Notes on Growth: Facts and Theories Intermediate Macroeconomics Spring 2006 Guido Menzio University 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 1.2 Percentage of World Population 1 0.8 0.6 0.4 0.2 0 10 20 30 40 50 60 70 80 90 100 110 Relative GDP per capita 2000
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
8 6 4 2 0-2 -4-6 Growth Rates ARG AUS BOL BRA COL EGY ESP ETH FRA GBR IND IRL ITA JPN KEN MAR MEX NGA PAK PER THA TUR USA VEN
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 rate countries 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 growth countries 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 GDP countries with growth rate that is 1 standard deviations above average, have per-capita GDP that is 0.15 standard deviations above average
40000 35000 GDP per capita 90 30000 25000 20000 15000 10000 5000 0 0-5000 10 20 30 40 50 60 70 Investm ent Rate 90
30000 G DP per capita 90 25000 20000 15000 10000 5000 0-3 -2-1 0 1 2 3 4 5 6-5000 -10000 Population Growth 90-00
30000 25000 GDP per capita 90 20000 150 0 0 10 0 0 0 5000 0-10 -5 0 5 10 15 GDP grow th 90-00
Growth Facts 3. Correlations with per-capita GDP growth per-capita GDP growth is positively correlated with the investment rate countries 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 growth countries with investment rate that is 1 standard deviations above average, have growth rate that is 0.17 standard deviations below average
GDP growth 90-00 12 10 8 6 4 2 0-2 -4-6 -8-10 0 5 10 15 20 25 30 35 40 45 Investm ent Rate 90
12 10 GDP growth 90-00 8 6 4 2 0-2 -1-2 0 1 2 3 4 5-4 -6-8 -10 Population growth 90-00
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. World GDP p er-cap ita 4500 4000 3500 3000 2500 2000 1500 1000 500 0 1500 1750 1870 1960 2000
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 Per-capita GDP is given by Y t = z t K ta N t 1-a, 0 < a < 1 y t = Y t / N t = z t (K t / N t ) a Apply the log operator to both sides of the per-capita production function. Then, we obtain that Similarly, we can show that (1) log y t = log z t + a log K t a log N t (2) log y t+1 = log z t+1 + a log K t+1 a log N t+1
Growth Accounting Subtracting (1) from (2), we find that log y t+1 log y t = (log z t+1 log z t ) + a ( log K t+1 log K t ) a (log N t+1 log N t ) Because of Taylor s theorem, the difference between the log of x t+1 and the log of x t is two numbers is approximately equal to the growth rate, i.e. log x t+1 log x t = (x t+1 x t ) / x t Therefore, the GDP per-capita growth rate can be decomposed as g y = g z + a [g K g N ] the growth rate in TFP: g z a times the difference between the growth rate of capital and labor: g K 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 Y t = z t F(K t, N t ) such that K t represents land and is in fixed supply, K t = K* N t represents labor and MP N is positive and decreasing in N t In period t, there are P t households in the economy and each of them supplies as much labor as possible (L - l t = L) whenever w t >0 [let L = 1] consumes all of its income c t = w t (L- l t ) + Π t / P t
Malthusian Growth Malthus Theoretical Model The market clearing condition in the labor market is P t (L- l t ) = N t The market clearing condition in the consumption good market is C t = c t P t = z t F(K t, P t ) 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 P t+1 = P t g (C t / P t )
Malthusian Growth Malthus Theoretical Model 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 Malthus Theoretical Model Solving for the competitive equilibrium in date-t consumption good and labor markets Labor market firm s labor demand is w t = MP(K*, N t ) household s aggregate labor supply is N t = P t L = P t for w t > 0, N t = 0 otherwise in equilibrium, supply equals demand and w t = MP(K*, P t ) > 0 N t = P t Consumption good market firm s supply is equal to Y t = z t F(K*, P t ) aggregate household s demand is equal to P t c t in equilibrium, supply equals demand and P t c t = Y t
Solving for the population dynamics Malthusian Growth Malthus Theoretical Model In period t+1, the number of households in the economy is given by P t+1 = P t g (C t / P t ) In equilibrium, C t /P t = c t = z t F (K t *, P t ) / P t, i.e. C t /P t is the average productivity of labor. Therefore, the equilibrium dynamics of population growth are given by the equation (1) P t+1 = P t g (z t F (K*, P t ) / P t ) Assume that the fundamentals of the model are such that the right hand side of (1) is an increasing and concave function of P t
Population Dynamics
Malthusian Growth Malthus Theoretical Model Steady State Analysis If z t 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 P t g (z F (K*, P t ) / P t ) 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*
The Effects of Technological Progress Malthusian Growth Malthus Theoretical Model 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 Malthus Theoretical Model 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
Consider an infinite horizon economy Solow Growth Model The Theoretical Model In period t, aggregate output is determined by a production function such that Y t = z t F(K t, N t ) K t represents capital and MP K is positive and decreasing in K t N t represents labor and MP N is positive and decreasing in N t there are constant returns to scale In period t, there are P t households in the economy and each of them supplies as much labor as possible (l t = 0) whenever w t >0 [let L = 1] consumes a fraction (1 - s) of its gross income c t = (1-s) [w t (L- l t ) + ( Π t + I t ) / P t ] saves a fraction s of its income
Solow Growth Model The Theoretical Model The market clearing condition in the labor market is P t (L- l t ) = N t The market clearing condition in the consumption good market is C t = c t P t = z t F(K t, P t ) - I t The law of motion for capital accumulation is K t+1 = K t (1 - d) + I t Finally, it is assumed that the population grows at a constant rate 1 + p, i.e. P t+1 = P t (1 + p)
Solow Growth Model The Theoretical Model The model conjectures that households save a constant fraction of their income. The conjecture is valid when the household s utility function is and the production function is Cobb-Douglass U (c t, c t+1,.) = log c t + b log c t+1 +. 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 Solving for the competitive equilibrium in date-t consumption good and labor markets Labor market firm s labor demand is w t = MP(K t, N t ) household s aggregate labor supply is N t = P t L = P t for w t > 0 in equilibrium, supply equals demand and w t = MP(K t, P t ) > 0 N t = P t Consumption good market firm s supply is equal to z t F(K t, P t ) - I t aggregate household s demand is equal to P t c t = P t (1 - s) (w t + (Π t + I t ) / P t ) = (1 - s) z t F(K t, P t ) in equilibrium, supply equals demand and (1 - s) z t F(K t, P t ) = z t F(K t, P t ) - I t
Solow Growth Model Solving for the population and capital dynamics In period t+1, the number of households in the economy is given by P t+1 = P t (1 + p) In period t+1, the stock of capital in the economy is given by K t+1 = K t (1 - d) + s z t F(K t, P t )
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 k t+1 = K t+1 / P t+1. Specifically, we have K t+1 / P t+1 = [K t / P t (1 - d) + s z t F(K t, P t ) / P t ] (P t / P t+1 ) k t+1 = [k t (1 - d) + s z t F(K t, P t ) / P t ] / (1 + p) and using the assumption of constant returns to scale, we finally obtain (SE) k t+1 = [k t (1 - d) + s z t F(k t, 1)] / (1 + p) Remark: the derivative of z t F(k, 1) with respect to k is MP K (k, 1).
The Per-Worker Production Function
Solow Growth Model Steady State Analysis If z t 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 k t+1 for every k t, 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 Comparative Statics #2 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 k t+1 for every k t, 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 Comparative Statics #3 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 k t+1 for every k t, 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
The golden rule for capital accumulation Solow Growth Model 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 is MP K = p + d
Steady State Consumption per Worker
The Golden Rule Quantity of Capital per Worker
Predictions of the Solow Growth Model International income inequality Solow Growth Model 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
Consider an infinite horizon economy Advanced Solow Growth Model The Theoretical Model In period t, aggregate output is determined by a Cobb-Douglas production function Y t = z t F( K t, N t )= z t K ta N 1-a t such that K t represents capital N t represents efficiency units of labor z t represents total factor productivity A t is defined as z 1/(1-a) t and represents the labor equivalent of TFP
Advanced Solow Growth Model The Theoretical Model In period t, there are P t 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 w t is positive consumes a fraction (1 - s) of its gross income c t = (1-s) [w t (1-u -l t ) h (u) + ( Π t + I t )/ P t ]
Advanced Solow Growth Model The Theoretical Model The law of motion for capital accumulation is K t+1 = K t (1 - d) + I t The law of motion for population growth is P t+1 = P t (1 + p) The law of motion for labor-augmenting technological progress is A t+1 = A t (1 + g)
Advanced Solow Growth Model Solving for the competitive equilibrium in date-t consumption good and labor markets Labor market firm s labor demand of efficiency units of labor is w t = MP(K t, N t ) = (1-a) A 1-a t K ta N -a t household s aggregate supply of efficiency units of labor is h (1-u) P t for w t > 0 in equilibrium, supply equals demand and w t = (1-a) A 1-a t K ta N -a t > 0 N t = h (1-u) P t Consumption good market firm s supply is equal to K ta (A t N t ) 1-a -I t aggregate household s demand is equal to P t c t = P t (1 - s) (w t (1-u) h + (Π t + I t ) / P t ) = (1 - s) K ta (h (1-u) A t P t ) 1-a in equilibrium, supply equals demand and I t = s K ta (h (1-u) A t P t ) 1-a
Advanced Solow Growth Model Solving for population, capital and technology dynamics In period t+1, the number of households in the economy is given by P t+1 = P t (1 + p) In period t+1, the labor equivalent of TFP is given by A t+1 = A t (1 + g) In period t+1, the stock of capital in the economy is given by K t+1 = K t (1 - d) + s K ta (h (1-u) A t P t ) 1-a
Advanced Solow Growth Model 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 k t = K t /(A t P t. ). After some algebraic manipulations, we obtain (ASE) k t+1 = k t (1 - d) / [(1 + p) (1+g)] + s k t a [h (1-u)] 1-a / [(1 + p) (1+g)] Dynamics of capital per efficiency unit of labor: if k t (d + p + g + pg) is smaller than s k t a [h (1-u)] 1-a, then capital per-efficiency unit increases, i.e. k t+1 > k t if k t (d + p + g + pg) is greater than s k t a [h (1-u)] 1-a, then capital per-efficiency unit increases, i.e. k t+1 < k t if k t (d + p + g + pg) is equal to s k t a [h (1-u)] 1-a, then capital per-efficiency unit remains constant
Advanced Solow Growth Model Steady State Analysis 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)
Advanced Solow Growth Model 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
Advanced Solow Growth Model 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 Advanced Solow Growth Model 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
Advanced Solow Growth Model 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