Question

Consider an economy with the following aggregate production function: Y = 3K1/3(AL)2/3
Capital grows through investment but also decays due to wear and tear at a constant rate δ per period. Assume that A is growing at the exogenous rate g, that L is growing at the exogenous rate n, and that households save a constant proportion s of their income.

Calculate the saving rate needed to reach the golden rule level of capital per effective worker.

Answer 1

The saving needed to reach the golden rule level of capita per effective worker should be above the saving rate at steady state level. The following calculations satisfy it:

Capital per worker equals to:

When k is in steady state:

$sk^*0.5=\delta k^*$

Output per worker equals to:

This gives us $k^*=(\frac {s}{\delta})^2$

let s=0.4

$y^*=(\frac {s}{\delta})=\frac {0.4}{\delta}$

Consumption per worker equals to:

$c^*=(1-s)(\frac{s}{\delta})=\frac {0.24}{\delta}$

Therefore:

When the economy is at the golden rule steady state, $MPK^*=\delta+n.$ Given that $f(k)=k^(\frac {1}{3})$ , this means that $(\frac {1}{3})^*k_G^*-\frac {1}{3}= \delta+n$.