Chapter 8, Problem 31C

Sub part (a):

To determine

Steady state level of output.

Explanation of Solution

The steady state level of output is calculated as follows:

$\begin{array}{c}{\text{Y}}^{\text{SS}}=\sqrt{{\text{K}}^{\text{SS}}\times \text{L}}\\ =\sqrt{400\times 100}\\ =\sqrt{40000}\\ =200\end{array}$

The steady state level of output is 200 units.

Economics Concept Introduction

Concept introduction:

Economic growth: The economic growth is the increase in the overall goods and services produced per head in the economy over a specific period of time.

Sub part (b):

To determine

Solow diagram to show short run impact.

Explanation of Solution

Figure 1 depicts Solow diagram which shows the short run impact of a 21% increase in the amount of labor available.

In figure 1, the horizontal axis represents the capital (K) and the vertical axis represents the output (Y). The initial production function ${\text{Y}}^{\text{SS}}=\sqrt{{\text{K}}^{\text{SS}}\text{L}}$ shows how much output is produced for every level of the capital stock K and labor L. For instance, when K=400, and L is 100, then 200 units $\left(\sqrt{400\times 100}\right)$ of output are produced. The 21% increase in the amount of labor available creates a short run effect which gives rise to new production function which is depicted as ${\text{Y}}_1=\sqrt{\left({\text{K}}^{\text{SS}}\right)\left(1.21\text{ L}\right)}$ or ${\text{Y}}_1=\sqrt{\left({\text{K}}^{\text{SS}}\right){\text{L}}_1}$ where L₁=1.21 L in figure 1. This implies for K=400 and L= 100 or L₁=121, the output produced is 54.77 $\left(\sqrt{400\times 121}\right)$ units.

Economics Concept Introduction

Concept introduction:

Economic growth: The economic growth is the increase in the overall goods and services produced per head in the economy over a specific period of time.

Solow growth rate The Solow growth rate is the rate of economic growth with given flexible price, and the existing real factors of capita, labor and knowledge. The Solow growth rate is an economy’s potential growth rate.

Sub part (c):

To determine

Steady state level of output.

Explanation of Solution

The new steady state level of output is calculated as follows:

$\begin{array}{c}{\text{Y}}_1=\sqrt{{\text{K}}^{\text{SS}}\times 1.21\times \text{L}}\\ =\sqrt{400\times 1.21\times 100}\\ =\sqrt{48400}\\ =220\end{array}$

The new steady state level of output is 220 units.

Economics Concept Introduction

Concept introduction:

Economic growth: The economic growth is the increase in the overall goods and services produced per head in the economy over a specific period of time.

Sub part (d):

To determine

The new steady state level of output in the diagram.

Explanation of Solution

The new steady state level of output is 220 units which is depicted in the figure 1.

Sub part (e):

To determine

Action to gain long lasting benefits from the increase in capital stock.

Explanation of Solution

The initial production function is algebraically represented as follows:

$\text{Y}=\sqrt{\text{KL}}$

Squaring both sides,

${\text{Y}}^2=\text{KL}$

And solving for K, we get

$\text{K}=\frac{{\text{Y}}^2}{\text{L}}$

Substitute K into the steady-state condition  $\text{ γY=δK}$ so that

$\text{γY=δ}\frac{{\text{Y}}^2}{\text{L}}$

 And solve for Y by multiplying L and dividing by $\text{δY}$ , we get

${\text{ Y}}_0^{\text{SS}}\text{=}\frac{\text{γ L}}{\text{δ}}$ (1)

Similarly, the new production function can be algebraically represented as follows:

$\begin{array}{c}\text{Y}=\sqrt{\left(\text{K}\right)\left(1.21\text{ L}\right)}\\ {\text{Y}}^2=\left(\text{K}\right)\times \left(1.21\times \text{ L}\right)\\ \text{K}=\frac{{\text{Y}}^2}{1.21\times \text{ L}}\end{array}$

Substitute K into the steady-state condition  $\text{ γY=δK}$ so that

$\text{γY=δ}\frac{{\text{Y}}^2}{1.21\times \text{ L}}$

And solve for Y by multiplying L and dividing by $\text{δY}$ , we get

${\text{ Y}}_1^{\text{SS}}\text{=}\frac{\text{γ(1}\text{.21 L)}}{\text{δ}}$ (2)

Equating (1) in (2) we get

${\text{Y}}_1^{\text{SS}}\text{= 1}{\text{.21Y}}_0^{\text{SS}}$ (3)

Substituting the values in equation (3) we get

$\begin{array}{c}{\text{Y}}_1^{\text{SS}}\text{= 1}\text{.21}\times 200\\ =242\end{array}$

Further, the percentage change in the output is depicted as follows:

$\begin{array}{c}\text{Percentage change in output =}\frac{{\text{Y}}_1^{\text{SS}}{\text{-Y}}_0^{\text{SS}}}{{\text{Y}}_0^{\text{SS}}}\times 100\\ =\frac{242-200}{200}\times 100\\ =\frac{42}{2}\\ =21\%\end{array}$

This implies the steady-state level of output will grow by 21% per cent in the long run.

Economics Concept Introduction

Concept introduction:

Production function: It is the relationship between the inputs employed by a firm and the maximum output the firm can produce with those inputs.

Sub part (f):

To determine

Output per worker.

Explanation of Solution

The output per worker in the initial steady state is calculated as follows:

$\begin{array}{c}{\text{Output per worker}}_{\text{initial steady state}}=\frac{{\text{Y}}_0^{\text{SS}}}{\text{L}}\\ \text{=}\frac{\frac{\text{γ L}}{\text{δ}}}{\text{L}}\\ =\frac{\text{γ }}{\text{δ}}\\ =\frac{200}{100}\\ =2\end{array}$

The output per worker in the initial steady state is 2 units per labor.

The output per worker in the short run is calculated as follows:

$\begin{array}{c}{\text{Output per worker}}_{\text{short run}}=\frac{{\text{ Y}}_1}{\text{1}\text{.21 L}}\\ \text{=}\frac{220}{\text{1}\text{.21 L}}\\ =\frac{220}{121}\\ =1.82\end{array}$

The output per worker in short run is 1.82 units per labor.

The output per worker in the initial steady state is calculated as follows:

$\begin{array}{c}{\text{Output per worker}}_{\text{long run}}=\frac{{\text{ Y}}_1^{\text{SS}}}{\text{1}\text{.21}\times \text{L}}\\ \text{=}\frac{\frac{\text{γ(1}\text{.21 L)}}{\text{δ}}}{\text{1}\text{.21}\times \text{L}}\\ =\frac{\text{γ }}{\text{δ}}\\ =\frac{242}{121}\\ =2\end{array}$

The output per worker in the long run is 2 units per labor.

Sub part (g):

To determine

New immigration policy and its effect.

Explanation of Solution

The citizens of the country are neither made worse off nor better off in the long run by a new immigration policy. This is because the new long-run level of output per worker compare with the initial level of output per worker remains unchanged which is 2 units per labor. This is unlike the short run effect which has lower output per worker. In the long run, the steady output level is determined largely by the depreciation and savings rate.

Economics Concept Introduction

Concept introduction:

Depreciation: Depreciation is the process of decreasing the value of an asset over time especially due to wear and tear.

Investment: The investment is the money invests in terms of assets and building by the individual for the future consumption and profit making.

Sub part (h):

To determine

New steady state level of capital.

Explanation of Solution

The new steady state level of capital is calculated as follows

We know $\text{ γY=δK}$ and $\text{γ}$ and $\text{δ}$ remains unchanged .

Also ${\text{Y}}_1^{\text{SS}}\text{= 1}{\text{.21Y}}_0^{\text{SS}}$.

Equating all these we get,

${\text{K}}_1^{\text{SS}}\text{= 1}{\text{.21K}}_0^{\text{SS}}$

Substituting the values we get

$\begin{array}{c}{\text{K}}_1^{\text{SS}}\text{= 1}{\text{.21K}}_0^{\text{SS}}\\ =1.21\times 400\\ =484\end{array}$

Thus the new steady state level of capital is 484, so that $\frac{\text{δ }}{\text{γ}}$remains unchanged.