b Find the golden rule quantity of capital per worker Is the

b. Find the golden rule quantity of capital per worker. Is there overinvestment or underinvestment in this economy, relative to golden rule? Show/ explain your work.

Solution

(a)

Let Y = F(K, N) = K^0.5N^0.5

Dividing both sides by N,

Y / N = (K / N)^0.5

y = f(k) = k^0.5 [where y: Output (income) per capita, k: Capital per capita]

In steady state,

s / (d + n + g) = k / f(k), Or

s / (d + n) = k / f(k) [Since g = 0 assumed]

Since f(k) = k^0.5 , we have

k / f(k) = k / k^0.5 = k^0.5

0.2 / (0.1 + 0.01) = k^0.5

k^0.5 = 0.11 / 0.2 = 0.055

Squaring both sides, k = 0.003

y = Income (output) per capita = k^0.5 = 0.055

Consumption per capita, c = 1 - s = 1 - 0.2 = 0.8

(b)

In golden state, MPK = d + n

MPK = dY / dK = 0.5 x (N / K)^0.5 = 0.5 / (k)^0.5

So,

0.5 / (k)^0.5 = d + n = 0.1 + 0.01 = 0.11

(k)^0.5 = 0.5 / 0.11 = 4.55

k* = 20.66

The golden-rule capital per worker is much higher than the steady-state capital per worker (= 0.003). So, there is overinvestment in golden rule state, or (in other words), under-investment in steady state level.