Price Discrimination Math — Practice & Proofs

Third-degree price discrimination is two independent monopoly problems joined by one $M C$ . Solve each market on its own. Set $M R_{i} = M C$ . Read each price off that market’s demand curve.

Tip

Two-market recipe

With $P_{A} = a_{A} - b_{A} Q_{A}$ , $P_{B} = a_{B} - b_{B} Q_{B}$ , and a common $M C = c$ :

Market A: $M R_{A} = a_{A} - 2 b_{A} Q_{A} = c \Rightarrow Q_{A} = (a_{A} - c) / (2 b_{A})$ , $P_{A} = (a_{A} + c) /2$ .

Market B: same algebra with $(a_{B}, b_{B})$ .

The market with the higher choke price $a$ ends up at the higher price; the slope $b$ does not affect price at all. For the same choke price, the market with the steeper slope $b$ ends up with the smaller quantity.

Worked Example

A cinema sells to adults and students. Adults: $P_{A} = 100 - 2 Q_{A}$ . Students: $P_{S} = 60 - Q_{S}$ . $M C = 10$ dollars per seat. Find prices and quantities for both groups.

Practice · 1 / 4

Wrong answers cost no XP

Drill the two-market algebra. Procedural variants re-roll.

Market A has $P_{A} = 70 - 2 Q_{A}$ , $M C = 15$ . Find $Q_{A}$ . (Round to 2 decimals.)

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