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Ex7_Solow_growth_model.pdf

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Department
Economics
Course
ECON 2400
Professor
Wai Ming Ho
Semester
Fall

Description
AP/ECON2400A (Fall 2012) W.Ho Example 7: The Solow Growth Model Consider the Solow growth model. In the current period, the output of an economy is Y = zF(K;N), it depends on the current input of capital, K, the current input of labor, N, and the current total factor productivity, z. The function F(K;N) has the properties of constant returns to scale and decreasing marginal returns to each factor of production. De▯ne the capital per worker as k = K . We can rewrite the production function as y = Y = zf(k). N N The population of the economy grows over time, the population in the future period 0 is given by N = (1 + n)N, where N is the population in the current period, and n > 0 is the growth rate of population, which is assumed to be constant over time. Consumers collectively will receive all current real output Y as income. It is assumed that there is no government sector and no taxes, and that consumers consume a constant fraction of income in each period, C = (1 ▯ s)Y , where C is current consumption and s is the aggregate saving rate. Hence, the capital stock of the economy in the next period is 0 given by K = (1▯d)K +I. d is the depreciation rate of capital, which is exogenously given and constant over time. I is the aggregate investment in the current period, where I = sY . The evolution equation of k is sz f(k) (1 ▯ d)k k = + : 1 + n 1 + n 1. Illustrate how to determine the steady state equilibrium value of capital per worker. 2. Illustrate how to determine the steady state equilibrium value of consumption per worker. 3. Suppose that the economy is initially in its steady state equilibrium. Suppose now that there is a permanent decrease in the population growth rate n. (a) How is the steady state equilibrium value of k a▯ected? Why? C (b) How is the steady state equilibrium value of N a▯ected? Why? Y (c) How is the steady state equilibrium growth rate of N a▯ected? Why? 4. Suppose that the economy is initially in its steady state equilibrium. Suppose now that in the current period, some of the nation’s capital stock is destroyed because of a natural disaster. (a) Determine the long-run e▯ects of this on the quantity of capital per worker, and on output per worker. (b) In the short run, will aggregate output grow at a rate higher or lower than the growth rate of the labor force? Explain your answer. AP/ECON2400A (Fall 2012) W.Ho { Example 7 2 1. (a) In the steady state equilibrium, the capital per worker will be constant, k = 0 ▯ k = k , (See Figure 1). Using the evolution equation of k, we can solve for the steady-state equilibrium value k from the equation, ▯ ▯ ▯ (1 ▯ d)k sz f(k ) ▯ ▯ k = + ) (n + d)k = sz f(k ): 1 + n 1 + n k0 ▯ ▯ 6 zf(k );(n + d)k 45 line 6 ▯ k▯ E▯ 0 szf(k)(1▯d)t y = zf(k ) ▯r k = 1+n+ 1+n k3 ▯ ▯ ▯ k2 ▯ y r (n + d)k ▯ 6 ## ▯ ▯ # ▯ c # s zf(k ) ▯ E # ▯ ▯ ▯ #r? ▯ s y # # ▯ # ▯ # ▯ - k r# - k▯ 0 k1 k2 k3k▯ 0 k▯ Figure 1 Figure 2 (b) Given that the steady-state equilibrium value of k is equal to k , the steady state equilibrium value of y =is equal to y , and the steady state equilibrium value N of c =C is equal to c , where y = zf(k ) and c = (1▯s)y = zf(k )▯(n+d)k▯ N (See Figure 2). (c) The initial steady state equilibrium is at point E, and the steady state equilibrium value of k is given by kNow there is a permanent decrease in the population o growth rate, n. (See ▯gure 3L n < n.) ▯ A decrease in n shifts the k =f(k+ (1▯d)equation upward proportionally. 1+n 1+n ▯ ▯ A decrease in n makes the breakeven investment (n + d)k line atter, while ▯ leaving the actual investment szf(k ) curve unchanged. i. A decrease in n raises the steady state equilibrium value of k. With a lower population growth
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