually cut resource consumption depends on the rateof time preferenceρ and the weight of the resource in the production function compared to theweight of labour. Note that for a meaningful solution we require thatρ > gY.

As an exercise, you should redo the above assuming capital depreciation at rateδ. You shouldfind that it makes no difference, since when capital depreciates we haveK/K = sY/K − δ. On ab.g.p. (withK/K constant) this again implies thatY/K is constant, henceY/Y= K/K, just as whenδ was zero. Since capital still grows at the same rate as overall production in this economy, weobtain the same result for the growth rate. Note however thatthe level of production at a giventime will be lower given depreciation, since the stock of capital will be lower. More precisely,assume an economy in whichδ = 0 is on its b.g.p. Now assume that, suddenly, the capital stockstarts to depreciate at rateδ > 0; then the economy will shift gradually towards a new b.g.p.onwhich (a) the level ofY at a given time is lower than on the previous b.g.p., and (b) the growth rateof Y is the same. This shift is illustrated in Figure 5.1.

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

years

log

Y

Figure 5.1. The transition from one b.g.p. to another (lower) b.g.p., afterδ in-creases from 0 to 0.1. Note that the figure is for a discrete-time economy withthe following parameters:α = 0.33;β = 0.05; θ = 0.1 (assumed exogenous andconstant);s= 0.2; gAL = 0.03. The economy starts with a capital stock such thatit is on the initial b.g.p.

5.3.4. Hotelling with a resource supply monopoly.Imagine you have a monopoly over theresource; you are the only supplier. This makes your problemmore complex, since you mustaccount for the effect of your own extraction on the price. To see what’s going on, we need toset up adynamic optimization problem. The simplest way to do this is to usediscrete time, in

36 5.3. A limited resource, costless to extract (Hotelling/DHSS)

which case we need to set up aLagrangian. We want to maximize discounted profits, subject tothe resource restriction. The profit function and the restriction can be written as follows.

π = wrtRt;∞∑

t=0

Rt ≤ S.

Herewrt is the resource price, andRt is the quantity extracted and sold in periodt. We can usethese two equations to set up the Lagrangian, which is

L =

∞∑

t=0

(

11+wm

)t

wrt Rt +λ(S−∞∑

t=0

Rt)(5.1)

Hereλ is theresource rentor (expressed in a more standard way) the shadow price of the resourcestock.2 Now take the f.o.c. inRt to obtain

(1+wm)t ∂L

∂Rt= wrt +Rt

∂wrt

∂Rt−λ(1+wm)t = 0.(5.2)

Now the first two terms together represent the change in revenue given a change in quantity:marginal revenue,MR. To find the Hotelling rule, consider the first-order condition in successiveperiods to derive

MRt+1

MRt= 1+wm.

Now let’s close the model by adding a demand function. For reasons that will become clear, weassume an inverse demand function of the form

wrt = A+BR−ǫt .

Then

MRt = wrt +Rt∂wrt

∂Rt= wrt (1− ǫ)+ ǫA,

and the rule is

wrt+1+ ǫA/(1− ǫ)wrt + ǫA/(1− ǫ)

= 1+wm.

To get a handle on the above result assume that our monopolistis actually a representativeresource owner with no market power. Then marginal revenue is simply price, and

wrt+1

wrt= 1+wm.

To link with our results above (in continuous time) note thatthe expression corresponding towR/wR is (wrt+1−wrt )/wrt . But we can rearrange the above result to yield

wrt+1−wrt

wrt= wm,

which is thus equivalent to the continuous time result we obtained earlier (see the mathematicalappendix, Chapter 17.2). So whenA = 0 the growth rate of prices is identical to the growth rategiven perfect markets; whenA< 0 prices grow more slowly given market power; and whenA> 0prices grow faster given market power. The intuition comes from the elasticities: whenA = 0we have constant elasticity demand, so as the price rises over time the elasticity of demand isunchanged. But whenA < 0 the elasticity of demand increases as price increases, making themonopolist more reluctant to increase prices, and whenA> 0 the elasticity of demand decreasesas price increases, encouraging the monopolist to raise prices further.

Note that we also require atransversality conditionto solve the model fully. It is not enoughto know the rate of change of prices, we must also know the level of the price as some point in time.There are two particularly simple possibilities: one is that the resource should only be exhaustedasymptotically, the other is that the price should be equal to some backstop price at the time ofexhaustion. For the case of exhaustion with a backstop, a slower rate of price increase implies (fora given transversality condition) that the initial price must be higher, and hence the initial pricepath will be higher and the date of exhaustion will be put back. We return to such questions laterin the book (in Chapter 7).

2This quantity has many names in the literature; two other common ones are theHotelling rentand thescarcity rent.

Chapter 5. Neoclassical growth and nonrenewable resources 37

5.3.5. Hotelling with extraction costs.Now we introduce extraction costs into the Hotellingmodel, simultaneously returning to continuous time. The simplest thing to do is to assume thatmarginal extraction costs are constant,c. Continue to write the price aswr , and the resource rentasλ. Then (in the case with perfect competition) the price is thesum of the resource rent and theextraction cost,wR = λ+c, and sincec is constant ˙wR = λ. The resource rent rises at the interestrate (λ/λ = wm) and the modified Hotelling rule is

wR

wR= wm ·

wR−cwR

.

So whenc= 0 this reduces to the original rule, but whenc approacheswR (implying that the re-source rent is only a small proportion of the total price), thenwR/wR approaches zero, i.e. constantprice.

Unfortunately the model with constant extraction costs is still all-too naive compared to realityin which a variety of factors affect extraction costs. Consider the picture below, and with its helptry to identify as many such factors as you can. Furthermore,categorize them according to whetherthey should make extraction costs rise or fall over time.

Wage RiseProductivity DeclineDepth Rise

The three key factors are the wages paid to workers, their productivity, and the depth of the re-source (where the latter should be interpreted broadly as the physical quality of resource deposits).Both wages and productivity should rise over time, and thesetrends may be expected to canceleach other out. (Why?) Depth should also rise over time, hence overall it seems that we should ex-pect resource prices to rise over time. Note also that if the ‘width’ of the resource stock increases,then the rate of increase in depth for a given extraction ratewill be lower. We return to resourceextraction in Chapter 7.

5.3.6. Hartwick. The majority of the older literature focused on the case withno technolog-ical progress, i.e.gAL = 0. Then we know straight away (from the Solow model) that long-rungrowth is not possible, since there are diminishing returnsto capital accumulation. This conclu-sion is strengthened when there are also resource constraints.

An even stronger result—which also follows directly—is that long-run sustainability (non-negative growth) is also impossible in the baseline case with a fixed resource stock free to extract.Given such a stock, the extraction rate must approach zero inthe long run, and the only way tocompensate is if the capital stock (and hence alsoK/Y) approaches infinity. But this is impossibleto sustain in the presence of depreciation.

The response to these results was not to reject the model withzero technological progress,but instead to do further violence to the original Solow set-up: having taken away technologicalprogress, the next step was to take away capital depreciation as well. In this set-up long-rungrowth remains impossible (due to diminishing returns to capital) but sustaining production andconsumption is possible: all that is required is a sufficiently high rate of capital investment. Howhigh is sufficiently high? This question is tied up with the famous Hartwick rule.

38 5.3. A limited resource, costless to extract (Hotelling/DHSS)

The Hartwick rule—Hartwick (1977), following Solow (1974)—has generated a huge amountof confusion, and many incorrect claims have been made.3 Here we state some of the correct re-sults in a simple way. The key result is that—in an economy with perfect markets, constant pop-ulation and constant technology—if the value of capital stocks is kept constant then productionwill also be constant.

What does the above result imply about sustainability? How difficult is it to maintain aconstant capital stock? If capital doesn’t depreciate thenall we need to do to keep the total stockconstant is to invest the proceeds of using up one stock of capital into boosting some other stockof capital. On the other hand, if capitaldoesdepreciate over time then we have a much harder taskholding the total stock of capital constant, indeed it will typically be impossible to do so.

Consider a DHSS economy such as those we have analysed above,and assume that invest-ment in capital equipmentK is determined by the Hartwick rule, implying that it is not determinedby market. In such an economy there are two capital stocks,K andR, and the rule is

wK K +wRR= 0.(5.3)

So we assume that a regulator lets the market determine resource extraction and price, and theninvests according to the rule.

The production function is

Y= (ALL)1−α−βKαRβ

and the capital accumulation equation is

K = sY− δK,

where we now treats as an unknown variable. Furthermore,wK and wR are the (respective)marginal revenue products ofK andR, hence

wK K = αY

and wRR= βY.

The first equation tells us that the marginal product of capital wK = αY/K. Consider now acapital owner hiring out a single unit capital, value 1 (recall that capital and the final good arethe same, and that the price of the final good is normalized to 1). The gross return (income flowper unit of capital hired out, value 1) is justwK , but the net return iswK − δ, because the capitaldepreciates (disappears!) at a constant rate ofδ. In equilibrium capital owners must be indifferentbetween hiring and selling capital, which implies that the interest rate—here we call itρ ratherthanwm—must be equal towK − δ. So we have

ρ = wK − δ = αY/K − δ.(5.4)

Now return towK K = αY and use the capital accumulation equation to write

wK K = αY/K(sY− δK).(5.5)

Now turn toR. SincewRR= βY and the Hotelling rule applies we know that

R/R= Y/Y−ρ,

whereρ (the discount rate) is determined by (5.4). Furthermore, from the production function weknow that

Y/Y= αK/K+βR/R,

hence R/R= αK/K+βR/R−ρ.

Rearrange to yield R/R=1

1−β

[

αK/K−ρ]

,

hence R/R=1

1−β[

αsY/K −αδ−ρ]

,

and (sincewRR= βY) wRR=β

1−β[

αsY/K −αδ−ρ]

Y.(5.6)

Now combine (5.5) and (5.6) with (5.3) to yield

sY= [ρ(β/α)+ δ]K,

and (substituting forρ) sY= βY+ δ(1−β/α)K.

When δ = 0 a Hartwick path is feasible: investment is a constant fraction of production, theinterest rate approaches zero from above, and total resource extraction is bounded. However,

3For rigourous support for this statement see Asheim et al. (2003).

Chapter 5. Neoclassical growth and nonrenewable resources 39

1900 1920 1940 1960 1980 2000

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Year

log1

0, n

orm

aliz

ed

1850 1900 1950 2000−0.5

0

0.5

1

1.5

Year

log1

0, n

orm

aliz

ed

Metalquantity

Metalexpenditure

Metal price

Global productGlobal product

Energyexpenditure

Energyprice

Energyquantity

(a) Global Metals (b) Global Energy

Figure 5.2. Long-run growth in consumption and prices, compared togrowthin global product, for (a) Metals, and (b) Primary energy from combustion.

whenδ > 0 then there is no feasible Hartwick path: on a feasible path,R must approach zerowhile K approaches infinity, but ifK →∞ then s must be greater than 1 (which is impossible),and furthermoreY/K will approach zero implying thatρ must be negative henceRmust grow, notdecline.

Finally, note that even if we observe in the present (a) that resource rents are reinvested inphysical capital, and (b) that physical capital does not depreciate, and (c) that all markets areperfect, this still does not guarantee sustainability, because we do not know the preferences basedon which the market interest rate is determined. One thing wecan say for sure is that if theinvestment rate is determined by the market and if the pure rate of time preference is non-zerothen the rule will not be followed in the long run, because this implies that the interest rate isbounded below at a strictly positive level even when the growth rate is zero, implying that there isa limit to capital accumulation.

Summing up, it remains to be demonstrated that the Hartwick rule is of anything other thanpurely academic interest.

5.3.7. DHSS and historical data.Recall that the key results of the DHSS model are that ona balanced growth path the following relationships should hold:

gY = gAL −ρβ/(1−α)

1−β/(1−α)and θ = ρ−gY,

while wR/wR = ρ.

Hereθ is the rate of decline of resource use over time,ρ is the pure rate of time preference, andwR is the resource price. Furthermore,α andβ are the respective factor shares of capital andthe resource in final-good production. The results show how the need for resource inputs (β > 0)puts a penalty on the growth rate that can be sustained in the economy. The size of the penaltyis increasing in the rate of time preference,ρ, and the weight of the resource in the productionfunction,β. Furthermore, we see that resource use declines at a constant rate, and that the resourceprice increases rapidly (recall thatρ > gY).

How do these results match up to the data for real economies? The answer to this questionis to be found in Figure 1.3, reproduced here as Figure 5.2. Here we see that—far from declin-ing exponentially—resource and energy use have risen exponentially, tracking global product.Furthermore, the prices of resources and energy have been remarkably constant in the long run,whereas according to the model they should grow faster than global product (the discount rate isalways faster than the growth rate in optimal growth models). So the model predictions seem tobe completely, hopelessly wrong.

Things are perhaps not as bad as they seem for the DHSS model. The reason is that theproblems can all be traced to the same root, which is the wildly incorrect model of the resourceextraction sector. Resources are not in reality available in a fixed known quantity, free to extract.Instead they are expensive to extract, and furthermore the costs of extraction vary in a complex

40 5.3. A limited resource, costless to extract (Hotelling/DHSS)

way due to factors such as technological change and cumulative extraction (as more is extractedwe must dig deeper, raising costs).

Recall our previous model in which resource stocks were unlimited and extraction costs con-stant. This model did a much better job of fitting the data, butat the expense of ignoring resourcescarcity, making it useless for very long run analyses. In the next chapter we explore models inwhich we allow for both extraction costs and scarcity,

5.3.8. DHSS, prediction, and policy for sustainability.The majority of modelling withinthe DHSS paradigm has been based on the ability of capital to substitute for resource flows, withzero technological progress. If the model is interpreted literally—with capital interpreted as re-cycled final product, and final product being unchanging—it makes no sense as a description ofthe economy. If it is interpreted flexibly—as suggested by Groth (2007), p.10–11—and capitalaccumulation is interpreted as a move towards clean technology, recycling, substitution betweeninputs, and changes in the composition of final output, then the interpretation of the model is sofar removed from its actual assumptions as to make the model meaningless and impossible to test.

Put differently, if we really think that moves towards clean technology, recycling, substitutionbetween inputs, and changes in the composition of final output are the key to sustainability, thenwe should build a model in which these processes are explicitly accounted for, and find waysof testing the strengths of these different processes and hence their ability to deliver sustainablegrowth.

This point is effectively conceded by the authors of the DHSS model themselves. Prior tothe development of the DHSS model, Solow (1973)—in an essay where he is unconstrained bythe need for mathematical formalism—argued for the abilityof the economy to adapt to resourcescarcity. In this context he sets out a series of mechanisms as key, without ranking them in impor-tance; ironically, the DHSS mechanism—Solow’s focus in subsequent quantitative modelling—is not mentioned at all. Regarding demand for resource inputs Solow sets out three mechanisms:

(1) Increase—through technological change—resource efficiency in production of one ormore product categories;

(2) Substitute on the consumption side away from product categories in which the produc-tion process is resource-intensive.

(3) Increase—through technological change—the efficiency of an alternative (substitute)resource in production of one or more product categories.

Solow pays less attention to the supply of resources, but there we have improving technology ofextraction and—working in the other direction—the impoverishment of deposits. He could alsohave mentioned recycling.

In the same vein, Dasgupta (1993) also discusses our abilityto substitute for physical re-sources in the production process. Dasgupta argues that there are nine ways in producers can sub-stitute for non-renewable resource inputs, of which 7 are different forms of technological progress,number 8 is switching to lower-grade inputs, and number 9 is the mechanism of the DHSS model,i.e. the substitution of capital for resources. Dasgupta admits (p.1115) that this last mechanism is‘limited’, indeed ‘beyond a point fixed capital in production is complementary to resources’.

Thus the critique of ecological economists such as Hermann Daly is correct. Consider Daly’scritique of Solow–Stiglitz [i.e. DHSS], Daly (1997) p. 263:

In the Solow–Stiglitz variant, to make a cake we need not onlythe cook andhis kitchen, but also some non-zero amount of flour, sugar, eggs, etc. Thisseems a great step forward until we realize that we could makeour cake athousand times bigger with no extra ingredients, if we simply would stir fasterand use bigger bowls and ovens.

This is essentially the same as our critique, that the literal interpretation of capital accumulationas being able to compensate for declining resource flows makes no sense.

We argue—following Solow—that the three mechanisms above are key to understandinghow the economy adapts in the long-run to changes in resourceor energy availability, or to policymeasures regarding resources or energy. How important are they relative to one another, and howdoes their existence affect optimal policy?

CHAPTER 6

Pre-industrial human development

We now apply some of what we learnt in the Chapter 5 to help us build a very simple model toanalyse pre-industrial human (economic) development. Theaim is to gain a primitive understand-ing of the trends shown in Figure 1.1 from the introduction, in which we observe accelerating(although slow) population growth and almost constant GDP per capita. A key element of themodel is that there is a fixed quantity of land—the surface of the globe—which is the habitat ofpre-industrial people.

6.1. Population dynamics

We begin by approaching the problem of long-run population and economic growth froma non-economic angle, focusing on population alone. Consider the population dynamics of anywild animal, prior to the era of human domination of the planet. Populations will of coursefluctuate due to random variations (for instance in climate)and interactions between species, butit is nevertheless reasonable to suppose that over timescales of thousands of years the populationsof many species will vary around some long-run ‘equilibrium’ level: when the population is abovethis level it will tend to decrease, and when it is below this level it will tend to decrease. Thereis certainly no reason to suppose that a randomly chosen species should—on average—have anincreasing population.

What then determines the equilibrium population level? By definition, it is the populationlevel at which births (assuming an animal species) equal deaths, on average. When the populationis above this level births tend to exceed deaths, whereas when the population is below this levelthe reverse tends to be true. The reason for this pattern is likely to be competition within thespecies for some resource the quantity of which is exogenous; this resource could be food, but itcould also be something else, such as nesting sites.1

Applying the above reasoning to prehistoric humans, the obvious question to ask is, why dowe see a slow increase in the human population in prehistorictimes, even prior to 5000 BCE?The reason must be either that exogenous changes are making the global environment more hos-pitable for the species, or that the species itself is developing in some way (possibly managing itsenvironment) such that a higher population is sustainable.

Put this way, it should be obvious why human population has risen over time since 20000BCE: over time we have learnt more and more about how to control and utilize our environment(the globe), and this has allowed us to increase in number, partly by populating our existing (20000BCE) range more densely, and partly by extending our range tohabitats that were previouslyinhospitable for us. Put another way, technological progress has allowed our population to grow.Consider the following quote from Sagoff (1995, p617).

[F]or almost all individuals of any species, nature is quitepredictable. It guar-antees a usually quick but always painful and horrible death. Starvation, para-sitism, predation, thirst, cold, and disease are the cards nature deals to virtuallyevery creature, and for any animal to avoid destruction longenough to reachsexual maturity is the rare exception rather than the rule (Williams, 1988).Accordingly, mainstream economists reject the idea [. . . ] that undisturbedecosystems, such as wilderness areas, offer better life-support systems thando the farms, suburbs, and cities that sometimes replace them. Without tech-nology, human beings are less suited to survive in nature than virtually anyother creature. At conferences, we meet in climate-controlled rooms, dependon waiters for our meals, and sleep indoors rather than al-fresco. Nature is notalways a cornucopia catering to our needs; it can be a place where you cannotget good service.

1There are other possible reasons for the pattern, includingthe number of predators, which may rise when the speciesis abundant, hence pushing numbers down again.

41

42 6.2. A simple model of Malthusian growth

−20000 −10000 0 1700

0

1

2

3

4

5

year

Average yearly production per capita

Global population

Lo

gsc

ale,

no

rmal

ized

Figure 6.1. Global product and population, historical (data from Brad DeLong).Both variables are normalized to start at zero. Population grows by a factor ofapproximately exp[5], i.e. about 150. Average yearly production per capita isclose to 100 USD throughout.

If technological progress has allowed population growth, has it also allowed us to becomericher? In recent times the answer is a resounding yes, but what about 20000 years ago? Accord-ing to Figure 1.1 in the introduction (reproduced here as Figure 6.1), income per person was infact almost unchanged between 20000 BCE and 1700 CE. Why might this be?

To understand this, return to the example of the population of a wild animal. Assume thatsomething happens which shifts the equilibrium populationof the animal upwards. This couldbe a change in the climate, the extinction of a predator, or the discovery by the species of anew technology.2 In the short run, the result of the change will be an increase in productionand consumption per capita in the wild population. However,in the long run the result will bean increase in equilibrium population, and consumption percapita will be back to its original(equilibrium) level.

Turning to the human population, it is evidently not reasonable to apply the above reasoningto the analysis of population trends in modern economies, however, if we go back in time farenough it seems likely that the same mechanisms should apply: that is, technological progressresults—in the long run—not in increased income per capita,but rather in increased populationper square kilometre.

6.2. A simple model of Malthusian growth

Now we consider how to use the above insights in an economic model to explain the datamore fully.

6.2.1. Aggregate production.Recall the basic Solowian insight that accumulating a singlefactor—such as capital—cannot give long-run growth when the accumulated factor must be com-bined with other factors in a constant-returns-to-scale (CRS) production function. In an economyin which land is an important input this means that accumulation of both labour and capital to-gether will not yield a corresponding increase in production; that is, if for instance both labourand capital are doubled, production will not double. The reason is that the greater number ofpeople implies that each person has to make do with less land.This is captured in a simple wayusing the aggregate production function familiar from Section 5.1:

Y= (ALL)1−α−γKαRγ.

Hereγ is a parameter which determines the factor share (and hence relative importance) of landR, which is fixed.

In modern industrialized economies it is not obvious whether or not this production functionis applicable. On the one hand, land is of course essential for the production of food, and for livingspace, recreation, and nature among other things. But it is possible that if the same populationwere crowded onto half the land the total value of productionmight go up rather than down,

2The latter idea may sound ridiculous to you. If it does, consider the broad definition of ‘technology’ in economics,and then look at, for instance, Nature News, 14 December 2007, “Unique orca hunting technique documented”.

Chapter 6. Pre-industrial human development 43

as denser living might encourage cooperation or the spread of knowledge. However, in oldereconomies in which agricultural production makes up a largeproportion of total GDP it seemsclear that the relationship described in the above equationshould hold.

Recall also that in a growing economy the quantity of capitaltends to track overall production;that is, on the equilibrium growth path we haveK = φAL, whereφ is a parameter. AssumingKtracksY we haveK = K0Y whereK0 is a parameter. Substitute this in to yield

Y = (ALL)1−α−γ(K0Y)αRγ;

Y1−α = (ALL)1−α−γKα0 Rγ;

Y= (ALL)(1−α−γ)/(1−α)K′0Rγ/(1−α).

With suitable definitions we can write the latter equation as

Y= (AL)1−βRβ.

So, technological progress drives growth, whereas increases in populationreduceproduction percapita:

Y/L = A1−β(R/L)β.

This is what we expect, since higher population dilutes the land available.

6.2.2. Population dynamics.For our model of the global economy we must add equationsdetermining the growth in population and the technology factor A. Regarding population, a stan-dard growth function from biology is the logistic:

Lt+1− Lt = θLt(1− Lt/L∗t ).

Hereθ is a parameter, andL∗t is equilibrium population, often called thecarrying capacity. Thisis normally thought of as a constant, but in our model it will of course be a variable: as humans’ability to control the environment increases, the same environment can support a larger population.Note thatθ determines how quickly actual population will react to changes inL∗. We choosea value ofθ such that changes in technology are slow compared to changesin population; theevidence from Figure 6.2 suggests extremely low rates of technological change up to 1700 CE,implying that population will always be close to its equilibrium level.

To find L∗t we simply assume a level of income per capitaY/L at which the population isstable. Denoting this level as ¯y, rearrange the aggregate production functionY/L = A1−β(R/L)β toyield

L∗t = (At/y)1/βR/At.

So asAt rises, so doesL∗t .

6.2.3. Technological progress.Finally, and crucially, the rate of technological change. Herewe simply take the result for aggregate technological progress derived in Chapter 4, i.e. that growthis proportional to the size of the labour force raised to the power ofφ, where we choseφ = 0.1 inour parameterization. Thus we have

At/At−1 = ΩLφt−1,

whereΩ is a parameter.

6.2.4. Results and discussion.This model is straightforward to simulate using softwaresuch as Excel or Matlab. Figure 6.2 shows the simulated global economy—compared to thedata presented previously—using the following parameters: y= 1, R= 1,β = 0.3, θ = 1.4,φ = 0.1,andΩ= 1.00014, with the starting pointA= 1, L= 1. (We have subsequently normalized to matchthe levels in the data.) Note that the model produces an excellent match to the data. Two pointsmust however be raised at this point: firstly, the data are very uncertain; secondly, the model isextremely simple and is only intended to show one way in whichpopulation, production, andtechnology can be linked in a dynamic model.

Note that we called the model Malthusian. The name is derivedfrom Thomas Malthus, andin particular his bookAn essay on the principles of population, first published in 1798. He ar-gued for the existence of precisely the mechanism captured in population model: that increasingproductivity from a fixed quantity of land leads to increasedpopulation and not increased incomeper capita. Hence humanity is permanently on the borderlineof survival, and the natural tendencyfor population to increase must be checked by death caused bydisease, starvation, or (perhaps)self-control.

Ironically, Malthus’ essay was written at almost preciselythe time at which the link betweenproductivity and population was broken, i.e. the time at which increased productivity actually

44 6.2. A simple model of Malthusian growth

−20000 −10000 0 1700

1

10

100

Average yearly production per capita, 1990 USD

Global population (baseline 1700)

Year

Lo

gar

ithm

icsc

ale

Figure 6.2. Global product and population. Continuous lines, model simula-tion; dashed lines, historical data from Brad DeLong.

led to permanent increases in welfare rather than increasesin population alone: the demographictransition. That is, the transition from a situation with high birth and death rates, to one with lowbirth and death rates. The transition typically occurs at the same time as a country’s economyindustrializes, although that does not mean the industrialization as such is the direct cause; likelydirect factors driving the transition include the strengthening of women’s rights and education, anda reduced probability that newborn children will die beforereaching adulthood. We do not discussit further here, but simply note that global population growth is not on an inherently unsustainablepath; indeed, global population is likely to level off at around 10 billion people (current population7 billion), i.e. an increase of less than 50 percent over today’s level. By contrast, average globalproduct per capita is likely to grow by around 50 percent every 20 years for the next 100 years ormore. (50 percent every twenty years corresponds to 2 percent per year.) We therefore leave outpopulation growth completely from our forward-looking models.

CHAPTER 7

Resource stocks

In this chapter we analyse past and future paths of price and extraction rate of exhaustible(non-renewable) resources, and link this to long-run economic growth and sustainability. First weestablish the fact that existing and historical price trends demonstrate that—according to marketagents—exhaustion of critical non-renewable resources isdefinitely not imminent. It is of coursepossible that market agents have got it wrong, but this wouldbe surprising if so, since the differ-ence between right and wrong predictions is worth billions of dollars, and the key variables andprocesses are rather predictable and observable, unlike (for instance) the driving forces of businesscycles. We go on to examine long-run scenarios based on economic modelling and assessment ofstocks and likely technological changes. To help us we develop a simple model of resources asinputs to production, resources which are extracted from inhomogeneous stocks. We skate oversome major aspects of the maths, focusing instead on the key results in simplified form.

7.1. The elephant in Hotelling’s room

In this section we show that—according to market agents—exhaustion of critical non-renewableresources is definitely not imminent. We borrow heavily fromHart and Spiro (2011): large sec-tions of the text are taken directly from that paper.

7.1.1. Theory. Market agents value resources in the ground as assets, as shown by Hotelling.If those resources are valuable, and their price is failing to rise, then agents will realize those assets,i.e. sell them. This will cause the price to fall to a lower level, from which it will (in equilibrium)rise.

In terms of Hotelling’s analysis, there are two possible reasons for the failure of prices torise: either (i) resource markets systematically fail to value resources in the ground accordingto the theory; or (ii) the scarcity rent is well-behaved, butmasked by other factors. In bothcases, the implication is that factors other than the scarcity rent are important in shaping theresource price. Failure to value resources correctly couldfor instance be due to a failure to foresee(stochastic) discoveries, leading to a fall in the rent eachtime a discovery is made; this is illustratedin Figure 7.1(b). However, as was shown as early as 1982 by Arrow and Chang, rational actorswill take account of the probability of new discoveries being made, and the effect of allowingfor stochasticity will be for the rent to fluctuate around theoriginal trend. Only in the case of aconstant series of surprises, all in the same direction, could new discoveries hold back the long-run growth in the rent. Imputing such a series of surprises boils down to assuming that actorson the resource market are not rational, hence the analysis is hoist by its own petard; that is, ifmarket actors are not rational then other fundamental elements of the market analysis also breakdown. Another reason for markets not to value resources according to the theory could be thatpolitico-economic factors play an important role, as argued by many authors in the resource curseliterature.1 Resources are frequently state-owned, and in such cases it is a reasonable conjecturethat there are other, non-market, mechanisms shaping extraction and price paths.

Turning to the possible masking of the scarcity rent, note first that other components of theprice may be a combination of extraction costs and rents due to market power, which we denote as‘resource costs’. Resource costs could mask a rising scarcity rent in two different ways, illustratedin Figure 7.1(c) and (d). In the first, which we denote ‘declining costs’, the rent is a significantcomponent of the price, but the fall in resource costs compensates for its rise. In the second,‘low scarcity’, the rent is only a tiny component of the overall price, and hence its rise has aninsignificant effect on the overall price. It is straightforward to demonstrate both cases in theory.For instance, a popular explanation for declining costs is that technological progress in extractionpushes down unit extraction costs; see for instance Lin and Wagner (2007). However, as Hart(2009) points out, this should only occur if technological progress in extraction outstrips progressin other sectors, since otherwise the prices of inputs (suchas labour) should rise in line withincreasing productivity.

1For a survey of this literature see Van der Ploeg (2011).

45

46 7.1. The elephant in Hotelling’s room

pp

pp

tt

tt

(a) Hotelling resource (b) Non-rational agents

(c) Declining costs (d) Low scarcity

Figure 7.1. Resource price as a function of time. The lightly shadedarea repre-sents the scarcity rent, the heavily shaded area representsother factors, i.e. thesum of extraction costs and rents due to market power.

Note that low scarcity rents can arise in a number of ways despite finite stocks, for example ifthere is a renewable substitute. For brevity, we follow Nordhaus (1973a) in the following argumentby assuming abackstop technology, i.e. a technology capable of substituting for the resourceat aprice which is independent of demand. Then if extraction costs are equal to the cost of a backstoptechnology, it is obvious that the scarcity rent will be zeroup to the point of exhaustion.2 Moresubtly, this will also be the case if extraction costs are expected to be equal to the backstop costat the time of resource exhaustion. Furthermore, it has long been known that given a sufficientdegree of market power prices will go straight to the backstop price and stay there, even in theabsence of extraction costs; see for instance Teece et al. (1993) or Dasgupta and Heal (1979).

7.1.2. Simulation. We report the results of some simple simulations for crude oil. The aimis not to prove the level of the scarcity rent, but instead to illustrate the necessary implications of ahigh scarcity rent. We assume that resource costs change at aconstant rate, while the scarcity rentrises at a constant rate (implying that the rate of return on holding the resource is constant). Underthese circumstances, if prices are known at three points in time—such as past, present, and at thetime of exhaustion—then if any one of the rate of return on theasset, the rate of cost decline, andthe current scarcity rent are known, then the other two are fixed by the model. Furthermore, thelower the rate of return, the higher the current scarcity rent, and the steeper the rate of cost decline.We illustrate this in Figure 7.2, in which we take the averageprices for 1980–1989 and 2000–2009,3 and assume that oil reserves will run out in 2050 at a backstopprice of 150 dollars/barrel,4

and illustrate two of the possible combinations which are consistent with the model.Figure 7.2 shows how a higher level of the current scarcity rent with a given backstop price

and time of exhaustion implies, ceteris paribus, that resource holders demand a lower rate of returnfor holding the resource in the ground, and that extraction costs are falling more steeply. However,we wish to focus on the relationship betweenR (the current percentage of the price made up bythe scarcity rent), the time of exhaustion, the backstop price, and the rate of return demanded byresource holders. To do so we plot level curves forR as a function of the backstop price and rateof return, for two different exhaustion dates (Figure 7.3).

In Figure 7.3 we see that the rate of return demanded onin situ resources is crucial to the levelof the scarcity rent obtained from the model. If resource holders are happy with a rate of returnsimilar to the average returns on bonds, around 3 percent,5 then in the baseline scenario (backstopprice 150, oil runs out 2050), the Figure shows that the scarcity rent makes up between 50 and90 percent of the current price. On the other hand, if resource holders demand returns similar to

2See Tahvonen and Salo (2001) for a model in which a renewable substitute plays the role of backstop, putting a capon resource prices.

355.4 dollars/barrel and 53.5 dollars/barrel, in constant 2009 dollars. For data see BP (2010).4The current reserve to production ratio is 46 implying 2055 to be the expected time of exhaustion if current produc-

tion continues. See BP (2010) for the data.5Average returns on 3-month UK treasury bonds are approximately 3 percent per annum.

Chapter 7. Resource stocks 47

150150

100100

5050

19851985 20052005 20502050

pri

ce,$/b

arre

l

pri

ce,$/b

arre

l

Scarcity rentR

Scarcity rentR

(a) Low scarcity rent, high returns (b) High scarcity rent, low returns

Figure 7.2. Graphs of price (dollars per barrel) against time for crude oil, show-ing alternative paths between the same observed prices for 1980–1989, 2000–2009, and in 2050 when the backstop is assumed to take over. The lightly shadedarea represents the scarcity rent, which grows at rater, and the darker area rep-resents resource costs, which shrink at rateγ. In (a)R2005= 5 percent, implyinga high rate of returnr = 8.5 percent/yr, and a slow decline in costs,γ = 0.4percent/yr; in (b) R2005= 50 percent, implying a lower rate of returnr = 3.8percent/yr, and a more rapid decline in costs,γ = 2.4 percent/yr.

R= 1%R= 1%4%

4% 10%

10% 25%

25% 50%

50% 90%

90%

00

00

22

44

66

88

1010

1212

1414

100100 200200 300300 400400 500500Backstop price at exhaustion, $/barrelBackstop price at exhaustion, $/barrel

Rat

eo

fret

urn

,per

cen

t/y

ear

Rat

eo

fret

urn

,per

cen

t/y

ear

(a) Cheap oil runs out 2050 (b) Cheap oil runs out 2040

Figure 7.3. Simulated level curves for the current scarcity rent asa percentageof current price of oil, where the variables are the final (backstop) oil price, andthe rate of return (in percent per year) demanded by resourceholders.

average returns on shares, around 7 percent, then the scarcity rent accounts for just 10 percent ofthe current price.

The question of what rate of return demanded by resource holders has received little attentionin the literature. Note however that Stollery (1983) finds byestimation of a CAPM model thatthe best fit comes from a rate of return of 14 percent for Canadian Nickel. We do not go deeplyinto the question here, but only note that commodity marketsin general, and the oil market inparticular, is considered volatile; see for instance Pindyck (2004). Thus it seems reasonableprimafacie to assume that crude oil holders demand expected returns of at least 7 percent onin situ oil,probably significantly higher. This in turn suggests a scarcity rent of up to 25 percent.

From Figure 7.3 we can see that the backstop price and time of exhaustion also have sig-nificant effects. There is of course a lot of uncertainty about both numbers. However, Lindholt(2005) use acurrent backstop price of $105/barrel (based on Manne et al., 1995), and assume asteady decline over time; it is common to assume declining backstop prices due to technological

48 7.2. An extraction model with inhomogeneous stocks

progress. Note that a reasonable first guess might be to assume that the backstop price falls at thesame rate as the resource cost. If we impose this further restriction, then we can instead plot levelcurves for Hotelling rent as a function ofcurrent backstop price and rate of return. The result(not shown graphically) is that the level curves shift down significantly, because a high currentscarcity rent implies rapidly falling resource costs, implying that the backstop price is also likelyto fall. Specifically, a current backstop price of over $500/barrel is required to yield a 25 percentHotelling rent assuming 7 percent rate of return, while if the backstop price is limited to a morereasonable $200/barrel then the current Hotelling rent is limited to 10 percent of the price.

The point of the above is that the sum of something increasingexponentially (and at a ratherhigh rate) and something else decreasing must also start to increase within a rather short timeunless the increasing part start off very very low. Since resource prices generally are not showinga clear upward trend this suggests strongly that the resource rent remains a small part of the price.

7.2. An extraction model with inhomogeneous stocks

In this section we develop a general equilibrium model of theglobal economy with very sim-ple models of final-good production and biased technological change—models which are broadlyin line with historical data and our understanding of the keyprocesses—and focus our attentionon extraction. We assume a representative resource owner with extraction rights to a contiguousmass of the resource underground. The resource owner hires labour inputs in order to extract theresource, and the rate of extraction per unit of labour depends on the productivity of the input,plus the depth of the marginal resource (an increasing function of cumulative extraction). Thealternative employment for labour is in final-good production; there is thus an arbitrage conditionbetween the sectors. Labour productivity in the two sectorsgrows exogenously at rates that maydiffer. We focus on three situations in which we can demonstrate the existence of a balancedgrowth path, and argue that the economy is likely to move through these situations in turn. Theanalysis is based on the paper of Hart (2016),Non-renewable resources in the long run.

7.2.1. The need for an extraction model with inhomogeneous stock. The optimal exploita-tion of an exhaustible resource is a classic problem in economics. However, the solution is well-characterized only for a narrow subclass of problems, i.e. those in which the resource stock ishomogeneous. Recall that in the baseline version of the DHSSmodel the production function isCobb–Douglas, and the resource is extracted from a known non-renewable stock at zero cost. Animmediate consequence of the Cobb–Douglas is that the resource factor share is constant, whichfits with very long-run observations for aggregate resources such as ‘metals’ and ‘energy’, asshown in Figure 1.3. On the other hand, the assumption of zeroextraction costs leads directlyto the prediction that resource price should rise at the interest rate while the consumption ratedeclines exponentially. This is totally contrary to the evidence, which is that prices show nolong-run trend while consumption rates tend to track long-run GDP (see Figure 1.3 again).

Stocks of exhaustible resources are in reality inhomogeneous and costly to extract, hencean obvious extension to the model is to include these characteristics. However, this area is rela-tively unexplored, despite the early contributions from Heal (1976) and Solow and Wan (1976).The resource stocks assumed by Heal and Solow and Wan are bothspecial cases: in the former,long-run extraction is from an infinite homogeneous stock, hence the scarcity rent is zero; in thelatter it is from a finite homogeneous stock, hence the scarcity rent grows at the interest rate (thesimple Hotelling result). Questions remain regarding the transition paths to these long-run states,but also (more generally) the long-run solution to the case in which remaining stocks are alwaysinhomogeneous. Hence there is a need for a more general modelof resource stocks to be incorpo-rated into a general equilibrium model of long-run economicdevelopment. The need for such ageneral equilibrium model is indirectly supported by the analysis of Livernois and Martin (2001),who review related work in partial equilibrium frameworks;Livernois and Martin [p.840] statethat their findings can easily be reconciled with ‘any kind ofbehaviour for scarcity rent and priceover time’, simply by introducing exogenous trends or shocks in variables such as extraction pro-ductivity (they could also have mentioned resource demand,or the prices of extraction inputs). Ingeneral equilibrium, trends in such variables can be endogenized.6

6Examples of partial-equilibrium models can be found in Levhari and Liviatan (1977), Hanson (1980), Krulce(1993), and Farzin (1992). Levhari and Liviatan (1977) and Hanson (1980) show that the scarcity rent declines overtime, Krulce (1993) derives sufficient conditions for increasing rent, and Farzin (1992) argues that rent may also follow anon-monotonic path. Livernois and Martin (2001) show that the results depend crucially on the nature of the instantaneousnet benefit functionπ(xt ,nt) wherex is the extraction rate andn is cumulative extraction. (Note that Livernois and Martinactually express the model in terms of remaining stock, but concavity ofπ is not affected by the switch.) For ‘clean’ results—in which the scarcity rent is always increasing, and approaches the interest rate—they show that the function should be

Chapter 7. Resource stocks 49

7.2.2. The basic environment.There is a constant population7 and utility is a linear functionof consumption of the aggregate goodY:

U =∫ ∞

0e−ρtYtdt.

The interest rate is thus constant and equal toρ.There is a unit continuum of resource owners, each of whom owns identical stocks of the

resource; we can thus consider the representative resourceowner and her stock.8 We define theextraction rate of the representative owner asxt and cumulative extraction asnt. We thus have

nt = xt.(7.1)

There is a technology parameteran associated with each unit of resource, reflecting the phys-ical difficulty of extracting that particular unit. We denote this parametereconomic depth. Therepresentative owner hires extraction labourlx as the sole input to extraction. The productivity ofextraction labour isax, and the resultant flow of extractionx is given by the following equation:

xt = lxtaxt/ant,(7.2)

whereant is the economic depth of the unit of resource extracted at time t. Due to discounting,resources are always extracted in order of increasingan, hence we can write

ant = F(nt),(7.3)

whereF is an increasing function. The functionF depends on the nature of the resource stocks,which we discuss further below.

Resources are sold to price-taking firms producing the final good, total quantityY. There is aunit continuum of such firms, hence (i) we can consider a representative price-taking firm, and (ii)the flow of resource inputs to the representative productionfirm is equal to the flow of resourceoutputx from the representative extraction firm. The representative firm’s production, denotedy,is a Cobb–Douglas function of labour and resource inputs, asfollows:

y= ayl1−αy xα.(7.4)

Hereα is a parameter between 0 and 1,ay is total factor productivity, andly andx are the respectivequantities of labour and resources used by the representative firm in production. We thus abstractfrom capital, which simplifies the model at little cost in terms of explanatory power since there isperfect foresight and the interest rate is constant; if capital were included its growth rate would beequal to the growth rate inY, the capital share would be constant, and the model results would notchange in essence.

The productivity indicesay andax grow exogenously at strictly positive rates, and total labourL is fixed:

ay/ay = θay; ax/ax = θax;

L = lx+ ly.(7.5)

We now return to equation 7.4 in order to find the market-clearing conditions. Definepx as theprice of extracted resource, andpl as the wage, and normalize the price of the final good to unity.Perfect competition in the final goods sector then implies the following equilibrium conditions(after eliminatingly using equation 7.5):

pl = (1−α)y/(L− lx)(7.6)

and px = αy/x,(7.7)

where y= ayxα(L− lx)1−α.(7.8)

7.2.3. Balanced growth paths in three cases.We now characterize three growth paths, onein the case with abundant resources, one in the case with increasing depth, and one in the casewith imminent exhaustion.

jointly concave inx andn. But in practice the shape of the function will depend on the nature of stocks, and there existhighly plausible stock characteristics—such as lower-grade resources being more abundant than higher-grade resources—which imply that the function will not be concave.

7This assumption can easily be relaxed, and we do so in the empirical application.8We make this assumption for conceptual simplicity. All the results follow if we instead assume the same overall

characteristics of the stock, but with heterogeneous ownership, as long as there are many owners at all possible resourcedepths.

50 7.2. An extraction model with inhomogeneous stocks

Abundant resources.Consider now a situation with a small, primitive economy (low ay andax), and vast resource stocks. In this situationant (resource depth) is constant, and hence thescarcity rent of the resource stock is effectively zero. Given this assumption it is straightforwardto show that the following equations describe a balanced growth path of the economy.

x/x= θax(7.9)

y/y= θay+αx/x(7.10)

px/px = θay− (1−α)x/x.(7.11)

To confirm that this is a b.g.p. assume a constant allocation of labour between extraction and pro-duction. Then the rate of increase of extraction will be equal to the rate of increase of productivityof extraction labour, hence the first equation. The rate of increase of production follows directlyfrom (7.4). And the rate of increase of the resource price follows from (7.7). Finally, we needto check that wages grow at equal rates in the two sectors. From 7.4, the wage to final-good pro-duction labour grows at the rateθay+αθax on our b.g.p. (with fixed labour allocation), whereasthe wage to extraction labour grows at ˙px/px+ θax. Confirm that these growth rates are equal toconfirm the existence of the b.g.p.

Now assume thatθax is slightly higher thanθay, so productivity grows slightly faster in the ex-traction sector than in the final-good production sector. This reflects the presence of (for instance)service industries in the production sector where productivity growth tends to be slower. Sinceα (the resource share) is small, this suggests that when resources are abundant the resource priceshould be constant or decline slowly, and the extraction rate should track the overall growth rate,or even grow slightly faster. The resource price is constantbecause the increasing productivityof the extraction input is cancelled out by its increasing price, and extraction increases as moreresource inputs are sucked in to the growing economy.

Increasing depth.Now assume that, over time, the economy and the extraction rate grow somuch that the initially superabundant resource stock starts to diminish. More specifically, extrac-tion workers must dig deeper to extract the resource, reducing their productivity. In terms of theequations above,an rises.

In order to solve for a b.g.p. under these circumstances we need to specify the relationshipbetween depth and the marginal stock of resources. We assumethe following relationship:

mm0=

(

an

an0

)ψ−1

.(7.12)

Herean is depth underground,m is the cross-sectional area of the resource,an0 andm0 are thedepth and cross-sectional area att = 0, andψ is a parameter which may take any (real) value.Although we denotean as depth, the model can encompass a situation in which the difficultyof extraction is a function of several variables, which may include depth but also grade (i.e. thepercentage of the resource by weight in the rock), and the size of the deposit (larger deposits willtypically have lower unit extraction costs). Later on we give an example of howan can be relatedto depth and grade in a specific case, extraction of copper.

The function is flexible, as shown in Figure 7.4, where four different cases are illustrated. Inthe first three cases we simply choose different values ofψ, allowing the cross-sectional area ofthe resource to either be constant, increase, or decrease with depth. In the fourth case we showhow a more complex stock may be approximated by building up the total stock piecewise fromsubstocks, and by putting a lower bound on depth.

Since extraction will always be performed in order of increasing depth, cumulative extractionn is as follows:

nt =

∫ ant

an0

m dan.(7.13)

Substitute form from equation 7.12 to obtain

nt =m0an0

ψ

(

ant

an0

)ψ

−1

.(7.14)

Note that whenψ ≥ 0 thenn→∞ whenan→∞, so if we want the stockn to be bounded we mustimpose a limit onan beyond which there are no further stocks; on the other hand, whenψ < 0thenan→∞ whenn→−(m0an0)/ψ, hence the total available stock att = 0 is−(m0an0)/ψ. Notealso the special case whenψ = 0, in which casean/an0 = exp(n/(m0an0)). To obtain the function

Chapter 7. Resource stocks 51

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

6

Dep

th,a

n

Cross-sectional area,m

(i) (ii) (iii) (iv)

rx0

rxt

ψ = 0

ψ = 0ψ = 1

ψ = 2

ψ = 2

Figure 7.4. The relationship between depthan and cross-sectional aream forfour alternative resource stocks. In each casean0= 1,ant = 2, andm0= 1; for thesecond part of stock (iv) we havean0 =m0 = 2.5. The dark shading representscumulative extraction at timet, nt, and the lighter shading represents remainingresources at timet. Resource stocks in the first three cases are infinite, but in(iv) there is an upper bound onan and the stock is finite.

F (recall equation 7.3,ant = F(nt)) we use equation 7.14 to obtain

F(nt) = ant = an0

(

1+ψnt

F0

)1/ψ

.(7.15)

Given this relationship between cumulative extractionn and depthan it is straightforward toshow that the following equations describe a balanced growth path.

x/x= ψ/(1+ψ)θax;(7.16)

y/y= pl/pl = θay+αx/x;(7.17)

px/px = λ/λ = θay− (1−α)x/x.(7.18)

λ

px=

11+ψ

1+(1−α)ψ1+ψ +

ρ−θayθax

(7.19)

We now discuss each of these equations in turn.

(31) If ψ is positive (implying an infinite stock) this implies increasing rates of resourceextraction over time, whereas ifψ is negative the stock is finite and resource extractiondecreases exponentially on the b.g.p., and whenψ = 0 resource extraction is constant onthe b.g.p.

(32) Production and the wage both grow at the rate of TFP growth plus the resource factorshare× resource input growth.

(33) The resource price and the scarcity rent both grow at therate of TFP growth minus thelabour share× resource input growth.

(34) We have now introduced the scarcity rentλ. Asψ→∞ the scarcity rent on the b.g.p. ap-proaches zero, whereas as (ρ− θay)/θax+ (1−α)ψ/(1+ψ)→ 0 the rent approaches 100percent of the price, also implying (equation 7.18) thatλ/λ = ρ; these results are eas-ily understood in terms of the nature of the resource stocks implied by the value ofψ.The scarcity rent is decreasing in the effective discount rate (ρ− θay)/θax, because thescarcity rent arises from the fact that today’s extraction raises future extraction costs,and the weight of this effect is small when the discount rate is high.

These results might seem complex, but they are highly intuitive. Whenψ is very large, im-plying that marginal resource stocks increase rapidly withincreasing depth, then the b.g.p. is ef-fectively identical to the b.g.p. with superabundant resources: the resource price remains constantor declining, and resource extraction continues to track growth. However, the more the marginalstock tends to narrow with depth, the lower the rate of increase of extraction on the b.g.p. Whenψ is negative, extraction must actually decline on the b.g.p., which is of course reasonable giventhat the stock is then finite.

52 7.2. An extraction model with inhomogeneous stocks

It may be argued that the idea of an infinite stock makes no sense and hence thatψ must benegative. However, as shown in Figure 7.4 there are other alternatives: we can divide the resourcestock into layers with different values ofψ, and we can also set a limit on depth beyond whichthere are no further stocks available.

Imminent exhaustion.We now consider the growth path when exhaustion of the resource isimminent, i.e. when a limit on depth such as that shown in Figure 7.4(iv) is approaching. Inthis case we have a simple Hotelling economy in which extraction costs are zero (since depth iseffectively constant, the rising extraction productivity pushes extraction costs towards zero) andthe resource price rises at the interest rateρ. We leave it to the reader to derive the equations forthe rates of change of the other variables in the model on thisgrowth path. Note however that thepath differs from the others in that labour allocation changes over time: as the resource price risesand extraction declines, extraction labour also declines,approaching zero. Resource extractionalso approaches zero.

7.2.4. Economic development in theory and practice.We now turn to the use of the modelto understand real economies. First we discuss transition paths in general, then we look at the spe-cific cases of copper and petroleum. The parameterizations are illustrative rather than strictlypredictive, the main reason being the great uncertainty concerning many of the assumptions. Nev-ertheless, the model succeeds in explaining observations from the last 100 years, and makes ap-parently reasonable predictions for the next several hundred in the cases of oil and copper.

Transition paths.The overall picture emerging from the model is straightforward. In the caseof a strictly limited resource which is essential for production then we expect the economy to passthrough three phases. In the initial or frontier phase resource depth is constant, extraction increasesrapidly, and the resource price is roughly constant. In the mature phase resource depth increase,the rate of increase of extraction is moderated, and the resource price rises. In the exhaustionphase the depth is again constant, the extraction rate approaches zero, and the price rises at theinterest rate.

The case of a strictly limited and essential resource is however not very realistic. The onlytruly non-renewable and essential resource is energy, but energy is not limited since we have theoption of harvesting the inflow of energy from the sun. On the other hand, although minerals suchas metals are of course in strictly limited supply they are not consumed in the production process,hence we have the option of recycling. Furthermore, if we focus on each metal separately thennone of them are essential to the production process: if one metal runs out (including options forrecycling) then we will of course do without it, substituting it with other materials.

Since there are—very generally—substitutes for non-renewable resources it is important toinclude this fact in the model. The simplest way to do so is to assume abackstop resource, i.e. asubstitute available in unlimited quantity at an exogenousprice. When the resource price hits thebackstop price extraction stops. Clearly at this point the value of remaining resource stocks is zero,implying that the the scarcity rentλ is zero. We can solve the model in this case, and the behaviourof the economy as the backstop price is approached depends very strongly on that price: if thebackstop price is very high then the behaviour of the economywill be similar to the exhaustionphase as the backstop price is approached, with rapidly increasing price and declining extraction.On the other hand, if the backstop price is ‘moderate’ then the mature b.g.p. may evolve seamlesslyinto the backstop economy, and if the backstop price is very low then the mature b.g.p. may neverbe approached: instead, the frontier economy may evolve directly into the backstop economy.

Copper. The parameterization of the model for copper is complex, since there are two keydimensions along which the quality of copper deposits varies. The first of these is grade (thefraction of copper in the rock, by mass), and the second is depth. Combining grade and depthinto one measure of ‘depth–grade’rx we obtain the relationship described in Figure 7.5. Herewe see that extraction so far is only scratching the surface of total stocks, and furthermore thatmarginal stocks are (for now) rapidly increasing in depth. Furthermore, our calculations suggestthat depth–graderx is only relatively weakly linked to ‘economic depth’an: an = r0.42

x . This leadsto the result that the upward pressure on copper prices due toincreasing depth is weak and willremain so for a long time into the future.

Feeding all this into the overall general-equilibrium model we obtain the results shown inFigure 7.6. The economy starts close to the first b.g.p. whichapplies for the initial stock, andprice declines by around 0.4 percent per year. Once the initial stock is used up in 2054, the rate ofincrease in depthan increases, and the economy starts moving towards the secondb.g.p. on whichprice rises by 0.8 percent per year. Finally, from around 2200 the scarcity rent starts to rise asexhaustion approaches, at least in the case with a high backstop price. With a low backstop price

Chapter 7. Resource stocks 53

0 0.02 0.04 0.06 0.08

0

100

200

300

400

500

600

7000 0.02 0.04 0.06 0.08

0

2000

4000

6000

8000

10000

12000

Cross-section,mCross-section,m

Dep

th,r

x

Dep

th,r

x

Figure 7.5. The relationship between the combined measure of depthand grade,rx, and cross-sectional aream for copper, based on our interpretation of theliterature (continuous line), and the parameterization ofour economic model(dashed line). Notice the difference in scale on the two panels. The shaded areashows extraction from 1900–2011, 5.75× 108 tons, and the dotted lines showthe relationship between depth and cross-section layer-by-layer.

the scarcity rent hardly rises, and exhaustion occurs a few years earlier. Note the close agreementbetween the model and observed trends in prices and extraction rates.

1900 1950 2000 2050 2100 2150 2200 2250 23000

20

40

60

1900 1950 2000 2050 2100 2150 2200 2250 2300

−2

0

2

4

Backstop 1

Backstop 2Price, 1 and 2P

rice

,103

US

D/to

n

Extraction rate, 1 and 2

log

no

rmal

Figure 7.6. Observed price and extraction rate of copper, and the paths of priceand extraction rate—up to the time of exhaustion—predictedby the model.Two model scenarios are shown, which differ in the assumed backstop price.Note that the extraction rate is plotted on a (natural) logarithmic scale, normal-ized by the rate in 2000. Prices are in 1998 USD.

Petroleum.If the copper simulation is an advertisement for the power ofthe model, the pe-troleum simulation highlights its weaknesses. There are two key aspects of the petroleum marketwhich the model cannot handle as it stands: firstly, the significance of market power in the petro-leum market, and secondly the inextricable links between petroleum and its substitutes, includingnatural gas, coal, and other energy sources such as nuclear power. Of course, market power and

54 7.2. An extraction model with inhomogeneous stocks

substitutes also exist in the market for copper, but their scale and influence is greater in the oilmarket. Concerning market power, consider for instance thefact that petroleum extraction occurssimultaneously from deposits for which marginal extraction costs differ by a factor of 5 or more(compare for instance the Ghawar field in Saudi Arabia to the Athabasca oil sands of Alberta).Concerning substitutes, petroleum demand is linked tightly to markets for coal and other energysources, and strongly affected by technological change. Consider for instance the substitutionfrom coal to oil driven by the development and refinement of the internal combustion engine.Given these problems—which are evident in Figure 7.8—the model calibration is at best illustra-tive, showing possible future scenarios and highlighting the effect of backstop energy sources.

The data regarding petroleum resources in the ground are uncertain. Furthermore, the dataregarding the cost of extraction of these resources are evenmore uncertain. The most frequentlycited paper on the subject is probably Rogner (1997). However, Rogner’s curve relating cumula-tive extraction to extraction cost (see for instance his Figure 6) shows estimated extraction costat the time of extraction. Its calculation must therefore involve (implicit or explicit) calculationsof (i) current extraction costs, (ii) expected decline in extraction costs, and (iii) expected rate ofextraction. Since we model the latter two, we need data on thefirst factor alone, i.e. unit extractioncosts for each type of deposit making up the reserves, if full-scale extraction were to be carried outtoday. This is estimated by the International Energy Agencyin their World Energy Outlook 2008(p.218). The data are very approximate, but can be broadly summarized as follows: consideringinitial resource stocks, there was a large rather homogeneous stock of easily accessible stocks,approximately 2000 billion barrels at an economic depth of around 18 USD/barrel. Regarding theremaining stocks—about 7000 billion barrels—economic depth an rises approximately linearlywith cumulative extraction, reaching approximately 115 USD/barrel for the deepest stocks. Wecapture this in the model by assuming an initial stock with low ψ (ψ = −2.2), so that the entirenear-homogeneous stock is at a depth of 10–20, switching to the deeper stock withψ = 1 from20–115. The cross-section of the second stock is determinedby its size (assumed to be 6.7×109

barrels), and the parameters for the first stock are then fixedby the limits on depth (10–20), thesize (2.3×109 barrels), and the need form to be continuous over the boundary between the stocks.The result is shown in Figure 7.7. Note that the curve shows unit extraction costs, in 2008 USDwith today’s technology, for all petroleum resources including the (hypothetical) current extrac-tion cost of resources already extracted. (Note that we ignore the fact that a significant proportionof cumulative extraction has been from deeper stocks.)

−400 −300 −200 −100 0 100 200 300 400

0

20

40

60

80

100

120

Cross-section,m

Eco

no

mic

dep

th,a n

Figure 7.7. The relationship between economic depth,an, and cross-sectionalaream for oil, based on our interpretation of the International Energy AgencyWorld Energy Outlook 2008. Depth is measured in 2008 USD, andcross-sectional area in billion barrels per USD. The shaded area shows extractionfrom 1900–2008, 1100 billion barrels.

Having parameterized the model, and given the assumption about the total stock of resources,the future development of prices and quantities predicted by the model depends on what we as-sume about the price of the backstop (i.e. the substitutes for oil that will take over when oil isexhausted or too expensive). Here we make two alternative assumptions to demonstrate the roleplayed by the backstop resource. In the first case we assume that a backstop is available at a fixedprice of 150 US dollars (2011); in the second case we assume that a backstop is availabletodayat

Chapter 7. Resource stocks 55

that price, and that this price will decline at the rateθax−θay; that is, the backstop price declines aslong as manufacturing productivity growth outstrips TFP growth. The result is that the backstopprice is around 65 USD at the time of exhaustion, rather than 150.

1900 1950 2000 2050 2100 21500

50

100

150

1900 1950 2000 2050 2100 2150

−6

−4

−2

0

2

Backstop 1

Backstop 2

Price, 1 and 2

Pri

ce,U

SD/

bar

rel

Extraction rate, 1 and 2

log

no

rmal

Figure 7.8. Observed price and extraction rate of petroleum, and the pathsof price and extraction rate—up to the time of exhaustion—predicted by themodel. Two model scenarios are shown, which differ in the assumed backstopprice. Note that the extraction rate is plotted on a logarithmic scale, normal-ized by the rate in 2000. Prices are in 2012 USD. Scenarios: continuous lines,backstop price 150 USD (year 2012); dashed lines, current backstop price 150USD, declining at a rateθax−θay per year. Price data from BP (2012), consump-tion data from Boden et al. (2012), assuming a linear relationship between CO2emissions and petroleum consumption.

The results are as follows. Note first that there is no market power in the model economy,hence the results are what the model predicts in an economy similar to the actual global economybut without the exercise of market power by oil producers. Turning now to the results, up to theexhaustion of the upper stock, depth is almost constant, thescarcity rent is close to zero, and pricedeclines at a rate equal to the difference between the growth rates of extraction productivityandlabour productivity in final-good production, i.e. 0.6 percent per year. However, as the upper stocknears exhaustion depth starts to rise at a significant rate, and the economy heads back towards theb.g.p. for the stock, for whichψ = 1; the mature extraction phase. On this b.g.p. we have—fromequations 7.16–7.19—that the growth rate of extraction is halved, the resource price rises by 0.6percent per year, and the scarcity rent makes up 21 percent ofthe price.9

In the latter half of the 21st century the price paths of the alternative backstop scenarios di-verge significantly: the upper path (high backstop price) isslightly above the b.g.p. price path,while the lower path is below it. Hence when the backstop price is fixed at 150 USD the scarcityrent rises above 21 percent of the total price as exhaustion approaches, whereas given the lowerbackstop price the rate of price increase slows down as exhaustion approaches, and the rent actu-ally declines as a proportion of the price.

7.2.5. Sensitivity of the model to assumptions.The above simulations are sensitive to theassumptions made, the most uncertain of which are those regarding future demand, and futuredevelopment of extraction productivity. On the one hand, our assumption about future demand isessentially at the upper bound of what is realistic, i.e. that demand per capita continues to growindefinitely at a similar rate to the rate observed over the last 100 years. At least two factors mightbe expected to lead to lower future demand: firstly, if globalgrowth slows in the long run, andsecondly if there is a transition from ‘early’ growth based on manufacturing and hence resources,

9Note that after the transition to the deeper stock withψ = 1 the economy approaches the mature b.g.p. for that stockfrom above, i.e. the state variablea is above its level in the steady state. As the economy approaches the new b.g.p.x1 fallsback, which is why prices rise quite steeply throughout the 21st century.

56 7.3. Concluding discussion on resource stocks in the verylong run

and ‘post-industrial’ growth based on services and hence labour.10 The effect of lower demandwould be to reduce extraction rates and hence also reduce thegrowth rate of prices predicted by themodel. On the other hand, our assumption regarding future development of extraction productivityis also an upper bound; again, we assume that it continues to increase indefinitely. This is unlikely,not least because in reality resource extraction requires energy, and there are physical limits to theefficiency with which this energy can be used. Since these limitsare already coming close insome cases, this implies that even if labour productivity continues to increase, energy productivitywill not do so and hence the proportion of the energy cost in the unit cost will rise, and the riseof overall extraction productivity will slow. This assumption therefore biases the results towardslower prices and higher extraction rates than are likely to be observed.

The effect of assuming both lower future demand and lower productivity growth rates in ex-traction is therefore that the price path is likely to be relatively unchanged, whereas the extractionpath will be lower. Furthermore, the proportion of the priceaccounted for by the scarcity rent willbe lower. Given a finite stock, the lower extraction path willlead to later exhaustion, and henceany price spike as exhaustion approaches is also likely to bedelayed.

7.3. Concluding discussion on resource stocks in the very long run

Regarding the very long run, we study the paper of Goeller andWeinberg (1976). They arguethat reserves of the majority of minerals—including staples such as iron and aluminium—are sovast that they can never realistically be consumed. With theclear exception of phosphorus theyargue that society can exist on these superabundant resources indefinitely. Of course, fossil fuelsare certain to run out in the not-too-distant future, but here there are obvious substitutes includingthe abundant inflow of energy from the sun.

Goeller and Weinberg are well aware that resource stocks areinhomogeneous. However, theyshow that extraction costs are relatively insensitive to grade and depth of the resource deposits,since the physical extraction and sorting of the mineral-rich material is only part of the process,and generally not the most expensive. Instead, the energy-intensive reduction of the metal oresto the pure form is typically a major part of the costs. This shows that we should—ideally—include energy in the extraction function for minerals, andhence that future energy prices will beimportant determinants of mineral prices. This applies particularly since the energy-efficiency ofthe reduction process is already close to the physical limitof what is possible.

When studying the very long run, the limiting effect of the surface area of the globe is crucial.This limit obviously puts a brake on indefinite (exponential) population growth, and also resourceextraction. Thus it is clear that the current trend of constant growth in global resource extraction—tracking global product—cannot continue indefinitely. What then will make extraction flattenout or even turn downwards, and when?

Before considering the above questions, we think about the likely nature of a (very) long-rungrowth path. Given the constant flow of energy inputs (from the sun) and the fixed resources onEarth, it seems reasonable to suppose that use of energy willbe approximately constant in the verylong run. This implies that some fixed proportion of land willbe devoted to energy harvesting.The remainder of the land will presumably be devoted—in fixedproportions—to other uses suchas agriculture, living space, production, recreation, and(hopefully) nature. Given a fixed energyflow, what about minerals? Since we are already close to the limits of the (energy) efficiency ofmineral extraction—and the energy costs of this extractionare large—it is clear that a long-rungrowth path must involve non-increasing mineral extraction.

So in the very long run we should expect an economy with roughly constant use of minerals,energy, and land. Furthermore, the productivity of energy in making goods such as motive powerand light will be constant, as will the productivity of land in making (or harvesting) energy. Onthe other hand, the productivity of labour may still be able to increase even in the very long run,as it is hard to envisage a limit on human ingenuity and hence our ability to generate more valuefrom given inputs. The key resource for energy production will be land, indeed land will be theultimate scarce resource, strictly limited and needed for harvesting energy, production (includingfood), recreation, and (hopefully) nature.

10To get a feel for the sizes of demand changes in the model, we consider each simulation in turn. For copper, theextraction rate peaks at around 50 times the observed rate inyear 2000. Compare this to the arbitrary assumption that theentire future global population consumes copper at the samerate as the average U.S. citizen in year 2000; this would leadto a global extraction rate approximately 6.3 times greater than that observed in 2000. If demand levels off in this waythen the copper stocks will last for many centuries rather than just two or three. For petroleum, the extraction rate in themodel peaks at around 3 times the year 2000 rate, which is lessthan the rate which would arise if all countries matchedthe U.S. per-capita rate from year 2000.

CHAPTER 8

Renewable natural resources and pollution as inputs

In this chapter we briefly consider the analysis of growth in aneoclassical framework when re-newable natural resources and pollution are treated as essential inputs. What are the consequencesfor growth, and for the environment? We conclude that for a more meaningful analysis we need toconsider substitution between different inputs (e.g. renewable and non-renewable), and we needto consider pollution as a biproduct of production rather than as an input.

8.1. Renewables

Note firstly that there are two very different types of renewable resource, which we denote‘stock renewables’ and ‘flow renewables’. A stock renewableis a resource with a stockS (ratherthan just a flow) and where the growth of the stockS is a function of the current stock amongother factors. On the other hand, a pure flow renewable has no stock. The intermediate case is arenewable which can be stored (so a stock may exist) but whereS is not a function of the size ofthe stock. Examples of stock renewables include forests, fish populations, etc. An example of apure flow renewable is sunshine, whereas an intermediate case is rainfall. The economic problemwhen faced by limited stock renewables is very different from the problem when faced by limitedflow renewables, hence we deal with them separately.

8.1.1. Flow renewables.In the case of flow renewables the problem is simply to allocate theflow between different uses (including leaving it for the benefit of nature). If the flow is exogenousand fixed in the long run—which is a good approximation for resources such as sunshine, wind,geothermal energy, etc—then the appropriate analysis is very similar to that for land in Chapter 5,Section 5.1: the exogenous, constant flow of the renewable resource is equivalent to the exogenous,constant quantity of land available.

So if the flow of the essential renewable resource isR, and there is exogenous labour-augmentingtechnological progress, we have

Y= (ALL)1−α−βKαRβ

AL/AL = gAL

K = sY− δK.

To characterize the development of the economy, assume thatthere exists a balanced growth pathon which all variables grow at constant rates. This implies that K/K is constant, which implies(from the capital-accumulation equation) thatsY/K − δ is constant, implying in turn that the ratioof productionY to capitalK is constant. This implies thatY/Y= K/K, and hence (differentiatingthe production function with respect to time and substituting)

Y/Y= (1−α−β)gAL +αY/Y.

Rearrange to obtain

gY = gy =1−α−β

1−αgAL .

This confirms the reasonableness of the original assumption, i.e. that there exists a balancedgrowth path on which all variables grow at constant rates.

So given labour-augmenting technological progress, whenβ > 0 (i.e. land is a non-negligiblefactor of production) the growth rate of production is slowed. The underlying reason why land inthe production function slows growth whereas capital does not is that land is non-reproducible (bycontrast to capital).

The price or the resourcewR can be obtained by considering the representative producer’sprofit-maximization problem,

maxπ = pY(ALL)1−α−βKαRβ− (wL L+wK K+wRR).

57

58 8.1. Renewables

Normalize the price of the final good,pY, to one. Then take the first-order condition inR to showthatwR = βY/R, and hence that

wR/wR = Y/Y.

So the price of the resource tracks growth in GDP.Now assume that there is some cost to harvesting the flow of therenewable resource. If this

cost is such that we choose to harvest all of the resource—implying thatwR = βY/R> xr , wherexr

is the unit extraction cost—then little changes. On the other hand, if the cost is such that not all ofthe resource flow is utilized then the analysis is similar to the case of an abundant nonrenewableresource which can be extracted at a fixed cost, Chapter 5.2.

As in 5.2 we denote final goods devoted to extraction asX, and we have

Y= (ALL)1−α−βKαRβ;

AL/AL = gAL;

K = s(Y−X)− δK;

R= φX.

Finally, note that total extraction costs are simplyX, since the price of the final good is normalizedto 1. And since we assume perfect markets, the resource pricewR is equal to unit extraction costsX/R, hence

wR= 1/φ.

Now, as previously, assume a b.g.p. and then characterize it. On a b.g.p.K/K is constant,which implies (from the capital-accumulation equation) that s(Y−X)/K− δ is constant, implyingin turn thatY, X, andK all grow at equal rates, and hence (from the extraction function) R alsogrows at this rate. Differentiate the production function with respect to time, andsubstitute forK/K andR/R to yield

Y/Y= (1−α−β)gAL+αY/Y+βY/Y.

Rearrange to yield

Y/Y= gAL .

So the need for the resource imposes no brake at all on the growth rate of production, sinceresource use tracks production (by contrast to land, which is fixed). Of course, this only appliesup to the point at which all of the resource flow is used, at which time the analysis switches to theprevious case.

8.1.2. Stock renewables.In the case of stock renewables we have a much more complexdynamic management problem including both the allocation between uses and the allocation ofuse across time, where a patient manager can harvest much larger cumulative quantities of the re-source than an impatient one. And where global economic growth must be added to the equation.However, the relevant question is perhaps more about whether long-run economic growth is com-patible with sustainable harvests of stock renewables, rather than whether unsustainable harvestsof stock renewables may lead to derailment of the growth process.

Consider for instance fish stocks. The complete destructionof global fish stocks would beunlikely to have much direct effect on global economic growth, since there are many substitutesources of nutrition. However, global economic growth clearly has implications for global fishstocks. As global population and global product increases,demand for fish also increases, asdoes our ability to catch fish. We can build a very simple modelbased on the models above, butabstracting from capital for simplicity.

Assume that agricultural productsg are available at pricepg, which is constant over time.Meanwhile, fishf are close substitutes for agricultural products. They are harvested using finalgoodsxf , and the rate of harvest isf = [xf − s(F) · f ]/pf , wheres(F) is a unit search cost whichis declining inF, the fish stock. So the unit cost of fish harvesting—and hence also the price offish given that fishers are price takers—is

pf = xf / f = pf + s(F).

Now assume that global production function is

Y= (ALL)1−αRα,

whereR is food production. Furthermore,R is a CES function of fish and agricultural products:

R= (σ f ǫ + (1−σ)gǫ)1/ǫ ,

Chapter 8. Renewable natural resources and pollution as inputs 59

whereǫ > 0. Take first-order conditions inf andg to show that the price of fish is

pf = pg(σ/(1−σ))(g/ f )1−ǫ.

More to come here. Use data fromhttp://www.fao.org/worldfoodsituation/foodpricesindex/en/,http://www.fao.org/3/a-i3720e/i3720e01.pdf,

http://download.springer.com/static/pdf/535/art%253A10.1007%252Fs10640-012-9611-1.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Fa

.

8.2. Pollution

The standard way to model pollution is as an input to production; the more pollution a firmemits, the more final-product it can produce while holding its other inputs (such as labour andcapital) constant. Thus there are strong parallels betweenthe analysis of an economy whichsuffers damage from polluting emissions and an economy in which resource inputs are scarce.Indeed, we can easily set up two limiting cases of the ‘pollution economy’ which are analyticallyequivalent to the resource economy. We begin by doing so.

The first limiting case of pollution is an economy in which flows of pollution up to some limitcause no damage, whereas flows above that limit cause the total collapse of the economy. This isequivalent to the economy with a homogeneous stock of land asan input; as long as not all of theland is being used, then we can freely increase the amount of landR used; but when all land isbeing used, we cannot increaseR at all. Thus the analysis from the ‘land’ example applies, andthe need to hold pollution flows under a fixed bound puts a brakeon the growth rate which canbe maintained, and the strength of the braking effect depends on the size ofβ, the factor share ofpollution in the production function:

gY = gy =1−α−β

1−αgAL .

The second limiting case is an economy in which (i) there is anabsolute limit on the totalstockof pollution which can be accommodated, (ii) pollution onceemitted never decays, and (iii)emissions up to the limit cause no damage. This case is of course equivalent to the case of a fixedand finite stock of resources, free to extract, and we have

gY =

(

1−β

1−α

)

gAL −βθ

1−α,

whereθ is the rate of decline of polluting emissions. Whenθ is low polluting emissions are lowand decline only slowly, allowing relatively rapid growth in GDP. Whenθ is high then pollutingemissions are initially high and must therefore decline rapidly, putting a powerful additional brakeon the growth rate.

These two cases are obviously highly stylized, or (in plain English) not relevant in practice.However, it is not obvious that the common assumption of a fixed stock free to extract is morerelevant when applied to non-renewable resources. Continuing the analogy, to generalize theresource case we need to allow for extraction costs which vary with cumulative extraction and alsowith extraction technology. To generalize the pollution case we need to allow for damage costswhich should vary depending both on the state of the economy and the state of the environment(i.e. previous emissions in a simple model). However, more importantly we need to recognizethat pollution flows typically originate from the use of non-renewable (or sometimes renewable)natural resources, and therefore that the pollution problem is actually inextricably linked to theproblem of supply and demand of non-renewable resources. Wereturn to this issue in Chapter 12.

Part 3

Demand for natural resources and energy

In this part of the book we turn from a focus on the supply of nonrenewable resources todemand. In Part 2 we assumed a Cobb–Douglas final-good production function for a single finalgood, with exogenous labour-augmenting technological change, which gave us the properties of (i)a constant resource factor share, and (ii) a constant growthrate of demand for the resource, more-or-less by assumption. This approach allowed us to achieve agood fit to historical aggregate data,but is clearly not adequate if we want to predict future demand for resources, and the effect of(for instance) policy measures restricting resource use, or scarcity driving up market prices. Forprediction we need a much deeper understanding of the driving forces behind the aggregate data.

Recall Solow’s three mechanisms for adaptation to resourcescarcity, page 40: (i) to increase—through technological change—resource efficiency in production of one or more product cat-egories; (ii) to substitute on the consumption side away from product categories in which theproduction process is resource-intensive; and (iii) to raise the efficiency of an alternative (sub-stitute) resource in production of one or more product categories. Looking at the historical pricedata, where the price of resources has tended to decline relative to labour, the first two mechanismsshould work in reverse, thus we should see a decline in resource efficiency relative to labour ef-ficiency, and substitution towards consumption categoriesthat are resource-intensive rather thanlabour intensive. The first two mechanisms are therefore candidates for explaining the constantfactor share of resources.

In Chapter 9 we test whether the forces of directed technological change—and hence thefalling relative price of resources leading to a fall in resource-efficiency relative to labour efficiencyin the production of individual products—are the cause of the increase in resource consumptionrates observed in the global economy. And then, turning to prediction of the future, we ask whetherincreasing resource prices would be likely to trigger a massive increase in resource efficiency, thushelping—a la Solow—to reduce the need for resources. We conclude that the answer is likely tobe ‘no’, on both counts.

The failure of the DTC model to account for the historical data puts the spotlight on the al-ternative, substitution on the consumption side. Has resource and energy consumption risen sorapidly because of substitution towards resource- and energy-intensive consumption categories, aform of structural change? If we accept the clear evidence from the energy sector that the energy-efficiency of individual products has indeed increased dramatically, the answer to the above ques-tion must be ‘Yes’, since these two alternatives exhaust allthe possibilities for explaining whyaggregate efficiency has failed to increase. Is this backed up by the evidence, and if so, what arethe implications? We tackle these questions in Chapter 10.

Solow’s third adaptation mechanism was to increase—through technological change—theefficiency of an alternative (substitute) resource in production of one or more product categories.In the final chapter of this part of the book we build a model allowing for such substitution, amodel that we develop further when investigating pollutionin Part 4.

CHAPTER 9

Directed technological change and resource efficiency

Recall Solow’s three mechanisms determining demand for resource inputs, page 40. In thischapter we focus on Solow’s first mechanism, i.e. that firms may adapt to resource scarcity byincreasing—through investment in technological change—the efficiency with which they useresources in the production of their goods, i.e. by increasing the level of resource-augmentingknowledge. We begin by showing how resource-efficiency can be represented in firms’ productionfunctions (and the aggregate production function of the economy if there is one). We go on to setup a model of economic growth and endogenous directed technological change, and use it to testthe idea that DTC can explain the long-run data on energy and natural-resource demand.

9.1. Resource efficiency in the production function

In this section we take a more detailed look at alternative ways of representing productionas a function of two inputs. This is important groundwork before beginning our analysis of DTC(directed technological change). We focus on a simple case in which widgets are produced usingflows of resources and labour. As discussed above we abstractfrom (i.e. ignore) capital. In essencethis implies that we assume that the capital stock grows at the same rate as the stock of effectivelabourALL, hence including capital would not affect the analysis.

9.1.1. Familiar cases: the general production function, and Cobb–Douglas.Assume aneconomy with one product, widgets, and two production inputs. For now we call them labourand resources. Denote flows of labour and resources asL andR respectively, units workers andtons/year. In general we can write

y= F(ALL,ARR).(9.1)

Recall then that the units ofL andRare widgets per worker per year and widgets per ton per year.So far in the book we have used the Cobb–Douglas function almost exclusively. That is, we

write the production function as

y= (ALL)α(ARR)1−α = ALαR1−α.

Thus the two separate factor-augmenting technology levelshave been subsumed into a singletechnology index and there is no role for directed technological change. For instance, if weraise resource-efficiency AR this serves to raiseA and hence raise production, but it does notreduce the demand for resource inputs. This paradoxical result follows from the high degree ofsubstitutability between the inputs: when resource efficiency rises the cost of resource inputs falls,causing producers to use more resources.

The Cobb–Douglas model fails when confronted by further evidence. For instance, we knowthat it is in fact very hard for producers to substitute for resource inputs using labour, at least in theshort run. This is clear from the very small short-run effect of increases in the resource price onresource demand: when resource prices rise dramatically, demand fails to fall because firms andconsumers are locked in to their demand for resources by the technology and capital they possess.To go deeper we therefore need a model in which there is low short-run elasticity of substitutionbetween labour and resources, but where there is also some mechanism through which they canbe substituted in the long run. In this chapter that mechanism is DTC.

9.1.2. The Leontief production function. The simplest way to specify a production func-tion with low substitutability between the inputs is through theLeontief production function, inwhich there is no substitutability whatsoever between the inputs. The function looks like this:

y=minALL,ARR.(9.2)

This equation reads as follows: productiony is equal to the smallest of the following set of quan-tities: effective labour inputsALL, and effective resource inputsARR.

To interpret the equation, consider for instance production of hammers from steel. Given thedesign of the hammers (which determinesAR), and assuming no steel is wasted, the productionrate of hammers will be limited by the flow of steel inputsR. If the design is such that each

63

64 9.1. Resource efficiency in the production function

hammer requires 1 kilo of steel, and the flow of steel inputs is103 tons per year, then no morethan 106 hammers can be produced by the firm per year. However, there isno guarantee thatthe firm will produce that many hammers; to do so, they must have enough (productive) labouremployed, whereAL is the productivity of the workers andL is their number. Note that we have—following Solow (1973)—left out capital; effectively, we assume that each worker has enoughcapital (machines) in order to work productively.

A profit-maximizing firm will of course make sure that it hiresjust enough workers, and buysjust enough resources, such thatALL = ARR. The alternative is that workers stand idle waiting forresource inputs, or that resources stand idle waiting for workers to use them.

Now assume thatAL —worker productivity—rises exponentially, whileAR remains unchanged.Furthermore, assume that equation (9.2) is the aggregate production function of the economy.Then assuming that resources are available, resource flows into production will also increase ex-ponentially, tracking the growth in labour productivity and global product. If the resource priceremains constant then the share of resources in global product will also remain constant. However,such exponential growth in physical inputs cannot continueindefinitely, and the model stronglysuggests that a crash is imminent when resources ‘run out’.

Superficially this example matches the evidence regarding the global economy (see for in-stance Figure 1.3). However, it should be clear that this model is far too simple to use as abasis for drawing conclusions and designing policy. The model is obviously grossly simplifiedin assuming that it is impossible to increase resource efficiency. If labour efficiency can increaseexponentially, why should resource efficiency be unable to increase likewise? Furthermore, as inthe neoclassical model, there is no allowance for shifts in consumption patterns. Finally, there isno model of the price of the resource input, which we expect tobe linked to scarcity of the inputand to affect demand for the input.

The model above shares crucial features with the famous—butnevertheless obscure—‘Lim-its to Growth’ model. The ‘Limits’ model dates to 1972 when the Club of Rome published a book,the Limits to Growth, which caused a sensation at the time. The analysis used a “system dynamicscomputer model to simulate the interactions of five global economic subsystems, namely: popula-tion, food production, industrial production, pollution,and consumption of non-renewable naturalresources”.1

The ‘Limits’ team programmed various scenarios, and all ended in disaster, typically byaround 2050. This was either due to resource exhaustion or excessive pollution. The reasonsfor this are not easy to elucidate as the model is not open to examination, however, the behaviourof the model can be approximated by a very simple economic model, as follows:

Y=minALL,ARR;(9.3)

AL/AL = g;(9.4)

S0 ≥

∫ ∞

0Rtdt.(9.5)

Furthermore, we have thatAL0L < ARS0. Thus we have a Leontief production function withexogenous growth inAL. If AR is constant then we get exponential growth in bothY and R,whereas ifAR is allowed to grow then the growth inR will be slower.2 But why doesAL grow?And what determines the growth rate ofAR? And what about the resource price, does that haveno effect whatsoever on the allocation? The model raises many questions, some of which we tryto answer in this part of the book.

9.1.3. The CES production function. An alternative functional form—much more flexiblethan both Cobb–Douglas and Leontief—is the CES production function:

y= [γ(ALL)ǫ + (1−γ)(ARR)ǫ ]1/ǫ .

Hereǫ ∈ (−∞,1). The Cobb–Douglas emerges from the CES as a special case when ǫ = 0, andLeontief whenǫ → −∞. Finally, whenǫ = 1 then the inputs are perfect substitutes, like 5 and10-dollar bills. However, typically we assume that labour and capital, or labour and resources, arepoorly substitutable for one another, that is they arecomplements, implying thatǫ < 0. When they

1According to the website http://www.clubofrome.org/?p=1161+, 3 Oct. 2012.2Note that we have simplified slightly here. The Limits model is actually built on an outdated growth model known

as theAK model in which labour productivityAL grows due to capital accumulation, which is possible due to ahighlycontrived mechanism where the greater the quantity of capital possessed by other firms, the more productive is any par-ticular firm (irrespective of how much capital that firm has).However, the overall effect is consistent with the equationspresented here.

Chapter 9. Directed technological change and resource efficiency 65

are complements, an increase in the quantity of one of the inputs available on the market leads toa large reduction in its price such that relative returns to that input factor decline.

To get a handle on the production function, consider the following example. Assume aneconomy with 10 people on an island, and 10 trees/week wash up on the beach. Furthermore, theislanders have a technology called ‘knives’ which allows them to cut the trees into planks, whichcan rapidly be made into houses (final product). They manage to make 0.01 houses per week.What do they need more of to boost their rate of housebuilding? Suggest values for the elasticityof substitution between the inputs, and knowledge levels. Explain briefly.

Now assume that the islanders invent a technology called ‘sawmills’ (and are somehow ableto obtain the necessary capital goods). What do they need more of now in order to boost their rateof housebuilding? Suggest new values for the knowledge levels. Explain briefly.

Some possible answers to the questions above follow. We haveL = 10 andR = 10, andy= 0.01. Presumably it is rather hard to substitute workers for trees, although not impossible. Forinstance, if there are a lot of trees the (fixed number of) workers could select the best trees to makeplanks out of, rejecting less suitable ones. This would allow them to produce planks faster. Soρ should definitely be negative, but we have to guess its value.For simplicity we chooseρ = −1.We drop the distribution parameterγ, incorporating it into the productivity factors. So we have

y=

(

110AL

+1

10AR

)−1

.

Sincey= 0.01 we can rearrange the equation to obtain 1000= 1/AL+1/AR. Finally, and crucially,note that it is clearly workers who are in short supply in thiseconomy, not trees; only a smallfraction of the 10 trees per week can be used considering thatthe 10 workers only have kniveswith which to work. Since the supply of workers is limiting, this implies thatAL should be smallcompared toAR.

More precisely, imagine that there is an abundance of trees such thatARR is very large. Thenwe can approximate 1/(ARR) = 0, and the production function becomesy= ALL. ThenAL = 0.001(sincey= 0.01 andL = 10). If we instead imagine that labour is abundant, thenAR becomes theminimum number of houses that can be made per tree, given thatinfinite care is taken to avoidwaste in making the planks. If this is 0.5 then the production function is

y=

(

10.001L

+1

0.5R

)−1

.

WhenL andRare both equal to 10, this gives 0.01 houses produced per week.3

When sawmills are invented, the productivity of trees in housemaking is presumably more-or-less unchanged; it still takes 2 trees to make a house. (Although we could imagineAR chang-ing: for instance, if the sawmills produced more waste such as sawdust thenAR would go down,whereas if they could cut thinner planks thus using the timber more efficiently then it would goup.) The productivity of labour on the other hand will go up enormously. Now we might imag-ine 10 people running a sawmill being able to cut up hundreds of trees each week. (Recall thatthe planks can ‘quickly’ be made into houses, implying that the time spent on this step can beignored.) If we assume a relatively rudimentary sawmill then we can setAL = 10, and the newproduction function is

y=

(

110L+

10.5R

)−1

.

Now trees are the limiting factor, and 4.8 houses can be made each week given that there are 10trees available per week and 10 workers.

9.2. A simple model of directed technological change

The CES production function gives us the ability to capture the effects of factor-specific(directed) technological change on production in a flexibleway. This is an essential ingredient inour models of growth and sustainability. However, this is not enough on its own; we must alsobe able to model changes in the relative productivities of the factors as an endogenous result ofother changes in the economy, such as changes in the availability of the factors. That is, we needa model of endogenous directed technological change, henceforth DTC.

To build such a model, we return to our model of endogenous technological change fromChapter 4 and add the need for a resource input, with an associated level of resource-augmentingknowledge. Furthermore, we simplify the model in some otherrespects. Because we are primarily

3More exactly, 0.00998 houses per week.

66 9.2. A simple model of directed technological change

interested in the direction of technological change ratherthan overall growth we simply fix thetotal quantity of research labour in the economy. And because we are primarily interested in thelong run, we remove capital from the model, and focus solely on labour and the resource. And,for simplicity, we assume that research investment occurs in the same period as its results are feltby the firm.

9.2.1. Utility and demand. First, let’s assume that consumers have the very simple utilityfunction

U =∞∑

t=0

βtyt,

whereβ is a parameter less than 1, andy is the aggregate good. So we have a constant, exogenousdiscount rate.

Next we assume that there is a fixed labour forceL, and fixed unit continua of possibleproducts and also firms, such that one firm produces each product. ResourcesR arrive in someexogenous flow, and are bought by firms on perfect markets. (Wecan relax this assumption lateron, when necessary.)

The quantity of the aggregate goody is as follows:

y=

(∫ 1

0yηi di

)1/η

,

whereη is a parameter less than 1. Thus there is monopolistic competition between the producersof the different goods. Normalize the price of the aggregate to 1. Then (by differentiating) we canobtain

pyi =∂y∂yi=

(

yyi

)1−η

.

Given symmetryy= yi so all the goods have price 1. But crucially, we can differentiate the abovewith respect to quantity to obtain the following expressionfor marginal revenue:

pyi +yi∂pyi

∂yi= ηpyi.

So whenη = 1 we have perfect competition, because the goods of the different firms are perfectsubstitutes. But asη falls below 1, marginal revenue falls as a proportion of the price due to thenegative effect of quantity on price.

9.2.2. A single firm’s knowledge production function. To begin the process of modellingDTC, we focus on the knowledge production function of an individual firm. Assume that the firmproducing producti has the production function

yi = [(Ali l i)ǫ + (Ari r i )ǫ ]1/ǫ .

The productivitiesAli and Ari are specific to the firm, and arise as a result of investments bythe firm,zli andzri . Furthermore, we assume all knowledge in periodt builds in some way on theexisting set of all knowledge in the economy,At−1, and that the elasticity of knowledge productionwith respect to investment is a positive constant (less than1),φ. Thus we have:

Alit = Fli (At−1)zφlit ;

Arit = Fri (At−1)zφrit .

Note thatAt−1 is a vector of all the different knowledge stocks of all firms (and potentially oth-ers) in the economy. An important feature of the production function is that firms do not buildtheir period-t knowledge on their existing private knowledge, rather theybuild on all existingknowledge in the entire economy. This simplifies the dynamics of the model immensely.

9.2.3. A single firm’s optimization problem. Now assume that the firm is small, thus it isa price-taker with regard to buying or hiring inputs. However, its product is unique, hence it isnotnecessarily a price-taker with regard to selling its product. Defining input prices aswz, wl andwr ,and the price of the product aspyi, and dropping thet subscripts for clarity, we can write downthe firm’s problem as a Lagrangian:

L = pi(yi)[(Ali l i)ǫ + (Ari r i )ǫ ]1/ǫ −wz(zli +zri )− (wl l i +wr r i)

−λli (Ali −Fl ·zφ

li )−λri (Ari −Fr ·zφ

ri ).

Hereλli andλri are the shadow prices of firm knowledge.

Chapter 9. Directed technological change and resource efficiency 67

To solve the problem we take first-order conditions inl i and r i to yield an expression forwL l/(wRr):

wL l iwRr i

=

(

Ali l iAri r i

)ǫ

.

This equation relates relative factor expenditures by the firm to the relative augmented quantitiesof the factors. (The augmented quantity is the physical quantity multiplied by the productivity.)Sinceǫ is negative, the expression shows that the firm spends more onthe factor which is relativelyscarce.

Now we take first-order conditions inAli andAri to yield an expression forλli Ali /(λri Ari ):

λli Ali

λri Ari=

(

Ali l iAri r i

)ǫ

.

We can actually write down this expression without the need for working through the first-orderconditions, using the symmetry implicit in the Lagrangian,where theλs play the role of thews,and theks play the role of theqs.

Finally, we take first-order conditions inzli andzri to yield a second expression forλli Ali/(λri Ari ):

zli

zri=λli Ali

λri Ari.

This simply states that firms invest in proportion to the value of the knowledge produced, whichis the shadow price of that knowledge times its quantity.

Now we can eliminate the shadow prices from the two expressions above to yield an expres-sion for relative investments as a function of prices and quantities of the inputs:

zli

zri=

(

Ali l iAri r i

)ǫ

=wL l iwRr i

.(9.6)

So firms invest in proportion to the resultant relative factor expenditures. If a firm spends twice asmuch on labour as it does on resources, it will invest twice asmuch in increasing the productivityof labour compared to resources!

9.2.4. General equilibrium. Having solved for a single firm, the big question is what hap-pens at the level of the overall economy, in the long run. To answer this question we need to addsome more elements to the model.

First we must specify the relationship between the economy-wide knowledgeA and the firm-specific knowledge stocksAli andAri . Here we divide the economy-wide knowledge stocks intotwo sets,AL andAR, which are the sum of the individual firms’ knowledge stocks:

AL =

∫ 1

0Ali di;

AR=

∫ 1

0Ari di.

So in symmetric equilibrium with a representative firm (for which we drop the subscriptis) wehaveAL = Al , andAR= Ar . Thus—recalling the knowledge production functions above—we canwrite

Alit = Fl(Alt−1,Art−1)zφlit ;

Arit = Fr (Alt−1,Art−1)zφrit .

Then we must specify the functionsFl andFr . We begin with the simplest possible specifica-tion, an extreme case, as explained below.

Definition 1. Independent knowledge stocks.Knowledge stocks develop independently when

Fl = Alt−1/ζL and

Fr = Art−1/ζR,

whereζL andζR are positive parameters. Thus labour-augmenting knowledge builds exclusivelyon existing labour-augmentingknowledge, and resource-augmenting knowledge builds exclusivelyon existing resource-augmenting knowledge.

Then (using the knowledge production functions above) we have

Alit /Arit = (Alt−1/Art−1)(ζR/ζL)(zlit /zrit )φ.(9.7)

68 9.2. A simple model of directed technological change

9.2.5. Solution to the model.Now to finalize the solution. To solve, recall equation 9.6, andassume symmetric equilibrium to show that

zl

zr=

(

ALLARR

)ǫ

=wl LwrR

.(9.8)

Now substitute for relative investments using equation 9.7to obtain(

Alt/Alt−1

Art/Art−1

ζR

ζL

)1/φ

=wlt Lt

wrtRt=

(

Alt Lt

ArtRt

)ǫ

.

We can then use this equation to obtain a single equation for the development of relative knowl-edge stocks either as a function of the relative quantities of the inputs, or the relative prices. Forconcreteness we assume that the relative prices of the inputs are exogenously given, i.e.wl/wr isgiven in each period, although it may vary. Then (after a bit of algebra) we have

Alt/Alt−1

Art/Art−1=

(

Alt−1

Art−1·wrt

wlt

)ǫφ/(1−ǫ(1+φ)) (ζL

ζR

)(1−ǫ)/(1−ǫ(1+φ))

.(9.9)

For balanced growth, the growth rates of each of the variables must be constant. This implies thatthe LHS of the equation is constant, since it is one constant divided by another. Which impliesin turn that the RHS is constant, so ifwl/wr is increasing at some fixed rate thenAL/AR must beincreasing at the same rate:θa = θw. So if the relative price of the resource is declining at somefixed rate, the relative productivity of the resource must decline at the same rate. This implies inturn that relative quantities will increase at the same rate, hence relative investments are constanton the b.g.p.

9.2.6. Stability of the b.g.p. Whatever the values of parameters, a bgp will exist. But willit be stable? In other words, will the economy approach the bgp over time, or will it head offsomewhere else? It turns out that the elasticity of substitution between the inputs is key.

The key is the result that relative investments are equal to relative factor shares. Assume thatwe are on a b.g.p. such that constant relative investments lead to constant factor shares. Thenassume that there is some perturbation to the system such that the ratio ofAL to AR moves offthe balanced growth path. Does this trigger the economy to move away from the path, or doesit return to the path? For concreteness, assume thatAR falls. Then we know from equation 9.6that the factor share of the resource will rise (sinceǫ is negative) causing investment inAR to rise,thus boosting the growth ofAR. This shows that the b.g.p. of the economy is stable, as long as ǫis negative; whenǫ > 0 the analysis and results change completely, as we see in Chapter 11. Thesituation is illustrated in Figure 9.1

State of technologyAugmentinglabour

Augmentingenergy

Market forces

Figure 9.1. Illustration of how relative prices (the shape of the economic land-scape) determine the relative levels of technology augmenting labour and en-ergy.

9.2.7. The long-run aggregate production function.We have established that if the ratioof the price of labour to the price of the resource displays a constant trend, then the economy willapproach a b.g.p. on which the ratio of the factor-augmenting knowledge stocks has exactly thesame trend. But from the first-order conditions in quantities we know that

wlLwrR=

(

ALLARR

)ǫ

,

henceLR=

(

AL

AR

)ǫ/(1−ǫ) ( wl

wr

)−1/(1−ǫ)

,

Chapter 9. Directed technological change and resource efficiency 69

which implies thatL/R has exactly the opposite trend toAL/AR andwl/wr , implying that bothaugmented inputs and the factor share are constant. So the long-run factor shares are constantdespite changes in the relative quantities of the inputs. This implies that the long-run aggregateproduction function is, in reduced form, Cobb–Douglas:

y= AL1−αRα.

So we are back to the production function of the DHSS model! (Although without capital.)The value ofα will be determined by the relative values ofζL andζR, and the relative growth

rates of input quantities (or prices). If it is easier (cheaper) to develop technologies augmentingresources than it is to develop technologies augmenting labour thenα will be small and the factorshare of resources will also be small. Just as in the DHSS model, sustainability should be noproblem as long as we manage resource stocks sensibly: thus we should gradually reduceR ifthere is a finite stock of the resource.

The key difference compared to the ‘Limits’ model is thatAR, resource-augmenting knowl-edge, is allowed to grow exponentially and without bound. This allows sustainable use of a finiteresource stock even when the resource and other inputs (in this case labour) are highly comple-mentary in the short run. Given such complementarity, andwithoutDTC, we are very close to thelimits model, and there is no way to achieve sustainability (if defined as any consumption levelthat can be maintained into the indefinite future).

The model is not really vulnerable to Daly’s criticisms of DHSS—that we are making morecake just by getting a bigger oven (page 40)—because the model is about technological change,not capital accumulation. Technological change does, demonstrably, allow us to get more product(value) out of given inputs, and there is no obvious limit to this process.4 But does the model standup to more detailed examination? We find out in the next section.

9.3. Evidence regarding the DTC model

9.3.1. Simulating the economy.We can easily simulate the economy in (for instance) excel.If we assume that prices (rather than quantities) are exogenous then the key equation is 9.9. Putinto simulation-friendly form, the equation is

Alt/Art = Alt−1/Art−1 ((Alt−1/Art−1)(wrt/wlt ))ǫφ/(1−ǫ(1+φ)) (ζL/ζR)(1−ǫ)/(1−ǫ(1+φ)) .

In simpler notation we can write

At = At−1(At−1/Wt)ǫφ/(1−ǫ(1+φ))ζ(1−ǫ)/(1−ǫ(1+φ)),

whereζ = ζL/ζR.To write the program, we must choose parameter values and assume a trend for the relative

priceswl/wr . Assume for instance thatwl/wr rises by a factorθ each period. Choose a value lessthan 1 forφ, and a negative value forǫ. SetζL andζR equal to one another initially, for simplicity.Choose an initial value for the relative knowledge stocks, and see what happens over time! Notethat once you know howK andW evolve over time, we can (and should) also find out how theother variables evolve. Assume (for instance) that labour is constant and thatAL grows at by thefactorθ per period.

You can find the program for the case of complementary inputs on the webpage. Plot somecurves (remember to take logarithms!) to get a better picture of what is going on.

9.3.2. Simulation based on real data.The above conclusions (based on the theoretical re-sults) are reflected in simulations based on real data, as shown in Figure 9.2. Here we see thatwe can parameterize the model outlined above such that when we feed in data on global energyprices the model predicts rates of energy use which match observations. The bottom panel showswhat lies behind the model results: global energy use has tracked global product because the levelof energy-augmenting knowledgeAR has failed to rise. The reasonAR has not risen is in turn thatenergy prices have not risen.

In order to test the model the simplest strategy is to look at the growth of energy-augmentingknowledge. More specifically, has production of specific products become more or less energy-efficient over time? Estimation of productivity changes is typically performed indirectly: a pro-ductivity increase is imputed as the residual to explain changes in the value of output per unitof time from given inputs. This approach leads to the conclusion that the growth rate of labour-augmenting knowledge should be approximately equal to the growth rate of per capita GDP; this

4Note however that in some cases thereareobvious limits, as with the efficiency of the production of artificial light.We return to this in Chapter 13 below.

70 9.3. Evidence regarding the DTC model

1880 1900 1920 1940 1960 1980 2000−0.5

0

0.5

1

1.5

2

1880 1900 1920 1940 1960 1980 2000

0

1

2

1880 1900 1920 1940 1960 1980 2000

0

0.5

1

1.5

2

GDP/cap, model

GDP/cap, data

R/L, model

R/L, datawr

AL, model

AR, model

Figure 9.2. The model parameterized to fit global data. The top row showsGDP (model and data); the middle row shows energy consumption (model anddata) and energy price; the bottom row shows the growth of theknowledgestocks. All axes are log-normalized.

result is in agreement with the simulation, where we see (bottom panel) thatAL rises at the samerate asY (upper panel).

However, in the case of energy we can use a more direct approach to measurement of factor-augmenting knowledge, since there is plenty of direct evidence about changes in our ability toextract specific physical outputs from measurable energy inputs. To illustrate we consider twoproducts, artificial light and motive power. Light is a convenient product category for analysissince light is a consumption good which is rather homogeneous and unchanging over very longtimescales, and the energy efficiency of its production is easily measured. Fouquet and Pearson(2006) study light production and consumption in the U.K. over seven centuries. They concludethat the efficiency of light production in the U.K. (measured by lumen produced per watt of energyused) increased 1000-fold from 1800 to 2000; the productivity of labour in the U.K. over the sameperiod rose by a factor of 12–15 (estimates vary). Light production is a convenient sector withinwhich to measure efficiency, but it is not very large. Now we turn to the productionof motivepower from fossil fuels, a very large sector. Motive power istypically an intermediate good ratherthan a final good, nevertheless increases in the efficiency with which energy inputs are used togenerate motive power are very likely to be reflected in the overall efficiency with which energyis used to generate final goods, as long as the final goods are homogeneous and do not changeover time. In the 19th century motive power was largely generated by steam engines, while overthe last 100 years we must consider electric power generation and the internal combustion engine.Regarding steam engines, sources such as Hills (1993) suggest that their efficiency in generatingpower from coal inputs increased steadily from their invention in the early 1700s up to 1900, andby a factor of around 20 over the entire period; this growth inefficiency is again more rapid thanthe growth in labour productivity over the same period. Subsequently, the efficiency of coal-firedpower stations has continued to increase but at a declining rate; see for instance Yeh and Rubin(2007) for detailed evidence.

The above evidence suggests that energy productivity over this period, far from being con-stant, actually increased faster than labour productivity. So the model with independent knowledgestocks matches energy demand at the expense of wildly incorrect predictions about the growth ofenergy-augmenting knowledge. On the other hand, if we lock knowledge stocks together thiscomes closer to the truth in its predictions about knowledgegrowth, but at the expense of nolonger being able to predict energy demand correctly. And whatever we assume about knowledge

Chapter 9. Directed technological change and resource efficiency 71

stocks, the model cannot match the data. For an analysis of what is reasonable to assume aboutlinks between knowledge stocks, see the next chapter on technology transitions.

9.4. Links between knowledge stocks

9.4.1. Why we should include links in our model.Recall that we assumed that knowledgestocks grow independently of one another. That is, a higher level of labour-augmenting knowledge(or overall productivity in the economy) does not help at allwhen it comes to performing researchto raise resource-augmenting knowledge. Is this reasonable? Arguments that it is not date atleast to Nordhaus (1973b), however Norhaus’s arguments appeal only to intuition, and it wouldbe reassuring if we had more direct microeconomic evidence.

The evidence based on intuition is nevertheless powerful. Consider the following thoughtexperiment. Assume there was no generation of power from wind between 1900 (when the lastwindmills were decommissioned) and 1990 (when electricitygeneration from wind started). Onwhat knowledge stock would the new wind power generators build? More broadly, is the idea ofindependent knowledge stocks defensible? It implies that technologies which allow us to makebetter use of raw energy inputs (the steam engine, the steam turbine for the generation of electricity,the internal combustion engine) are developed completely independently of other technologicaladvances in the economy. This seems to be an indefensible idea: such technologies are developedhand-in-hand with advances in mathematics, physics, engineering etc., advances which are alsorelevant to augmenting inputs of labour–capital. Thus stocks of knowledge augmenting energy andstocks of knowledge augmenting labour–capital are intimately linked, overlapping and feedingoff one another. So, summing up, it is hardly surprising that themodel fails when confronted byevidence.

There is also direct evidence that (for instance) both resource-augmentingand labour-augmentingknowledge draw on a common pool of general or fundamental knowledge, and that advances inthis general knowledge thus drive advances both inal andar? Such evidence can be found in stud-ies of patent data. Very direct evidence can be found in Trajtenberg et al. (1992), who study patentcitations and show that patenting firms cite patents both within their own three-digit industry, butalso outside it.5

Popp (2002) provides more indirect—but no less convincing—evidence, in that he findsevidence for diminishing returns to investment in specific technologies over time; a rise in energy-prices induces a surge in patenting activity within energy sectors such as wind or solar, but theeffect peters out rather rapidly. Within our framework we can interpret this result as follows:Discoveries of potential relevance to energy-augmentation are made frequently in other (muchlarger) research sectors; For instance, think of the invention of the computer. It takes time andresearch effort for the benefits of these discoveries to be incorporated in the energy sector. Ifthere is a surge of research in the energy sector then initially there will be many potentially usefultechnologies available which have not yet been applied in that sector, but over time these ‘low-hanging fruits’ will be picked and the productivity of energy-augmenting research will fall.

9.4.2. The effect of including links. Developing and solving a model with links betweenknowledge stocks is beyond the scope of this book. We simply note that it would be more rea-sonable to assume that knowledge stocks are linked, such that if (for instance) there is a lot oflabour-augmenting knowledge this makes it easier to accumulate resource-augmenting knowl-edge. Given a model with links we expect the knowledge stocksto grow together (although notnecessarily at exactly the same rate). What would be the results of such a change in the model?

The result of linking knowledge stocks would be to yield a better fit to the data on inputproductivity, at the expense of the model’s ability to matchthe data on factor shares: if knowledgestocks grow together while the price of resources falls relative to the price of labour, then weexpect the resource share to fall over time. Consider the simple case when knowledge stocks arelocked together:Ar = γAl . Then the aggregate production function is

y= Al [Lǫ + (γR)ǫ]1/ǫ .

If the inputs are strongly complementary (ǫ is large and negative) this implies that the inputs willbe bought by firms in almost fixed proportions, irrespective of their relative prices. Thus resourceuse will track the size of the labour force rather than overall growth. This is directly contradictedby the evidence, for instance the data shown in Figure 1.3.

5They score patents as follows: within 3-digit scores zero, within 2-digit scores 0.33, within 1-digit scores 0.67 andoutside 1-digit scores 1. The average score is 0.31. A three-digit industry is a relatively narrowly defined industrial sector,according to the Standard Industrial Classification. Two- and one-digit industries are successively more broadly defined.

72 9.4. Links between knowledge stocks

To be more precise, assume perfect markets and take first-order conditions inL andR to showthat

wlLwrR=

(LR

)ǫ

=

(

wl

wr

)−ǫ/(1−ǫ)

.

Sinceǫ is large and negative this shows that when resource inputs rise relative to labour inputs theresource share should decline steeply, and that when the resource price falls relative to the wagethe resource share should decline steeply.

CHAPTER 10

Structural change and energy demand

In the single-sector model developed in the previous chapter, if resource use tracks growthit must be because resource-augmenting knowledge has not risen. In order to add alternativeexplanations we must include multiple final goods, the production of which differs in resourceintensity. Given multiple goods resource consumption may track growth even when resourceefficiency increases, if consumers switch towards resource-intensive products. This switch maybe an endogenous result of increase in resource efficiency, in which case it is calledrebound.Alternatively, the switch may be caused by other factors, such as income growth. In this chapterwe look first at some of the broad evidence which demonstratesthat structural change must becentral to the analysis of long-run demand for energy and resources, before turning to the keyquestions of what drives structural change and what the policy implications are.

10.1. Introduction to structural change

1880 1900 1920 1940 1960 1980 20000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Year

logG

DP

/cap

ita

Figure 10.1. Long-run growth in real GDP per capita: U.S. (upper line) and China.

10.1.1. Growth and structural change.The growth rate of GDP per capita in the leading in-dustrial economy has been astonishingly constant in the industrial era. On the other hand, growthrates of other economies have varied enormously; Figure 10.1.1 The reason is that growth in percapita production is primarily driven by new technology: leading-edge technology seems to growat a remarkably constant rate, whereas the distance of an economy from the leading edge mayvary dramatically over time. The U.S. has remained at or close to the leading edge for well over100 years, hence its constant growth rate. China on the otherhand moved further and further awayfrom the leading edge for many decade, before this trend was turned upside down starting in the1970s.

Note that growth—driven by new technology—is primarily about production of different,more valuable, final goods over time. Only a small proportionof our expenditure today goes ongoods that were available in the same form 50 years ago. And the proportion becomes very smallif we go back 100 or 150 years. Consider food. One hundred years ago food made up a large part

1Jones (2005) pointed this out. Data from Angus Maddison website, Statistics on World Population.

73

74 10.1. Introduction to structural change

of household budgets, everywhere. However, as we have become richer and richer our expenditureon food has changed relatively little, whereas our expenditure on other goods has risen.

Figure 10.1 shows that GDP/capita is more than 10 times higher today than in 1870; is thatgrowth due to U.S. citizens today consuming 10 times more of the same products today thatthey consumed in 1870? Consider car ownership. In 1870 therewere no cars. Car ownershipsubsequently grew rapidly, but between 1970 and 2008 it was constant at 0.44 per capita; in themeantime, GDP per capita had more than doubled.2 Meanwhile, ownership of home computersand mobile phones was effectively zero in 1970, whereas today it is more-or-less universal. Now,do we consume cars and smartphones today because we work longer hours, or have saved up morecapital, but with the same skills and the same machines as we had in 1870? Obviously not: it isthe arrival of new products, and increases in the quality of existing products, which is the majorand fundamental driver of long-run growth.

Finally, note that the transformation of the economy is not simply about the addition of newproducts, made in new ways, i.e. increasing variety. It is better characterized as a process in whichnew ideas—new technologies—transform existing processesand products, as well as addingcompletely new ones. The most important of these new technologies are sometimes denotedgeneral purpose technologies, or GPTs; see David (1990) and Bresnahan and Trajtenberg (1995)for classic discussions of such technologies. Examples include the steam engine, the electricmotor, semiconductors, and the internet. Such inventions lead, over time, to fundamental changesthroughout many areas of the economy, or indeed throughout the entire economy, transformingthe way many existing goods or services are produced, and opening up previously unimaginablepossibilities for new goods and services.

10.1.2. Resources.Global resource and energy use has shown an astonishing tendency totrack global product in the long run. At the same time, priceshave been constant. This is shownin Figure 10.2 for metals and energy, reproduced from Figure1.3. Note that the result of constantprices and consumption tracking global product is that theshareof metals in global product isalso constant.3

1900 1920 1940 1960 1980 2000

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Year

log1

0, n

orm

aliz

ed

1850 1900 1950 2000−0.5

0

0.5

1

1.5

Year

log1

0, n

orm

aliz

ed

Metalquantity

Metalexpenditure

Metal price

Global productGlobal product

Energyexpenditure

Energyprice

Energyquantity

(a) Global Metals (b) Global Energy

Figure 10.2. Long-run growth in consumption and prices, compared to growthin global product, for (a) Metals, and (b) Primary energy from combustion.

How do we explain the data of Figure 10.2? The easy part is to rule out the one-sector neoclas-sical model, based on the analysis of the previous chapter (see for instance Section 9.3). Recallthat, for the model to be capable of explaining the data, there should have been no significantincrease—since 1900—in the resource efficiency with which individual goods are produced inthe economy. In reality efficiency has increased across the board, so the explanation for the failureof resource use to fall relative to global product must be sectoral shifts towards resource-intensivesectors.

2Sources: Popn. data from US census, and car-ownership data from the Bureau of Transportation Statistics.3The figures show normalized prices, quantities, etc., so they show how the factor shares of resources change over

time, but nothing about the absolute levels of the factor costs compared to the value of global product. The absolute shareof resources is significant but not large. For instance, the factor share of crude oil in the global economy in 2008 was 3.6percent, whereas the factor share of the 17 major metals was just 0.7 percent.

Chapter 10. Structural change and energy demand 75

Recall Daly’s critique of the neoclassical model (page 40):can we bake more and morecake with the same quantity of ingredients? Can we, for example, produce more and morelight using the same energy inputs? Can we travel further andfurther using the same energyinputs? Can we keep our houses warm. . . ? Etc. It turns out thatwe can! Consider for instanceFouquet and Pearson (2006) on the history of light production. They conclude that the efficiencyof lighting in the U.K. (measured by lumen produced per watt of energy used) increased 1000-fold from 1800 to 2000. That’s a lot more cake! Regarding the production of motive power fromfossil fuels, historically this concerns the efficiency of steam engines, while over the last centurywe must consider electric power generation and the internalcombustion engine. Regarding steamengines, sources such as Hills (1993) suggest that their efficiency in generating power from coalinputs increased steadily from their invention in the early1700s up to 1900, and by a factor ofaround 20 over the entire period; this growth in efficiency is at least equal to the growth in labourproductivity over the same period. Subsequently, the efficiency of coal-fired power stations hascontinued to increase but at a declining rate; see for instance Yeh and Rubin (2007) for detailedevidence.

Does the above discussion mean Daly was wrong in using his cake analogy to criticize DHSS?In fact it does not, since in the DHSS model it is capital accumulation which allows us to bakemore cake from the same ingredients, but in the above examples it is technological progressthathas allowed us to do so.

We consume a vast range of products. But the limits and one-sector DTC models lump themall together into one aggregate,Y, and never try to deconstruct that aggregate. (DHSS does thesame; it is based on the one-sector neoclassical growth model.) Is that OK? It would be OK ifour consumption of all the different products increased at the same rate, or (less restrictively) ifwe could collect products into groups which were similar in arelevant sense (for instance, labour-intensive and resource-intensive products) and show that aggregate consumption of the groups ofproducts increased at the same rate. It turns out that we cannot do so.

Return to light. We already know from Fouquet and Pearson (2006) that the efficiency of light-ing in the U.K. increased 1000-fold from 1800 to 2000. In the same period, GDP per capita roseby a factor of 15. Meanwhile, consumption of artificial lightper capita rose by a factor of 7000.So we have a massive substitution towards (energy-intensive) light production and consumption.

Regarding transport, the situation is complicated by the fact that the cost of personal transportis not just financial, it is also measured in time. Furthermore, transport varies in quality as well asquantity; faster is, ceteris paribus, better. The result isthat rising income is correlated with morerapid forms of transport, and a greater distance travelled per person–year, but not with more timespent travelling. Schafer (2006) shows that world travel (in terms of person-kilometres travelledper year) has grown more rapidly than global product per capita. Furthermore, rising income iscorrelated with a successive shift from non-motorized transport→ public transport→ light-dutyvehicles→ high-speed transport modes (such as flying).

See also Knittel (2011), who analyses technological changeand consumption patterns in theU.S. automobile industry, and shows that for a vehicle of fixed characteristics in terms of weightand engine power, then fuel economy would have increased by 60 percent over the period 1980–2006 due to technological change, but that actual average fuel economy increased by just 15percent; the difference is due to countervailing increases in the weight and power of vehicles.Rebound?

10.2. Preliminary analysis

10.2.1. Why we need structural change to explain the data.Before analysing structuralchange, we revisit why we need it to explain the data in Figure1.3.

Throughout this chapter, we consider an economy with two inputs, labour and energy, andtreat the quantity and productivity of labour inputs as exogenous, and the price of primary energyinputs as exogenous. We can think of primary energy—in the form of fossil fuels such as coal, orrenewables such as wind—being extracted or harvested at an exogenous cost per unit of energy.4

The analysis when there is a single product in the economy (a widget) is familiar from previouschapters. Since there are only two inputs (L andR) and one product (Y), we must be able to writethe production function as

Y= F(ALL,ARR).

4More specifically, each unit of final goods devoted to extraction/harvest yields an exogenously determined quantityof primary energy.

76 10.2. Preliminary analysis

Recall thatAL and L are treated as exogenous here. Furthermore, we treatAR as exogenous,and model the evolution ofR over time. To simplify the analysis we go further and assumea CES production function with low elasticity of substitution (labour and energy are stronglycomplementary), thus we have

Y= [(ALL)ǫ + (ARR)ǫ ]1/ǫ ,

whereǫ < 0. Assuming perfect markets we can model the choices of a representative final-goodproducer whose profit-maximization problem is

maxL,R

π = [(ALL)ǫ + (ARR)ǫ ]1/ǫ −wLL−wRR.

Take the first-order condition inR to show that

RY=

(

AR

wR

)1/(1−ǫ) 1AR.

Now differentiate with respect to time to show that

RR=

YY−−ǫ

1− ǫAR

AR−

11− ǫ

wR

wR.

Now the historical data shows thatwR is approximately constant in the long run, while resourceuse tracks global product. Sinceǫ is large and negative (the inputs are strongly complementary)this implies thatAR/AR= 0, i.e. there has been no resource-augmenting technological change. Butthis is strongly contradicted by the evidence of the Chapter9, especially in the case of energy.

Since the resource efficiency of individual products has increased, the only way toexplainthe failure of overall resource efficiency to increase is through a shift in consumption patternsover time, from products of low intrinsic resource intensity towards products of higher intrinsicresource intensity. Such a shift is known asstructural changein macroeconomics, and in order tounderstand it and predict the future we must build and test models of structural change.

10.2.2. Driving forces of structural change.We know from the aggregate data that theremust have been a shift in consumption patterns over the last 100 years or more, from labour-intensive to energy-intensive goods. What might have driven this shift? There are essentiallytwo possible economic explanations: a substitution effect and an income effect. When there is asubstitution effect, consumers switch to energy-intensive goods because they got cheaper relativeto labour-intensive; when there is an income effect, they switch because they got richer, and richpeople prefer energy-intensive goods. To get a better handle on these two effects, we now setup a simple consumption model with just two goods, one labour-intensive and the other energy-intensive.

Consider an economy with two final goods,X andY. We do not study the production side,but simply assume thatX is energy-intensive andY is labour-intensive. The prices of the goodsarepx andpy respectively, and total income (to be divided between the goods) ism. Markets areperfect. Now consider the following two (alternative) utility functions.

Our first utility function is a simple CES:

U = [αXǫ + (1−α)Yǫ]1/ǫ ,

whereα ∈ (0,1) andǫ ∈ (−∞,1). This implies that

pxXpyY=

α

1−α

(XY

)ǫ

, andXY=

(

α

1−α

py

px

)1/(1−ǫ)

.

So there are no income effects: the ratio ofX to Y (energy-intensive to labour-intensive consump-tion) is unaffected by incomem.

What about the substitution effect? This depends onǫ, and in particular theelasticity ofsubstitutionbetween the goods, 1/(1− ǫ). The higher is the elasticity of substitution (implyingǫclose to 1), the greater the effect on relative consumption of a change in relative prices, i.e. thestronger the substitution effect.

Both substitution and income effects are illustrated in Figure 10.3, where we show threeindifference curves and three budget restrictions, given parameter valuesǫ = 0.6 andα = 0.5. Inthe baseline restrictionb1 we havem= 2 andpx = py = 1. In b2 we have doubled income withoutchanging relative prices, and inb3 we again havem= 4 butpx = 0.7 while py = 1.3. When incomeincreases but relative prices are unchanged we see that relative consumption remains unchanged.However, whenpx falls relative topy there is substantial substitution towards goodX and awayfrom goodY. This is a result of the high degree of substitutability between the goods (ǫ = 0.6).

Chapter 10. Structural change and energy demand 77

So if this model is a reflection of the real economy, then the shift towards energy-intensiveconsumption goods has occurred due to a fall in the price of these goods relative to less energy-intensive alternatives. In the next section (10.3) we will look at whether such a fall has occurred,and whether the consequent substitution effect might be large enough to account for the data.

0 1 2 3 4 5 60

1

2

3

4

5

6

b1

b2

b3

(x1,y1)

(x2,y2)

(x3,y3)

Figure 10.3. Substitution and income effects in the first model

0 1 2 3 4 5 60

1

2

3

4

5

6

b1

b2

b2p

b3

(x1,y1)(x2,y2) (x3,y3)

Figure 10.4. Substitution and income effects in the second model

We now set up a second utility function which allows for both substitution and income effectsas drivers of structural change. We do so in the simplest way possible by assuming a form of lexi-cographic preferences: consumers demand a certain quantity Z of one good as a precondition forstarting consumption of the second good. Furthermore, oncethey haveZ, they switch additionalconsumption entirely to the second good. We assume that the first goodZ is a composite of goodsX andY, while the second good is simplyX, the energy-intensive good. The utility maximizationproblem is

maxU = X−X∗,

subject to (X∗)αY1−α ≥ Z.

SoX is still total production of the first final good, of whichX∗ goes to making goodZ, andX−X∗

remains.We are only interested in the case when income is sufficient to allow the restriction to be

satisfied. Assume that the restriction is satisfied and all income is used up, soU = 0. Furthermore,we know that

(X∗)αY1−α = Z,pxX∗

pyY=

α

1−α, and pxX

∗+ pyY=m.

78 10.3. Structural change without income effects

We setα = 0.5,m= 2, 4, and 4 (as above),px = 1, 1, and 0.7 (as above), andpy = 1, 1, and 1.3 (asabove). Finally,Z = 1 so the restriction can be satisfied whenm= 2, with nothing left for furtherconsumption. We then obtain Figure 10.4.

Now we see that an increase in income leads to a very large change in relative consumption,as all of the additional income is spent on goodX, and none on goodY. Furthermore, the changein prices leads to both an income and a substitution effect, where the income effect is much larger;the income effect is the shift from (x2,y2) to the empty circle, and the substitution effect is thefurther shift from the empty circle to (x3,y3).

10.2.3. Why is it important? If we know that structural change is happening, it is veryimportant for policy to know what is causing it. There are several reasons for this, of whichwe discuss two. The first reason we need to know the causes of structural change is in orderto predict future energy demand: if we want to predict futureenergy demand, then we need tounderstand the drivers of energy demand. Furthermore, prediction may also be important foroptimal environmental policy: if demand is likely to rise steeply in the future, this will implyhigher carbon emissions and this may in turn lead to higher marginal damage costs of carbonemissions today, and hence higher optimal taxes.

The second reason we need to know the causes of structural change is that it may be directlyrelevant to the choice of policy instruments in second best.When the only market imperfection isthe failure to price carbon emissions then we know from Pigou(1920) that the optimal allocationcan be achieved by applying a Pigovian tax on those emissions, i.e. a tax set at the level of mar-ginal damages caused by the emissions.5 But when there are multiple market imperfections, thesituation is unlikely to be so simple, since some of these imperfections may be difficult or impossi-ble to correct, and this may affect the efficacy of emissions taxes. For instance, in an internationalcontext it may be difficult for countries to agree on a uniform tax rate (or a global trading system).Under these circumstances, if one country applies emissions taxes unilaterally, this may lead toleakage, i.e. the tax may cause emissions to shift out of thatcountry and into other countries. Thismay be caused by energy-intensive industries relocating away from the taxing country.

If a tax is counterproductive in this way, an attractive alternative may be to subsidize invest-ment in energy-augmenting technology, thus making energy-intensive industries more efficientand (hopefully) reducing emissions. But if the elasticity of substitution between energy-intensiveand labour-intensive goods is high, the policy may not give the desired result: an increase in en-ergy efficiency will reduce the price of energy-intensive goods, causing consumers to substitutetowards such goods. This effect is known asreboundand will be discussed at length below. Onthe other hand, if the elasticity of substitution is low—implying that demand for energy-intensivegoods is inelastic—then a Pigovian tax will have little direct effect on consumption, and its maineffect is likely to be on technology. Under these circumstances, technology subsidies (either toenergy-efficiency, or to reducing the costs of clean energy production)may be a good option.6

10.3. Structural change without income effects

10.3.1. Rebound and consumption patterns.Rebound is frequently ignored in theoreticalliterature, perhaps because of the common assumption that the economy consists of just a singlesector. However, the idea has a long history, starting with Jevons (1865), who argued that futurescarcity of coal would be exacerbated, not alleviated, by innovations increasing the efficiencyof technologies based on coal use, the reason being that suchinnovations would lead to a largeincrease in the use of coal-based technologies. The idea hasbeen picked up more recently byenergy and ecological economists (see for instance Binswanger, 2001, and citations), where it hasbeen named the rebound effect.

To define rebound, assume an economy in which total energy useis R, and focus on produc-tion of goodi using (among other inputs) augmented energy flowAri Ri . Rebound is present whenan increase in energy-efficiencyAri by a factorx leads to a reduction ofRby less thanRi(1−1/x).Note that according to this definition rebound may occur within the production process itself:if the producer of the good has access to a more energy-efficient technology, the producer maychoose to use more augmented energy and less augmented labour or capital in the productionprocess. However, we generally think of rebound as occurring on the consumption side of theeconomy. Given the definition above, we can then think of the producer as having a Leontiefproduction function with no substitutability between augmented energy and other inputs.

5The same result can of course be achieved through tradable permits as well.6Note that if the elasticity of substitution is low, this implies that structural change must have been driven by income

effects, i.e. consumers choosing more energy-intensive goodsas they got richer.

Chapter 10. Structural change and energy demand 79

Both income and substitution effects may contribute to rebound: an increase inAri leads (ce-teris paribus) to a fall in the price of goodi, which raises the purchasing power of consumers(income effect) and induces them to substitute towards consumption of good i (substitution ef-fect). Given the small factor share of energy, the income effect of increases in energy-augmentingtechnology is likely to be small; on the other hand, given themuch higher energy share of someproducts, the substitution effect of increases of the energy-efficiency of such products may besubstantial.

The evidence for rebound effects is reviewed by Sorrell (2007), who finds that they are sig-nificant but generally much less than 100 percent, implying that increases in energy efficiency ofspecific products do lead to large reductions in energy use associated with consumption of thoseproducts. A key reason for this is that the substitutabilitybetween energy-intensive and other prod-ucts is far from perfect, just as intuition would suggest.7 This evidence suggests that rebound alonecannot explain the shift towards consumption of energy-intensive goods, implying that income ef-fects (driven by rising labour productivity) must also havea part to play. Although microeconomicstudies of rebound abound, there have only been a few attempts to build macroeconomic modelsin the literature: see for instance Saunders (1992, 2000).

10.3.2. A general rebound model.To analyse rebound we must have a general equilibriummodel. Since we want to focus on the substitution between products on the consumption sidein the simplest possible context we assume two productsy1 andy2, both of which are producedcompetitively by a representative firm using a Leontief technology and inputs of labour and aresource. The key equations describing the production sideof the economy are as follows:

y1 =min(Al l1,Ar1r1);(10.1)

y2 =min(Al l2,Ar2r2);(10.2)

l1+ l2 = L;(10.3)

r1+ r2 = R.(10.4)

The first two equations are the production functions fory1 andy2, and the second two equationsshow total labour and resource inputs. We assume that the labour forceL is fixed, whereas re-sourcesRare provided at a pricewr which is fixed relative to the wagewl , i.e.wr/wl = ψ whereψis fixed. Both labour and the resource are traded on competitive markets.

Define p1 as the price of goody1, and p2 as the price ofy2. Furthermore, let the wagebe the numeraire, normalized to be equal to labour productivity Al : wl = Al . Since markets arecompetitive then price must be equal to marginal cost. Sincewe have Leontief then marginal costis the same as average cost, and

r1 = l1(Al/Ar1); r2 = l2(Al/Ar2).

So total costs for good 1 are

c1 = Al l1+ψAl l1(Al/Ar1),

and similarly for good 2. Furthermore, production of the goods is

y1 = Al l1 and y2 = Al l2,

hence average costs, and hence prices, are given by

p1 = 1+ψ(Al/Ar1);

p2 = 1+ψ(Al/Ar2).

Finally, useL = l1+ l2 and the expressions forr1 andr2 above to show that

R= r1+ r2 = Al [l1/Ar1+ (L− l1)/Ar2].

To investigate the rebound effect we assume thatAr1 increases. What happens? From theinformation we have, we know thatr1/l1 will decline, andp1 (the price of goody1) will alsodecline. In general we expect this to lead to an increase in the quantityy1 demanded, and hencealso an increase inl1, labour employed on the production of good 1.

Mathematically we want to find the elasticity of total resource demandR to an increase inAr1:we denote this elasticityηr . Furthermore, for convenience—since we do not yet wish to specifythe demand side of the model—we define the elasticity ofl1 w.r.t. the change inAr1 asηl . Giventhese definitions, differentiate the above expression forR with respect toAr1 to show that

ηr = −r1

r1+ r2

[

1−

(

1−Ar1

Ar2

)

ηl

]

.(10.5)

7For the first analysis of rebound see Jevons (1865), and for another useful presentation see Binswanger (2001).

80 10.3. Structural change without income effects

To understand equation (10.5), assume first thatηl = 0, implying that the elasticity of substi-tution between the products on the consumption side is zero.Then the elasticity of total energydemand with respect to an increase inAr1 is simply equal to the share of product 1 in total energydemand. That is, there is zero rebound.

Now assume instead thatηl > 0, implying that the price reduction in product 1 causes somereallocation of consumption (and hence also production) towards that product. Thus the term insquare brackets may differ from 1. However, as a baseline case note that whenAr1 = Ar2 then thisterm is still 1, andηr is still equal to the share of product 1 in total energy demand. The reason isthat when the products (1 and 2) are of equal energy intensitythen a reallocation of consumptionbetween them does not affect total energy demand. Thus the rebound effect is zero.

Now assume thatηl > 0 andAr1 > Ar2, implying that product 1 is less energy-intensive thatproduct 2. Now the term in square brackets is greater than one, implying that the rebound effectis negative, i.e. the increase inAr1 causes agreaterreduction in total energy demand than wouldbe expected on the basis of a naive analysis. The reason is that the reallocation of consumptiontowards product 1 occurs at the expense of product 2, and product 2 is—by assumption—moreenergy-intensive than product 1. Therefore this reallocation leads to a reduction in energy demandover and above that which is caused directly by the efficiency increase in production of good 1.

Finally assume thatηl > 0 andAr1 < Ar2, implying that product 1 is more energy-intensivethat product 2. Now the term in square brackets is less than one, implying that the rebound effectis positive, i.e. the increase inAr1 causes a smaller reduction in total energy demand than wouldbe expected on the basis of a naive analysis. The reason is that the reallocation of consumptiontowards product 1 causes a net increase in energy demand because product 2 is—by assumption—less energy-intensive than product 1. Therefore this reallocation leads to an increase in energydemand, partly or even completely cancelling out the reduction which is caused directly by theefficiency increase in production of good 1.

So we can conclude that rebound is caused by the reallocationof labour (and other inputsother than the resource) between sectors, triggered by an increase in resource efficiency in onesector. When that sector is of low initial resource-intensity then this reallocation is a furtherbenefit, increasing the reduction in resource demand. In other words, the rebound effect is negative.When that sector is energy intensive then the reallocation diminishes—or may even reverse—thedirect effect of the efficiency increase.

10.3.3. A very simple specified model.We now set out to specify our rebound model (above)and try to calibrate it to match data. One property we would like our specified model to possessis that it should yield the constant resource share we observed in Figure 1.3. We begin by testinga model that mirrors the simple assumption of a Cobb–Douglasproduction function of a singlegood with two inputs (labour and the resource), but where we now have two goods each producedwith one input, and the substitutability is on the consumption side. Having checked the modelagainst data we go on to develop a slightly more sophisticated variant.

In our simplest model the production functions fory1 andy2 are as follows:

y1 = γlkl l;(10.6)

y2 = γrkr r.(10.7)

Thus labour is the only input toy1, and energy is the only input toy2. In equilibrium,l = L andr =R. To complete the overall picture we assume that aggregate consumptionY is a CES functionof the two aggregate productsy1 andy2, hence (at the level of the representative firm) we define aparameterǫ ∈ (−1,∞), and

y= (αy−ǫ1 + (1−α)y−ǫ2 )−1/ǫ .(10.8)

Thus whenǫ is positive the two aggregate products are complements in the sense that if a productbecomes increasingly scarce then its factor share rises.

As above, the energy price iswr and the wagewl , andwr/wl = ψ. We want to test the abilityof the model economy here to reproduce the data seen in Figure1.3. To do so we let labourL andthe ratio of the input prices,wr/wl , evolve exogenously, and derive total energy useR from themodel.

The solution is straightforward. Briefly, derive two different expressions for the ratio of theprices of the aggregate goods: firstly by comparing their marginal contribution toy, and secondlyby comparing their unit production costs. Use these two expressions to eliminate the price ratio,

Chapter 10. Structural change and energy demand 81

and rearrange to show that

RL=

1−αα

(

γrkr

γlkl

)−ǫ (wr

wl

)−1

1/(1+ǫ)

.

Hence the aggregate elasticity of substitution between energy and labour is 1/(1+ ǫ).Now setǫ = 0. This implies that equation 10.8 is Cobb–Douglas, and the aggregate elasticity

of substitution between energy and labour is 1. Thus we have the constant-share result and 100percent rebound (energy demand is not affected by the direction of technological change)! Theresult is intuitive: we have two products, one made entirelyusing labour, the other using onlyenergy. When the products are combined in a Cobb–Douglas function on the consumption sidethe products take constant shares, and therefore labour andenergy must also take constant shares.

This model tackles two of the weaknesses of the Cobb–Douglasmodel: the unrealisticallyhigh substitutability between the inputs on the productionside, and assumption that all productshave equal energy intensity and are perfect substitutes on the consumption side. However, itreplaces them with an equally troubling characteristic, i.e. the assumption that final goods areproduced either using pure labour or pure energy. To see how problematic this is, consider Figure10.5, where we illustrate how energy intensity varies across sectors. In Figure 10.5(a) we seethat if we divide consumption into two equal parts, one energy-intensive the other not, then thelow-energy-intensity consumption accounts for just under20 percent of energy consumption. In10.5(b) we see the energy intensity and expenditure share ofdifferent consumption categories:different types of services—of low energy-intensity—account for more than half of expenditure,while the two major energy-intensive categories are habitation and motor transport, and the finalcategory (with highest intensity but only a small expenditure share) is air transport. The figureshows that dividing consumption into two input-specific products does not follow naturally fromthe data, which suggests a continuum of products of gradually increasing intensity. Furthermore,the energy share of the most energy-intensive product (air transport) is only around 14 percent,nothing like to 100 percent intensity assumed in the model.8

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

Cumulative expenditureCumulative expenditure

Cu

mu

lativ

een

erg

y

En

erg

yin

ten

sity

,per

cen

t

Figure 10.5. Cumulative energy use and energy intensity plotted against cumu-lative expenditure when consumption products are sorted inorder of increasingenergy intensity. All the axes are normalized. Regarding energy intensity, weonly have data on relative intensities, and we normalize to give an average in-tensity of 4 percent.

10.3.4. A slightly more general model.Now assume two aggregate productsY1 andY2,where the former is labour-intensive in production and the latter is energy-intensive. Each areproduced in a constant-returns production function in which technology is exogenous. We can thusassume that they are produced by perfectly competitive firmsand hence there are no aggregationproblems within the two production sectors. We therefore focus on productiony1 andy2 from therepresentative firm in each sector.

8The data for Figure 10.5 are from Mayer and Flachmann (2011).The products—in order of increasing energyintensity—are Education services; Health services; Health services and social work; Other services; Cultural and sportservices; Retail and wholesale trade; Hotel and restaurantservices; Office and electrical machinery; Paper and publish-ing; Water transport; Auxiliary transport services; Otherland transport; Furniture, jewellery, musical instruments etc.;Other products; Textiles and furs; Food and tobacco; Agricultural products; Transport via railways; Habitation; Chemicalproducts, rubber, and plastic; Motor transport; Air transport.

82 10.3. Structural change without income effects

Because we want to build a simple model, and because labour takes more than 95 percent ofreturns (energy less than 5 percent), we assume thaty1 is produced using only labour, whereasy2

uses both. The production function fory2 is Leontief: as discussed above, the short-run elasticityof substitution between labour and energy is very low, and inthe context of this model it can beignored. Finally, we assume that technological change is completely unbiased, so we have a singleknowledge stockk that grows exogenously and boosts the productivity of all inputs equally. Wetherefore have

y1 = γl1kl1;(10.9)

y2 = kmin(γl2l2,γr2r2);(10.10)

l1+ l2 = L;(10.11)

r2 = R.(10.12)

The remaining parameters are analogous to those of the DTC model: L andR are total labourand total resource use respectively. To complete the overall picture we assume that aggregateconsumptionY is a CES function of the two aggregate productsYl andYr , hence (at the level ofthe representative firm) we define a parameterǫ ∈ (−1,∞), and

y= (αy−ǫ1 + (1−α)y−ǫ2 )−1/ǫ .(10.13)

Thus whenǫ is positive the two aggregate products are complements in the sense that if a productbecomes increasingly scarce then its factor share rises.

As above, the energy price iswr and the wagewl . We want to test the ability of the modeleconomy here to reproduce the data seen in Figure 1.3. To do sowe let labourL and the ratio ofthe input prices,wr/wl , evolve exogenously, and derive total energy useR from the model.

To solve, first note that (from the Leontief production function) γl2l2 = γr2r2. Then derivetwo different expressions for the ratio of the prices of the aggregate goods: firstly by comparingtheir marginal contribution toy, and secondly by comparing their unit production costs. Usethesetwo expressions to show that

l1l2=

[

α

1−α

(

γl1

γl2

)−ǫ

(1+ω)

]1/(1+ǫ)

,

where

ω =wr

wl

γl2

γr2,

and represents the ratio of the energy price to the wage (in efficiency units). Use the expressionfor l1/l2 and the restriction on total labour to findl2, and henceRusingγl2l2 = γr2r2:

RL=γl2

γr2

1+

[

α

1−α

(

γl1

γl2

)−ǫ

(1+ω)

]1/(1+ǫ)

−1

.

Finally show that the elasticity of substitution between energy and labour is as follows:

ηs =WQ∂Q∂W=

ω

1+ω1

1+ ǫ

[

α1−α

(

γl1γl2

)−ǫ(1+ω)

]1/(1+ǫ)

[

α1−α

(

γl1γl2

)−ǫ(1+ω)

]1/(1+ǫ)+1

.

So when the price of energy declines relative to labour (as ithas done historically), whathappens to the energy share in the model economy? Start with the case ofǫ = 0, in which caseequation 10.13 reduces to the Cobb–Douglas. Then we have

ηs =ω

1+ω

α1−α (1+ω)α

1−α (1+ω)+1

Soηs is less than 1, i.e. when the price of energy relative to labour declines by 1 percent, energyuseR (relative to labour) should rise by less than 1 percent, hence the energy share declines.

However, there is a special case in which the above result does not hold, and that is whenω→∞, implying (in the limit) that the production function fory2 is simply

y2 = γr2kr2.

Then we haveηs= 1, i.e. the factor share of energy is constant. This is to be expected, as this caseamounts to reducing the model to our previous model with input-specific products.

The special case sheds light on the general cases. When both labour and energy are used inproducing the energy-intensive good then—if we holdǫ at zero—the effect of declining energyprice on the relative price of that good is not so great, hencethe shift in consumption patterns

Chapter 10. Structural change and energy demand 83

caused by the price shift is not so great, hence the energy share declines as the relative price ofenergy declines (ηs < 1). Furthermore, as the energy share of the energy-intensive good declinesthe elasticityηs declines, approaching zero in the very long run asω approaches zero, at whichpoint the energy share is zero.

Whenǫ < 0—indicating a very high degree of substitutability between the labour-intensiveand the energy-intensive goods—then forω sufficiently high the model can deliverηs = 1 andhence a constant energy share. However, again, the decline in the relative price of energy causesω to decline, and asω declines thenηs will decline. That is, the constancy of the energy share isonly temporary, and in the long run the energy share will—farfrom being constant—approachzero.

We do not parameterize the model formally, rather we look forevidence suggesting reason-able values for the parameters. Consider first Figure 10.5. Firstly, the figure shows that dividingconsumption into just two products of differing energy intensity does not follow naturally fromthe data, which suggests a continuum of products of gradually increasing intensity. Secondly, thespread of energy intensities in the data is not very great: half of consumption expenditure goeson products with energy intensity between 25 and 50 percent of the average level, 49 percent ofexpenditure goes on products with energy intensity between63 and 250 percent of the averagelevel, and even the most energy-intensive good (air transport) at 1 percent of expenditure is just3.5 times more energy-intensive than the average good.

The data suggests that if we must lump consumption into two goods, and assuming that aver-age energy intensity is 5 percent, then we could choose one good with intensity 2 percent whichaccounts for 50 percent of consumption, and the other with intensity 8 percent accounting for theother 50 percent. Given the even more restrictive set-up of the model—with one good havingzero energy intensity—then we could think of the second goodas accounting for 50 percent ofconsumption at 10 percent intensity. If this represents thesituation today, could we have got there(in the model economy) via a long-run path with constant energy share?

It is straightforward to show that the above parameterization implies thatω = 0.12.9 The onlyway to achieve a sufficiently high elasticity of substitution between labour andenergy would thenbe to assume thatǫ were close to−1, i.e. labour-intensive goods such as services, and energy-intensive goods such as transport and housing should be almost perfect substitutes. The data doesnot support such a parameterization.

10.3.5. The failure of models without income effects. Consider for instance microeco-nomic studies of rebound effects. Rebound is present when an increase in the energy efficiencyof some process in the economy by a factorx (wherex > 1) leads to a reduction of total rate ofenergy use in the economy by less thanQ(1−1/x), whereQ was the original rate of energy usein that process. There are potentially several reasons for rebound in a real economy, but in themodel above there is just one: that an increase in the efficiency with which the energy-intensiveproduct is produced leads to a reallocation of production labour towards that product, thus reduc-ing the potential saving on energy use. The evidence for rebound effects is reviewed by Sorrell(2007), who finds that they are significant but generally muchless than 100 percent: increases inenergy efficiency of specific products do lead to large reductions in energy use associated withconsumption of those products. In terms of the model, this indicates that the substitutability be-tween energy-intensive and other products is relatively far from perfect, just as intuition wouldsuggest.10

The rebound evidence shows that the price-elasticity of demand for specific products is insuf-ficient to account for the historically observed aggregate elasticity of substitution between energyand labour. But the evidence over the period 1870–1970—rapidly increasing energy-efficiencyin the production of specific goods (such as artificial light and motive power), combined with therise in energy use tracking the rise in global product—nevertheless demonstrates that there musthave been a shift in consumption patterns towards energy-intensive goods. Direct evidence on con-sumption patterns confirms that such shifts have occurred. Regarding consumption of light, forinstance, Fouquet and Pearson find that per capita consumption of artificial light in the U.K. roseby a factor of 7000 between 1800 and 2000. This factor should be compared to the approximately15-fold increase in per-capita GDP over the same period; without shifts in consumption patterns,consumption of all products should have risen by this factorover the period. Regarding transport,

9If the energy share of the second good is 10 percent while its expenditure share is 50 percent then 44 percentof labour must be employed in making the second good, i.e.l2/l1 = 0.8. Furthermore, if the overall energy share is5 percent thenwl (l1 + l2) = 19wr r2. Eliminate l1 to show thatwl l2 = 8.4wr r2. But we know thatγl2l2 = γr2r2, hencewl/γl = 8.4wr/γr2, hence we haveωr = 0.12.

10For the first analysis of rebound see Jevons (1865), and for another useful presentation see (Binswanger, 2001).

84 10.4. A model of endogenously expanding variety

Knittel (2011) analyses technological change and consumption patterns in the U.S. automobile in-dustry, and shows that—for a vehicle of fixed characteristics in terms of weight and engine power—fuel economy would have increased by 60 percent over the period 1980–2006 due to technolog-ical change; this is approximately on a par with increases inlabour productivity. Furthermore, healso shows that actual average fuel economy increased by just 15 percent, the difference being dueto countervailing increases in the weight and power of vehicles. Thus we have efficiency improve-ment leading to a fall in unit costs of energy services, combined with a countervailing increase inconsumption of these services. We can thus summarize the rebound model as follows.

The above evidence supports the idea that substitution between products of differing energyintensity is important to take into account when determining energy policy, and demonstrates howsuch substitution has the potential to undermine efforts to reduce energy demand through increasesin energy efficiency. However, the failure of our simple model above showsthat we need a lessrestrictive model in order to capture the key mechanisms: two mechanisms which might be rele-vant are (i) substitution due to income effects as well as substitution effects, and (ii) substitutiontowards new products rather than between existing products. These two mechanisms are relatedto one another, as we can see by considering the historical data. Consider for instance transport:during the 20th century technological progress drove both rising incomes and the appearance ofnew energy-intensive consumption goods such as automobileand air transport. Such goods areluxuries to low-income households (and hence also to any household if we go back in time suf-ficiently) hence their income-elasticity of demand was initially high, and rising incomes causeda substitution towards such goods from less energy-intensive alternatives. In the next chapter wepresent a model which envisions this process of technological change and substitution towardsnew, more energy-intensive goods as a continuous process which will only be broken by changesin the relative price trend of energy to labour.

10.4. A model of endogenously expanding variety

We now build a model which includes both substitution and income effects, and furthermoreallows for the possibility of an endogenously expanding variety of goods.

10.4.1. The model.In the model there is an infinite continuum of goods which can poten-tially be made, indexed byi. The goods are made by perfectly competitive firms using labour andenergy, so price is equal to marginal cost. There are constant returns to scale in labour and energy,so returns to labour and energy are equal to revenue. The production function is Leontief, thus wehave—for firmi —

yi =minγli kli l i ,γri kri r i.

For tractability we restrict the production function further, with regard to labour and energy pro-ductivity: we assume that labour productivityγli kli is equal across all the goods, and denote itAl ; and we assume that energy productivityγri kri is a declining function ofi, Ar/i. ProductivityindicesAl andAr evolve exogenously. Thus we have the following equations:

yi =minAl l i , (Ar/i)r i;

R=∫ ∞

0r idi;

L =∫ ∞

0l idi.

The representative consumer’s utility is a CES function of the quantities of the goods consumedby that agent—where each good has the same weight—but adjusted by a constant factor ¯y whichensures that no good is essential even though the goods are complements (ǫ > 0):

u=

∫ ∞

0[(yi/L+ y)−ǫ − y−ǫ ]di

−1/ǫ

.

It follows from the Leontief production function that when firms optimize

yi = Al l i ,

and r i = (Al/Ar )il i .

Sincepi (the price ofyi) is equal to marginal cost we have

pi = wl/Al +wr i/Ar = wl/Al(1+ωi),(10.14)

where ω = wr/Ar/(wl/Al).

Chapter 10. Structural change and energy demand 85

As previously,wl andwr are the wage and the energy price, andω is the ratio of the energy priceto the wage, in efficiency units. We continue to assume that the ratio of the energy price to thewage,wr/wl , evolves exogenously. Since productivities are also exogenous this implies thatω isexogenous.

10.4.2. Solution to the expanding-variety model.The solution of the model is straightfor-ward but somewhat long-winded. For the full working the reader is referred to Hart (2014). Notehowever that for tractability we assumeǫ = 1. The results are as follows:

N =1ω

[

1+ A]2−

1ω

;(10.15)

y0 = LyA;(10.16)

yi =y0+ Ly

(1+ωi)1/2− Ly;(10.17)

r i = yi i/Ar ;(10.18)

andwrRwl L=

23

A

(

1+A2

4

)

,(10.19)

where A=

(

Alω

y

)1/2

.(10.20)

So the factor share of energy is increasing inA, andA is increasing inAl (labour productivity) andω (the relative energy price in efficiency units). Furthermore, since we know that the factor shareof energy is small—less than 5 percent—this tells us thatA is small (around 0.05) and henceA2/4≪ 1, so we can approximate

wrRwl L≈

2

3y1/2(Alω)1/2 .(10.21)

So if the price of energy (in efficiency units) declines at the same rate as labour productivityincreases then the energy share is constant. This implies that if the energy pricewr is constant,and the productivitiesAl andAr grow at equal rates, then the factor share of energy is constant.On the other hand, since the elasticity of the energy share tothe energy price is 0.5, an increasingenergy price leads to a slow increase in the energy share. Finally, if energy efficiencyAr growsfaster then labour efficiencyAl this reduces the energy share.

10.4.3. Parameterization.We parameterize the model to fit the data of Figures 1.3 and 10.5.However, it turns out that there is almost nothing to parameterize. We normalize (without lossof generality) the initial levels of the productivitiesAl andAr , and the energy pricewr , then sety to match the energy factor share;y0 is then determined by (10.16). We set the energy shareat 4 percent and normalizeAr = Al = L = wr = 1, implying thaty = 278 andy0 = 16.7. Puttingthese numbers into the model we obtain the pattern of expenditure and energy intensity shownin Figure 10.6, where we compare the pattern derived from themodel to the pattern found inthe data previously shown (Figure 10.5). Note that the modeldoes a reasonable job of matchingthe observed spread of expenditure across goods of differing energy intensity, despite its lack offlexibility.

Turning to the time-series data, we now have a little more flexibility because we can testdifferent assumptions about the growth ofAr . In Figure 10.7 we set the growth rate ofAr at 10percent higher than the growth rate ofAl , where the latter is imputed from growth in GDP. Wemake no attempt to adapt the model to match the short-run data. Again we see that the model doesa reasonable job in matching the data, although given the very simple pattern in the data this isperhaps not a very startling achievement. The main failure of the model (apart from not matchingshort-run fluctuations) is that it predicts a steep increasein energy consumption following theprice declines after 1980, an increase not found in the data.We discuss possible reasons for thisdivergence in the next section.

10.4.4. Implications of the expanding variety model.The key result of the expanding va-riety model is (10.21), implying that

RL≈C ·Al

(

1wr

)1/2 (

Al

Ar

)1/2

,(10.22)

whereC is a constant. The model matches the long-run data fairly well. Furthermore, it is consis-tent with the data of Figure 10.5, showing the spread of production across energy intensities; theparameterization is broadly consistent with evidence regarding the bias of technological change,

86 10.4. A model of endogenously expanding variety

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

Cumulative expenditureCumulative expenditure

Cu

mu

lativ

een

erg

y

En

erg

yin

ten

sity

,per

cen

t

Figure 10.6. Cumulative energy use and energy intensity plotted against cumu-lative expenditure when consumption products are sorted inorder of increasingenergy intensity: the model economy (dashed lines) and datacompared. Dataas in Figure 10.5

1850 1900 1950 2000−0.5

0

0.5

1

1.5

2

2.5

3

Energy Price... Quantity, observed... Quantity, modelled

1880 1900 1920 1940 1960 1980 2000−0.5

0

0.5

1

1.5

2

2.5

3

Energy Price... Quantity, observed... Quantity, modelled

ln,n

orm

aliz

ed

Figure 10.7. Long-run growth in observed energy consumption compared tomodelled energy consumption: (a) U.K.; (b) Globally. Data as in Figure 1.3

i.e. that there has been some bias towards the growth of energy-augmenting technology; and themodel includes both income and substitution effects in its explanation for the long-run failure ofthe energy share to decline. Finally, the expansion of consumption into energy-intensive sectors isclearly relevant: the two most energy-intensive sectors inFigure 10.5 are motor transport and airtransport. These sectors rose from scratch during the 20th century as a result of a combination ofincome growth and progress in energy-augmenting technology, exactly as described by the modeleconomy. We therefore focus the remainder of the discussionon this model.

In order to better understand the implications of the expanding variety model we need aclearer picture of the structure of the economy and the nature of structural change, which is givenby Figure 10.8. In Figure 10.8 we havei —i.e. the index of different goods—on thex-axis, and onthey-axis we show energy user i (upper panel) and three different elasticities (lower panel). Thefirst thing to note from the figure is that most energy use comesfrom products in the middle ofthe energy-intensity range, which is to be expected since production of the most energy-intensivegoods approaches zero.

Assume now that a regulator has policy tools (such as technology subsidies) with which toselectively boost specific productivity levelsAri andAli , or the overall productivity levelsAr andAl . This could be through a mechanism similar to that which applies in the DTC model. Firstassume that the regulator induces a general increase in productivity (Al andAr both increase bythe same factor, while relative priceswr/wl are constant), and define the elasticity ofr i w.r.t. theproductivity increase asηriA. From (10.22) we know that the elasticity of total energy usetothe TFP increase is 0.5, but now we want to study individual products. The top panelof Figure10.8 shows that ther i curve is flattened and shifted to the right as the variety of goods producedincreases. This leads to a fall in the total energy use from low-intensity goods, and steep rises inenergy use from high-intensity goods. The lower panel of Figure 10.8 shows thatηriA is −1 wheni = 0, and approaches infinity wheni → N. The reason for the latter result is that wheni is closeto N the goods are on the cusp of affordability, hence their price elasticity of demand is extremelyhigh.

Chapter 10. Structural change and energy demand 87

0 0.02 0.04 0.06 0.08 0.1 0.120

0.1

0.2

0.3

0.4

0.5

0.6

0 0.02 0.04 0.06 0.08 0.1 0.12

−2

0

2

4

6

r i baseliner i raisedAr i raisedAr

i

ηriAηriArηRAri

Figure 10.8. Energy consumption for individual productsr i as a function ofi.The top panel shows the baseline curve and the effect of 10 percent increases inA andAr respectively. The lower panel shows the elasticity of energy consump-tion with respect to overall technological progress,ηriA, energy-augmentingtechnological progress,ηriAr , and product-specific energy-augmenting techno-logical progress,ηRAri.

Secondly, assume instead a general increase inAr alone, holding prices constant, and definethe elasticity ofr i w.r.t. the increase inAr alone asηriAr . Again we can use (10.22) to find theelasticity of total energy use to such a change, which is−0.5, and again we see from Figure 10.8that ther i curve is flattened and shifted to the right. The difference is thatηriAr is now negativefor the majority of the products, because the rise inAr has a strong direct downward effect onenergy use. However, it also causes a relatively strong substitution effect towards energy-intensiveproducts, hence the rise in total energy use for highly high-intensity products.

Thirdly we investigate the effect on total energy consumption of an increase inAri , the energy-efficiency of a specific product. The elasticity ofR w.r.t. Ari is zero since producti is one of aninfinite number and any change in just one product will have a negligible effect on total energyconsumption. Therefore we define the ‘elasticity’ηRAri as dR/r i/(dAri/Ari ), hence it shows thetotal change in energy consumption as a percentage ofr i , divided by the percentage change inAri .The curve is broadly similar to the curve forηriAr . A rise in the energy-efficiency of a medium-intensity product yields a reduction in energy use per unit of that product made, and also causessubstitution towards that product. But because the productis of medium energy-intensity thissubstitution has little effect on total energy, hence the total effect of the rise inAri is negative. Fora high-intensity product, on the other hand, the model indicates backfire: an increase inAri leadsto an increase in total energy consumptionR.

Finally, the effect of a rise inwr (not illustrated) is almost a mirror-image of the effect ofthe rise inA: consumption is pushed back towards low-intensity products, leading to increasesin total energy use associated with these products, at the same time as energy use in producinghigh-intensity products declines dramatically.

Note that the results forηRAri are most relevant to the discussion of the rebound effect, as theyrelate to the effect on total energy consumption of product-specific improvements in energy effi-ciency. They show that the effect is very strongly dependent on the energy-intensity of the productin question: for products of low energy-intensity the rebound effect isnegative, that is (recallingthe discussion in the introduction), an increase inAri by a factorx leads to a reduction ofR bymore thanRi(1− 1/x). The reason is that substitution by consumers towards products of lower-than-average energy intensity makes a positive contribution towards reducing total energy demand.However, for products of the highest energy intensity we have insteadbackfire, i.e. increases inAri lead to increases in total energy consumption.

88 10.5. Evidence for income effects in energy demand

10.4.5. Relevance of the model to real economies.The expanding variety model capturestwo mechanisms which are not captured by our previous, simpler models: (i) substitution dueto income effects as well as price effects; and (ii) substitution towards new products rather thanbetween existing products. These two mechanisms are related to one another, as we can see byconsidering the historical data. Consider for instance transport: during the 20th century technolog-ical progress drove both rising incomes and the appearance of new energy-intensive consumptiongoods such as automobile and air transport. Such goods are luxuries to low-income households(and hence also to any household if we go back in time sufficiently) hence their income-elasticityof demand was initially high, and rising incomes caused a substitution towards such goods fromless energy-intensive alternatives.

The model has a number of weaknesses that limit our ability touse it to draw conclusionsabout real economies. Here we discuss the following: the failure of the model to match recentdata; the high price-elasticity of demand for goods and hence the unrealistically powerful reboundeffects for energy-intensive goods; the lack of flexibility in treating income effects.

The model does a poor job in predicting the rather constant global energy consumption after1980. Can we then trust its long-run predictions? There are several possible explanations for theapparent failure of the model after 1980, of which we mentiontwo. The first possible explanationis that the price shocks of the 1970s had a profound and long-lived effect on investment decisions,due in turn to a change in long-run price expectations: pre-1974 the dominant expectation mayhave been the prices would continue their slow decline, whereas post-1974 it seems reasonablethat expectations about future prices may have been reviseddramatically upwards. Such a changecould lead to an increase in the growth rate ofAr relative toAl , thus holding energy consumptiondown. The second possible explanation is that the energy crisis of 1974 led to a series of energy-related policies—not fully captured by the energy price—which amount to implicit rises in theprice of energy paid by firms and consumers, even though theseincreases have not been capturedin our data.

There are also several possible reasons why the model could be more fundamentally wrong.The key reason is that the model works because demand for individual goods is highly price-elastic, which allows consumers to substitute quite freelybetween goods of low and high energy-intensity, depending on their relative prices. The other side of the coin is that income effects—while present in the model—are limited in power. Due to the need for tractability we cannot cal-ibrate the model to raise the strength of income effects on demand for energy-intensive products;neither can we calibrate consumers’ willingness to pay for expanding variety. If we could lessenthe substitutability between goods and strengthen income effects we could potentially continue tofit the historical data, while weakening the rebound effects predicted by the model. More gener-ally, the balance between income and substitution effects is crucial to the predictions and policyconclusions of the model. In the model this balance is stable, by construction. However, it ispossible that the income-elasticity of demand for energy-intensive products declines with risingincome, and is very low or even negative when incomes are veryhigh (i.e. in rich industrializednations). This is a very optimistic scenario as it suggests that energy demand will tail off endoge-nously without the need for price rises or technology policy. This vision of a clean, post-industrialglobal economy is tempting, but little convincing evidencehas been presented that it will material-ize without powerful policy interventions. It is contradicted by the evidence presented by Knittel(2011).

A further weakness of the model is the need to assume a representative consumer, implyingthat a product on the cusp of affordability for one consumer is on the cusp of affordability for allconsumers. This causes the demand elasticities for such products to be seriously overestimated:in reality the good may be inferior for richest consumers, and completely out of reach for thepoorest. A useful extension to the model would therefore be to disaggregate consumers intogroups at different income levels and treat their demand for goods separately, as is done in muchof the literature on rebound effects for individual products. In such a model it is possible thatincome effects would dominate, and (in the most optimistic scenario) that in the very long runenergy consumption will level off or decline as the richest consumers substitute away from energy-intensive goods. The approach developed by Boppart (2014) could be applied to this problem.

10.5. Evidence for income effects in energy demand

In this section we look for evidence for income effects in cross-country data. The work so faris highly preliminary.

We know that energy prices differ greatly between countries, mainly due to differences in tax-ation; in some countries energy taxes may be rather high, whereas in others energy consumption

Chapter 10. Structural change and energy demand 89

0 2 4 6 8 10 12

x 104

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5

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15

20

25

30

35

40

45

GDP

CO

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issi

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5 6 7 8 9 10 11−10

−9.5

−9

−8.5

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−7.5

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−6.5

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lnGDP

lnC

O2/

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ABW

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TCDSGPMLI

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CAF NORFRADNKZMBBDIISLHKGVUTCOG URYGAB AUTBFA IRLESPBRA ITAEMU LUXTLS JPNCRICPVETH GBRSDNUGA NZLDEUMDG EUUMWI PRTCOD BELNLDNPLERI CYPCIV PRYLKACOMLAONERGIN FINCOLSLE GTMSWZGHA WSMGNB DMANAM OECTZA SLV OEDGRCSLBBTN BLZ ISRVCTHPCMOZ BHS CANLVA MLTHTICMR LAC SVN AUSPNG HICCHLLCNLCAGRD ECSNGA KNAFSMOSS HRVALB PANLDC BRBAGO LTU NACKHM PER USABWA GNQPSSKEN HUNDOMFJI TURPHL SVKMUSTON MEXKIR ATGARM ROUECUTGO ARGSTPGMB MDVTJK HND VENAFG WLDNIC GEOJAMSURSENLIC LBN KORBGD CZE QATMAR ARETUNMRT IDNLBR SSFSSA MNEMHL POLSSTYEM SYCEAS BRNGUY ARBDZA KWTBOL JORBEN NOCMEAMDA IRQECAAZEMYS SAULMYUMCMICPAKLMC THAEGY BGR BHROMNESTMNASYRSAS PLWBLR RUSCSSMKDIND SRBEAP ZAFZWEKGZ IRNCHNVNM

KAZBIHMNGUKRTKM TTOUZB

Figure 10.9. Data on GDP and emissions from 2010. Source: World Bank,World development indicators.

is actually subsidized by the government either directly orindirectly. A way for a governmentto subsidize indirectly could be to extract the resources itself and then sell them domestically ata price below the world-market price. Given this variation in prices we have a way of investi-gating substitution and income effects: if substitution effects are insignificant (energy demand isprice-inelastic but rises with income) then we should see a strong correlation between income andemissions, and a weak correlation between prices and emissions. Unfortunately we don’t haveeasy access to price data, but we can plot emissions as a function of income (GDP), and alsoemissions per unit of GDP as a function of GDP: Figure 10.9.

In Figure 10.9 we see that the emissions-intensity of GDP varies greatly between countries,and that any relationship between emissions intensity and GDP is unclear; there may howeverbe a tendency for emissions intensity to rise as GDP rises from a very low level, and then fallagain at very high levels. So there is some evidence for income effects of the type described inthe model, where energy is initially a luxury good but becomes an inferior good at high incomelevels. However, the ‘inverted U’ appearance may be driven by outliers; the four countries in thebottom-right corner are Bermuda, Switzerland, Singapore,and Macau. In all four countries GDPis inflated by a very high level of trade and financial services, which (it could be argued) pullsdown the CO2/GDP ratio artificially.

On the other hand, the data is suggestive of a relationship between emissions intensity andenergy price: the seven countries with the highest emissions intensity are (in reverse order) Kaza-khstan, Bosnia and Herzegovina, Mongolia, Ukraine, Turkmenistan, Trinidad and Tobago, andUzbekistan. These are energy-rich low-income countries with low energy prices. Note alsoAtkeson and Kehoe (1999), who find—based on a cross–country study—that the price-elasticityof demand for energy is around 1. So price effects seem to be important in explaining emissionsintensity across countries.

Part 4

Substitution between resource inputs, andpollution

In this part of the book we turn from the analysis of overall demand for nonrenewable re-sources and energy, to the analysis of demand for alternative—substitutable—resource inputs.Such inputs may be alternatives such as iron and aluminium, or coal and gas. But they couldalso be coal and wind, or coal and solar. Clearly, such substitution has major implications for theanalysis of polluting emissions, and after an initial (moregeneral) analysis, we turn our focus tothe prediction and regulation of polluting emissions. We show that rising income drives substitu-tion from dirty inputs, such as coal, to cleaner inputs such as gas or renewables. This helps us tounderstand the observed tendency of polluting emissions tofirst rise over time, and then decline:the rises are driven by the increase in overall resource demand, driven by growth; and the declinesare driven by switches to cleaner inputs.

CHAPTER 11

Substitution between alternative resource inputs

Return now to Goeller and Weinberg (1976), which we discussed at the end of Chapter 7. Re-call that they argue that reserves of the majority of minerals—including staples such as iron andaluminium—are so vast that they can never realistically be consumed. With the clear exception ofphosphorus they argue that society can exist on these superabundant resources indefinitely. How-ever, even though the total supply of minerals is unimaginably vast, it is still of course possiblethat the supplies some particular minerals may be exhausted, or at least that they may increasevery steeply in price. Under these circumstances we need to consider not just the possibility ofadaptations which help us to do without mineral inputs in general—as above—but we shouldalso consider the substitution of abundant or cheap minerals for those that are becoming scarce orexpensive. We investigate such substitution in this chapter.

We begin by building and testing a very simple model of resource substitution, in whichtwo alternative resources are available which are substitutable for one another, and technologicalchange in the resource sector is unbiased. We show that the model can do a reasonable job ofaccounting for aggregate data in two cases. We then go on to consider Solow’s third adaptationmechanism to resource scarcity (page 40), which was to increase—through technological change—the efficiency of an alternative (substitute) resource in production of one or more product cat-egories. The idea here is that when there is a need to switch toan alternative resource, directedinvestments lead to an increase in the efficiency with which we can use the alternative resource.This idea is related to the concepts of path dependence and lock-in discussed by Arthur (1989)among others, and the idea that we are ‘locked in’ to fossil-fuel use by history dates at least toUnruh (2000). More recently, Acemoglu et al. (2012) have proposed a model in which a regulatorcan transform lock-in to break-out: through a massive but short-run effort the regulator can setin train a technology transition from dirty to clean energy,a transition which will then continuewithout the need for further regulation. We analyse—and question—the relevance of this modelin the next part of the book, on pollution.

11.1. A simple model with alternative resource inputs

11.1.1. The basic model.Recall from Part 2 that the simple Cobb–Douglas production func-tion with labour, capital, and resources does a decent job ofmatching the aggregate long-run data,with constant factor shares for labour, capital, and resources.1 Furthermore, in a long-run contextwith perfect information and relatively constant growth wecan abstract from capital without anymajor loss of relevance for the model. We then have the following aggregate production function:

Y= (ALL)1−α(ARR)α.

The units ofY arewidgets year−1, the units ofL are simplyworkers, hence the units ofAL arewidgets worker−1 year−1. The units ofR, the resource input, are nowres year−1, where res is ameasure of resource services, the meaning of which will become clearer below.

The production function implies (as we saw above) that returns to the resource relative tolabour are constant, irrespective of relative prices, directed technological change, etc. To see thisset up the representative producer’s problem—

maxπ = py(ALL)1−α(ARR)α−wlL−wrR

—and differentiate w.r.t.R to findwr :

wr = αY/R

and wrR/Y= α.

1Note that we know from the previous chapter that the diversity of products made in real economies is importantfor understanding resource demand. However, here we make the assumption that it can be ignored—at least for the timebeing—when modelling demand for alternative resources.

93

94 11.1. A simple model with alternative resource inputs

Now we return toR, resource services. We have two physical resource inputs (units tonsyear−1), Xc andXd, and the production function for resource services is

Rt =[

(γcActXct)ǫ + (γdAdtXdt)ǫ]1/ǫ

,(11.1)

where the elasticity of substitution between the two is 1/(1− ǫ), andǫ ∈ (0,1). So the two inputsare highly substitutable in the production of resource servicesR. Consider for example iron andaluminium, or coal and oil. The units are shown in Table 11.1,where (for instance) ideasC meansideas augmenting resource inputC.

Table 11.1. Units in the production function for resource services

Quantity Unit Quantity Unit

γc res ton−1C idea−1

C γd res ton−1D idea−1

DAC ideasD AD ideasDC tons ofC year−1 R tons ofD year−1

Note that the production function for resource services implies that if we have only one re-source input (perhaps only iron and no aluminium) then the function is very simple. For instance,if we have only inputC then

Rt = γcActXct,

and likewise for inputD. Now assume thatγcActXct andγdAdtXdt are both equal toR/2. Thenwhen they are combined in the production function we have

R= 2(1−ǫ)/ǫR,

which is greater thanR. Because the inputs are imperfect substitutes they therefore complementeach other to some extent, implying that the whole is more than the sum of the parts.

Now we turn to technological change. In this section we simply assume that technologicalchange in the resource production function is unbiased and exogenous, implying thatAC andAD

grow at equal rates. Furthermore, we assume thatAR is constant, since resource services arealready augmented by technologiesAC andAD. Finally, AL grows exogenously at a constant rateg, which is also the growth rate ofAC andAD, and population grows at raten. Without loss ofgenerality we normalizeAR = 1, andAC = AD = AL = A. The two resources are extracted usingfinal goods, and unit extraction costs arewct andwdt respectively. So we have

Yt = (AtLt)1−αRαt ,

Rt = At[

(γcXct)ǫ + (γdXdt)ǫ]1/ǫ

,

and Ct = Yt − (wctXct+wdtXdt),

whereC is aggregate consumption. If the resources are scarce then resource price will be extrac-tion cost plus scarcity rent; however, for now we simplify byassuming that price equals extractioncost, and account for scarcity informally by allowing priceto rise. When testing the model empir-ically we only have data on price (and hence no data on extraction cost and scarcity rent).

11.1.2. The solution.The solution to the model is straightforward. We already have that

wrR= αY.

Consider now production ofR. Set up the producer’s profit-maximization problem as follows,

π = wrt At[

(γcXct)ǫ + (γdXdt)

ǫ]1/ǫ −wctXct−wdtXdt,

and take first-order conditions to show that

wcXc = wr (R/A)1−ǫ(γcXc)ǫ

and wdXd = wr (R/A)1−ǫ(γdXd)ǫ ,

and hence

wcXc = w1/(1−ǫ)r (R/A)(γc/wc)

ǫ/(1−ǫ)

and wdXd = w1/(1−ǫ)r (R/A)(γd/wd)ǫ/(1−ǫ).

Chapter 11. Substitution between alternative resource inputs 95

Because we have perfect markets, price equals unit cost so

wr = (wcXc+wdXd)/R

=

A/[(γc/wc)ǫ/(1−ǫ)+ (γd/wd)ǫ/(1−ǫ)]

ǫ/(1−ǫ).

And sincewrR= αY we have

wr = α(AL/R)1−α,

and we can eliminatewr to yield

R= AL

α[(γc/wc)ǫ/(1−ǫ)+ (γd/wd)ǫ/(1−ǫ)]1/(1−α)

.

So if wc andwd are both constant thenR grows at the same rate asY, i.e. g+n, the sum of thegrowth rates of labour productivity and population. And therelative factor shares of the resourcesare

wcXc

wdXd=

(

γc/wc

γd/wd

)ǫ/(1−ǫ)

.

This implies that the resource that is cheaper per efficiency unit takes the larger factor share, andthe advantage is bigger the higher is the substitutability between the resources (i.e. whenǫ→ 1).

11.1.3. Empirical tests.To test the explanatory power of this very simple model we taketwo pairs of resources, oil–coal and iron–aluminium. In each case we first present data on pricesand factor expenditure, and compare the expenditure to GGP (gross global product). We then plottotal expenditure on the two factors together against GGP, and check whether the ratio of the twois approximately constant. (Recall from the model that thisratio should be constant,α.) Finallywe plot the ratio of expenditures on the two factors, and compare this to the same ratio obtainedusing the price data and the parameterized model. In the model economy (immediately above) theratio is

wcXc

wdXd=

(

γc/wc

γd/wd

)ǫ/(1−ǫ)

.

We take the price data (i.e.wc andwd), then find parametersǫ andγc/γd to fit the factor-share dataas well as possible.2

1900 1950 2000−18

−16

−14

−12

−10

−8

−6

−4

−2

0

Oil price

Coal price

Oil exp

Coal exp

GGP

log

1900 1950 2000−10

−8

−6

−4

−2

0

GGPC+O exp, OBSC+O exp, MOD

1900 1950 2000−4

−3

−2

−1

0

1

2

3

shares, OBSshares, MOD

Figure 11.1. Long-run growth in prices and factor expenditure, compared togrowth in global product, for crude oil and coal, and a test ofthe model. Inthe left-hand figure we see observed prices and expenditures, with expenditurescompared to global product. In the middle figure we see observed total expen-diture on coal and oil, compared to the model prediction (based on the prices).And in the right-hand figure we see the observed relative factor shares of coaland oil, compared to the model prediction. In the calibratedmodel we haveα = 0.02,γc/γd = 0.55, andǫ = 0.76.

In Figure 11.1 we see that the Cobb–Douglas does a reasonablejob of approximating the ag-gregate global production function in this case, although the factor share of oil and coal combinedhas (according to the data) actually risen during the 140 years for which we have data, rather than

2Note that at this stage we simply eyeball the graphs, Figures11.1 and 11.2 to determine the goodness of fit.

96 11.2. Technological change

being constant. This can be seen most clearly in the middle panel. In the left panel we see that theprice trend of oil relative to coal was a very slow decline up to 1973, after which prices becamemuch more volatile, and the gap decreased. The model (smoothed) can easily match the increasein the oil share as the oil price declines (right-hand panel), but it doesn’t do quite such a good jobof accounting for what happens after the oil crisis, where the oil share continues to grow despitethe increasing price of oil relative to coal. Nevertheless,considering the simplicity of the modelit does a remarkably good job of matching the data.

1900 1950 2000−18

−16

−14

−12

−10

−8

−6

−4

−2

0

Al price

Fe price

Al exp

Fe exp

GGP

log

1900 1950 2000

−10

−8

−6

−4

−2

0

2

4

GGPC+O exp, OBSC+O exp, MOD

1900 1950 2000

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

shares, OBSshares, MOD

oil

Figure 11.2. Long-run growth in prices and factor expenditure, compared togrowth in global product, for iron and aluminium, and a test of the model. Inthe left-hand figure we see observed prices and expenditures, with expenditurescompared to global product. In the middle figure we see observed total expen-diture on iron and aluminium, compared to the model prediction (based on theprices). And in the right-hand figure we see the observed relative factor sharesof iron and aluminium, compared to the model prediction. In the calibratedmodel we haveα = 0.002,γc/γd = 50, andǫ = 0.55.

In Figure 11.2 we see that the Cobb–Douglas does an excellentjob of approximating the ag-gregate global production function when we assume that the inputs are labour (or labour–capital),iron, and aluminium (middle panel). In the left panel we see that the price trend was for a steadydecline in the price of aluminium relative to iron, while thetrend in factor share was a correspond-ing increase in the share of aluminium. The model (smoothed)also does a remarkably good jobof matching the increase in factor share triggered by the decline in relative price of aluminium,although it could be argued that this is due to a lack of variability in the data (if the long-run trendof relative prices were more complex this would be a more discriminating test of the model).

11.2. Technological change

In the above model we assumed unbiased technological changein the resource sector. We thusrule out by construction the mechanism discussed by Solow (1973) and modelled by Acemoglu et al.(2012) whereby a resource which increases in importance (factor share) attracts more investment,and therefore increases in efficiency also. The success of our model without DTC suggests thatthis mechanism may be of limited importance, but it is intuitively appealing and potentially im-portant for prediction and policy, hence we investigate further here, and again in the next part ofthe book.

11.2.1. The basic DTC model.Now to our model of DTC. We could specify and solve thesame DTC model that we developed in the previous chapter. However, for convenience we simplyborrow the results of that model and apply them here. Recall that we showed there (equation 9.8)that relative investments are equal to relative factor shares,

zlt

zrt=

wlt Lt

wrtRt=

(

Alt Lt

ArtRt

)ǫ

.

This shows relative investments in labour-augmenting and resource-augmenting technologies. Acorresponding equation applies within the resource-intensive sector when there are two competing

Chapter 11. Substitution between alternative resource inputs 97

resource inputs,C andD:

zct

zdt=

wctCt

wdtDt=

(

ActCt

AdtDt

)ǫ

.(11.2)

The resource and labour inputs are complements, implying thatǫ < 0 and hence an abundant factorearns a low share. However, the alternative resource inputsin equation 11.2 are substitutes, henceǫ is positive and an abundant factor earns a big share. Think ofspruce and pine trees in the lattercase; if there is a lot of spruce and little pine the price of pine might be a little bit higher thanotherwise, but the share of expenditure on spruce will be high compared to pine (spruce won’t bethat cheap).

If we add the assumption that knowledge stocks grow independently then we have (corre-sponding to equation 9.7)

Act/Act−1

Adt/Adt−1=

(

zct

zdt

)φ (ζd

ζc

)

.

Putting it all together we have

Act/Act−1

Adt/Adt−1=

(

wctCt

wdtDt

)φ (ζd

ζc

)

.(11.3)

In order to say more about the development of the economy we must specify how the inputsCandD are supplied. For instance, we could assume that they are supplied in fixed (exogenous)quantities, and that the price is then determined by the market. Alternatively, we could assumethat they are supplied at fixed prices, with the quantity thendetermined by the market. Or wecould assume supply functions (linking price and quantity).

If we assume that quantities are exogenous, then all we need to do is to find an expression forrelative factor costs as a function of quantities and the state of knowledge. But we already havesuch an expression, equation 11.2. Substitute this into 11.3 to obtain

Act/Act−1

Adt/Adt−1=

(

ActCt

AdtDt

)ǫφ (ζd

ζc

)

.

Finally, multiply both sides by(

Act/Act−1Adt/Adt−1

)−ǫφto obtain

Act/Act−1

Adt/Adt−1=

(

Act−1Ct

Adt−1Dt

)ǫφ (ζd

ζc

)

1/(1−ǫφ)

.

Sinceǫ > 0 this implies that the factor which starts offmore abundant earns a greater share, henceits abundance (after allowing for factor-augmenting knowledge) tends to increase! This processaccelerates over time, so the economy heads for a corner in which the initially abundant factordominates completely. (Note that we have ignored the role ofthe relative productivities of re-search. There is no obvious reason to suppose that these should differ, and if they do not differthen the termζd/ζc disappears.)

Now assume instead that prices are exogenous. Thinking about non-renewable resources,this assumption makes more sense than the assumption of exogenous quantities; recall that itimplies that the alternative resources are available to thefinal-good sector (in any quantity) atsome exogenous price level. Of course, in the short-run we know that a sudden increase in demandwill lead to a steep rise in price, but in the long run it is reasonable to suppose that prices are closeto unit extraction costs in most cases, and that unit extraction costs do not vary greatly in totalquantity.3

When prices are exogenous we have, from 11.2, that

wctCt

wdtDt=

(

Act

Adt

)ǫ/(1−ǫ) (wct

wdt

)−ǫ/(1−ǫ)

.

Substitute this into 11.3—and assume that the research productivities are equal—to obtain

Act/Act−1

Adt/Adt−1=

(

Act/wct

Adt/wdt

)ǫφ/(1−ǫ)

.

3For instance, the cost of extracting a ton of coal from the Blackwater mine in Queensland, Australia is not affectedby the global extraction rate of coal.

98 11.3. An alternative to lock-in: The Fundamentalist economy

Finally, multiply both sides by(

Act/Act−1Adt/Adt−1

)−ǫφ/(1−ǫ)to obtain

Act/Act−1

Adt/Adt−1=

(

Act−1/wct

Adt−1/wdt

)ǫφ/(1−ǫ(1+φ))

.

Now the input with the higher ratio of productivity to price (effectively, the cheaper input) takesthe higher factor share, and thus attracts more investment,and thus dominates more and more overtime, and the economy heads to a corner in which only one of theinputs is used.

In terms of the balanced growth path determined in the previous chapter (forǫ < 0), such apath also exists in the case ofǫ > 0, but it is not stable. On a b.g.p. we have

(

ActCt

AdtDt

)ǫφζd

ζc= 1,

implying that the more abundant input is harder to augment. However, the situation is as illustratedin the picture below when the ball is balanced at the top of thehill; the slightest disturbance and itwill start rolling one way or the other. See Figure 11.3

Market forces

TECH

State of technologyDirty Clean Environmental friendliness of technologyDirty Clean Environmental friendliness of technologyDirty Clean

Figure 11.3. Illustration of how relative prices (the shape of the economic land-scape) determine the relative levels of technology augmenting clean and dirtyinputs in the model, and the role of a regulator.

Recall the simulation of the economy withǫ < 0. You should now redo the simulation forǫ > 0 (but less than 1). Plot some curves (remember to take logarithms!) to get a better picture ofwhat is going on.

11.2.2. Implications of the DTC model.The basic DTC model with independent knowl-edge stocks and fixed exogenous prices implies that the economy should head to a corner in whichone or the other input dominates completely. Furthermore, even if prices vary (exogenously) theeconomy is still likely to head for a corner. Once the economyis close to a corner—such that oneinput is very dominant technologically—then even a radicalfall in price of the other input willnot shift the economy towards the other corner, as long as thetechnological advantage of the firstinput is larger.

Now assume that the first input—which has dominated for 100 years—is found to be runningout. This causes its price to rise steeply, and potentially without bound. The second input must beused. However, it will take 100 years of investment to bring the second input’s productivity up tothe level of the first input, if we assume that the total quantity of research investment is constantover time. So the switch from one input to the other will be enormously costly in terms of lostproduction. Finally, if a completely new resource appears on the market, there will be no marketfor it, since there will be no technology complementing thatresource initially, implying that itsprice in efficiency units (the price of a unit ofAcC for instance) will be infinite, there will be zerodemand for the resource, and hence also zero investment in technology augmenting that resource.

Fortunately, and clearly, the simple model with independent knowledge stocks is not applica-ble to empirical cases. This is demonstrated by the data presented in Section 11.1, where we seethat substitute resources coexist rather than outcompeting one another. Furthermore, when a newresource appears (oil, aluminium) it rapidly takes market share rather than being ‘locked out’ bythe incumbent resource. Thus we must refine the DTC model if weare still convinced that theDTC approach is potentially relevant.

11.3. An alternative to lock-in: The Fundamentalist economy

The above aggregate evidence is consistent with a model in which knowledge stocks arelinked: new knowledge boosting the productivity of alternative energy inputs is produced notsimply using existing knowledge regarding that input, but also using overall general knowledge,and also knowledge which is specific to the use of other energyinputs. Recall the discussionof the previous chapter (Section 9.4) regarding links between knowledge stocks and wind power.

Chapter 11. Substitution between alternative resource inputs 99

When the interest in wind power rose during the late 20th century, researchers did not turn towindmill designs from the 19th century. Neither did they perform their research into new designsusing 19th-century techniques. They harnessed the power ofthe technological progress madebetween 1900 and 1990 in order to make very rapid progress regarding the productivity of powergeneration from wind. Much of the knowledge they used was general to the whole economy—forinstance computers—whereas some would have been rather specific to other alternative powersectors, such as electric turbines.

A simple alternative to the lock-in/break-out economy is an economy in which relative knowl-edge stocks grow at equal rates, based on growth in overall general knowledge. This alternativehas an obvious drawback, however, which is that is seems to rule out the technology transitionswhich we do actually observe, such as towards the use of oil and aluminium in the 20th century.The following model—the Fundamentalist economy—capturessuch transitions in a simple way.

In the Fundamentalist economy we introduce a new distinction between knowledge stockskc andkd and productivitiesAc andAd, and the dynamics are driven by the difference betweenresource-specific parameterskc andkd, which represent the degree of technological sophisticationrequired to make use of each resource. Production ofyc is a function of input productivityAc,

yc = γcAcqc,

and input productivity is a function of input-related knowledgekc andkc,

Ac = kc(1− kc/kc)1/ωc for kc > kc, otherwisekc = 0.

The parameterωc > 0. Symmetric expressions apply for inputD. For simplicity we completelyshort-circuit the process of DTC by assuming that there is only one type of investmentzr , and itboosts both types of knowledge equally. Sincekc andkd are equal we definekct = kdt = krt , and inequilibrium

krt+1 = krtzφ

rt+1/ζr .(11.4)

In this economy there will therefore be no path-dependence or lock-in.The dynamics of resource productivity are as follows. Ifkc < kc the productivity of inputC

is zero; technology is too primitive to make any use of the input. However, sincekc rises at aconstant rateθ then at some point we havekc = kc, and the productivity of the input rises abovezero beyond this point. The initial rate of increase will be very large, approachingθ asymptoticallyfrom above. The rate of approach will depend onωc; in the limit asωc approaches infinity,Ac

jumps straight tokc as soon askc > kc, whereas whenωc→ 0 the rate of approach becomes slow.The productivity of resourceD will follow a similar pattern, but the timing will be different ifkc , kd.

The model based on technological fundamentals does a much better job of explaining the datathan the lock-in/ break-out model. According to this model, oil and aluminiumdemand a higherlevel of technology in order to be used productively, and once the overall economy has reachedthis level then they rapidly take their place alongside the other inputs (including coal and iron).The role for directed investments is limited.

If the relative productivities are governed by a process such as that in the Fundamentalisteconomy then policies to encourage directed research efforts are likely to be a waste of time.Assume for instance that solar PV is a technology with a high technology thresholdkpv, but thatthis threshold has now been passed and hence thatapv is approachingkpv, and productivity growthin the sector is high. Now, ifkpv is high enough then solar power will soon take over from fossilpower; on the other hand, ifkpv is not high enough then solar power will remain small relativeto fossil power as long as the price of fossil power is not raised relative to solar, through forinstance taxation of CO2 emissions. So in the fundamentalist economy emissions taxes are thekey instrument to yield a technology transition, not research subsidies.

Finally, note that we have in no way proved the suitability ofthe fundamentalist model as adescription of the economy and basis for policy. We have simply presented evidence suggestingthat it is a much more promising alternative than the DTC model with independent knowledgestocks.

CHAPTER 12

The EKC and substitution between alternative inputs

We now return to the simple model of substitution between inputs from the previous chapter,and apply it to the analysis of polluting flows, which are linked to use of natural-resource inputs.Growth in pollution flows is driven by increasing demand for such inputs (the prices of whichhave no long-run trend), while dramatic falls in pollution occur when firms switch to cleanerinputs or production processes. These switches are triggered by increasing marginal pollutiondamages, linked to increasing income. We investigate the relevance of the model and explorepossible extensions using evidence regarding emissions ofSO2 and CO2.

12.1. Introduction

Global and local environmental quality are dramatically affected by flows of pollutants gen-erated by human activity. Country-level data on pollution and GDP show that there is often a ten-dency for flows of individual pollutants to grow initially and then decline as GDP grows over time,a phenomenon which Panayotou (1993) described as the environmental Kuznets curve (henceforthEKC). In this paper we present a simple explanation for the observed pattern, which—by contrastto existing explanations—is based on a model in which polluting emissions are linked to use ofnatural-resource inputs, and demand for these inputs is a function of their prices and the priceof emissions (assumed equal to the social cost). We show thatthe curve emerges as a generalresult when we make empirically grounded assumptions aboutinput and emissions prices, and wedemonstrate the applicability of the model to specific empirical cases.

The mechanisms behind the EKC observations are of great relevance to environmental policy.For instance, if it is found that growth causes a demand for environmental quality, which is thensatisfied through stricter policy, then this would support the idea that environmental policies areeffective and justified; but if instead it is found that growth itself leads to better environmentalquality in the long run then environmental regulations might be bad for the environment, if theyslow growth.1 Our model is consistent with the former explanation. Emissions are driven up by ascale effect; a bigger economy involves more resource use and more emissions. But given risingincome, the damage cost of a unit of emissions tends to rise, while the abatement cost of a unitof emissions remains constant. When the damage cost exceedsthe abatement cost, regulations oreconomic instruments are introduced which cause emissionsto fall rapidly.

We now very briefly review the EKC literature, making three claims. The first claim is thatthere is general agreement that the inverted U is a common if not ubiquitous feature when pollutionflows are plotted against time at country level, but that the idea that different countries all haveturning points at the same level of GDP is not supported. The second claim is that flows ofpolluting emissions are inextricably linked to flows of natural-resource inputs. And the thirdclaim is that it follows that models aiming to shed light on the mechanism should link pollutionflows to natural-resource inputs, but existing models fail to do so.

Regarding our first claim—about the inverted U—the most cited work on the empirics ofthe EKC is by Grossman and Krueger (1995), who find that the idea of an inverted U is supportedby the historical data, but are careful to point out that theysay nothing about the mechanismbehind the observations, and that without knowledge of thismechanism conclusions about policycannot be drawn. Many other authors find the inverted U pattern; for instance, Selden et al. (1999)(discussed further below) analyse emissions of six pollutants in the US, all of which show someform of inverted U.

For Stern (2004) and many others (see references therein) the key question is not whetheremissions have tended to rise and then fall in growing economies, but rather the question iswhether it is the passage of time or increasing GDP which is responsible for the fall. He concludes

1Growth might lead directly to higher environmental qualitydue to structural change away from polluting manu-factured goods and towards non-polluting services, or because the technological change that drives growth in advancedeconomies is intrinsically pollution-saving. For a famousstatement of the case see Beckerman (1992).

101

102 12.1. Introduction

[p1435] that ‘[t]here is little evidence for a common inverted U-shaped pathway that countries fol-low as their income rises’, arguing that the time trend of emissions per unit of GDP is downwards(across countries), while the link between GDP and emissions is typically increasing at countrylevel.

Regarding our second claim, the inextricable link between polluting emissions and use ofnatural-resource inputs is demonstrated by much research both within and outside the EKC litera-ture. Within the EKC literature, studies in which the main aim is to identify the direct causes ofchanges in emissions over time invariably put the use of natural resource inputs at the centre oftheir analysis. For instance Selden et al. (1999)—mentioned above—study emissions in the USof PM, SOx, NOx, VOC, CO, and Pb, and find that each is closely linked to the use of natural-resource inputs in the production process, especially (butnot only) the combustion of fossil fuels.2

Outside the EKC literature, the link between natural-resource inputs and pollution is frequentlyobvious and indisputable. There are countless examples, the most important of which is perhapsthat CO2 emissions are primarily caused by the burning of fossil fuels, with other sources beingthe clearing of forest land for agriculture and the heating of calcium carbonate in making cement.

In order to predict the future path of polluting emissions, and how this path can be affected bypolicy, we must use the historical data to help us construct and calibrate a microfounded model.3

It follows that analysis of emissions should involve analysis of demand for natural-resource inputs.But the most well-known theoretical models of the EKC do not include natural resources; instead,pollution itself is typically treated as an input. This approach has recently been criticized byMurty et al. (2012) in the analysis of efficiency and productivity; they demonstrate that pollutingemissions should be treated as a byproduct of the use of ‘pollution-causing inputs’—i.e. naturalresources—rather than as inputs in their own right.

Of the theoretical models, our main interest is in those based on a general equilibrium frame-work in which there is underlying growth in production in theeconomy—driven by increases inTFP—which is the driver of increasing pollution flows, and wefocus on three of the most well-known models of this type: Stokey (1998), Andreoni and Levinson (2001), and Brock and Taylor(2010).4 In all three models there is—implicitly—a Leontief production function of effectiveinputs of labour–capital and gross (pre-abatement) polluting emissions. Using an adapted neoclas-sical framework we can writeY=minF(ALL,K),D, whereAL is labour productivity,L is labour,K is capital, andD is gross polluting emissions. So asF rises—due to population growth, labour-augmenting technological progress, and capital accumulation—pollutionD rises at the same rate.The possibility for emissions to turn down whileY continues to grow is generated by adding thepossibility of abating polluting flows, where abatementA is the difference between gross pollutionD and net pollution reaching the environmentP: A= D−P.

Brock and Taylor (2010) take abatement effort, population growth, rates of technologicalchange, and capital investment as exogenous, and assume that abatement efficiency grows fasterthan effective labour, so in balanced growth emissions decline. They then assume that the econ-omy starts far from a b.g.p., with very little capital in relation to labour productivity, and the risingportion of the curve is driven by super-rapid capital accumulation. By contrast, Stokey (1998) andAndreoni and Levinson (2001) both seek to explain the EKC as the result of (optimally) increas-ing abatement effort over time: Stokey (1998) shows that if the marginal utility of consumptiondeclines sufficiently strongly with growth then an EKC is observed, and Andreoni and Levinson(2001) show that if the marginal abatement cost (MAC) curve shifts down sufficiently rapidlywith growth then an EKC is observed.5 Andreoni and Levinson argue persuasively that the MAC-curve is likely to shift downwards in growing economies, by reference to a series of examples in

2Particulate matter, sulphur oxides, nitrogen oxides, volatile organic compounds, and lead.3For a general statement of the case see Lucas (1976). According to Lucas, we cannot expect to predict the effects

of a policy experiment based only on patterns in aggregate historical observations, because the policy experiment willchange the rules of the game and therefore the old patterns may no longer apply. We must instead understand the rules ofthe game and how it is played—in economic terms we must know agents’ preferences, the technologies available to them,what resources they are endowed with, etc.—in order to understand and predict the effect of a policy intervention.

4An example of a class of models not of this type is models in which the pollution path is directly linked to timerather than growth in GDP; for instance, the rise in pollution may be due to the introduction of new technology, and thesubsequent decline due to a lagged introduction of regulations or technology to control it. See Smulders et al. (2011) foran example of such a model. These models help us to see a potential cause of the time trend, but for a full understandingthe model of the time trend should be incorporated into a general equilibrium model with growth.

5In both Stokey, and Andreoni and Levinson’s worked example,utility is a separable function of consumption andpollution; Lopez (1994) demonstrates that the key to the EKCin this class of model is the combined strength of the twoeffects, declining marginal utility of consumption and declining marginal cost of abatement. See Figueroa and Pasten(2013) for further discussion.

Chapter 12. The EKC and substitution between alternative inputs 103

all of which the link between pollution and use of natural-resource inputs is obvious. (Recall thatnatural-resource demand is not included in their model.)

We now move on from the EKC literature to other literature on the economics (and dynamics)of polluting emissions. We start with a brief look at two of the many papers on optimal environ-mental policy in which pollution is linked to natural-resource use, limiting the discussion to paperswhere there is an explicit cost to using the resource (e.g. extraction cost or scarcity rent).6 Thereare many papers in which there is a nonrenewable resource available in finite quantity which isan input to production, but the contribution of which is attenuated by concomitant polluting emis-sions. Typically in these papers the problem is when rather than whether to extract the resource,and in the simplest of them—when there are no extraction costs, as in Schou (2000)—the pol-lution has no effect on the optimal extraction path and there is no need for environmental policy.The price of the resource rises at the interest rate in these papers—i.e. steeply and exponentially—and the path of emissions is monotonically declining.7 Since these resource price and extractiontrends are completely at odds with empirical observations,the empirical relevance of papers ofthis type is limited. In more sophisticated papers—such as Grimaud and Rouge (2008)—featuressuch as extraction costs and directed technological changeare added. In these papers extraction istypically too high initially in laissez faire, which can be corrected through a decliningad valoremtax on the resource. Again, these papers typically predict monotonically declining extraction ratesand steeply rising resource prices in laissez faire, and hence are of limited direct relevance to theunderstanding of empirical observations.

The papers most relevant to this one are integrated assessment models built to determinethe optimal rate of taxation of fossil fuels, such as Golosovet al. (2014) and Traeger (2016).Golosov et al. (2014) are closest to this paper in their approach to modelling emissions, and wefollow them in the following three respects: they assume that polluting emissions arise as a directconsequence of the use of natural-resource inputs; they model demand for these inputs using anested function in which the elasticity of substitution between labour–capital and the energy in-termediate is 1 (Cobb–Douglas) whereas the elasticity of substitution between alternative primaryenergy sources may be higher; and they assume that the primary energy sources differ both inprice and pollution intensity. However, the focus of Golosov et al. (2014) is not the EKC and thetime-path of polluting emissions, but rather the current level of the optimal carbon price. Herewe show that models of the type developed by Golosov et al. (2014) yield an EKC when optimalpolicy is applied over time, and that an EKC arises under a wide range of empirically relevantcircumstances (i.e. not just in the case of carbon). Furthermore, we do not need to appeal topoorly understood processes of directed technological change, although adding such processes toour model—if they can be empirically supported—may make it even more powerful, explainingthe EKC as driven by GDP growth in leading countries, but witha time dimension in followers.

We build a simple, Ricardian model in which an increase in effective labour inputs—quantity× productivity—drives growth in production, and also growthin demand for a natural-resourceinput the use of which leads to polluting emissions. We do notmodel directed technologicalchange, but treat technological change as exogenous and unbiased. Furthermore, our main focus ison the first-best optimal path in the regulated economy, so wedo not allow for any lag between theemission of pollutants, the realization that these emissions are damaging, and the implementationof regulatory policy. Finally, we assume that there are no pollution stocks, only flows. All ofthe above—directed technological change, policy lags, andstock pollution—could be added asextensions to the model which would enrich it without changing the fundamental properties.

We show that an EKC is likely to arise under a wide range of circumstances, and that the pat-tern may be repeated more than once for the same pollutant. There are two keys to the mechanism.The first is that the social cost of polluting is the sum of (i) the cost of the polluting input, and (ii)the damage cost of the concomitant emissions. At low income,input costs dominate; since theinput price is constant, the price of pollution is approximately constant and hence pollution tracksgrowth. As income grows, the input price shows no trend, while unit damage cost (WTP to avoida unit of pollution) rises. When income is high, pollution costs dominate the input cost, the socialcost of polluting tracks the growth rate, and polluting emissions level off.

The second key to the mechanism—which delivers the observeddecline in pollution—isthat alternative, cleaner inputs or production processes are available, at a cost. For clarity assumetwo inputs which are perfect substitutes, the second of which is more expensive but clean. Whenincome is low, the cheaper (dirty) input is preferred. But asincome rises, at some point the sumof unit extraction and damage costs for the dirty input exceeds the unit cost of the clean input.

6See Grimaud and Rouge (2008) for a more thorough review of this literature.7Note that these results are typically not mentioned explicitly in these papers.

104 12.2. The model

At this point the clean input takes over, leading to a precipitous decline in pollution. In practicethe clean input may be a completely different input (wind instead of fossil fuel, for instance), butit may also be the addition of end-of-pipe abatement such as FGD equipment to remove sulphuremissions from coal-fired power stations; the important thing is that the switch from dirty to cleancomes at a cost that is approximately constant per unit of production.

Based on the model there should be no expectation that the downward portions of the EKCshould occur at the same income level for different pollutants, unless these pollutants are all linkedto the same resource input. And—given a set of heterogeneouscountries and a single local pollu-tant—there should be no expectation that each country should enter a downward phase simultane-ously irrespective of relative income (which we might expect if the discovery of clean technologywere key), nor even that each country should enter a downwardphase at the same income level(which we might expect if increasing returns in the abatement function were key); differences inpollution damages due to differences in population density (among other possible factors) shouldlead to differences between countries, and furthermore the relative prices of alternative resourceinputs or production processes may differ significantly between countries and over time.

12.2. The model

12.2.1. The environment.Consider a closed economy on an area of landH with populationL spread uniformly over this area. There is a unit mass of competitive firms—also uniformlyspread—producing a single aggregate final good the price of which is normalized to 1. Theproduction function of the representative firmi hiring labourLi and resource inputRi is

Yi (t) = [AL(t)Li(t)]1−αRi(t)αe−[P(t)/H]φ ,(12.1)

whereα is the resource share, which is small,P is the aggregate flow of pollution, andφ is aparameter greater than 1. So pollution damages destroy a proportion of what would otherwise beproduced, and this proportion is an increasing function of the concentration of pollutant; further-more, the second derivative is strictly positive, so a doubling in concentration more-than doublesthe proportion of production lost.8

In equilibriumLi = L andRi = R, whereL is aggregate labour andR is the aggregate flow ofresource inputs. BothAL andL are exogenously given. Note that, for clarity, we abstract fromcapital in the production function. Furthermore, we assumethat ALL, effective labour, growsexogenously over time at a constant rateg:

AL(t)/AL(t)+ L(t)/L(t) = g.

From now on we omit the time index whenever possible.Aggregate resource inputsR are the sum of inputs from 2 different sources,C andD, which

are perfect substitutes in production, so

R=C+D.

The use of resource quantityD leads to emission of pollutionψDD, whereψD > 0, and similarly forinputC. Each resource is extracted competitively from a large homogeneous stock, and each unitextracted requireswj units of final good as input (wherej is eitherC or D). The price of resourcej in laissez faire is thuswj . Note that we can interpret alternative resourcesC andD literally asdescribed, for instance low- and high-sulphur coal for electricity generation. However, a thirdresourceE could bescrubbedhigh-sulphur coal; then the pricewE would bewD plus the unit costof scrubbing, and unit emissionsψE would beψD× the fraction remaining after scrubbing.

We denote aggregate production net of extraction costs asX, so

X = (ALL)1−α (C+D)αe−[(ψDC+ψDD)/H]φ − (wCC+wDD).(12.2)

This expression highlights the fact that we treat pollutionas affecting gross aggregate productionY, rather than net production or utility.

8Note that this formulation is consistent with damages that directly affect humans or the economy, for instancethrough damage to human health or man-made capital. If on theother hand damages are to nature (so there is a willingnessto pay to avoid damages to the natural environment, and this WTP is increasing in income) then we would expect WTP toavoid damage to be increasing in the land area, linked to the quantity of nature. Assuming that there is also a direct effect,the damage term could be exp−[P(t)/H]φ(1+ θH).

Chapter 12. The EKC and substitution between alternative inputs 105

12.2.2. The solution.In solving the model we focus throughout on the social planner’s so-lution, and mostly state results regarding optimal taxation without proof, since they are straight-forward to derive given the planner’s allocation. The planner choosesC andD to maximizeX(equation 12.2). For expositional clarity we begin by assuming that only a single input,D, isavailable, son= 1. Take the first-order condition inD to show that

αY/D = wD+φ(ψD/H)φDφ−1Y.(12.3)

This equates the marginal benefits ofD to its costs, which are extraction costswD and marginaldamagesφ(ψD/H)φDφ−1Y. It follows directly that the optimal (Pigovian) emissionstax τ in aregulated economy is given by

τ = φ(ψD/H)φDφ−1Y.

Furthermore, we can derive the following proposition aboutthe optimal path.

Proposition 1. Assume there is only one input, D. Then from any given initialstate (definedby AL(0)L(0), effective labour at t= 0), P increases monotonically and approaches a limit ofP= (α/φ)1/φH. If we let AL(0)L(0) approach zero then the initial growth rate of P approaches gfrom below, and P(0) approaches(α/wd)1/(1−α) AL(0)L(0).

Proof. Proposition 1 can be proved in many ways. Here we show a proofbased on graphicalreasoning. Rewrite (12.3) as

α/D1−α = wDeψDD/(ALL)1−α+ψDDα,(12.4)

and denote

LHS= α/D1−α

and RHS= wDeψDD/(ALL)1−α+ψDDα.

Plot LHS and RHS againstD. LHS declines monotonically and is independent of (ALL)1−α,approaching infinity whenD→ 0, and approaching 0 whenD→∞. RHS increases monotonicallyin D, starting atwD/(ALL)1−α when D = 0 and approaching infinity whenD → ∞. And thecurve shifts down to the right when (ALL)1−α increases, and in the limit atALL→∞ the curveapproaches RHS= ψDDα. It follows directly that there is a unique solution forD, which isincreasing inALL; furthermore,D→ 0 whenALL→ 0, andD→ α/ψD whenALL→∞.

Finally, the solution forD/D whenALL is small. Since whenALL→ 0 we know thatD→ 0we can rewrite (12.4) as (in the limit)

α/D1−α = wD/(ALL)1−α.

Hence, if we letAL(0)L(0) approach zero thenD(0) approaches(α/wD)1/(1−α) AL(0)L(0). It fol-lows directly that the growth rates ofD andALL are equal in the limit.

The intuition is as follows. The shadow price of the polluting input to the social planner is thesum of extraction cost and marginal damages. The extractioncost is constant, whereas marginaldamages increase linearly inY. So whenY is small the shadow price is approximately equalto the constant extraction cost, hence both resource use andpolluting emissions track growth.As Y increases, marginal damages increase and hence the shadow price of using the pollutinginput increases, braking the growth in its use. WhenY is large marginal damages dominate theextraction cost, the shadow price of using the input grows atthe overall growth rate, and emissions(and input use) are constant. So we have a transition from emissions tracking growth towards (inthe limit) constant emissions.

Now assume two inputs,C andD, and thatwC > wD, whileψC < ψD; so the second input ismore expensive but cleaner. Furthermore, to keep the expressions concise, we normalizeH = 1.To reintroduceH explicitly—as we must in a multicountry setting—we can simply rewriteψ j asψ j/H.

Givenn= 2 the first-order condition in inputj —a necessary condition for an internal opti-mum—yields

αY/(C+D) = wj +φ(ψCC+ψDD)φ−1ψ jY.(12.5)

So when the use of inputj is greater than 0, the marginal revenue product of an increase in inputj should be equal to the marginal social cost, which is the sum of the extraction costwj and thedamage costφ(ψCC+ψDD)φ−1ψ jY. Recall that the inputs are perfect substitutes, so we mightexpect that only one of them will be used (the one with lower marginal social costs), but if themarginal social costs of the two inputs are equal then they may be used simultaneously. WhenYis low, the damage cost is small, and inputD will be used alone. But asY andD rise, damages

106 12.2. The model

rise until the marginal social costs are equal. At this pointinputC enters, and then gradually takesover from input 1. IfψC > 0 then inputD is eventually abandoned completely. So asALL grows(driving growth inY) there is a transition from inputD to inputC, as described in Proposition 2.

Proposition 2. Assume inputs C and D, wD < wC, andψD > ψC > 0, while (φ−1)(1+α)/α >ψ2

D(wC−wD)/(wCψD−wDψC). Furthermore, assume an initial state AL(0)L(0) such that onlyinput 1 is used. Then D increases monotonically up to time T1a, while C= 0. Between T1a andT1b (where T1b > T1a), D decreases monotonically while C increases monotonically. For t ≥ T1b,D = 0 and C increases monotonically. Furthermore,

T1a =1g

log

[

D(T1a)AL(0)L(0)

wDe(ψDD(T1a))φ

α−φ(ψDD(T1a))φ

1/(1−α) ]

,(12.6)

and T1b =1g

log

[

C(T1b)AL(0)L(0)

wCe(ψCC(T1b))φ

α−φ(ψCC(T1b))φ

1/(1−α) ]

,

(12.7)

where D(T1a) =1ψD

(

αψD

φ

wC−wD

wCψD−wDψC

)1/φ

and C(T1b) =1ψC

(

αψC

φ

wC−wD

wCψD−wDψC

)1/φ

.

(12.8)

If ψC = 0 then equation 12.6 still applies, but T1b is not defined; instead, as t→∞, D→ 0.

Proof. The proof is beyond the scope of this book. See Hart (forthcoming), ‘The rise and fallof polluting emissions’ for details.

Finally, we present the results of an extended model withn inputs,D1, D2, D3, . . . , Dn.

Definition 2. Assume alternative inputs j= 1, . . .n, with associated parameters wj andψ j .Define the cheapest input as input1, and the cleanest input as input m. And define input k+1—where k∈ (1, . . .m−1)—as the input such that

wk+1−wk

wk+1ψk−wkψk+1<

wk+m−wk

wk+mψk−wkψk+m(12.9)

for all m∈ (2, . . . ,n−k). Since wj andψ j are continuous variables, we rule out the possibility thatequation 12.9 may be satisfied with equality.

To understand Definition 2, considerk= 1. Then input 2 must satisfy

w2−w1

w2ψ1−w1ψ2<

w1+m−w1

w1+mψ1−w1ψ1+m

for all m∈ (2, . . . ,n−k). From equation 12.8, this implies that no other input is adopted prior to theadoption of input 2. And the corresponding equation fork= 2 implies that input 3 is adopted afterinput 2, and so on. Once inputm (the cleanest input) has been adopted there will be no furthertransitions, and the remainingn−m inputs are never used. Then we have Proposition 3.

Proposition 3. Assume that we have n inputs ordered as in Definition 2, and that (φ−1)(1+α)/α > ψ2

j (wj+1−wj)/(wj+1ψ j −wjψ j+1) for j = 1, . . . ,m−1. Then—given sufficiently low initialproductivity ALL—there is a series of transitions from input j to input j+1, starting from input1and ending with input m. Each of these transitions proceeds analogously to the transition from1to 2 described in Proposition 2. The remaining n−m inputs are never used.

Proof. During a transition fromk to k+1, equation 12.5 shows that

wk+φψkPφ−1Y= wk+1+φψk+1Pφ−1Y,

and hence Pφ−1Y=1φ

wk+1−wk

ψk−ψk+1.

So during the transition,Pφ−1Y is constant. Now assume that the transition tok+2 startsduringthe transition fromk to k+1. Then (analogously to equation 12.5) we have

wk+φψkPφ−1Y= wk+1+φψk+1Pφ−1Y= wk+2+φψk+2Pφ−1Y

But sincePφ−1Y is constant during the transition to inputsk+ 1 andk+ 2 should have startedsimultaneously, which is ruled out by our earlier assumption that equation 12.9 is never satisfiedwith equality. So the transition tok+2 can only start after the transition tok+1 is complete, whichcompletes the proof.

Chapter 12. The EKC and substitution between alternative inputs 107

The implication of Proposition 3 is that polluting emissions P tend to rise over time due toa scale effect (i.e. the direct effect of economic growth on demand for the resource input), butthat periods of gradual increase are punctuated by rapid falls when increasing marginal damagesjustify switches to cleaner technologies or inputs. In the long run the economy will switch to azero-emissions technology if one is available.

0 50 100 150 200 250 3000

0.02

0.04

0.06

0.08

0.1

(ALL)∗

P

P= (α/φ)1/φ

Figure 12.1. Pollution flowP compared to the limit,P = α, and normalizedeffective labour (ALL)∗, where (ALL)∗ = ALL/[AL(0)L(0)] · P(0). Parameters:g = 0.02; AL(0)L(0) = 1; φ = 1.3; ψ1 = 0.0072,ψ2 = ψ1/6, ψ3 = 0; α = 0.05;w1 = α, w2 = 2α, w3 = 2.5α. The dotted lines show pollution paths in case onlyone of the inputs is available.

In Figure 12.1 we illustrate the development of the economy in a specific case in whichm= n= 3, so there are three inputs all of which are used, implying that there are two transitions.Since the third input is perfectly clean, the third transition is completed asymptotically, since theuse of the second input approaches zero in infinite time. In Figure 12.1 we show the paths ofeffective labourALL, and pollutionP, and the pollution limitP. We also show—using dottedlines—the paths ofP which would be followed if (respectively) only input 1 and input 2 wereavailable.

12.3. The elements of the model

In this section we seek to establish the following three points: firstly, that the individualelements of the model are good approximations in a wide rangeof cases; secondly that the modelhelps us to understand cross-country observations in the case of sulphur emissions to air; andthirdly that we can (and should) adapt and extend the model toanalyse specific cases (where a‘case’ means a specific pollutant and country or region).

12.3.1. Empirical relevance of the elements of the model.We divide the model into fiveelements; these elements could be seen as five key assumptions which together make the patternof rise and fall at country level inevitable. They are that pollution is directly linked to the useof natural resources; that resource use tends to increase linearly with GDP; that there are perfectsubstitutes available to resource inputs that differ in price and polluting emissions; that thesesubstitute inputs are available at constant relative prices; and that marginal damages from pollutionrise linearly in GDP.

The link between pollution and resource use. Is pollution directly linked to the use of naturalresources? We argue that it is. Start with the six pollutantsanalysed by Selden et al. (1999): PM,SOx, NOx, VOC, CO, and Pb. All of these pollutants originate in the use of natural resourcesas inputs: PM (particulate matter) emissions come from combustion of fossil fuels (especially inindustry), plus cement, mining, and mineral-processing sectors; SOx and NOx come from combus-tion of fossil fuels; VOC (volatile organic compounds) comefrom petroleum refining, industrialchemicals, plastics, and paper; CO comes from combustion; and Pb also comes from combus-tion. Moving beyond Selden et al. (1999), other major pollutants include CO2 which comes fromcombustion, and nitrogen and phosphorous leaching (from fertilization in agriculture).

108 12.3. The elements of the model

The link between resource use and economic growth. We now turn to the link between re-source use and economic growth. In the simple model we assumethat the resource price is con-stant, and given the Cobb–Douglas production function (implying constant factor shares) resourceuse tracks overall production. Are these assumptions broadly consistent with the data? We arguethat at the global level they are, whereas at the national level the picture is often significantly morecomplex, and trade and structural change may be important toinclude in an explanatory model.

1900 1950 2000−1

0

1

2

3

4

1850 1900 1950 2000−1

0

1

2

3

4

1960 1970 1980 1990 2000−1

0

1

2

Global productExpenditurePriceQuantity

Nat

ura

llo

gsc

ale

Figure 12.2. Long-run growth in total consumption and prices, compared togrowth in total global product, for (a) primary energy from combustion, (b)metals, and (c) cement.

1800 1850 1900 1950 2000

0

1

2

3

4

5

6

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1900 1950 2000

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1960 1970 1980 1990 2000

−0.5

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US GDPQuantity

Nat

ura

llo

gsc

ale

Figure 12.3. Long-run growth in US consumption compared to growth in USGDP, for (a) primary energy from combustion, (b) metals, and(c) cement.

At the global level there is a clear pattern to the data on consumption rates and prices ofprimary inputs—see Figure 12.2—and this pattern is consistent with the model; there is no cleartrend in resource prices, while resource consumption tracks gross global product.9 At countrylevel things are more complex—see Figure 12.3 for US data—and the pattern is not alwaysconsistent with the model.10 In Figure 12.3 we see that US primary energy consumption—byfar the most important natural resource input in terms of factor share—has almost matched GDPgrowth over a very long period. We only have price data since 1970, and when we include thesedata we find a substantial rise in factor share since 1970; on the other hand, if we have accurate

9Note: Global product data from Maddison (2010). Metals: Al, Cr, Cu, Au, Fe, Pb, Mg, Mn, Hg, Ni, Pt, Rare earths,Ag, Sn, W, Zn. All metals and cement data from Kelly and Matos (2012). Energy: Coal, oil, natural gas, and biofuel. Fossilquantity data from Boden et al. (2012). Oil price data from BP(2012). Coal and gas price data from Fouquet (2011); notethat these data are only for average prices in England; we make the (heroic) assumption that weighted average global pricesare similar. Biofuel quantity data from Maddison (2003). Biofuel price data from Fouquet (2011); again, we assume thatthe data are representative for global prices, and we extrapolate from the end of Fouquet’s series to the present assumingconstant prices. The older price data is gathered from historical records and is not constructed based on assumptions aboutelasticities of demand or similar. Sensitivity analysis shows that the assumptions are not critical in driving the results.

10GDP data from Maddison (2010). All metals and cement data from Kelly and Matos (2012) (same metalsas above). Primary energy data from the EIA, History of energy consumption in the United States, 1775–2009,http://www.eia.gov/todayinenergy/detail.cfm?id=10.

Chapter 12. The EKC and substitution between alternative inputs 109

data prior to 1970 it would likely show a very slow decline, delivering no clear trend in the shareover the entire period.11 Summing up, in the crucial primary energy sector we find broadsupportfor the model from US data. Turning to metals and cement, however, we see a weakly risingconsumption trend in the case of cement, and a declining trend for metals since 1974. Price trends—which are omitted for clarity—do not explain these shifts.Based on the analysis of Selden et al.(1999), the reason for these trends is likely to be structural change (their ‘composition effect’).In the case of cement we could speculate that it is due to the rise of the consumption of servicesrather than manufactured goods, whereas in the case of metals an important contributory factormay be the rise of imported manufactured goods such as automobiles, and hence the decline ofdomestic manufacturing.12

The availability of perfect substitutes. A crucial elementof the model—which determinesthe shape of the abatement cost function—is the availability of close substitutes for pollutingresources (in the basic model developed above, perfect substitutes). We take three examples toshow that this is frequently a good approximation, althoughin each case there are complicatingfactors. The examples are lead emissions to air, sulphur emissions to air, and carbon dioxideemissions to air.

The vast majority of emissions of lead to air have come—historically—from lead added tomotor fuel (gasoline) which allows improvements in the efficiency of engines.13 Lead-free fuel iseffectively a perfect substitute for leaded fuel, since there is a fixed cost to switching to unleadedper unit of output (which in this case is motive power from theengine, or the extra cost of thealternative additives which replace lead). However, note that there are complicating factors: firstly,there are fixed costs of adapting engines, therefore phase-out may be preferred to an instant ban;there are synergies with reductions in other pollutants, because catalytic converters to removeunburnt or partially burnt fuel are choked by lead in the fuel; and finally, knowledge about thedamage caused by lead emissions may improve over time. The latter two of these factors maycause low-income countries to adopt lead-free fuel at lowerincomes than high-income countries,since they face lower costs and are more aware of the benefits.

Turning to sulphur, the major source of sulphur emissions toair historically has been theburning of coal and fuel oil (for heating, electricity generation, power generation, and in indus-trial process such as smelting iron). In each case, alternatives are available which lead to muchlower emissions. For each individual case the assumption ofperfect substitutability is generallyreasonable, but the unit cost of the transition to the cleaner input or technology varies between thecases, so at the aggregate level the inputs should be treatedas perfect substitutes. And if the priceof emissions rises gradually (with GDP) then we should see a series of steps down in emissionsas the switch to clean technology is made in different sectors. On the other hand, if one sector—such as electricity generation from coal—is dominant in a given country then the assumption ofperfect substitutability will fit well in that country.

Focusing on coal burning, there are several alternative technologies which can reduce sulphuremissions at an approximately fixed unit cost (thus implyingperfect substitutability). The simplestof these is to switch to low-sulphur coal. A closely related alternative is to switch to an alternativefuel—typically natural gas—which leads to sulphur emissions very close to zero. The thirdalternative is to continue to use high-sulphur coal, but to clean up the emissions, most commonlythrough flue-gas desulphurization (FGD). This technology typically removes the large majority ofsulphur emissions—between 80 and 90 percent—at a rather constant cost per ton.14

Finally, the analysis of carbon dioxide emissions to air is similar to that for sulphur: sector-by-sector the assumption is reasonable, but the switch from dirty to clean inputs is optimal at differentrelative prices in different sectors (power generation, domestic heating, rail transport, road trans-port, air transport, sea transport, etc.), so in the aggregate we do not observe a discrete switch ata particular relative price, but switches will occur at different times in different sectors. Further-more, in some cases the assumption of perfect substitutability may not be justified, for instance in

11We know that energy prices tended to decline over the period 1870–1970, as illustrated by the well-known pricetrend for crude oil. On the other hand, there was a countervailing switch from cheaper energy sources (coal) to moreexpensive (oil and gas) over the same period.

12A feature of the cement data is the extraordinary sensitivity of demand to the business cycle, with small downturnsin GDP associated with very large downturns in cement consumption. This is presumably linked to the well knownsensitivity of the construction sector to the business cycle.

13See for instancehttps://www.epa.gov/lead-air-pollution for background information on lead pollution.14Evidence for constant marginal costs can be found in many places in the literature. One example is the RAINS

model, in which abatement of SO2 is based on FGD, and the model yields marginal costs that are almost constant up toabatement of 85 percent of total emissions. See for instanceFigure 4.6 in Amann et al. (2004). Furthermore, the US EPA(2003) quotes cost estimates for FGD from large power plantswhich are constant per ton of pollutant removed.

110 12.3. The elements of the model

the case of wind substituting for fossil in electricity generation; the more wind generation capac-ity there is available, the lower the value of the marginal unit of capacity, because the difficultiesassociated with the variability and unpredictability of supply increase as supply increases.

The relative prices of substitutes. Given the existence of perfect substitutes, are their relativeprices constant? Here we distinguish two classes of case: firstly, where the substitute technologiesare both mature (typically implying that the clean technology is a substitute input), and secondlywhere the substitute technology is not mature.

Typically, when the substitutes are alternative inputs andboth technologies are mature (coaland gas, high-sulphur and low-sulphur coal, etc) a better general assumption than constant relativeprices might be that the prices of each follow a random walk. For instance... coal transport costsdown in US ... gas prices down relative to coal in UK ... but then back up again!

In the second case, when the clean technology is not mature, there is obviously a likelihoodthat the costs of the substitute will be higher for early adopters than for later adopters. Thus,according to the model, if we have a global economy consisting of two countries, one of which isthe leader in terms of GDP per capita, then the leader will adopt the clean input first (because ofits higher valuation of damages), but the follower will—ceteris paribus—adopt at a lower levelof GDP than the leader, because of the learning in the leader country reducing that unit cost of theswitch.

Willingness to pay. In the model we assume that marginal damages are linear in gross in-come, so the higher is income the higher is willingness to payto avoid pollution damage. Thuswe assume that environmental quality is a normal good. An alternative assumption with essen-tially equivalent results is that a given level of pollutionleads to a loss of a fixed proportion ofproduction. Remarkably little research has been performedon this question, but the assumption isstandard in (for instance) climate models such as Nordhaus (2008) and Golosov et al. (2014), andit is also made by Finus and Tjotta (2003) in their study of SO2 policy. See Jacobsen and Hanley(2009) for one study on this question.

A second assumption about the damage function is that marginal damages are at least weaklyincreasing in the rate of polluting emissions (or if we had a stock pollutant, the stock of pollu-tion).15 Here also there is remarkably little concrete evidence in the literature for most pollutants—but note papers such as Levy et al. (2009). The exception is again with regard to CO2, where alot of effort has been put into estimating damages as a function of concentration. Nordhaus (2008)arrives at a rather flat function, implying that ourφ should be close to 1, whereas others—suchas Acemoglu et al. (2012)—argue for greater curvature, translating into a higher value ofφ in ourmodel.

12.3.2. Extensions suggested by the existing literature.In this section we consider the fol-lowing extensions or adaptations to the model: increasing returns to scale in abatement; directedtechnological change; weaker substitutability between inputs; and structural change. In each casewe conclude that these features could be added to our model and thus contribute to its ability toexplain and predict trends in particular sectors.

Increasing returns to abatement. The most influential explanation in the literature for the EKC-type observations is that of Andreoni and Levinson (2001), i.e. that there are increasing returns toscale in abatement, so as the economy grows pollution first increases due to the scale effect ofeconomic growth, and then decreases due to declining abatement costs. In the present paper thereare no such increasing returns. Here we argue that increasing returns may be relevant for somepollutants, in which case our model can easily be extended toincorporate this feature.

We set up a comparison between the models, by deriving equations for marginal abatementcosts and benefits both in Andreoni and Levinson (2001) and inour model. First we clarify def-initions. By marginal abatement benefits, MAB, we mean the decrease in consumption—at themargin—which would leave the representative household indifferent given a reduction in pol-lution. And by marginal abatement costs, MAC, we mean the rate—at the margin—at whichconsumption must be reduced in order to reduce pollution. Inan empirical application bothMABandMAC might have units of USD/ton.

Here we calculate expressions forMACandMAB in the respective models, and plot the curveswith pollutionP on thex-axis andMACandMABon they-axis. We start with our own model. Forthis purpose we assume an economy for whichm= 2, andψ2 = 0. So we have a single transitionfrom the dirty to the clean input. We denoteALL = M throughout this section. First rewrite

15Recall that this is necessary to ensure a tractable solution, Proposition 2: (φ − 1)(1 + α)/α >

ψ21(w2−w1)/(w2ψ1−w1ψ2).

Chapter 12. The EKC and substitution between alternative inputs 111

equation 12.2 in terms ofP andD2,

X = M1−α(P/ψ1+D2)αe−Pφ − (w1P/ψ1+w2D2),

and take the first-order condition inP to show that up to the transition to input 2 we have

MAC= αY/P−w1/ψ1,

and MAB= φPφ−1Y.

During the transition, the expression forMABstill applies, butMAC= (w2−w1)/ψ1, since the costper unit of input switched isw2−w1, and the reduction in emissions isψ1.

Now we turn to Andreoni and Levinson (2001). In their ‘simpleexample’ (Section 2) theyassume separable utility. However, in Section 3 they consider a more general utility function. Toaid the comparison to our paper, we choose a utility functionas close as possible to our own, so anydifferences between the abatement cost and benefit curves is due to factors other than the choiceof utility function. We therefore have (adapting the Andreoni and Levinson notation slightly to beconsistent with our own)

Y= Me−Pφ

and X = Me−Pφ −E, whereP= ψ[M−E− (M−E)αEβ] andE ≥ 0.

SoMAC= (∂X/∂E)/(∂P/∂E), while MAB= −∂X/∂P. So we have

MAC=1ψ

[

1+Eβ(M−E)α(

β

E−

α

M−E

)]−1

,

and MAB= φPφ−1Y.

We now simulate, using the same parameter for the utility function in each case:φ = 1.3. Andin each case we plot five pairs of curves for increasing valuesof M.16

0 0.02 0.04 0.06 0.08 0.1 0.120

2

4

6

8

10

0 0.02 0.04 0.06 0.08 0.1 0.120

5

10

15

20

PollutionPPollutionP

MA

Can

dM

AB

Figure 12.4. Marginal costs and benefits of abatement plotted against pollutingemissions in our model compared to Andreoni and Levinson.

We plot the results in Figure 12.4. The figure illustrates thedifferent mechanisms at work inthe two models. In our model, there is an upper limit on the marginal abatement cost, which is thecost of switching to the clean input. As marginal abatement benefits increase with increasing in-come, they eventually reach this limit and then emissions decline steeply. In Andreoni and Levinson(2001), theMAC curve moves outwards for low levels of abatement, just as in our model. How-ever, it also changes shape, rotating downwards at higher levels of abatement, because of theassumption of increasing returns. Declining marginal abatement costs lead to steeply decliningemissions after the turning point, a decline that is accentuated by increasing marginal benefits ofabatement. AsALL approaches infinity marginal abatement costs approach zeroat all levels ofabatement (right back to zero emissions).

The interplay between changes in theMAB andMAC curves is essential to the pattern ofemissions in our model, whereas this is not the case in Andreoni and Levinson (2001). To seethis clearly, eliminate this interplay by assuming a simpleseparable utility function such thatthe MAB curve is a horizontal line at a fixed benefit level. Then our model delivers pollutingemissions which track growth in the long run—so no EKC—whereas the Andreoni and Levinson

16The values ofM are as follows:M = M(0), 2M(0), 4M(0), 8M(0), and 16M(0), whereM(0) = 1.76. Otherparameters: our model,α = 0.05,w1 = α, w2 = 2α, ψ1 = 0.0072; Andreoni and Levinson,α = 0.8, β = 0.6,ψ = 0.01.

112 12.4. Case studies of pollution reduction

(2001) model still delivers an EKC because the marginal abatement cost curve approaches they-axis whenALL approaches infinity, implying that the total cost of abatingall polluting emissionsapproaches zero.17

Andreoni and Levinson (2001) argue persuasively that in some industries there are signifi-cant increasing returns to scale in abatement; loosely, when the industry becomes larger, plantsbecome larger, and larger plants have lower average abatement costs. (Incidentally, their analysisof this question supports our model, since they discuss changes in technology in order to reduceemissions which follow from the use of natural-resource inputs.) However, we argue that it isequally possible to find industries in which abatement yieldsdecreasingreturns to scale, the obvi-ous example being power generation. At small scales clean power sources—such as hydropower—may be available in sufficient quantity to meet demand. However, as the scale of the economyincreases these clean options become more expensive relative to fossil, because of (in the case ofhydro) the limited flow of precipitation in a given geographical area.

Endogenous learning. In the model as it stands, overall productivity increases exogenously,and there is no other learning in the model. Based on the literature (such as Smulders et al. (2011)and Acemoglu et al. (2012)) we know that a switch to a new, clean technology in one country maylead to a reduction in costs for that technology, thus makingadoption more attractive for othercountries. Consider a global economy consisting of two countries, one of which is the leader interms of GDP per capita, where a clean technology is available at a price premium. If the countriesare identical in other respects then the leader will adopt the clean input first (because of its highervaluation of damages), but the follower will—ceteris paribus—adopt at a lower level of GDPthan the leader, because of the learning in the leader country reducing that unit cost of the switch.

It may be relevant to include processes of directed technological change in the model. Suchprocesses may help to explain the time dimension in the EKC observations, but again the DTCmechanism strengthens our model rather than challenging it.

In the second case, when the clean technology is not mature, there is obviously a likelihoodthat the costs of the substitute will be higher for early adopters than for later adopters. Thus,according to the model, if we have a global economy consisting of two countries, one of which isthe leader in terms of GDP per capita, then the leader will adopt the clean input first (because ofits higher valuation of damages), but the follower will—ceteris paribus—adopt at a lower levelof GDP than the leader, because of the learning in the leader country reducing that unit cost of theswitch.

Weaker substitutability between alternative inputs.Structural change.

12.4. Case studies of pollution reduction

In this section we consider 6 cases of polluting emissions. In each case we begin with abrief look at empirical data, before discussing the role of technology. The evidence shows thatwhen a high price is put on pollution, drastic reductions in emissions are generally achieved. Is thisdone through efficiency (generating less pollution through better technology) or substitution (usinganother input to substitute for pollution)? Or is it simply aquestion of investment, i.e. increasingthe quantity of capital in order to reduce polluting emissions, in line with the neoclassical (DHSS)model. The answer seems to vary from case to case.

12.4.1. Sulphur emissions from power generation.Coal contains sulphur, and coal-firedpower stations emit sulphur oxides to the atmosphere. Thesecause respiratory problems locally,and acid rain regionally. Sulphur emissions in Europe and the U.S. rose rapidly during industri-alization, and have fallen even more rapidly since the early1970s. See Figure 12.5. One of themajor reasons for the fall was the installation of so-called‘scrubbers’ in the chimneys of coal-firedpower stations. These scrubbers remove sulphur oxides fromthe exhaust flow of the power station,i.e. from the smoke.

To the extent that the technology has always existed (and according to Biondo and Marten(1977) the first systems were installed in the 1930s) then their inclusion in power stations is a sim-ple case of capital (the cost of the scrubber) substituting for pollution (the emissions). However,to the extent that the use of scrubbers stimulates increasesin their efficiency and hence decreasesin cost then we can talk of directed technological change playing a role. Thus if we imagine aconstant price being put on the polluting emissions, when scrubber technology becomes cheaperdue to DTC then adoption of the technology will increase. In reality however emissions rose incost from zero to a high level, causing the universal adoption of scrubbers. Or in some cases the

17We assume that theMABcurve is not so high that emissions are always zero in our model.

Chapter 12. The EKC and substitution between alternative inputs 113

1850 1900 1950 2000−2

−1.5

−1

−0.5

0

0.5

1

GDP

Sulphur Clean air act

Nat

ura

llo

gsc

ale

Figure 12.5. UK Sulphur emissions compared to GDP. Both curves normalizedto zero in 1956, the date of introduction of the first of a long series of regulationsrestricting emissions. Data: Maddison (GDP), David I. Stern (Sulphur). Twoanomalous points in the sulphur data have been altered.

adoption of scrubbers was simply mandated by the national orinternational authorities. Substitu-tion between inputs has also played an important role in reducing sulphur emissions from powergeneration generally, in that power generators switch to cleaner types of coal, or from coal to gasor renewables.

In all cases, the key driver of sulphur emissions reduction has been a dramatic increase in theprice of emissions either explicitly (emissions actually were priced, as they are in an emissionstrading program) or implicity, i.e. when regulations are imposed which mandate the use of certainemissions-reducing technologies or banning the use of other technologies.

12.4.2. Smokeless fuel in urban areas.The first Clean Air Act was an Act of the Parliamentof the United Kingdom passed in 1956, following the so-called ‘Great Smog’ of 1952.18 Throughthe act it effectively became illegal to emit significant quantities of smoke in and around London.This necessitated, in the short run, a switch of inputs to smokeless fuels for many agents, includinghouseholds which heated their homes by burning coal. The drastic nature of the price increase(implicitly, the price of emissions rose to infinity) necessitated an input switch rather than simplyefficiency improvements. Again, it is not clear to what extent the initial input switch was followedby (directed) technological change to make the new inputs (such as smokeless fuel) cheaper overtime.

12.4.3. Chlorofluorocarbons.Chlorofluorocarbons (CFCs) are highly stable, relatively non-toxic organic compounds which were widely used as refrigerants. However, once discharged tothe atmosphere they migrate to the upper atmosphere (the stratosphere) where they remain fordecades, catalysing the destruction of ozone. In 1987 the Montreal protocol was signed, effectivelyensuring a rapid phase-out in CFC production and use. It was subsequently strengthened, makingthe phase-out more rapid than first planned. Thus, again, we have a dramatic rise in the cost ofemitting pollution, from zero to infinity (or close to infinity). Not surprisingly, pollution emissionsfall very rapidly following such a change.

Again, the main mechanism for the fall is substitution. Sucha dramatic cost increase meansit is not feasible to increase the efficiency with which CFCs are used in order to comply withregulations, they must be replaced by a substitute. Again, it is not clear the extent to which theforced substitution led to DTC making the new (substitute) input cheaper over time.

12.4.4. Dichlorodiphenyltrichloroethane (DDT). DDT is a chemical first synthesized in1874, but not used as an insecticide before WW2. It was—initially—extraordinarily effectiveat wiping out insects, and it was soon noted that this could cause ecological imbalances; becauseDDT eliminated predatory insects as well as pests, application could trigger population explosionsof fast-breeding species.

18For the act itself, seehttp://www.legislation.gov.uk/ukpga/Eliz2/4-5/52/enacted.

114 12.4. Case studies of pollution reduction

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

Glob.prod.

CFC

Montreal protocol

Figure 12.6. Global production of CFCs, (CFC11+CFC12) compared to globalproduct, both curves normalized to zero in 1987, the date of signing of theMontreal protocol. Data: Maddison (GDP), AFEAS (CFCs).

Over time, other problems emerged. Apart from resistance evolving among target species,it was also found the DDT accumulated in the bodies of specieshigher up in the food chain—such as the bald eagle and peregrine falcon—causing death orinability to reproduce. This washighlighted by Rachel Carson’sSilent Spring, (Carson, 1962). Following publication of the bookuse of DDT in the United States declined rapidly, and it was banned in 1972.

The graph of use against time—Figure 12.7—does not follow the usual pattern. Use growsvery rapidly from 1939, but then quickly reaches a plateau before falling gradually. The banin 1972 is the final death knell, although use does not cease completely until 1975. There arevarious reasons for the difference from the usual pattern. One is that DDT was used primarilyin agriculture, which is not a growing sector, hence we do notexpect a rising trend for inputs.Another is the growth in resistance. And a third is the gradual realization of the harmful effects.19

1940 1945 1950 1955 1960 1965 1970 19750

20

40

60

80

DD

Tu

se,m

illio

nlb/

year

Figure 12.7. Consumption of DDT in the U.S.

Again, the disappearance of DDT seems to be mainly due to substitution of other inputsrather than a price increase causing greater efficiency in its use. The substitutes used depend onthe circumstances. They include alternative pesticides (agriculture) and alternative methods ofmosquito control (malaria).

12.4.5. Toxins in car exhaust.Car exhausts emit a cocktail of dangerous pollutants, includ-ing nitrogen oxides (often known as NOx), volatile organic compounds, ozone, carbon monoxide,and particulates. Furthermore, between around 1920 and 1980 large quantities of lead were emit-ted. Finally, of course, large quantities of carbon dioxide, but we consider that in the next section.

Emissions per kilometre driven have fallen drastically. Why? In the case of lead, the answeris simple: adding lead to petrol was banned during the 1980s and 1990s. This led to cars running

19For more information, and data, see ‘NIOSH special occupational hazard review:DDT’, 78-200.

Chapter 12. The EKC and substitution between alternative inputs 115

on lower-octane fuel, or alternative technologies were used to boost the octane rating of fuel.However, the role of technological change in the process seems to be limited.

Regarding the other pollutants, the reasons for their decline are more complex. On the onehand, when engines become more efficient (giving an economic benefit regardless of the cost ofpollution) polluting emissions per kilometre tend to fall.On the other hand, catalytic converters—mandatory in most countries—drastically reduce these emissions. As with lead, the only reasonfor a firm to introduce such technology is regulation, or pressure from altruistic consumers. Thebasic technology existed for 15 years or more before it was first introduced into mass-producedcars in the US during the 1970s. So again, regulation has driven the adoption and refinement ofexisting technology.

A fascinating and topical case is that of emissions from diesel-engined cars. Diesel enginesproduce more power output per unit of CO2 emitted than petrol engines (all else equal) becauseof the higher heat content of the fuel, higher burning temperature, and higher compression ratio.However, the high temperature also causes oxidation of nitrogen in the air to nitric oxide andnitrogen dioxide (NO and NO2) commonly known as NOx. NOx, once formed, is hard to getrid of. It can be converted catalytically in a process known as SCR. However, this requires (inaddition to the catalyst) a flow of a chemical such as urea intothe reaction chamber, and thischemical needs a tank and regular refills. This is already applied to many heavy goods vehiclesand buses, but has not (yet) spread to the automobile market.An alternative to SCR is to preventthe creation of NOx in the first place, through EGR. EGR lowersthe combustion temperatureof the fuel, thus reducing NOx formation. However, this alsoreduces power output per unit offuel burnt. What then is the solution of the unscrupulous carmaker? Program the car’s computer(engine management system) to (a) detect when the car is in a test environment rather than on theroad, and (b) react by putting the EGR system into overdrive.On the other hand, when the car ison the road the system is shut down and the engine delivers high power—and high emissions.20

12.4.6. Carbon dioxide in car exhaust.Carbon dioxide emissions from car exhausts con-tinue to rise, despite increases in efficiency. Income and price effects have led to increased travel,and the choice of more powerful, heavier vehicles, which have offset the efficiency increases. Seefor instance Knittel (2011). Furthermore, internal combustion engines are approaching the theo-retical limits of efficiency; not much improvement remains technically possible. Given these facts,it is clear that there are only two ways of drastically reducing such emissions: either a change offuel, or a drastic reduction in car use.

Regarding a change of fuel, the potential for DTC to play a role is clear. It is possible thatinvestment in (for instance) the technology necessary for electric vehicles may lead to such a greatimprovement in that technology that it actually becomes preferable to fossil-based technologywithout the need for subsidies. Our previous analysis givessome pointers to how likely such ascenario really is.

12.5. Conclusions

Return to Selden et al. (1999). They sorted changes in polluting emissions in the US intocategoriesscale, composition, energy intensity, energy mix, andother technology.

The scale effect is simply the increase in emissions which is to be expected if emissions trackGDP; so if GDP doubles, the scale effect indicates a doubling in emissions. To the extent thatemissions did not track GDP during the period of study, the other effects must be present.

The composition effect is Solow’s shifting between consumption categories. Ifthis effect isdriving down emissions, it is because we are shifting the balance of our consumption towardscategories which have low pollution-intensity. The energy-intensity effect is the effect of energy-efficient technology reducing energy requirements per unit of production, whereas the energy mixeffect is about switching from one (polluting) energy source toanother (clean) one. So the first ofthese effects is related to our discussion of DTC and rebound, and the second is related to lock-inand break-out. The ‘other technology’ effect typically refers to technology changes which reduceemissions per unit of fuel burned; typical examples are catalytic converters on car exhausters, orscrubbers on smoke stacks (chimneys).

Selden et al. find that the key effect is ‘other technology’: emissions have not been reducedby changes in the composition of consumption, nor by changesin the energy mix, nor by energy-efficiency. Instead, they have been driven by technologies specifically introduced to cut those very

20For more on this case see ‘The VW scandal exposes the high techcontrol of engine emissions’ onhttp://theconversation.com/.

116 12.5. Conclusions

emissions. Crucially, such technologies are introduced due to the introduction of regulations (or,in some cases, pollution pricing).

The result fits naturally with the observation that CO2 emissions have continued to rise: suchemissions can only be reduced by composition changes, changes in the energy mix, or increasedefficiency. They cannot be reduced by end-of-pipe solutions, barring the successful introductionof carbon capture and storage.

We have presented a simple explanation for the rise and fall of polluting emissions basedon a model in which such emissions are linked to use of natural-resource inputs, and alternativeinputs differ in pollution intensity. We conclude by making three claims based on the analysis inthe paper. Our first claim is that the mechanism through whichpolluting emissions rise and fallmay be applicable to a large number of relevant empirical cases. We have discussed the casesof SO2 and CO2, but could equally well have taken dioxins, lead, NOx, or many other cases inwhich the key criteria for our mechanism seem to be fulfilled:that marginal damages increasein income, that emissions are linked to the use of natural resources in production, that the pricesof the relevant natural resources display little or no long-run trend, and that alternative inputs orproduction processes are available—at a cost—resulting inlower emissions.

Our second claim is that phenomena not in the model—such as directed technological change—may add important dimensions to the analysis and require specific policy instruments if anoptimal allocation is to be achieved; however, the relevance of such phenomena does not renderthe basic model irrelevant, on the contrary DTC should be incorporated in a framework in whichemissions are linked to input use, and the costs of alternative inputs (such as clean and dirty)are anchored by reality rather than simply being functions of cumulative research effort in therespective sectors.

Thirdly, our analysis is congruent with the analysis of Stern (2004) and many others: weshould not expect the exact pattern for one pollutant and location to be repeated for anotherpollutant or location; key factors in the model which may be expected to vary—both betweenpollutants, and over time and space for the same pollutant—include the relative prices of alterna-tive resource inputs or production processes, and the valuations of environmental quality at givenincome and pollution levels. For instance, both natural gasand coal are relatively expensive totransport, hence in some locations coal may have a price advantage over gas, whereas in otherlocations the reverse may be true. Regarding the time dimension, clean production processes(e.g. solar PV technology) may fall in price over time due to research effort; in such cases, coun-tries which are lower on the growth ladder may adopt these processes at lower levels of GDP thanthe richest countries.

Finally, a word about future research. As it stands, the model implies that the more success-ful we are at finding clean technologies, the greater will be the long-run rate of resource use. Apromising extension not discussed above would be to add resource constraints (most fundamen-tally a finite area of land) to the model, and investigate the very long run development of theglobal economy. See Goeller and Weinberg (1976) for an earlyanalysis focused only on resourcescarcity.

CHAPTER 13

Production, pollution and land

In the previous chapter we assumed a simple trade-off between pollution and production, andthat zero pollution could be achieved per unit of productionat a fixed unit cost. We then found that—given continued income growth—households would increasingly find this cost worth paying (atthe aggregate level), and therefore their elected representatives would enact legislation to ensurethat zero-emission technologies were used.

A problem with this analysis is that the trade-off is not simply between pollution and final-good production. There is also a third factor, which is land.And in many cases there is alsoa trade-off between land and pollution: polluting technologies—such as fossil-fuel burning andintensive agriculture—often use less land than their cleaner substitutes, such as wind and solarpower, or low-input agriculture. But the total quantity of land is limited, and furthermore there aremany other competing uses for land, such as recreation and leisure, and not least as the substrateon which the natural (non-human) world can thrive. Given increasing income, willingness to payfor these goods (leisure and thriving nature) will also increase, just as WTP to avoid pollutionincreases. But we may not be able to find technologies which use zero (or close to zero) landand emit zero pollution. Consider for instance energy generation. We therefore expect a trade-offin the long run between land, pollution, and production. In this chapter we analyse this tradeoff.In doing so we also take a deeper look at the nature of pollution, in particular analysing stockpollution rather than just flows.

Limits to productivity of energy production for instance, and of production of some goodssuch as motive power.

More to come . . .

117

Part 5

Policy for sustainable development

Here we consider policy for sustainable development. We take a step back and consider theimplications of the analysis presented earlier in the book,and also put it into a broader context.We examine the role of green consumerism, and also consider arguments for government policiesto encourage shorter working hours and thus less productionand consumption in the economy.Section under construction!

CHAPTER 14

Is unsustainability sustainable?

Assume that the following statement is true.

• We are getting richer and healthier, but trashing the planet.

Given the above, consider the following two hypotheses.

(1) • We rely on services provided by the planet, and by trashing the planet we are de-stroying the planet’s ability to provide these services in the future.• Therefore, if we carry on trashing the planet the loss of these services will lead to

us getting poorer and sicker.(2) • We are adaptable and ingenious.

• Therefore we can carry on both trashing the planet and getting richer and healthier,indefinitely.

Now assume that you care about the planet, and would like to help redress the balance be-tween the pursuit of material wealth and care of the planet. What to do? Consider the followingtwo strategies.

(1) Find evidence for hypothesis 1, or try to convince othersof its veracity.(2) Persuade others to care too: either more about the planet, or less about material wealth!

There are of course other strategies. For instance:

(3) Demonstrate that the system (e.g. ‘global capitalism’)is going to crash anyway, irre-spective of the state of the planet. And argue that since it’sgoing to crash we might aswell slow it down gently and save the planet at the same time.

The first strategy is fine as long as hypothesis 1 holds. But what if it doesn’t? If you’ve pinnedall your arguments onto it, and it turns out not to be true, you’re in trouble. Maybe following thesecond strategy would be a better idea? In this chapter I willsuggest that hypothesis 2 is probablytrue, and therefore that strategy 2 is much preferable to strategies 1 and 3, which are both veryrisky.

14.1. Lessons from historical adaptation

The idea here is to argue that the regulated market economy has shown a remarkable abilityto adapt and react to crises when they arise, including environmental crises.

On the other hand, we know that environmental crises may often have far-reaching conse-quences for nature, and sometimes for human welfare. And when the consequences areonly fornature, not a lot tends to get done. Consider for instance theBaltic Sea, or bird populations inEurope.

Finally, there are examples of civilizations that have collapsed, apparently due to environmen-tal collapse. E.g. Easter Island. What lessons are there here? E.g. Brander and Taylor (1998).

14.2. Financial and other crises

We know that financial crises—especially large-scale ones across many countries—typicallyhave severe and long-lasting effects. However, such a crisis does not signal the death-throes ofcapitalism, the collapse of the system under the weight of contradictions. We know why the recentglobal financial crisis occurred, and we know why recovery from it is so slow. The reason is thelack of confidence in the future which is widespread among agents, a lack of confidence which isrational for each individual in the knowledge that everyoneelse lacks confidence. It is a giganticcoordination problem, the solution to which is either some massive shock (such as WW2 in 1939)or gradual, inch-by-inch progress.

14.3. Uncertainty and future crises

So the global economy will very likely be able to keep going asit has up to now, for decadesor even centuries to come. Growing, triggering environmental problems and even catastrophes,and then solving them. All the while, the space for the non-human or ‘natural’ world is likely to

121

122 14.3. Uncertainty and future crises

be circumscribed ever-more by our thirst for consumption, consumption of everything from foodto wilderness experiences.

CHAPTER 15

Growth and environment

In this chapter we discuss policies which may help to combinecontinued rapid growth inper-capita production with improving environmental quality and respect for the natural world.

15.1. Taxation contra Command and Control

The idea here is to show that there is an important role for command and control.

15.2. Precaution

The precautionary principle is open to varying interpretations. But it is clear that if we aremore careful this will slow long-run growth while potentially avoiding future crises.

15.3. Lessons from observations of structural change

Maybe we (consumers) are very flexible about what we consume.And maybe there is a herdinstinct which leads us to choose similar options. Considerfashion. Maybe the shift towardsenergy- and resource-intensive goods can be turned around at minimal cost in utility?

123

CHAPTER 16

The no-growth paradigm

16.1. Consumerism

Firms want to sell more stuff, employ more people, etc. The market economy has its owndynamic. Or something.

16.2. ‘Green’ consumerism

Green consumerism is a tricky business. The key problem is rebound. If you don’t consumeone thing, but your income is unchanged, you will consume something else instead, or invest incapital which may be just as bad. It is an impossible task for individual consumers to weigh upthe environmental effects of their actions. We need consumers to elect politicians who enact lawswhich (a) lead to external effects being internalized in the prices of goods (in borderline cases),and (b) lead to highly damaging or unnecessary practices being banned (in black-and-white cases).A recent example of the latter is the ban on incandescent light bulbs in both the US and the EU.

16.3. Conspicuous consumption, labour, and leisure

16.3.1. Background chat, very preliminary. Are we prioritizing consumption of goods toohighly, and preservation of nature too low? But we elect governments in a democratic process, andthey choose policies which determine these priorities.

But what about the Easterlin paradox, the claim that increasing average income in a countryis only weakly correlated with increasing happiness and well-being? Does this not suggest thatthe all-out effort to work harder and produce more is a rat-race with no winner?

This idea links with the ideas of Thorstein Veblen (see for instance Veblen, 1899), who coinedthe term ‘conspicuous consumption’ to capture the idea thatwe consume in order to be seen toconsume, and by being seen to consume we raise our status which is gives us utility. However, inorder to consume we must earn income, and in order to earn income we must work, and in orderto work we must sacrifice leisure, and the sacrifice of leisurereduces our utility.

Putting Veblen’s ideas into a modern economic context, if weconsume to gain status, thenthere is a consumption externality: one person’s higher status is her neighbours’ lower status,driving down their utility. Therefore each individual’s choice to work long hours to earn moremoney and consume more has a negative external effect on that individual’s neighbours. And, inthe global village, that might mean everyone else in the global population.

Consider climate negotiations. Is it possible that a major stumbling block in these negotiationsis not the desire of China and India to get richer, but rather the desire of China and India tocatchup with the OECD countries in terms of wealth and income per capita? And, by the same token,the desire of the richest countries that China and India should not catch up with them in terms ofwealth and income per capita, implying that China and India (by virtue of their higher populations)would wield much more economic power than USA and the EU?

Why do politicians constantly exhort their citizens to adopt growth-friendly policies so thatthey do not lose out in the ‘global race’? There is no global race: countries which adopt newtechnology less aggressively—and countries whose populations work less and take more leisuretime—have lower GDP per capita than other countries, but they have neither higher unemploy-ment nor lower welfare. Why then the political obsession with growth and productivity? Could itbe that citizens compare their consumption rates across borders, and furthermore that politiciansgain utility from observing that ‘their’ economies are larger and more powerful than those of theirneighbours?

If the above is true (or partly true) then it is highly likely that we worktoo hardand taketoo little leisure time for our own good. More precisely, if everyone worked less and took moreleisure time, everyone would be better off! And this applies irrespective of spin-off benefits forthe environment and nature. In the next section we build and solve a simple model to demonstratethe mechanism.

125

126 16.3. Conspicuous consumption, labour, and leisure

16.3.2. A very simple model.Assume a population of identical households indexed byiwhere the utility of a given household is described by the following function:

ui = cα1i r1−α1−α2

i (ci/c)α2 .

Hereci is consumption,r i is leisure, and ¯c is average consumption across all households. Further-more,α1 andα2 are parameters the sum of which is less than 1. For convenience define

α = α1+α2.

Production is a linear function of labour, but consumption is reduced by an income tax at a flatrateτ, and boosted by a lump-sum transfer of public goods which is equal to average productionl multiplied by the tax rate. Thus we have

ci = l i(1− τ)+ lτ.

Leisurer (for recreation) is equal to total timeRminus labour time, i.e.

r i = R− l i .

Now take the tax as exogenous and work out how much each household chooses to work. Todo so, substitute into the utility function to yield

u=[

l i (1− τ)+ lτ]α

(R− l i)1−α/cα2.

Take the first-order condition inl i and solve to show that

l i = αR− (1−α)lτ/(1− τ).

So when income tax is zerol i = αR: as labour income dominates the utility function (α high)households devote more of their time to labour and less to leisure.

Now assume a symmetric equilibrium such that average labourl is equal to the labour sup-plied by householdi, l i . Inserting this into the above result we have

l i = l =αR

1+ (1−α)τ/(1− τ).

Now the question for a regulator is, what level of taxτ maximizes utility for households?Economic theory tells us that if markets are perfect then theoptimal tax should be zero, hencel i = αR. But if there is a consumption externality—i.e. ifα2 > 0—then this no longer holds.

To solve the problem, we insert the expression forl i as a function ofτ into the utility function—noting that in symmetric equilibriumci = c and (as already stated)l i = l —to obtain

u=

[

αR1+ (1−α)τ/(1− τ)

]α1[

R−αR

1+ (1−α)τ/(1− τ)

]1−α

.

Simplify to obtain

u= R1−α2αα1ω−(1−α2)(ω−α)1−α,

where ω = 1+ (1−α)τ/(1− τ).

Take the first-order condition inω to solve for the optimalω, and then use the definition ofω tosolve for the optimal tax:

τ =α2

α1+α2.

So, the stronger the weight of ‘conspicuous consumption’ inutility, the more labour income shouldbe taxed.

How big is the effect? Assuming conspicuous consumption has equal weight to consumptionin utility then labour income should be taxed at 50 percent. The effect of the tax is to reduce laboursupply by a factorω. And if leisure has 50 percent weight in utility (implying thatα = 0.5 so inlaissez-faire the individuals would work 8 hours and have 8 hours of leisure time, assuming that 8hours are needed for sleep) thenω = 1.5, so labour supply is reduced by one third.

Chapter 16. The no-growth paradigm 127

16.3.3. Conclusions on conspicuous consumption.The above model should not be takentoo seriously: it is based on assumptions which are plucked more-or-less out of thin air rather thanbacked by careful argument and empirical evidence. Nevertheless, it demonstrates that there mayexist sound economic arguments for governments to discourage labour and encourage leisure evenin the absence of environmental damage from production. However, if internationalconsumptionexternalities are important then it is only rational for national governments to impose such policiesif their neighbours do the same. Perhaps this is why Europeaneconomies have (collectively) beenable to hold down or even reduce working hours over recent decades, whereas the US has lurcheddramatically in the opposite direction.1

1For a reference to give an introduction to the field, see Aronsson and Johansson-Stenman (2008). On internationalcomparison of working hours see Prescott (2004).

CHAPTER 17

Mathematical appendix

17.1. Growth rates

What is meant by a growth rate? What is a constant rate of growth? And how is growth bestrepresented graphically? Assume we are interested in GDP, denotedY. A constant rate of growthin Y implies thatY grows exponentially, such that (for instance) the time it takes for global productto double is constant. Mathematically we haveY= Y0egt, whereY0 is global product at time zero,g is the growth rate, andt indicates time.

Consider for instance an economy growing by 3 percent per year, and with initial GDP of1× 109 USD/year. We then havey = y0egt wheret is time measured in years,g = 0.03, andy0

is initial GDP. If we ploty against time we get the familiar exponential form; Figure 17.1(a).However, this is a poor way to organize data as it is hard for the eye to interpret: given a curve ofincreasing slope it is not generally possible to determine by eye whether it represents a constant,increasing, or decreasing growth rate. For instance, compare the curve to Figure 17.1(b), whichplots the functiony= y0(1+ t2.5/5300). This is not exponential growth, but this is far from obviousfrom the figure.

0 20 40 60 80 1000

5

10

15

20

0 20 40 60 80 1000

5

10

15

20

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

GD

P,1

09U

SD/y

ear

GD

P,1

09U

SD/y

ear

lnG

DP,

109

US

D/y

ear

lnG

DP,

109

US

D/y

ear

yearsyears

yearsyears

Figure 17.1. Two different growth patterns, illustrated using a linear scale ((a)and (b)) and a logarithmic scale ((c) and (d)).

In order to see the true growth trend, take the (natural) logarithm of the equations. In the caseof exponential growth we have

ln(y/y0) = gt.(17.1)

So we have a straight line through the origin, the slope of which isg; Figure 17.1(c). The secondequation results in the curve shown in Figure 17.1(d), wherewe can see that the growth rate isinitially zero, subsequently increases, and then declines.

Another advantage to plotting the logarithm of a growing variable is that the relative sizesof fluctuations are also shown in proportion. Consider for instance the data showing USA’s GDPfrom 1870 to 2008, Figure 17.2. Consider the size of the fluctuations in GDP before and after thegreat depression and WW2. Have the fluctuations increased over time, decreased, or remained

129

130 17.2. Continuous and discrete time

the same? From 17.2(a) it is very hard to tell; a naive interpretation of the curve would be thatthe fluctuations have increased. However, when we plot the logarithm (17.2(b)) we see that thefluctuations in GDP in the second period were smaller than in the first, as a proportion of GDP atthe time.

1880 1900 1920 1940 1960 1980 20000

5

10

1880 1900 1920 1940 1960 1980 2000

0

1

2

3

GD

P/G

DP 0

lnG

DP/

GD

P 0

years

years

Figure 17.2. Two different representations of U.S. growth. The logarithmicscale helps us to see both that the trend growth rate is very constant, and thatthe fluctuations are smaller in the latter period (after WW2).

Finally, note that another way to illustrate growth data is through plotting the growth rateY/Yagainst time. Given a constant growth rateg we haveY/Y= g hence this yields a horizontal lineat heightg.

17.2. Continuous and discrete time

Dynamic economic models may be set up either in continuous ordiscrete time. In continuoustime we can think of time flowing, and also physical goods (such as natural resources or machines)should flow in the sense that their quantities change gradually rather than suddenly jumping fromone level to another. In discrete time we divide time up into discrete periods, and the state ofthe system jumps between one period and the next. Models in continuous time are often moremathematically elegant, whereas models in discrete time may be more practical in some circum-stances. Empirical (economic) data is typically collectedat discrete intervals, so we measure forinstance total production and natural resource use in a quarter (three months), rather than the flowof production and resource use at each instant. In some economic models the assumption of dis-crete time is crucial because it is assumed, for instance, that firms all invest simultaneously andperiodically.

Mathematically the appearance of models in discrete and continuous time is typically verydifferent, but the results are typically essentially the same (as long as it is practically possibleto model the system using both). Assume a process of exponential growth in continuous time.Following the examples of the previous section we assume

Y/Y= g.

Chapter 17. Mathematical appendix 131

Now consider measuringY at discrete intervals of one year, indexed byt, starting att = 0 whenwe assume thatY= 1. What isY at t = 1?

dYdt= gY,

∫ Yt

Y0

(1/Y)dY=∫ 1

0gdt,

log(Yt/Y0) = gt,

Yt/Y0 = egt.

So att = 1 we haveY= eg, at t = 2 we haveY= e2g, and so on. Furthermore, we can writeYt+1−Yt

Yt= eg

andYt+1

Yt= 1+eg.

So if we take measurements in discrete time we find agrowth factor1+ eg, or 1+ θ, whereasin continuous time we have agrowth rate g. Note that wheng is small thenθ ≈ g, whereaswheng becomes largeθ becomes significantly greater thang because compound growth duringeach period becomes significant. Finally, if we let the period length approach zero theng alsoapproaches zero, andθ approachesg.

17.3. A continuum of firms

In macroeconomic modelling it is very common to assume acontinuumof firms (and indeedhouseholds). Why do we do this, and what does it mean?

To understand this, assume instead that we have just one firm.This is nice and simple inone sense, because aggregate production and aggregate demand for inputs is the same as theproduction and demand of the single firm. However, there is a big problem in that this single firmmust then have market power, both on the market for inputs (e.g. the labour market) and on themarket for the final good. Of course, we know that market powerdoes exist in the real economy,but to start off with a model of only one firm is unsatisfactory: typically we want to start ourmodels assuming the simplest possible case, i.e. competitive markets, and then introduce marketpower later when it is relevant; furthermore, even when market power is relevant we are unlikelyto have pure monopsony or monopoly (only one buyer or seller).

In order to generate competitive markets in our models we need to add more firms. But howmany? In a competitive market the actions of any one firm have no effect on prices, firms areprice takers. But this implies that each firm must be infinitesimally small, its production must benegligible compare to the total. Thus we need an infinite number of very small firms. So firmsare no longer countable, but we can measure their mass; and insome circumstances we may wantto claim that the mass of firms has grown (e.g. doubled), even though thenumberof firms is nota meaningful concept. However, in most circumstances we simply assume that there is aunitmassof firms, or a continuum of firms measure 1. This has the advantage that, in symmetricequilibrium, production per firm is the same as aggregate production from all the firms (and thesame holds for input demand).

To see this mathematically, index firms byi and assume thati ∈ (0,1). Production from firmi is denotedYi , and total production from all firms isY. All firms produce the same good. Thentotal production is simply the integral across all the firms,

Y=∫ 1

0yidi.

Now, if the equilibrium is symmetric so all firms produce the same amount thenyi is simplyconstant (it does not vary asi runs from 0 to 1), and

Y= y,

wherey is production by any onerepresentative firm.Note that in more complex models the goods produced by the firms may differ, and the math-

ematics is slightly more complicated. These cases are analysed in detail in the main text whenthey arise.

Bibliography

Acemoglu, D., Aghion, P., Bursztyn, L., Hemous, D., 2012. The environment and directed techni-cal change. American Economic Review 102, 131–166.

Aghion, P., Howitt, P., 1992. A model of growth through creative destruction. Econometrica 60,323–51.

Aghion, P., Howitt., P. W., 2009. The economics of growth. MIT.Amann, M., Cofala, J., Heyes, C., Klimont, Z., Mechler, R., Posch, M., Schopp, W., 2004. The

regional air pollution information and simulation (RAINS)review. Tech. rep., IIASA.Andreoni, J., Levinson, A., 2001. The simple analytics of the environmental Kuznets curve. Jour-

nal of Public Economics 80, 269–286.Aronsson, T., Johansson-Stenman, O., Jun. 2008. When the Joneses’ consumption hurts: Optimal

public good provision and nonlinear income taxation. Journal of Public Economic 92 (5-6),986–997.

Arrow, K. J., Chang, S., Mar. 1982. Optimal pricing, use, andexploration of uncertain naturalresource stocks. Journal of Environmental Economics and Management 9 (1), 1–10.

Arthur, W., 1989. Competing technologies, increasing returns, and lock-in by historical events.Economic Journal 99, 116–131.

Asheim, G. B., Buchholz, W., Withagen, C., Jun. 2003. The Hartwick rule: Myths and facts.Environmental and Resource Economics 25 (2), 129–150.

Atkeson, A., Kehoe, P. J., Sep. 1999. Models of energy use: Putty-putty versus putty-clay. Ameri-can Economic Review 89 (4), 1028–1043.

Beckerman, W., 1992. Economic growth and the environment: whose growth? whose environ-ment? World Development.

Binswanger, M., 2001. Technological progress and sustainable development: What about therebound effect? Ecological Economics 36 (1), 119–132.

Biondo, S. J., Marten, J. C., 1977. A history of flue gas desulphurization systems since 1850.Journal of the Air Pollution Control Association 27, 948–961.

Boden, T. A., Marland, G., Andres, R. J., 2012. Global, regional, and national fossil-fuel CO2emissions. Tech. rep., CDIAC, Oak Ridge National Laboratory, U.S. Department of Energy,Oak Ridge, Tenn., U.S.A.

Boppart, T., 2014. Structural change and the Kaldor facts ina growth model with relative priceeffects and non-Gorman preferences. Econometrica 82 (6), 2167–2196.

Boulding, K. E., 1966. The economics of the coming spaceshipearth. In: Jarrett, H. (Ed.), Envi-ronmental Quality in a Growing Economy. Resources for the Future/Johns Hopkins UniversityPress, pp. 3–14.

BP, 2012. Statistical review of world energy. Annually published data, British petroleum.Brander, J. A., Taylor, M. S., Mar. 1998. The simple economics of Easter Island: A Ricardo–

Malthus model of renewable resource use. American EconomicReview 88 (1), 119–138.Bresnahan, T., Trajtenberg, M., 1995. General purpose technologies: Engines of growth? Journal

of Econometrics, Annals of Econometrics 65, 83–108.Brock, W. A., Taylor, M. S., 2010. The green Solow model. Journal of Economic Growth 15 (2),

127–153.Carson, R., 1962. Silent Spring. A Fawcett Crest Book. Houghton Mifflin.Cass, D., 1965. Optimum growth in an aggregative model of capital accumulation. Review of

Economic Studies 32 (3), 233–240.Daly, H. E., 1997. Georgescu-Roegen versus Solow/Stiglitz. Ecological Economics 22, 261–266.Dasgupta, P., 1993. Natural resources in an age of substitutability. In: Kneese, A. V., Sweeney,

J. L. (Eds.), Handbook of natural resource and energy economics. Vol. 3. Elsevier, Ch. 23.Dasgupta, P., Heal, G., 1974. The optimal depletion of exhaustible resources. The Review of

Economic Studies 41, 3–28.Dasgupta, P., Heal, G., 1979. Economic Theory and Exhaustible Resources. Cambridge University

Press, Cambridge.

133

134

David, P. A., May 1990. The dynamo and the computer: An historical perspective on the modernproductivity paradox. American Economic Review 80 (2), 355–61.

Farzin, Y. H., Jul. 1992. The time path of scarcity rent in thetheory of exhaustible resources.Economic Journal 102 (413), 813–830.

Figueroa, E., Pasten, R., Apr. 2013. A tale of two elasticities: A general theoretical framework forthe environmental Kuznets curve analysis. Economics Letters 119 (1), 85–88.

Finus, M., Tjotta, S., Sep. 2003. The Oslo Protocol on sulfurreduction: the great leap forward?Journal of Public Economics 87 (9-10), 2031–2048.

Fouquet, R., 2011. Divergences in long-run trends in the prices of energy and energy services.Review of Environmental Economics and Policy 5 (2), 196–218.

Fouquet, R., Pearson, P. J. G., 2006. Seven centuries of energy services: The price and use of lightin the United Kingdom (1300–2000). Energy Journal 27, 139–177.

Goeller, H. E., Weinberg, A. M., 1976. The age of substitutability. Science 191 (4228), 683–689.Golosov, M., Hassler, J., Krusell, P., Tsyvinski, A., 2014.Optimal taxes on fossil fuel in general

equilibrium. Econometrica 82 (1), 41–88.Grimaud, A., Rouge, L., Dec. 2008. Environment, directed technical change and economic policy.

Environmental and Resource Economics 41 (4), 439–463.Grossman, G. M., Krueger, A. B., May 1995. Economic growth and the environment. Quarterly

Journal of Economics 110 (2), 353–377.Groth, C., 2007. A new-growth perspective on non-renewableresources. In: Bretschger, L., Smul-

ders, S. (Eds.), Sustainable Resource Use and Economic Dynamics. Springer, pp. 127–163.Hanson, D. A., 1980. Increasing extraction costs and resource prices—some further results. Bell

Journal of Economics 11 (1), 335–342.Hart, R., 2009. The natural-resource see-saw: Resource extraction and consumption with directed

technological change. Presented at EAERE 2009.Hart, R., 2014. Rebound, directed technological change, and aggregate demand for energy. Paper

in submission.Hart, R., 2016. Non-renewable resources in the long run. Journal of economic dynamics and

control 71, 1–20.Hart, R., Spiro, D., 2011. The elephant in Hotelling’s room.Energy Policy 39 (12), 7834–7838.Hartwick, J. M., 1977. Intergenerational equity and the investing of rents from exhaustible re-

sources. American Economic Review 67, 972–974.Heal, G., 1976. The relationship between price and extraction cost for a resource with a backstop

technology. The Bell Journal of Economics 7 (2), 371–378.Hills, R. L., 1993. Power from Steam: A history of the stationary steam engine. Cambridge Uni-

versity Press.Hotelling, H., 1931. The economics of exhaustible resources. Journal of Political Economy 39,

137–175.Jacobsen, J. B., Hanley, N., Jun. 2009. Are there income effects on global willingness to pay for

biodiversity conservation? Environmental & Resource Economics 43 (2), 137–160.Jevons, W. S., 1865. The coal question; an inquiry concerning the progress of the Nation, and the

probable exhaustion of our coal-mines. Macmillan.Jones, C., 2005. Growth and ideas. In: Aghion, P., Durlauf, S. (Eds.), Handbook of Economic

Growth. Elsevier.Kelly, T. D., Matos, G. R., 2012. Historical statistics for mineral and material commodities in the

United States. Tech. rep., U.S. Geological Survey, Data Series 140.URL http://pubs.usgs.gov/ds/2005/140/

Knittel, C. R., Dec. 2011. Automobiles on steroids: Productattribute trade-offs and technologicalprogress in the automobile sector. American Economic Review 101 (7), 3368–3399.

Koopmans, T. C., 1963. On the concept of optimal economic growth. Cowles foundation discus-sion paper.

Krulce, D. L., 1993. Increasing scarcity rent – a sufficient condition. Economics Letters 43 (2),235–238.

Levhari, D., Liviatan, N., 1977. Notes on Hotelling’s economics of exhaustible resources. Cana-dian Journal of Economics – revue Canadienne D Economique 10(2), 177–192.

Levy, J. I., Baxter, L. K., Schwartz, J., 2009. Uncertainty and variability in health-related damagesfrom coal-fired power plants in the United States. Risk Analysis 29 (7), 1000–1014.

Lin, C.-Y. C., Wagner, G., Jul. 2007. Steady-state growth ina Hotelling model of resource extrac-tion. Journal of Environmental Economics and Management 54(1), 68–83.

Chapter 17. Bibliography 135

Lindholt, L., Sep. 2005. Beyond kyoto: backstop technologies and endogenous prices on co2permits and fossil fuels. Applied Economics 37 (17), 2019–2036.

Livernois, J., Martin, P., 2001. Price, scarcity rent, and amodified r per cent rule for non-renewable resources. Canadian Journal of Economics 34 (3),827–845.

Lopez, R., Sep. 1994. The environment as a factor of production: the effects of economic growthand trade liberalization. Journal of Environmental Economics and Management 27 (2), 163–184.

Lucas, R. E., 1976. Econometric policy evaluation: A critique. Carnegie-Rochester ConferenceSeries on Public Policy 1 (0), 19–46.

Maddison, A., 2003. Growth accounts, technological change, and the role of energy in Westerngrowth. In: Economia e Energia, secc.XIII-XVIII. IstitutoInternazionale di Storia Economica“F. Datini” Prato, Florence.

Maddison, A., 2010. Historical statistics of the world economy: 1–2008 ad. Tech. rep., Groningengrowth and development centre.

Malthus, T., 1798. An essay on the principle of population. Oxford World Classics, reprinted 2008.Oxford University Press.

Manne, A., Mendelsohn, R., Richels, R., Jan. 1995. A model for evaluating regional and globaleffects of GHG reduction policies. Energy Policy 23 (1), 17–34.

Mayer, H., Flachmann, C., 2011. Extended input-output model for energy and greenhouse gases.Tech. rep., Federal Statistical Office Germany.

Murty, S., Russell, R. R., Levkoff, S. B., 2012. On modeling pollution-generating technologies.Journal of Environmental Economics and Management 64 (1), 117–135.

Nordhaus, W., 2008. A Question of Balance: Weighing the Options on Global Warming Policies.Yale University Press.

Nordhaus, W. D., 1973a. The allocation of energy resources.Brookings Papers on EconomicActivity 4 (1973-3), 529–576.

Nordhaus, W. D., 1973b. Some skeptical thoughts on theory ofinduced innovation. QuarterlyJournal of Economics 87 (2), 208–219.

Orwell, G., 1958. The Road to Wigan Pier. Harcourt, Brace, and company, New York.Panayotou, T., 1993. Empirical tests and policy analysis ofenvironmental degradation at different

stages of economic development. ILO Working Papers 292778,International Labour Organiza-tion.

Parfit, D., 1984. Reasons and Persons. OUP Oxford.Pigou, A., 1920. The Economics of of Welfare, 4th Edition. Macmillian.Pindyck, R. S., 2004. Volatility and commodity price dynamics. Journal of Futures Markets

24 (11), 1029–1047.Popp, D., 2002. Induced innovation and energy prices. American Economic Review 92 (1), 160–

180.Prescott, E. C., July 2004. Why do Americans work so much morethan Europeans? Federal

Reserve Bank of Minneapolis Quarterly Review, 2–13.Ramsey, F. P., 1928. A mathematical theory of saving. Economic Journal 38, 543–559.Rawls, J., 1971. A theory of justice. Harvard University Press, Cambridge.Rogner, H. H., 1997. An assessment of world hydrocarbon resources. Annual Review of Energy

and the Environment 22, 217–262.Romer, P. M., 1990. Endogenous technological change. Journal of Political Economy 98 (5), S71–

102.Sagoff, M., 1995. Carrying capacity and ecological economics. Bioscience 45 (9), 610–620.Saunders, H. D., 1992. The Khazzoom–Brookes postulate and neoclassical growth. The Energy

Journal 13 (4), pp. 131–148.Saunders, H. D., Jun. 2000. A view from the macro side: Rebound, backfire, and Khazzoom-

Brookes. Energy Policy 28 (6-7), 439–449.Schafer, A., 2006. Long-term trends in global passenger mobility. The Bridge, 24–32.Schou, P., Jun. 2000. Polluting non-renewable resources and growth. Environmental & Resource

Economics 16 (2), 211–227.Selden, T. M., Forrest, A. S., Lockhart, J. E., 1999. Analyzing the reductions in u.s. air pollution

emissions: 1970 to 1990. Land Economics 75 (1), 1–21.Smulders, S., Bretschger, L., Egli, H., May 2011. Economic growth and the diffusion of clean

technologies: Explaining environmental Kuznets curves. Environmental & Resource Econom-ics 49 (1), 79–99.

136

Solow, R. M., 1956. A contribution to the theory of economic growth. Quarterly Journal of Eco-nomics 70, 65–94.

Solow, R. M., 1973. Is the end of the world at hand? Challenge,39–46.Solow, R. M., 1974. Intergenerational equity and exhaustible resources. Review of Economic

Studies, Symposium on the economics of exhaustible resources 41, 29–45.Solow, R. M., Wan, F. Y., 1976. Extraction costs in the theoryof exhaustible resources. Bell

Journal of Economics 7 (2), 359–370.Sorrell, S., 2007. The rebound effect: an assessment of the evidence for economy-wide energy

savings from improved energy efficiency. UKERC.Stern, D. I., 2004. The rise and fall of the environmental Kuznets curve. World Development

32 (8), 1419 – 1439.Stern, D. I., 2005. Global sulfur emissions from 1850 to 2000. Chemosphere 58, 163–175.Stern, N., 2006. Stern Review on the Economics of Climate Change. UK Treasury.Stiglitz, J. E., 1974. Growth with exhaustible natural resources: Efficient and optimal growth

paths. Review of Economic Studies, Symposium on the Economics of Exhaustible Resources41, 123–37.

Stokey, N. L., 1998. Are there limits to growth? International Economic Review 39, 1–31.Stollery, K. R., 1983. Mineral depletion with cost as the extraction limit – a model applied to the

behavior of prices in the nickel industry. Journal of Environmental Economics and Management10 (2), 151–165.

Tahvonen, O., Salo, S., Aug. 2001. Economic growth and transitions between renewable andnonrenewable energy resources. European Economic Review 45 (8), 1379–1398.

Teece, D. J., Sunding, D., Mosakowski, E., 1993. Natural resource cartels. In: Kneese, A.,Sweeney, J. (Eds.), Handbook of Natural Resource and EnergyEconomics. Vol. III. Elsevier,Ch. 24, pp. 1131–1165.

Traeger, C. P., 2016. ACE – an analytic climate economy (withtemperature and uncertainty). In:SURED conference 2016.

Trajtenberg, M., Henserson, R., Jaffe, A., 1992. Ivory tower versus corporate lab: An empiricalstudy of basic research and appropriability. NBER working paper 4146.

Unruh, G., 2000. Understanding carbon lock-in. Energy Policy 28 (12), 817–830.US EPA, 2003. Air pollution control technology fact sheet. EPA-452/F-03-034.Van der Ploeg, F., 2011. Natural resources: Curse or blessing? Journal of Economic Literature

Forthcoming.Veblen, T., 1899. The theory of the leisure class: An economic study in the evolution of institu-

tions. Macmillan.Williams, G. C., 1988. Huxley’s evolution and ethics in sociobiological perspective. Zygon 23 (4),

383–407.Yeh, S., Rubin, E. S., 2007. A centurial history of technological change and learning curves for

pulverized coal-fired utility boilers. Energy 32, 1996–2005.