Monopoly: Marginal Revenue and Deadweight Loss

Look at a market with only one seller. That single firm chooses price and quantity alone.

The monopolist slides down the demand curve to set the price. Selling one more unit means cutting the price, so marginal revenue falls twice as fast as demand. It stops where marginal revenue meets marginal cost. Producing less than competition leaves welfare destroyed. That triangle is the deadweight loss. The markup over cost is measured by the Lerner index below.

For example, say the neighborhood has exactly one snack 🏪 shop. The owner sets the price of tteokbokki alone. Charge too much and friends stop buying. Charging a bit more and selling less can still raise profit. But the friends who go without feel the loss. That loss is the deadweight loss.

MR(Q)=A−2BQ,Qm=A−c2BMR(Q) = A - 2BQ, \qquad Q_m = \frac{A-c}{2B}
Pm−cPm=1∣ε∣\frac{P_m - c}{P_m} = \frac{1}{|\varepsilon|}

Part of these learning paths

Examples & Exercises

With demand P=10−QP = 10 - Q and marginal cost c=2c = 2, compute by hand the monopolist's profit-maximizing quantity, price, profit, and deadweight loss.

MR(Q)=10−2Q,MR=MCMR(Q) = 10 - 2Q, \qquad MR = MC
Qm=A−c2B,Pm=A+c2Q_m = \frac{A-c}{2B}, \quad P_m = \frac{A+c}{2}
A=10A = 10, c=2c = 2 — the monopoly point M and the DWL triangle.
Show proof

  1. Marginal revenue is MR=10−2QMR = 10 - 2Q. From MR=MCMR = MC, 10−2Q=210 - 2Q = 2, so Qm=4Q_m = 4, and on the demand curve Pm=10−4=6P_m = 10 - 4 = 6.

  2. Profit is (6−2)⋅4=16(6-2) \cdot 4 = 16. The competitive quantity is Qc=8Q_c = 8, so deadweight loss is (6−2)(8−4)/2=8(6-2)(8-4)/2 = 8 — welfare the monopoly destroys without capturing.

    πm=16,DWL=8=(Pm−c)(Qc−Qm)2\pi_m = 16, \qquad DWL = 8 = \frac{(P_m - c)(Q_c - Q_m)}{2}

In the same example, verify numerically the monopoly inverse-elasticity rule P−MCP=1∣ε∣\frac{P - MC}{P} = \frac{1}{|\varepsilon|}.

Elasticity at Qm=4Q_m = 4, Pm=6P_m = 6.
Show proof

  1. At the monopoly point the price elasticity of demand is ε=−dQdPPQ=−(−1)⋅64=1.5\varepsilon = -\frac{dQ}{dP}\frac{P}{Q} = -(-1)\cdot\frac{6}{4} = 1.5.

  2. The markup is (6−2)/6=2/3(6-2)/6 = 2/3, and 1/∣ε∣=1/1.5=2/31/|\varepsilon| = 1/1.5 = 2/3 — an exact match. The more elastic demand (larger ∣ε∣|\varepsilon|), the less room the monopolist has to raise price.

    Pm−cPm=23=11.5\frac{P_m - c}{P_m} = \frac{2}{3} = \frac{1}{1.5}

11. Market Power