Chapter 8

Consider a Solow growth model with no population growth. Assume that the aggregate production function has the form [tex]Y = K^{3/1}L^{3/2}[/tex]. People save 20% of their income, so their savings rate [tex]s = 0.20[/tex] and capital depreciates at rate [tex]\delta = 0.10[/tex]. Let per-worker output be [tex]y = \frac{Y}{L}[/tex], and per-worker capital be [tex]k = \frac{K}{L}[/tex].

a. What is the per-worker production function? (Express per-worker output [tex]y[/tex] as a function of per-worker capital [tex]k[/tex].)

b. What is savings per worker if capital per worker today is 2?

c. How much capital depreciates per worker if capital per worker today is 2?

d. What is the net change in capital? Is the economy at the steady-state when per-worker capital is 2? If not, what is the steady-state level of capital?

e. Draw a well-labeled graph of savings per worker and depreciation of capital per worker with per-worker capital [tex]k[/tex] on the x-axis. On the graph, clearly indicate where the steady-state level of capital per worker is.

The per-worker production function [tex]y = k^1/3[/tex]. Savings per worker is 0.20 times per-worker output. Capital depreciation per worker is 0.10 times per-worker capital. The steady-state level of capital is reached when net change in capital is zero. The graph shows the relationship between savings per worker, capital depreciation per worker, and per-worker capital.

Explanation:

a. To find the per-worker production function, we substitute the given aggregate production function Y = K1/3 L2/3 into the equation for per-worker output y = LY. Therefore, y = k1/3.

b. To find savings per worker, we multiply the saving rate (s = 0.20) with per-worker output. Therefore, savings per worker is 0.20 times y.

c. To find capital depreciation per worker, we multiply the depreciation rate (δ = 0.10) with per-worker capital. Therefore, capital depreciation per worker is 0.10 times k.

d. The net change in capital is equal to savings per worker minus capital depreciation per worker. If per-worker capital today is 2, we can use the per-worker production function to find per-worker output (y) and then calculate savings per worker and capital depreciation per worker. The steady-state level of capital is reached when net change in capital is zero.

e. To draw the graph, we plot the savings per worker and capital depreciation per worker against per-worker capital (k) on the x-axis. The steady-state level of capital per worker is where the savings per worker curve intersects with the capital depreciation per worker curve.

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