Price discrimination with differentiated products: definition and identification

Price discrimination with differentiated products: definition and identification

Sofronis K. Clerides

I. INTRODUCTION

Price discrimination is said to exist when the same product is sold to different consumers at different prices. Most economists also agree that price discrimination is at work when similar–but not identical–products are sold at prices that do not reflect differences in costs. Economics textbooks are rife with examples, such as student or senior citizen discounts, hardcover and paperback versions of books, dinner versus lunch prices at restaurants, airfares with various restrictions, and price spreads of retail gasoline products. On the other hand, some economists have criticized the identification of such apparent “price anomalies” as price discrimination. For example, Lott and Roberts (1991) argue that there are usually cost-based explanations for these phenomena and propose such explanations for the cases of airfare, gasoline, and restaurant prices. Nevertheless, the authors do not challenge the conventional wisdom that price differences that cannot be explained by cost differences are discriminatory.

In recent years a number of papers have set out to empirically assess instances of alleged price discrimination. The literature goes to great lengths to control for potential sources of cost variation among different products and thus–to the extent that it is successful–is not subject to the Lott-Roberts critique. Almost all studies conclude that price discrimination is practiced in the particular market they analyze. A skeptic, of course, could always argue that some sources of cost variation have not been accounted for and thus dismiss the results as erroneous.

Despite the mild controversy over the methodology and its effectiveness, the basic premise seems universally accepted: The existence of price variation that cannot be explained by cost differences constitutes price discrimination. This is not, however, a formal definition. In fact there is no single, widely accepted definition of price discrimination when products are not homogeneous. Some authors define price discrimination to exist when price-cost margins (absolute differences) between differentiated products are unequal, whereas others prefer to compare price-cost markups (percentage differences). Few authors have discussed the relative merits of one measure over the other. (1) The relationship between the two has never been spelled out, and the choice among them remains arbitrary.

In this article I aim to shed some light on this issue and provide some guidelines to empirical researchers interested in price discrimination. I start by providing an overview of the current state of the literature in section II. I introduce the two competing definitions of price discrimination and briefly discuss the justification that has been provided for each. I then review the empirical literature that uses one or the other definition as the basis of testing for price discrimination. I present the basic methodology and discuss the definitional choices made by researchers investigating different market environments. The picture that emerges from this overview is murky and provides little guidance to the empirical practitioner.

The absence of controversy on the choice of definition might simply reflect the fact that the two commonly used definitions are equivalent. In section III I show that this is not the case. Specifically, it is possible for one definition to indicate the presence of price discrimination and for the other one to reject it. It is also possible for the two measures to take opposite signs; that is, one product may have a higher markup and a lower margin than the other. I also show that, in general, the margin criterion is more likely to indicate the presence of price discrimination. I conclude that the choice of definition has important implications and should be given careful consideration.

The challenge is to come up with a consistent way of thinking about price discrimination that can easily be applied in a variety of settings. In section IV I argue that pricing policies should be designated as discriminatory if they cannot be sustained in the presence of arbitrage. This implies that the cost of engaging in arbitrage should determine whether margins, markups, or some other metric is the right way to establish discriminatory conduct. If the cost of arbitrage is a fixed per-unit cost, nondiscriminatory prices will feature equal margins, lf the cost of arbitrage depends on product characteristics, nondiscrimination requires equal markups, lf both components are important, neither metric is sufficient. In all cases it is a good idea to report results using both measures as a simple robustness check.

II. THEORETICAL DEFINITIONS AND EMPIRICAL TESTS

This section provides an overview of the current state of the literature. It starts with a discussion of two theoretical definitions of price discrimination. It then outlines the empirical methodology that has been used to test for price discrimination and highlights its link to the theoretical definitions. A review of the empirical literature follows that reveals no systematic pattern of usage for the competing definitions.

Definitions

Consider the examples of airfare and books. The price of first-class passenger service is much higher than that of economy service; similarly, hard-cover books are typically priced at more than twice the price of their paperback versions. The cost of providing the two different products in each example is not the same, yet the additional cost associated with the higher-quality product appears to be very small compared to the premium the consumer is asked to pay for it. This is the basic premise of the general notion of price discrimination.

To formulate this idea into a rigorous definition one needs to be precise about what it means for price differences to not be explained by cost differences. Should price differences reflect cost differences one-to-one to be considered nondiscriminatory? Or should prices deviate from costs by a fixed percentage? Existing definitions follow from an affirmative response to one or the other question. Formally, suppose that two varieties of a certain good are sold at prices [p.sub.1] and [p.sub.2] and marginal costs of production are equal to [c.sub.1] and [c.sub.2]. The first definition requires that cost differences be reflected in price differences one-to-one:

DEFINITION 1. Price discrimination occurs whenever price-cost margins of two varieties of the same good are not the same; that is, whenever [p.sub.2] – [c.sub.2] [not equal to] [p.sub.1] – [c.sub.1].

Phlips (1983) is the foremost advocate of this definition. He goes through a fairly lengthy explanation of his position, but he never directly compares this definition with the next one. Notable authors like Tirole (1988) and Norman (1999) have adopted the margin definition in their treatments of price discrimination. Stigler (1987), on the other hand, proposed a different definition:

DEFINITION 2. Price discrimination occurs whenever price-cost markups of two varieties of the same good are not the same; that is, whenever [p.sub.2]/[c.sub.2] [not equal to] [p.sub.1]/[c.sub.1] (or, equivalently, whenever ln [p.sub.2] – ln [c.sub.2] [not equal to] ln [p.sub.1] – ln [c.sub.1]).

Varian (1989) and Stole (2001) adopted the markup definition. It also seems to be the definition that Machlup (1952) had in mind, although he did not state it formally. Stigler (1987, 210) justifies his preference for it as follows:

The proportionality definition has the merit of

separating a monopolist’s behavior into two parts:

(1) the simple restriction of output such that price

is greater than marginal cost; and (2) the

misallocation of the two or more goods among

buyers when they are charged different prices,

which is zero if prices are proportional to marginal

costs.

Margins can demonstrate the output restriction as well as markups do, but they do not capture the misallocation aspect. It is easy to see how misallocation can arise when consumers demand multiple units of a homogeneous good. (2) It is less clear how that works when consumers have unit demands for differentiated products. This distinction will be shown to be important in section IV. It is also worth noting that comparing the markups of two products is equivalent to comparing their respective Lerner indices: A higher markup implies a higher Lerner index. (3) This may give some appeal to the markup definition because the Lerner index is a well-known quantity that is often used as a measure of market power.

These observations are useful, but they are not general enough to settle the debate. In fact, many of the authors cited have themselves stressed the difficulty of coming up with a precise and comprehensive definition. (4) The premise of this article is that even though we may not be able to formulate an all-encompassing definition, we should at least be aware of the general relationship between the two definitions and of the different implications they may have in specific settings.

The rest of this section outlines the basic methodology used to test for price discrimination and reviews some of the empirical studies. A word on terminology is in order. Stole (2001) points out that economists’ notion of price discrimination has diverged from Pigou’s (1920) traditional taxonomy of first-, second-, and third-degree price discrimination. Even though Pigou’s terminology has been retained, the classification given in most textbooks today is different from what he had suggested. To avoid any confusion, I follow Stole (2001) in classifying price discrimination techniques as direct versus indirect. Direct price discrimination (third-degree price discrimination in Pigou’s classification) is exercised on the basis of observable characteristics, such as age, gender, and location. Indirect price discrimination (which was not really considered by Pigou) sorts consumers by offering menus of products that differ in quality or quantity. The crucial distinction is that when discrimination is direct, each consumer is offered only one product; with indirect discrimination consumers have a choice among the available products.

Empirical Tests

The most direct way to test for price discrimination is to use information on the prices different consumers pay for the same good (as, for example, in Goldberg 1996). Because such detailed data are usually not available, researchers have had to infer discriminatory practices from price observations in different markets. Inference is based on a simple and intuitive econometric methodology that tests for price discrimination on the basis of one of the two definitions presented. I describe a basic version of this methodology while noting that different variants of it have been employed.

Consider the case of two products, indexed by 1 and 2. Typically this pair of products is observed in a number of different markets. Available data are {[p.sub.ij], [x.sub.ij]; i = 1,2; j = 1, … J}, where [p.sub.ij] denotes price, [x.sub.ij] is a vector of observed cost shifters, and i and j index products and markets, respectively. Suppose that the researcher has decided to test for price discrimination by comparing markups. Nondiscriminatory pricing under the markup definition implies that the difference in log-prices must be equal to the difference in log-costs; that is, ln[p.sub.2] – ln[p.sub.1] = ln[c.sub.2] – ln[c.sub.1]. If marginal costs are observed, then all one has to do is compute the markups and compare them. But costs are usually not observed. The solution is to specify cost econometrically as a function of observed product characteristics and estimate it. For i = 1, 2 (and temporarily dropping j subscripts), let ln[c.sub.i] = [x’.sub.i][beta] + [[omega].sub.i], where [[omega].sub.i] is an unobserved cost shifter. If there is no price discrimination, ln[p.sub.2] – ln[p.sub.1] = ln[c.sub.2] – ln[c.sub.1] = ([x.sub.2]'[beta] + [[omega].sub.2]) – [x.sub.1]’ [beta] + [[omega].sub.1]) = ([x.sub.2] – [x.sub.1])'[beta] + [epsilon], where [epsilon] = [[omega].sub.2] – [[omega].sub.1]. Letting [DELTA] denote differences, the corresponding regression equation is

(1) [DELTA]ln[p.sub.j] = [alpha] + [DELTA][x.sub.j]'[beta] + [[epsilon].sub.j], j = 1, … J.

The test for price discrimination amounts to a statistical test of the null hypothesis [alpha] = 0. Failure to reject the hypothesis means that price differences are fully explained by cost differences and there is no price discrimination. Rejection of the hypothesis indicates that demand differences induce price differences, and thus price discrimination is said to exist.

The same idea can be applied to the margin definition. Nondiscrimination requires that absolute price differences be equal to absolute cost differences: [p.sub.2] – [p.sub.1] = [c.sub.2] – [c.sub.1]. Letting [c.sub.i] = [x.sub.i]'[beta] + [[omega].sub.i], no price discrimination implies that [p.sub.2] – [p.sub.1] = [c.sub.2] – [c.sub.1] = ([x.sub.2] – [x.sub.1])’ [beta] + [epsilon]. The regression equation is

(2) [DELTA][p.sub.j] = [alpha] + [DELTA][x.sub.j]'[beta] + [[epsilon].sub.j], j = 1, … J.

A comparison of equations (1) and (2) reveals the empirical analog of the markup versus margin dilemma. Adopting a definition of price discrimination is equivalent to deciding whether to use differences in prices or differences in log-prices as the dependent variable in the regression. Either way, it is important to control for all possible variation in cost. Any cost variation that is not accounted for will be captured in the constant [alpha], which will falsely indicate the presence of price discrimination. The prime challenge for empirical researchers is to come up with ways of controlling for cost differences.

Two pioneering studies of gasoline pricing showed the way. Borenstein (1991) formulates a test for price discrimination by exploiting the varying availability of leaded and unleaded gasoline. He finds that margins for leaded gasoline are higher, a result that is consistent with the fact that competition is less strong in that market. He justifies his use of margins on industry conventions. He also argues that the margin is a better indicator of market power in his example because market power derives from the buyer’s transportation cost of switching dealers. Discrimination in Borenstein’s study is direct (third-degree) because it is based on an observable characteristic, the car’s engine type.

The second pioneering study is Shepard (1991), which compares gas prices at stations with both full-service and self-service pumps (multiproduct stations) versus those that offer only one of the two options (single-product stations). The theoretical model predicts that in a competitive market margins should be the same in all stations, whereas in the presence of market power margins will differ between full-service and self-service stations. The intuition is the following. Assume that the cost of selling gasoline in a multiproduct station is the same as the cost of doing so in a single-product station. Then, when there is no price discrimination, the price of each type of gasoline should be the same regardless of station type. She finds that this is not the case. The price of full-service gasoline is higher in multiproduct stations, and the price of self-service is lower. This implies that multiproduct stations are exploiting the presence of two products to price discriminate. Discrimination in this example is indirect: Consumers have the option of going to any station and sort themselves based on location and brand preference.

In another study of indirect price discrimination, Borenstein and Rose (1994) look at variation in fares paid by different passengers on the same flight. The authors prefer using markups on the grounds that “cost-based differences, such as peak-load pricing, hold markups constant.” They find that price variation cannot be fully explained by cost differences. Verboven (2002), on the other hand, uses margins to compare gasoline- and diesel-fueled automobiles in Europe. He argues that cars with diesel engines can be considered higher quality than autos of similar features running on gasoline because of the lower cost of diesel fuel and its favorable tax treatment. He finds that as theory would predict, the higher-quality product (diesel-engine cars) sells at a higher margin. Clerides (2002) uses actual measures of marginal cost to compare both margins and markups of hardcover and paperback versions of books. He finds that both measures are higher for hardcovers than for paperbacks. In line with corollary (4), his conclusion is stronger when margins are compared.

Two recent papers examine quantity-based indirect discrimination. Cohen (2000) looks at the pricing of different package sizes of paper towels. He uses margins as the benchmark for comparison and finds substantial price dispersion that cannot be explained by cost variation. Busse and Rysman (2002), on the other hand, prefer markups as the relevant measure in their study of Yellow Pages advertising prices. They find that the price per square inch is lower for larger advertisements, which therefore have lower markups. The authors argue that cost factors cannot explain this difference.

Finally, some researchers have looked for price discrimination across geographically distinct markets. Verboven (1996) and Goldberg and Verboven (2001) compare automobile prices across European countries. They adopt markups as the basis of comparison and find substantial variation across countries. Giulietti and Waterson (1997) compare prices of several products across Italian supermarkets. These authors declare themselves to be agnostic on the markup versus margin issue and use both in their analysis. Both measures indicate the presence of price discrimination.

If any picture emerges from this review, it is one of inconsistency. Most studies do not go to great lengths to justify their choice of definition. They simply select one or the other (or both) and cite one or more of the earlier studies in support of their choice. Table 1 summarizes the key aspects of the studies reviewed in this section. The lack of any systematic pattern is apparent. Different measures of price discrimination are used to evaluate pricing policies in seemingly similar situations. This would not be an issue if the two measures were equivalent. In the next section, I show that they are not.

III. DO MARGINS AND MARKUPS TELL US THE SAME THING?

The ultimate purpose of the empirical literature is to test the null hypothesis of no price discrimination. If margins and markups always deliver the same verdict on this test, they can be thought of as being qualitatively equivalent. In other words, margins and markups are qualitatively equivalent if sgn[[DELTA]MARGIN] = sgn[[DELTA]MARKUP], where [DELTA]MARGIN = ([p.sub.2] – [c.sub.2]) – ([p.sub.1] – [c.sub.1]) and [DELTA]MARKUP = ([p.sub.2]/[c.sub.2]) – ([p.sub.1]/[c.sub.1]). To determine whether this is the case, I first establish an algebraic relationship between [DELTA]MARGIN and [DELTA]MARKUP:

PROPOSITION 1. Assume [c.sub.1], [c.sub.2] > 0. Then,

(3) [DELTA]MARGIN = [c.sub.1] x [DELTA]MARKUP + [([p.sub.2] – [c.sub.2])([c.sub.2] – [c.sub.1])]/[c.sub.2].

All derivations are relegated to the appendix. The main implications of this proposition are laid out in a series of corollaries. I start with the simplest case:

COROLLARY 1. If [c.sub.2] = [c.sub.1] then sgn [[DELTA]MARGIN] = sgn[[DELTA]MARKUP].

The corollary states that the two definitions are qualitatively equivalent when there are no cost differences between the two products. In this case the choice between them is immaterial. The next corollary deals with the case where cost differences do exist:

COROLLARY 2. If [c.sub.2] [not equal to] [c.sub.1], both definitions will reject the presence of price discrimination if and only if price equals marginal cost for both products.

This is an important point: Anything that is not marginal cost pricing will be identified as price discrimination by at least one of the two definitions. The significance of selecting a definition ex ante and for the right reasons is obvious. Otherwise one could look at the data and pick the definition that delivers the “right” answer. A simple example will help illustrate this point. Consider a firm that monopolizes two markets. The costs of serving those markets are [c.sub.1] and [c.sub.2], with [c.sub.2] > [c.sub.1]. If the firm decides to price at equal margins, the high-priced product will have a lower markup than the cheaper product. If, on the other hand, the firm prices at equal markups, the more expensive product will have a higher margin. Anything the firms does (other than price at marginal cost) will be deemed discriminatory by one of the two definitions. Suppose further that the good can be freely transported across markets at zero cost. Then the firm will be forced to set [p.sub.2] = [p.sub.1], and both margins and markups will differ across markets. That is, both measures will indicate the presence of price discrimination even though the firm does not have the ability to segment the markets.

For the next two corollaries, suppose that without loss of generality, good 2 is no cheaper than good 1; that is, [p.sub.2] [greater than or equal to] [p.sub.1]. Consider first the case [c.sub.2] < [c.sub.1]. This is not a common occurrence, but examples of it have been reported in the literature, notably in the context of "damaged goods." (5)

COROLLARY 3. Suppose [p.sub.2] [greater than or equal to] [p.sub.1] and [p.sub.j] > [c.sub.j], j = 1,2. If in addition, [c.sub.2] 0 and [DELTA]MARGIN > 0.

The corollary states that the practice of selling damaged goods is always discriminatory according to both measures. This is in accordance to our intuition; few economists would disagree with the notion that charging a higher price for a good that is less costly to produce is discriminatory.

The most important and empirically relevant scenario is the case [c.sub.2] > [c.sub.1]:

COROLLARY 4. Suppose [p.sub.2] [greater than or equal to] [p.sub.1] and [p.sub.j] > [c.sub.j], j = 1,2. If [c.sub.2] > [+c.sub.1] then,

(i) [DELTA]MARKUP = 0 [??] [DELTA]MARGIN >0.

(ii) [DELTA]MARGIN = 0 [??] [DELTA]MARKUP<0.

(iii) If further, [c.sub.1] [greater than or equal to] 1, then [DELTA]MARGIN [greater than or equal to] [DELTA]MARKUP.

(iv) If [p.sub.2]>[p.sub.1] then we may have [DELTA]MARGIN > 0 and [DELTA]MARKUP < 0.

The first two statements indicate the ways different outcomes can emerge from the two definitions. The third statement states that if [c.sub.1] > 1 we can actually rank the two measures: [DELTA]MARGIN will always be at least as high as the [DELTA]MARKUP. This implies that as long as [c.sub.1] > 1, margins are more likely to indicate the presence of price discrimination than markups.

The fourth statement points out that the two quantities could take opposite signs. That is, one product may have a higher margin, whereas the other has a higher markup. Because neither definition restricts the direction of the inequality, mechanical application of the two criteria will tell us that both find price discrimination. Nonetheless, the direction of price discrimination should be of some interest. Although usually it is not formally stated, price discrimination embodies a notion of someone being discriminated against, in the sense of paying more than someone else. If that is the case, what are we to make of situations where our two measures point in opposite directions? Consider an example where [p.sub.2] = 12, [c.sub.2] = 8, [p.sub.1] = 8, [c.sub.1] = 5, and therefore [DELTA]MARGIN > 0 and [DELTA]MARKUP < 0. Who is paying more in this case? Although there is no clear-cut answer, I am inclined to go with the principle that whoever pays a higher price is "paying more." After all, how often do people paying a lower price complain that they are being discriminated against because they pay a higher margin? If we accept this logic, we should use margins as an indicator of discrimination because the more expensive product will rarely have a lower margin.

The data on hard-cover and paperback books used in Clerides (2002) provide a useful illustration of the results in Corollary 4. In Figure 1, the horizontal axis plots [DELTA]MARGIN and the vertical axis plots [DELTA]MARKUP for the hard-cover and paperback versions of 71 books. A book that lies on the horizontal axis and to the right of the vertical axis will have a higher margin for the hardcover version but equal markups. The implications of each statement of Corollary 4 can be clearly seen: (i) all titles with nonnegative [DELTA]MARKUP have positive [DELTA]MARGIN; (ii) the five observations in the southwest quadrant have (nearly) equal margins and lower markups for the more expensive product; (iii) all observations are below the diagonal; and (iv) four observations (those in the southeast quadrant) take opposite signs for [DELTA]MARKUP and [DELTA]MARGIN.

[FIGURE 1 OMITTED]

In summary, the results from this section indicate that the two definitions will coincide only when the high-price product is no more costly to produce than the low-price product. When this is not the case, the choice of definition is important and should be carefully considered. It is important to note that these conclusions are based solely on algebraic considerations. The only economic restriction imposed is that price is above marginal cost. One would like to know what economic conditions, if any, might give rise to each of the cases outlined in Corollary 4. For example, visual inspection of equation (3) suggests that [DELTA]MARGIN and [DELTA]MARKUP will diverge when prices deviate substantially from marginal costs and cost differences are high. It is difficult, however, to make general statements. Economic restrictions on possible combinations of margins and markups must be justified by a theoretical model. Any such model would invariably have to be tailored to the particular market under examination, and its conclusions will be difficult to generalize.

IV. DISCRIMINATION AND THE COST OF ARBITRAGE

Identification of price discrimination requires the existence of a nondiscriminatory pricing policy that will serve as a benchmark for comparison. In the case of homogeneous products, this benchmark is uniform pricing: When all consumers pay the same price, there is no price discrimination. When products are differentiated, however, there is no obvious pricing policy that serves this purpose. Consider, for example, a monopolist serving two geographically distinct markets that have different demands and different costs of serving. Or consider a firm that produces two versions of the same product that differ in quality and production cost. What is the nondiscriminatory pricing policy in those cases? Does such a policy require equal margins, equal markups, or something completely different?

Feasibility and Cost of Arbitrage

A natural way to obtain a benchmark is to relax the conditions that make discrimination possible. The two most important requirements for price discrimination are the existence of market power and the impossibility of resale. In the homogeneous product case, the difference between the discriminatory and nondiscriminatory policies stems from the impossibility of resale. Market power by itself is not enough to enable the firm to discriminate. It seems reasonable to extend this intuition to the case of differentiated products and designate as nondiscriminatory those policies that survive the introduction of arbitrage. This idea is not new; it underlies many existing treatments of price discrimination (see, for example, Phlips 1983; Tirole 1988). The contribution of this article is to make this interpretation more concrete and show how it can be implemented in empirical work.

The case of quantity discounts illustrates this well. Suppose a product is sold individually at a price of $10 per unit and in packages of two at a price of $15 per package. This pricing policy cannot survive the introduction of arbitrage: One can purchase the package, break it up and sell the individual items for a net gain. This threat will force the monopolist to price linearly, at a constant unit price. If the marginal cost of the package is (roughly) twice the cost of a single item, linear pricing implies equal markups but unequal margins (unless price equals marginal cost). Hence, markups are the preferred indicator of price discrimination when comparing packages of different sizes. (6)

Arbitrage works in a straightforward way in this case because a package is just a combination of individual units. When products are physically differentiated, however, arbitrage is no longer a simple matter of resale. One of the products must be somehow transformed into the other to be resold. The transformation process may simply involve transporting the good to a different location, or it may require something more elaborate, like upgrading the product. Either way, there is some cost involved in the process. I argue that the form of the arbitrage cost function should determine the nondiscriminatory policy. Broadly speaking, if the cost of arbitrage is a fixed per-unit cost, nondiscrimination requires margin equality. If the cost of arbitrage depends on product characteristics, the requirement should be markup equality.

The case of geographic differentiation provides a nice illustration of this point. Consider a car manufacturer selling his product in two countries. The good is manufactured in country 1, where it is priced at [p.sub.1]. It can be transported to country 2 at a cost of T = f + [tau][p.sub.1] per unit. Freight cost may depend on price for a number of reasons; for example, insurance costs during transportation are typically a function of the good’s value. In the absence of trade, the firm would be able to set the profit-maximizing price for each market. When arbitrage is possible, on the other hand, the two prices cannot deviate by more than the freight cost, t. The nondiscriminatory price in country 2 is [p.sub.2] = [p.sub.1] + T = [p.sub.1] + f + [tau][p.sub.1] = f + (1 + [tau])[p.sub.1]. (7) If freight cost is fixed (f > 0 and [tau] = 0), nondiscrimination requires [p.sub.2] = [p.sub.1] + f. Given that there are no other cost differences, this implies identical margins. If, on the other hand, freight cost is variable (f = 0 and [tau] > 0), nondiscriminatory pricing requires [p.sub.2] = (1 + [tau])[p.sub.1], implying identical markups.

As another example, consider the case of vertically differentiated products (quality discrimination). What is the nondiscriminatory pricing policy when a firm sells high- and low-quality versions of some product? Arbitrage in this case involves purchasing the low-quality product, upgrading it to high quality, and reselling it. The analog to freight cost is the cost of upgrading. If the cost of upgrading is mostly fixed, margins are the better measure of price discrimination. If the cost depends on the value of the good, markups should be preferred. Consider the example of books. Binding a book in hard cover certainly has a variable component: Longer books (in terms of number of pages) require more materials. But, as Clerides (2002) notes, most of the cost is that of the cover itself, which is independent of the length of the book. Thus nondiscriminatory pricing of books requires equal margins. (8)

The argument carries through in exactly the same way when products are horizontally differentiated. A market definition problem may however arise if products are “too” differentiated. When can two products be considered different varieties of the same good (in which case we can talk about price discrimination) and when are they two completely different goods? Consider a food establishment that sells pizzas and sandwiches. Suppose that the costs of preparing an individual serving of each are the same (both items require similar ingredients: flour, tomatoes, cheese, etc.). Should they be priced the same? Is it price discrimination if they are not? I would say that the answer to both questions is yes, but it is easy to see how this logic can take us down a slippery path. (9)

Empirical Implementation

As seen in the two examples, choosing the correct measure requires knowledge of the arbitrage cost function. This may not always be easy. The problem is further confounded when both the fixed and the value-dependent component of arbitrage cost are important: Neither metric would be a good indicator of price discrimination in that case. Consider implementation of the methodology described in section II using data on prices and characteristics (cost shifters) of a number of different car models in two countries. To test margin equality we can estimate equation (2), where f is parameterized by x’ [beta]. If arbitrage cost is T = f + [tau][p.sub.1], the relationship between the prices in the no discrimination case is [p.sub.2] – [p.sub.1] = f + (1 + [tau])[p.sub.1]. Estimation of equation (2) does not control for the (1 + [tau])[p.sub.1] part. The latter will show up in the constant, incorrectly indicating the presence of price discrimination.

Adding [p.sub.1] as an explanatory variable in the regression can correct that problem, but only at the risk of creating a different one. Suppose the arbitrage cost is fixed (T = f) and the firm is discriminating by charging [p.sub.2] – [p.sub.1] = f + (1 + [epsilon])[p.sub.1], with [epsilon] > 0. If we run a regression of price differences on cost shifters and [p.sub.1], the coefficient on the latter will capture all residual price variation that is unexplained by cost factors. This will lead us to conclude erroneously that there is no discrimination. The problem is that the coefficient on [p.sub.1] is capturing two things. One is price discrimination; the other is the extent to which the cost of arbitrage depends on the value of the good. Without more information, I cannot separately identify the two effects.

A different identification problem may arise when demand is unknown. Suppose the car manufacturer monopolizes two countries that have identical demands, but one is more costly to serve than the other. Suppose also that we know that the cost of arbitrage is fixed, so that nondiscrimination requires equal margins. Because demand is identical, a profit-maximizing (and price-discriminating) firm will set a higher price in the high-cost market. But what are margins going to look like? Unfortunately, there is no single answer; it all depends on the shape of the demand curve. Margins will be higher in the high-cost market if demand is log-linear, they will be lower if demand is linear, and they will be equal if demand is exponential. If the latter happens to be the case. we will fail to identify price discrimination. (10) This suggests that demand information is also an important element in the identification of discriminatory practices.

Where does this all leave us? Exploiting the idea of arbitrage as a way of establishing discriminatory policies is useful in situations where the cost of arbitrage is easy to determine. When this is not possible, assessing pricing policies may require a complete model of consumer demand, firm behavior, and arbitrage activity. In any case, it is good practice to always report results with both measures; they do, after all, contain different information. If they concur, the researcher can be quite confident of his or her conclusions. If not, this is something the reader should know about.

V. CONCLUDING REMARKS

In this article I examine the properties of the two commonly used definitions of price discrimination with differentiated products. I show that unless costs are equal, anything that is not marginal cost pricing will be identified as price discrimination by at least one of the two definitions. Moreover, the two measures are qualitatively different: It is possible for one to accept price discrimination and the other to reject it.

Clearly, choosing the right definition is important. But how does one choose? Markups are the conservative choice because they are generally less likely to indicate the presence of price discrimination than margins. They are also unit-free and correspond directly to the widely used Lerner index. On the other hand, markups may often be lower for the more expensive product, making interpretation difficult. I propose a systematic way of thinking about price discrimination in terms of the feasibility of arbitrage. I argue that the form of the arbitrage cost function determines the right measure to use in testing for price discrimination. Given that the cost of arbitrage is not typically observed, empirical implementation of this idea will not always be straightforward. But it can, at the very least, help researchers sort out the different sources of variation in the data and guide modeling approaches.

It is important to keep in mind that identification of price discrimination becomes increasingly tenuous the further one moves away from a world of homogeneous products and equal costs. Despite this caveat, this is a line of research worth pursuing. The analysis of firm strategy and pricing practices is an interesting topic in and of itself. Perhaps more important, prices are also an important source of information about the nature of demand in different industries; firms, after all, set prices in response to consumer preferences. Hopefully this study will prove useful to empirical practitioners in this area.

APPENDIX: DERIVATIONS

Proposition 1

[DELTA]MARGIN = ([p.sub.2] – [c.sub.2]) – ([p.sub.1] – [c.sub.1]) = [c.sub.1] [[p.sub.2]/[c.sub.1] – [c.sub.2]/[c.sub.1] – [p.sub.1]/[c.sub.1] + 1] = [c.sub.1] [[p.sub.2]/[c.sub.1] – [c.sub.2]/[c.sub.1] – [p.sub.2] /[c.sub.2] +[p.sub.2]/[c.sub.2] – [p.sub.1]/[c.sub.1] + 1] = [c.sub.1] [[DELTA]MARKUP + [p.sub.2]/[c.sub.1] – [c.sub.2]/[c.sub.1] – [p.sub.2]/[c.sub.2] + 1]

where line 1 is the definition; in line 2 we get rid of the parentheses and multiply and divide by [c.sub.1]; in line 3 we add and subtract [p.sub.2]/[c.sub.2]; and in line 4 we replace [DELTA]MARKUP = [p.sub.2]/[c.sub.2]-[p.sub.1]/[c.sub.1] and reorder terms. It is easy to show that the last four terms in the parentheses can he rewritten as

[p.sub.2]/[c.sub.1] – [c.sub.2]/[c.sub.1] – [p.sub.2]/[c.sub.2] + 1 = [([p.sub.2] – [c.sub.2])([c.sub.2] – [c.sub.1])]/[c.sub.1] [c.sub.2].

Substituting this in yields the desired relationship:

[DELTA]MARGIN = [c.sub.1] x [DELTA]MARKUP + [([p.sub.2] – [c.sub.2]) ([c.sub.2] – [c.sub.1])]/[c.sub.2].

Corollary 1

Trivial.

Corollary 2

There are two directions to the statement.

([??]) Clearly, if [p.sub.2] = [c.sub.2] and [p.sub.1] = [c.sub.1], we have [DELTA]MARGIN = [DELTA]MARKUP = 0.

([??]) Suppose both products reject the presence of price discrimination; that is, [DELTA]MARGIN = [DELTA]MARKUP = 0. It follows that the second term on the right-hand side of equation (3) must also equal zero. Because [c.sub.1] [not equal to] [c.sub.2], it must be that [p.sub.2] = [c.sub.2]. Given that and the fact that [DELTA]MARGIN = 0, it must be that [p.sub.1] = [c.sub.1].

Corollary 3

Trivial.

Corollary 4

Statements (i), (ii), and (iii) follow immediately from the fact that the last term in equation (3) is positive. Statement (iv) follows from statement (ii) by a continuity argument: it is always possible to increase [p.sub.2] by a small enough amount so that [DELTA]MARGIN becomes positive and [DELTA]MARKUP remains negative.

TABLE 1.

A Summary of Empirical Studies of Price Discrimination

Study Market Analyzed Type of PD

Borenstein (1991) Gasoline types Direct

Shepard (1991) Gas stations Indirect

Borenstein and Rose (1994) Airfare Indirect

Verboven (1996) Automobiles Direct

Giulietti and Waterson (1997) Supermarkets Direct

Verboven (2002) Automobiles Indirect

Clerides (2002) Books Indirect

Cohen (2000) Paper towels Indirect

Busse and Rysman (2002) Advertising Indirect

Criterion Used

Study Basis of PD Margin Markup

Borenstein (1991) Engine type [check]

Shepard (1991) Quality [check]

Borenstein and Rose (1994) Quality, capacity [check]

Verboven (1996) Location [check]

Giulietti and Waterson (1997) Location [check] [check]

Verboven (2002) Quality [check]

Clerides (2002) Quality [check] [check]

Cohen (2000) Quantity [check]

Busse and Rysman (2002) Quantity [check]

(1.) The exceptions are two short papers by Kahana and Spiegel (1988) and Liebowitz (1988). The papers focused on specific examples and did not make an attempt to generalize the debate or address empirical issues.

(2.) It occurs, for example, if the incremental price of the last unit paid by a buyer of large quantities is lower than what a buyer of a small quantity is willing to pay for an additional unit.

(3.) The Lerner index L [equivalent to](p – c)/p = 1 – c/p, which implies that p/c = 1/(1 – L). Hence, if [p.sub.2]/[c.sub.2] > [p.sub.1] /[c.sub.1], then 1/(1 – [L.sub.2]) > 1/(1 – [L.sub.1]), which implies that [L.sub.2] > [L.sub.1].

(4.) Some indicative quotations: Machlup (1952, 136) says that “comprehensive definitions of price discrimination will always be clumsy”; Phlips (1983, 5) notes that “the more one thinks of price discrimination, the harder it is to define”; Tirole (1988, 134) states that “it is difficult to offer an all-encompassing definition.

(5.) The term, coined by Deneckere and McAfee (1996), refers to situations in which firms incur additional cost to produce lower-quality versions of their main product.

(6.) The margin criterion will practically always indicate the presence of price discrimination when packages of different size are available. Margins will only be equal when the incremental price of the second unit is equal to marginal cost. This will rarely by the case. Alternatively, one might prefer to compare price per unit across packages: nondiscrimination requires that price per unit not depend on package size.

(7.) This is free-on-board (fob) pricing: Consumers are charged a uniform price plus freight cost. Most economists will agree that fob pricing is nondiscriminatory but freight absorption policies (“free shipping” offers) are discriminatory.

(8.) Even though the theoretical literature treats quality and quantity as interchangeable, this article recommends different criteria for price discrimination in those two cases. The fundamental difference is that with quantity discounts the incremental price per unit is decreasing, whereas with vertical differentiation the price per additional unit of quality is increasing. This does not change the mathematical formulation of the model, but it does change the way arbitrage is implemented.

(9.) I am grateful to Simon Anderson for posing this question.

(10.) It is easy to construct other examples where the opposite is true: We will identify price discrimination when it does not exist. This point has been recognized in the extensive literature on exchange rate pass-through and pricing-to-market; see Goldberg and Knetter (1997) for a survey.

REFERENCES

Borenstein, S. “Selling Costs and Switching Costs: Explaining Retail Gasoline Margins.” Rand Journal of Economics, 22(3), 1991, 354-69.

Borenstein, S., and N. L. Rose. “Competition and Price Dispersion in the U.S. Airline Industry.” Journal of Political Economy, 102(4), 1994, 653-83.

Busse, M., and M. Rysman. “Competition and Price Discrimination in Yellow Pages Advertising.” Unpublished manuscript, Boston University, 2002.

Clerides, S. K. “Book Value: Intertemporal Pricing and Quality Discrimination in the U.S. Market for Books.” International Journal of Industrial Organization, 20(10), 2002, 1385-408.

Cohen, A. “Package Size and Price Discrimination in Paper Towels.” Unpublished manuscript, University of Virginia, 2000.

Deneckere, R., and R. P. McAfee. “Damaged Goods.” Journal of Economies and Management Strategy, 5(2), 1996, 149-74.

Giulietti, M., and M. Waterson. “Multiproduct Firms’ Pricing Behaviour in the Italian Grocery Trade.” Review of Industrial Organization, 12(516), 1997, 817-32.

Goldberg, P. K. “Dealer Price Discrimination in New Car Purchases: Evidence from the Consumer Expenditure Survey.” Journal of Political Economy, 104(3), 1996, 622-54.

Goldberg, P. K., and M. M. Knetter. “Goods Prices and Exchange Rates: What Have We Learned?” Journal of Economic Literature, 35(3), 1997, 1243-72.

Goldberg, P. K., and F. Verboven. “The Evolution of Price Dispersion in the European Car Market.” Review of Economic Studies, 68(4), 2001, 811-48.

Kahana, N., and U. Spiegel. “On the Definition of Price Discrimination.” Economic Inquiry, 26(4), 1988, 775-77.

Liebowitz, S. J. “Price Differentials and Price Discrimination: Reply and Extensions.” Economic Inquiry, 26(4), 1988, 779-83.

Lott, J. R., and R. D. Roberts. “A Guide to the Pitfalls of Identifying Price Discrimination.” Economic Inquiry, 29(1), 1991, 14-23.

Machlup, F. The Political Economy of Monopoly: Business, Labor and Government Policies. Baltimore: Johns Hopkins University Press, 1952.

Norman, G. “Introduction,” in The Economics of Price Discrimination, edited by G. Norman. Cheltenham: Edward Elgar, 1999.

Phlips, L. The Economics of Price Discrimination. Cambridge: Cambridge University Press, 1983.

Pigou, A. C. The Economics of Welfare. London: Macmillan, 1920.

Shepard, A. “Price Discrimination and Retail Configuration.” Journal of Political Economy, 99(1), 1991, 30-53.

Stigler, G. J. The Theory of Price. New York: Macmillan, 1987.

Stole, L. A. “Price Discrimination in Competitive Environments.” Unpublished manuscript, University of Chicago, 2001.

Tirole, J. The Theory of Industrial Organization. Cambridge, MA: MIT Press, 1988.

Varian, H. R. “Price Discrimination,” in Handbook of Industrial Organization, Volume I, edited by R. Schmalensee and R. D. Willig. Cambridge: Elsevier Science Publishers, 1989.

Verboven, F. “International Price Discrimination in the European Car Market.” Rand Journal of Economics, 27(2), 1996, 240-68.

–. “Quality-Based Price Discrimination and Tax Incidence: Evidence from Gasoline and Diesel Cars.” Rand Journal of Economics, 33(2), 2002, 275-97.

SOFRONIS K. CLERIDES, I have greatly benefited from discussions with Marc Rysman and Frank Verboven and from the comments of two anonymous referees. Participants at the 2001 conferences of the European Association for Research in Industrial Economics and the European Network on Industrial Policy also provided useful feedback. I am solely responsible for all errors.

Clerides: Assistant Professor. Department of Economics, University of Cyprus, P.O. Box 20537, CY-1678 Nicosia, Cyprus. Phone +357 2289 2450, Fax +357 2289 2432, E-mail sofronis@ucy.ac.cy

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