价格歧视(AER)

Monopoly Price Discrimination and Demand Curvature Iñaki Aguirre, Simon Cowan and John Vickers� Abstract This paper presents a general analysis of the e¤ects of monopolis- tic third-degree price discrimination on welfare and output when all markets are served. Su¢ cient conditions – involving straightforward comparisons of the curvatures of the direct and inverse demand func- tions in the di¤erent markets – are presented for discrimination to have negative or positive e¤ects on social welfare and output. (JEL D42, L12, L13) This paper develops general conditions that determine whether third- degree price discrimination by a monopolist serving all markets reduces or raises output and social welfare, de…ned as the sum of consumer surplus and pro…t. A …rm practising third-degree price discrimination uses an exogenous characteristic, such as the age or location of the consumer or the time of �Aguirre: Departamento de Fundamentos del Análisis Económico I, University of the Basque Country, Avda. Lehendakari Aguirre 83, 48015-Bilbao, Spain (e-mail: in- aki.aguirre@ehu.es); Cowan: Department of Economics, University of Oxford, Manor Road Building, Manor Road, Oxford OX1 3UQ, UK (e-mail: simon.cowan@economics.ox.ac.uk); Vickers: All Souls College, Oxford OX1 4AL, UK (e-mail: john.vickers@all-souls.ox.ac.uk). Financial support from the Ministerio de Ciencia y Tecnología and FEDER (SEJ2006- 05596), and from the Departamento de Educación, Universidades e Investigación del Go- bierno Vasco (IT-223-07) is gratefully acknowledged. We are grateful to a referee who, along with very helpful comments, recommended that we combine three working papers (available on request) into one, as well as to three referees for comments on one of the papers. We would also like to thank Mark Armstrong, Christopher Bliss, Ian Jewitt, Meg Meyer, Volker Nocke, Ignacio Palacios-Huerta, Joe Perkins, John Thanassoulis and Michael Whinston for their comments. 1 purchase, to divide customers into separate markets. The monopoly price can then be set in each market if discrimination is allowed. Moving from non- discrimination to discrimination raises the …rm’s pro…ts, harms consumers in markets where prices increase and bene…ts the consumers who face lower prices. The overall e¤ect on welfare can be positive or negative. The main aim of this paper is to provide conditions based on the shapes of the demand functions to determine the sign of the welfare e¤ect. We also address the classic question of the e¤ect of discrimination on total output, and the paper combines new …ndings with existing results in a uni…ed framework. The e¤ect of discrimination on welfare can be divided into a misalloca- tion e¤ect and an output e¤ect. With discrimination output is ine¢ ciently distributed because consumers face di¤erent prices in di¤erent markets. This negative feature of discrimination may, however, be o¤set if there is an in- crease in total output, which is socially valuable since prices exceed marginal costs. Arthur Pigou (1920) proved that if all demand functions are linear and all markets are served at the non-discriminatory price then total output remains at the no-discrimination level, in which case discrimination is bad for welfare. Joan Robinson’s (1933) pioneering analysis, taken forward by Richard Schmalensee (1981), showed how the curvature of demands deter- mines the sign of output e¤ect. Hal Varian (1985) proved very generally that a necessary condition for welfare to rise with discrimination is that total output increases (see also Marius Schwartz, 1990). In this paper we explore the welfare e¤ect directly using the technique developed by Schmalensee (1981) and Thomas Holmes (1989) to analyze the output e¤ect.1 Throughout it is assumed that at the non-discriminatory price all markets are served with positive quantities, so price discrimination does not open up new markets.2 To simplify the exposition, but without loss of generality, we explore the case with two markets. The …rm is supposed initially to be required to set the same price in both markets –i.e. the price di¤erence is constrained to be zero. As this constraint is relaxed, the …rm moves towards the laissez-faire outcome with price discrimination. As this happens, output and welfare will increase in what Joan Robinson (1933) termed the ‘weak’market, where the discriminatory price is below the non- discriminatory price, and decrease in the ‘strong’market. The question is 1Earlier applications of the method are by Wassily Leontief (1940) and Eugene Silber- berg (1970). 2See Jerry Hausman and Je¤rey Mackie-Mason, 1988, Stephen Layson, 1994, and Victor Kaftal and Debashi Pal, 2008, for analyses of price discrimination that opens new markets. 2 how overall welfare and total output vary as the price-di¤erence constraint is relaxed. Central to our analysis is a (commonly met) condition on demand func- tions – the increasing ratio condition –which ensures that welfare varies monotonically with the price-di¤erence constraint, or else has a single in- terior peak. Given the increasing ratio condition, discrimination is shown to reduce welfare if the direct demand function in the strong market is at least as convex as that in the weak market at the non-discriminatory price. Second, welfare is higher with discrimination if the discriminatory prices are not far apart and the inverse demand function in the weak market is locally more convex than that in the strong market: total output then rises while the misallocation e¤ect is relatively small. Outside these cases, welfare …rst rises but then falls as the price-di¤erence constraint is relaxed, so an inter- mediate degree of discrimination would be optimal, and the overall e¤ect on welfare of unfettered discrimination can be positive or negative. Its sign can however be determined in important special cases: (i) when inverse demand curvature is constant, welfare falls with discrimination if curvature is su¢ - ciently below unity and rises if curvature is su¢ ciently above unity, and (ii) when demands have constant elasticities, although total output rises with discrimination (Iñaki Aguirre, 2006), welfare falls if the di¤erence between the elasticities is no more than one. In parallel to the welfare analysis, we also obtain new results on how discrimination a¤ects total output, which rises if both inverse and direct demand in the weak market are more convex than those functions in the strong market, but not if both inverse and direct demand in the strong market are at least as convex as those in the weak market.3 The broad economic intuition for why the di¤erence between the curva- tures of demand in weak and strong markets is important for welfare and output is as follows. A price increase when demand is concave has relatively little e¤ect on welfare (the extreme form of concavity is when the demand function is rectangular and there is no deadweight loss from monopoly pric- ing). If at the same time price falls in a market with relatively convex de- mand, there is a large increase in output and thus in welfare in that market. This is the insight of Robinson (1933), who showed that total output rises 3These output results build on, and encompass, those of Robinson (1933), Schmalensee (1981), Jun-ji Shih, Chao-cheng Mai and Jung-chao Liu (1988) and Francis Cheung and Xinghe Wang (1994). 3 when discrimination causes prices to rise in markets with concave demands and prices to fall in markets with convex demands, and David Malueg (1994) explored further the relationship between the curvature of the demand func- tion and the deadweight loss from monopoly pricing.4 The paper is organized as follows. Section I presents the model of monopoly pricing with and without third-degree price discrimination. Section II con- tains the welfare analysis using the price-di¤erence technique. The e¤ect of discrimination on total output is considered in Section III. Section IV presents the results of the welfare analysis using a restriction on how far quantities can vary from their non-discriminatory levels, and considers the important special case where demands have constant elasticities. Conclusions are in Section V. I. The Model of Monopoly Pricing Amonopolist sells its product in two markets and has a constant marginal cost, c � 0. The assumption of two markets is made for simplicity – all the results can be generalized to the case of more than two markets and the method for doing this is discussed later. Utility functions are quasi- linear. Demand in a representative market with price p is q(p), which is twice-di¤erentiable, decreasing and independent of the price in the other market. (To avoid notational clutter we omit subscripts where it is not necessary to indicate which market is which.) The price elasticity of demand is � � �pq0=q. The pro…t function in a market is � = (p � c)q(p). Assume that �00(p) = 2q0 + (p� c)q00 = � 2 + (p� c)q 00 q0 � q0 < 0; so the expression in square brackets is positive and the pro…t function is strictly concave.5 With strict concavity the second-order conditions hold for the maximization problems that follow. De…ne �(p) � �pq00=q0 as the convexity (or curvature) of direct demand, which is analogous to relative risk aversion for a utility function and is the elasticity of the slope of demand. The 4See Glen Weyl and Michal Fabinger (2009) for a general analysis of demand curvature and social welfare with imperfect competition. 5Part A of the Appendix discusses conditions that ensure strict concavity. See Babu Nahata, Krzysztof Ostaszewski and Prasanna K. Sahoo (1990) for an analysis of price discrimination when pro…t functions are not concave in prices. 4 Lerner index, the mark-up of price over marginal cost, is L(p) � (p�c)=p and 2 + (p� c)q00=q0 = 2�L� > 0 by strict concavity. Similarly the curvature or convexity of the inverse demand function p(q) is �(q) � �qp00=p0 = qq00=[q0]2. The two curvature measures are related to the price elasticity by � = �=�. The values of � and of � play key roles in the analysis. When the …rm discriminates, the …rst-order condition for its problem in each market is �0(p�) = q(p�) + (p� � c)q0(p�) = 0; where p� > c is the pro…t-maximizing price and the star denotes the value that applies with full discrimination. From the …rst-order condition comes the Lerner condition for monopoly pricing L� = 1=��. Thus L��� = ��=�� = �� and, with strict concavity, 2�L��� = 2��� > 0. The subscript w denotes the weak market, where the discriminatory price is below the non-discriminatory one (see below), and subscript s denotes the strong market, where the price is higher with discrimination. The classi…cation of a market as strong or weak is endogenous. It is assumed that both markets are served at the non- discriminatory price –a su¢ cient condition for this is that qw(p�s) > 0. When the …rm cannot discriminate it chooses the single price p that maximizes aggregate pro…t, which is de…ned by the …rst-order condition �0w(p) + � 0 s(p) = 0. The …rst-order condition and the assumption that both markets are served at the non-discriminatory price imply that �0w(p) = qw(p)[1 � L(p)�w(p)] < 0 and �0s(p) = qs(p)[1 � L(p)�s(p)] > 0, so �w(p) > �s(p). The weak market has the higher elasticity at the non-discriminatory (or uniform) price. With strict concavity of each pro…t function it follows that p�s > p > p � w: Social welfare W is the sum of consumer surplus and pro- ducer surplus (or gross utility minus cost) so the marginal e¤ect of price on social welfare in a market is dW dp = (p� c)q0(p), i.e. the e¤ect on the quantity multiplied by the price-cost margin. II. The E¤ect of Discrimination on Welfare The method used by Schmalensee (1981), Holmes (1989) and Lars Stole (2007) to consider the output e¤ect is adopted here to analyze the welfare e¤ect. In the following section we use it to re-examine the output e¤ect. Initially the …rm is not allowed (or is unable) to discriminate and thus sets the uniform price p. Then the constraint on the …rm’s freedom to discrimi- nate is gradually relaxed until the …rm can discriminate as much as it likes. 5 Our approach is to calculate the marginal e¤ect on welfare of relaxing the constraint; if this keeps the same sign as more discrimination is allowed, then the overall e¤ect of discrimination can be found. In particular, we assume that the …rm chooses its prices to maximize pro…t subject to the constraint that ps� pw � r where r � 0 is the degree of discrimination allowed. The objective function is �w(pw)+�s(pw+r) and the …rst-order condition is �0w(pw) + � 0 s(pw + r) = 0 when the constraint binds. When r = 0 the …rm sets the non-discriminatory price. As r rises more discrimination is allowed, the price in the weak market falls and that in the strong market rises: (1) p0w(r) = ��00s �00w + �00s < 0; p0s(r) = �00w �00w + �00s > 0: When the constraint does not bind the …rm sets the discriminatory prices. The marginal change in social welfare W as more price discrimination is allowed is (2) W 0(r) = (pw � c)q0w(pw)p0w(r) + (ps � c)q0s(ps)p0s(r): A relaxation of the constraint alters prices and thus the quantities demanded, and each additional unit of output has social value equal to the price-cost margin in that market. For r > r� = p�s � p�w the marginal welfare e¤ect is zero because the prices remain at the discriminatory levels. De…ne W 0(0) and W 0(r�) as right- and left-derivatives respectively. The marginal e¤ect on total output is Q0(r) � q0wp0w + q0sp0s so, following Schmalensee (1981), (2) may be written as: (3) W 0(r) = (pw � p)q0w(pw)p0w(r) + (ps � p)q0s(ps)p0s(r)| {z } Misallocation e¤ect + (p� c)Q0(r)| {z } : (Value of) output e¤ect The …rst two terms equal zero at r = 0 and are negative for r > 0. Together they represent the marginal misallocation e¤ect. The …nal term is the value of the change in output. At the non-discriminatory price, because there is no misallocation e¤ect, the marginal welfare e¤ect is proportional to the marginal change in aggregate output. Integrating (3) over [0; r�] gives the total welfare e¤ect as two negative terms (the total misallocation e¤ect) plus 6 (p� c) times the total output change. This con…rms that an output increase is necessary for social welfare to rise.6 In our analysis a crucial role is played by (4) z(p) � (p� c)q 0(p) 2q0 + (p� c)q00 = p� c 2� L�; the ratio of the marginal e¤ect of a price increase on social welfare to the second derivative of the pro…t function. Substituting the comparative statics results for prices, (1), into (2) and using (4) gives the marginal welfare e¤ect: (5) W 0(r) = � ��00w�00s �00w + �00s � | {z } >0 [zw(pw(r))� zs(ps(r))] : The marginal welfare e¤ect thus has the same sign as [zw(pw(r))� zs(ps(r))]. The following assumption is made for the three propositions in this section. The increasing ratio condition (IRC): z(p) is increasing in p in each market. This holds for a very large set of demand functions. These include: functions that are linear; inverse demands with constant positive curvature, including the exponential and constant-elasticity functions; direct demand functions with constant curvature (whether positive or negative); probits and logits (derived from the normal and logistic distributions respectively); and demand functions derived from the lognormal distribution. Part B of the Appendix presents su¢ cient conditions for the condition to hold and gives a fuller list of the demand functions to which it applies. While the increasing ratio condition holds very commonly – and always it holds locally in the region around marginal cost –it is not universally applicable. For example when inverse demand has constant negative curvature the condition does not hold for high enough prices and di¤erent techniques are necessary to deal with this case. 6It should be noted that the decomposition of the total welfare e¤ect into an output e¤ect and a misallocation e¤ect is not unique. See Aguirre (2008) for a graphical analysis based on a di¤erent decomposition. 7 Lemma Given the IRC, if there exists br such that W 0(r^) = 0 then W 00(r^) < 0: Proof. From (5) W 00(r) = � ��00w�00s �00w + �00s � [z0wp 0 w � z0sp0s] + [zw � zs] d dr � ��00w�00s �00w + �00s � ; which is negative if W 0 = 0 because z0wp 0 w < 0 and z 0 sp 0 s > 0, and zw = zs where W 0 = 0. The IRC therefore implies thatW (r) is strictly quasi-concave, and thus is monotonic in r or has a single interior peak. Only three outcomes are possi- ble: either welfare, as a function of r, is everywhere decreasing, or everywhere increasing, or it …rst rises then falls. Which holds depends on the signs of W 0(0) and W 0(r�). First, if W 0(0) � 0, then W (r) is decreasing for r > 0 and discrimination therefore reduces welfare. Proposition 1 Given the IRC, if the direct demand function in the strong market is at least as convex as that in the weak market at the non- discriminatory price then discrimination reduces welfare. Proof. The Lemma implies that discrimination reduces welfare ifW 0(0) � 0. At the non-discriminatory price, where r = 0, pw�c = ps�c and Lw = Ls. So from (5), [zw(p)� zs(p)] and henceW 0(0) have the sign of [�w(p)��s(p)], the di¤erence in curvatures of direct demand, which is non-positive under the condition stated in the proposition. The condition on the di¤erence in the demand curvatures implies that locally output does not increase, and since at the non-discriminatory price the marginal misallocation e¤ect is zero a local output e¤ect that is negative or zero implies that the welfare e¤ect has the same sign. The IRC then extends this local result to all additional increases in the amount of discrimination, and thus acts as a sign-preserver. Proposition 1 encompasses the results of Simon Cowan (2007), who has demand in the strong market being an a¢ ne transformation of demand in the weak market, i.e. qs(p) =M +Nqw(p) whereM and N are positive (and demand in both markets is zero at a su¢ ciently high price). At the same 8 price the direct demand functions, by construction, have the same curvature. This is analogous to the result in expected utility theory that the coe¢ cients of absolute and relative risk aversion, at a given income level, are invariant to positive a¢ ne transformations of the utility function. An example is when the direct demand functions have constant and common curvature, �, and a special case is when both demand functions are linear (� = 0). Proposition 1 is more general because it allows the demand functions to have di¤erent parameters or di¤erent functional forms, as in the following example. Example 1: exponential and linear demands. Demand in market 1 is q1(p) = Be �p=b (with B and b positive), so �1 = 1; �1 = �1 = p=b > 0 and p�1 = b + c. Demand in market 2 is q2(p) = a � p so �2(p) = p=(a � p); �2 = �2 = 0 and p�2 = (a + c)=2. Proposition 1 applies if b > (a � c)=2, which is the condition for market 1 to be the strong one. The weak market is served with non-discriminatory pricing if (but not only if) a > b+ c. If discrimination is to raise welfare, given the IRC, direct demand in the weak market must be strictly more convex than demand in the strong market at the non-discriminatory price. Only then does a small amount of discrimination causes total output to rise. This is a local version of the condition that for welfare to rise total output must increase. Figure 1 shows, in a standard monopoly diagram, that as demand in the weak market becomes more convex the welfare gain in this market from dis- crimination rises. Initially inverse demand is the linear function p1(q) and its associated marginal revenue curve is MR1(q) = p1(q) + qp01(q). The non- discriminatory quantity is q and the discriminatory quantity is q1. Suppose that demand becomes more convex, while retaining the same slope and po- sition at the no