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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). • Output can be used for consumption or investment. • The Solow Growth Model: – 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 Capital Accumulation • Goods invested for the future determines the accumulation of capital. • Capital accumulation equation: Next year’s capital This year’s capital Investment Depreciation rate Case Study: An Example of Capital Accumulation • To understand capital accumulation, we must assume the economy begins with a certain amount of capital, K0. • Suppose: – The initial amount of capital is 1,000 bushels of corn. – The depreciation rate is 0.10. • Depreciation 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 • Thus: • The change in the stock of capital is investment subtracted by the capital that depreciates in production. Labor • To keep things simple, labor demand and supply not included • The amount of labor in the economy is given exogenously at a constant level. 2 Investment • Farmers eat a fraction of output and invest the rest. Fraction Invested • Therefore: – Consumption is the share of output we don’t invest. 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 • Stock – A quantity that survives from period to period. • tractor, house, factory • Flow – A quantity that lasts a single period • meals consumed, withdrawal from ATM • 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. 3 • 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 • Saving – The difference between income and consumption 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. • Combine the investment allocation and capital accumulation equation. Depreciation 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: • We have reduced the system into two equations and two unknowns (Yt, Kt). - The real interest rate equals the rental price of capital which equals the MPK. 4 • 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. • 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. 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. 5 Solving Mathematically for the Steady State • In the steady state, investment equals depreciation. • Sub into the production function • 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. • Solve for K* 5.5 Looking at Data through the Lens of the Solow Model The Capital-Output Ratio • 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 • Plug K* into the production function to get Y*. • Plug in our solved value of K*. • 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. 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: • Higher steady-state production – Caused by higher productivity and investment rate • Lower steady-state production – Caused by faster depreciation From Chapter 4 See figure 5.3 (next slide) 6 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 • Empirically, however, economies appear to continue to grow over time. – Thus, we see a drawback of the model. • According to the model: • We find that the factor of 66 that separates rich and poor countries’ income per capita is decomposable: – TFP differences – Investment differences 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. – Depreciation is not diminishing as capital increases. • Eventually, net investment is zero. – The economy rests in steady state. – 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 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. 7 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. 5.8 Some Economic Experiments • The Solow model: – Does not explain long-run economic growth – Does help to explain some differences across countries Case 1. An Increase in 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 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. Depreciation rate • Case 3. Capital • Case 4. Productivity Present the results using the following: 1. Solow diagram; 2. Dynamics over time graph; 3. In words. 8 • 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**. Case 2. A Rise in the Depreciation 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 What can we say about the rate of growth during the transition to the new steady state, K** and Y**? •After s has Investment, Depreciation, Income increased to s’, we are still at K* temporarily Y** •∆K = sY – dK is the vertical distance between red and green and is now very large Y* • 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**. •So growth is rapid! •As K increases, etc. K* K** that distance shrinks, growth slackens Capital, Kt 9 The graph of Yt against time shows a discontinuous jump in Yt brought on by the destruction of K t Yt In the old steady state before the calamity, Yt is constant at Y* Growth slackens as we return to the new steady state at Y* Y* At time t0, the destruction reduces Kt, and Yt falls immediately to Y’ < Y* Y’ Growth, shown by the slope, accelerates during recovery t0 Case 3. Destruction of capital • • Why would we be interested in this application? What are examples of destruction of capital? Important (but tragic) world events involve capital destruction: Time, t Case 4. Increase in Productivity • • Try to solve on your own. Will be reviewed in class. – 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? Destruction of capital per se does not change any fundamentals like s or d; it just removes K •Prior to the Investment, Depreciation, Income destruction, we are in steady state at K* and Y* Y* •The destruction moves us from K* to K’ < K* as capital is destroyed Y’ Destruction •But since the fundamentals haven’t changed, the steady state is still at K* and Y*, and we transition back toward it K’ K* 5.9 The Principle of Transition Dynamics • 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 Capital, Kt 10 The Principle of Transition Dynamics • If an economy is below steady state – It will grow. The Solow Diagram graphs these two pieces together, with Kt on the x-axis Investment, Depreciation • If an economy is above steady state. – Its growth rate will be negative. So what? • When graphing this, a ratio scale is used. At this point, dKt = sYt, so – 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 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 Capital, Kt Suppose the economy starts at this K0: •We see that the red line is above the green there: Investment, Depreciation •Saving = investment is greater than depreciation •So ∆Kt > 0 because •Then since ∆Kt > 0, Kt increases from K0 to K1 > K0 K0 What we’ve learned so far Capital, Kt K1 Now imagine if we start at a K0 here Investment, Depreciation •There, the green is above the red •Saving = investment is now less than depreciation • The key equations of the Solow Model are these: – The production function – And the capital accumulation equation •So ∆Kt < 0 because • 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 •Then since ∆Kt < 0, Kt decreases from K0 to K1 < K0 K1 K0 Capital, Kt 11 Transition dynamics: Transitioning from any Kt toward the economy’s steady-state K*, ∆Kt = 0 Investment, Depreciation No matter where we start, we’ll transition to K*! At this value of K, dKt = sYt, so K* Capital, Kt We can see what happens to output, Y, and thus to growth if we rescale the vertical axis •Saving = investment Investment, Depreciation, Income and depreciation now appear here •Now output can be Y* 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* Capital, Kt 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 Case Study: South Korea and the Philippines • South Korea – 6 percent per year – Increased from 15 percent of U.S. income to 75 percent • Philippines – 1.7 percent per year – Stayed at 15 percent of U.S. income • Transition dynamics predicts – South Korea must have been far below its steady state. – Philippines is already at steady state. 12 • 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 5.10 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 • 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. 13 • As the economy accumulates more capital – Depreciation 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. • 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. • 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. This concludes the Lecture Slide Set for Chapter 5 Macroeconomics Second Edition by Charles I. Jones W. W. Norton & Company Independent Publishers Since 1923 • 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 14