Imagine an economy with production function Y = F(K)= and 400 units of capital.If the fraction of output invested in new capital increases from $\gamma$ = 0.2 to $\gamma$ = 0.5 and the depreciation rate is $\delta$ = .05,what is the new steady-state amount of capital?
A) 16 units
B) 64 units
C) 100 units
D) 256 units

Question

Quizplus · Accepted Answer

To find the new steady-state amount of capital, we can use the formula K* = ((sY)/(δ + n))^(1/(1-α)), where s is the fraction of output invested in new capital (0.5), Y is the level of output determined by the production function, δ is the depreciation rate (0.05), n is the population growth rate (not given), and α is the share of capital in the production function (not given).

Since we are given Y = F(K) and K = 400, we can substitute to get Y = F(400). However, we do not know the functional form of F(K), so we cannot find Y directly. Instead, we can use the fact that the change in steady-state capital is proportional to the change in the investment rate. Specifically, if the investment rate doubles from 0.2 to 0.4, then the steady-state capital stock will also double. Similarly, if the investment rate triples from 0.2 to 0.6, then the steady-state capital stock will triple.

In our case, the investment rate increases from 0.2 to 0.5, which is an increase of 0.3. Therefore, we can estimate that the steady-state capital stock will increase by roughly 1.5 times (since 0.3 is half of 0.6). This gives us a new steady-state capital stock of K* = 1.5*400 = 600.

Therefore, the answer is C) 100 units.

Imagine an Economy with Production Function Y = F(K)= and 400