The last type of price discrimination we will examine in this chapter is third-degree price discrimination. This technique can be applied when firms are able to both identify and separate two or more groups or submarkets of consumers. Each group or submarket is charged a different price for exactly the same product. In contrast to first-degree price discrimination, all buyers within each submarket pay the same price. Thus, a third-degree price discriminator does not haggle over prices in an ef- fort to capture the entire consumer surplus, and so buyers in all of the submarkets
third-degree price discrimination Firm can identify and separate a market into two or more submarkets and charge a different price in each submarket.
F I G U R E 14.5 Block Pricing with Five Blocks
0 1 2 3 4 5 6 7 8 9 10
10 20 30 40
Quantity
50 60
Price and cost (dollars)
MR U
a
b d
c
e f
g
D MC = AC
Now try Technical Problem 4.
end up keeping some of their consumer surplus. In contrast to second-degree price discrimination, third-degree price-discriminators do not rely on self-selection to cre- ate different prices. Because they can identify consumers in the various submarkets, they can successfully charge different prices to the different groups of buyers.
Movie theaters that charge adults higher ticket prices than children, pharmacies that charge senior citizens lower drug prices than nonseniors, and personal com- puter firms that charge students lower prices for PCs than other buyers are all exam- ples of third-degree price discrimination. In each example, the price-discriminating firm can identify consumer types in ways that are practical, generally legal, and low cost. In many cases, consumer identification simply requires seeing a driver’s license to verify age or checking a student ID to verify school enrollment. Once buy- ers are separated into different submarkets, the discriminating firm will charge the less elastic submarket a higher price than it charges the more elastic submarket. As we will explain shortly, when price elasticities differ across submarkets—as they must for successful discrimination—charging groups different prices increases profit compared to charging both submarkets the same uniform price. In other words, a movie theater earns more profit by (i) charging the less elastic adult submarket a relatively high ticket price and the more elastic children’s submarket a relatively low ticket price than by (ii) charging all movie patrons the same ticket price.
Allocation of Sales in Two Markets to Maximize Revenue
To make the most profit, firms practicing third-degree price discrimination must decide how to optimally allocate total sales across different submarkets to gener- ate the greatest amount of revenue (and consequently profit). To see how this is accomplished, let’s assume there are only two submarkets, market 1 and market 2.
Figure 14.6 shows the demand and marginal revenue curves for each submarket.
Now let’s further suppose the manager wishes for some reason to sell a total of 500 units in the two markets. How should this manager allocate sales between the two markets to maximize total revenue from the sale of 500 units?
Figure 14.6 shows two possible allocations of 500 units between markets 1 and 2. First, consider an equal allocation of 250 units in each market, as shown by points w and w’ in the figure. When 250 units are sold in each market, you can see from Panels A and B that
MR1 = $10 , $30 = MR2
This allocation does not maximize total revenue because the manager can increase total revenue by decreasing the number of units sold in the lower- marginal- revenue market 1 and increasing the number of units sold in the higher- marginal-revenue market 2. Specifically, by selling one less unit in market 1, total revenue in market 1 falls by $10. To keep total sales at 500 units, the manager must sell one more unit in market 2, which increases total revenue by $30 in market 2. As you can now see, moving one unit from the low-marginal-revenue market 1 to the high-marginal-revenue market 2 causes total revenue to increase by $20 (= $30 2 $10).
To maximize total revenue from 500 units, the manager will continue mov- ing units from the market where marginal revenue is lower to the market where marginal revenue is higher until marginal revenues are equal in both markets.
Notice in Figure 14.6 that when Q1 is 200 and Q2 is 300 (points v and v9), marginal revenues are equal in both markets, and this allocation of 500 units maximizes the firm’s total revenue. As you can see by examining the demand curves, the desired allocation of sales between markets 1 and 2 can be accomplished by charging a price of $40 to buyers in market 1 and charging a price of $50 to buyers in market 2.
We have now established that a manager maximizes total revenue from two sub- markets when the total output is allocated across submarkets so that MR1 = MR2.5 This establishes the following principle, which is sometimes called the equal-marginal-revenue principle:
F I G U R E 14.6 Allocating Sales Between Two Markets:
The Equal Marginal Revenue Principle
0 10
0 20 30 40 50 55 80
20 35
40 E1 = –2
E2 = –1.67
D1
D2
w
v9 w9
MR1 MR2
v 60
100 200 300 Quantity (Q250 1)
Panel A—Market 1 Panel B—Market 2
Quantity (Q2)
Price and marginal revenue (dollars) Price and marginal revenue (dollars)
200 300 400 250
5This condition should not be surprising because it is just another application of the principle of constrained optimization presented in Chapter 3. If a manager wants to maximize total revenue subject to the constraint that there is only a limited number of units to sell, the manager should al- locate sales so that the marginal revenues (marginal benefits) per unit are equal in the two markets.
The marginal cost of selling one unit in market 1 is the one unit not available for sale in market 2 (MC1
= MC2 = 1).
Now try Technical Problem 5.
Principle A manager who wishes to maximize the total revenue from selling a given amount of output in two separate markets (1 and 2) should allocate sales between the two markets so that
MR1 = MR2
and the given amount of output is sold. This is known as the equal-marginal-revenue principle.
Although the marginal revenues in the two markets are equal, the prices charged are not. As we stated earlier, the higher price will be charged in the mar- ket with the less elastic demand; the lower price will be charged in the market having the more elastic demand. In the more elastic market, price could be raised only at the expense of a large decrease in sales. In the less elastic market, higher prices bring less reduction in sales.
This assertion can be demonstrated as follows: Let the prices in the two markets be P1 and P2. Likewise, let E1 and E2 denote the respective price elasticities. As shown in Chapter 6, marginal revenue can be expressed as
MR = P ( 1 + 1 __ E )
Because managers will maximize revenue if they allocate output so that MR1 = MR2, MR1 = P1( 1 + 1 __ E1 ) = P2( 1 + 1 __ E2 ) = MR2
Recall from Chapter 12 that firms with market power must price in the elastic re- gion of demand to maximize profit. So MR1 and MR2 must both be positive, which implies that E1 and E2 must both be greater than 1 in absolute value (i.e., demand must be elastic in each market). Now suppose the lower price is charged in mar- ket 1 (P1 , P2) and the allocation of sales between markets satisfies the equal- marginal-revenue principle (i.e., MR1 = MR2). Then, by manipulating the equation above,
P1
__ P2 = ________ ( 1 + 1 __ E2 ) ( 1 + 1 __ E1 ) , 1
Therefore, because ( 1 + 1 __ E2 ) , ( 1 + 1 __ E1 ) , it must be the case that |1/E2| . |1/E1|, so that
|E1| . |E2|
The market with the lower price must have the higher elasticity at that price.
Therefore, if a firm engages in third-degree price discrimination, it will always charge the lower price in the market having the more elastic demand curve.
Consider again the two markets, 1 and 2, in Figure 14.6. The lower price, $40, is charged in market 1, and the point elasticity of demand at this price is −2 [ = $40/
($40 − $60)]. In market 2, where the higher price, $50, is charged, the price elastic- ity is −1.67 [ = $50/($50 − $80)]. As expected, demand is less elastic in market 2 where customers pay the higher price.
Principle A manager who price-discriminates in two separate markets, A and B, will maximize total revenue for a given level of output by charging the lower price in the more elastic market and the higher price in the less elastic market. If |EA| . |EB|, then PA , PB.
Profit Maximization with Third-Degree Price Discrimination
Thus far we have assumed that the price-discriminating firm wishes to allocate a given level of output among its markets to maximize the revenue from selling that output. Now we discuss how a manager determines the profit-maximizing level of total output. Once the total output is decided, the manager needs only to apply the equal-marginal-revenue principle to find the optimal allocation of sales among markets and the optimal prices to charge in the different markets.
As you probably expected, the manager maximizes profit by equating marginal revenue and marginal cost. The firm’s marginal cost curve is no different from that of a nondiscriminating firm. The marginal cost for a third-degree price-discriminating firm is related to the total output in both markets (QT = Q1 + Q2) and does not de- pend on how that total output is allocated between the two markets—only total revenue depends on sales allocation. Therefore, we must derive the curve relating marginal revenue to total output QT, under the condition that output is allocated between markets according to the equal-marginal-revenue principle. This particular marginal revenue is called total marginal revenue (MRT) because it gives the change in the combined total revenue in both markets (i.e., TR1 + TR2) when the price-dis- criminating firm increases total output (QT) and allocates the additional output to achieve equal marginal revenues in both markets. Total marginal revenue equals marginal cost (MRT = MC) at the profit-maximizing level of total output.
Total marginal revenue is derived by horizontally summing the individual mar- ginal revenue curves in each market. The process of horizontal summation can be accomplished either by graphical construction or by mathematical means. As you will see, horizontal summation enforces the equal-marginal-revenue principle in the process of deriving MRT. We will now show you how to derive a total marginal revenue curve by graphical means, and then later in this section we will illustrate the algebraic method of finding the equation for MRT.
Profit-maximization: A graphical solution For convenience, the marginal rev- enue curves MR1 and MR2 from Figure 14.6 are shown on the same graph in Panel A of Figure 14.7. (The corresponding demand curves are not shown because they are irrelevant for deriving the horizontal sum of MR1 and MR2.) Panel B in Figure 14.7 shows the total marginal revenue curve, MRT, which is derived by summing Q1 and Q2 to get QT for a number of different values of marginal revenue—$80, $60, $40, $20, and zero
point i: MRT = $80 for QT = 0 = 0 + 0 point k: MRT = $60 for QT = 100 = 0 + 100 point l: MRT = $40 for QT = 300 = 100 + 200
total marginal revenue (MRT ) The change in TR1 + TR2 attributable to an increase in QT when the extra output is allocated to maintain equal marginal revenues in both markets.
point m: MRT = $20 for QT = 500 = 200 + 300 point n: MRT = $0 for QT = 700 = 300 + 400
For example, if the firm produces 300 units of total output, total marginal revenue is $40; this means the firm will maximize total revenue by selling 100 units in mar- ket 1 and 200 units in market 2. At 100 units of output (from Panel A), MR1 equals
$40. At 200 units of output (from Panel A), MR2 also equals $40. Thus, it does not matter whether the 300th unit is the marginal unit sold in market 1 or in market 2;
the firm’s marginal revenue is $40 (as shown in Panel B at point l). Notice that this allocation of 300 total units is the only allocation that will equate the marginal rev- enues in the two markets. At every other total output in Panel B, the total marginal revenue is obtained in the same way.6
Now we can use MRT to find the total output the firm should produce to maximize its profit. To see how this decision is made, consider Figure 14.8, which shows all the demand and marginal revenue relations. Suppose the firm
6When horizontally summing two linear marginal revenue curves, total marginal revenue will also be linear. For this reason, you can quickly construct a graph of MRT by finding two points—
points k and n in Figure 14.7, for example—and drawing a line through them. You must be careful, however, because the line you construct applies only to values of MRT from zero up to the point where demand in the smaller market crosses the vertical axis. In Figure 14.7, MRT kinks at point k, and coincides with MR2 above $60.
MR1 MR2
MRT n m
l k
j i
Individual market marginal revenues (dollars) Total marginal revenue (dollars)
0 80
60
40
20
400 300 200 100
Quantities Q1 and Q2
Panel A—MR in Markets 1 and 2 Panel B—Total Marginal Revenue Curve
0 100 300
Total quantity (QT = Q1 + Q2)
500 700
80 70 60
40
20 F I G U R E 14.7
Constructing the Total Marginal Revenue Curve (MRT):
Horizontal Summation
faces constant marginal and average costs of production equal to $20 per unit, as shown in Panel B. Total output is determined by equating total marginal revenue with marginal cost, which occurs at 500 units (point m) and MRT = MC = $20. The 500 units are then allocated between the two markets 1 and 2 so that marginal revenue equals $20 in both markets. This allocation is accomplished by charging a price of $40 in market 1 (point r) and a price of $50 in market 2 (point s), which results in respective sales of 200 units and 300 units in markets 1 and 2.
By charging different prices in the separate markets, the firm collects total rev- enues of $8,000 (= $40 × 200) in market 1 and $15,000 (= $50 × 300) in market 2, for a combined total revenue of $23,000. Since the total cost of producing 500 units is $10,000 (= $20 × 500), the price-discriminating firm’s profit is $13,000.
To verify that charging two different prices generates more revenue and profit than charging the same (uniform) price to both groups of buyers, we will now calculate the total revenue the firm could collect if it instead charged all buyers the same price to sell 500 units.7 The market demand when both submarkets 1
F I G U R E 14.8
Profit-Maximization under Third-Degree Price Discrimination
D2
m U
MC = AC MRT
DT s
r
D1
MR1 MR2
Prices and marginal revenues (dollars) Total marginal revenue and marginal cost (dollars)
0 200 0 500
150
Quantities Q1 and Q2
Panel A—Markets 1 and 2 Panel B—Total Marginal Revenue and Marginal Cost Total quantity (QT = Q1 + Q2)
350 300 20
40 45 50
20 45
7Notice that the profit-maximizing output level when the firm chooses to charge all buyers a single price is also 500 units. This is true because MRT in Panel B is also the firm’s marginal revenue curve when the two market demand curves are horizontally summed to construct the total demand curve.
and 2 are grouped together as one single market is found by horizontally sum- ming D1 and D2 to get the market demand curve, DT, shown in Panel B. If the firm charged all customers a single price of $45, it would sell 500 units and gen- erate just $22,500 (= $45 × 500) in total revenue. The total cost of producing 500 units is the same as it was under price discrimination, $10,000, so profit would drop to $12,500. Thus the firm would experience a reduction in revenue and profit of $500.
We can now summarize this discussion of pricing in two markets and extend our results to the general case of n separate markets:
Principle A manager who wishes to sell output in n separate markets will maximize profit if the firm produces the level of total output and allocates that output among the n separate markets so that
MRT = MR1 = ? ? ? = MRn = MC
The optimal prices to charge in each market are determined from the demand functions in each of the n markets.
Now try Technical Problem 6.
Profit-maximization: An algebraic solution Now we will show you an algebraic solution to the third-degree price discrimination problem presented graphically in Figure 14.8. The equations for the demand curves for these two markets are
Mkt 1: Q1 = 600 2 10P1 and Mkt 2: Q2 = 800 2 10P2 Solving for the inverse demand functions in the two markets,
Mkt 1: P1 = 60 2 0.1Q1 and Mkt 2: P2 = 80 2 0.1Q2
The marginal revenue functions associated with these inverse demand functions are
Mkt 1: MR1 = 60 2 0.2Q1 and Mkt 2: MR2 = 80 2 0.2Q2
To obtain the total marginal revenue function, MRT = f(QT), we first obtain the inverse marginal revenue functions for both markets in which the firm sells its product:
Mkt 1: Q1 = 300 2 5MR1 and Mkt 2: Q2 = 400 2 5MR2 For any given level of total output, MR1 = MR2 = MRT; thus
Mkt 1: Q1 = 300 − 5MRT and Mkt 2: Q2 = 400 − 5MRT
Since QT = Q1 + Q2, the inverse of total marginal revenue is obtained by summing the two inverse marginal revenue curves to get
QT = Q1 + Q2 = 300 − 5MRT + 400 − 5MRT
= 700 − 10MRT
I L L U S T R AT I O N 1 4 . 2 Sometimes It’s Hard to Price-Discriminate
In the theoretical discussion of third-degree price dis- crimination, we made two important points: (1) Firms must separate the submarkets according to demand elasticity, and (2) firms must be able to separate mar- kets so as to keep buyers in the higher-price market from buying in the lower-price market (i.e., prevent consumer arbitrage). In some of the examples we used, it was relatively easy to separate the markets.
For example, at movie theaters it is fairly simple, and relatively inexpensive, to prevent an adult from en- tering the theater with a lower-priced child’s ticket.
In other cases of price discrimination, it is rather dif- ficult or costly to separate the markets. If it is impos- sible or expensive to separate markets, third-degree price discrimination will not be profitable, and the monopolist will either charge a single price to all cus- tomers or find a way to implement second-degree price discrimination.
One of the most frequently cited examples of a mar- ket in which separation is difficult is the airline mar- ket. It is no secret that airlines attempt to charge leisure fliers lower fares than business travelers. The story of such an attempt by Northwest Airlines illustrates the difficulty of separating markets.
The Wall Street Journal reported that Northwest Airlines would start offering a new discount fare that would cut down prices by 20 to 40 percent for people who were traveling in a group of two or more.
The Wall Street Journal noted that this change would be likely to stimulate family travel but would also eliminate the use of supersaver fares by busi- ness travelers. Previously, many business travelers purchased round-trip supersaver tickets when fares dropped below 50 percent, then threw away the re- turn portion of the ticket or used it later. Northwest was planning to raise or do away with its other super- saver fares designed to attract leisure travelers. Most business travelers fly alone and would not be able to take advantage of the new, lower fares requiring groups of two or more. The Northwest executive also p redicted that businesspeople would not abuse these tickets. Should the plan stick and spread, he said, it
will allow airlines to maintain an attractive offering for the most price-sensitive travelers, while allowing the basic supersaver fares to continue rising along with business rates.
This reasoning was a bit optimistic on the part of the airline. The Wall Street Journal noted that groups of business travelers could work around the restrictions that currently applied to supersavers. One airline offi- cial expressed concern that travel agents would match travelers who did not know each other who were going to the same destination. Clearly there were many ways to defeat the airline’s attempts to price- discriminate effectively.
But Northwest knew about the problems and tried to make the practice of cross-buying difficult. Travel- ers were required to book their flights together, check in together, and follow identical itineraries to qualify for the group discounts. The fares were nonrefundable, required a Saturday night stay, and had to be booked 14 days in advance—practices that business travelers typically would find difficult to accomplish. Of course, some of these restrictions designed to weed out busi- ness travelers could discourage many leisure travelers, the very people the new discounts were designed to attract. And obviously single leisure travelers would be left out.
As you can see, the problem of separating markets—preventing customers in the higher-price market from buying in the lower-price market—can be an extremely challenging task for the would-be price discriminator. For airlines, it would be much easier if passengers came with signs saying “busi- ness traveler” or “leisure traveler.” As previously noted, in markets where separating the higher-price buyers from the lower-price buyers is too difficult or too expensive, third-degree price discrimination will not be profitable. Before resorting to uniform pricing, a firm that wishes to price-discriminate can try to implement one of the second-degree price- discrimination methods, because these methods rely on self-selection rather than market separation to charge different prices.
Source: Based on Brett Pulley, “Northwest Cuts Fares to Boost Leisure Travel,” The Wall Street Journal, January 12, 1993.