Question

Suppose that the production function for output in an economy is given by Yt=Kt0.25N0.75

The number of workers, N, is constant. The saving rate is s, and the depreciation rate of physical capital is δ.

a) Write down the equation showing the evolution of physical capital stock over time. Explain in words.

b)    Derive the steady state levels of capital per worker and output per worker in terms of the saving rate (s) and the depreciation rate (δ).

Answer 1

Yt=Kt0.25N0.75

We know, output per worker, y = Y/N = (Kt0.25N0.75) / N = (K/N)0.25 = k^0.25 = f(k) ---equation 1

Capital per worker, k = K/N

Differentiating both sides and dividing by k

/k = /K - /N

/k = /K - n

n = population growth rate

/k = (sY - δK) / K - n

/k = sY/K - δ - n

/k = (s*f(k) / k) - δ - n

= s*f(k) - (δ - n)*k

kt+1 - kt = s*f(k) - (δ - n)*k : this equation shows evolution of physical capital over time

b)

n = population growth rate = 0

At steady state, = 0

s*f(k) - (δ - n)*k = 0

s*(k^0.25) = δk (using equation 1 and given that n = 0

s/δ = k^0.75

k* = (s/δ)^1.33 : steady state level of captial per worker in terms of s and δ

y* = k^0.25

y* = ((s/δ)^1.33)^0.25

y* = (s/δ)^0.33 : steady state level of output per worker in terms of s and δ

**if you liked the answer, then please upvote. Would be motivating for me. Thanks.