Question 3 (a)
Cartel of two firms: joint-profit-maximising outputs, price and profit
9 marks
- Time to write
- About 18 minutes
Question
Question 3 (a) · 9 marksFirms A and B form a cartel for maximization of joint profit. The cost function of firms A and B, respectively, is C₁ = 110 + 16 x₁ + 2 x₁² and C₂ = 56 + 40 x₂ + 2 x₂², where x₁ and x₂ are the outputs of firms A and B respectively. Market demand is given by x = 50 - 0.5 p, where x = x₁ + x₂. Obtain the number of units of the product each firm should produce to maximise joint profit. Compute the maximum joint profit and the price the firms charge?
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Read from the question
- ComputeFormula first, then the working, then the answer with its unit (₹, %, units).
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Joint profit function 2 marks 4 min
- 2 First-order conditions and outputs 3 marks 5 min
- 3 Second-order conditions 2 marks 4 min
- 4 Price and maximum profit 2 marks 3 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Try it first. About 18 minutes for 9 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
Joint profit function. Market demand x = 50 − 0.5p gives p = 100 − 2x, with x = x₁ + x₂.
Total revenue R = px = (100 − 2x₁ − 2x₂)(x₁ + x₂).
Joint profit π = R − C₁ − C₂ = 100(x₁ + x₂) − 2(x₁ + x₂)² − (110 + 16x₁ + 2x₁²) − (56 + 40x₂ + 2x₂²).
First-order conditions (partial derivatives set to zero; maxima of a function of two variables without constraints, as the DU guideline sets it):
∂π/∂x₁ = 100 − 4(x₁ + x₂) − 16 − 4x₁ = 84 − 8x₁ − 4x₂ = 0 ∂π/∂x₂ = 100 − 4(x₁ + x₂) − 40 − 4x₂ = 60 − 4x₁ − 8x₂ = 0
From these, 2x₁ + x₂ = 21 and x₁ + 2x₂ = 15, so x₁ = 9 and x₂ = 3 (total output x = 12).
π₁₁ = −8 < 0, π₂₂ = −8 < 0, π₁₂ = −4; π₁₁π₂₂ − π₁₂² = 64 − 16 = 48 > 0. So (9, 3) is a maximum.
Price and maximum joint profit
Price p = 100 − 2 × 12 = ₹76.
| Item | ₹ |
|---|---|
| Total revenue (76 × 12) | 912 |
| Cost of firm A: 110 + 16(9) + 2(81) | 416 |
| Cost of firm B: 56 + 40(3) + 2(9) | 194 |
| Maximum joint profit | 302 |
Joint profit = 912 − 416 − 194 = ₹302.
Check (cartel rule): MR = 100 − 4x = 52 at x = 12; MC₁ = 16 + 4x₁ = 52 and MC₂ = 40 + 4x₂ = 52. Each firm produces where its marginal cost equals the common marginal revenue.
Firm A should produce 9 units and firm B 3 units; the cartel charges ₹76 and earns a maximum joint profit of ₹302.
Where the 9 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Joint profit function | 2 |
| 2 | First-order conditions and outputs | 3 |
| 3 | Second-order conditions | 2 |
| 4 | Price and maximum profit | 2 |
| Total | 9 |
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