Aggregate Growth Models Economics 448walker/wp/wp-content/... · growth) savings doesnotproduce long{run growth. I In the long run income per capita is constant and equal to the steady

Aggregate Growth ModelsEconomics 448

J.R. WalkerUW-Madison

Spring 2014

UW-Madison Agg Growth

Sources of Economic Growth

1. Trade. Ricardo comparative advantage. Smith division oflabor.

2. Economies of scale. Increasing returns to scale; DOL.

3. Investment: Increase capital per worker.

4. Technological Change: Knowledge.


Stocks and Flows

I Absolutely critical to keep stocks and flows distinct.

I An asset (physical or financial) is a stock; exists at point intime.

I Income is a flow. Amount per unit of time. Time period mustbe defined. E.g., Annual earnings, or wage per hour.

I Flows can be accumulated into stocks.

I Stocks can be deaccumulated to produce flows. E.g., Live offsavings for a month.


Main Message

1. Investment (increase capital per worker) increases income percapita. Accumulation increases income, but does not producesustained economic growth.

2. Technical change, innovation, produces sustained economicgrowth.

3. That’s the message; make sure you understand why (1) and(2) are correct.


Difference between Accumulation and Innovation

I Increasing the number of shovels from 1 to 3 is accumulation.

I Shifting from a feathered quill (as a writing instrument) tometal quill is innovation.

I Buying 10 white cotton t–shirts is accumulation.

I Replacing a white cotton t–shirt by a Under Armourmicrofiber shirt is technological change.

I Common for technical change to be embodied in the machine.Ford Model–S (1905–1909) gave way to Ford Model–T(1908–1927).


Standard Notation

I Discretize time, t = 0, 1, 2, 3, . . .

I Capitals denote aggregate variables.I Y total outputI K total capital stockI S total savingsI I total investment

I Lower case denotes per capita variablesI y = Y (t)/P(t)I k = K (t)/P(t)I etc.


Aggregate Models

I Construct analytical framework, similar and indeed simpler insome ways than Findlay’s model.

I Findlay: 3 countries, study connections among.

I Solow: One country, investigate determinants of income percapita over time.

I Hence, Will think of economy in the aggregate.I National Income and Product Accounts. (I = S)I Conceptualize economy as production function Y = F (K , L).


Realistic versus Relevant Models

I Will study Solow model of economic growth. It is the centerof the universe for economic growth models.

I Will see that Solow’s model is simple yet it remains highlyrelevant for economic growth.

I Its simplicity means that it is not realistic. To say it leaves outa lot is an understatement.

I We will use the Solow model as our trusted guided throughthe land of growth and development economics.


Review: Macroeconomic Balance

I In its simplest terms, accumulation is the result of abstentionfrom current consumption.

I Commodity production creates income which creates demandfor those very commodities.

I Two groups of commodities:

1. consumption goods produced to satisfy human wants.2. capital goods produced for the purpose of producing other

commodities.

I Generally, households buy consumption goods; firms buycapital goods.

I Households need not spend all their income. HH savings usedto finance purchase of capital goods.

I Macroeconomic balance Investment demand equals thesavings of households, I = S .


Accumulation

Focused on the accumulation of (physical) capital (K ).

Accumulation occurs ∆K = K (t + 1)− K (t) > 0,


Basic Ingredients and Approach

I To determine equilibrium quantities: Y ,K , L,S , I .

I Have economy represented by production function

I Combine macroeconomic balance S = I .

I And, equation for the change in capital stock.

I 3 equations, five unknowns =⇒ need to make someassumptions.


Solow I

I Assume savings is a constant fraction of Income

S(t) = sY (t)

I Capital Stock next period:

K (t + 1) = (1− δ)K (t) + I (t),

where δ equals depreciations.

I Assume Population grows at constant rate n

P(t + 1) = (1 + n)P(t)


Solow Model II

I Equilibrium: I (t) = S(t).

I Return to equation for capital stock, and substitute foraggregate Investment I (t) = S(t) = sY (t) to yield:

K (t + 1) = (1− δ)K (t) + sY (t)

I Divide by Population P(t) to place equation on per capitaterms

K (t + 1)

P(t)=(1− δ)

K (t)

P(t)+ s

Y (t)

P(t)

=(1− δ)k(t) + sy(t)


Solow Model III

I Realize that K(t+1)P(t) 6= k(t + 1)

I Rather k(t + 1) = K(t+1)P(t+1)

I So, multiply LHS by one, written as 1+n1+n .

(1 + n)K (t + 1)

(1 + n)P(t + 1)=(1− δ)k(t) + sy(t)

(1 + n)k(t + 1) =(1− δ)k(t) + sy(t) (3.9)


Interpretation

I Right hand side (RHS) two parts depreciated per capitacapital and current per capita savings.

I If n = 0 (no pop growth) together the two pieces give us thenew per capita capital stock.

I Population growth, n > 0 puts downward drag on per capitacapital stock; spread stock over larger population.

I The larger the rate of population growth, the lower is the percapita capital stock the next period.


Figure 3.3

0

0

Capital per Capita (k)

Out

put p

er C

apita

l

Output--Capital Ratios

Production function y=f(k)


Figure 3.4

k

k

(1+n)k

(1-∆)k+sy

k*k(0)


Steady State

1. if k(0) < k∗

2. if k(0) > k∗


Long Run Growth in Solow Model?

I In Solow model the savings rate determines level ofsteady–state income per capita but does not producesustained (long–run) economic growth.

I Accumulation produces higher (average) income, but notpersistent growth.


Population Growth

Higher population growth, lowers the steady–state level of percapita income.

But the total income must grow faster as a result.

Economy converges to a SS level of per capita income, which isimpossible unless long–run growth of total income equals the rateof population growth.

Labor is both an input in production and a consumer of finalgoods. First raise total output and drives higher rate of growth oftotal income; second lowers savings and investment and bringsdown the SS level of per capita income.


Summary Solow Model [Pop Growth]

I The simplest Solow model (i.e., with exogenous populationgrowth) savings does not produce long–run growth.

I In the long run income per capita is constant and equal to thesteady state value. Hence, need to extend the model togenerate long–run income growth as observed for the last twohundred (or so) years.

I We know from Global Economic History sustained economicgrowth requires technical progress, ∆k(t) > 0.


Incorporate Technical Progress into Solow Model

I Distinguish between accumulation (k∗ ↑) and innovation.

I We’ve seen that accumulation is not sufficient to generateeconomic growth in the presence of diminishing marginalproductivity. Need something to offset.

I Can think of k as physical capital stock (machines) whiletechnical progress is better and more advanced methods ofproduction. Knowledge.

I Increase in knowledge can offset diminishing marginal returnsto production. If so, economic growth (y) can increaseindefinitely.


Exogenous Technical Progress

I Assume that technical progress contributes to efficiency or(economic) productivity of labor.

I Make distinction now between working population P(t) andeffective population, L(t).

L(t) = E (t)P(t)

where E (t) is a scale of efficiency units that translatesworking population into units of effective population

I Thus with an increase in knowledge the population can bemore efficient and thus represent a larger stock of labor.

I Assume innovation occurs at rate π each year.

E (t + 1) = (1 + π)E (t)

where, π is the rate of technical progress.


Labor Saving Technical Progress

I Equation (3.8) of accumulation remains unchanged (caps),

K (t + 1) = (1− δ)K (t) + sY (t)

I Before divided by P(t) to express in per capita terms. Nowdivide by effective population E (t)P(t), multiply by 1 (thistime as 1+π

1+π )

(1 + n)(1 + π)k(t + 1) = (1− δ)k(t) + sy(t)

I where the carrot x above a variable means per effectivepopulation.


Steady State

Figure 3.6Looks like Fig 3.4 with carrots


Steady State

I Same logic as before (population growth) applies.I Convince yourself

1. That k∗ is the Steady State. What is the economicinterpretation?

2. That the capital per efficiency unit converges to a stationarysteady state (k∗(t + 1) = k∗(t)). But the per capita capitalstock (k∗) increases. Indeed, the long–run increase in percapita income takes place precisely at the rate of technicalprogress!


Message: Solow Growth Model

Solow model with technological progress yields sustained percapita growth of capital and income.


Empirical Evidence: Solow Model

I The empirical tests of the Solow model center on testingconvergence.

I As you might expect, convergence comes in two forms:

1. Unconditional2. Conditional (on savings and population growth rates)


Unconditional Convergence

This is the strongest prediction (with the fewest assumptions) andthe easiest to refute.

Suppose that countries, in the long run, have no tendency todisplay differences in the rates of technical progress savings,population growth, and capital depreciation.

The Solow model predicts then in all countries, capital per capitaconverges to the common value k∗, and this happens regardless ofthe initial state of each economy, as measured by their startinglevels of per capita income (or equivalently per capita capitalstock).


Meaning of Unconditional Convergence

I If the parameters governing the evolution of the economy aresimilar, then history in the sense of different initial conditionsdoes not matter.

I Initial conditions is not some long ago level, but rather k(0) isthe level of the per capita capital stock that we can firstreasonably measure. In the long run, the starting point of theprocess does not matter. All possible histories converge at thesteady state k∗.

I If empirically true this would be huge.


Illustration of Unconditional Convergence

Time

Log

Per

Capi

ta In

com

e

A

B

C


Assessment

The model prediction of unconditional convergence soundlyrejected by the data.

With free trade and the open exchange of ideas there are reasonsto believe the rate of technological change should be the sameacross countries.

Yet, not obvious why countries have the same rate of populationgrowth or saving level.

These considerations lead to the notion of conditional convergence.


Conditional Convergence

Unconditional Convergence: assumes that across all countries, thelevel of technical knowledge (and its change), rate of savings, rateof population growth, and the rate of depreciation are the same.

Countries differ in most if not all factors.

Gives rise to the notion of conditional convergence: the growthrate of per capita income will be the same (in the long–run).

Assume that knowledge flows freely across countries.

We allow other parameters such as the rate of population growthand rate of savings to differ across countries.


Assessment

Many other studies obtain results similar to MRW (1992)

Some evidence (at least in terms of direction) in support of Solowgrowth model.

But we can’t rest assuming that savings and population should beequal and opposite in magnitude. And find consistently that thisassumption is false.

We can assume the problem away and say that differences are dueto preferences, to save or procreate, or perhaps differences due toculture of social differences. Empty.

What are the economic incentives and determines for savings ratesand population growth rates to have different effects acrosscountries?


T.W.Schultz and Human Capital

T. W. Schultz pioneered the idea of “human capital” investment inhuman beings.

I Interestingly, the importance of human capital (late 1940s)came to him as he realized that models of economic growthdidn’t explain differences in per capita income (acrosscountries). The view of labor was limited and considered(following A. Marshall) that labor only in terms of quantity.

I Schultz recognized the diversity of workers. Obvious now, butat that time “labor” was just a lump, a homogenous factorinput.


Human Capital

Any form of investment, embodied in people.

I Schooling

I Training programs

I Experience (on the job training)

I Health

I Migration – an investment to leave a poor labor market andmove to a good labor market. Pay fixed cost today for higherwages, earnings “tomorrow”.

I Premarket and pre schooling investments by parents (e.g.,child care, Head Start). The Rage today.


Growth & Development Accounting

Will discuss the nature of productivity first and then (briefly)discuss how it is measured.

Four key questions:

1. How much does productivity vary among countries?

2. How much of the variation in the income per capita amongcountries is explained by productivity differences?

3. How much does productivity growth differ among countries?

4. How much variation in growth rates among countries isexplained by variation in productivity growth, and how muchby variation in factor accumulation?


Define Terms:

Same basic idea used in growth and development accounting.

Growth Accounting used with time series data (e.g., annualinformation on a single country).

Development Accounting used to compare two countries at thesame point in time. Typically use cross–sectional data (oncountries, geographical regions).


Basic Idea

Production is composed of two parts:

Output = Productivity× Factors of Production

Does the USA produces more than UK because of (a) greaterproductivity; (b) accumulated more factors (physical & humancapital) or (c) both?

In comparing two countries will want to decompose differences inoutput into differences in productivity and differences inaccumulation; factors of production.

Make comparison for any set of countries.


Growth Acct, with Cobb Douglas PF

Use Cobb Douglas (per capita) production function :

y(t) = A(t)k(t)αh(t)1−α

where A(t) is a general productivity term

k(t)αh1−α composite term of two factors (physical & humancapital)


Development Accounting

Start with basic idea:

Output = Productivity× Factors of production

Assume each country i = 1, 2 has Cobb–Douglas productionfunction

drop time subscript as doing calculation at the same t

Yi = AiKαi N

(1−α)i

where Ni is working population or human capital in country i

Ai measure of productivityKαi N

1−αi composition factor of production


Development Accounting (cont)

Divide p.f of country 1 by p.f. country 2:

y1

y2=

A1Kα1 N

1−α1

A2Kα2 N

1−α2

y1

y2=

[A1

A2

](Kα

1 N1−α1

Kα2 N

1−α2

)

Q = P × F

or

P =QF

=y1/y2

Kα11 N1−α

1 /Kα22 N1−α2

2


Example:

Table : Data to Compare Productivity

Country y k h

1 24 27 82 1 1 1


Example - Calculation

Assume that countries have same technology with income share ofcapital α = 1/3 and 1− α = 2/3 the income share of humancapital.

A1

A2=

241

271/3×82/3

11/3×12/3

=24

3×41

= 2.

Hence, Country 1 has twice the productivity of Country 2.


Developmental Growth Accounting (2005)

Output Phys Human Factors Productivity

Country Y /P K/P h/P k1/3h2/3 A

USA 1.00 1.00 1.00 1.00 1.00Norway 0.92 1.08 0.97 1.01 0.92UK 0.76 0.69 0.97 0.87 0.87Canada 0.75 0.86 1.01 0.96 0.79Japan 0.69 1.10 0.99 1.02 0.67S.Korea 0.54 0.73 0.93 0.86 0.63Mexico 0.29 0.27 0.79 0.56 0.52Peru 0.14 0.12 0.82 0.44 0.32India 0.13 0.10 0.74 0.38 0.35Cameroon 0.13 0.036 0.58 0.23 0.44Zambia 0.034 0.032 0.65 0.24 0.14

Source: Weil (2009) Economic Growth, 2nd Ed. p.193


Developmental Growth Accounting (2009)

Output Phys Human Factors Productivity

Country Y /P K/P h/P k1/3h2/3 A

USA 1.00 1.00 1.00 1.00 1.00Norway 0.92 1.32 0.98 1.08 1.04UK 0.82 0.68 0.87 0.80 1.03Canada 0.80 0.81 0.96 0.91 0.88Japan 0.73 1.16 0.98 1.04 0.70S.Korea 0.62 0.92 0.98 0.96 0.64Turkey 0.37 0.28 0.78 0.55 0.68Mexico 0.35 0.33 0.84 0.61 0.56Brazil 0.20 0.19 0.78 0.48 0.42India 0.10 0.089 0.66 0.34 0.31Kenya 0.032 0.022 0.73 0.23 0.14Malawi 0.018 0.029 0.57 0.21 0.087

Source: Weil (2012) Economic Growth, 3rd Ed. p.186


Growth Accounting

Now want to make comparison over time to compare rates ofchange of output, factor accumulation and productivity.Once again start with the per capita Cobb–Douglas productionfunction

y(t) = A(t)kα(t)h(t)1−α

Take logs to yield:

ln y(t) = lnA(t) + α ln k(t) + (1− α) ln h(t)


Time Derivatives of ln(z(t))

Recall that the time derivative of ln(z(t)) is:

d ln(z(t)

dt=

1

z

dz

dt


Growth Accounting

Take derivative w.r.t. time t

1

y

dy

dt=

1

A

dA

dt+ α

1

k

dk

dt+ (1− α)

1

h

dh

dt

Represent time derivative by a dot above the variable, z = dzdt .

Use “carrot” to denote a percent change zz = z .

y = A + αk + (1− α)h

Recall that α is the income share of capital while 1− α is theincome share of human capital.


Growth Accounting

Thus the rate of growth of output is the sum of productivitygrowth and the share weight sum the growth of factors ofproduction.

We observe: y , k , h. Requires effort and much attention to detail.Calculation where the devil is in the details.

Direct measurement of the rate of growth of productivity is notcredible. (You could try, but no matter the estimate, no one wouldbelieve it.)

Hence, “measure” growth rate of productivity as residual

A = y − αk − (1− α)h


Growth Accounting

The above formulation assumes data on education (to measureHC) is available.

Show for yourself that if the production function is:

Y (t) = A(t)K (t)αP(t)1−α

then the growth accounting equation is:

y = αk + (1− α)P + A


Comparison with Textbook

y = αk + (1− α)P + A

Textbook:

∆Y (t)

Y (t)= σk(t)

∆K (t)

K (t)+ σP(t)

∆P(t)

P(t)+ TFPG(t)

TFPG = A

Ray’s formulation allows income shares of capital and labor to varyover time.


Comments on TFP

Important P(t) should be the working population. Sometimes wellapproximated by total population, sometimes times not .

Total population not accurate for labor force if major changes inlabor force composition (entry by women, or longer schoolingperiod or declining retirement age).


TFP Growth

I Units of A are arbitrary so level of A is meaningless. What’simportant is the rate of change of TFP.

I Assumed production function exhibits constant returns toscale. where assumed?

I If production function exhibits increasing return to scale theobserved factor shares underestimate the true productivity offactors. Which implies we overestimate the rate of technicalprogress.


Documents

Aggregate Growth Models Economics 448walker/wp/wp-content/... · growth) savings doesnotproduce long{run growth. I In the long run income per capita is constant and equal to the steady