Price Discrimination and Investment Incentives

Transcription

1 Price Discrimination and Investment Incentives Alexei Alexandrov and Joyee Deb University of Rochester and New York University March 12, 2010 Abstract We examine a model of a monopolist supplier selling to two segments of consumers, who have different preferences for quality (or some product characteristic). We show that if the firm is unable to price discriminate between the segments, then there is less investment in quality (the product characteristic). We find that both consumer segments, and the society overall, may suffer if the firms are unable to price discriminate. We extend the model to duopoly competition in the upstream market, and find that our results still hold. 1 Introduction The effect of price discrimination on social welfare had been a topic of interest among economists for a long time, at least since Pigou (1920) who conjectured that price discrimination decreases welfare if the total output decreases at the same time. Back in 1936, the Robinson-Patman Act was passed to disallow price discrimination in the intermediary markets in the United States, and price discrimination between consumers is routinely a source of public relations problems for companies. The central question we ask is how the ability to price discriminate affects a firm s incentives to invest in quality (or any other product characteristic), and in turn how this affects consumer and social welfare. We are concerned with markets where different segments of consumers have different valuations for a given product characteristic (we call it quality from now on). A business owner cares much more about her hard drive not failing than a consumer with nothing irreplaceable on that hard drive. Business travelers care much more about their flight landing on time than leisure travelers on the same flight going for a week-long vacation with no particular plans. A sick patient who might have a tumor cares much more about the kind of MRI machine in the hospital than a healthy person. Hospitals and welders care about oxygen quality much more than oxygen bar owners. Rocket fuel producers care about helium purity much more than kids balloon fillers. In Keywords: price discrimination, investment, input markets, Robinson-Patman Act, oligopoly. JEL Codes: L42, L13, D92. Assistant Professor of Economics and Management, Alexei.Alexandrov@Simon.Rochester.edu and Assistant Professor of Economics, joyee.deb@nyu.edu 1

2 such markets, firms must make both pricing decisions as well as investment decisions for quality, and their incentives to invest depend critically on whether they are allowed to price discriminate between the two segments. We study a model of a monopolist supplier choosing both quality and prices. The firm operates in a market where consumers have different preferences for quality. There are two segments discerning consumers who care about quality and undiscerning ones who do not. We examine the firm s choices under two regimes one in which the firm is allowed to price discriminate and another where it is not. We find that the supplier invests less in quality when it cannot price discriminate between the two consumer segments. We also find that the undiscerning segment that does not care about quality always suffers if price discrimination is not allowed. We show also that, somewhat surprisingly, the discerning segment that cares about quality may suffer as well. It turns out that if the the investment cost function is not too convex, all consumers are worse off in a regime without price discrimination. We extend our model to duopoly competition (differentiated Bertrand competition), and find that our results still hold with strategic competition (see the Appendix). What is the intuition behind the results? When price discrimination is allowed, the monopolist ensures that only the discerning segment pays more because of a higher investment. Without price discrimination, both segments pay for it through the uniform price. The undiscerning segment is the weak segment which gets stuck with a higher price because of the monopolist s inability to price discriminate, and this results in a lower welfare for this segment. The discerning segment is the strong segment which gets a lower price with uniform pricing. 1 However, the lower price does not necessarily imply an increase in welfare. While price is lower, investment decreases as well in a regime with no price discrimination. Thus, depending on how much the investment decreases, consumers can be worse off, and the magnitude of the drop in investment depends on the shape of the investment cost function. The effects of price discrimination on welfare had been studied before. The literature has focused on two questions. First, if there is a monopolist who might discriminate between two consumer segments, does increased output mean that the social welfare increased too? Second, if there is a monopolist supplier and more and less efficient downstream firms, does the society suffer from prohibition of price discrimination in the intermediate market? Katz (1987) examines a setting where one of the firms can vertically integrate backwards, and thus receives a lower price from suppliers. There, price discrimination decreases welfare unless it prevents inefficient vertical integration. DeGraba (1990) studies investment choice of downstream producers with a supplier who might price discriminate. He finds that the investment is lower with price discrimination, and moreover with a specific functional form, welfare is always lower with price discrimination. Yoshida (2000) makes the model richer by allowing the amount of output to change, and finds, in contrast to the line of literature starting with Pigou (1920) and culminating with Schwartz (1990), that increasing output is a sufficient condition for welfare to decrease. Inderst and Valetti (2009) find 1 See Schmalensee (1981) for more on weak and strong segments. 2

3 that with demand substitution downstream, a ban on price discrimination decreases the incentives of the downstream firms to invest in cost reduction. Inderst and Shaffer (2009) find that if two-part tariffs are available, then more efficient downstream firms pay lower wholesale price, and thus a ban on price discrimination might lead to higher wholesale prices for all downstream firms, similar to our finding that both segments of consumers might suffer. See Stole (2007) for a review of price discrimination literature in oligopoly settings. This paper departs from earlier literature in that we study upstream firm(s) that can each invest in quality. The consumer segments can be viewed either as final consumers or small downstream firms. Unlike existing work on investment incentives of downstream firms, in our setting, the investment decision and the pricing decision are made by the same player. The rest of the paper is organized as follows. We present the model and main results in the next section. We focus on monopoly here. Section 3 concludes. In the appendix, we present an extension of our results to a duopoly. 2 Monopoly 2.1 Model There is a unit measure of customers. Customers belong to two segments. A proportion σ of customers are discerning, and the remainder 1 σ undiscerning. Discerning consumers care about the quality of the product, and undiscerning consumers do not. There is a monopolist serving the market. A firm can identify which segment any consumer belongs to. (An example of this setting could be a hospital dealing directly with patients. Consumers are either sick or healthy. The hospital can distinguish whether a consumer is sick or healthy. A sick consumer cares more about the quality of hospital facilities than a healthy one.) The firm s demand from discerning consumers is D discerning = D(p) + sm, (1) where p is the price, and M is the quality of the product. Undiscerning people do not care about quality, so for them s = 0. The prices do not have to be the same. If the prices are different, we denote the price for the discerning segment by p d, and the price for the undiscerning segment by p u. The standard profit maximization problem requires ŜOC = 2D (p) + pd (p) < 0, (2) for the second order conditions to be satisfied. We have to impose stronger assumptions, and assume that ŜOC is sufficiently negative (see below). The monopolist sets quality level and price simultaneously. For the firm to provide quality M, it needs to incur a cost of c(m), where c( ) is sufficiently convex relative to ŜOC. We assume 3

4 ŜOC < s2 σ c ( ). This is a technical assumption needed for the monopolist s second order condition to be satisfied. Note that we assume that the preference for quality is separable - this is mainly for tractability. Also, in our setting the monopolist s cost function comprises only a fixed cost that depends on the level of quality he chooses. In many applications, it may be the case that total cost depends both on the quality level chosen and the quantity produced. This would not make any qualitative difference to our results. 2.2 No Price Discrimination Assume first that the firm has to charge the same price to the two segments p = p u = p d. Then the profit function of the firm is: Π = (D(p) + σsm) p c(m). (3) Proposition 1 When the monopolist is unable to price discriminate, he sets price p = D(p )+σsm D (p ), and provides a quality level M, such that c (M ) = σsp. Proof. Differentiating the profit function with respect to price and quality: Π NoP D = D(p) + D (p)p + σsm (4a) Π NoP D M = σsp c (M). (4b) For the second order conditions to be satisfied, we need to have the determinant of the Hessian to be [ positive (all other ] conditions are clearly satisfied) ŜOC σs σs c. (M) The determinant is positive iff ŜOC < s2 σ 2 c ( ), which is weaker than our assumption (by the extra σ in the numerator). We get the result in the proposition from the first order conditions. 2.3 Price Discrimination Assume now that the firm can price discriminate between segments, and charge prices p u and p d to undiscerning and discerning consumers respectively. For a given set of prices and quality level M, the demands in two segments are Demand d = D(p d ) + sm, Demand u = D(p u ), (5a) (5b) 4

5 and the profit is Π P D = σ (D(p d ) + sm) p d + (1 σ)d(p u )p u C(M). (6) Proposition 2 Monopolist charges p d = D(p d )+sm D (p d ), p u = D(p u ) D (p u ), and provides a quality level M, such that c (M ) = σsp d. Proof. It is clear that monopolist has two unrelated maximization problems: maximizing Π d = σ (D(p d ) + sm) p d C(M) (7) with respect to p d and M, and maximizing Π P D = (1 σ)d(p u )p u (8) with respect to p u. The second problem is the standard monopolist profit maximization problem with the answer given in the proposition, and its second order conditions are satisfied if ŜOC < 0. For the first problem, differentiating profit equation (7) with respect to the choice variables (price for the discerning segment and quality level), we get (dropping the price subscript): Π d = σ ( D(p) + sm + D (p)p ) Π d M = σsp c (M). (9a) (9b) For the second order conditions to be satisfied, we need to have the determinant of the Hessian to be [ positive (all other] conditions are clearly satisfied) σŝoc σs σs c. (M) The determinant is positive iff ŜOC < s2 σ c ( ), which we have assumed. We get the result in the proposition from the first order conditions. As expected, when monopolist can price discriminate, price for the discerning segment depends on product quality and price for the undiscerning segment is independent of product quality. 2.4 Regime Comparison Proposition 3 Equilibrium price without price discrimination is lower than the price for the discerning segment, and higher than the price for the undiscerning segment, p u < p NoP D < p d. Proof. Let s examine the price absent price discrimination: p = D(p) + σsm D, (10) (p) where C (M) = σps, or M = C 1 (σps), where C 1 is the inverse of the marginal cost function. 5

6 If p increases in σ, then price for the discerning segment (σ = 1) is higher than the uniform price (σ (0, 1)), which is in turn higher than the price for the undiscerning segment (σ = 0). Implicitly differentiating with respect to σ (and dropping all the unnecessary subscripts and arguments, which simplifies to ( D ) + pd = D + sm + ( σs2 C 1) ( p + σ ), (11) = By the inverse function theorem, ( C 1) = 1 C, thus ( sm + pσs2 C 1 ) ŜOC σ 2 s 2 (C 1 ). (12) = pσs2 sm + C. (13) ŜOC σ2 s 2 C The numerator is clearly positive. Using our earlier assumption to satisfy the second order condition, the denominator is positive as well. This is the expected result from the literature on price discrimination one of the segments is charged more than the other, and the optimal price without price discrimination is in between (see for example Schmalensee (1981), or DeGraba (1990) for a similar result with intermediary markets). It would make sense for the prices to be arranged as in the proposition above, and we assume from now on that σ < M. Corollary 1 The quality of the product is higher when price discrimination is allowed, M > M NoP D. Proof. C (M) = σps, and since price increases with σ and marginal cost is an increasing function, we get the result in the corollary. Corollary 2 Consumer welfare of the undiscerning segment is higher in a regime with price discrimination, than in a regime without price discrimination. Proof. The price for the undiscerning segment with price discrimination is lower than the equilibrium price without price discrimination. Lemma 1 Consumer welfare of the discerning segment increases with price discrimination if and only if the output increases. Proposition 4 Consumer welfare of the discerning segment is higher with price discrimination if the cost function of quality investment is not too convex. Proof. For the consumer welfare to increase if price discrimination is allowed, from the previous lemma: Output = D(p) + sm (14) 6

7 must increase in σ, where C (M) = σps, or M = C 1 (σps), where C 1 is the inverse of the marginal cost function. By another application of the inverse function theorem, Output = D ( + s2 C p + σ ) > 0, (15) which is true if and only if C < s 2 p + σ. (16) D Note that despite our initial assumption on sufficient convexity, there is still a range of values for which the cost function is convex enough to satisfy our assumption and concave enough to cause the discerning segment to suffer as a result of a ban on price discrimination. The result is reminiscent of other papers on nonlinear investment cost and standard comparative statics reversing: for example, Alexandrov (2008) shows that for similar conditions oligopolistic firms profits decline with increased differentiation, if firms can invest in making their product more attractive to consumers. Corollary 3 The parameter region where the condition above and all the other assumptions are satisfied is not empty if and only if ŜOC D ŜOC < σ p. Proof. From the proof of the proposition above, and the second order condition assumptions, the region is not empty if ( C s 2 We can re-arrange that to is between the following values): σ ŜOC < σ D ŜOC ŜOC p + σ D. (17) < p σ which can be simplified to the expression in the statement of the corollary., (18) The expression on the right hand side is the elasticity of price with respect to the proportion of discerning customers. The numerator of the left hand side is negative by our assumptions. We cannot sign the denominator based on our assumptions, but if it is positive or not too negative, then the inequality is satisfied. Corollary 4 Sufficiently concave investment cost (as in the proposition above) is a sufficient condition for social welfare to be higher with price discrimination. 7

8 3 Conclusion We have examined investment incentives of a firm which might or might not be able to price discriminate between two consumer segments (with only one of the segments caring about investments). We have found that if the investment cost function is not too convex, then both consumer segments and the monopolist are worse off if price discrimination is prohibited. The unexpected result is that the discerning segment might suffer. There are two effects for this segment - price decreases, however the investment decreases as well, and depending on the shape of the cost function the discerning consumers would prefer paying a higher price, but getting a much higher investment. Our results suggest that in certain industries price discrimination should be encouraged, including the industries where the investing firm sells to downstream firms that in turn sell to the final consumer. Potential empirical tests of this result crucially depend on measuring the amount of investment and investment costs our model differs from existing ones only because some consumers care about the firm s investment. References [1] Alexandrov, Alexei, 2008, Fat Products, Journal of Economics & Management Strategy, 17(1), [2] DeGraba, Patrick, 1990, Input Market Price Discrimination and the Choice of Technology, American Economic Review, 80(5), [3] Inderst, Roman and Greg Shaffer, 2009, Market Power, Price Discrimination, and Allocative Efficiency in Intermediate-Goods Markets, Rand Journal of Economics, 40(4), [4] Inderst, Roman and Tommaso Valletti, 2009, Price Discrimination in Input Markets, Rand Journal of Economics, 40(1), [5] Katz, Michael, 1987, The Welfare Effects of Third-Degree Price Discrimination in Intermediate Goods Markets, American Economic Review, 77(1), [6] Pigou, Arthur, 1920, The Economics of Welfare, London: Macmilian. [7] Schmalensee, Richard, 1981, Output and Welfare Implications of Monopolistic Third-Degree Price Discrimination, American Economic Review, 71(1), [8] Schwartz, Marius, 1990, Third-Degree Price Discrimination and Output: Generalizing a Welfare Result, American Economic Review, 80(5), [9] Stole, Lars A., 2007, Price Discrimination and Imperfect Competition, in M. Armstrong and R. Porter, editors, Handbook of Industrial Organization: Volume III, North-Holland, Amsterdam. [10] Yoshida, Yoshihiro, 2000, Third-Degree Price Discrimination in Input Markets: Output and Welfare, American Economic Review, 90(1),

9 Appendix Differentiated Bertrand Duopoly We switch from the linear demand assumption in the monopoly section to a differentiated Bertrand formulation. There are two segments of consumers, discerning (σ of them) and undiscerning (1 σ). Discerning consumers care about the quality of the product, and undiscerning consumers do not. A firm can identify which segment any consumer belongs to. (An example of this setting could be a market with two hospitals. Consumers are either sick or healthy. A hospital can distinguish whether a consumer is sick or healthy. A sick consumer cares more about the quality of hospital facilities than a healthy one.) The demands of the discerning consumers for two firms are q i discerning = A bp i + cp j + sm i tm j, q j discerning = A bp j + cp i + sm j tm i. (19a) (19b) where A is the reservation utility of the product, same for both firms, p i is firm i s price, and M i is the quality of firm i s product. Undiscerning people do not care about quality, so for them s = t = 0. Undiscerning consumers demand is thus a standard differentiated Bertrand demand. Firms set quality levels and prices simultaneously. For a firm to provide quality M, they need to incur a cost of c(m), where c( ) is sufficiently convex. 2 To cut down on the number of cases that we have to examine, we assume that own-price coefficient is high relative to the cross-price coefficient. 3 We also assume that both firms want to cater to both markets regardless of whether price discrimination is allowed. Price Discrimination Assume first that firms can price discriminate between segments, and charge prices p ui and p di to undiscerning and discerning consumers respectively. Then each firm decides on the quality level and two prices one for undiscerning consumers, another for the discerning ones. Proposition 5 In the symmetric equilibrium, p d = σsp d. A 2b c + s t 2b c M, p u = A 2b c, and c (M P D ) = Proof. The undiscerning consumers are a separate market, which does not care about M. Thus we can treat them separately. With the standard differentiated Bertrand demands, we know that p u = A 2b c. The profit of firm i from discerning consumers, and accounting for the costs of quality, is: Π i = p di σ (A bp i + cp j + sm i tm j ) c(m i ). (20) 2 Same as in the monopoly case, we assume c ( ) > s2 σ this is a technical assumption needed for second order 2b conditions to hold. From the monopoly case, ŜOC = 2b, so the assumption here is the same as in the monopoly case. 3 s More precisely, we assume that b > c. The assumption is trivially satisfied for low σ or if s < 2t. We 2(s σ(s t)) need this assumption Proposition 7. 9

10 Differentiating with respect to the two remaining choice variables of firm i, we get: Π i di = σ (A 2bp i + cp j + sm i tm j ) Π i M i = σsp di c (M i ). (21a) (21b) For the second order conditions to be satisfied, we need to have c (M) > s2 σ 2b, which we have assumed for the monopoly case already. In the symmetric equilibrium, p di = p dj, p ui = p uj, and M i = M j. From the first order conditions we get the results of the proposition. Corollary 5 Price for the discerning segment depends on product quality and is higher than the price for the undiscerning segment. Price for the undiscerning segment is independent of product quality. No Price Discrimination Now assume that the firms have to charge the same price to the two segments p ui = p di. The profit function of firm i becomes: Π i = (A bp i + cp j + σ(sm i tm j )) p i c(m i ). (22) Proposition 6 Without price discrimination, in the symmetric equilibrium, p NoP D = A 2b c + σ s t 2b c M and c (M ) = σsp. Proof. Differentiating (22) with respect to firm i s price and quality: Π i = A 2bp i + cp j + σ(sm i tm j ) i (23a) Π i = σsp di c (M i ). M i (23b) To satisfy the second order conditions we need c (M) > s2 σ 2 2b, which follows from our assumptions on sufficient convexity of the cost function. In the symmetric equilibrium p i = p j and M i = M j. We get the result in the proposition from the first order conditions. Regime Comparison Proposition 7 Equilibrium price without price discrimination is between the equilibrium prices for two segments with price discrimination, p u < p NoP D < p d. Proof. As in the monopoly case, we examine function p = A 2b c + σ s t M, (24) 2b c 10

11 where C (M) = σps, or M = C 1 (σps), where C 1 is the inverse of the marginal cost function. If p increases in σ, then price for the discerning segment (σ = 1) is higher than the uniform price (σ (0, 1)), which is in turn higher than the price for the undiscerning segment (σ = 0). Implicitly differentiating with respect to σ (and dropping all the unnecessary subscripts and arguments, and using the inverse function theorem, which simplifies to = s t ( s M + σ 2b c C (M) = M + 2b c s t sσp C (M) sσ2 C (M) ( p + σ )), (25). (26) The numerator is clearly positive. Using our earlier assumption to satisfy the second order condition, the denominator is in the worst case 2b c. That is positive if and only if which we have assumed as well. b > s t 2bσ s s c, (27) 2(s σ(s t)) This is similar to the result that we got in the monopoly case, and what we expected. Corollary 6 The equilibrium quality of the product is higher when price discrimination is allowed, M > M NoP D. Corollary 7 Consumer welfare of the undiscerning segment is higher in a regime with price discrimination, than in a regime without price discrimination. Lemma 2 Consumer welfare of the discerning segment increases with price discrimination if and only if the output for this segment increases. Proposition 8 Consumer welfare of the discerning segment is higher ( with price discrimination )) if and only if the cost function of quality investment is not too convex c ( ) < σs(s t) b c + 1. Proof. The proof is similar to the proof of Proposition 4. For the consumer welfare to increase if price discrimination is allowed, from the previous lemma: ( σ p Output = (s t)m (b c)p (28) must increase in σ, where C (M) = σps, or M = C 1 (σps), where C 1 is the inverse of the marginal cost function. By another application of the inverse function theorem, Output = (s t)s C ( p + σ ) (b c) which is true if and only if the expression in the proposition is satisfied. > 0, (29) 11

12 Corollary 8 The range of parameters for which the condition above and all the other assumptions are satisfied is not empty if and only if b(2t s) cs 2b(s t) < 1. (30) σ p In particular, it is satisfied if s > 2t. Overall, the exact values and some of the proof intuition might be different, but all the qualitative results are the same in the duopoly case as in the monopoly case. 12