Price Discrimination showed a seller charging different groups different prices. If the seller can tell the groups apart and stop resale between them, the markup rule tells it exactly what to charge each one.
Key idea
With one marginal cost and separate markets, set $$MR_1(q_1) = MR_2(q_2) = MC.$$ Each market's price follows the markup rule $p_i\left(1 - \dfrac{1}{|\varepsilon_i|}\right) = MC$, so the market with the less elastic demand pays the higher price.
Why it works
Suppose $MR_1 > MR_2$. Selling one unit fewer in market 2 and one more in market 1 leaves cost unchanged and raises revenue by $MR_1 - MR_2$. So at the optimum the marginal revenues must be equal, and each must equal MC (otherwise changing total output pays). Equal MRs do not mean equal prices: a market where buyers are less sensitive to price gets a bigger markup.
Worked example: Two linear markets
Students: $p_1 = 40 - q_1$. Other adults: $p_2 = 80 - 2q_2$. $MC = 10$, no resale.
- Students: $MR_1 = 40 - 2q_1 = 10$, so $q_1 = 15$ and $p_1 = 25$.
- Adults: $MR_2 = 80 - 4q_2 = 10$, so $q_2 = 17.5$ and $p_2 = 45$.
- Elasticities at those points: students $\dfrac{25}{15} \approx 1.67$, adults $\dfrac{1}{2}\cdot\dfrac{45}{17.5} \approx 1.29$. The adults' demand is less elastic, and they pay more.
Answer: Students \$25, adults \$45.
Worked example: Constant elasticities
MC = \$6. Market A has $\varepsilon = -2$ and market B has $\varepsilon = -4$.
- A: $p\left(1 - \tfrac{1}{2}\right) = 6$, so $p = 12$.
- B: $p\left(1 - \tfrac{1}{4}\right) = 6$, so $p = 8$.
Answer: \$12 in A, \$8 in B.
Common mistake
Setting the prices, rather than the marginal revenues, equal across markets, or charging more where demand is more elastic. Equal prices throw away the profit from tailoring each price; a high price where buyers are very price-sensitive drives them away. Check: compute MR in each market at your quantities. They must be equal to each other and to MC.
Three conditions make this possible: market power, a way to tell groups apart (age, location, time of purchase), and no resale. Whether discrimination raises or lowers total surplus compared with one price depends on the case: it can open a market that a single price would leave unserved, or it can cut total output.