Answer to Question #222832 in Macroeconomics for Gorgee

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.

 

a. Find the steady state level of the capital per effective worker (k*), output per effective worker (y*) and consumption per effective worker (c*) - in terms of the parameters of the model.

 

b. What is the level of k (k**) that maximizes consumption? 

d. To move to the level of capital that maximizes consumption, how should the saving rate be changed? Explain. 

 

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

Answer 1

A.

capital per worker

When k is in steady state:

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

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

output per worker

let s=0.4

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

consumption per worker

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

B.

The level of k that maximizes consumption is when $MPK^*=\delta +n$

C.

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

This helps to compare the steady state capital stock with the golden rule level.

D.

To move to the level of capital that maximizes consumption, the saving rate should be increased. This will result to more investments because more capital Stock will be raised.

E.

The saving rate needed to reach the golden rule level of capital per effective worker should be above the saving rate at steady state level.