Lecture 6: The neo-classical growth model

February 23, 2009

1 The main sources of divergence in GDP percapita across countries

• A first source of difference in GDP per capita is the stock of physicalcapital. Physical capital consists in the all the non-labor inputs. e.gmachines, vehicles, buildings, and other pieces of equipment. For example,in 2000 the average US worker had $148,091 worth of capital to work with;in Mexico in the same year, the capital per worker was $42,991 and in Indiait was $6.270.

−→ Question: where do the differences in physical capital stocks comefrom?

−→ Answer: from differences in investment rates, which themselves comefrom differences in savings (savings rates, initial income)

• Second source of difference in GDP per capita: productivity, i.e how muchoutput is produced with each unit of capital.

1. First source of productivity difference: technology, i.e the amount ofknowledge accessible to producers in different countries. Technologyis acquired through innovations (R&D investments) and imitation oftechnologies from more advanced countries.

2. Second source of productivity difference: efficiency, which relatesmore to the organization of the economy, institutions, and so on.

• Of course, to get a good understanding of the differences in GDP per capitaand of growth rates across countries, one must go further and understandwhat ultimately determines savings, R&D investments, technological dif-fusion, productive efficiency in firms and markets.

2 The neo-classical growth model• his lecture presents a capital-based theory of why countries differ in theirlevels of income per capita, and why less advanced countries may or maynot converge (in per capita GDP) towards more advanced countries.

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2.1 The nature of capital

• Five main characteristics of (physical) capital:

1. Capital is productive: it raises the amount of output a worker canproduce

2. Capital is itself produced. The process of producing capital is calledinvestment. Because it is produced, capital requires sacrifice of someconsumption. In the US, in 2000, 1.76 trillion dollars, i.e 18%, werespent on investment.

3. Capital is ”rival” in its use: only a limited amount of people can usea given piece of capital at a given moment in time; this distinguishescapital from ideas, which in turn are produced through another kindof investment: R&D investment.

4. Capital earns a return from renting it. However some capital goodslike roads and ports are built and owned by governments.

5. Capital depreciates, both due to physical obsolescence and also be-cause the arrival of new technologies make capital goods that embodyold technologies, become obsolete.

2.2 The first equation of the Solow model

• The first equation you already know:

Y = AKαL1−α, (1)

where the left-hand side is the current flow of output, and the right handside is equal to a technology parameter A times the capital contributiontimes the labor contribution.

• Remember that:K ×MPK = αY,

so that

α =K ×MPK

Y

is the capital’s share of output. Figure 3.3 shows that share for a sampleof 53 countries. The average is about 1/3.

2.3 The second equation of the Solow model

• Growth rate in discrete time, between t and t+ 1 :

gt (z) =zt+1 − zt

zt.

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• Growth rate in continuous time:

gt(z) =dz

dt/z =

z

z= bz;

• Second equation of the Solow model:

dK

dt= sY − δK,

that is: net current capital accumulation is equal to total current invest-ment (itself equal to total savings in equilibrium of the goods market)minus total current depreciation of the capital stock.

2.4 Steady state

• Fix labor supply at L = 1. The steady-state level of capital is simplydetermined by:

dK

dt= 0,

or equivalentlysY = sAKα = δK,

or equivalently again:

K = Kss = (sA

δ)

11−α .

• This steady state is stable, in the sense that starting from a level of capitalK < Kss, capital will accumulate until the capital stock reaches its steady-state level K = Kss; similarly, starting from a level of capital K > Kss,capital will decumulate until the capital stock reaches K = Kss.

• To this steady-state level of capital corresponds a steady state level ofoutput:

Y ss = A(Kss)α = A1

1−α (s

δ)

α1−α .

• Remarks:

1. This latter equation provides us with a theory of the sources of (long-run) income differences across countries: in particular (per capita)GDP across two countries may differ either due to differences in pro-ductivity as captured by A, or because of differences in savings orinvestment rate rate as captured by s. Thus, suppose, that we ab-stract from differences in technologies and concentrate in differencesin savings or investment rates, and compare the predicted ratio ofincome per worker in each country to income per worker in the US,to the ratio of actual incomes between that country and the US. Weare very far from the 450 line. In particular, Uganda should have a

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GDP per capita of around 33% of US GDP per capita, however thetrue ratio is only 3%. Why? Maybe Uganda still lies very far fromits steady-state, whereas US lies close to its steady-state.

2. The Solow model predicts no growth of GDP per capita in the long-run, since output per capita converges to a fixed value Y ss. Thereason for this no-growth in steady state result, is the assumption ofdecreasing returns to capital accumulation and therefore of decreas-ing savings to output ratio, whereas depreciation occurs at a constantrate; so, eventually, depreciation catches up with savings.

2.5 Convergence

• The Solow model predicts cross-country convergence: namely, a countryvery far from its steady-state will grow very fast, whereas a country veryclose from its steady-state will grow very slowly. For example, supposethat all countries have the same savings rate, productivity parameter, anddepreciation rate, but the only difference between them is that they startfrom different levels of capital stocks (and therefore from different incomelevels). The growth rate of capital stock is equal to:

bK =dK

dt/K = sAKα−1 − δ,

thus sinceα < 1,

we see that the higher the current level of capital, the lower the growthrate of the capital stock, and therefore the lower the growth rate of outputsince bY = α bK.

Thus the country will lower current level of its capital stock will grow fasterthan the country with the higher level of its current capital stock. Notealso that the Solow model leads to the following additional predictions:

1. if two countries have the same rate of investment but different levelsof income, then the country with lower income will grow faster;

2. if two countries have the same level of income but different rates ofinvestment, then the country with a higher rate of investment willhave higher growth;

3. a country that raises its level of investment will experience an increasein its growth rate of income.

• While the Solow model predicts convergence to the same steady state forall countries or regions with similar savings rates s, depreciation rates δand productivity parameter A, it does not explain why some countriesmanage to converge towards the most advanced countries and why others

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stagnate and do not converge. This phenomenon we refer to as ”clubconvergence”. The Solow model also does not explain why some countriesstarted to grow very fast and then stopped converging.

• While the convergence prediction appears to be (partly) verified by cross-country (or by cross-state data within the US), the prediction of zerolong-run growth is at odds with the evidence that growth in developedcountries has been sustained at 2 to 3% per year. So, how can extend theSolow model to account for positive long-run growth?

1. allow for technical progress in the form of a permanent growth in pro-ductivity A. However, the problem is that we cannot explain how thistechnological progress will be remunerated, especially since the Eulertheorem implies that once you pay capital and labor at their mar-ginal productivities, you simply exhaust total output Y and thereforethere is nothing left to pay ”innovators”.

2. allow for population growth; however this will affect the growth oftotal output, not the steady-state result on output per capita: sup-pose

Y = AKαL1−α,

withL = ent

=⇒y = Akα,

where y = Y/L and k = K/L. Also, one can show that equation (2)implies:

dk

dt= sy − δk.

So, we are back to the previous model, which implies that k convergesto a steady-state level kss and y converges to a steady-state level yss.This implies that in the long run, total GDP Y grows at same rateas population, that is at rate n, but per capita GDP stops growingin the long-run.

2.6 A first glimpse at the AK model

• A first attempt at ”endogeneizing” the long-run growth rate, was to saythat even though individual firms may experience decreasing to capitalaccumulation at an individual level, yet, because of externalities in learn-ing by doing across firms (or externalities in capital accumulation acrossfirms), at the aggregate level the production function might exhibit con-stant returns to capital only, namely:

Y = A0K.

This view of the world identifies knowledge accumulation with capitalaccumulation by all firms simultaneously.

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• Now, let us replace the first equation in the Solow model by this equation,and put it together with the second equation of the Solow model; we have:

dK

dt= sA0K − δK,

and therefore: bK = sA0 − δ.

• We thus get a positive long-run rate of growth, which depends positivelyon the savings rate and negatively on the depreciation rate. However, ifwe restore positive long-run growth, we also lose the convergence result ofthe Solow model since all countries no matter their current capital stockgrowth at the same rate. Therefore less advanced countries can no longercatch up with more advanced countries since they do not grow faster thanmore advanced countries.

• Note that the learning-by-doing externality must be exactly right in orderto obtain an aggregate AK function. More generally, externalities will leadus to an aggregate function of the form:

Y = A0Kβ,

whereβ > α.

If β > 1, then we get explosive growth ( bK increases with K), whereas ifβ < 1 we obtain zero growth in the long-run as we are back to the Solowmodel but with a higher coefficient of capital.

• Early attempts at justifying the AK model on empirical grounds, reliedon the observation that the Solow model with α = 1/3 predicts too fast aspeed of convergence compared to what we seem to observe across coun-tries or across US states. However, next time we shall see that there isa simple way to extend the Solow model so as to deal with this problem:namely, by adding human capital on top on physical capital and labor asa third factor of production.

• In any case, from this subsection we have seen that growth models basedon capital accumulation can not explain convergence and long-run growthsimultaneously. And they do not account for several important aspects ofconvergence/divergence across countries.

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