Chapter 6, Problem 17P

 (a)

To determine

The returns to scale.

Explanation of Solution

The returns to scale shows the changes in the amount of output in response to a proportional increase or decrease in all of the inputs. In this case, the given production function is $\text{Q}=10{\text{K}}^{0.75}{\text{L}}^{0.25}$. If L is 1 unit and K is 1 unit, then the output can be calculated as follows:

$\begin{array}{c}\text{Q}=10{\left(1\right)}^{0.75}{\left(1\right)}^{0.25}\\ =10\end{array}$

If L is 2 unit and K is 2 unit, then the output can be calculated as follows:

$\begin{array}{c}\text{Q}=10{\left(2\right)}^{0.75}{\left(2\right)}^{0.25}\\ =20\end{array}$

Therefore, the changes of inputs by the same proportion also changes the output by the same proportion. Hence, this is a constant returns to scale.

Economics Concept Introduction

Constant returns to scale: The constant returns to scale describes a production function for which changing all inputs by the same proportion changes the quantity of output by the same proportion.

 (b)

To determine

The returns to scale.

Explanation of Solution

In this case, the given production function is $\text{Q}={\left({\text{K}}^{0.75}{\text{L}}^{0.25}\right)}^2$. If L is 1 unit and K is 1 unit, then the output can be calculated as follows:

$\begin{array}{c}\text{Q}={\left(1^{0.75}{1}^{0.25}\right)}^2\\ =1\end{array}$

If L is 2 unit and K is 2 unit, then the output can be calculated as follows:

$\begin{array}{c}\text{Q}={\left(2^{0.75}{2}^{0.25}\right)}^2\\ =4\end{array}$

Therefore, the changes of inputs by the same proportion also changes the output by more than the proportion. Hence, this is an increasing returns to scale.

Economics Concept Introduction

Increasing returns to scale: An increasing returns to scale describes a production function for which changing all inputs by the same proportion changes output more than proportionally.

 (c)

To determine

The returns to scale.

Explanation of Solution

In this case, the given production function is $\text{Q}={\text{K}}^{0.75}{\text{L}}^{0.75}$. If L is 1 unit and K is 1 unit, then the output can be calculated as follows:

$\begin{array}{c}\text{Q}=1^{0.75}{1}^{0.75}\\ =1\end{array}$

If L is 2 unit and K is 2 unit, then the output can be calculated as follows:

$\begin{array}{c}\text{Q}=2^{0.75}{2}^{0.75}\\ =2.828\end{array}$

Therefore, the changes of inputs by the same proportion also changes the output by more than the proportion. Hence, this is an increasing returns to scale.

Economics Concept Introduction

Increasing returns to scale: An increasing returns to scale describes a production function for which changing all inputs by the same proportion changes output more than proportionally.

 (d)

To determine

The returns to scale.

Explanation of Solution

In this case, the given production function is $\text{Q}={\text{K}}^{0.25}{\text{L}}^{0.25}$. If L is 1 unit and K is 1 unit, the output can be calculated as follows:

$\begin{array}{c}\text{Q}=1^{0.25}{1}^{0.25}\\ =1\end{array}$

If L is 2 unit and K is 2 unit, then the output can be calculated as follows:

$\begin{array}{c}\text{Q}=2^{0.25}{2}^{0.25}\\ =1.414\end{array}$

Therefore, the changes of inputs by the same proportion also changes the output by less than the proportion. Hence, this is a decreasing returns to scale.

Economics Concept Introduction

Decreasing returns to scale: The decreasing returns to scale describes a production function for which changing all inputs by the same proportion changes output less than proportionally.

 (e)

To determine

The returns to scale.

Explanation of Solution

In this case, the given production function is $\text{Q}=\text{K}+\text{L}+\text{KL}$. If L is 1 unit and K is 1 unit, then the output can be calculated as follows:

$\begin{array}{c}\text{Q}=1+1+\left(1\times 1\right)\\ =3\end{array}$

If L is 2 unit and K is 2 unit, then the output can be calculated as follows:

$\begin{array}{c}\text{Q}=2+2+\left(2\times 2\right)\\ =8\end{array}$

Therefore, the changes of inputs by the same proportion also changes the output by more than the proportion. Hence, this is an increasing returns to scale.

Economics Concept Introduction

Increasing returns to scale: An increasing returns to scale describes a production function for which changing all inputs by the same proportion changes output more than proportionally.

 (f)

To determine

The returns to scale.

Explanation of Solution

In this case, the given production function is $\text{Q}=2{\text{K}}^2+3{\text{L}}^2$. If L is 1 unit and K is 1 unit, then the output can be calculated as follows:

$\begin{array}{c}\text{Q}=2{\left(1\right)}^2+3{\left(1\right)}^2\\ =5\end{array}$

If L is 2 unit and K is 2 unit, then the output can be calculated as follows:

$\begin{array}{c}\text{Q}=2{\left(2\right)}^2+3{\left(2\right)}^2\\ =20\end{array}$

Therefore, the changes of inputs by the same proportion also changes the output by more than the proportion. Hence, this is an increasing returns to scale.

Economics Concept Introduction

Increasing returns to scale: An increasing returns to scale describes a production function for which changing all inputs by the same proportion changes output more than proportionally.

 (g)

To determine

The returns to scale.

Explanation of Solution

In this case, the given production function is $\text{Q}=\text{KL}$. If L is 1 unit and K is 1 unit, then the output can be calculated as follows:

$\begin{array}{c}\text{Q}=1\times 1\\ =1\end{array}$

If L is 2 unit and K is 2 unit, then the output can be calculated as follows:

$\begin{array}{c}\text{Q}=2\times 2\\ =4\end{array}$

Therefore, the changes of inputs by the same proportion also changes the output by more than the proportion. Hence, this is an increasing returns to scale.

Economics Concept Introduction

Increasing returns to scale: An increasing returns to scale describes a production function for which changing all inputs by the same proportion changes output more than proportionally.

 (h)

To determine

The returns to scale.

Explanation of Solution

In this case, the given production function is $\text{Q}=\min \left(\text{3K, 2L}\right)$. If L is 1 unit and K is 1 unit, then the output can be calculated as follows:

$\begin{array}{c}\text{Q}=\min \left(\left(\text{3}\times \text{1}\right)\text{, }\left(\text{2}\times \text{1}\right)\right)\\ =\min \left(3,2\right)\end{array}$

If L is 2 unit and K is 2 unit, then the output can be calculated as follows:

$\begin{array}{c}\text{Q}=\min \left(\left(\text{3}\times 2\right)\text{, }\left(\text{2}\times 2\right)\right)\\ =\min \left(6,4\right)\end{array}$

Therefore, the changes of inputs by the same proportion also changes the output by the same the proportion. Hence, this is constant returns to scale.

Economics Concept Introduction

Constant returns to scale: The constant returns to scale describes a production function for which changing all inputs by the same proportion changes the quantity of output by the same proportion.