Market Participation under Delegated and Intrinsic Common Agency Games (Preliminary) David Martimort Lars A. Stole November 25, 2003 *We are grateful to Wouter Dessein, Bruno Jullien and Canice Prendergast for helpful comments and discussions. Any errors are our own. University of Toulouse (IDEI-GREMAQ) and Institut Universitaire de France. University of Chicago, GSB.
Abstract: In a stylized model of common agency with two competing principals and a privately informed agent, we study how competition in nonlinear pricing affects the measure of agents who participate. Two forms of competition are examined. When participation is restricted to all-or-nothing (what has been called intrinsic agency), the agent must choose between both principal s contracts or selecting her outside option; when the agent is afforded the additional possibilities of choosing only one principal s contract ( delegated agency), competition operates on a more intense level. Unlike previous analyses of common agency that focused on marginal allocation distortions and found that the delegated and intrinsic agency games have identical outcomes, we find that two games have distinct predictions for participation. We derive the following results: When the principals contracting variables are substitutes on the intensive margin, we find that equilibrium outcome in the intrinsic agency game always induces more distortion in participation relative to the monopoly outcome, and the monopoly outcome induces still more participation distortion relative to the outcome under delegated agency. In the case where the contracting variables are complements on the intensive margin, the participation outcome is the same in the delegated and the intrinsic games, and this outcome exhibits a greater participation distortion than what a monopolist would choose. The intuition for these results is closely tied with the observation that intrinsic agency is strategically equivalent to delegated agency when the contracting variables are Leontief complements in the agent s utility function on the extensive margin. Seen in this light, such complementarity on the extensive margin generates distortions in the extensive consumption decision (i.e., participation rate) as expected.
1 Introduction This is a paper about common agency games between principals using self-selection contracts targeted at a distribution of privately informed agents. Many interesting economic applications fit into this setting. For example, when two non-cooperating regulatory bodies regulate the same privately-informed firm but on different dimensions (e.g., output and pollution), the outcome can be modeled as an equilibrium to a common agency game. As a second example, when two firms sell non-homogenous goods to the same consumer using nonlinear pricing as a price discrimination strategy, the price schedules which arise can also be modeled as an equilibrium to a common agency game. There is a critical difference between these two examples which we explore in detail. In the first example, the regulated firm does not have a choice to be regulated by only one regulator; that is, the firm can choose to leave the industry and face no regulation, or it can choose to abide by both sets of regulations; hence, common agency and non-participation are the only outcomes which can arise and common agency is intrinsic to the game. In the second example, it is natural to allow the consumer the option of purchasing from only one firm (providing that the agent s participation choice vis-a-vis the other principal cannot be contracted upon). Hence, the consumer may conceivably purchase from only one firm in equilibrium (i.e., exclusive agency arises), and so common agency is no longer intrinsic to the game but rather it is an outcome that is delegated to the agent. In this paper, we explore both settings. We are particularly interested in how these two variations in common agency games impact the allocation distortions that are inherent in self-selection settings, and further how these allocations compare to the outcome where principals cooperate in their contract design. Early efforts by Martimort (1992, 1996) and Stole (1991), as well as most subsequent applications and theoretical extensions to common agency with asymmetric information, have been in the context of intrinsic 1 common agency games games in the form of our regulation example. 2 In the canonical form of this common agency game, an agent learns some private preference parameter regarding the margins of two economic activities, say q 1 and q 2. Both principals simultaneously and non-cooperatively offer selection contracts with 1 This common agency game is called intrinsic by Bernheim and Whinston (1986) because common (as opposed to exclusive) agency is intrinsic to the game when the agent does not have the option to contract exclusively with a single principal. 2 See Mezzetti (1997), Laffont and Tirole (1993, ch. 17), Bond and Gresik (1997), Ivaldi and Martimort (1994), Olsen and Torsvik (1993, 1995) and Olsen and Osmudsen (2001) among others. 1
the restriction that principal i s contract cannot depend upon activity q j, j i. Following this stage of the intrinsic game, the agent must decide between accepting or rejecting both contracts; the agent is not allowed to accept one contract and reject the other. Activities are subsequently chosen and payoffs are awarded in accord with the activities and the contracts. Such a modeling is naturally appropriate in regulation since the agent is then a regulated firm and the principals are distinct regulatory bodies, each with authority over a mutually exclusive set of activities. A firm may decide to exit the industry (i.e., reject the regulatory contracts), but the firm may never decide to accept one set of regulations and reject the other. Alternatively, we could allow the agent the extra options of accepting a proper subset of the principals contract offers. Following Bernheim and Whinston (1986), we call this variation a delegated common agency game because, unlike its intrinsic counterpart, common agency endogenously emerges in equilibrium by the choice of the agent rather than being an exogenously fixed institution that is forced upon the agent. The presence of additional options for the agent generally increases the rents the agent earns. The utilities from these additional additional options, however, depend upon the agent s private information, and as consequence delegated agency games introduce type-dependent participation constraints. 3 To the best of our knowledge, no one has studied the full economic consequences of common agency (both intrinsic and delegated) on distortions in contractual activities and participation. In our early papers, Martimort (1992) and Stole (1991), we demonstrated that understanding equilibrium outcomes to the easier-to-compute intrinsic agency game perfectly translates over the case of delegated agency games when contracting activities are complements; and we further argued that, when the activities are substitutes, intrinsic outcomes still provide considerable insight into delegated agency games but that one has to pay closer attention to the agent s rents and participation constraints. Calzolari and Scarpa (1999) explored non-intrinsic common agency in greater depth and proved that that in the case of substitutes, the agent obtains greater rents in a non-intrinsic game but that otherwise the productive allocations are identical. While their claim is true when all agent types participate in the intrinsic agency equilibrium, full participation is a critical qualifier for their result. The assumption of full participation greatly reduces the complexity of the 3 Laussel and Lebreton (2001) have shown that delegated common agency may lead to outcomes where the agent gets a positive rent even under complete information if the number of principals is greater than two. Our focus on asymmetric information and two principals can thus be seen as complementary to theirs. 2
problem, but it misleads one to conclude that there are no efficiency differences between intrinsic and non-intrinsic common agency games and that only the level of rents left to agents is altered. When the market is not covered, as we show below, an important set of distortions emerge which typically depend upon whether the common agency game is delegated or intrinsic. Hence, the first contribution of the present paper is as analysis of the two forms of distortions intensive margins (activity levels) and extensive margins (participation) and in relating the direction and magnitudes of these distortions to the underlying game form and the preferences of the possibly common agent. A second contribution of this paper is in exploring a richer structure of demand preferences that allows for variations in substitution and complementarity on both the intensive and extensive margins, and how this structure directly relates to the outcomes of a variety of game forms (intrinsic, delegated, monopoly and perfect competition). To understand this more clearly, suppose that a delegated agency game is being played by two firms selling distinct consumer goods to a class of consumers. For concreteness, suppose that the agent s utility is u(q 1, q 2, θ) P 1 (q 1 ) P 2 (q 2 ), where q i is the quality of good i purchased from firm i for price P i (q i ). We say that the goods are (intensive) substitutes if u q1 q 2 (q 1, q 2, θ) < 0 for all positive values of q 1 and q 2. But it may also be the case that without the purchase of good i (of any quality), that good j has no value. In this case, we say that the goods are substitutes on the intensive margin but perfect complements on the extensive or base margin. As an illustration, it is easy to imagine a monopoly seller of software and a monopoly supplier of CPUs in this situation: both the software and the hardware products are Leontief complements at the base level, but it may be that faster CPU speed is a substitute for higher quality (i.e., faster) software. In this setting, the intrinsic common agency game and the delegated game and the game are economically equivalent. This conclusion follows because the additional options of contracting with only one seller has no value and therefore is strategically irrelevant to the consumer. Of course, it is at least as easy to imagine that our software and hardware monopolists sell goods which are complementary on both the extensive and intensive margins: e.g., both a CPU and a software package are necessary for any utility, but the faster CPU is more valuable on the intensive margin if one also uses the feature-packed high-quality version of the software. Here too we will find that delegated and intrinsic agency games are economically equivalent. There are still other preferences to consider. There is the possibility that the base goods are perfect (exclusive) substitutes, meaning that only one of the two goods can 3
be ultimately consumed. Now, we will find that not only is common agency never an equilibrium, but that perfect competition ensues and exclusive agency with marginal cost pricing emerges in any pure-strategy equilibrium. Finally, there is also the possibility that the goods are independent at the extensive margin. Here, intrinsic and delegated agency differ in economically important ways. To (partially) foreshadow of our conclusions, the second contribution of this paper is in generating and discussing the following table of outcomes for delegated agency games under eight possible demand preference configurations. 4 The question marks will be replaced in the course of the analysis. Figure 1: Equivalences of Equilibrium Outcomes in Delegated Common Agency Games Extensive (base) preferences Exclusive substitutes Independent Perfect complements Substitutes perfect competition? intrinsic outcome Intensive γ > 0 (exclusive agency) (?) (common agency) (marginal) Independent perfect competition monopoly intrinsic outcome preferences γ = 0 (exclusive agency) (common agency) (common agency) Complements? intrinsic outcome γ < 0 (?) (common agency) Section 2 presents the notation and description of the common agency games. Section 3 reviews the one-principal (monopoly) nonlinear pricing game as a reference point. Section 4 provides a few results on incentive compatibility and indirect utility functions that are useful for studying common agency contracting games in general. Section 5 solves for the equilibrium set in the intrinsic common agency game. Section 6 solves for the equilibrium set in the delegated common agency game and makes comparisons to the case of intrinsic agency and monopoly. Our main theorem regarding participation distortions is then presented. All proofs are relegated to the appendix. 4 For the case of extensive perfect substitutes and intensive complements, demand is not well defined. If it were, we would have a setting in which the base goods are perfect substitutes (and hence only consumption of one good is of value) but the intensive margins are complementary and hence have indirect value, implying that the base units have value as they are necessary for the intensive margins. Thus, there are only eight relevant cells to consider. 4
2 The common agency game In the multi-principal game we consider, each firm simultaneously offers a continuous price schedule, defined over a compact set of available outputs, to the privately-informed consumer(s): {P i ( ), Q i }. Specifically, firm i offers to sell any q i Q i for a price of P i (q i ). Upon observing the posted price schedules, the consumer decides whether or not to participate and, conditional upon participation, how much to consume from each firm. In the intrinsic common agency game, the consumer must decide between joint consumption and no purchase at all. In the delegated common agency game, the consumer can choose between joint consumption, consuming exclusively from firm 1, exclusively from firm 2, or no consumption at all. For concreteness, assume throughout that the agent is a consumer, potentially purchasing from two principals (i.e., firms) i = 1, 2. The consumer has a privately known type, θ, which is distributed uniformly on [0, θ]. On occasion, when we wish to make our statements more general, we will use F (θ) and f(θ) to represent general cumulative distribution and positive continuous density functions of θ on the support [0, θ]. Our stated propositions, however, will make use of the simplicity of the uniform distribution to provide closed-form results. 5 In this paper, we focus on symmetric, utility functions that are quadratic in the consumptions of the two products, (q 1, q 2 ). More specifically, we assume that the consumer s derived demand curves for consumption are symmetric, linear in prices, and that θ appears only in the demand intercepts. 6 In such a setting, demand can be characterized by q i = α + θ βp i + γp j, for i j, i = 1, 2 with β > γ. This statement is equivalent to assuming a quadratic utility 5 1 F (θ) Our results can be generalized to the case of distributions with a linear hazard rate = λ( θ θ) f(θ) where λ > 0 with only minor changes to the proofs. 6 Once we assume that θ only appears in the demand intercepts, it is without further loss of generality to assume that demand is linear in θ. Assuming in tandem that θ is uniformly distributed does impose some additional structure on the problem, however. 5
function of the form: u(q 1, q 2, θ) = α + θ β γ (q 1 + q 2 ) β 2(β 2 γ 2 ) (q2 1 + q 2 2) γ β 2 γ 2 q 1q 2 + w, and w is the numeraire. These assumed preferences are restrictive to the extent that actual demand is nonlinear in prices or that θ affects the own or cross-price derivatives of the demand function. 7 Note that this representation provides that u θ > 0 and the standard single-crossing property u qi θ > 0 is satisfied for each good. If γ (0, β), the goods are demand substitutes in the traditional sense, while the goods are demand complements if γ ( β, 0). We consider both variations below. We assume that firm s have symmetric, quadratic, convex costs of production: C(q) = c 0 + c 1 q + c 2 2 q 2. Because the consumer s preferences are quadratic, it is without loss of generality to suppose that c 2 = 0 and c 0 = 0, and fold these components into the consumer s preferences. We can also perform the same simplification with respect to c 1, but we choose not to do so in order to provide a convenient place-holder for marginal cost. Thus, we assume, without further loss of generality, that costs are represented by C(q) = cq, and each firm s profit (as a function of q i ) is given by π i (q i ) = P i (q i ) cq i. Finally, define q fb (θ) θ + α (β γ)c to be the first-best, full-information allocation of output. We assume that q fb (0) > 0; i.e., it is socially efficient for all consumers to purchase some positive levels of (q 1, q 2 ). Given our assumptions of quadratic preferences, we consider the class of symmetric Bayesian Nash equilibria whose allocations are linear in θ, and we use the term equilibrium with these symmetry and linearity conditions left implicit. 8 7 Following Martimort (1992), one could instead posit that the utility function satisfies a separability restriction: u qi (q 1, q 2, θ) = ũ 1 q 1 (q 1, φ(q 2, θ)) = ũ 2 q 2 (q 2, φ(q 1, θ)), where u i (q i, φ) is assumed to satisfy a single-crossing property in (q i, φ). With the restriction to quadratic preferences in consumption, this separability restriction is equivalent to the requirement that demand curves are linear in prices and θ only enters through the intercepts. 8 In the case of (intensive) complements with symmetric quadratic preferences, Martimort (1992) and Stole (1991) demonstrated that there exists a continuum of equilibria; in the case of (intensive) substitutes, 6
In short, we have assumed that all preferences are quadratic (e.g., demand is linear), that the heterogeneity parameter θ appears as a demand intercept shifter, and that θ is uniformly distributed. This model is tractable, while allowing us to obtain closed-form solutions, providing a complete study of the effects of substitutes and complements (captured by γ) and of the nature of competition (captured in the intrinsic versus delegated games) on intensive and extensive marginal distortions. 3 Review of the monopoly model It is useful to first review the classic monopoly nonlinear pricing problem (e.g., Mussa and Rosen (1978), and Maskin and Riley (1984)). We do this using a general utility function and distribution of θ before applying the results to the quadratic-uniform setting. To this end, suppose that consumer preferences are characterized by a function, v(q, θ) and that the consumer type, θ [0, θ], is distributed according to the distribution function F (θ), with corresponding density f(θ). The monopolist wishes to design a nonlinear price function, P (q), to maximize expected profits. Given an offered nonlinear price schedule that generates a unique choice of output for each type θ, we can define a consumer s optimal choice as a function of his type, θ: q(θ) arg max q Q v(q, θ) P (q), and we can define the associated indirect utility function as U(θ) max q Q v(q, θ) P (q) = v(q(θ), θ) P (q(θ)). the unique equilibrium is linear. As such, it is with loss of generality to restrict attention to linear equilibria when the goods are complements. Nonetheless, because linear allocations are supported with quadratic price schedules which are tractable and provide reasonable approximations to more complex nonlinear pricing schedules, such equilibria are naturally appealing. It is also comforting that the unique optimal allocation under monopoly is linear and hence within the same class, as is the first-best allocation. Furthermore, if one is prepared to restrict firms to offering quadratic price schedules, focusing on linear equilibrium allocations is without loss of generality. Fortunately, the economic insights behind the participation distortion for complements generalize to nonlinear equilibria as well, although additional complications arise in the analysis and we are not able to derive closed-form solutions. 7
The firm maximizes θ θ 0 (P (q(θ)) C(q(θ))) f(θ)dθ, where the interval [θ 0, θ] is the set of consumer types who purchase from the firm in equilibrium. maximizing Using our definitions of q(θ) and U(θ), this objective function is equivalent to θ θ 0 (v(q(θ), θ) C(q(θ)) U(θ)) f(θ)dθ. As such, this maximization can proceed by choosing {q(θ), U(θ), θ 0 } that is implementable by some nonlinear price schedule, rather than optimizing over P (q) directly. This reparameterization, of course, is the essence of the revelation principle. At this point, we make use of the single-crossing property that is assumed, a priori, to hold for v(q, θ): namely, v qθ (q, θ) > 0. Given this assumption on preferences, it is easily established 9 that the firm can design a nonlinear price schedule, P (q), to implement the allocation q(θ) if and only if q(θ) is nondecreasing in θ. Furthermore, for any implemented allocation q(θ), the corresponding indirect utility function is absolutely continuous and satisfies θ U(θ) = U(θ 0 ) + v θ (q(s), s)ds. θ 0 This implementation result allows us to replace the U(θ) function in the firm s maximand. After integrating U(θ) by parts, this objective simplifies to θ θ 0 ( v(q(θ), θ) C(q(θ)) 1 F (θ) ) v θ (q(θ), θ) U(θ 0 ) f(θ)dθ. f(θ) The monopolist maximizes this expression over q(θ), U(θ 0 ) and θ 0, subject only to the condition that q(θ) is nondecreasing and that the consumer wishes to participate: i.e., U(θ) 0 for every θ θ 0. We also make use of an assumption that v θ (q, θ) 0, so that higher types have greater utilities of consumption. Given that U (θ) = v θ (q(θ), θ) > 0, it follows that U(θ) 0 for θ θ 0 if and only if U(θ 0 ) 0. Hence, a profit-maximizing monopolist will always set U(θ 0 ) = 0. What remains to be determined are θ 0 and the allocation function, q(θ). 9 See, e.g., Fudenberg and Tirole (1992, ch. 7). 8
To this end, define the firm s virtual profit function as Λ(q, θ) v(q, θ) C(q) 1 F (θ) v θ (q, θ). f(θ) The firm s optimization program can be restated as maximizing with respect to θ 0 and q(θ) the objective θ θ 0 Λ(q(θ), θ)f(θ)dθ subject to q(θ) nondecreasing. Fundamental to reaching this remarkable simplification is the single-crossing property that allowed us to replace U(θ) with a representation involving only q(θ). Without single-crossing, designing optimal nonlinear price schedules is largely intractable. 10 To facilitate optimization, additional assumptions on v, C and F are typically made so as to ensure that Λ(q, θ) is strictly quasi-concave in q and that Λ qθ (q, θ) 0 (i.e., Λ(q, θ) is supermodular in (q, θ)). Such assumptions allow us to proceed with pointwise optimization and ignore the requirement that q(θ) is nondecreasing. For example, if F (θ) = θ/ θ (i.e., θ is uniformly distributed) and the social surplus function, S(q, θ) v(q, θ, θ) C(q), is strictly concave in q, then the additional assumptions that v θqq (q, θ) 0 and v θθq (q, θ) 0 are sufficient for Λ qq (q, θ) < 0 and Λ θq (q, θ) > 0. Regardless of how it is established, if Λ qq (q, θ) < 0 and Λ θq (q, θ) > 0, it follows that the pointwise optimization of Λ(q, θ) is uniquely determined from the first-order condition, Λ q (q(θ), θ) = 0, and that this optimal q(θ) function is increasing: q (θ) = Λ θq /Λ qq > 0. Hence, the optimal allocation function, q(θ), is entirely determined by the equation, Λ q (q(θ), θ) = 0, θ [θ 0, θ]. Simultaneously, θ 0 is optimally chosen (using Leibniz s rule to differentiate the limits of the integral) to satisfy the first-order condition, Λ(q(θ 0 ), θ 0 ) = 0, when a root θ 0 [0, θ] can be found and θ 0 = 0 otherwise. The associated second-order condition for θ 0 is that Λ θ (q, θ) > 0. In the present setting, this is satisfied because Λ θ (q, θ) = v θ (q, θ) > 0, by assumption. Once q(θ) and θ 0 have been thus determined, P (q) can be recovered from the differential 10 See Araujo and Moreira (2000) for a discussion and a first approach to designing optimal nonlinear price schedules without single-crossing. 9
equation P (q(θ)) = u q (q(θ), θ) and the boundary condition that v(q(θ 0 ), θ 0 ) P (q(θ 0 )) = 0 (i.e., U(θ 0 ) = 0). When Λ(q, θ) fails to be smooth and strictly quasi-concave, the optimal allocation is not necessarily uniquely determined by the first-order condition. Furthermore, when Λ(q, θ) is not supermodular in (q, θ), the pointwise optimum of Λ(q, θ) is decreasing over some interval, and hence it is not implementable. Specialized control-theoretic techniques of ironing are then needed to guarantee incentive compatibility. The monopoly screening literature has largely avoided these issues by making reasonable restrictions on preferences. Under common agency, we will find that similar reasonable restrictions guarantee that these problems can be avoided. To summarize, providing that v(q, θ) satisfies single-crossing, and that it is sufficiently regular so as to ensure that Λ(q, θ) is strictly quasi-concave in q and supermodular in (q, θ), the optimal allocation is determined from Λ q (q(θ), θ) = 0 and the marginal consumer is determined by Λ(q(θ 0 ), θ 0 ) = 0. Hence, for θ θ 0, q(θ) satisfies v q (q(θ), θ) C (q(θ)) = 1 F (θ) v θ (q(θ), θ) 0. f(θ) Consumption is distorted downward for all but the highest type consumer, θ = θ. In economic terms, in the simple one-good one-principal monopoly setting, a principal designs a schedule of prices to trade off the costs of inefficient consumption distortions against the gain such distortions provide in increased rent extraction. When we examine the multi-principal game, we will find that the presence of two principals complicates matters considerably because of the rent-extraction externalities the principals generate on each other s relationship with the consumer. In order to isolate this strategic effect on allocations, we need to consider the monopoly benchmark when two goods are present and there are no strategic interactions. Introducing two goods creates no new difficulties when preferences are symmetric. Redefine Λ as Λ(q 1, q 2, θ) u(q 1, q 2, θ) C(q 1 ) C(q 2 ) 1 F (θ) u θ (q 1, q 2, θ). f(θ) It is straightforward to show that the multi-product monopolist chooses q i (θ), i = 1, 2 and θ 0 10
to maximize θ θ 0 (Λ(q 1 (θ), q 2 (θ), θ)) f(θ)dθ. Implementability does not require that both q 1 and q 2 both be nondecreasing, but if both are nondecreasing then the allocations are implementable. Assuming that Λ is globally and strictly quasi-concave in (q 1, q 2 ), that Λ qi θ > 0 and that Λ(q 1, q 1, θ) is symmetric in (q 1, q 2 ), the allocations will each be nondecreasing. Hence, the optimal allocations are symmetric and satisfy Λ q1 (q m (θ), q m (θ), θ) = Λ q2 (q m (θ), q m (θ), θ) = 0 θ [θ 0, θ], and θ 0 is determined by Λ(q m (θ 0 ), q m (θ 0 ), θ) = 0 when such a root exists and θ 0 = 0 otherwise. Applying this preliminary result to our quadratic-uniform setting, we have our benchmark proposition. Proposition 1 The monopoly equilibrium allocation is given by q m (θ) = q fb (θ) ( θ θ) for θ [θ m 0, θ]. The marginal consumer under monopoly, θ m 0, is given by θ m 0 { = max 0, 1 ( θ q (0)) } fb. 2 We will see that both the multi-product monopoly allocation, q m (θ), and the monopoly participation region, [θ m 0, θ], depart from the equilibrium outcomes of the intrinsic and delegated common agency games due to competitive (indirect) externalities. 4 Indirect (residual) utility and incentive compatibility Before characterizing the equilibrium outcomes under intrinsic and delegated common agency games, it is useful to develop some general tools for competitive contracting situations with private information. Consider principal 1 s problem of designing the optimal price schedule, P 1 (q 1 ). For clarity, we proceed from the point of view of principal 1; given 11
symmetry, identical virtual profit functions, indirect utility functions, and best-response functions will arise in equilibrium. If u(q 1, q 2, θ) were additively separable in q 1 and q 2 i.e., u(q 1, q 1, θ) = g(q 1, θ)+h(q 2, θ) then firm 1 could proceed as in the monopoly case, suppressing the component of utility related to q 2. By assumption, u q1 q 2 0 (i.e., γ 0), so firm 1 cannot ignore the presence of q 2 in the consumer s utility function. Furthermore, q 2 is endogenous, as it depends upon (q 1, θ) and the offered nonlinear price schedules. Herein lies the difficulty of multi-principal problems. There is a useful method to deal with this endogeneity problem, however. Firm 1 can replace q 2 in the consumer s utility function with an indirect utility function that depends only upon q 1 and θ (and the offered scheduled, P 2 (q), which we suppress in the notation): v(q 1, θ) max q 2 Q 2 u(q 1, q 2, θ) P 2 (q 2 ). (In equilibrium, v(q, θ) will also apply to firm 2 s maximization program, so we do not distinguish this function, or firm 1 s virtual profit function Λ(q 1, θ), below, by firm.) Given this residual utility representation, v(q, θ), we can proceed as if we were in the classic monopoly setting. There are two remaining difficulties: determining whether or not v(q 1, θ) satisfies the single-crossing property, and whether or not it satisfies sufficient regularity conditions to ensure that the associated common-agency virtual surplus function for each firm, Λ(q 1, θ) v(q 1, θ) cq 1 (1 θ)v θ (q 1, θ), is strictly quasi-concave in q 1 and supermodular in (q 1, θ) in equilibrium. 4.1 Single-crossing property and v(q, θ) When the goods are demand complements (i.e., u q1 q 2 (q 1, q 2, θ) > 0), it is straightforward to demonstrate that the single-crossing property in v(q i, θ) is satisfied. When u q1 q 2 (q 1, q 2, θ) < 0 and the goods are demand substitutes, however, this is not true for any nonlinear price 12
schedule offered by the other firm. To understand this, define the optimal level of consumption of q 2, conditional on θ and consuming q 1 : q 2(q 1, θ) = arg max q 2 Q 2 u(q 1, q 2, θ) P 2 (q 2 ). Providing P 2 is continuous over the compact set Q 2, the correspondence q2 is well-defined. As we will see, in the multi-principal equilibrium, u(q 1, q 2, θ) P 2 (q 2 ) will be strictly quasiconcave in q 2 and therefore q2 (q 1, θ) will be a continuous function. Because u qi θ > 0, it follows that q2 (q 1, θ) is nondecreasing in θ. For the case of complements, γ < 0, it must also be that q2 (q 1, θ) is nondecreasing in q 1 ; while in the case of substitutes, γ > 0, it follows that q2 (q 1, θ) is nonincreasing in q 1. In either setting, the monotonicity of q2 (q 1, θ) implies that it is almost everywhere differentiable in each argument. By the envelope theorem, v θ (q 1, θ) = u θ (q 1, q 2(q 1, θ), θ) > 0. Differentiating again, v q1 θ(q 1, θ) = u q1 θ(q 1, q 2(q 1, θ), θ) + u q2 θ(q 1, q 2(q 1, θ), θ) q 2 (q 1, θ) q 1. In the case of demand complements, both terms are positive, and the single-crossing property is satisfied. In the case of demand substitutes, unfortunately, the first term is positive while the second term is negative. Single-crossing only obtains if the substitution effect is not too great. Given that preferences are quadratic, this amounts to the requirement that q 2 (q 1,θ) q 1 > 1. Because q2 (q 1, θ) is an equilibrium construction, we will proceed as if the single-crossing property is guaranteed and we will check ex post that the equilibrium does indeed generate the requisite single-crossing. Note that if v(q, θ) did not satisfy singlecrossing, we would be left with an intractable optimization problem, just as in the classic monopoly setting. 13
4.2 Quasi-concavity and v(q, θ) We must also be careful that the shape of v(q, θ) does not introduce convexities into the firm s optimization program or generate nondecreasing allocations that fail to be implementable. As with single-crossing, we will proceed with the assumption that the relevant virtual profit function, Λ(q 1, θ), is strictly quasi-concave in q and supermodular in (q 1, θ) over the relevant range of q 1, and then we will check ex post that this is indeed the case for the candidate equilibrium. Note that the elimination of q 2 via q2 (q 1, θ) and the indirect utility function, v(q, θ), introduces a subtle problem when the nonlinear price schedule is only offered over the set of equilibrium allocations, Q i = [q i (θ 0 ), q i ( θ)], rather than over a larger set of outputs. The problem is that an outward kink is introduced in the firm s virtual profit function, Λ(q i, θ), which may render the first-order condition Λ qi (q 1 (θ), q 2 (θ), θ) = 0 invalid for determining the optimum. To understand this problem with kinks, assume that firm 2 only offers outputs that are chosen by some type in equilibrium; i.e., Q 2 = [q 2 (θ 0 ), q 2 ( θ)]. In the case of demand substitutes, for each θ there is a critical value of q 1 denoted by ˆq 1 (θ) such that for q 1 < ˆq 1 (θ), q2 (q 1, θ) = q 2 ( θ). If principal 1 offers such a small quantity to the common agent, it follows that the agent prefers to consume the highest possible quantity from principal 2. The function q2 (q 1, θ) is thus constant in q 1 on the left neighborhood of ˆq 1 (θ). For q 1 ˆq 1 (θ), q2 (q 1, θ) belongs instead to Q 2 = [q 2 (θ 0 ), q 2 ( θ)] (at least for q 1 not too high so that the common agent does not choose the lower bound of Q 2 ) and that function is strictly decreasing in q 1 to the right of ˆq 1 (θ). In particular, this means that the slope of firm 1 s virtual profit function is given by Λ q (q 1, θ) = u q (q 1, q 2, θ) C (q 1 ) u q (q 1, q 2 ( θ), θ) C (q 1 ) 1 F (θ) f(θ) ( ) u q1 θ(q 1, q2, θ) + u q 2 θ(q 1, q2, θ) q 2 (q 1,θ) q 1, if q 1 > ˆq 1 (θ); 1 F (θ) f(θ) u q1 θ(q 1, q 2 ( θ), θ), if q 1 < ˆq 1 (θ). Because q 2 (q 1,θ) q 1 < 0 in the case of substitutes, as one lowers q 1 across the critical point 14
ˆq 1 (θ), the marginal virtual profit of q 1 increases. Thus, Λ( ) is locally non-concave in q when only equilibrium consumption allocations are offered by firm 2. To understand the consequence of this kink, suppose that we have a symmetric and differentiable equilibrium: q 1 (θ) = q 2 (θ) = q(θ) and q2 (q(θ), θ) = q(θ) for all θ. For an agent of type θ, we therefore must have the kink occur precisely at the equilibrium allocation: ˆq( θ) = q( θ). Now consider a downward deviation by firm 1. By offering q( θ) ε, the rent reduction to the highest type is fully obtained by principal 1. In some circumstances, this deviation is attractive and the first-order approach fails. In this case, out-of-equilibrium offers become essential for the existence of pure-strategy, differentiable equilibria. 11 Fortunately, by extending the offered consumption domain beyond the equilibrium value q i ( θ), the failure of concavity around q i ( θ) can be pushed out to a region of Λ(q, θ) where the virtual profit is sufficiently low that the kink can generate only a local maximum. Also fortunate, close inspection indicates that the corresponding kink in Λ(q 1, θ) at the low end of consumption (for q 1 in the neighborhood defined by q2 (q 1, θ) = q 2 (θ 0 )), is inward, and so local concavity is unaffected. Hence, for the case of substitutes, we need only consider extensions at the high end of consumption. A similar analysis of the case of complements reveals that the outward kink in Λ(q 1, θ) also occurs for q 1 in the neighborhood defined by q2 (q 1, θ) = q 2 ( θ), but not for the neighborhood given by q2 (q 1, θ) = q 2 (θ 0 ). Thus highend extensions in the nonlinear price schedule entirely address the problems of concavitydestroying kinks in Λ(q 1, θ). In the present paper, we accomplish this by considering tariffs extended over the positive real line, so there is no difficulty with kinks in the principal s virtual profit functions. 12 To summarize, we can proceed by constructing candidate equilibria with extended tariffs, assuming that v(q, θ) satisfies single-crossing and is sufficiently regular such that the firms virtual profit functions are strictly quasi-concave and supermodular in (q i, θ). We then check that the candidate equilibria induce these conditions in the indirect utility function, v(q, θ). 11 As an aside, this is an example where the restriction of each firm to direct revelation mechanisms over the communication space of θ [0, θ] is with loss of generality. See Martimort and Stole (2002) for more details of this failure, and approaches to rectifying the problem. 12 Alternatively, one can show that if the underlying concavity in the agent s utility is sufficiently great, (in the present context, the precise condition is γ ( β, 1 β) so the goods cannot be overly strong substitutes), 2 these kinks are not problematic because they are weighted by 1 F (θ) f(θ) and the concavity of the full-information program swamps the effects of the kink for θ < θ. In such a case, the corresponding virtual surplus function is strictly quasi-concave in q, even though it fails to be concave in the neighborhood of the kink. 15
5 Intrinsic agency Recall that in the intrinsic agency game, the agent must choose to consume either from both firm s price schedules, or from neither firm. Choosing to consume from a single firm is either ruled out by law or by an assumption on preferences. Given that preferences are quadratic and the agent s type is uniformly distributed, we will see that a differentiable equilibrium exists in which q i (θ) is a linear (affine) function of θ. Looking at the consumer s maximization program, it is immediate that such linear allocations arise if and only if each firm chooses a quadratic tariff schedule defined over the set of q i that are chosen in equilibrium. Hence, we may proceed by characterizing firm 1 s best response, given that firm 2 offers a quadratic price schedule. We then demonstrate that firm 1 s best response is also a quadratic schedule. Using symmetry and firm 1 s best response function (mapping from firm 2 s quadratic price schedule to firm 1 s quadratic price schedule), we find the equilibrium fixed point in nonlinear price schedules. It is important to note at the outset some issues regarding equilibrium selection. First, we exclude from consideration the trivial equilibrium that always exists in which each firm chooses to sell nothing by posting arbitrarily high prices. This equilibrium represents a coordination failure by the principals and is largely uninteresting. Second, we should note that by restricting attention to quadratic price schedules over the positive real line (rather than just over the equilibrium set of outputs), we may be implicitly selecting an equilibrium level of rents to the agent. This is indeed true when we consider delegated agency in section 6. Fortunately, this equilibrium selection is not an issue for the present case of intrinsic agency. Suppose that firm 2 s offered tariff is P 2 (q 2 ) = a 0 + a 1 q + a 2 2 q 2, defined over the set Q = [0, ). It follows that q2 (q 1, θ) is linear in its arguments and the single-crossing property of v(q 1, θ) reduces to the following requirement: 2 v(q 1, θ) q θ = 1 + a 2(β + γ) β + a 2 (β 2 γ 2 ) 0. More simply, the single-crossing property is satisfied if and only if the equilibrium tariff component, a 2, satisfies a 2 1 β + γ, (1) 16
or in words that the nonlinear tariff is not too concave. We will check that the equilibrium a 2 does indeed satisfy this condition. Putting aside issues of single-crossing, we construct firm 1 s best-response function as though firm 1 were a monopolist and the consumer s utility function were given by v(q 1, θ). Quadratic preferences and quadratic equilibrium price schedules ensure that v(q, θ) is itself quadratic, and hence we have sufficient regularity to guarantee that each firm s virtual profit function is strictly quasi-concave and supermodular in (q i, θ) whenever v(q, θ) satisfies single-crossing. If, in addition, 2v θ (q, θ) ( θ θ)v θθ (q, θ), then the virtual profit function is nondecreasing in θ as well, Λ θ (q, θ) 0; this later assumption will be checked ex post to ascertain the concavity of the θ 0 -optimization program. Applying the results from the monopoly model, define Λ(q 1, θ) v(q 1, θ) cq 1 1 F (θ) v θ (q 1, θ). f(θ) If q 1 (θ) maximizes this function pointwise in θ and the resulting q 1 (θ) is nondecreasing (which is guaranteed by v(q 1, θ) quadratic and satisfying single-crossing), then q 1 (θ) is a best-response allocation to firm 2 s nonlinear price schedule. The optimal participation cutoff for intrinsic agency, θ I 0, is determined by the requirement that Λ(q 1 (θ I 0 ), θi 0 ) = 0 (and the second-order condition, Λ θ (q, θ) 0) if such a root exists in [0, θ], and θ I 0 = 0 otherwise). Having computed q 1 (θ) and θ0 I, firm 1 constructs its nonlinear price schedule from the differential equation P 1(q 1 (θ)) = v q (q 1 (θ), θ), and the boundary condition that v(q 1 (θ0 I), θi 0 ) = P 1(q 1 (θ0 I )). Solving the differential equation and imposing symmetry generates a fixed point that identifies the equilibrium price schedule parameters a 1 and a 2. Given a 1 and a 2, the boundary condition determines a 0 (and simultaneously, θ I 0 ). We need only to check that a 2 is sufficiently large so as to satisfy the single-crossing property given above in equation (1). In the appendix, we prove the following proposition and demonstrate that the equilibrium value of a 2 does indeed satisfy single-crossing. Proposition 2 Under the intrinsic common agency game, there exists an equilibrium with quadratic price schedules where the (symmetric) equilibrium allocation of output is linear in 17
type and, for all participating consumer types, satisfies ( ) ( ) q I (θ) = q fb 4γ (θ) ( θ θ) 1 β + = q m 4γ (θ) + ( θ θ) β 2 + 8γ 2 β +, β 2 + 8γ 2 for all θ [θ0 I, θ]; the marginal intrinsic consumer, θ0 I, is determined by the root of Λ(q I (θ I 0), θ I 0) = 0 when θ I o [0, θ], and θ I 0 = 0 otherwise. The marginal intrinsic consumer earns zero consumer rent: U(θ I 0) = 0. As noted in Martimort (1992) and Stole (1991), if γ = 0, the utility function is fully separable in q 1 and q 2, and the allocation function q I (θ) is identical to the multi-product monopolist s optimal allocation, q m (θ). When γ < 0 and the goods are complements, q I (θ) < q m (θ) < q fb (θ) for all θ < θ, and thus the distortion is greater with competing principals relative to the multi-product monopolist; when γ (0, β) and the goods are substitutes, q fb (θ) > q I (θ) > q m (θ) for all θ < θ and the consumption distortion is smaller with competing principals relative to the multi-product monopolist. In fact, as γ β, the goods become perfect substitutes, and q I (θ) converges pointwise to q fb (θ). We summarize these statements in a corollary. Corollary 1 Among purchasing consumers, when goods are demand substitutes, the downward consumption distortion is reduced by competition among the firms; when goods are demand complements, the downward consumption distortion is increased by competition among the firms. The result for complements is similar in spirit to results going back to at least as far as Cournot (1838), who observed that competition when goods are complements actually worsens welfare, as each firm separately introduces a distortion that reduces the demand for the other firm s product, and hence its profitability. An integrated monopoly, on the other hand, would introduce a smaller distortion. Remarkably, the same is true when strategy spaces are larger and allow for nonlinear price schedules. Cournot s ideas generalize to 18
strategy sets larger than uniform prices. This is the primary insight of Martimort (1992) and Stole (1991). For the purpose of this paper, we are most interested in the effect of competition on participation by consumers. Not only does common agency affect the allocation functions, q(θ); it also affects the set of purchasing consumers. To see this, note that in equilibrium Λ(q, θ) v(q, θ) cq 1 F (θ) v θ (q, θ) f(θ) = [u(q, q, θ) P (q)] cq 1 F (θ) u θ (q, q, θ) f(θ) = Λ(q, q, θ) [P (q) cq] Λ(q, q, θ). Note that the price schedule, P (q), is the equilibrium schedule under intrinsic common agency. The last inequality follows from the fact that P (q) cq; it will be a strict inequality whenever P (q) > cq since in equilibrium, the principals serve only customers for whom they make a positive profit. This last inequality will be an equality for the case where P (0) = 0 and q(θ 0 ) = 0. Using the result that Λ(q, θ) Λ(q, q, θ), in tandem with the facts that q m (θ) = arg max q Λ(q, q, θ) and q m (θ) q I (θ) whenever γ 0, it follows that Λ(q m (θ), q m (θ), θ) > Λ(q I (θ), q I (θ), θ) Λ(q I (θ), θ). Since at θ I 0 a principal makes a positive profit (zero if θi 0 is interior), we must haveλ(qi (θ I 0 ), θi 0 ) 0. It follows that, whenever γ 0, it must be that Λ(q m (θ0 I), qm (θ0 I), θi 0 ) > 0. From this statement we conclude that θ m 0 < θi 0 if θi 0 > 0 because d dθ Λ(qm (θ), q m (θ), θ) = Λ θ (q m (θ), q m (θ), θ) > 0 and θ m 0 = 0 if θi 0 = 0. Strikingly, the monopolist introduces a smaller participation distortion than competing firms under intrinsic agency, regardless of whether the goods are substitutes or complements on the margin (i.e., γ > 0 or γ < 0). To summarize: Proposition 3 Consumer participation is at least as great under monopoly as under competing principals in the intrinsic agency setting. θ m 0 θ I 0, 19
and participation is strictly greater under monopoly when a positive measure of consumers choose not to purchase under intrinsic agency and γ 0 (i.e., the goods are either substitutes or complements). This result is surprising given that we normally think of competition as increasing efficiency, except when the goods are demand complements. Here, on the other hand, is a setting where inefficient exclusion is more pronounced under competition even when the goods are substitutes on the intensive margin. Upon reflection, however, the participation distortion is consistent once we recognize that intrinsic agency is equivalent to unfettered delegated agency with goods that are perfect complements at the base level (extensive margin). What this result tells us is that perfect complementarity on the extensive margin (base units of consumption) implies that competition generates greater extensive (participation) distortions relative to monopoly. The nature of preferences on the intensive margins is not relevant. In this sense, perfect complementarity on the extensive margin is the source of participation inefficiencies, whether this complementarity is legally required (as in a setting with multiple authorities regulating a single firm) or naturally arises (as in the case of monopoly hardware and software vendors). 6 Delegated agency In the present version of this paper, we focus on delegated agency games in which principals cannot make their contract depend upon the relationship the agent has with the rival principal. [Results for the discriminatory form of delegated agency, in which each principal s contracts can be made conditional on the acceptance decision of the agent with the rival, will be available in the a later version of this paper.] As with intrinsic common agency, we can use the identical virtual surplus function, Λ(q, θ) v(q, θ) cq 1 F (θ) v θ (q, θ), f(θ) to determine the equilibrium allocation, q D (θ). Nothing has changed on the (intensive) margin, so allocations are unchanged for all participating consumers. The consumer s outside option under delegated common agency has changed, however, and so we will find that 20
the firms equilibrium participation choices, θ0 D, and the marginal consumer s rent, U(θD 0 ), will frequently differ from the outcomes under either monopoly or intrinsic common agency. Consider first the simplest case of perfect (extensive) substitutes. When the two goods are perfect substitutes on the extensive (base) margin, the firm are Bertrand competitors in utility space. It is immediate that there is a unique equilibrium where each principal offers marginal cost pricing and earns zero profits and the agent chooses one principal (randomly) with whom to exclusively contract. When the goods are not perfect substitutes on the extensive (base) margins, however, we may discard the possibility of an exclusive agency equilibrium outcome. For such an exclusive-dealing outcome to be an equilibrium, following standard Bertrand arguments, it must be that each principal makes zero profits for each type served, otherwise one principal could respond with a modified tariff and steal some positive measure of profitable consumer types from the other principal. It follows immediately, therefore, that P (q) = cq is the only possible equilibrium contract. It is equally immediate, however, that this contract encourages consumers to purchase from both firms when the goods are not perfect substitutes. Hence, there does not exist a pure-strategy exclusivedealing equilibrium outcome. While this simple argument does preclude the possibility that there may exist equilibria with a region of common agency and a region of exclusive dealing, we demonstrate below using a similar argument that hybrid equilibria also fail to exist. Hence, the only equilibrium possibility is for all types either to contract with both principals or to reject participation entirely. The off-the-equilibrium path possibility of exclusive dealing, however, does change the nature of the equilibrium outcome in an important and interesting way, as this possibility fundamentally alters the participation constraint relative to intrinsic agency. Recall that the participation constraint under intrinsic agency was simply: v(q i (θ), θ) P i (q i (θ)) 0, for i = 1, 2. Under delegated agency, the consumer has the option of consuming from both firms, neither firm, or either firm exclusively. This increased set of options raises the consumer s requirements for participation. Now, the agent s participation constraint vis-a-vis firm i is v(q i (θ), θ) P i (q i (θ)) max{0, v(0, θ)}, 21