2 marksMicroeconomicsMonopoly and price discriminationMonopoly equilibrium and marginal revenueHardabout 3 min

GATE 2023 Question 65: Monopoly and price discrimination

A monopolist is facing the demand function $Q = \dfrac{100}{(P-1)}$ , where $Q$ is the quantity demanded and $P$ is the price per unit of the good ( $P>1$ ). The average variable cost for the monopolist is $\dfrac{4}{\sqrt{Q}}$ and the fixed cost is 10. The profit maximizing price is ______ (in integer).

Answer and worked solution

Answer: 26

Step 1 — Invert the demand function. $Q=\dfrac{100}{P-1} \Rightarrow P-1=\dfrac{100}{Q} \Rightarrow P = 1+\dfrac{100}{Q}$ .

Step 2 — Total and marginal revenue.

$TR = PQ = \left(1+\frac{100}{Q}\right)Q = Q+100$

$MR = \frac{d(TR)}{dQ} = 1$

Step 3 — Variable cost and marginal cost. $VC = AVC \times Q = \dfrac{4}{\sqrt Q}\times Q = 4\sqrt Q = 4Q^{1/2}$ .

$MC = \frac{d(VC)}{dQ} = 4 \times \frac{1}{2}Q^{-1/2} = \frac{2}{\sqrt Q}$

Step 4 — Set MR = MC.

$1 = \frac{2}{\sqrt Q} \quad\Rightarrow\quad \sqrt Q = 2 \quad\Rightarrow\quad Q=4$

Step 5 — Find the profit-maximizing price.

$P = 1+\frac{100}{4} = 1+25 = 26$

Therefore the profit-maximizing price is 26. (Note: the fixed cost of 10 doesn't affect the profit-maximizing price/quantity choice at all, since it drops out of the MR=MC first-order condition — it only affects the resulting profit level.)

The trap: 25

A common slip is stopping at $Q=4$ and then either reporting $Q$ itself or the '100/Q' term alone (=25) as the final answer, forgetting to add back the '+1' from the inverted demand function when computing the actual profit-maximizing price.

Plain-text version

A monopolist is facing the demand function Q = 100/((P-1)), where Q is the quantity demanded and P is the price per unit of the good (P>1). The average variable cost for the monopolist is 4/√Q and the fixed cost is 10. The profit maximizing price is (in integer).

Also written as: A monopolist is facing the demand function Q = 100/((P-1)), where Q is the quantity demanded and P is the price per unit of the good (P>1). The average variable cost for the monopolist is 4/sqrt(Q) and the fixed cost is 10. The profit maximizing price is (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.

TestStop has 43 past-paper questions on this subtopic; the nearest in year are below.

Take this paper as a timed mock More on Monopoly equilibrium and marginal revenue

Question from the GATE 2023 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.