Answer to Question #222461 in Macroeconomics for Ike

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? 

(C) Given a depreciation rate of 7%, population growth rate of 2%, technological progress of 1% and a saving rate of 30%, calculate the steady state levels of k, y and c.

(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)

Let's assume here A and L are growing at exogenous rate, household saving remains constant then

capital per worker

When k is in steady state:

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

 gives $k^*=(\frac {s}{\delta})^2$

output per worker

let s=0.4

$y=(\frac {s}{\delta})$

$y=\frac {0.4}{\delta}$

B)

The level of k that maximizes consumption is when

$MPK^*=\delta +n$

C)

In this condition when 

$MPK^*=\delta+n.$ 

we can see the difference between the steady 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 lead to attract more investments because more capital will be raised by the company.

e)

$MPK=\frac{1}{3}k^{-\frac{2}{3}}=\frac{1}{3}\times4^{-\frac{1}{2}}=0.25>n+\delta=0.10$