Issues in Price Discrimination: A Comment on and Addendum to "Teaching Price Discrimination," by Carroll and Coates.

Jeitschko, Thomas D.

Thomas D. Jeitschko [*]

1. Introduction

There is a large variety of treatments of the topic of price discrimination in the most widely used undergraduate principles and intermediate microeconomic textbooks as well as industrial organization texts. In a recent Southern Economic Journal article, Carroll and Coates (1999) set out to bring into focus the issues that ought to be discussed and made clear when teaching price discrimination to undergraduates. Their paper concludes with a list of six points that instructors should follow to spark students' interest and create as little confusion as possible when covering the topic.

While in several ways their article is an improvement over many a textbook discussion, I think their contribution can also be improved upon, as it contains some misconceptions and mistakes. Specifically, there are two concerns regarding the coverage of price discrimination in general and both of these affect the paper by Carroll and Coates in particular. The first is mostly technical, but has led to widespread misconceptions about the relationship between the price elasticity of demand and third-degree price discrimination. The other is mainly pedagogical and concerns what can (or should) actually be taken away from the classroom after a discussion of price discrimination, especially regarding implications of price discrimination on economic efficiency.

2. Price Elasticity of Demand

Point (ii) on Carroll and Coates' list intended to prevent students' confusion on price discrimination states that one should cover the three necessary conditions for price discrimination to occur. These are: (i) The firm must have some market power; (ii) there can at best be imperfect arbitrage opportunities for consumers; and (iii) "consumers have different price elasticities of demand" (p. 471).

The third point (repeated again on p. 472, where it is stated that price discrimination would not be possible "if buyers' price elasticities were identical"), however, needs some qualification. First of all, a difference in the price sensitivity of the quantity demanded is not a necessary condition for either first- or second-degree price discrimination--and there is no confusion about this in the most common textbook treatments on price discrimination. Second, when it comes to third-degree price discrimination, the role of the price elasticity of demand is frequently mischaracterized. [1] In particular, the common (and seemingly intuitive) statement that "if the firm knows that elasticity is related to some identifiable group characteristic, then it can use third-degree price discrimination to induce the price-insensitive buyers to pay a high price and price-sensitive buyers to pay a low price" (p. 471) is hard to verify in practice, and is indeed frequently misleading or false. The reason for this is that what constitutes a "price-insensitive buyer" can only be determined by comparing the demands of two consumers, whereas an elasticity criterion is often only applied to a particular quantity demanded.

The following example (taken from Jeitschko and Thon 1999) illustrates this point.

Example 1 A firm with zero cost can price discriminate between the two following demands.

[Q.sub.1] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [Q.sub.2] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The inverse demands for the firm's product in the two markets are given by

[P.sub.1] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [P.sub.2] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

resulting in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In both submarkets there are two critical points, (P, Q), where [MR.sub.i] = MC(=0). In market 1 these are (50, 10) and (15, 30); in market 2 these are (52.5, 10.5) and (17.5, 35). Since AC = 0, the firm maximizes revenue. Hence, in market 1, the optimum is at [[P.sup.*].sub.1] = 50, whereas in market 2 it is at [[P.sup.*].sub.2] = 17.5. Thus, the price in market 1 is higher than in market 2, that is, [[P.sup.*].sub.1] [greater than] [[P.sup.*].sub.2].

Consider now the elasticity in the two respective markets. Given that the firm has zero cost, at the two distinct optimal prices the two respective price elasticities must both be equal to 1. Specifically.

[[eta].sub.1]([[P.sup.*].sub.1] [equivalent] -[[P.sup.*].sub.1]/[Q.sub.1]([[P.sup.*].sub.1]) [Q'.sub.1]([[P.sup.*].sub.1]) = -50/10(-1/5) = 1 and

[[eta].sub.2]([[P.sup.*].sub.2]) [equivalent] -[[P.sup.*].sub.2]/[Q.sub.2]([[P.sup.*].sub.2]) [Q'.sub.2]([[P.sup.*].sub.2]) = -17.5/35(-2) = 1.

Of course this does not suggest that consumers in both markets are equally "price sensitive." [2] Indeed, such a suggestion would rely solely on the firm's cost structure instead of demand considerations!

Instead, consider the price elasticity of demand in the two respective markets at a uniform price: For instance at the price P = 50, where

[[eta].sub.2](50) = -50/[Q.sub.2](50) [Q'.sub.2](50) = -50/11(-1/5) = 10/11 [less than] 1 = [[eta].sub.1](50);

or at the price P = 17.5, where

[[eta].sub.2] (17.5) = -17.5/[Q.sub.1](17.5) = -17.5/25(-2) = 7/5 [greater than] 1 = [[eta].sub.2](17.5).

For these two prices the price elasticity of demand in market 1 is higher despite the fact that it also has the higher price.

In general, given [[eta].sub.i](P) = [-P/[Q.sub.i](P)][Q'.sub.i](P),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so [[eta].sub.2] [less than] [[eta].sub.1] for virtually any price (including the nondiscriminatory price of 16.25). [3] Consequently, the market with the lower price is the less elastic market.

The above elasticity ranking is based on point elasticity, so the question arises whether other notions of elasticity--in particular, arc elasticity--yield a different interpretation of the price sensitivity of demand in the two markets. Unfortunately, the concept of arc elasticity does not allow for an unambiguous ranking of the two markets: For small price intervals around the kink in the demand curves the arc elasticity ranks the elasticity of the first market lower, but for larger intervals and for intervals that do not contain that price, in particular, for intervals around either of the critical prices at which the firm may consider pricing, the ranking is in line with the results from the use of the more common point elasticity.

The point of the example is thus that while elasticity is a concept tied to a price (or, in the case of arc elasticity, a price interval), the pricing decision of the firm may have to account for a much wider range of prices. Consequently, the two notions of pricing and elasticity are not as intimately related as is often suggested. Indeed, as long as prices in both markets are fairly close to one another, one will usually detect the inverse relationship between price and elasticity in that price range, but if prices are vastly different the relationship may not hold.

Although Example 1 is clearly constructed to make a theoretical point, It yields some real-world insight. Consider the following scenario: A beverage producer develops a new sports soft drink that appeals both to students and professionals, the latter being reflected by the high intercepts and steep slopes in the example. The firm sells its product in two separate markets, one an expensive college town in a resort-like environment (market 2), the other a large city with a small college (market 1). Suppose that the residents of the small resort-like college town are ceteris paribus wealthier than the city dwellers and have a lower elasticity of demand. If the relative size of the student population is larger in the college town, then prices may be determined largely by this group, whereas in the city prices may be set with professionals in mind. Thus, the price may be lower in the wealthy resortlike college town, even though the price elasticity of demand for any type of individual there is smaller when compa red to the city. [4]

3. Efficiency

Point (iv) in the paper by Carroll and Coates suggests discussing the economic efficiency of different types of price discrimination, possibly by use of the numeric example they provide. It is clearly important to discuss economic efficiency when discussing price discrimination. There are two points worth making concerning the discussion given in the paper. The first concerns the numeric example given. The second concerns what can be said in general about price discrimination and market efficiency. I comment on the second point in the Concluding Remarks.

First, the numeric example used is incorrect. Given the postulated demands of the two types of consumers, total demand is not linear, but piecewise linear and given by

[Q.sub.T](P) = [Q.sub.1](P) + [Q.sub.2](P) = {125 - 4.5P, if 0 [less than or equal to] P [less than or equal to] 25, 25 - 0.5P, if 25 [less than or equal to] P [less than or equal to] 50,

resulting in

P = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

MR = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This leads to quantitatively different results from those reported in the article. Despite this incorrect algebraic derivation of market demand, a piecewise linear inverse demand is depicted in Figure 1, p. 474, yet the slope on the lower branch of the curve should be flatter than the slope of demand in market 1. Moreover, in that same figure, an incorrect marginal revenue curve is depicted, which (depending on marginal cost of the firm), if used in an analysis, would also lead to qualitatively incorrect results. Last, a discrepancy between total welfare under perfect competition and first-degree price discrimination appears in the results (pp. 478-9).

The following example serves for an analysis and comparison of the standard models, namely, perfect competition, nondiscriminating firm with market power, and first-and third-degree price discrimination. The example preserves the goals of "Teaching Price Discrimination," as it is easily tractable for use in undergraduate classes and avoids ambiguities due to rounding.

Example 2 Demand of two different groups of consumers are given by

[Q.sub.1] = 100- 5[P.sub.1] and [Q.sub.2] = 32 - [P.sub.2],

so that inverse demand for the two groups are

[P.sub.1] = 20 - 1/5[Q.sub.1] and [P.sub.2] = 32 - [Q.sub.2].

The technology used to produce the good or service in question implies

AC = MC = 10.

Figure 1 depicts the individual market demands with their respective marginal revenue curves--market 1 on the right and market 2 on the left of the top panel, as well as the aggregated market--bottom panel.

Perfect Competition (PC)

Since the technology yields a constant average cost, the long-run supply curve is given by P = AC = 10. Hence,

[P.sup.PC] = 10.

Output and consumer surplus can either be calculated for the entire market, or for the two groups individually and then added to yield the same result. The latter is easier, so, given the demands for the two groups and [P.sup.PC] = 10,

[[Q.sup.PC].sub.1] = 50 and [[Q.sup.PC].sub.2] = 22,

so consumer surplus is

[[CS.sup.PC].sub.T] = [[CS.sup.PC].sub.1] + [[CS.sup.PC].sub.2] = (20 - [P.sup.PC])[[Q.sup.PC].sub.1]/2 + (32 - [P.sup.PC])[[Q.sup.PC].sub.2]/2 = 250 + 242 = 492.

Moreover, profit for any and all firms ([pi]) and dead weight loss (DWL) are both zero, that is,

[[pi].sup.PC] = 0 and [DWL.sup.PC] = 0.

Nondiscriminating Monopolist (ND)

Given the two groups of consumers, the joint demand is given by

[Q.sub.T](P) = [Q.sub.1](P) + [Q.sub.2](P) = {132 - 6P, if 0 [less than or equal to] P [less than or equal to] 20,

{32 - P, if 20 [less than or equal to] P [less than or equal to] 32,

resulting in inverse demand and marginal revenue of

P = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and MR = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first-order profit maximizing conditions (MC = MR) hold at two levels of total output, namely [Q.sub.T] = 11 and [Q.sub.T] = 36. It is easily verified that the latter is the global maximum. That is,

[[Q.sup.ND].sub.T] = 36 and [P.sup.ND] = 16.

Given the price [P.sup.ND] = 16, consumer surplus can be calculated as above per individual group and summed, or, yielding the same result, calculated for the aggregated market. The former is easier, so noticing that a price of [P.sup.ND] = 16 the first group purchases [[Q.sup.ND].sub.1] = 20 units and the second group purchases [[Q.sup.ND].sub.2] = 16 units,

[[CS.sup.ND].sub.T] = [[CS.sup.ND].sub.1] + [[CS.sup.ND].sub.2] = (20 - [P.sup.ND])[[Q.sup.ND].sub.1]/2 + (32 - [P.sup.ND])[[Q.sup.ND].sub.2]/2 = 40 + 128 = 168.

Profit is given by

[[pi].sup.ND] = ([P.sup.ND] - AC)[[Q.sup.ND].sub.T] = 216.

Similar to calculating consumer surplus, dead weight loss can be calculated per consumer group and then summed. That is,

[DWL.sup.ND] = [[DWL.sup.ND].sub.1] + [[DWL.sup.ND].sub.2] = ([P.sup.ND] - MC)([[Q.sup.PC].sub.1] - [[Q.sup.ND].sub.1])/2 + ([P.sup.ND] - MC)([[Q.sup.PC].sub.2] - [[Q.sup.ND].sub.2])/2 = 90 + 18 = 108.

First-Degree Price Discrimination (PD1)

In the case of first-degree price discrimination by a single firm supplying both markets, all consumers are charged their reservation prices leading to the efficient outcome in which all gains from trade accrue to the firm. Consequently the allocation of goods is identical to the case of perfect competition, yet all surplus is in the form of profit. Specifically,

[[Q.sup.PD1].sub.T] = 72 (= [[Q.sup.PC].sub.T]); [CS.sup.PD1].sub.T] = 0 (= [[pi].sup.PC]); [[pi].sup.PD1] = 492 (= [[CS.sup.PC].sub.T]); [DWL.sup.PD1] = 0 (= [DWL.sup.PC]).

Third-Degree Price Discrimination (PD3)

Under third-degree price discrimination the firm is able to charge different prices to the two groups of consumers. Marginal revenues in the two submarkets are given by

[MR.sub.1] = 20 - 2/5[Q.sub.1] and [MR.sub.2] = 32 - 2[Q.sub.2].

The first-order sufficient conditions ([MR.sub.i] = MC) yield outputs of

[[Q.sup.PD3].sub.`] = 25 and [[Q.sup.PD3].sub.2] = 11,

implying

[[P.sup.PD3].sub.1] = 15 and [[P.sup.PD3].sub.2] = 21.

Hence, consumer surplus is

[[CP.sup.PD3].sub.T] = [[CS.sup.PD3].sub.1] + [[CS.sup.PD3].sub.2] = (20 - [[P.sup.PD3].sub.2] - AC)[[Q.sup.PD3].sub.2] = 125 + 121 = 246;

and dead weight loss is given by

[DWL.sup.PD3] = [[DWL.sup.PD3].sub.1] - [[DWL.sup.PD3].sub.2] = ([[P.sup.PD3].sub.1] - MC)([[Q.sup.PC].sub.1] - [[Q.sup.PD3].sub.1])/2 + ([[P.sup.PD3].sub.2] - MC)([[Q.sup.PC].sub.2] - [[Q.sup.PD3].sub.2])/2 = 125/2 + 121/2 = 123.

Table 1 summarizes these results.

4. Concluding Remarks

Carroll and Coates caution that after pointing out the economic efficiency result of perfect competition and first-degree price discrimination, it is important to emphasize that economic efficiency is not in general associated with price discrimination. In particular, moving from a nondiscrimination pricing scheme to a form of price discrimination need not improve economic efficiency (decrease dead weight loss). Indeed, as Carroll and Coates mention, a move to third-degree price discrimination may either increase or decrease total welfare-depending on the structure of costs and demand. The implications of second-degree price discrimination may be even harder to assess. As a consequence, I would strongly discourage the use of the linear demands and constant marginal cost model to derive general efficiency implications. Instead, the message concerning price discrimination and economic efficiency should be: Compared with a nondiscriminating firm, when a firm engages in third-degree price discrimination it attem pts to capture more of the potential gains from trade that exist in the market--be this through a transfer of consumer surplus or dead weight loss to the firm's profit. To the degree that it captures what is otherwise dead weight loss, efficiency increases. However, as Carroll and Coates point out, because the marginal consumers (the last consumer willing to purchase) in the two markets have differing values, third-degree price discrimination necessarily entails an inefficiency. Consequently the net effect is not clear and is negative whenever output is not increased (as is the case in the examples given).

In a more advanced course it may be worth illustrating these points using contrasting examples, such as nonconstant marginal costs in which output decreases and only one market is serviced under third-degree price discrimination (e.g., [P.sub.1] = 200 - 5[Q.sub.1] [P.sub.2] 300 - 5[Q.sub.2]; and TC = 5[Q.sup.2] on the one hand, and some examples from the fairly comprehensive treatment of the issue presented by Nahata, Ostaszewski, and Sahoo (1990). Last, the 1991 paper by Lott and Roberts is still a very good source for "real-world" pricing policies that are often mistaken for price discrimination.

(*.) Department of Economics, Texas A&M University, College Station, TX 77843, USA; E-mail thosd@econ.tamu.edu.

(1.) Indeed. it is possible for a firm to have differing optimal pricing schemes across markets even when the elasticities of demand in those markets are the same. For a simple example of pricing meals in restaurants, please contact the author.

(2.) Nevertheless, it is this type of comparison that leads to the common misconception concerning the relation of price elasticity and price differences. Indeed, the basis for the verbal assertion is a formal argument that relates the firm's first-order conditions to the point elasticities. See, e.g., Varian (1999, pp. 440-441), or virtually any other text discussing price discrimination, for that matter.

(3.) Notice that this price is lower than the discriminatory prices in both markets. For a fuller discussion of when discriminatory prices deviate in the same direction from the nondiscriminatory price in all submarkets, see Nahata, Ostaszewski, and Sahoo (1990).

(4.) Of course, as Carroll and Coates point Out in their reply, whenever possible, the firm would like to further price discriminate between students and professionals in both markets.

References

Carroll, Kathleen, and Dennis Coates. 1999. Teaching price discrimination: Some clarification Southern Economic Journal 66:466-80.

Jeitschko, Thomas D., and Dominique Thon. 1999. Third degree price discrimination and price elasticities. Working paper, Bodo Graduate School of Business, Bodo Norway.

Lott, John R., and Russell D. Roberts. 1991. A guide to the pitfalls of identifying price discrimination. Economic Inquiry 29:14-23.

Nahata, Babu. Krzysztof Ostaszewski, and P. K. Sahoo. 1990. Direction of price changes in third-degree price discrimination. American Economic Review 80:1254-8.

Varian, Hal R. 1999. Intermediate microeconomics: A modern approach. 5th edition. New York: W.W. Norton & Co.

Table 1. Prices (P), Market Output ([Q.sub.T]), Consumer Surplus (CS),
Profits ([pi]), and Dead Weight Loss (DWL) for The Cases of Perfect
Competition (PC), Nondiscriminating Firm with Market Power (ND),
First-Degree Price Discrimination (PD1), and Third-Degree Price
Discrimination (PD3)
 P [Q.sub.r] CS [pi] DWL
PC 10 72 492 0 0
ND 16 36 168 216 108
PD1 N/A 72 0 492 0
PD3 [P.sub.1] = 15, 36 123 246 123
 [P.sub.2] = 21

[Graph omitted]

[Graph omitted]