Question

A firm produces output according to the production function $\mathbf{Q}\mathbf{=}\mathbf{F}\mathbf{(}\mathbf{K}\mathbf{,}\mathbf{L}\mathbf{)}\mathbf{=}\mathbf{4}\mathbf{K}\mathbf{+}\mathbf{8}\mathbf{L}\mathbf{.}$

a. How much output is produced when$\mathbf{K}\mathbf{=}\mathbf{2}$and$\mathbf{L}\mathbf{=}\mathbf{3}$?

b. If the wage rate is$\mathbf{\backslash (}\mathbf{60}$per hour and the rental rate on capital is$\mathbf{\backslash )}\mathbf{20}$per hour, what is the cost-minimizing input mix for producing 32 units of output?

c. How does your answer to part b change if the wage rate decreases to$\mathbf{\backslash (}\mathbf{20}$per hour but the rental rate on capital remains at$\mathbf{\backslash )}\mathbf{20}$per hour?

Short answer

1. The output produced when$\mathrm{K}=2$ and $\mathrm{L}=3$is 32 units
2. Since thecost of using $8\mathrm{K}$is lower, the firm will use $(8\mathrm{K},0\mathrm{L})$ to produce $32$ of units output.
3. Since cost of using $4\mathrm{L}$is lower, the firm will use $(0\mathrm{K},4\mathrm{L})$to produce $32$units of output.

Step-by-step solution

Finding the Output

Production function:$\mathrm{F}(\mathrm{K},\mathrm{L})=\mathbf{4}\mathbf{}\mathbf{K}\mathbf{+}\mathbf{8}\mathbf{}\mathbf{L}$

1. When$\mathbf{K}\mathbf{=}\mathbf{2}$ and$\mathbf{L}\mathbf{=}\mathbf{3}\mathbf{,}$

$\begin{array}{l}\mathrm{Q}=\mathbf{4}\mathbf{}\mathbf{K}\mathbf{+}\mathbf{8}\mathbf{}\mathbf{L}\\ \mathrm{Q}=\mathbf{4}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{+}\mathbf{8}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\\ \mathrm{Q}=\mathbf{32}\mathrm{units}\end{array}$

Finding the cost minimizing input

b.

The firm can either use If $\mathbf{23}\mathbf{/}\mathbf{4}\mathbf{=}\mathbf{8}\mathbf{}\mathbf{K}$or$\mathbf{32}\mathbf{/}\mathbf{8}\mathbf{=}\mathbf{4}\mathbf{L}$

Cost of$\mathbf{8}\mathbf{}\mathbf{K}\mathbf{=}\mathbf{8}\mathbf{\left(}\mathbf{20}\mathbf{\right)}\mathbf{=}\mathbf{\$}\mathbf{160}$

Cost of$\mathbf{4}\mathbf{}\mathbf{L}\mathbf{=}\mathbf{4}\mathbf{\left(}\mathbf{60}\mathbf{\right)}\mathbf{=}\mathbf{\$}\mathbf{240}$

Since cost of using$\mathbf{8}\mathbf{K}$ is lower, the firm will use $\mathbf{(}\mathbf{8}\mathbf{}\mathbf{K}\mathbf{,}\mathbf{0}\mathbf{}\mathbf{L}\mathbf{)}$to produce $\mathbf{32}$units of output.

Finding the cost minimizing output mix

c.

The firm can either use$\mathbf{23}\mathbf{/}\mathbf{4}\mathbf{=}\mathbf{8}\mathbf{}\mathbf{K}$ or$\mathbf{32}\mathbf{/}\mathbf{8}\mathbf{=}\mathbf{4}\mathbf{L}$

Cost of$\mathbf{8}\mathbf{}\mathbf{K}\mathbf{=}\mathbf{8}\mathbf{\left(}\mathbf{20}\mathbf{\right)}\mathbf{=}\mathbf{\$}\mathbf{160}$

Cost of$\mathbf{4}\mathbf{}\mathbf{L}\mathbf{=}\mathbf{4}\mathbf{\left(}\mathbf{20}\mathbf{\right)}\mathbf{=}\mathbf{\$}\mathbf{80}$

Since cost of using$4\mathrm{L}$ is lower, the firm will use$(0\mathrm{K},4\mathrm{L})$ to produce 32 units of output.