A firm uses a single plant with costs C = 160 + 16Q + .1Q2 and faces the price equation P = 96 - .4Q.
a. Find the firm’s profit-maximizing price and quantity. What is its profit?
b. The firm’s production manager claims that the firm’s average cost of production is minimized at an output of 40 units. Furthermore, she claims that 40 units is the firm’s profit-maximizing level of output. Explain whether these claims are correct.
c. Could the firm increase its profit by using a second plant (with costs identical to the first) to produce the output in part (a)? Explain.
a. To find the firm's profit-maximizing price and quantity, we need to first calculate the firm's total cost (TC) and total revenue (TR). Then, we can determine the quantity (Q) at which profit is maximized.
The firm's total cost (TC) can be calculated using the cost equation:
TC = C = 160 + 16Q + .1Q^2
The firm's total revenue (TR) can be calculated using the price equation:
TR = P x Q = (96 - .4Q) x Q
To maximize profit, we need to find the quantity (Q) that maximizes the difference between total revenue and total cost: Profit = TR - TC.
Profit = TR - TC
Profit = (96 - .4Q) x Q - (160 + 16Q + .1Q^2)
To find the quantity (Q) at which the profit is maximized, we can take the derivative of the profit function and set it equal to zero. This will give us the critical points, one of which will be the maximum profit point.
d(Profit)/dQ = 0
(96 - .4Q) - (160 + 16Q + .1Q^2) = 0
Simplifying this equation will result in a quadratic equation that can be solved to find the quantity (Q) at which profit is maximized. Substituting the value of Q back into the price equation (P = 96 - .4Q) will give us the profit-maximizing price.
Once we have the profit-maximizing quantity and price, we can calculate the profit by substituting these values into the profit equation (Profit = TR - TC).
b. The firm's production manager's claim that the firm's average cost of production is minimized at an output of 40 units, and that 40 units is the firm's profit-maximizing level of output can be verified by determining the average cost at different levels of output.
The average cost (AC) can be calculated as:
AC = TC / Q
By calculating the average cost at different output levels, such as 40 units, we can compare it to the average cost at other levels. If the average cost is indeed minimized at 40 units, then the manager's claim is correct. However, this does not necessarily mean that the profit is maximized at 40 units.
To determine the profit-maximizing level of output, we need to follow the steps outlined in part a, which involves maximizing the difference between total revenue and total cost.
c. Whether the firm can increase its profit by using a second plant to produce the output in part a depends on the economies of scale and the additional costs associated with setting up and operating a second plant.
If the firm can achieve economies of scale (e.g., decreased costs per unit of output due to increased production), then it may be able to increase its profit by operating a second plant. However, if the costs of setting up and operating a second plant outweigh the benefits of increased production, then the firm may not be able to increase its profit.
To make a conclusive determination, the firm would need to analyze the cost and revenue implications of operating a second plant and compare it to the profits achieved with a single plant.
a. To find the firm's profit-maximizing price and quantity, we need to set the marginal cost equal to the marginal revenue.
The marginal cost (MC) is the derivative of the cost function C with respect to quantity (Q), which is given by:
MC = dC/dQ = 16 + 0.2Q
The marginal revenue (MR) is the derivative of the price function P with respect to quantity (Q), which is given by:
MR = dP/dQ = -0.4
Setting MC equal to MR, we have:
16 + 0.2Q = -0.4
0.2Q = -16.4
Q = -16.4 / 0.2
Q = -82
Since quantity cannot be negative, we discard this value. It means that no valid quantity exists where MC equals MR. Therefore, we cannot directly find the profit-maximizing quantity using the given cost and price equations.
b. The production manager's claim that the firm's average cost of production is minimized at an output of 40 units is unrelated to the profit-maximizing level of output. Average cost is minimized where marginal cost equals average cost, not at the output level that maximizes profit.
Based on the information provided, we cannot determine the profit-maximizing level of output without further analysis.
c. To determine if the firm could increase profit by using a second plant, we need to consider the additional cost of operating a second plant and compare it to the potential increase in revenue.
Since the given cost equation represents the cost of a single plant, let's analyze the scenario using the given equation for Q = 82 (rounded up from -82) as a reference.
For Q = 82, the cost is:
C = 160 + 16Q + 0.1Q^2
C = 160 + 16(82) + 0.1(82)^2
C ≈ $3,680
Now, let's consider the cost of using two plants with the same cost equation and the quantity from part a:
Total cost with two plants = 2C
Total cost with two plants ≈ 2 * $3,680
Total cost with two plants ≈ $7,360
If the price equation remains the same with a quantity of Q = 82, the revenue will be:
P = 96 - 0.4Q
P = 96 - 0.4(82)
P ≈ $64.80
Profit with one plant ≈ Revenue - Cost
Profit with one plant ≈ $64.80 - $3,680
Profit with one plant ≈ -$3,615.20
Profit with two plants ≈ Revenue - Cost
Profit with two plants ≈ $64.80 - $7,360
Profit with two plants ≈ -$7,295.20
Comparing the profits, we can see that using a second plant would result in even higher losses. Therefore, the firm would not increase its profit by using a second plant to produce the output in part a.
In summary:
a. The profit-maximizing price and quantity cannot be determined with the given information.
b. The claims of the production manager are incorrect. The average cost minimized output does not necessarily coincide with the profit-maximizing level of output.
c. The firm would not increase its profit by using a second plant.