Worth: Mankiw Economics 5e 182 PART III Growth Theory: The Economy in the Very Long Run The Supply of Goods and the Production Function The supply of goods in the Solow model is based on the now-familiar production function, which states that output depends on the capital stock and the labor force Y=F(K, L) Solow growth model assumes that the production function has constant re to scale. This assumption is often considered realistic, and as we will see shortly, it helps simplify the analysis. Recall that a production function has con- stant returns to scale if for any positive number z. That is, if we multiply both capital and labor by z,we also multiply the amount of output by z Production functions with constant returns to scale allow us to analyze all quantities in the economy relative to the size of the labor force. To see that this is true, set z=1/L in the preceding equation to obtain Y/L= F(K/L, 1) equation shows that the amount of output per worker Y/L is a function he amount of capital per worker K/L.(The number "1"is, of course, constant and thus can be ignored. )The assumption of constant returns to scale implies that the size of the economy-as measured by the number of workers-does not affect the relationship between output per worker and capi tal per worker Because the size of the economy does not matter, it will prove convenient to denote all quantities in per-worker terms. We designate these with lowercase let ters, so y=YL is output per worker, and k= k/L is capital per worker. We can then write the production function as where we define f(l )= F(k, 1). Figure 7-1 illustrates this production function The slope of this production function shows how much extra output a worker produces when given an extra unit of capital. This amount is the marginal prod uct of capital MPK. Mathematically, we write MPK=f(k+ 1)-f(k) Note that in Figure 7-1, as the amount of capital increases, the production func tion becomes fatter, indicating that the production function exhibits diminish- ing marginal product of capital. When k is low, the average worker has only a little capital to work with, so an extra unit of capital is very useful and produces lot of additional output When k is high, the average worker has a lot of capital, so an extra unit increases production only slightly. The Demand for Goods and the Consumption Function The demand for pods in the Solow model comes from consumption and investment. In other User JoENA: Job EFFo1423:6264_ch07: Pg 182: 26799#/eps at 100sl ed,Feb13,20029:484MUser JOEWA:Job EFF01423:6264_ch07:Pg 182:26799#/eps at 100% *26799* Wed, Feb 13, 2002 9:48 AM The Supply of Goods and the Production Function The supply of goods in the Solow model is based on the now-familiar production function, which states that output depends on the capital stock and the labor force: Y = F(K, L). The Solow growth model assumes that the production function has constant re￾turns to scale. This assumption is often considered realistic, and as we will see shortly, it helps simplify the analysis. Recall that a production function has con￾stant returns to scale if zY = F(zK, zL) for any positive number z.That is, if we multiply both capital and labor by z, we also multiply the amount of output by z. Production functions with constant returns to scale allow us to analyze all quantities in the economy relative to the size of the labor force.To see that this is true, set z = 1/L in the preceding equation to obtain Y/L = F(K/L, 1). This equation shows that the amount of output per worker Y/L is a function of the amount of capital per worker K/L. (The number “1” is, of course, constant and thus can be ignored.) The assumption of constant returns to scale implies that the size of the economy—as measured by the number of workers—does not affect the relationship between output per worker and capi￾tal per worker. Because the size of the economy does not matter, it will prove convenient to denote all quantities in per-worker terms.We designate these with lowercase let￾ters, so y = Y/L is output per worker, and k = K/L is capital per worker.We can then write the production function as y = f(k), where we define f(k) = F(k,1). Figure 7-1 illustrates this production function. The slope of this production function shows how much extra output a worker produces when given an extra unit of capital.This amount is the marginal prod￾uct of capital MPK. Mathematically, we write MPK = f(k + 1) − f(k). Note that in Figure 7-1, as the amount of capital increases, the production func￾tion becomes flatter, indicating that the production function exhibits diminish￾ing marginal product of capital. When k is low, the average worker has only a little capital to work with, so an extra unit of capital is very useful and produces a lot of additional output.When k is high, the average worker has a lot of capital, so an extra unit increases production only slightly. The Demand for Goods and the Consumption Function The demand for goods in the Solow model comes from consumption and investment. In other 182 | PART III Growth Theory: The Economy in the Very Long Run