A firm has two factories, for which costs are given by:
Factory #I: C1(Q1) = 10Q12
Factory #2: C2(Q2) = 20Q22
The firm faces the following demand curve:
P = 700 – 5Q
where Q is total output, i.e., Q = Q1 + Q2.
On a diagram, draw the marginal cost curves for the two factories, the average and marginal revenue curves, and the total marginal cost curve (i.e., the marginal cost of producing Q = Q1 + Q2). Indicate the profit-maximizing output for each factory, total output, and price.
Calculate the values of Q1, Q2, Q, and P that maximize profit.
Suppose labor costs increase in Factory #I but not in Factory #2. How should the firm adjust (i.e., raise, lower, or leave unchanged): Output in Factory #1? Output in Factory #2? Total output? Price?
The Correct Answer and Explanation is:
Solution:
Step 1: Given
- Factory 1 Cost: C1(Q1)=10Q12C_1(Q_1) = 10Q_1^2 → MC1=dC1dQ1=20Q1MC_1 = \frac{dC_1}{dQ_1} = 20Q_1
- Factory 2 Cost: C2(Q2)=20Q22C_2(Q_2) = 20Q_2^2 → MC2=dC2dQ2=40Q2MC_2 = \frac{dC_2}{dQ_2} = 40Q_2
- Demand: P=700−5QP = 700 – 5Q, where Q=Q1+Q2Q = Q_1 + Q_2
- Total Revenue: TR=P⋅Q=(700−5Q)Q=700Q−5Q2TR = P \cdot Q = (700 – 5Q)Q = 700Q – 5Q^2
- Marginal Revenue (MR): MR=dTRdQ=700−10QMR = \frac{dTR}{dQ} = 700 – 10Q
Step 2: Profit Maximization Condition
The firm maximizes profit when:
- MR=MC1=20Q1MR = MC_1 = 20Q_1
- MR=MC2=40Q2MR = MC_2 = 40Q_2
So: 20Q1=700−10Q(1)40Q2=700−10Q(2)20Q_1 = 700 – 10Q \quad \text{(1)} \\ 40Q_2 = 700 – 10Q \quad \text{(2)}
Since Q=Q1+Q2Q = Q_1 + Q_2, substitute this into both:
From (1):
Q1=700−10(Q1+Q2)20Q_1 = \frac{700 – 10(Q_1 + Q_2)}{20}
From (2):
Q2=700−10(Q1+Q2)40Q_2 = \frac{700 – 10(Q_1 + Q_2)}{40}
Now solve these equations simultaneously.
From (1):
20Q1=700−10(Q1+Q2)20Q_1 = 700 – 10(Q_1 + Q_2)
→ 20Q1+10Q1+10Q2=70020Q_1 + 10Q_1 + 10Q_2 = 700
→ 30Q1+10Q2=70030Q_1 + 10Q_2 = 700
→ (a) 3Q1+Q2=703Q_1 + Q_2 = 70
From (2):
40Q2=700−10(Q1+Q2)40Q_2 = 700 – 10(Q_1 + Q_2)
→ 40Q2+10Q1+10Q2=70040Q_2 + 10Q_1 + 10Q_2 = 700
→ 10Q1+50Q2=70010Q_1 + 50Q_2 = 700
→ (b) Q1+5Q2=70Q_1 + 5Q_2 = 70
Solving (a) and (b):
Multiply (a) by 5:
→ 15Q1+5Q2=35015Q_1 + 5Q_2 = 350
Now subtract (b):
15Q1+5Q2−(Q1+5Q2)=350−7015Q_1 + 5Q_2 – (Q_1 + 5Q_2) = 350 – 70
→ 14Q1=28014Q_1 = 280
→ Q1=20Q_1 = 20
Substitute into (a):
3(20)+Q2=703(20) + Q_2 = 70
→ Q2=10Q_2 = 10
Then:
- Total Output: Q=20+10=30Q = 20 + 10 = 30
- Price: P=700−5(30)=550P = 700 – 5(30) = 550
Answer Summary:
- Q₁ = 20
- Q₂ = 10
- Q = 30
- P = 550
Adjustment to Labor Cost Increase in Factory 1:
If labor costs increase in Factory 1, its marginal cost curve shifts upward. Thus, for any given price, Factory 1 should produce less, and Factory 2 should produce more, since its cost structure remains unchanged. However, total output should fall, and price should rise slightly, because the overall marginal cost of production has increased.
Explanation
To determine the firm’s optimal output and pricing strategy, we analyze marginal revenue (MR) and marginal costs (MC) for both factories. Each factory’s cost structure is quadratic, so their MCs are linear: MC1=20Q1MC_1 = 20Q_1, MC2=40Q2MC_2 = 40Q_2. Given a market demand curve P=700−5QP = 700 – 5Q, the firm’s total revenue is TR=P⋅Q=700Q−5Q2TR = P \cdot Q = 700Q – 5Q^2, making MR = 700−10Q700 – 10Q.
For profit maximization, the firm allocates output so that MR equals the marginal cost of each factory. Setting MR=MC1=20Q1MR = MC_1 = 20Q_1 and MR=MC2=40Q2MR = MC_2 = 40Q_2 leads to a system of equations that, when solved, yields Q1=20Q_1 = 20, Q2=10Q_2 = 10, and total output Q=30Q = 30. Plugging into the demand curve, the corresponding price is P=550P = 550.
If labor costs increase in Factory 1, the cost and thus the marginal cost of producing each unit in Factory 1 rises. This shifts MC1MC_1 upward. To maintain profit-maximization (where MR = MC), Factory 1 should reduce output. The firm will compensate by increasing production in Factory 2, up to the point where MR=MC2MR = MC_2 again. However, since Factory 2 has a steeper MC curve, total output will decrease overall. Lower total supply leads to a higher market-clearing price.
In summary:
- Output in Factory 1: Decrease
- Output in Factory 2: Increase
- Total Output: Decrease
- Price: Increase
