section iii: real world application (21 points)\n3. (21 points) a firm is currently producing their profit…

section iii: real world application (21 points)\n3. (21 points) a firm is currently producing their profit - maximizing quantity of 50 units of output using 40 hours of labor and 25 hours of capital. the marginal product of labor is 10 units of output per hour and the marginal product of capital is 30 units of output per hour.\na. if the wage rate for labor is $10 per hour and the rental rate for capital is $7 per hour what is the firm’s total cost given their current allocation of capital and labor? be sure to incorporate this information into your graph in part c. (5 points)\n(40*10)+(7*25)=575 tc = $575\nvc fc\nb. what is the equation to determine profit? show all components of the equation. (5 points) profit = tr - tc=(p*q)-(wages*labor + rental rate*capital)\ntr tc\nc. determine whether the firm is profit - maximizing given their allocation of capital and labor. make any necessary recommendations to the firm if their allocation of capital and labor should be altered. you must include a well - labeled diagram to support your conclusion. hint: at a tangency, mpₗ/mpₖ = w/r. (11 points)\n10/7≠10/30 not profit - maximizing\nthe firm is not profit - maximizing because the isoquant and isocost do not meet at a point of tangency

section iii: real world application (21 points)\n3. (21 points) a firm is currently producing their profit - maximizing quantity of 50 units of output using 40 hours of labor and 25 hours of capital. the marginal product of labor is 10 units of output per hour and the marginal product of capital is 30 units of output per hour.\na. if the wage rate for labor is $10 per hour and the rental rate for capital is $7 per hour what is the firm’s total cost given their current allocation of capital and labor? be sure to incorporate this information into your graph in part c. (5 points)\n(40*10)+(7*25)=575 tc = $575\nvc fc\nb. what is the equation to determine profit? show all components of the equation. (5 points) profit = tr - tc=(p*q)-(wages*labor + rental rate*capital)\ntr tc\nc. determine whether the firm is profit - maximizing given their allocation of capital and labor. make any necessary recommendations to the firm if their allocation of capital and labor should be altered. you must include a well - labeled diagram to support your conclusion. hint: at a tangency, mpₗ/mpₖ = w/r. (11 points)\n10/7≠10/30 not profit - maximizing\nthe firm is not profit - maximizing because the isoquant and isocost do not meet at a point of tangency

Answer

Explanation:

Step1: Calculate total cost

The total - cost formula is $TC = wL + rK$, where $w$ is the wage rate, $L$ is the amount of labor, $r$ is the rental rate of capital, and $K$ is the amount of capital. Given $w = 10$, $L = 40$, $r = 7$, and $K = 25$. $TC=(10\times40)+(7\times25)=400 + 175=575$

Step2: Write profit - equation

Profit $\pi=TR - TC$, where $TR$ is total revenue and $TC$ is total cost. $TR = P\times Q$ (price times quantity) and $TC=wL + rK$. So, $\pi=(P\times Q)-(wL + rK)$

Step3: Check profit - maximization condition

The profit - maximization condition for a firm in the input market is $\frac{MP_{L}}{MP_{K}}=\frac{w}{r}$, where $MP_{L}$ is the marginal product of labor, $MP_{K}$ is the marginal product of capital, $w$ is the wage rate, and $r$ is the rental rate of capital. Given $MP_{L}=10$, $MP_{K}=30$, $w = 10$, and $r = 7$. $\frac{MP_{L}}{MP_{K}}=\frac{10}{30}=\frac{1}{3}$, and $\frac{w}{r}=\frac{10}{7}$. Since $\frac{1}{3}\neq\frac{10}{7}$, the firm is not profit - maximizing. The firm should adjust its input combination. It should increase the use of capital and decrease the use of labor because the marginal product per dollar of capital ($\frac{MP_{K}}{r}=\frac{30}{7}\approx4.29$) is greater than the marginal product per dollar of labor ($\frac{MP_{L}}{w}=\frac{10}{10} = 1$)

Answer:

a. The firm's total cost is $$575$. b. The profit equation is $\pi=(P\times Q)-(wL + rK)$ c. The firm is not profit - maximizing. It should increase the use of capital and decrease the use of labor.