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When graduate microeconomics homework starts mixing monopoly pricing, inverse demand curves, and segmented markets, this is one of the places students get stuck fast. The confusion usually comes from seeing one firm charge different prices in different markets while still using one production decision. If you have been trying to understand why the firm does not simply set one common price everywhere, this question is really about how profit maximization works when demand conditions differ across groups. For broader theory support, see Microeconomics Tutoring NYC.
Answer First
Third-degree price discrimination sets marginal revenue equal to marginal cost in each served market because the monopolist chooses output across segments to maximize total profit subject to a single cost function. If one extra unit sold in Market A produces more marginal revenue than one extra unit sold in Market B, the firm can raise profit by shifting sales toward Market A and away from Market B. At the optimum, that arbitrage condition disappears, so the firm chooses quantities such that
\[ MR_1 = MR_2 = \cdots = MR_n = MC. \]
Prices can still differ across markets because each market has its own demand curve, so the quantity where marginal revenue equals marginal cost implies a different price in each segment.
Problem Setup
Suppose a monopolist can separate consumers into distinct markets and prevent resale between them. Let market \(i\) have inverse demand \(P_i(Q_i)\). Total output is the sum of segment outputs:
\[ Q = \sum_{i=1}^{n} Q_i. \]
Total revenue is
\[ TR = \sum_{i=1}^{n} P_i(Q_i)Q_i, \]
and profit is
\[ \pi = \sum_{i=1}^{n} P_i(Q_i)Q_i – C\!\left(\sum_{i=1}^{n} Q_i\right). \]
The key point is that revenue is market-specific, but cost depends on total production. That is why the firm must choose the allocation of output across markets as well as total output.
Step-by-Step Solution
1. Write the profit function
For two markets, profit is
\[ \pi = P_1(Q_1)Q_1 + P_2(Q_2)Q_2 – C(Q_1+Q_2). \]
2. Differentiate with respect to each market quantity
The first-order conditions are
\[ \frac{\partial \pi}{\partial Q_1} = MR_1 – MC = 0 \]
and
\[ \frac{\partial \pi}{\partial Q_2} = MR_2 – MC = 0. \]
So the optimal allocation satisfies
\[ MR_1 = MC \quad \text{and} \quad MR_2 = MC. \]
Combining them gives
\[ MR_1 = MR_2 = MC. \]
3. Interpret the condition
If \(MR_1 > MR_2\), the monopolist gains by shifting one unit of output from Market 2 to Market 1. Revenue rises more in Market 1 than it falls in Market 2, while total cost is essentially unchanged for that reallocation at the margin. So unequal marginal revenues cannot be optimal.
4. Explain why prices differ even when MC is the same
The equality condition is about marginal revenue, not price. In each market, marginal revenue lies below price when demand slopes downward. Since each segment has a different demand elasticity, the price associated with the quantity satisfying \(MR_i=MC\) will generally differ across markets.
A common way to express this is the Lerner-style relationship:
\[ \frac{P_i – MC}{P_i} = \frac{1}{|\varepsilon_i|}, \]
where \(|\varepsilon_i|\) is the absolute value of demand elasticity in market \(i\). Markets with less elastic demand get higher markups.
5. Numerical intuition
Imagine the firm can sell in two markets. If one additional unit in Market A adds \(20\) dollars of marginal revenue while one additional unit in Market B adds only \(12\), then selling more in A and less in B raises profit until the two marginal revenues are equalized. The final stopping point is where both equal marginal cost.
This is why the condition is not “set the same price everywhere.” It is “allocate output so the last unit sold in every market contributes the same marginal payoff after accounting for cost.”
Intuition
Third-degree price discrimination is really an output-allocation problem. The monopolist has one pool of productive capacity and several customer groups. Profit maximization requires sending units to the places where they earn the most revenue at the margin. As long as one market values the next unit more than another market does, the firm has not finished optimizing. The equal-marginal-revenue rule is the condition that says there is no further profitable reallocation left.
Common Exam Mistakes
- Confusing price equalization with marginal revenue equalization. Prices usually differ across markets.
- Writing \(P=MC\). That is competitive pricing, not monopoly price discrimination.
- Forgetting that the no-resale assumption is necessary. If consumers can arbitrage, segmented pricing collapses.
- Setting \(MR_1=MR_2\) but forgetting to also set them equal to \(MC\).
- Missing the role of elasticity: less elastic markets support higher prices.
Why This Matters
This result shows up everywhere in graduate microeconomics, industrial organization, and applied pricing problems. It is the backbone of segmented airline pricing, student discounts, region-specific pricing, and many monopoly regulation questions. Once you understand why third-degree price discrimination sets \(MR\) equal across markets and equal to \(MC\), many longer proofs and comparative statics questions become much easier.
Final Summary
A monopolist practicing third-degree price discrimination chooses quantities in each market to maximize total profit. Because total cost depends on total output, the firm must allocate production so that the last unit sold in every market generates the same marginal revenue, and that common marginal revenue equals marginal cost:
\[ MR_1 = MR_2 = \cdots = MR_n = MC. \]
Different prices across markets are consistent with this rule because each market has a different demand curve and therefore a different relationship between price and marginal revenue.
If this topic is showing up in homework, exam review, or a graduate micro problem set, Talk Directly to a Tutor, Not a Marketer. You can also call or text 510-398-0006 or email tutor@mytutornyc.com.