growth: facts and...

Notes on

Growth: Facts and Theories

Intermediate MacroeconomicsSpring 2006

Guido MenzioUniversity of Pennsylvania

Growth

• In the last part of the course we are going to study economic growth, i.e. the secular dynamics of GDP per-capita and the international differences in GDP per-capita

• In particular, we are going to– identify the main stylized facts about economic growth– study three major theories of economic growth and discuss their ability to

match the stylized facts• Malthusian growth theory• Solow growth theory• Endogenous growth theory

• In the end, we won’t be able to provide a satisfactory and unified explanation to all the evidence on growth

Growth Facts

1. Income Inequality

• In 2000….

– 33 percent of the world population lives in countries with a GDP per capita that is less than 1/10 of GDP in the US (Ethiopia 2%, Nigeria 2.3%, Bangladesh 5%, India 8%)

– 65 percent of the world population lives in countries with a GDP per capita that is less than 1/5 of GDP in the US (China 11%, Indonesia 11%, Egypt 12%, Colombia 16%, Thailand 18%)

– 84 percent of the world population lives in countries with a GDP per capita that is less than 1/2 of GDP in the US (Brazil 21%, Mexico 27%, Argentina 32%, Korea 41%)

Income Distribution

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Growth Facts

2. International and Intertemporal Variation in Growth Rates

• There is large variation in growth rates across countries and over time

• In the Nineties– High growth economies: China 7%, Ireland 6%, India 4%, Argentina 4%,

Thailand 3.5% – Low growth economies: Cameroon -1%, Ecuador -0.7%, Venezuela -1%

• From the Fifties to the Nineties– Slowing down economies: Japan, France, Italy, Spain, Brazil, Venezuela,

Nigeria– Accelerating economies: Ireland, Thailand, Korea, China, India, Argentina– Stable growth economies: US, UK, Mexico

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I do not see how one can look at figures like these without seeing them as representing possibilities. Is there some action a government of India could take that would lead the Indian economy to grow like Indonesia’s or Egypt’s? If so, what exactly? If not, what is it about the “nature of India” that makes it so? The consequences for human welfare involved are simply staggering: Once one starts to think about them, it is hard to think about anything else. (Lucas, 1988)

Growth Facts

3. Correlations with per-capita GDP

• Per-capita GDP is

– positively correlated with the investment ratecountries with investment rate that is 1 standard deviations above average, have per-capita GDP that is 0.65 standard deviations above average

– negatively correlated with population growthcountries with investment rate that is 1 standard deviations above average, have per-capita GDP that is 0.61 standard deviations below average

– almost uncorrelated with the growth rate of per-capita GDPcountries with growth rate that is 1 standard deviations above average, have per-capita GDP that is 0.15 standard deviations above average

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Growth Facts

3. Correlations with per-capita GDP growth

• per-capita GDP growth is

– positively correlated with the investment ratecountries with investment rate that is 1 standard deviations above average, have growth rate that is 0.24 standard deviations above average

– negatively correlated with population growthcountries with investment rate that is 1 standard deviations above average, have growth rate that is 0.17 standard deviations below average

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Growth Facts

4. Factor Shares

• (Nicholas Kaldor’s facts) Over the last century in the US and in most other industrialized countries

– output per worker and capital per worker grow over time at relatively constant rates

– the ratio between capital and output is relatively constant over time

– the income share of labor is constant at 2/3 and the income share of capital is constant at 1/3

Growth Facts

5. Historical Perspective

• Before the Industrial Revolution, output per capita differed little over time and across countries.

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Growth Accounting

Solow (1957) has proposed a way of decomposing economic growth into factor growth and technological progress

• Assume that aggregate technology can be represented by the Cobb-Douglass production function

Yt = zt Kta Nt

1-a , 0 < a < 1

• Per-capita GDP is given by yt = Yt / Nt = zt (Kt / Nt) a

• Apply the log operator to both sides of the per-capita production function. Then, we obtain that

(1) log yt = log zt + a log Kt – a log Nt

• Similarly, we can show that(2) log yt+1 = log zt+1 + a log Kt+1 – a log Nt+1

Growth Accounting

• Subtracting (1) from (2), we find that

log yt+1 – log yt = (log zt+1 – log zt) + a ( log Kt+1 – log Kt ) – a (log Nt+1 – log Nt )

• Because of Taylor’s theorem, the difference between the log of xt+1 and the log of xt is two numbers is approximately equal to the growth rate, i.e.

log xt+1 – log xt = (xt+1 – xt ) / xt

• Therefore, the GDP per-capita growth rate can be decomposed as

gy = gz + a [gK – g N]

the growth rate in TFP: gz

a times the difference between the growth rate of capital and labor: gK – g N

Malthusian Growth

• In 1798, Thomas Malthus wrote “An Essay on the Principle of Population,” where he argues that any advances in technology would eventually lead to an increase in the population and a fall in the standards of living towards the subsistence level.

Malthusian Growth

Malthus’ Theoretical Model

• Consider an infinite horizon economy

• In period t, aggregate output is determined by a production function

Yt = zt F(Kt, Nt)such that

– Kt represents land and is in fixed supply, Kt = K*– Nt represents labor and MPN is positive and decreasing in Nt

• In period t, there are Pt households in the economy and each of them– supplies as much labor as possible (L - lt = L) whenever wt>0 [let L = 1]– consumes all of its income ct = wt (L- lt ) + Πt/ Pt

Malthusian Growth


• The market clearing condition in the labor market is

Pt (L- lt) = Nt

• The market clearing condition in the consumption good market is

Ct = ct Pt = zt F(Kt, Pt)

• Finally, it is assumed that consumption per capita determines the growth rate of population. Specifically, there exists an increasing and concave function g(.) such that

Pt+1 = Pt g (Ct / Pt)

Malthusian Growth


• The model assumes that households do not want to save, either because the technology for producing capital goods is very inefficient or because households are very impatient.

• Given the no-savings assumption, we can solve the competitive equilibrium of the infinite horizon economy as a sequence of one-period equilibria that are linked only through the population growth equation.

Malthusian Growth


Solving for the competitive equilibrium in date-t consumption good and labor markets

• Labor market– firm’s labor demand is wt = MP(K*, Nt)– household’s aggregate labor supply is Nt = Pt L = Pt for wt > 0, Nt = 0 otherwise – in equilibrium, supply equals demand and

wt = MP(K*, Pt) > 0Nt = Pt

• Consumption good market– firm’s supply is equal to Yt = zt F(K*, Pt) – aggregate household’s demand is equal to Pt ct

– in equilibrium, supply equals demand and Pt ct = Yt

Malthusian Growth


Solving for the population dynamics

• In period t+1, the number of households in the economy is given by

Pt+1 = Pt g (Ct / Pt)

• In equilibrium, Ct/Pt = ct = zt F (Kt*, Pt) / Pt, i.e. Ct/Pt is the average productivity of labor. Therefore, the equilibrium dynamics of population growth are given by the equation

(1) Pt+1 = Pt g (zt F (K*, Pt) / Pt)

• Assume that the fundamentals of the model are such that the right hand side of (1) is an increasing and concave function of Pt

Population Dynamics

Malthusian Growth


Steady State Analysis

• If zt is constant, the economic system eventually reaches a stable steady state where population, per-capita consumption and per-capita output are all constant over time.

• The stable steady-state level of population P* is the point where the Pt g (z F (K*, Pt) / Pt) function intersects the 45 degree line with a slope smaller than 1.

• At the stable steady-state, P* g (z F (K*, P*) / P*) = P* and therefore

g (z F (K*, P*) / P*) = 1

• At the stable steady-state, consumption and output per-capita are equal to

c* = z F (K*, P*) / P*

Malthusian Growth


The Effects of Technological Progress

• Suppose that a new agricultural technology increases total factor productivity from z to z’, where z’ > z

• On impact, the technological improvement drives up consumption per-capita and leads to an increase in population. Eventually, the economic system reaches a new steady state.

• In the new steady state g (z’ F (K*, P*’) / P*’) = 1– the population P*’ is higher than P* – the consumption per-capita c*’ = z’ F (K*, P*’) / P*’ is equal to c*

• In a Malthusian economy, technological progress immediately leads to higher per-capita, but eventually only leads to higher population.

Adjustment to the Steady State when z Increases

Malthusian Growth


Are the predictions of the Malthusian model correct?

• Before the Industrial Revolution– evidence of significant technological progress– no major improvement in per-capita GDP

• In western countries, after the Industrial Revolution – systematic technological progress– almost constant growth in GDP per-capita

• What did Malthus’ theory miss?– the number of children per household is increasing in income for low starting levels of

income– there is significant accumulation of capital

Solow Growth Model

The Theoretical Model


• In period t, aggregate output is determined by a production function

Yt = zt F(Kt, Nt)such that

– Kt represents capital and MPK is positive and decreasing in Kt

– Nt represents labor and MPN is positive and decreasing in Nt

– there are constant returns to scale

• In period t, there are Pt households in the economy and each of them– supplies as much labor as possible (lt = 0) whenever wt>0 [let L = 1]– consumes a fraction (1 - s) of its gross income ct = (1-s) [wt (L- lt ) + ( Πt+ It ) / Pt]– saves a fraction s of its income

Solow Growth Model


• The market clearing condition in the labor market is

Pt (L- lt) = Nt

• The market clearing condition in the consumption good market is

Ct = ct Pt = zt F(Kt, Pt) - It

• The law of motion for capital accumulation is

Kt+1 = Kt (1 - d) + It

• Finally, it is assumed that the population grows at a constant rate 1 + p, i.e.

Pt+1 = Pt (1 + p)

Solow Growth Model


• The model conjectures that households save a constant fraction of their income. The conjecture is valid when the household’s utility function is

U (ct, ct+1, ….) = log ct + b log ct+1 + ….

and the production function is Cobb-Douglass

• Given the constant savings rate assumption, we can solve for the competitive equilibrium of the infinite horizon economy as a sequence of one-period equilibria that are linked through the capital accumulation and population growth equations.

Solow Growth Model


• Labor market– firm’s labor demand is wt = MP(Kt, Nt)– household’s aggregate labor supply is Nt = Pt L = Pt for wt > 0– in equilibrium, supply equals demand and

wt = MP(Kt, Pt) > 0Nt = Pt

• Consumption good market– firm’s supply is equal to zt F(Kt, Pt) - It

– aggregate household’s demand is equal to Pt ct = Pt (1 - s) (wt + (Πt+ It) / Pt) = (1 - s) zt F(Kt, Pt)

– in equilibrium, supply equals demand and (1 - s) zt F(Kt, Pt) = zt F(Kt, Pt) - It

Solow Growth Model

Solving for the population and capital dynamics


Pt+1 = Pt (1 + p)

• In period t+1, the stock of capital in the economy is given by

Kt+1 = Kt (1 - d) + s zt F(Kt, Pt)

Solow Growth Model

Solving for the population and capital dynamics

• Combining the law of motion for capital and population, we can derive the law of motion for per-capita output kt+1 = Kt+1 / Pt+1. Specifically, we have

Kt+1 / Pt+1 = [Kt / Pt (1 - d) + s zt F(Kt, Pt) / Pt] (Pt / Pt+1)

kt+1 = [kt (1 - d) + s zt F(Kt, Pt) / Pt] / (1 + p)

and using the assumption of constant returns to scale, we finally obtain

(SE) kt+1 = [kt (1 - d) + s zt F(kt , 1)] / (1 + p)

• Remark: the derivative of ztF(k , 1) with respect to k is MPK (k, 1).

The Per-Worker Production Function

Solow Growth Model


• If zt is constant, the economic system eventually reaches a stable steady state where per-capita consumption, per-capita capital and per-capita output are all constant over time.

• The stable steady-state level of per-capita capital is the point k* where the [k (1 - d) + s z F(k,1)] / (1 + p) function intersects the 45 degree line with a slope smaller than 1.

• At the stable steady-state, per-capita GDP and per-capita consumption are respectively given by

y* = z F (k*, 1)

c* = (1-s) z F (k*, 1)

Determination of the Steady State Quantity of Capital per Worker

Solow Growth Model

Comparative Statics #1

• Consider an economy at its stable steady-state of per-capita capital. At date T, the growth rate of population n increases permanently

• The increase in the population growth rate leads to an decrease in the per-capita capital kt+1for every kt, i.e. the function [k (1 - d) + s z F(k,1)] / (1 + p) becomes flatter

• Per-capita capital decreases to the new steady-state level k*’ < k*• Per-capita output decreases to the new steady-state level y*’ < y*• Per-capita consumption decreases. The new steady-state level c*’ is lower than c*.• In the new steady-state aggregate variables grow at a higher rate than before.

• Remark: an increase in the population growth rate leads to lower per-capita income but to faster aggregate growth

Solow Growth Model


• Consider an economy at its stable steady-state of per-capita capital. At date T, the total factor productivity z increases permanently

• The increase in TFP leads to an increase in the per-capita capital kt+1 for every kt, i.e. the function [k (1 - d) + s z F(k,1)] / (1 + p) becomes steeper

• Per-capita capital increases to the new steady-state level k*’ > k*• Per-capita output increases to the new steady-state level y*’ > y*• Per-capita consumption increases to the new steady-state level c*’> c*.• At the new steady-state, aggregate output, consumption and capital grow at rate p

Solow Growth Model


• Consider an economy at its stable steady-state of per-capita capital. At date T, the savings rate s increases permanently

• The increase in the savings rate leads to an increase in the per-capita capital kt+1 for every kt, i.e. the function [k (1 - d) + s z F(k,1)] / (1 + p) becomes steeper

• Per-capita capital increases to the new steady-state level k*’ > k*• Per-capita output increases to the new steady-state level y*’ > y*• At first, per-capita consumption decreases. The new steady-state level c*’ might be higher or

lower than c*.• At the new steady-state, aggregate output, consumption and capital grow at rate n

• Remark an increase in the savings rate leads to faster growth only during the transition phase

Solow Growth Model

The golden rule for capital accumulation

• As we observed in the previous comparative statics exercise, an increase in the savings rate may increase or decrease per-capita consumption in steady-state. This begs the question: what is the savings rate that maximizes the steady-state level of per-capita consumption.

• Formally, we want to solve the maximization problem

max{s,k} (1- s) z F(k, 1) s.t.

(1 + p) k = (1-d) k + s z F(k, 1)

• Using the constraint, we can reformulate the maximization problem as

max{k} z F(k, 1) – (p + d) k

s = (p + d) k / z F(k, 1)

• The optimality condition isMPK = p + d

Steady State Consumption per Worker

The Golden Rule Quantity of Capital per Worker

Solow Growth Model

Predictions of the Solow Growth Model

• International income inequality – countries that have a higher savings rate should be richer, everything else being equal– countries that have a higher population growth should be poorer, everything else being

equal

• Kaldor’s facts– in steady-state per-person capital and per-person output grow at constant rates– in steady-state the output/capital ratio is constant – the labor share of output is constant

• Economic growth– without technological progress, per-capita GDP grows along the transition path towards

steady-state– without technological progress, per-capita GDP cannot grow indefinitely

Advanced Solow Growth Model



• In period t, aggregate output is determined by a Cobb-Douglas production function

Yt = zt F( Kt, Nt)= zt Kta Nt

1-a

such that– Kt represents capital– Nt represents efficiency units of labor– zt represents total factor productivity– At is defined as z t1/(1-a) and represents the labor equivalent of TFP



• In period t, there are Pt households in the economy and each of them

– is endowed with 1 unit of time to be divided between schooling, work and leisure

– spends u units of time in schooling and receives h(u) efficiency units of labor per unit of time

– works 1-u units of time whenever the wage per efficiency unit of labor wt is positive

– consumes a fraction (1 - s) of its gross income ct = (1-s) [wt (1- u - lt) h (u) + ( Πt+ It )/ Pt]



• The law of motion for capital accumulation is Kt+1 = Kt (1 - d) + It

• The law of motion for population growth isPt+1 = Pt (1 + p)

• The law of motion for labor-augmenting technological progress isAt+1 = At (1 + g)



• Labor market– firm’s labor demand of efficiency units of labor is wt = MP(Kt, Nt) = (1-a) At

1-a Kta Nt

-a

– household’s aggregate supply of efficiency units of labor is h (1-u) Pt for wt > 0– in equilibrium, supply equals demand and

wt = (1-a) At1-a Kt

a Nt-a > 0

Nt = h (1-u) Pt

• Consumption good market– firm’s supply is equal to Kt

a (AtNt )1-a- It

– aggregate household’s demand is equal to Pt ct = Pt (1 - s) (wt (1-u) h + (Πt+ It) / Pt) = (1 - s) Kt

a (h (1-u) At Pt )1-a

– in equilibrium, supply equals demand and

It = s Kta (h (1-u) At Pt )1-a


Solving for population, capital and technology dynamics


Pt+1 = Pt (1 + p)

• In period t+1, the labor equivalent of TFP is given by

At+1 = At (1 + g)

• In period t+1, the stock of capital in the economy is given by

Kt+1 = Kt (1 - d) + s Kta (h (1-u) At Pt )1-a


Solving for population, capital and technology dynamics

• Combining the law of motion for capital, population and TFP, we can derive the law of motion for capital per efficiency unit of labor kt = Kt /(At Pt.).

• After some algebraic manipulations, we obtain

(ASE) kt+1 = kt (1 - d) / [(1 + p) (1+g)] + s kta [h (1-u)]1-a / [(1 + p) (1+g)]

• Dynamics of capital per efficiency unit of labor:– if kt (d + p + g + pg) is smaller than s kt

a [h (1-u)]1-a, then capital per-efficiency unit increases, i.e. kt+1 > kt

– if kt (d + p + g + pg) is greater than s kta [h (1-u)]1-a, then capital per-efficiency unit

increases, i.e. kt+1 < kt

– if kt (d + p + g + pg) is equal to s kta [h (1-u)]1-a, then capital per-efficiency unit remains

constant



• The economic system eventually reaches a stable steady state where capital per efficiency unit of labor is constant.

• At the steady-state– per-capita capital grows at the gross rate (1+g) – per-capita output grows at the gross rate (1+g) – per-capita consumption grows at the gross rate (1+g)

• At the steady-state– aggregate capital grows at the gross rate (1+p) (1+g)– aggregate output grows at the gross rate (1+p) (1+g) – aggregate consumption grows at the gross rate (1+p) (1+g)


Comparative Statics

• Savings Rate– a higher savings rate leads to higher capital per efficiency unit of labor in steady-state– a higher savings rate leads to higher per-capita GDP– savings rate does not affect per-capita GDP growth

• Population Growth– higher population growth rate leads to lower steady-state capital-labor ratio– higher population growth rate leads to lower per-capita GDP– population growth does not affect per-capita GDP growth

• Education– higher education leads to higher capital per efficiency unit of labor in steady-state – higher education leads to higher per-capita GDP– education does not affect economic growth in steady-state


Predictions of the Advanced Solow Growth Model

• International income inequality – countries that have a higher savings rate should be richer, everything else being equal– countries that have a higher population growth should be poorer, everything else being

equal– countries that have higher education should be richer, everything else being equal

• Kaldor’s facts– in steady-state per-person capital and per-person output grow at constant rates– in steady-state the output/capital ratio is constant – the labor share of output is constant

• Economic growth– in steady-state, per-capita output grows at (1+g)– while the capital-technology ratio is growing towards the steady-state, per-capita output

grows faster than (1+g)


Testing the Advanced Solow Growth Model

• International income inequality – in country x, assume that the capital share a to 1/3– in country x, let h (x) = exp (0.1 * u (x)), where u (x) is the average number of schooling

years – in country x, let the sum of technological growth and depreciation rate to be g + d = .075– in country x, let p (x) be the population growth rate– estimate the ratio between the steady-state per-capita ratio between country x and the US


Testing the Advanced Solow Growth Model

• International growth inequality

– absolute convergence hypothesis: (Baumol, 86) for countries with the same investment rate, population growth rate and education, the model predicts that growth should be higher the lower is the capital-technology ratio (and therefore output per-capita) with respect to the steady-state level

– conditional convergence hypothesis: (Mankiw Romer & Weil, 91) more generally, the model predicts that growth should be higher the lower is the capital-technology ratio (and therefore output per-capita) with respect to the steady-state level

growth: facts and...

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