Question

According to The Wall Street Journal, Mitsubishi Motors recently announced a major restructuring plan in an attempt to reverse declining global sales. Suppose that as part of the restructuring plan Mitsubishi conducts an analysis of how labor and capital are used in its production process. Prior to restructuring Mitsubishi’s marginal rate of technical substitution is $\mathbf{0}\mathbf{.}\mathbf{12}$ (in absolute value). To hire workers, suppose that Mitsubishi must pay the competitive hourly wage of$\mathbf{1800}$ . In the study of its production process and markets where capital is procured, suppose that Mitsubishi determines that its marginal productivity of capital is$\mathbf{0}\mathbf{.}\mathbf{8}$ small cars per hour at its new targeted level of output and that capital is procured in a highly competitive market. The same study indicates that the average selling price of Mitsubishi’s smallest car is$\mathbf{1}\mathbf{,}\mathbf{200}\mathbf{,}\mathbf{000}$ . Determine the rate at which Mitsubishi can rent capital and the marginal productivity of labor at its new targeted level of output. To minimize costs Mitsubishi should hire capital and labor until the marginal rate of technical substitution reaches what proportion

Short answer

The marginal rate of technical substitution should be $\mathbf{0}.\mathbf{0019}.$

Step-by-step solution

Differentiating the function K and L

The profit function of the firm is given as follows

$\mathrm{\pi }=\mathrm{PF}(\mathrm{K},\mathrm{L})-\mathrm{wL}-\mathrm{rK}---\left(1\right)$

Where P is the price of car, $P_K$ is the price of capital and $P_L$ is the price of labor, r is the rent of capital and $\mathrm{F}(\mathrm{K},\mathrm{L})$is the production function of the car.

By differentiating the function with respect to $\mathrm{K},$we get,

$\begin{array}{c}\frac{\mathrm{d\pi }}{\mathrm{dK}}=\mathrm{P}\frac{\mathrm{F}(\mathrm{K},\mathrm{L})}{\mathrm{dK}}-\mathrm{r}\\ 0={\mathrm{PM}}_{\mathrm{K}}-\mathrm{r}\\ {\mathrm{PM}}_{\mathrm{K}}=\mathrm{r}---\left(2\right)\end{array}$

By differentiating the function $\pi$with respect to $\begin{array}{l}\frac{\mathrm{d\pi }}{\mathrm{dL}}=\mathrm{P}\frac{\mathrm{F}(\mathrm{K},\mathrm{L})}{\mathrm{dL}}-\mathrm{w}\\ 0={\mathrm{PM}}_{\mathrm{L}}-\mathrm{w}\\ {\mathrm{PM}}_{\mathrm{L}}=\mathrm{w}---\left(3\right)\end{array}$we get$\begin{array}{c}\frac{\mathrm{d\pi }}{\mathrm{dL}}=\mathrm{P}\frac{\mathrm{F}(\mathrm{K},\mathrm{L})}{\mathrm{dL}}-\mathrm{w}\\ 0={\mathrm{PM}}_{\mathrm{L}}-\mathrm{w}\\ {\mathrm{PM}}_{\mathrm{L}}=\mathrm{w}---\left(3\right)\end{array}$

Substituting the given value

After substituting the given values in equation (2), we get

$\begin{array}{c}{\mathrm{PM}}_{\mathrm{K}}=\mathrm{r}\\ 1,200,000\cdot 0.8=\mathrm{r}\\ \mathrm{r}=960,000\end{array}$

Hence, the rent of capital is $\mathbf{960}\mathbf{,}\mathbf{000}$

After substituting the given values in equation (3), we get

$\begin{array}{c}{\mathrm{PM}}_{\mathrm{L}}=\mathrm{w}\\ 1,200,000\cdot {\mathrm{M}}_{\mathrm{L}}=1,800\\ {\mathrm{M}}_{\mathrm{L}}=\frac{1,800}{1,200,000}\\ {\mathrm{M}}_{\mathrm{L}}=0.0015\end{array}$

Hence, the marginal product of capital is $\mathbf{0}\mathbf{.0015}$

The following equation can be used to calculate the marginal rate of technical substitution.$\mathrm{MRTS}=\frac{{\mathrm{M}}_{\mathrm{L}}}{{\mathrm{M}}_{\mathrm{K}}}$

After substituting the given values in equation (3), we get

$\begin{array}{l}\text{MRTS}=\frac{{\mathrm{M}}_{\mathrm{L}}}{{\mathrm{M}}_{\mathrm{K}}}\\ \text{MRTS}=\frac{0.0015}{0.8}\\ \text{MRTS}=0.0019\end{array}$

Hence, the marginal rate of technical substitution should be $\mathbf{0}.\mathbf{0019}.$