Question

. Using the Solow model, illustrate and explain an economy at the Golden Rule level of capital per worker. Note: you must include the derivation of the mathematical condition in your answer.

Answer 1

Let's take a cobb douglas production function,

Y = K^ L^1-​​​​​​

Now we need to convert production function into per capita output by dividing output Y by L,

Y/L = K^ L^1-/L

Y/L = K^ L^1 - - 1

Y/L = K^ L^-​​​​​​

Let's denote the output per capita by y,

y = (K/L)^​​​​​​

And let's denote capital per capita that is K/L by k

y = k^

Now the golden rule level of steady state is defined as the steady state where the consumption per capita is maximized.

c = income per capita - savings per capita

c = y - sy

The steady state level of capital per person is defined as the capital stock per person where the investment in capital is equal to depreciation of capital.

Investment in solow model is equal to,

= sy

Here s = saving rate

y = Output per person

In steady state savings is equal to investment. So putting sy = (n + g + ​​​​​​)k

Here n = population growth rate

g = technological growth rate

= depreciation rate of capital.

c = (k)^ - (n + g + ​​​​​​)k

Now maximizing c with respect to k, by simply derivating c by k and then putting it equal to 0.

0 = d(k)^/dk - d(n + g + ​​​​​​) ​​​k/dk

0 = k^-1 - (n + g+ ​​​​​​)

n + g+ = k^-1

(n + g + ​​​​​​)/ = k^-1

Now taking (1/-1) raised to the power both the sides we get,

[(n + g + ​​​​​​)/]^1/-1 = k

So this is the value of golden rule level of steady state per person in solow model.

I hope I was able to help you, thank you.