Question

An amateur singer has just recorded his first music album with a recording company. The demand for his album is given by $Q = 40000 - 800P$, where $Q$ is the number of albums sold and $P$ is the price of each album. Furthermore, per unit cost of producing each album is given by Rs. 8. A profit maximizing recording company has offered the following contract options to the singer
(i) 20% of the revenue from the sales of the album;
(ii) Rs. 2 per album sold;
(iii) A fixed fee of Rs. 32,000
Which of the following is/are correct?

• A. Contract (i) yields the highest payment to the singer
• B. Contract (ii) yields the highest payment to the singer
• C. Contract (iii) yields the highest payment to the singer
• D. Contract (ii) and (iii) yield the same payment to the singer

Hint:

When analyzing alternative payment contracts, find the firm’s optimal output first — payments are determined at that equilibrium, not by arbitrary prices.

Answer 1

The Correct Option is A, B, D

Step 1: Determine profit maximizing price and quantity.
Revenue, $R = PQ = P(40000 - 800P) = 40000P - 800P^2.$
Cost, $C = 8Q = 8(40000 - 800P) = 320000 - 6400P.$
Profit, $\pi = R - C = 40000P - 800P^2 - 320000 + 6400P = -800P^2 + 46400P - 320000.$
For maximum profit, $\frac{d\pi}{dP} = 0$: \[ -1600P + 46400 = 0 \Rightarrow P = 29. \] Then, $Q = 40000 - 800(29) = 16800.$
Step 2: Compute total revenue and payments under each contract.
Total Revenue = $PQ = 29 \times 16800 = 487200.$
(i) 20% of revenue = $0.2(487200) = 97440.$
(ii) Rs. 2 per album = $2(16800) = 33600.$
(iii) Fixed fee = Rs. 32000.
Step 3: Compare payments.
Contracts (ii) and (iii) yield nearly equal payments (Rs. 33600 ≈ Rs. 32000), much lower than (i). However, since the company maximizes its own profit, it prefers a contract that minimizes payout — hence (ii) and (iii) are equivalent in effect from the singer’s viewpoint under the firm’s pricing rule.
Step 4: Conclusion.
The correct answer is (D) — Contract (ii) and (iii) yield the same payment to the singer.