Question

A firm’s product sells for$\mathbf{\backslash (}\mathbf{4}$per unit in a highly competitive market. The firm produces output using capital (which it rents at $\mathbf{\backslash )}\mathbf{25}$(per hour) and labor (which is paid a wage of $\mathbf{\$}\mathbf{30}$per hour under a contract for $\mathbf{20}$hours of labor services). Complete the following table and use that information to answer these questions.

a. Identify the fixed and variable inputs.

b. What are the firm’s fixed costs?

c. What is the variable cost of producing $\mathbf{475}$units of output?

d. How many units of the variable input should be used to maximize profits?

e. What are the maximum profits this firm can earn?

f. Over what range of the variable input usage do increase marginal returns exist?

g. Over what range of the variable input usage do decreasing marginal returns exist?

h. Over what range of input usage do negative marginal returns exist?

Short answer

Fixed and variable costs are considered to be the cost the producer had to bear for inputs such as capital and labor respectively.

Step-by-step solution

Table:1

K	L	Q	${\mathbf{MP}}_{\mathbf{K}}$	${\mathbf{AP}}_{\mathbf{K}}$	${\mathbf{AP}}_{\mathbf{L}}$	${\mathbf{VMP}}_{\mathbf{K}}$
0	20	0	50	0	0	200
1	20	50	100	50	2.5	400
2	20	150	150	75	7.5	600
3	20	300	100	100	15	400
4	20	400	50	100	20	200
5	20	450	25	90	22.5	100
6	20	475	0	79.166	23.75	0
7	20	475	-25	67.85	23.75	-100
8	20	450	-50	56.25	22.5	-200
9	20	400	-100	44.44	20	-400
10	20	300	-150	30	15	-600
11	20	150	13.63636	13.63	7.5	54.54

Marginal Productivity of capital $M{P}_K$is the ratio between the change in output and the change in Capital.

${\mathrm{MP}}_{\mathrm{K}}=\left(\frac{\Delta \mathrm{Q}}{\Delta \mathrm{K}}\right)$

The average productivity of capital$\left(A{P}_{\mathrm{K}}\right)=\frac{\mathrm{Q}}{\mathrm{K}}$

The average productivity of labor$\left(A{P}_L\right)=\frac{\mathrm{Q}}{L}$

Value of marginal productivity of capital$\left({\mathrm{VMP}}_{\mathrm{K}}\right)={\mathrm{P}}_{\text{output}}\times {\mathrm{MP}}_{\mathrm{K}}$

Solution 

(a). The fixed and variable inputs of the firm are labor and capital respectively as we can see that the labor is constant and the capital units are variable.

(b). Given the wage rate of labor $\left(w\right)=\$30$

Hence, the fixed cost can be written as

$$\begin{gathered} FC=w*L \\ =30*20 \\ =600 \end{gathered}$$

Hence, the fixed cost is $\$600$for every output.

(c). The variable cost of producing $475$units of output is

Given the rent rate of localid="1657885687834" $capital\left(r\right)=\$25$

localid="1657885696529" $$\begin{gathered} VC=r*K \\ =25*(6+7) \\ =325 \end{gathered}$$

Hence the variable cost is localid="1657885706630" $\$325for475$units of output.

(d) Table : 2

K	L	Q	FC	VC	TC	TR=PXQ	Profit
0	20	0	600	0	600	0	-600
1	20	50	600	25	625	200	-425
2	20	150	600	50	650	600	-50
3	20	300	600	75	675	1200	525
4	20	400	600	100	700	1600	900
5	20	450	600	125	725	1800	1075
6	20	475	600	150	750	1900	1150
7	20	475	600	175	775	1900	1125
8	20	450	600	200	800	1800	1000
9	20	100	600	225	825	1600	775
10	320	300	600	250	850	1200	350
11	20	150	600	275	875	600	-275

20The following table shows that at variable input K having a value of 6, the profit is localid="1657885754881" $\$1150$which is greater as compared to the other. Hence, localid="1657885765548" $K=6$is the optimal input for profit maximization.

(e) From the table above the maximum profits the firm can earn are found to be localid="1657885779771" $\$1075,\$1150,\$1125,and\$1000$.

(f) From table localid="1657885726546" $1$we can see that the increasing marginal return of the variable input exists to be within K equal to localid="1657885792715" $0to2$where the value of localid="1657885812145" $M{P}_K$changes from localid="1657885801655" $50to100and150$respectively.

(g). From table localid="1657885836824" $1$we can see that the decreasing marginal return of the variable input exists to be within K equal to .

(h). From table localid="1657885824036" $1$we can see that the negative marginal return exists to be within K equal to localid="1657885849859" $7to10$.