Question

2. Prove each of the following statements about the steady state of the Solow model with population growth and technological progress. (5 Points) The capital-output ratio is constant. (5 Points) Capital and labor each earn a constant share of an economy's income. [Hint: Recall the definition MPK = f(k+1), f(k).] (5 Points) Total capital income and total labor income both grow at the rate of population growth plus the rate of technological progress, n+g. (5 Points) The real rental price of capital is constant, and the real wage grows at the rate of technological progress 8. (Hint: The real rental price of capital equals total capital income divided by the capital stock, and the real wage equals total labor income divided by the labor force.)

Answer 1

The Solow model is the exogenous growth model and it explains the change in the output of the economy over a period of time due to the changes in technology, population and saving rate. The steady-state in the Solow model talking about the state at which capital per worker does not change.

Part A

Under the steady state model we can say that,

sy = (8+g+n)k

this shows,

y (8 +g+n)

Here,

$\small s,\delta ,g$ and $\small n$ are constant. it shows, $\small \frac{k}{y}$ is constant. Then,

$\small \frac{k}{y}=\frac{(\frac{K}{LE})}{(\frac{Y}{LE})}=\frac{k}{y}$. on this backgroud we can say that capital-output ratio is constant.

Part B

Here, the capital share of income is,

= MPK

Based on our first part, we understood that the capital-output ratio is constant. Also, MPK is a function of k, this constant under steady state. Then we can say that MPK is also constant. Hence, the capital share of income is also constant. Now the labor share of income,

= 1- (capital'sshare)

Suppose the capital share of income is constant, then the labor share of income is also assumed to be constant.

Part C

Under the steady-state, the income is growing at the rate of population and technological change
(n+g)
. Based on our second part, we understood that both capital share and labor share of income is constant. These both are constant, then income grows at the rate of
(n+g)
, so, both labor income and capital income grows at
(n+g)
.

Part D

The real rental price of capital is,

(MPK XK)

Under the steady-state, the MPK is constant because the effective capital per worker is constant. So, we can say that the real rental price of capital is also constant under the steady-state model.

Now, we can say that the real wage is grows at the rate of change in technology,

where, TLI is the total labor income and L is the labor.

TLI

From the equation, we can say that change in real wage plus the change in labor force equals the changes in total labor income. If the labor force grows at $\small n$ , then total labor income changes at $\small n+g$ . in sum, we can say that real wages grow at $\small g$ .