CHAPTER SEVEN - Eight Economic Growth 1 The Solow Growth Model is designed to show how: growth in the capital stock, growth in the labor force, and advances in technology interact in an economy, and how they affect a nation s total output of goods and services. How does the model treat the accumulation of capital? 2 1
Supply and demand for goods: How much output is produced at any given time? How is this output allocated among alternative uses? It represents the transformation of inputs (labor (L), capital (K), production technology) into outputs (final goods and services for a certain time period). The algebraic representation is: The Production Function zy = F ( z K, z L ) Income is some function of our given inputs Chapter Key Seven Assumption: The Production Function has constant returns to scale. 3 Let s analyze all quantities relative to the size of the labor force. Let s set z = 1/L. Y/ L = F ( K / L, 1 ) Output the amount of is some function of Per worker capital per worker Constant returns to scale imply: that the size of the economy as measured by the number of workers does not affect the relationship between output per worker and capital per worker. Let s denote all quantities in per worker terms in lower case letters. The production function: where f(k)=f(k,1). y = f ( k ) 4 2
MPK = f (k + 1) f (k) y 1 MPK k f(k) The production function shows: how the amount of capital per worker (k) determines the amount of output per worker y=f(k). The slope of the production function is the marginal product of capital: if k increases by 1 unit, y increases by MPK units. 5 1) y = c + i 2) consumption per worker c = (1-s)y depends on savings rate (between 0 and 1) Output per worker consumption per worker 3) investment per worker y = (1-s)y + i 4) i = s y Investment = savings. The rate of saving s is the fraction of output devoted to investment. 6 3
Here are two forces that influence the capital stock: Investment: expenditure on plant and equipment. Depreciation: wearing out of old capital; causes capital stock to fall. Investment per worker: i = s y. Output: y = f (k), (the production function), thus Investment per worker is a function of the capital stock per worker: i = s f(k) This equation relates the existing stock of capital k to the accumulation of new capital i. 7 The saving rate s determines the allocation of output between consumption and investment. For any level of k, output is f(k), investment is s f(k), and consumption is f(k) sf(k). y y (per worker) Output, f (k) c (per worker) Investment, s f(k) i (per worker) k 8 4
Impact of investment and depreciation on the capital stock: k = i δ k Change in Capital Stock Remember investment equals savings so, it can be written: k = s f(k) δ k Investment (i = s y) δ k Depreciation δ k Depreciation is proportional to the capital stock. k 9 Investment and Depreciation i* = δk* At k*, investment equals depreciation and capital will not change over time. Investment, s f(k) Depreciation, δ k Below k*, investment exceeds depreciation, so the capital stock grows. k 1 k* k 2 Above k*, depreciation exceeds investment, so the capital stock shrinks. Capital per worker, k 10 5
Investment and Depreciation The Solow Model shows that If the saving rate is high (s 2 ), the economy will have a large capital stock and high level of output. If the saving rate is low (s 1 ), the economy will have a small capital stock and a low level of output. Depreciation, δ k i* = δk* Investment, s 2 f(k) Investment, s 1 f(k) An increase in the saving rate causes the capital stock to grow to a new steady state. k 1 * k 2 * Capital per worker, k 11 The steady-state value of k that maximizes consumption is called the Golden Rule Level of Capital. To find the steady-state consumption per worker, the national income accounts identity: y - c + i and rearrange it as: c = y - i. This equation holds that consumption = output - investment. Steady-state output per worker is y = f (k*) where k* is the steady-state capital stock per worker. Because the capital stock is not changing in the steady state, investment is equal to depreciation: i= δ k*. Substituting f (k*) for y and δ k* for i, we can write steady-state consumption per worker as c*= f (k*) - δ k*. 12 6
c*= f (k*) - δ k* Steady-state consumption is what s left of steady-state output after paying for steady-state depreciation. An increase in steady-state capital has two opposing effects on steadystate consumption: 1. More capital means more output. 2. More capital means that more output must be used to replace capital that is wearing out. The economy s output is used for δ k δ k consumption or investment. In the steady state: investment equals depreciation. Output, f(k) Steady-state consumption is maximized at the Golden Rule steady state. c * gold The Golden Rule capital stock is denoted k* gold, and the Golden Rule consumption is c* gold. k* gold k 13 Conditions for the Golden Rule level of capital The slope of the production function is the marginal product of capital MPK. The slope of the δ k* line is δ. These two slopes are equal at k* gold, the Golden Rule can be described by the equation: MPK = δ. At the Golden Rule level of capital, the marginal product of capital equals the depreciation rate. Note: the economy does not automatically gravitate toward the Golden Rule steady state. If we want a particular steady-state capital stock, such as the Golden Rule, we need a particular saving rate to support it. δ k k* gold c * gold δ k Output, f(k) k 14 7
The basic Solow model shows that capital accumulation, alone, cannot explain sustained economic growth: high rates of saving lead to high growth temporarily, but the economy eventually approaches a steady state in which capital and output are constant. Expand the Solow model to incorporate the other two sources of economic growth: population growth technological progress. Assumption: the population and labor force grow at a constant rate n. Population (and labor force) growth means that capital stock per worker shrinks. 15 Investment, break-even investment If n is the rate of population growth and δ is the rate of depreciation, then (δ + n)k is break-even investment, which is the amount necessary to keep constant the capital stock per worker k. Break-even investment, (δ δ + n)k Investment, s f(k) For the economy to be in a steady state investment s f(k) must offset the effects of depreciation and population growth (δ + n)k. An increase in the saving rate causes the capital stock to grow to a new steady state. k* Capital per worker, k 16 8
Investment, break-even investment Let s increase the rate of population growth: from n 1 to n 2 New steady state: lower level of capital per worker, k* 1 >k* 2. Economies with higher rates of population growth will have lower levels of capital per worker and thus lower incomes. (δ δ + n 2 )k (δ δ + n 1 )k k* 2 k* 1 Capital per worker, k Investment, s f(k) An increase in the rate of population growth from n 1 to n 2 reduces the steady-state capital stock from k* 1 to k* 2. 17 In the steady-state: the positive effect of investment on the capital per worker just balances the negative effects of depreciation and population growth. Once the economy is in the steady state, investment has two purposes: 1) Some of it, (δk*), replaces the depreciated capital, 2) The rest, (nk*), provides new workers with the steady state amount of capital. sf (k) Break-even investment,(δ + n') k Break-even Investment,(δ + n) k The Steady State δ k An increase in the rate Investment, s f (k) of growth of population will lower the level of output per worker. k*' k* Capital per worker, k 18 9
The change in the capital stock per worker is: k = i (δ+ δ+n)k Now, let s substitute sf(k) for i: k = sf(k) (δ+ δ+n)k. New investment increases k, whereas depreciation and population growth decrease k. Final Points on Saving In the long run, an economy s saving determines the size of k and y. The higher the rate of saving, the higher the stock of capital and the higher the level of y. An increase in the rate of saving causes a period of rapid growth, but eventually that growth slows as the new steady state is reached. Conclusion: although a high saving rate yields a high steady-state level of output, saving by itself cannot generate persistent economic growth. 19 Adding the Efficiency of Labor "E" The Production Function is now written as: Y = F (K, L E) The term L E measures the number of effective workers. This takes into account the number of workers L and the efficiency of each worker E. Increases in E are like increases in L. 20 10
Technological progress causes E to grow at the rate g, and L grows at rate n so the number of effective workers L E is growing at rate n + g. Now, the change in the capital stock per worker is: k = i (δ+n +g) k, where i = s f(k) s f(k) The Steady State k* Investment, s f(k) Capital per worker, k (δ + n + g) k Note: k = K/(L E) and y=y/(l Ε). So, y=f(k) is now different. Also, when the g term is added, g k is needed to provide capital to new effective workers created by technological progress. 21 Labor-augmenting technological progress at rate g affects the Solow growth model in much the same way as did population growth at rate n. Now that k : = the amount of capital per effective worker, increases in the number of effective workers (due to techn progress) decrease k. In the steady state, investment sf(k) exactly offsets the reductions in k (by depreciation, population growth, and technological progress.) Capital per effective worker is constant in the steady state. So y = f(k) output per effective worker is constant. But the efficiency of each actual worker is growing at rate g. Output per worker, (Y/L = y E) also grows at rate g. Total output Y = y (E L) grows at rate n + g. 22 11
The criterion for the Golden Rule: The Golden Rule level of capital: the steady state that maximizes consumption per effective worker. c*= f (k*) - (δ + n + g) k* Steady-state consumption is maximized if MPK = δ + n + g, rearranging, MPK - δ = n + g. At the Golden Rule level of capital, the net marginal product of capital, MPK δ = the rate of growth of total output, n + g. As actual economies experience both population growth and technological progress, we must use this criterion to evaluate the Golden Rule steady state. Convergence Hypothesis: Among countries that have the same steady state, the convergence hypothesis should hold: poor countries should grow faster on average than rich countries. 23 Solow growth model Steady state Golden rule level of capital Efficiency of labour Labour augmenting technological change 24 12