Third-Degree Price Discrimination: Optimal Pricing and Profit Maximization

Introduction

In our previous lesson, we analyzed first-degree price discrimination, the theoretical ideal where a firm captures all consumer surplus by knowing each buyer's maximum willingness to pay. While this provided a crucial benchmark, its assumption of perfect information is rarely met in reality.

Today, we move to a far more common and practical strategy: third-degree price discrimination. This lesson will equip you to calculate the optimal prices and profits a firm can achieve by segmenting its market into distinct groups. Instead of charging every individual a different price, the firm identifies groups of consumers with different demand characteristics and sets a profit-maximizing price for each group.

This topic directly builds on your analytical skills. You will treat each market segment as a separate monopoly problem, solve for its equilibrium, and then aggregate the results to determine the firm's overall profit. This process of breaking down a complex problem into independent, solvable parts should be familiar and satisfying.

The Logic of Market Segmentation

Third-degree price discrimination occurs when a firm can:

Exercise market power.
Divide its customers into two or more distinct groups based on observable characteristics (e.g., age, location, income level, time of purchase).
Prevent arbitrage, meaning consumers in the low-price group cannot easily resell the product to consumers in the high-price group.

The core principle is to charge a different price to each group based on its specific price elasticity of demand. Groups that are less sensitive to price changes (more inelastic demand) will be charged a higher price, while groups that are more sensitive (more elastic demand) will be charged a lower price.

A classic example is airline ticketing.

This diagram illustrates how an airline might price discriminate for a San Francisco-New York flight. On the left, last-minute travelers (often business flyers) have inelastic demand (PED < 1) and face a high price of \$300. On the right, leisure travelers who book in advance have more elastic demand (PED > 1) and are offered a lower price of \$150. The firm analyzes each market separately to maximize its total profit.

This is a strategy you might have implicitly considered in your lingerie design hobby. Selling a standard bikini design to students (who are typically more price-sensitive) might require a lower price point than selling the same item in a boutique frequented by high-income professionals. The key is identifying these separate demand curves.

To understand the concept more formally, please read the introductory section of the following document.

[PDF] Third-Degree Price Discrimination - Wsu - Washington State University

This document from Washington State University provides a concise definition of third-degree price discrimination.

Read the two paragraphs under the heading "Third-Degree Price Discrimination". Focus on how a firm can split its market into identifiable groups with different demand functions.

The Profit-Maximizing Rule for Each Market

How does a firm decide the optimal price and quantity for each group? It follows a simple extension of the monopolist's rule we've already learned. A firm practicing third-degree price discrimination will maximize its total profit by producing a quantity for each market segment where the marginal revenue from that segment equals the firm's overall marginal cost.

If the firm has two market segments, 1 and 2, the condition is:

M R_{1} = M C and M R_{2} = M C

This implies that the marginal revenue from the last unit sold in each market must be equal:

M R_{1} = M R_{2} = M C

Think about it logically: if $M R_{1} > M R_{2}$ , the firm could increase its profit by reallocating a unit from market 2 to market 1. It would lose $M R_{2}$ but gain $M R_{1}$ , a net positive. The firm stops reallocating only when the marginal revenues are equalized and are both equal to the marginal cost of production.

A Step-by-Step Guide to Calculation

The process of finding the optimal prices and profits involves treating each market segment as a separate monopoly problem, linked only by the common marginal cost.

The following video provides an excellent, clear walkthrough of a numerical problem, demonstrating the exact steps you'll need for your exam.

Third-Degree Price Discrimination: How to Solve

This video, "Third-Degree Price Discrimination: How to Solve," from the channel Economics in Many Lessons, breaks down the calculation process into manageable steps.

Please watch the video from the beginning until the profit calculation. Pay close attention to the sequence: Deriving inverse demand for each market. Finding marginal revenue for each market. Setting MR = MC in each market to find the optimal quantity and then the price for each. Calculating total profit by summing the revenues and subtracting the total cost.

Worked Example

Let's consolidate this method with a fresh example. A cinema operator has identified two customer segments for its movie screenings: adults and students. The market demands and the firm's costs are:

Adult Demand: $Q_{A} = 200 - P_{A}$
Student Demand: $Q_{S} = 160 - 2 P_{S}$
Total Cost: $T C = 1000 + 20 Q$ , where $Q = Q_{A} + Q_{S}$ .

From the total cost function, we can see that marginal cost ( $M C$ ) is constant at £20.

Step 1: Find the Inverse Demand for Each Market.

Adults: $P_{A} = 200 - Q_{A}$
Students: $2 P_{S} = 160 - Q_{S} ⟹ P_{S} = 80 - 0.5 Q_{S}$

Step 2: Find the Marginal Revenue for Each Market.
Using the rule for linear demand (same intercept, twice the slope):

Adults: $M R_{A} = 200 - 2 Q_{A}$
Students: $M R_{S} = 80 - Q_{S}$

Step 3: Set $M R = M C$ in Each Market to Find Quantities.

Adult Market: $M R_{A} = M C ⟹ 200 - 2 Q_{A} = 20$ $180 = 2 Q_{A} ⟹ Q_{A} = 90$
Student Market: $M R_{S} = M C ⟹ 80 - Q_{S} = 20$ $Q_{S} = 60$

The cinema will sell 90 tickets to adults and 60 tickets to students.

Step 4: Find the Price in Each Market.
Substitute the quantities back into their respective inverse demand functions:

Adult Price: $P_{A} = 200 - 90 = £ 110$
Student Price: $P_{S} = 80 - 0.5 (60) = 80 - 30 = £ 50$

Step 5: Calculate Total Profit.

Total Revenue (TR):
$T R = (P_{A} \times Q_{A}) + (P_{S} \times Q_{S})$
$T R = (110 \times 90) + (50 \times 60) = 9900 + 3000 = £ 12, 900$
Total Quantity (Q):
$Q = Q_{A} + Q_{S} = 90 + 60 = 150$
Total Cost (TC):
$T C = 1000 + 20 (150) = 1000 + 3000 = £ 4, 000$
Profit ( $π$ ):
$π = T R - T C = 12, 900 - 4, 000 = £ 8, 900$

The cinema's maximum profit from this strategy is £8,900.

The Role of Elasticity

The calculation above is mechanical, but the underlying driver is elasticity. As a statistician, you'll appreciate the formal relationship between price, marginal cost, and elasticity ( $ϵ$ ), which we saw previously as the Lerner Index. For a profit-maximizing firm:

P = \frac{M C}{1 + ( 1/ ϵ )} or \frac{P - M C}{P} = - \frac{1}{ϵ}

When practicing third-degree price discrimination, this rule applies to each market:

P_{1} = \frac{M C}{1 + ( 1/ ϵ _{1} )} and P_{2} = \frac{M C}{1 + ( 1/ ϵ _{2} )}

Since $M C$ is the same for both, we can see that if market 1 is less elastic than market 2 (i.e., $∣ ϵ_{1} ∣ < ∣ ϵ_{2} ∣$ ), then $P_{1}$ will be greater than $P_{2}$ . The firm charges a higher price in the less elastic market.

The following reading formally derives this result.

[PDF] Third-Degree Price Discrimination - Wsu - Washington State University

This section of the WSU paper proves the relationship between elasticity and price in third-degree price discrimination.

Read from the heading "The result where price decreased..." to the end of that paragraph. This demonstrates mathematically why the subgroup with more inelastic demand pays a higher price.

Conclusion

In this lesson, we have moved from the theory of perfect price discrimination to the practical, calculable strategy of third-degree price discrimination. You are now equipped to determine how a firm with market power can enhance its profits by segmenting its customers and charging different prices to each group.

Key Takeaways:

Third-degree price discrimination involves dividing consumers into groups with different demand elasticities and charging each group a different price.
The profit-maximization rule is to set output for each group such that marginal revenue in each market equals the firm's marginal cost ( $M R_{1} = M R_{2} = ... = M C$ ).
The calculation process involves solving a separate monopoly problem for each market segment to find its specific price and quantity.
Total profit is the sum of revenues from all segments minus the total cost of producing the total output.
The group with the more inelastic demand will always be charged a higher price.

Now that we can calculate the profit from price discrimination, a natural question arises: how much better is this than simply charging everyone the same price? Our next lesson will address this directly by having you compute and compare total profit under price discrimination versus a single uniform price, quantifying the gains from this strategic pricing.

Recommended resources