Economic Growth: The Solow Model

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

SOME EMPIRICAL FACTS Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-3

U.S. Per Capita Income Growth In the United States, growth in per capita income has not strayed far from 2% per year (excepting the Great Depression and World War II) since 1900. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-4

Figure 1 Natural Log of Real Per-Capita Income in the United States, 1869–2005 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-5

Real Per Capita Income and the Investment Rate Across countries, real per capita income and the investment rate are positively correlated. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-6

Figure 2 Real Income Per Capita vs. Investment Rate Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-7

Real per capita income and the rate of population growth Across countries, real per capita income and the population growth rate are negatively correlated. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-8

Figure 3 Real Income Per Capita vs. the Population Growth Rate Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-9

Real per capita income and per capita income growth 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. 6-10

Figure 4 Growth Rate in Per Capita Income vs Figure 4 Growth Rate in Per Capita Income vs. Real Income Per Capita for the Countries of the World Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-11

All Key Facts Together US real p.c. income grew steady at 2% per year Across countries, real per capita income is: positively correlated with investment rate negatively correlated with population growth rate Rich countries don’t grow faster than poor, and vice-versa Rich countries are more alike w.r.t. growth rates than poor ones 6-12

SOLOW GROWTH MODEL Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-13

Solow Growth Model Want a model to capture and explain these facts  Solow model This is a key model which is the basis for the modern theory of economic growth. A key prediction is that technological progress is necessary for sustained increases in standards of living. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-14

Population growth Variables with ′ sign denote the value in the next period (N′ is population next year) In the Solow growth model, population is assumed to grow at a constant rate n. where n > −1 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-15

Consumption-Savings Behavior Consumers are assumed to save a constant fraction s of their income, consuming the rest: If we allow for endogenous savings, model gets much harder, but the predictions do not change a lot Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-16

Representative Firm’s Production Function (P.F.) Firms have a standard production function: Assume it has constant returns to scale Divide both sides by N to get: Will work with a per-capita P.F.: 6-17

Example: Cobb-Douglas P.F. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-18

Evolution of the Capital Stock Future capital equals the capital remaining after depreciation, plus current investment. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-19

Figure 5 The Per-Worker Production Function Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-20

Solving the Solow Model, 1 Consumption market must clear: In equilibrium, future capital equals total savings S (= I ) plus what remains of current K. since S = sY (rest gets consumed) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-21

Solving the Solow Model, 2 Substitute for output from the P.F.: Rewrite in per-worker form: where: Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-22

Solving the Solow Model, 3 Re-arrange to get the main equation: Determines the evolution of per-capita (or per worker) capital in the model Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-23

Example: Cobb-Douglas P.F. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-24

Finding the Steady State Steady state – situation when per worker capital stock does not grow, i.e. k′ = k Think about steady state as the long-run equilibrium level Will discuss several ways of finding it 2 graphical ways and analytic way Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-25

Discuss adjustment dynamics to the steady state Figure 6 Determination of the Steady State Quantity of Capital per Worker Discuss adjustment dynamics to the steady state Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-26

Finding Steady State Level of Capital Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-27

Figure 7 Determination of the Steady State Quantity of Capital per Worker Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-28

Analyzing the Steady State Now, we found k*, what’s its growth rate? Zero, by definition of steady state Hence y* = z f(k*) doesn’t grow either! So do we have a growth model that predicts growth will stop eventually? Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-29

Growth in the Steady State Of course not! Recall that k = K/N and Even though steady-state capital per worker k* doesn’t grow, total capital stock K= k N does! In fact, K and Y (and hence all other aggregate variables) grow at rate n in the steady state! Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-30

RUNNING EXPERIMENTS Solow Model: 6-31 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-31

Increase in the Savings Rate, s Having the steady state, can do experiments Suppose savings rate s goes up. Effects? 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 some intertemporal dynamics Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-32

Figure 8 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. 6-33

Figure 9 Effect of an Increase in the Savings Rate at Time T Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-34

Summary of Effects from Increase in s On per-capita variables (k, y) New steady state level (higher) No change in steady state growth rate (zero) On aggregate variables (K, Y) No change in steady state growth rate (n) What about consumption? That’s what consumers usually care about! Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-35

Figure 10 Steady State Consumption per Worker Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-36

Golden Rule Level of Savings GR level of s is the one that maximizes consumption per worker in the steady state c* and hence aggregate consumption C * To find it, write down the expression for c* : c* = (1­s) z f(k*) And maximize it Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-37

Finding the sGR, 1 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-38

Finding the sGR, 2 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-39

Increase in the Population Growth Rate, n Another experiment – increase the population growth rate n Results: Capital per worker and output per worker decrease Growth rates of aggregate variables go up (since they are equal to n) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-40

Figure 11 Steady State Effects of an Increase in the Labor Force Growth Rate Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-41

Limits to Growth So per capita income y will increase if: savings rate s goes up or if population growth rate n goes down (showed if n goes up, y goes down) But there are limits to impact of these factors! How can we have sustained increases in per- capita income? Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-42

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. 6-43

Figure 12 Increases in Total Factor Productivity in the Solow Growth Model Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-44

CONVERGENCE THEORY Solow Model: 6-45 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-45

Convergence in the Solow Growth Model Is there a tendency for poor countries to catch up with the rich ones? The Solow growth model says “yes”! As always, need some assumptions: Two identical countries (same z, n, s) The “rich” country initially has a higher level of capital per worker k. (Consequently, it also has a higher output per worker y) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-46

Predictions of the Solow Model If two countries are initially rich and poor, but identical in all other respects, they will converge in the long run to the same level and rate of growth of per-capita income. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-47

Figure 13 Rich and Poor Countries and the Steady State Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-48

Figure 14 Convergence in Income per Worker Across Countries in the Solow Growth Model Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-49

Figure 15 Convergence in Aggregate Output Across Countries in the Solow Growth Model Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-50

Convergence in the Data Unfortunately, we don’t see evidence supporting the convergence theory in cross-country data: while there seems to be some convergence among the rich countries there is nothing like this for the poor ones Here are the diagrams: Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-51

Figure 16 Hardly Any Convergence Across Countries Want negative correlation, not sure it’s here 6-52

Figure 17 Some Convergence Across Rich Countries Looks like negative correlation here 6-53

Figure 18 No Convergence Across Poor Countries Almost zero correlation here 6-54

Why No Convergence? If model predictions are wrong, usually this means some assumptions aren’t correct Here, assumed same z for all countries What if countries have access to different technologies? say, due to some government policies that forbid imports of technologies Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-55

Figure 19 Differences in Total Factor Productivity Can Explain Disparity in Income per Worker Across Countries Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-56

What Should Poor Countries Do? Some things could be done, it depends on the government to get things right Promote greater competition among firms: Absence of monopolies creates the incentive to innovate Promote free trade This kind of policy seemed to have worked for Japan after WWII Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-57