2 marksMicroeconomicsMonopoly and price discriminationMonopoly equilibrium and marginal revenueHardabout 4 min

GATE 2021 Question 36: Monopoly and price discrimination

Suppose the demand for a new pharmaceutical drug, on which the manufacturer has a patent monopoly, is given by: $Q=(100-P)A^{0.5}$ ; where $Q$ is output, $P$ is the price and $A$ is advertising expenditure. Production cost of the patented drug is given by: $C(Q)=60Q$ . At the firm's optimal choices, the ratio of advertising expenditure to sales revenue for the pharmaceutical product will be 1:______ (in integer).

Answer and worked solution

Answer: 8

Profit: $\pi = PQ - 60Q - A = (P-60)(100-P)\sqrt A - A$ .

FOC with respect to $P$ : treating $A$ as given,

$\frac{\partial \pi}{\partial P} = \sqrt A\left[(100-P)-(P-60)\right] = \sqrt A(160-2P) = 0 \Rightarrow P^*=80$

FOC with respect to $A$ : treating $P$ as given,

$\frac{\partial \pi}{\partial A} = (P-60)(100-P)\cdot\frac{1}{2\sqrt A} - 1 = 0$

Substituting $P=80$ : $(80-60)(100-80) = 20\times20=400$ , so:

$\frac{400}{2\sqrt A} = 1 \Rightarrow \sqrt A = 200 \Rightarrow A^* = 40{,}000$

Compute equilibrium quantity and sales revenue:

$Q^* = (100-80)\sqrt{40{,}000} = 20\times200 = 4{,}000$

$\text{Sales Revenue} = P^*Q^* = 80\times4{,}000 = 320{,}000$

Advertising-to-revenue ratio:

$\frac{\text{Revenue}}{\text{Advertising}} = \frac{320{,}000}{40{,}000} = 8$

So Advertising : Revenue $= 1:8$ .

Therefore the answer is 8.

The trap: 6

Students sometimes solve only one of the two first-order conditions (e.g., finding $P^*$ correctly but then guessing or approximating $A^*$ instead of solving its own FOC properly), or make an arithmetic slip in the somewhat involved algebra of the joint optimization, landing on a plausible but incorrect ratio.

Plain-text version

Suppose the demand for a new pharmaceutical drug, on which the manufacturer has a patent monopoly, is given by: Q=(100-P)A⁰·⁵; where Q is output, P is the price and A is advertising expenditure. Production cost of the patented drug is given by: C(Q)=60Q. At the firm's optimal choices, the ratio of advertising expenditure to sales revenue for the pharmaceutical product will be 1: (in integer).

Also written as: Suppose the demand for a new pharmaceutical drug, on which the manufacturer has a patent monopoly, is given by: Q=(100-P)A^(0.5); where Q is output, P is the price and A is advertising expenditure. Production cost of the patented drug is given by: C(Q)=60Q. At the firm's optimal choices, the ratio of advertising expenditure to sales revenue for the pharmaceutical product will be 1: (in integer).

Monopoly and price discrimination has come up 128 times in 7 exams since 2004.

What this question tests

This question is on Monopoly equilibrium and marginal revenue, part of Monopoly and price discrimination in Microeconomics.

How a monopolist sets price and output where marginal revenue equals marginal cost.

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