Find the golden rule quantity of capital per worker Is there

Find the golden rule quantity of capital per worker. Is there overinvestment or underinvestment in this economy, relative to golden rule?

Consider a numerical example using the Solow growth model. Suppose that F (K, N) = K0.3N0.7, with d = 0.07, s 0.2, n = 0.01, and z = 1.

Find the current steady state kss. Per worker production function is y = k^0.3

At te steady state we have

k* /y* = s/(n + d)

k*/k*^0.3 = 0.2/(0.01 + 0.07)

k* = 3.702

Now at the golden rule kg, the cosumption is maximum. Find the consumption function and maximize the same

c = y - sy

c = k*0.3 - (n+d)k

dc/dk = 0.3k^(-0.7) - (0.01 + 0.07)

For maximization, dc/dk = 0

0.3k^(-0.7) = (0.01 + 0.07)

This gives kg = 6.6 which is the golden rule quantity of capital per worker.

Since kss < kg, there is currently underinvestment.