5.1 Introduction. The Solow Growth Model. Additions / differences with the model: Chapter 5. In this chapter, we learn:

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1 Chapter 5 The Solow Growth Model By Charles I. Jones Additions / differences with the model: Capital stock is no longer exogenous. Capital stock is now endogenized. The accumulation of capital is a possible engine of long-run economic growth. Media Slides Created By Dave Brown Penn State University 5.1 Introduction In this chapter, we learn: How capital accumulates over time. How diminishing MPK explains differences in growth rates across countries. The principle of transition dynamics. The limitations of capital accumulation, and how it leaves a significant part of economic growth unexplained. 5.2 Setting Up the Model Production Start with the previous production model Add an equation describing the accumulation of capital over time. The production function: Cobb-Douglas Constant returns to scale in capital and labor Exponent of one-third on K Variables are time subscripted (t). The Solow Growth Model: Output can be used for consumption or investment. Builds on the production model by adding a theory of capital accumulation Was developed in the mid-1950s by Robert Solow of MIT Was the basis for the Nobel Prize he received in 1987 Consumption Investment Output This is called a resource constraint. Assuming no imports or exports 1

2 Goods invested for the future determines the accumulation of capital. Capital accumulation equation: Next year s capital Capital Accumulation This year s capital Investment rate Case Study: An Example of Capital Accumulation To understand capital accumulation, we must assume the economy begins with a certain amount of capital, K 0. Suppose: The initial amount of capital is 1,000 bushels of corn. The depreciation rate is rate The amount of capital that wears out each period Mathematically must be between 0 and 1 in this setting Often viewed as approximately 10 percent Change in capital stock defined as Labor Thus: To keep things simple, labor demand and supply not included The change in the stock of capital is investment subtracted by the capital that depreciates in production. The amount of labor in the economy is given exogenously at a constant level. 2

3 Investment Farmers eat a fraction of output and invest the rest. Stock A quantity that survives from period to period. tractor, house, factory Therefore: Fraction Invested Flow A quantity that lasts a single period meals consumed, withdrawal from ATM Consumption is the share of output we don t invest. A change in stock is a flow of investment. 5.3 Prices and the Real Interest Rate If we added equations for the wage and rental price, the following would occur: The MPL and the MPK would pin them. Omitting them changes nothing. Case Study: Some Questions about the Solow Model Differences between Solow model and production model in previous chapter: Dynamics of capital accumulation added Left out capital and labor markets, along with their prices Why include the investment share but not the consumption share? No need to it would be redundant Preserve five equations and five unknowns The real interest rate - The amount a person can earn by saving one unit of output for a year - Or, the amount a person must pay to borrow one unit of output for a year - Measured in constant dollars, not in nominal dollars 3

4 Saving The difference between income and consumption Combine the investment allocation and capital accumulation equation. Investment Is equal to investment A unit of investment becomes a unit of capital Substitute the fixed amount of labor into the production function. - The return on saving must equal the rental price of capital. Thus: - The real interest rate equals the rental price of capital which equals the MPK. We have reduced the system into two equations and two unknowns (Y t, K t ). 5.4 Solving the Solow Model The model needs to be solved at every point in time, which cannot be done algebraically. Two ways to make progress Show a graphical solution Solve the model in the long run We can start by combining equations to go as far as we can with algebra. The Solow Diagram Plots the two terms that govern the change in the capital stock New investment looks like the production functions previously graphed but scaled down by the investment rate. 4

5 Output and Consumption in the Solow Diagram As K moves to its steady state by transition dynamics, output will also move to its steady state. Consumption can also be seen in the diagram since it is the difference between output and investment. Using the Solow Diagram If the amount of investment is greater than the amount of depreciation: The capital stock will increase until investment equals depreciation. here, the change in capital is equal to 0 the capital stock will stay at this value of capital forever this is called the steady state If depreciation is greater than investment, the economy converges to the same steady state as above. Notes about the dynamics of the model: When not in the steady state, the economy exhibits a movement of capital toward the steady state. At the rest point of the economy, all endogenous variables are steady. Transition dynamics take the economy from its initial level of capital to the steady state. Solving Mathematically for the Steady State In the steady state, investment equals depreciation. Sub into the production function 5

6 Solve for K* The steady-state level of capital is Positively related with the investment rate the size of the workforce the productivity of the economy Negatively correlated with the depreciation rate 5.5 Looking at Data through the Lens of the Solow Model The Capital-Output Ratio Recall the steady state. The capital to output ratio is the ratio of the investment rate to the depreciation rate: Investment rates vary across countries. It is assumed that the depreciation rate is relatively constant. Plug K* into the production function to get Y*. Plug in our solved value of K*. Higher steady-state production Caused by higher productivity and investment rate Lower steady-state production Caused by faster depreciation Differences in Y/L The Solow model gives more weight to TFP in explaining per capita output than the production model. We can use this formula to understand why some countries are so much richer. Take the ratio of y* for two countries and assume the depreciation rate is the same: From Chapter 4 See figure 5.3 (next slide) Finally, divide both sides of the last equation by labor to get output per person (y) in the steady state. Note the exponent on productivity is different here (3/2) than in the production model (1). Higher productivity has additional effects in the Solow model by leading the economy to accumulate more capital. 6

7 We find that the factor of 108 that separates rich and poor countries income per capita is decomposable: TFP differences Investment differences Empirically, however, economies appear to continue to grow over time. Thus, we see a drawback of the model. According to the model: Capital accumulation is not the engine of long-run economic growth. After we reach the steady state, there is no long-run growth in output. Saving and investment are beneficial in the short-run do not sustain long-run growth due to diminishing returns 5.6 Understanding the Steady State The economy reachs a steady state because investment has diminishing returns. The rate at which production and investment rise is smaller as the capital stock is larger. Also, a constant fraction of the capital stock depreciates every period. is not diminishing as capital increases. Eventually, net investment is zero. The economy rests in steady state. Meanwhile, Back on the Family Farm Harvest starts with a small stock of seed. Grows larger each year, for a time Settles down to a constant level Diminishing returns A fixed number of farmers cannot harvest huge amounts of corn. Growth eventually stops. 5.7 Economic Growth in the Solow Model Important result: there is no long-run economic growth in the Solow model. In the steady state, growth stops, and all of the following are constant: Output Capital Output per person Consumption per person Case Study: Population Growth in the Solow Model Can growth in the labor force lead to overall economic growth? It can in the aggregate. It can t in output per person. The presence of diminishing returns leads capital per person and output per person to approach the steady state. This occurs even with more workers. 7

8 5.8 Some Economic Experiments The Solow model: Does not explain long-run economic growth Does help to explain some differences across countries An Increase in the Investment Rate The capital stock increases by transition dynamics to reach the new steady state this happens because investment exceeds depreciation The new steady state is located to the right investment exceeds depreciation Economists can experiment with the model by changing parameter values. Explain what happens to the economy according to the Solow model when some fundamentals change (Increase/Decrease): Case 1. Saving/Investment rate Case 2. rate Case 3. Capital Case 4. Productivity Present the results using the following: 1. Solow diagram; 2. Dynamics over time graph; 3. In words. Case 1. An Increase in the Saving/Investment Rate Suppose the investment rate increases permanently for exogenous reasons. The investment curve rotates upward The depreciation curve remains unchanged The capital stock increases by transition dynamics to reach the new steady state this happens because investment exceeds depreciation The new steady state is located to the right investment exceeds depreciation What happens to output in response to this increase in the investment rate? The rise in investment leads capital to accumulate over time. This higher capital causes output to rise as well. Output increases from its initial steadystate level Y* to the new steady state Y**. 8

9 What can we say about the rate of growth during the transition to the new steady state, K** and Y**?, Income Y** Y* K* etc. K** After s has increased to s, we are still at K* temporarily K = sy dk is the vertical distance between red and green and is now very large So growth is rapid! As K increases, that distance shrinks, growth slackens What happens to output in response to this increase in the depreciation rate? The decline in capital reduces output. Output declines rapidly at first, and then gradually settles down at its new, lower steady-state level Y**. Case 2. A Rise in the Rate Suppose the depreciation rate is exogenously shocked to a higher rate. The depreciation curve rotates upward The investment curve remains unchanged The capital stock declines by transition dynamics until it reaches the new steady state this happens because depreciation exceeds investment The new steady state is located to the left 9

10 Experiments on Your Own The graph of Yt against time shows a discontinuous jump in Yt brought on by the destruction of Kt Try experimenting with all the parameters in the model: Figure out which curve (if either) shifts. Follow the transition dynamics of the Solow model. Analyze steady-state values of capital (K*), output (Y*), and output per person (y*). Yt Y* Y In the old steady state before the calamity, Yt is constant at Y* t0 At time t0, the destruction reduces Kt, and Yt falls immediately to Y < Y* Growth slackens as we return to the new steady state at Y* Growth, shown by the slope, accelerates during recovery Time, t Case 3. Destruction of capital Case 4. Increase in Productivity Why would we be interested in this application? What are examples of destruction of capital? Important (but tragic) world events involve capital destruction: Warfare destroys productive capital World War II resulted in the virtual obliteration of German and Japanese industrial capacity. And 9/11 resulted in the destruction of a lot of physical capital. (In both cases, the value of human lives lost surely dwarfed the value of lost capital.) Natural disasters do too One recent example in this country is Hurricane Katrina in 2005, which destroyed capital and killed people in New Orleans and along the Gulf Coast How do we think about the destruction of capital in the Solow Model? What does it imply about initial effects and recovery? Try to solve on your own. Will be reviewed in class., Income Y* Y Destruction of capital per se does not change any fundamentals like s or d; it just removes K K Destruction K* Prior to the destruction, we are in steady state at K* and Y* The destruction moves us from K* to K < K* as capital is destroyed But since the fundamentals haven t changed, the steady state is still at K* and Y*, and we transition back toward it Case Study: Wars and Economic Recovery Hiroshima and Nagasaki Returned close to their original economic position in just a few decades Vietnam In both villages that were bombed or left untouched, poverty, literacy, and consumption were similar 30 years after the war. Implications of Solow growth model? 10

11 5.9 The Principle of Transition Dynamics What we ve learned so far 1. Direction of movement the economy will always move toward the steady state. 2. Speed of movement the farther from the study state, the faster the growth The key equations of the Solow Model are these: The production function And the capital accumulation equation How do we solve this model? We graph it, separating the two parts of the capital accumulation equation into two graph elements: saving = investment, and depreciation The Principle of Transition Dynamics If an economy is below steady state It will grow. If an economy is above steady state. Its growth rate will be negative. When graphing this, a ratio scale is used. Allows us to see that output changes more rapidly if we are further from the steady state As the steady state is approached, growth shrinks to zero. The Solow Diagram graphs these two pieces together, with Kt on the x-axis At this point, dkt = syt, so So what? The principle of transition dynamics The farther below its steady state an economy is, (in percentage terms) the faster the economy will grow The farther above its steady state the slower the economy will grow Allows us to understand why economies grow at different rates Suppose the economy starts at this K0: We see that the red line is above the green there: Saving = investment is greater than depreciation So Kt > 0 because Then since Kt > 0, Kt increases from K0 to K1 > K0 K0 K1 11

12 Now imagine if we start at a K0 here K1 K0 There, the green is above the red Saving = investment is now less than depreciation So Kt < 0 because Then since Kt < 0, Kt decreases from K0 to K1 < K0 Understanding Differences in Growth Rates Empirically, for OECD countries, transition dynamics holds: Countries that were poor in 1960 grew quickly. Countries that were relatively rich grew slower. Looking at the world as whole, on average, rich and poor countries grow at the same rate. Two implications of this: most countries have already reached their steady states countries are poor not because of a bad shock, but because they have parameters that yield a lower steady state Transition dynamics: Transitioning from any Kt toward the economy s steady-state K*, Kt = 0 No matter where we start, we ll transition to K*! At this value of K, dkt = syt, so K* We can see what happens to output, Y, and thus to growth if we rescale the vertical axis, Income Y* Saving = investment and depreciation now appear here Now output can be graphed in the space above We still have transition dynamics toward K*, So we also have dynamics toward a steady-state level of income, Y* K* 12

13 Case Study: South Korea and the Philippines South Korea 6 percent per year Increased from 10 percent of U.S. income to 75 percent Philippines 1.6 percent per year Stayed at 10 percent of U.S. income Transition dynamics predicts South Korea must have been far below its steady state. Philippines is already at steady state Strengths and Weaknesses of the Solow Model The strengths of the Solow Model: It provides a theory that determines how rich a country is in the long run. long run = steady state The principle of transition dynamics allows for an understanding of differences in growth rates across countries a country further from the steady state will grow faster Assuming equal depreciation rates The long-run ratio of per capita incomes depends on The ratio of productivities (TFP levels) The ratio of investment rates The weaknesses of the Solow Model: It focuses on investment and capital the much more important factor of TFP is still unexplained It does not explain why different countries have different investment and productivity rates. a more complicated model could endogenize the investment rate The model does not provide a theory of sustained long-run economic growth. Summary The starting point for the Solow model is the production model. The Solow model Adds a theory of capital accumulation. Makes the capital stock an endogenous variable The capital stock today Is the sum of past investments Consists of machines and buildings that were bought over the last several decades 13

14 The goal of the Solow model is to deepen our understanding of economic growth, but in this it s only partially successful. The fact that capital runs into diminishing returns means that the model does not lead to sustained economic growth. The second major accomplishment of the Solow model is the principle of transition dynamics. The farther below its steady state an economy is, the faster it will grow. Transition dynamics Cannot explain long-run growth Provide a nice theory of differences in growth rates across countries. Increases in the investment rate or TFP Increase a country s steady-state position and growth for a number of years As the economy accumulates more capital rises one-for-one Output and therefore investment rise less than one-for-one because of the diminishing marginal product of capital Eventually, the new investment is only just sufficient to offset depreciation. The capital stock ceases to grow. Out capital stock ceases to grow. The economy settles down to a steady state. In general, most poor countries have Low TFP levels Low investment rates, the two key determinants of steady-state incomes If a country maintained good fundamentals but was poor because it had received a bad shock It would grow rapidly This is due to the principle of transition dynamics. The first major accomplishment of the Solow model is that it provides a successful theory of the determination of capital. Predicts that the capital-output ratio is equal to the investment-depreciation ratio Countries with high investment rates Should thus have high capital-output ratios This prediction holds up well in the data. Additional Figures for Worked Exercises 14

15 Additional Solow graph examples from previous editions of slides The Solow Diagram graphs these two pieces together, with K t on the x-axis: At this point, dkt = syt, so This concludes the Lecture Slide Set for Chapter 5 Suppose the economy starts at this K 0 : Macroeconomics Third Edition by Charles I. Jones We see that the red line is above the green at K0 Saving = investment is greater than depreciation So Kt > 0 because Then since Kt > 0, Kt increases from K0 to K1 > K0 W. W. Norton & Company Independent Publishers Since 1923 K0 K1 15

16 Now imagine if we start at a K 0 here: At K0, the green line is above the red line Saving = investment is now less than depreciation So Kt < 0 because Then since Kt < 0, Kt decreases from K0 to K1 < K0 K1 K0 We call this the process of transition dynamics: Transitioning from any K t toward the economy s steady-state K*, where K t = 0 No matter where we start, we ll transition to K*! At this value of K, dkt = syt, so K* We can see what happens to output, Y, and thus to growth if we rescale the vertical axis:, Income Y* Saving = investment and depreciation now appear here Now output can be graphed in the space above in the graph We still have transition dynamics toward K* So we also have dynamics toward a steady-state level of income, Y* K* 16