An economy has the per-worker production function y = k 1/3 where y is output per worker and k is capital per worker

An economy has the per-worker production function

y = k^(1/3)

where y is output per worker and k is capital per worker.

Capital accumulation follows:

?k = sy − (n + δ)k..........(1)

(e) The economy is not in the steady state.

We have to determine the growth rate of capital per worker (k?/k).

Hence,

k?/k = s.(y/k) - (n + δ)

Here, y = k^(1/3)

Hence, y/k = [k^(1/3)/k] = k^(-2/3)

Hence,

k?/k = s.[k^(-2/3)] - (n + δ)........(2)

This is the growth rate of capital per worker as function of k, s, n and δ.

Now, if k?/k is positive i.e.

k?/k > 0

or, s.[k^(-2/3)] - (n + δ) > 0

or, k^(-2/3) > (n + δ)/s

or, k^(2/3) < s/(n + δ)

or, 

Under this condition, k?/k is positive.

Now, from equation (2) we write

k?/k = s.[k^(-2/3)] - (n + δ)

Here, we are given that,

s = 0.2, δ = 0.04 and n = 0.01

Putting the values of n, δ and n we get

k?/k = (0.2).[k^(-2/3)] - (0.01+0.05)

or, k?/k = 0.2[k^(-2/3)] - 0.05

This is the relationship between k?/k and k.

Hope the solutions are clear to you my friend.