Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Chapter 6 Economic Growth: Solow Model

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-2 Chapter 6 Topics Economic growth facts Solow growth model Growth accounting

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-3 Growth Facts Before the Industrial Revolution (about 1800), standards of living differed little over time and across countries. Since the Industrial Revolution, per-capita income growth has been sustained in the riches countries. Ex: in the United States, average annual growth in per- capita income has been about 2% (excepting the Great Depression and World War II) since 1900.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-4 Figure 6.1 Natural Log of Real Per- Capita Income in the United States, 1869–2005 A straight line is a good fit to the natural log of per-capita income over most of the period

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-5 Facts – cont’ Across countries, real per-capita income and the investment rate are positively correlated. Across countries, real per-capita income and the population growth rate are negatively correlated. Difference in per-capita incomes increased dramatically among countries of the world between 1800 and 1950, with the gap widening between rich (like OECD) countries and the rest of world. –Divergence between richest and poorest countries.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-6 Real per capita income and per capita income growth There is essentially no correlation across countries between the level of output per capita in 1960 and the average rate of growth in output per capita for the years –No correlation between initial wealth and average growth in per-capita income. –Ex: there is no tendency for rich countries to grow faster than poor countries, and vice-versa. Rich countries are more alike in terms of rates of growth than are poor countries.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-7 Figure 6.4 Growth Rate in Per Capita Income vs. Real Income Per Capita for the Countries of the World 1.Over the period of , one did not see that poor countries grow faster than rich ones. (No negative relationship) 2. There is much wider vertical scatter in the points on the left-hand side than on the right-hand side. (Much more disparity in growth rates among poor countries than rich ones)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-8 Solow Growth Model (Exogenous Growth) This is a key model which is the basis for the modern theory of economic growth. It provides interesting answers to following questions: –Source of economic growth –Why living standards rise over time –What happens to the level and growth rate of income when saving rate or population growth rate changes –What happens to the living standards across countries over time?

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-9 Equation 6.9: Population growth In the Solow growth model, population is assumed to grow at a constant rate n. Two periods: current and future periods. Exogenous population growth. n>0, positive growth; n<0, negative growth.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Labor Supply Every consumer is endowed with one unit of time in each period. Consumers do not value leisure. So only consumption goods are valued in utility function. Consumers supply labor inelastically. That is, in each period, they allocate all time endowment in work. Hence, the population equals labor force. N is the number of labor force and n is the growth rate of labor force.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 6.10: Consumption- Savings Behavior Consumers are assumed to save a constant fraction s of their income, consuming the rest. Y: all income available to consumers No government, so no taxes. Consumption-saving problem.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 6.11: Representative firm’s production function Constant returns to scale in K, N

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 6.12 Constant returns to scale implies: Define y=Y/N, and k=K/N. y= z f ( k ), where f ( k ) = F( k, 1 )

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 6.12 The Per-Worker Production Function 1. Slope of prod. Function is MP k. 2. Since MP k is diminishing in k, Production function is concave in k.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 6.13: Evolution of the capital stock Future capital equals the capital remaining after depreciation, plus current investment.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Competitive Equilibrium Two markets: –Labor market: trade consumption goods for labor –Capital market: trade consumption goods for capital Markets clear in CE –Wage is adjusted so that firms are willing to hire N workers (fixed labor supply). –S = I (capital supply = capital demand).

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 6.14: Income- Expenditure Identity The income expenditure identity holds as an equilibrium condition. S = I (capital market clears), and S = Y – C (consumer’s budget)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 6.15 In equilibrium, future capital equals total savings (= I ) plus what remains of current K.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 6.16 Substitute for output from the production function.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 6.17 Rewrite in per-worker form.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 6.18 Re-arrange, to get:

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 6.13 Determination of the Steady State Quantity of Capital per Worker

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 6.19 Equation determining the steady state quantity of capital per worker, k*:

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 6.14 Determination of the Steady State Quantity of Capital per Worker – Solow Diagram

Copyright © 2008 Pearson Addison-Wesley. All rights reserved An increase in the savings rate, s In the steady state, this increases capital per worker and real output per capita. In the steady state, there is no effect on the growth rates of aggregate variables.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 6.15 Effect of an Increase in the Savings Rate on the Steady State Quantity of Capital per Worker

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 6.2 Real Income Per Capita vs. Investment Rate

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Transitional Dynamics

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 6.16 Effect of an Increase in the Savings Rate at Time T

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 6.17 Steady State Consumption per Worker

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 6.18 The Golden Rule Quantity of Capital per Worker

Copyright © 2008 Pearson Addison-Wesley. All rights reserved An increase in the population growth rate, n Capital per worker and output per worker decrease. There is no effect on the growth rates of aggregate variables.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 6.19 Steady State Effects of an Increase in the Labor Force Growth Rate

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 6.3 Real Income Per Capita vs. the Population Growth Rate

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Increases in Total Factor Productivity, z Sustained increases in z cause sustained increases in per capita income.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 6.20 Increases in Total Factor Productivity in the Solow Growth Model

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Convergence For countries that have identical total factor productivity z, labor force growth rate n, and saving rate s, poor country (with low k) grows faster than rich country (with high k). In the long run, both countries will converge to the same level of capital per worker and output per worker.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Transitional Dynamics

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Empirical Evidence

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Empirical Evidence

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Conditional Convergence The further an economy is “below” its steady state, the faster the economy should grow. The further an economy is “above” its steady state, the slower the economy grows.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Empirical Evidence

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Growth Accounting An approach that uses the production function and measurements of aggregate inputs and outputs to attribute economic growth to: (i) growth in factor inputs; (ii) total factor productivity growth.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 6.20: Cobb-Douglas Production Function

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 6.21 A labor share in national income of 64% gives:

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 6.22 The Solow residual is calculated as:

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 6.21 Natural Log of the Solow Residual, 1948–2005

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Table 6.1 Average Annual Growth Rates in the Solow Residual

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Table 6.2 Measured GDP, Capital Stock, Employment, and Solow Residual

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Table 6.3 Average Annual Growth Rates

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Table 6.4 East Asian Growth Miracles (Average Annual Growth Rates)