Chapter 8, Problem 1PA

(a)

To determine

The nature of the production function.

Explanation of Solution

The production function exhibits constant returns to the scale when the output increases in a constant proportion to the increase in the factor inputs. In other words, a production function exhibits constant returns to the scale if a two times increase in the input factors of production increases the output by two times. A constant production function takes the following form, where z is any positive number.

$\text{zY}=\text{F}\left(\text{zK,zL}\right)$

The given production function can be written as follows:

$\begin{array}{c}\text{F}\left(\text{zK,zL}\right)={\left(\text{zK}\right)}^{\frac{1}{3}}{\left(\text{zL}\right)}^{\frac{2}{3}}\\ =\text{z}\left({\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{2}{3}}\right)\\ =\text{zY}\end{array}$

Thus, it is evident that the given production function shows constant returns to the scale.

Economics Concept Introduction

Constant returns to scale: Constant returns to scale implies that as the scale of production of the firm increases, the cost of production per unit remains unchanged.

(b)

To determine

The production function per worker.

Explanation of Solution

The per worker production function can be calculated using Equation (1) as follows:

$\text{Output per worker}=\frac{\text{Total output}}{\text{Number of workers}}$ (1)

The output per worker can be calculated by substituting the respective values in Equation (1) as follows:

$\begin{array}{c}\text{Output per worker}=\frac{{\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{2}{3}}}{\text{L}}\\ \frac{\text{Y}}{\text{L}}={\left(\frac{\text{K}}{\text{L}}\right)}^{\frac{1}{3}}\end{array}$

The given production function exhibits constant returns to scale.

$\begin{array}{c}\frac{\text{F}\left(\text{K,L}\right)}{\text{L}}=\frac{\text{LF}\left(\frac{\text{K}}{\text{L}},1\right)}{\text{L}}\\ =\text{F}\left(\frac{\text{K}}{\text{L}},1\right)\end{array}$

The ratio output per labor is $\frac{\text{Y}}{\text{L}}=\text{y}$ and the ratio capital per labor is $\frac{\text{K}}{\text{L}}=\text{k}$, the per worker production function can be expressed as follows:

$\begin{array}{c}\text{y}=\text{f}\left(\text{k}\right)\\ ={\text{k}}^{\frac{1}{3}}\end{array}$

Thus, the production function per worker is ${\text{k}}^{\frac{1}{3}}$.

Economics Concept Introduction

Output per worker: Output per worker is the ratio of the total output produced to the total workers employed.

(c)

To determine

The steady state level of capital per worker, income per worker and the consumption per worker.

Explanation of Solution

Given that the country faces a depreciation, $\text{δ}=0.2$, the savings of Country A, ${\text{S}}_{\text{A}}=0.1$ and the savings rate of Country B is ${\text{S}}_{\text{B}}=0.3$.

In the steady state, the condition for the capital per worker is given as follows:

$\begin{array}{c}\text{sf}\left(\text{k}\right)=\text{δk}\\ \text{s}{\left(\text{k}\right)}^{\frac{1}{3}}=\text{δk}\\ {\text{k}}^{\frac{2}{3}}=\frac{\text{s}}{\text{δ}}\end{array}$

$\text{k}={\left(\frac{\text{s}}{\text{δ}}\right)}^{\frac{3}{2}}$ (2)

The steady state level of capital per worker in Country A is calculated by substituting the respective values in the Equation (2) as follows:

$\begin{array}{c}{\text{k}}_{\text{A}}={\left(\frac{{\text{s}}_{\text{A}}}{\text{δ}}\right)}^{\frac{3}{2}}\\ ={\left(\frac{0.1}{0.2}\right)}^{\frac{3}{2}}\\ =0.35\end{array}$

The steady state level of capital per worker in Country A is 0.35.

The steady state level of capital per worker in Country B is calculated by substituting the respective values in the Equation (2) as follows:

$\begin{array}{c}{\text{k}}_{\text{B}}={\left(\frac{{\text{s}}_{\text{B}}}{\text{δ}}\right)}^{\frac{3}{2}}\\ ={\left(\frac{0.3}{0.2}\right)}^{\frac{3}{2}}\\ =1.84\end{array}$

The steady state level of capital per worker in Country B is 1.84.

The steady state level of income can be calculated as follows:

$\begin{array}{c}\text{y}={\text{k}}^{\frac{1}{3}}\\ ={\left(\frac{\text{s}}{\text{δ}}\right)}^{\frac{1}{2}}\end{array}$ (3)

The steady state level of income in Country A is calculated by substituting the respective values in the Equation (3) as follows:

$\begin{array}{c}{\text{y}}_{\text{A}}={\left(\frac{{\text{s}}_{\text{A}}}{\text{δ}}\right)}^{\frac{1}{2}}\\ ={\left(0.35\right)}^{\frac{1}{2}}\\ =0.59\end{array}$

The steady state level of income in Country A is 0.59.

The steady state level of income in Country B is calculated by substituting the respective values in the above equation.

$\begin{array}{c}{\text{y}}_{\text{B}}={\left(\frac{{\text{s}}_{\text{B}}}{\text{δ}}\right)}^{\frac{1}{2}}\\ ={\left(1.84\right)}^{\frac{1}{2}}\\ =1.36\end{array}$

The steady state level of income in country B is 1.36.

The consumption is the difference between income and savings. The consumption per worker can be calculated using the Equation (4) as follows:

$\text{c}=\left(1-\text{s}\right)\text{y}$ (4)

The steady state level of consumption per worker in Country A is calculated by substituting the respective values in the Equation (4) as follows:

$\begin{array}{c}{\text{c}}_{\text{A}}=\left(1-{\text{s}}_{\text{A}}\right){\text{y}}_{\text{A}}\\ =\left(1-0.1\right)0.59\\ =0.53\end{array}$

The steady state level of consumption per worker in Country A is 0.53.

The steady state level of consumption per worker in Country B is calculated by substituting the respective values in Equation (4) as follows:

$\begin{array}{c}{\text{c}}_{\text{B}}=\left(1-{\text{s}}_{\text{B}}\right){\text{y}}_{\text{B}}\\ =\left(1-0.3\right)1.36\\ =0.95\end{array}$

The steady state level of consumption per worker in Country B is 0.95.

Economics Concept Introduction

Output per worker: Output per worker is the ratio of the total output produced to the total workers employed.

(d)

To determine

The income per worker and the consumption per worker.

Explanation of Solution

 If the capital per worker in both countries is equal to one, then it is given as follows:

  $\begin{array}{c}{\text{y}}_{\text{A}}={\text{y}}_{\text{B}}\\ =1\end{array}$

 The consumption per worker in Country A is calculated as follows:

 $\begin{array}{c}{\text{c}}_{\text{A}}=\left(1-{\text{s}}_{\text{A}}\right){\text{y}}_{\text{A}}\\ =\left(1-0.1\right)1\\ =0.9\end{array}$

 The consumption per worker in Country A is 0.9.

 The consumption per worker in Country B is calculated as follows:

 $\begin{array}{c}{\text{c}}_{\text{B}}=\left(1-{\text{s}}_{\text{B}}\right){\text{y}}_{\text{B}}\\ =\left(1-0.3\right)1\\ =0.7\end{array}$

 The consumption per worker in Country B is 0.7.

(e)

To determine

The comparison of consumption in Country A and Country B.

Explanation of Solution

 Table 1 shows the change in capital stock in Country A calculated using a spreadsheet.

 Table 1

Year	k	y = k1/3	c = (1 – sA)y	i = sAy	δk	Δk = i – δk
1	1	1	0.9	0.1	0.2	−0.10
2	0.9	0.97	0.87	0.1	0.18	−0.08
3	0.82	0.93	0.84	0.09	0.16	−0.07
4	0.75	0.91	0.82	0.09	0.15	−0.06
5	0.69	0.88	0.79	0.09	0.14	−0.05
6	0.64	0.86	0.78	0.09	0.13	−0.04
7	0.6	0.84	0.76	0.08	0.12	−0.04

 Table 2 shows the change in capital stock in Country B calculated using a spreadsheet.

 Table 2

Year	k	y = k1/3	c = (1 – sA)y	i = sAy	δk	Δk = i – δk
1	1	1	0.7	0.3	0.2	0.1
2	1.1	1.03	0.72	0.31	0.22	0.09
3	1.19	1.06	0.74	0.32	0.24	0.08
4	1.27	1.08	0.76	0.32	0.25	0.07
5	1.34	1.1	0.77	0.33	0.27	0.06
6	1.4	1.12	0.78	0.34	0.28	0.06
7	1.46	1.13	0.79	0.34	0.29	0.05

 It is evident from the table values that it takes 5 years for Country B to have a higher consumption per worker than Country A. In the 6 th year the consumption per worker in Country B is 0.784 which exceeds the consumption per worker in Country A which is 0.775 in absolute terms.