Chapter 4, Problem 1E

(a)

To determine

Explain whether the production function exhibits increasing, decreasing, or constant returns to scale.

Explanation of Solution

If the capital (K) and labor (L)are doubled, then the production function is as follows:

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

If the capital and labor are doubled, then the output will double and there are constant returns to scale.

(b)

To determine

Explain whether the production function exhibits increasing, decreasing, or constant returns to scale.

Explanation of Solution

If the capital (K) and labor (L) are doubled, then the production function is as follows:

$\begin{array}{c}\text{Y}={\text{K}}^{\left(\frac{2}{3}\right)}{\text{L}}^{\left(\frac{2}{3}\right)}\\ ={\left(2\text{K}\right)}^{\left(\frac{2}{3}\right)}{\left(\text{2L}\right)}^{\left(\frac{2}{3}\right)}\\ =2^{\frac{2}{3}+\frac{2}{3}}{\text{K}}^{\left(\frac{1}{2}\right)}{\text{L}}^{\left(\frac{1}{2}\right)}\\ =2^{\frac{4}{6}}{\left(\text{KL}\right)}^{\left(\frac{2}{3}\right)}\end{array}$

If the capital and labor are doubled, then the output will increase the returns to scale.

(c)

To determine

Explain whether the production function exhibits increasing, decreasing, or constant returns to scale.

Explanation of Solution

If the capital (K) and labor (L) are doubled, then the production function is as follows:

$\begin{array}{c}\text{Y}={\text{K}}^{\left(\frac{1}{3}\right)}{\text{L}}^{\left(\frac{1}{2}\right)}\\ ={\left(2\text{K}\right)}^{\left(\frac{1}{3}\right)}{\left(\text{2L}\right)}^{\left(\frac{1}{2}\right)}\\ =2^{\frac{1}{3}+\frac{1}{2}}{\text{K}}^{\left(\frac{1}{3}\right)}{\text{L}}^{\left(\frac{2}{3}\right)}\\ =2^{\frac{5}{6}}{\text{K}}^{\left(\frac{1}{3}\right)}{\text{L}}^{\left(\frac{2}{3}\right)}\end{array}$

If the capital and labor are doubled, then the output will increase the returns to scale.

(d)

To determine

Explain whether the production function exhibits increasing, decreasing, or constant returns to scale.

Explanation of Solution

If the capital (K) and labor (L) are doubled, then the production function is as follows:

$\begin{array}{c}\text{Y}=\text{K+L}\\ =\text{2K}+\text{2L}\\ =2\left(\text{K+L}\right)\end{array}$

If the capital and labor are doubled, then the output will double and there are constant returns to scale.

(e)

To determine

Explain whether the production function exhibits increasing, decreasing, or constant returns to scale.

Explanation of Solution

If the capital (K) and labor (L) are doubled, then the production function is as follows:

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

If the capital and labor are doubled, then the output will decrease the returns to scale.

(f)

To determine

Explain whether the production function exhibits increasing, decreasing, or constant returns to scale.

Explanation of Solution

If the capital (K) and labor (L) is doubled, then the production function is as follows:

$\begin{array}{c}\text{Y}={\text{K}}^{\left(\frac{1}{3}\right)}{\text{L}}^{\left(\frac{2}{3}\right)}+\overline{\text{A}}\\ ={\left(2\text{K}\right)}^{\left(\frac{1}{3}\right)}{\left(\text{2L}\right)}^{\left(\frac{2}{3}\right)}+\overline{\text{A}}\\ =2^{\frac{1}{3}+\frac{2}{3}}{\text{K}}^{\left(\frac{1}{3}\right)}{\text{L}}^{\left(\frac{2}{3}\right)}+\overline{\text{A}}\\ =2{\text{K}}^{\left(\frac{1}{3}\right)}{\text{L}}^{\left(\frac{2}{3}\right)}+\overline{\text{A}}\end{array}$

If the first term is doubled, the output $\overline{\text{A}}$ remains unchanged. Thus, the output is less than the doubled values, which implies decreasing returns to scale.

(g)

To determine

Explain whether the production function exhibits increasing, decreasing, or constant returns to scale.

Explanation of Solution

If the capital (K) and labor (L) are doubled, then the production function is as follows:

$\begin{array}{c}\text{Y}={\text{K}}^{\left(\frac{1}{3}\right)}{\text{L}}^{\left(\frac{2}{3}\right)}-\overline{\text{A}}\\ ={\left(2\text{K}\right)}^{\left(\frac{1}{3}\right)}{\left(\text{2L}\right)}^{\left(\frac{2}{3}\right)}-\overline{\text{A}}\\ =2^{\frac{1}{3}+\frac{2}{3}}{\text{K}}^{\left(\frac{1}{3}\right)}{\text{L}}^{\left(\frac{2}{3}\right)}-\overline{\text{A}}\\ =2{\text{K}}^{\left(\frac{1}{3}\right)}{\text{L}}^{\left(\frac{2}{3}\right)}-\overline{\text{A}}\end{array}$

If the capital and labor are doubled, then the function exhibits constant returns to scale.