Problem Set # 3
Chapter 7 #2
a) The production function in the Solow growth model is Y = f(K,L), or expressed in terms of output per worker, y = f(k). If a war reduces the labor force through casualties, the L falls but Capital-labor ratio k = K/L rises. The production function tells us that total output falls because there are fewer workers. Output per worker increases, however, since each worker has more capital.
b) The reduction in the labor force means that the capital stock per worker is higher after the war. Therefore, if the economy were in a steady state prior to the war, then after the war the economy has a capital stock that is higher than the steady-state level. This is shown in the figure below as an increase in capital per worker from k1 to k2. As the economy returns to the steady state, the capital stock per worker falls from k2 back to k1, so output per worker also falls.
y = Y/L
y = f(k)
k = K/L
Chapter 7 #4
Suppose the economy begins with an initial steady-state capital stock below the Golden Rule level. The immediate effect of devoting a larger share of national output to investment is that the economy devotes a smaller share to consumption; that is, “living standards” as measured by consumption fall. The higher investment rate means that the capital stock increases more quickly, so the growth rates of output and output per worker rise. The productivity of workers is the average amount produced by each worker – that is, output per worker. So productivity growth rises. Hence, the immediate effect is that living standards fall but productivity growth rises.
In the new steady state, output grows at rate n+g, while output per worker grows at rate g. This means that in the steady state, productivity growth is independent of the rate of investment. Since we begin with an initial steady-state capital stock below the Golden rule level, the higher investment rate means that the new steady state has a higher level of consumption, so living standards are higher
Thus, an increase in the investment rate increases the productivity growth rate in the short run but has no effect in the long run. Living standards, on the other hand, fall immediately and only rise over time. That is, the quotation emphasizes growth, but not the sacrifice required to achieve it.
Chapter 7 #5
As in the text, let k = K/L stand for capital per unit of labor. The equation for the evolution of k is:
Δk = Savings – (δ + n)k
The book tells us that this economy consumes all wage income and saves all capital income. That is, in this economy, savings only come from the capital income earned by the owners of capital. If capital earns its marginal product (see problem description), then savings equals MPK x k, which is equivalent to saying that each piece of capital per worker k earns a rent r equal to the MPK. We can substitute the above into the original equation to find: Δk = (MPK x k) – (δ + n)k
In the steady state, capital per efficiency unit of capital does not change, so Δk = 0. From the above equation, this tells us that
(MPK x k) = (δ + n)k
MPK = (δ + n)
In this economy’s steady state, the net marginal product of capital, MPK – δ, equals the rate of growth of output, n. But this condition describes the Golden Rule steady state. Hence, we conclude that this economy reaches the Golden Rule level of capital accumulation.
Chapter 8, #2
To solve this problem, it is useful to establish what we know about the U.S. economy: • A Cobb-Douglas production function has the form y = kα , where α is capital’s share of income. The question tells us that α = 0.3, so we k now that the production function is y = k.3
• In the steady state, we know that the growth rate of output equals 3%, so we know that (n+g) = .03
• The depreciation rate δ = .04
• The capital-output ratio K/Y = 2.5. Because k/y = [K/(LxE)]/[Y/(LxE)] = K/Y, we also know that k/y = 2.5. (That is, the...
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