MPRA Munich Personal RePEc Archive Multiproduct Pricing Made Siple Mark Arstrong and John Vickers Departent of Econoics, University of Oxford 8. January 2016 Online at https://pra.ub.uni-uenchen.de/68717/ MPRA Paper No. 68717, posted 8. January 2016 14:27 UTC
Multiproduct Pricing Made Siple Mark Arstrong and John Vickers January 2016 Abstract We study pricing by ultiproduct firs in the context of unregulated onopoly, regulated onopoly and Cournot oligopoly. Using the concept of consuer surplus as a function of quantities (rather than prices), we present siple forulas for optial prices and show that Cournot equilibriu exists and corresponds to a Rasey optiu. We then present a tractable class of deand systes that involve a generalized for of hoothetic preferences. As well as standard hoothetic preferences, this class includes linear and logit deand. Within the class, profit-axiizing quantities are proportional to effi cient quantities. We discuss cost-passthrough, including cases where optial prices do not depend on other products costs. Finally, we discuss optial onopoly regulation when the fir has private inforation about its vector of arginal costs, and show that if the probability distribution over costs satisfies an independence property, then optial regulation leaves relative price decisions to the fir. Keywords: Multiproduct pricing, hoothetic preferences, Cournot oligopoly, onopoly regulation, Rasey pricing, cost passthrough, ultidiensional screening. JEL Classification: D42, D43, D82, L12, L13, L51. 1 Introduction The theory of ultiproduct pricing is a large and diverse subject. Unlike the single-product case, a ultiproduct fir ust decide about the structure of its relative prices as well as its overall price level. Classical questions include the characterization of optial pricing by a ultiproduct onopolist seeking to axiize profit or, as with Rasey pricing, the ost effi cient way to generate a specified level of profit when its choice for one price ust Both authors at All Souls College and the Departent of Econoics, University of Oxford. We are grateful to Konrad Stahl (who discussed an early version this paper at the University of Mannhei workshop on Multiproduct firs in industrial organization and international trade, 23-24 October, 2015) and to Jonas Müller-Gastell for helpful coents. 1
take into account its ipact on deand for other products. Additional coplexities arise in oligopoly, when a ultiproduct fir needs to choose prices to reflect both intra-fir substitution (or copleentarity) features and inter-fir interactions. Optial regulation of a ultiproduct fir with private inforation about its costs, say, ust take into account not just its likely average cost across all products but its pattern of relative costs. In this paper we show how these issues can be illuinated by studying consuer preferences in ters of consuer surplus considered as a function of quantities (rather than the ore failiar function of prices). 1 In section 2 we show how profit-axiizing and other Rasey prices, as well as prices in syetric Cournot equilibriu, can be expressed as a arkup over arginal costs proportional to the derivative of this surplus function. In particular, a product s optial price is below arginal cost when consuer surplus decreases with the supply of this product. We also show how a Cournot equilibriu corresponds to an appropriate Rasey optiu, and vice versa, which enables us to construct and deonstrate existence of Cournot equilibriu in any cases. A well-known feature of Rasey pricing is that when required departures of optial quantities fro effi cient quantities are sall, then optial quantities are approxiately proportional to the effi cient quantities. Thus, a reasonable rule of thub is often to scale down quantities equiproportionately relative to effi cient quantities, rather than to increase prices equiproportionately above arginal cost. For larger departures of prices fro costs, though, optial quantities are generally not proportional to effi cient quantities. In section 3, we specialise the deand syste so that consuer surplus is a hoothetic function of quantities, which iplies that relative quantities (or relative price-cost arkups) do not depend on the weight placed on profit in the Rasey objective. As shown in section 3.2, this is quite a flexible class of deand systes (uch broader than the class where consuer surplus is hoothetic in prices), and as well as the obvious case of gross utility being hoothetic in quantities it includes linear and logit deands. In section 3.3, we show that this property, together with assuing constant returns to scale, siplifies the analysis by allowing a ultiproduct proble to be decoposed into two steps: first calculate the effi cient quantities which correspond to arginal cost pricing, and second solve for the scale factor by which to reduce the effi cient quantities. This siplifies 1 Regarding consuer surplus as a function of quantities is apparently uncoon in the literature. However, in the single-product context Bulow and Kleperer (2012) show this to be a valuable perspective. (They regard consuer surplus as the area between the deand curve and the arginal revenue curve, which is the sae thing.) 2
coparative static analysis, such as how onopoly (and oligopoly) prices vary with cost paraeters. In soe leading exaples there is zero cross-cost passthrough e.g., the ost profitable price of each product depends only on its own cost and ore generally there are siple forulas for the size and sign of cross-cost effects. The fact that a profit-axiizing fir has effi cient incentives with respect to the pattern, though of course not the level, of quantities has iplications, oreover, for regulation of ultiproduct onopoly (section 3.4). It suggests that, in our class of deand systes, it ight be optial for regulation to allow the onopolist considerable discretion over the pattern of relative quantities (or prices). If the probability distribution over costs is such that relative costs and average costs are stochastically independent, this intuition is precisely correct and it is optial for the choice of relative quantities to be delegated to the fir. Related literature: Bauol and Bradford (1970), and the any references therein, discuss the econoic principles of Rasey pricing. They suggest (p. 271) that it is plausible that the daage to welfare resulting fro departures fro arginal cost pricing will be iniized if the relative quantities of the various products sold are kept unchanged fro their arginal cost pricing proportions. One ai of our paper is to ake this intuition precise in a broad class of deand systes. Goran (1961) described a class of utility functions such that incoe expansion paths (or Engel curves) for quantities deanded were linear. This resebles our class of utility functions, for which Rasey quantities are equiproportional; that is, where the quantity vector which axiizes consuer utility subject to a profit constraint expand linearly as the profit requireent is relaxed. Goran s preference faily was such that the consuer s expenditure function took the for, e(p, u) = a(p)+ub(p), where a and b are hoogeneous degree 1, while Proposition 2 below shows that our faily has gross utility of the for u(x) = h(x) + g(q(x)) where h and q are hoogeneous degree 1. Cournot oligopoly is studied in a rich literature on single-product firs see Vives (1999, chapter 4) for an overview of existence, uniqueness and coparative statics of Cournot equilibria. Soeties a Cournot oligopoly operates as if it axiizes an objective. Bergstro and Varian (1985) observe that a syetric oligopoly axiizes a Rasey objective, while Slade (1994) and Monderer and Shapley (1996) note that oligopolists soeties axiize a ore abstract potential function. This is useful as it converts 3
the fixed-point proble of calculating equilibriu quantities into a sipler optiization proble. In Proposition 1 we extend this analysis to cover ultiproduct cases and, like Bergstro and Varian (1985), show that oligopolists axiize an appropriate Rasey objective. Weyl and Fabinger (2013) discuss the passthrough of costs to prices and its various applications in settings of onopoly and iperfect copetition with single-product firs. Within the arketing literature on retailing, a ajor thee is the extent to which wholesale cost shocks (such as teporary prootions) are passed through into retail prices. Besanko et al. (2005) epirically exaine the patterns of cost passthrough in a large superarket chain. They find that own-cost proportional passthrough is ore than 60% for ost product categories (and soeties ore than 100%), while cross-cost passthrough can take either sign. Moorthy (2005) analyzes a theoretical odel where two retailers copete to supply two products to consuers, and as well as cost passthrough within a retailer he discusses how cost shock to the rival affects fir s prices. The sign of ost of the passthrough effects depends in an opaque way on the features of various atrices. In section 3.3, our deand syste yields soe relatively siple ultiproduct passthrough relationships. The optial regulation of ulti-product onopoly is analyzed by Laffont and Tirole (1993, chapter 3). In their ain odel, cost outturns are observable but the regulator cannot observe cost-reducing effort or the fir s underlying cost type. If the cost function is separable between quantities on the one hand and the fir s effort and type on the other, then the incentive-pricing dichotoy holds pricing should not be used to provide effort incentives. If there is a social cost of public funds, Rasey pricing is therefore optial, as characterized by super-elasticity forulas for arkups. The analysis of regulation in section 3.4 below does not consider effort incentives, but is for the situation studied by Baron and Myerson (1982) where the regulator cannot observe the fir s costs. We extend this odel to cover ultiproduct situations where a vector of arginal costs is unobserved by the regulator. Building on the approach in Arstrong (1996) and Arstrong and Vickers (2001), we describe a tractable class of situations in which it is optial to control only the fir s average output, leaving it free to choose relative outputs to reflect its relative costs. 4
2 A general analysis of ultiproduct pricing Suppose there are n 2 products, where the quantity of product i is denoted x i and the vector of quantities is x = (x 1,..., x n ). Consuers have quasi-linear utility, which iplies that deand can be considered to be generated by a single representative consuer with gross utility function u(x) defined on a (full-diensional) convex region R R n + which includes zero, where u(0) = 0 and u is increasing and concave. (We ight have R = R n +, so that utility is defined for all non-negative quantity bundles.) We suppose that u is twice continuously differentiable in the interior of R, although arginal utility ight be unbounded as soe quantities tend to zero. Faced with price vector p = (p 1,..., p n ), the consuer chooses quantities x R to axiize u(x) p x. (Here, a b n i=1 a ib i denotes the dot product of two vectors a and b.) The price vector which induces interior quantity vector x R to be deanded, i.e., the inverse deand function p(x), is p(x) u(x), where we use the gradient notation f(x) ( f(x)/ x 1,..., f(x)/ x n ) for the vector of partial derivatives of a function f. To ensure we can invert p(x) to obtain the deand function x(p), assue that the atrix of second derivatives of u, which we write as Dp(x), is non-singular (and hence negative-definite). The revenue generated fro quantity vector x is r(x) = x u(x), while the surplus retained by the consuer fro x is s(x) u(x) r(x) = d dk ku(x/k). (1) k=1 One of this paper s ais is to show the usefulness of the function s(x) for analyzing ultiproduct pricing. We next discuss soe features of s(x). First, the right-hand side of expression (1) shows that s( ) is related to the elasticity of scale of u( ) evaluated at x, and a ore concave u allows the consuer to retain ore surplus. 2 Second, the derivative of s can be expressed as s(x) = d dk p(x/k), (2) k=1 2 If all quantities x are increased by 1 per cent, then u increases by (1 s(x)/u(x)) per cent. 5
so that an equiproportionate contraction in quantities x oves the price vector in the direction s(x), i.e., noral to the surface s(x) = constant. To see (2), note that s(x) = [u(x) r(x)] = x j p j (x) = x j p i (x) = d x i x i x j i x j j dk p i(x/k), k=1 where the third equality follows fro the syetry of cross-derivatives of p(x). Unlike consuer surplus expressed as a function of prices which is necessarily a decreasing function of prices here s(x) can increase or decrease with x i. 3 Fro (2), a suffi cient condition for s to increase with x i is that p i decrease with all x j, which is the case if products are gross substitutes (see Vives (1999, section 6.1)). Note, though, that the above expressions iply that for x 0 we have d dk s(kx) = x Dp(x) x > 0, (3) k=1 where the inequality follows fro the atrix Dp(x) being negative-definite. Thus consuer surplus increases as all quantities are increased equiproportionately. The Rasey onopoly proble: Now suppose that these products are supplied by a onopolist with differentiable cost function c(x). To sidestep issues of fixed costs and the potential undesirability of producing at all, both with onopoly and in the later analysis of Cournot oligopoly, we suppose that c(0) = 0 and c(x) is convex. (4) Consider the Rasey proble of choosing quantities to axiize a weighted su of profit and consuer surplus. If α 1 is the relative weight on consuer surplus, the Rasey objective is [r(x) c(x)] + αs(x) = u(x) c(x) (1 α)s(x). (5) This includes as polar cases profit axiization (α = 0) and total surplus axiization (α = 1). Standard coparative statics shows that optial consuer surplus, s(x), in this Rasey proble weakly increases with α, while optial profit [r(x) c(x)] weakly decreases with α. Total surplus is axiized at quantities x w which involve prices equal to arginal costs, so that p(x w ) = c(x w ), and assuption (4) ensures that the fir breaks 3 Likewise, while consuer surplus is necessarily convex as a function of prices, even in the single-product case it is abiguous whether s(x) is convex or concave (or neither) as a function of quantities. 6
even with arginal-cost pricing. More generally, the Rasey proble with weight α has first-order condition for optial quantities given by p(x) = c(x) + (1 α) s(x). (6) Thus, when α < 1 the optial departure of price fro arginal cost is proportional to s(x). In particular, the Rasey price for product i is above its arginal cost if surplus s(x) increases with x i at optial quantities, while using the product as a loss leader is optial if s(x) decreases with x i. As we will see later, there are also natural cases where s depends only on the quantities of a subset of products, in which case (6) indicates that the reaining products should be priced at arginal cost. Thus, the function s succinctly deterines when it is optial to set a price above, below, or equal to arginal cost. 4 When α is close to 1 then choosing x = αx w approxiately solves the Rasey proble (5) when the cost function c(x) is hoogeneous degree 1. To see this, note that when α 1 expression (2) iplies in which case p(αx) p(x) (1 α) s(x), (7) p(αx w ) p(x w ) + (1 α) s(x w ) = c(x w ) + (1 α) s(x w ) = c(αx w ) + (1 α) s(x w ) c(αx w ) + (1 α) s(αx w ), so that x = αx w approxiately satisfies condition (6). (Here, the first strict equality follows fro the effi ciency of quantities x w while the second follows fro the hoogeneity of c( ).) In su, in the Rasey proble with constant returns to scale and α 1, the effi cient quantities should be scaled back equiproportionately by the factor α. Without aking further assuptions, there is little reason to expect that this insight for α 1 extends to the situation where a onopolist axiizes profit (α = 0), and in general profit-axiizing quantities are not proportional to effi cient quantities. 4 Expression (6) is an alternative and arguably ore transparent forulation of the insight in Bauol and Bradford (1970, section VIII) that the gap between price and arginal cost should be proportional to the gap between arginal revenue and arginal cost, where arginal revenue takes into account how increasing the supply of one product affects prices for other products. 7 To
illustrate, the bold curve on Figure 1 depicts Rasey quantity vectors as the weight on consuer surplus varies fro α = 0 to α = 1. 5 As shown, these Rasey quantities are the vectors the contract curve between consuers and the fir where iso-profit contours (which are curves centred on profit-axiizing quantities) are tangent to isowelfare contours (centred on effi cient quantities). As discussed above, when α 1 optial quantities are approxiately proportional to effi cient quantities, and so when α = 1 the bold line is tangent to the dashed ray fro the origin. x2 Figure 1: Rasey quantities as α varies fro 0 to 1 x1 Cournot oligopoly: A natural developent of our fraework is to the Cournot oligopoly setting with syetric firs that each supply the full range of products and have the cost function c( ) satisfying (4). Our ain result in this context is that equilibriu in this -player gae is closely related to an appropriate Rasey optiu. Assuption (4) iplies that the least-cost way for the industry to supply total quantity vector x is to split this quantity equally between the firs so that total cost is c( 1 x). In this case the Rasey objective (5) becoes u(x) c( 1 x) (1 α)s(x), (8) so that the corresponding first-order condition for the optial vector of total quantity x is p(x) = c( 1 x) + (1 α) s(x). (9) 5 The figure depicts for x 1, x 2 2 5 the exaple where u(x) = x 1 + x 2 1 3 (x 1) 3 2 3 (x 2) 3 2 and c(x) = 0. 8
Consider a candidate syetric equilibriu in which each fir supplies quantity vector 1 x (so that x is total supply). Then a fir ust axiize its profit π(y) = y p( 1 x + y) c(y) by choosing y = 1 x, which fro expression (2) has the first-order condition p(x) = c( 1 x) + 1 s(x). (10) Coparing with (10) with (9) reveals that a syetric Cournot equilibriu, if it exists, has the sae first-order conditions as the Rasey proble (8) when the weight on consuers is α = 1. The following result establishes the existence and syetry of Cournot equilibriu: Proposition 1 Suppose there are Cournot copetitors, each of which supplies all n products and has the sae cost function c(x) satisfying (4). Then there exists a syetric Cournot equilibriu in which quantities axiize the Rasey objective (8) with α = 1. There are no asyetric equilibria. If in addition r(x) is concave there is only one syetric equilibriu. Proof. We first rule out asyetric equilibria. Consider a (possibly asyetric) candidate equilibriu in which fir j supplies quantity vector x j, where x = Σ j x j is the total supply. At an asyetric equilibriu, at least one fir ust have x j 1 x. In this equilibriu fir j ust axiize its profit by choosing y = x j. π j (y) = y p(σ i j x i + y) c(y) In particular, it cannot be profitable to deviate fro supplying y = x j to supplying y = x j + ε(x x j ), where ε is a scalar. Evaluating the derivative of π j (x j + ε(x x j )) with respect to ε at ε = 0 therefore yields 0 = [p(x) c(x j ) + x j Dp(x)] (x x j ) = [p(x) c(x j ) 1 (x xj x) Dp(x)] (x x j ) [p(x) c(x j ) + 1 x Dp(x)] (x xj ), (11) where the inequality (11) follows fro the negative-definiteness of the atrix Dp(x). This inequality is strict if x j 1 x, which is the case for soe fir in an asyetric equilibriu, 9
and so suing (11) across the firs we obtain 0 > j ( 1 x xj ) c(x j ) j [c(x j ) c( 1 x)] 0, (12) which is a contradiction. To see the second inequality in (12), note that the convexity of c( ) iplies c( 1 x) c(xj ) ( 1 x xj ) c(x j ), while the third inequality in (12) follows directly fro the convexity of c( ). We deduce that x cannot be equilibriu supply unless it is syetrically shared between firs. Turning to equilibriu existence, note that the Rasey objective (8) can be written (1 α)r(x)+αu(x) c( 1 x). Suppose that quantity vector x solves this Rasey proble when α = 1. Since c is convex, it follows that choosing xj = 1 x for each j axiizes the function (1 α)r(σ j x j ) + αu(σ j x j ) Σ j c(x j ). In particular, choosing y = 1 x axiizes the function ρ(y) 1 r( 1 x + y) + 1 1 u( x + y) c(y). Now consider Cournot copetition and a fir s best response when its rivals each supply quantity vector 1 x. This fir chooses its quantity vector y to axiize its profit π(y) y p( 1 = ρ(y) 1 x + y) c(y) { u( 1 x + y) + ( 1 ρ(y) 1 u(x), 1 x y) p( x + y)} where the inequality follows fro the concavity of u. Since π( 1 x) = ρ( 1 therefore have π( 1 x) π(y) ρ( 1 x) ρ(y) 0 1 x) u(x), we where the final inequality follows since y = 1 x axiizes ρ(y). We deduce it is a Cournot equilibriu for each fir to supply 1 x. Finally, consider the uniqueness of equilibriu. We have already shown there are no asyetric equilibria, while expression (10) shows that any syetric equilibriu satisfies the first-order conditions for axiizing the Rasey objective (8) with α = 1. This Rasey objective can be written as 1 1 r(x)+ u(x) c( 1 x), and given that u is strictly concave and c is convex, this is strictly concave if r(x) is concave. In this case, there is a 10
unique quantity vector x which satisfies the first-order condition (10), and hence a unique syetric equilibriu. Thus, with syetric convex cost functions there are no asyetric Cournot equilibria, and there exists a syetric Cournot equilibriu in which total quantities axiize the Rasey objective (8) with α = 1. If revenue r is concave, there is a unique Cournot equilibriu, which coincides with the (unique) optiu for the Rasey objective (8) with α = 1. In this sense, the Cournot proble and the (appropriately weighted) Rasey proble are the sae. This generalizes the second reark in Bergstro and Varian (1985) that a syetric single-product Cournot oligopoly can be considered to axiize a Rasey objective to the ultiproduct context. When c(x) is hoogeneous degree 1, the nuber of suppliers has no ipact on industry costs and the Rasey objectives (5) and (8) coincide. In this case, since consuer surplus in the Rasey proble (5) increases, and profit decreases, with α, we deduce that as the nuber of copetitors increases, a syetric Cournot equilibriu delivers ore surplus to consuers and involves lower industry profit, and with any firs the equilibriu quantities are approxiately 1 xw (where x w is the effi cient quantity vector). One can also study how equilibriu prices depend on arginal costs by studying the siple Rasey proble, as we do below in section 3.3. The analysis in Proposition 1 assues firs are syetric. Aong other issues, this assuption eans one cannot study the ipact of fir-specific cost shocks, for instance, but only industry-wide cost shocks. When Cournot equilibria exist in asyetric settings it is straightforward to obtain first-order conditions for equilibriu prices. For exaple, suppose that each fir has constant arginal costs, and fir j has the arginal cost vector c j = (c j 1,..., c j n). Then if all firs supply all products in equilibriu, an arguent siilar to (10) shows that equilibriu prices satisfy p(x) = 1 Σ jc j + 1 s(x) (13) where 1 Σ jc j is the industry average vector of arginal costs. 6 Thus, the Cournot equilibriu here corresponds to a the Rasey optiu with weight on consuers α = 1 6 This generalizes the first reark in Bergstro and Varian (1985) that equilibriu industry output depends only on the average arginal cost in the industry, not its distribution to ultiple products. One can show that this Cournot equilibriu exists if (i) inverse deands p i (x) are each weakly concave and (ii) that the cost vectors c j are close enough that each fir supplies all products in equilibriu. 11 and
a hypothetical onopolist with cost function c(x) = 1 x [Σ jc j ]. Another way to allow for fir asyetries is discussed in the following Bertrand odel. Bertrand oligopoly: Although it is not the focus of this paper, consider briefly one way to odel Bertrand copetition in this fraework. Bliss (1988) and Arstrong and Vickers (2001, section 2) suggested a odel where consuers buy all products fro one fir or another, so there is one-stop shopping, and firs therefore copete in ters of the surplus they offer their custoers. Each consuer has the sae gross utility, u(x), when they purchase quantity vector x fro a fir, and this utility function is the sae at all firs. Firs copete by offering linear prices, so that a consuer obtains surplus s(x) when they buy quantity x fro a fir via linear prices, while fir i, say, obtains profit r(x) c i (x) fro each custoer where c i (x) is this fir s constant-returns-to-scale cost function. (Unlike the previous Cournot odel, here it is straightforward to allow firs to have different cost functions.) Consuers differ in their brand preferences for the various firs, say due to the distances they ust travel to reach the, and the nuber of custoers a fir attracts increases with the surplus s it offers and decreases with the surplus its rivals offer. In this fraework, each fir s strategy can be broken down into two steps: (i) choose the ost profitable way to deliver a given surplus to a custoer, and (ii) choose how uch surplus to offer its custoers. Step (i) is just the Rasey proble as discussed above, and a fir s optial prices take the for (6) where α now will reflect the fir s copetitive constraints in step (ii) rather than concern for consuer welfare. In equilibriu there is intra-fir effi ciency, but with cost differences across firs there will not in general be industry-wide effi ciency (i.e., industry profits are not axiized subject to an overall consuer surplus constraint). The general analysis in this section has introduced the consuer surplus function s(x) and shown its usefulness in analyzing the Rasey onopoly proble and, by extension, the syetric Cournot oligopoly proble. In the rest of the paper we develop the analysis of onopoly and oligopoly by supposing that s is hoothetic in x. This specification includes a nuber of failiar ultiproduct deand systes, and has notably convenient properties. In particular, the feature of equiproportionate quantity reduction that appeared locally (for α 1 in the Rasey proble or large in the Cournot equilibriu) in the analysis above, holds globally. 12
3 A faily of deand systes 3.1 Hoothetic consuer surplus The faily of deand systes on which we now focus is characterized by the property that consuer surplus s(x) is a hoothetic function of quantities x. We first describe which deand systes have this property: Proposition 2 Consuer surplus s(x) is hoothetic in x if and only if utility u(x) can be written in the for u(x) = h(x) + g(q(x)) (14) where h( ) and q( ) are hoogeneous degree 1 functions and g( ) is concave with g(0) = 0. Proof. First, note that we ust have g(0) = 0 and g concave in q given that u(0) = 0 and u( ) is concave in x. (Since u is concave, when u takes the for (14) for given x the function k kh(x) + g(kq(x)) is too, so that g( ) is concave.) To show suffi ciency, note that (14) iplies that inverse deand is p(x) = h(x) + g (q(x)) q(x). (15) Revenue is therefore r(x) = x p(x) = h(x) + g (q(x))q(x), (16) where we used the fact that x h(x) h(x) for a hoogenous degree 1 function. Consuer surplus s(x) is then s(x) = g(q(x)) g (q(x))q(x), (17) which is hoothetic since s(x) is an increasing function of the hoogenous function q(x). (Since g is concave, g(q) g (q)q is an increasing function.) To show necessity, suppose that consuer surplus s(x) is hoothetic, so that s(x) G(q(x)) for soe increasing function G and soe function q(x) which is hoogeneous degree 1. We can write G as G(q) g(q) qg (q) for another function g( ). 7 Then s( x/k) = g(q( x)/k) q( x) k g (q( x)/k) = d kg(q( x)/k). (18) dk 7 Given any function G( ), one can generate the corresponding g( ) using the procedure g(q) = q q [G( q)/ q 2 ]d q. This function g(q) is concave given that G(q) is increasing. 13
Note that (1) can be generalized slightly so that s( x/k) = d ku( x/k), and so (18) can be dk integrated to yield ku( x/k) = h( x) + kg(q( x)/k) for soe constant of integration h( x). Writing x = x/k this becoes u(x) = h(kx) k + g(q(x)). Since this holds for all k we deduce that u(x) = h(x) +g(q(x)), where h(x) is hoogeneous degree 1. This result iplies that the set of deand systes in which consuer surplus is hoothetic in quantities is broader than that where consuer surplus is hoothetic in prices. Expressed as a function of prices, consuer surplus is the convex function v(p) = ax x 0 {u(x) p x}. Duality iplies that u(x) can be recovered fro v(p) using the procedure u(x) = in p 0 {v(p) + p x}, and if v(p) is hoothetic in p then u(x) = in p 0 {v(p)+p x} is hoothetic in x. Thus, the utility functions such that consuer surplus is hoothetic in prices are siply the hoothetic utility functions, i.e., where h 0 in (14), which is a subset of the faily of preferences we study. In section 3.2 we discuss failiar instances of the faily (14) which do not have hoothetic u( ). For the reainder of the paper we assue that utility u(x), as well as being increasing and concave, can be written in the for (14). Soe iediate observations on this preference specification are: For a specific utility function u(x) it ay not be obvious a priori whether it accords with the for (14). However, Proposition 2 iplies that this is the case whenever consuer surplus, s(x) u(x) x u(x), is hoothetic, which in practice is easy to check. Expression (15) iplies that an equiproportionate change in quantities oves the price vector along a straight line in the direction q(x). In geoetric ters, then, quantity vectors on the ray joining x to the origin correspond to price vectors which lie on the straight line starting at p(x) pointing in the direction q(x). If u satisfies (14), then the odified environent in which a subset of these products are reoved also satisfies (14). That is, if a subset of products have quantities x i set 14
equal to zero, the utility function u defined on the reaining products continues to satisfy (14). Since g is concave, the function g(q) g (q)q in (17) is an increasing function, so surplus s increases with x i and the Rasey price for product i is above arginal cost in (6) if and only if q( ) increases with x i. When utility takes the for (14), the revenue function (16) takes a siilar for, with the sae h and q (but with qg (q) replacing g(q)). For this reason, a ultiproduct onopolist s proble of axiizing profit discussed below in section 3.3 is closely connected to the consuer s proble of axiizing surplus, where prices in the consuer s proble correspond to arginal costs in the fir s proble. There are three degrees of freedo when choosing a deand syste within the class q(x), h(x) and g(q) and expression (14) provides a useful toolkit for constructing tractable ultiproduct deand systes with particular desired properties. For this purpose it is useful to know conditions for the resulting utility function u to be concave. Suffi cient conditions to ensure that u in (14) is concave are that h and g are concave and either: (i) q is concave and g is increasing; (ii) q is convex and g is decreasing, or (iii) q is linear in x (which allows g to be non-onotonic). For the reainder of this subsection we discuss in ore detail the iplications of this utility specification for the corresponding deand syste, denoted x(p). Given prices p, the consuer with utility (14) can axiize her surplus with a siple two-step procedure. We can write quantities x in the for x q(x) x q(x). (19) Here, x/q(x) is hoogeneous degree zero and depends only on the ray fro the origin on which x lies, while q(x) is hoogeneous degree 1 and easures how far along that ray x lies, and so the decoposition (19) represents a generalized for of polar coordinates for the quantity vector x. (The coordinate x/q(x) lies on the (n 1)-diensional surface q 1.) Henceforth we refer to q(x) as coposite quantity and x/q(x) as the relative quantities. We know already that (axiized) consuer surplus, s(x), depends on x only via coposite quantity q(x). More generally, consuer surplus with arbitrary quantities x 15
and prices p can be written in ters of the coordinates in (19) as h(x) + g(q(x)) p x = g(q(x)) q(x) p x h(x) q(x). (20) (Since the function p x h(x) is hoogeneous degree 1, (p x h(x))/q(x) depends only on the relative quantities x/q(x).) Since consuer surplus in (20) is decreasing in (p x h(x))/q(x), the consuer should choose relative quantities to iniize this ter, regardless of her choice for coposite quantity. Therefore, write φ(p) in x 0 : p x h(x) q(x), (21) which is an increasing and concave function of p. The envelope theore iplies that its derivative is the optial choice of relative quantities, so if we write x (p) φ(p) the consuer facing prices p chooses relative quantities x (p). 8 Given the relative quantities, x (p), the optial choice of coposite quantity, say Q, is easily derived. Consuer surplus in (20) with the optial relative quantities is the concave function g(q) Qφ(p). Write ˆQ(φ) for the coposite quantity which axiizes g(q) Qφ, which is necessarily weakly downward-sloping, so that ˆQ(φ(p)) is the deand for coposite quantity given the price vector p. Price vectors with the sae φ(p) induce the consuer to choose the sae coposite quantity Q, and so φ(p) is the coposite price which corresponds to coposite quantity q(x). Since the consuer chooses relative quantities x (p) and coposite quantity ˆQ(φ(p)), fro (19) the vector of quantities deanded at prices p is x(p) = ˆQ(φ(p)) x (p). (22) Here, the function g( ) deterines the shape the coposite deand function ˆQ( ), while the functions h( ) and q( ) cobine to deterine the for of coposite price function φ( ). Expression (22) iplies that cross or own-price deand effects are x i p j = ˆQ(φ)φ ij + ˆQ (φ)φ i φ j. (23) (Here, recall that x (p) = φ(p), while subscripts to φ denote its partial derivatives.) This is akin to the Slutsky Equation fro classical deand theory. The first ter in (23) represents the substitution effect while staying on the sae coposite quantity (or 8 There is a unique vector of relative quantities which solves (21) provided that h and q are quasi-concave with one or both of the strictly so. 16
consuer surplus) contour, and the second ter represents the ipact of a price rise on the coposite quantity deanded. This second ter is negative, while the first ter is negative if j = i and could be positive or negative when j i. For instance, if utility is hoothetic then φ(p) is positive and hoogenous degree 1, and expression (23) has the sign of φφ ij φ ˆQ (φ). Here the first ter is the elasticity of substitution of deand and φ i φ j ˆQ(φ) the second ter is the elasticity of coposite deand, and the relative sizes of these two elasticities deterines the sign of cross-price effects. Since inverse deand p(x) in (15) induces deand x, it follows that p = h(x) + g (Q) q(x) x(p) = Q x q(x). (24) In particular, for fixed Q any price vector of the for p = h(x) + g (Q) q(x) induces the sae coposite deand Q, and hence the sae consuer surplus s = g(q) Qg (Q) and coposite price φ(p) = g (Q). Conversely, since deand x(p) induces inverse deand p, substituting (22) into the expression for inverse deand (15), and recalling that for positive deand we have g ( ˆQ(φ)) φ, reveals that p h(x (p)) + φ(p) q(x (p)). (25) (Alternatively, this expression is the first-order condition for proble (21).) Expression (25) is the analogue for prices of the change of coordinates for quantities in (19), and decoposes the price vector p into coposite price, φ(p), and relative prices which in this context we define to be x (p), i.e., the relative quantities which are optial with prices p. Fro (24), prices which induce relative quantities x lie on the straight line φ : h(x ) + φ q(x ), which is not necessarily a ray fro the origin, while the coordinate φ(p) deterines how far along such a line the price vector lies. 3.2 Special cases If u( ) is itself hoothetic for instance, if deand takes the failiar CES for then (14) is trivially satisfied by setting h 0. In this case, expression (21) iplies that φ(p) is hoogeneous degree 1, and x (p) = φ(p) is hoogeneous degree zero. More generally, hoothetic deand is an instance of the subclass of (14) where h takes the linear for h(x) = a x, when φ in (21) is a function that is hoogenous degree 1 in the adjusted price vector (p a). 17
Linear deand: Another instance of this subclass with linear h( ) is linear deand, where utility u(x) takes the quadratic for u(x) = a x 1 2 x M x (26) for constant vector a > 0 and (syetric) positive-definite atrix M. Here, inverse deands are p(x) = a Mx, and utility takes the for (14) by writing h(x) = a x, q(x) = x M x and g(q) = 1 2 q2. Here, q(x) = Mx and so expression (24) iplies that the set of price vectors which correspond to the sae relative quantities takes the for p = a tm x for scalar t, which are rays originating fro the vector of choke prices a. It ay be that q(x) and therefore s(x) decrease with x i when off-diagonal eleents of M are negative (which corresponds to products being copleents). Logit deand: Suppose that consuer deand takes the logit for x i (p) = e a i p i 1 + j ea j p j, (27) where a = (a 1,..., a n ) is a constant vector. It follows that inverse deand is p i (x) = a i log x i 1 q(x), (28) where q(x) j x j is total quantity. This inverse deand function (28) integrates to give the utility function u(x) = a x j x j log x j (1 q(x)) log (1 q(x)). (As with any deand syste resulting fro discrete choice, the utility function is only defined on the doain Σ i x i 1. 9 ) This utility can be written in the required for (14) as u(x) = a x + x i log q(x) i x }{{ i } h(x) + g(q(x)). (29) Here, h(x) as labelled is hoogenous degree 1, as is total output q(x), while g(q) is equal to the entropy function g(q) = q log q (1 q) log(1 q), which is concave in 0 q 1. Since g (q) = log(1 q) log q, deand for coposite quantity as a function of coposite 9 If one wishes only to consider non-negative prices, fro (28) one should further restrict attention to quantity vectors which satisfy x i (1 Σ j x j )e ai for i = 1,..., n. 18
price takes the logistic for. 10 With logit, as with hoogeneous goods, consuer surplus is a function only of total quantity, and product differentiation is reflected separately in the h(x) ter. Since q(x) (1,..., 1), the set of prices which correspond to the sae relative quantities takes the for p + (t,..., t), as can be seen directly fro (27). This contrasts with the subclass with linear h(x), where these lines were not parallel but eanated fro a point. More generally, with any deand syste in the subclass with linear q(x) = b x, the set of prices which correspond to quantity vectors on a given ray fro the origin take the for of parallel straight lines. Systes of strictly copleentary products: A coon situation is where consuers purchase a single unit of a base product, and then cobine this with variable quantities of one or ore copleentary products. For instance, a consuer ay need to gain entry to a thee park before they can go on the rides (Oi, 1971), or needs to buy a printer along with a suitable quantity of ink in order to print. To illustrate how these situations soeties fit into our fraework, suppose there is a continuu of consuers indexed by scalar θ, where the type-θ consuer has gross utility U(y) + θ if she consues quantity y of the cobined service. (The following discussion also applies if y is a vector of ultiple services.) Adding over the population of consuers iplies that aggregate gross utility if x 1 consuers (those with the highest value of θ) each consue quantity y of cobined service takes the for x 1 U(y) + g(x 1 ) for soe increasing concave function g( ), where g is deterined by the distribution of θ. If x 2 denotes the quantity of cobined service across all consuers, so that x 2 = x 1 y, it follows that aggregate utility in ters of the quantity vector (x 1, x 2 ) is u(x 1, x 2 ) = x 1 U(x 2 /x 1 ) + g(x 1 ). (30) Clearly, this utility function fits into our faily (14), where h(x) = x 1 U(x 2 /x 1 ) and coposite quantity is just q(x) = x 1. This is another instance of the subclass with linear q(x), but here s(x) is a function only of x 1, the nuber of active consuers. The set of prices which correspond to the sae relative quantities i.e., the sae usage per active consuer are horizontal lines with p 2 constant. 10 Anderson et al. (1988) have previously noted the connection between logit deand and entropy. The entropy function akes it diffi cult to obtain closed-for solutions for optial prices or quantities with logit deand. However, if we odify (29) slightly so that g(q) q(1 q), deand for coposite quantity is linear rather than logistic and explicit forulas can be obtained. 19
3.3 Analysis In this section we discuss how to axiize welfare and profit, as well as calculate oligopoly outcoes, when the deand syste satisfies (14) and the cost function satisfies c(x) is convex and hoogeneous degree 1. (31) Consider again the proble of axiizing a weighted su of profit and consuer surplus. If 0 α 1 is the relative weight on consuer surplus, the Rasey objective (5) is [r(x) c(x)] + αs(x) = αg(q(x)) + (1 α)g c(x) h(x) (q(x))q(x) q(x) q(x) Here, we used the forulas (16)-(17).. (32) As with the consuer s proble in section 3.1, the Rasey proble can conveniently be solved by eans of the change of variables (19). Expression (32) shows how the Rasey objective can be written in ters of coposite quantity q(x) and the relative quantities x/q(x). Expression (32) is decreasing in the ter (c(x) h(x))/q(x), and so relative quantities should be chosen to iniize this ter. As in (21), write κ = in x 0 : c(x) h(x) q(x), (33) which is solved by choosing relative quantities x = x, say. (Since the quantity vector which iniizes (33) is indeterinate up to a scaling factor, as in section 3.1 we noralize x so that q(x ) = 1. 11 ) We deduce that axiizing any Rasey objective involves choosing the sae relative quantities x, in contrast to the case depicted in Figure 1. In particular, profit-axiizing quantities (α = 0) are proportional to the effi cient quantities corresponding to arginal-cost pricing (α = 1). That is, the unregulated fir has an incentive to choose its relative quantities in an effi cient anner, and the sole ineffi ciency arises fro it supplying too little coposite quantity. Given this choice for relative quantities, the optial choice for coposite quantity Q is easily derived. Expression (32) with relative quantities x is the function αg(q) + (1 α)g (Q)Q κq. (34) 11 There is a unique vector of relative quantities which solves (33) provided that (c h) is quasi-convex and q is quasi-concave with one of the strictly so. To illustrate this analysis, suppose that q(x 1, x 2 ) = x 1 x 2, h(x 1, x 2 ) = 0 and c(x 1, x 2 ) = c 2 1 x2 1 + c2 2 x2 2. Then one can check that x = ( c2 c 1, c1 c 2 ) and κ = 2c 1 c 2. 20
(A suffi cient condition for (34) to be concave in Q for all α is that coposite revenue g (Q)Q be concave.) The vector of quantities which solves the Rasey proble is then Qx, where Q axiizes expression (34). The optial coposite quantity Q increases with α and decreases with κ, and satisfies the Lerner forula g (Q) κ g (Q) = (1 α)η(q), (35) where η(q) Qg (Q) g (Q) is the elasticity of inverse deand for coposite quantity. (36) Since relative quantities are the sae in all Rasey probles, so too are relative pricecost argins. As discussed in section 3.1, this is because an equiproportionate reduction in effi cient quantities causes the price vector to ove in a straight line away fro the vector of arginal costs. In ore detail, the optial quantities for the Rasey proble are Qx, where Q satisfies (35), and in particular let the coposite quantity which axiizes total surplus (i.e., when α = 1) be denoted Q w, so that g (Q w ) = κ. Then the price-cost argins in the Rasey proble with coposite quantity Q are p(qx ) c(qx ) = p(qx ) c(q w x ) = p(q w x ) c(q w x ) + [g (Q) g (Q w )] q(x ) = [g (Q) κ] q(x ). (37) (Here, the first equality follows fro c being hoogeneous degree zero, the second follows fro (15), and the final equality follows since prices equal arginal costs and g = κ when Q = Q w.) These argins are proportional to q(x ), and shrink equiproportionately when Q is larger. Product i is used as a loss leader, in the sense that its price is below arginal cost, in each Rasey proble when coposite quantity q decreases with x i at x. By virtue of the Rasey-Cournot result in Proposition 1, these properties extend to syetric Cournot oligopoly. To suarise: Proposition 3 Suppose that utility takes the for (14) and cost takes the for (31). As ore weight is placed on consuer surplus in the Rasey proble, the coposite quantity increases, the coposite price decreases, each individual quantity increases equiproportionately, and each price-cost argin contracts equiproportionately. The sae is true in syetric Cournot equilibriu as the nuber of firs increases. 21
An iportant special case involves constant arginal costs, so that c(x) c x for a constant vector of arginal costs c = (c 1,..., c n ). In this case, κ in (33) is siply φ(c) where the function φ( ) is defined in (21), while x = x (c). In this context, consider how optial prices relate to the fir s costs. This analysis is ost transparent using the change of variables for prices (and costs) in (25), so that φ(p) is the coposite price and x (p) are relative prices (and siilarly for the cost vector). As discussed earlier, in any Rasey proble it is optial to choose relative quantities equal to the relative quantities which correspond to effi cient arginal-cost pricing. This iediately iplies that it is optial to choose relative prices equal to the fir s relative costs, so that x (p) = x (c). The optial arkup of coposite price over coposite cost is then given by (35), and the optial coposite price, φ(p), decreases with the weight on consuer surplus, α, and increases with coposite cost, φ(c). 12 What does this ean for price-cost relationships: how does p i depend on c j? Expressions (35) and (37) iply that optial prices satisfy p c = (1 α)η(q)g (Q) q(x (c)). (38) Consider first the subclass where h takes the linear for h(x) = a x. quantities are Qx (c), expression (15) shows that prices satisfy Since optial Putting (38) and (39) together iplies that p c = p a = g (Q) q(x (c)). (39) (1 α)η(q) (c a). (40) 1 (1 α)η(q) In particular, when preferences are hoothetic, so that a = 0, we obtain the failiar result that proportional price-cost arkups are the sae across products. In the iso-elastic case where η is constant, expression (40) iplies that the optial price for product i depends only on c i, not on any other product s cost, and so there is no cross-cost passthrough in prices, even though there ay be substantial cross-price effects in the deand syste. Moreover, provided that the consuer can obtain positive 12 Since the profit-axiizing fir s choice of coposite quantity falls with φ(c), and since consuer surplus, s, is an increasing function of coposite quantity, we deduce that the fir necessarily offers a lower level of consuer surplus when unit cost c i increases. Our faily of deand systes therefore excludes the possibility explored by Edgeworth (1925) that iposing a linear tax on a product supplied by a ultiproduct onopolist could reduce all of its prices. 22
utility with a subset of products (e.g., if ρ > 0 in the CES specification), the optial price for one product is unaffected if the fir is restricted to offer a subset of products (or even just that product). 13 For instance, if u is hoothetic and g(q) = 1 γ Qγ, where 0 < γ < 1, then η 1 γ and (40) iplies that the ost profitable prices (i.e.,when α = 0) are p = 1 γ c. Likewise, with linear deand we have η 1 and expression (40) iplies that the profitaxiizing prices are 14 p = 1 (a + c). (41) 2 More generally within this subclass with linear h, expression (40) and the fact that the ost profitable Q decreases with each cost iplies that all cross-cost passthrough ters for p i have the sae sign as (a i c i )η (Q). 15 Alternatively, consider the subclass where coposite quantity takes the linear for q(x) = b x. Then (38) iplies that p c = (1 α)η(q)g (Q) b. (42) In the exaple with copleentary products where utility is (30) and b = (1, 0), this expression iplies there is arginal-cost pricing for usage (p 2 = c 2 ), and in particular changes in c 1 have no ipact on p 2. However, there is cross-cost passthrough in the other direction: since the optial Q decreases with both costs, it follows that p 1 increases or decreases with c 2 according to whether η(q)g (Q) decreases or increases with Q. In the logit exaple utility is (29) and b = (1,..., 1), so the price-cost argin p i c i is the sae for 13 Shugan and Desiraju (2001) discuss these points in the context of linear deand with two products. In the context of product line pricing, Johnson and Myatt (2015) explore when it is that a fir s optial price for one product variant can be calculated by supposing that the fir only supplies that variant. (They consider both onopoly and Cournot settings.) 14 It usually akes sense only to consider non-negative quantities, in which case (41) is only valid if a and c are such that the optial quantities, x = 1 2 M 1 (a c), are positive. In the case of substitutes, where the atrix M necessarily has all non-negative eleents, a necessary condition for this is that each a i c i. (When M has all non-negative eleents, the operation x Mx takes R n + into itself.) However, with copleents, it is possible to have all x i positive and soe a i c i negative. In such cases, (41) indicates that p i < c i for those products with a i < c i. 15 One can analyze how the optial quantity supplied of one product is affected by cost changes to other products in a siilar anner to the consuer deand expression (23). For instance, since profitaxiizing quantities satisfy the first-order condition r(x) = c, where c is the vector of constant arginal costs, there will be a dependence of one product s supply on another product s cost unless r(x) is additively separable in quantities, which is (essentially) only the case if deand for one product does not depend on other prices. 23