Participation constraints in discontinuous adverse selection models
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Martimort, David; Stole, Lars A. Article Participation constraints in discontinuous adverse selection models Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Martimort, David; Stole, Lars A. (2022) : Participation constraints in discontinuous adverse selection models, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 17, Iss. 3, pp. 1145-1181, https://doi.org/10.3982/TE3030 This Version is available at: https://hdl.handle.net/10419/296382 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/
Theoretical Economics 17 (2022), 1145–1181 1555-7561/20221145 Participation constraints in discontinuous adverse selection models Dav i d Martimort Paris School of Economics and EHESS Lars A. Stole Booth School of Business, University of Chicago We present a set of necessary and sufficient conditions for a class of optimal control problems with pure state constraints for which the objective function is linear in the state variable but the objective function is only required to be upper semicontinuous in the control variable. We apply those conditions to economic environments in contract theory where discontinuities in objectives prevail. Examples of applications include nonlinear pricing of digital goods and nonlinear pricing under competitive threat. Keywords. Optimal control, nonsmooth optimization, convex analysis, typedependent participation constraints, principal–agent models. JEL classification. D82, D86. 1. Introduction The textbook treatment of optimal screening contracts typically takes the agent’s outside option as a fixed constant, independent of type.1More complex settings, which allow for competition by rival principals, nontrivial ownership rights on productive assets, and type-dependent fixed costs require a departure from this restrictive assumption. Lewis and Sappington (1989) initiated the seminal study of screening contracts in this more general setting by constructing the solution to a class of optimal control problems with type-dependent participation constraints. This class of problems was further enriched by Maggi and Rodriguez-Clare (1995) with the most general statement of the problem and its solution culminating in the analysis offered by Jullien (2000). David Martimort: [email protected] Lars A. Stole: [email protected] We are especially thankful to John Birge for many helpful discussions. We also thank Simon Board, three referees for very useful comments and suggestions, and seminar participants at the 2019 Summer Meetings of the North American Econometric Society (Seattle) and the European Econometric Society (Manchester). A less general version of the main theorem in this paper appeared in an earlier, unpublished note, “Necessary and sufficient conditions for nonsmooth linear-state optimal control problems” (2009). The present paper provides the more general result along with a geometric intuition for its proof and several relevant applications. The usual disclaimer applies. 1See Laffont and Martimort (2002, Chapter 3) for instance. ©2022 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE3030
1146 Martimort and Stole Theoretical Economics 17 (2022) These techniques have allowed modelers to apply the optimal contracting paradigm to more general economic contexts, unveiling new features of optimal contracts. Applications have spread through many fields including the design of nonlinear prices under the threat of bypass (Curien, Jullien, and Rey (1998), Biglaiser and Mezzetti (1993)), competition in nonlinear prices (Martimort and Stole (2009), Stole (1995,2003), Calzolari and Denicolo (2013)),tradepolicyinopeneconomiesBrainard and Martimort (1997), regulation of privately-owned monopolies Caillaud (1990), and optimal contracting under liability constraints Ollier and Thomas (2013) and in dynamic contracting environments Deb and Said (2015), to name a few examples. Unfortunately, the existing techniques also have their own limits. In particular, the need for tractability has led authors to restrict their analysis to economic environments which are sufficiently smooth. In many circumstances, such as when firms face nontrivial fixed costs or sunk investments, the environment is inherently discontinuous. In other settings, such as when principals compete against one another using payment schedules, equilibria may emerge, which exhibit discontinuities in each player’s payoff function Martimort and Stole (2015). More broadly, discontinuities in principal-agent problems may directly come from the surplus function in the principal’s objective (Section 3.1) or, in a more subtle manner, from nondifferentiability in how the agent’s information rent depends on some control variables (Section 3.2). In such cases, we are left uncertain about the consequences of discontinuities for optimal contracts and the generality of results when environments (and equilibria) are not assumed to be smooth apriori. Important economic insights may go unnoticed because of such restricted attention. Developing the required techniques for nonsmooth environments and showing how they apply in practice are the purposes of this paper. First, we present a set of necessary and sufficient conditions for a class of optimal control problems with pure state constraints and an objective function (linear in the state variable) that may exhibit kinks or discontinuities in the control variable. Second, we apply these techniques to quite natural contracting environments where existing techniques have previously restricted focus. Examples of applications include nonlinear pricing of digital goods and nonlinear pricing in the presence of competitive threat. Section 2presents our main result: A theorem that characterizes solutions to optimal control problems in which the objective function is only required to be upper semicontinuous. This theorem builds on earlier work by Vinter and Zheng (1998), but refines its application to the case of quasilinear objectives, which is prevalent in contract theory. While Vinter and Zheng (1998) focus on necessary conditions for optimality, we prove that these conditions are also sufficient in our context. We also discuss to what extent this theorem extends the existing literature and especially the work by Jullien (2000)under much weaker conditions. Section 3develops applications of our framework, deriving economic insights that would not have been available without the use of new techniques. The contains the proofs of our theorem and of the various results related to our economic applications, but also provides a brief overview of optimization techniques in nonsmooth environments, illustrating the geometric intuition for our main theorem.
Theoretical Economics 17 (2022) Participation constraints 1147 2. The theorem We will consider control problems in which the state variable, u, is restricted to be an absolutely continuous function on the interval =[θ,θ];AC (,R)denotes the set of such functions. As a motivation, in the context of principal-agent models the state variable is typically the agent’s information rent as a function of his type. In such settings, incentive compatibility naturally implies absolute continuity.2We focus attention on problems in which that state variable must satisfy a nonnegativity constraint: u(θ)≥0∀θ∈≡[θ,θ].(1) Using again our motivation of principal-agent problems, the nonnegativity constraint (1) corresponds to a participation constraint that ensures the agent will accept an offer rather than take an outside option, which is here normalized at zero. When the state variable uis both absolutely continuous and nonnegative, it is said admissible. We are interested in the following pure-state control program: (P): Maximize u∈AC(,R)θ θsθ,˙ u(θ)−u(θ)f(θ)dθ subject to (1), where we use the standard control-theoretic notation of ˙ u(θ)to denote the derivative of u(θ)with respect to θ. Although expressed in abstract terms at this stage, readers accustomed with the principal-agent literature will recognize the structure of such problem. The integrand features the familiar rent-efficiency trade-off. Below, we will push this analogy even further by demonstrating how principal-agent models expressed in more traditional terms can be transformed so as to apply the general methodology we now present. Sections 3.1 and 3.2 provide explicit examples of this transformation. On the technical side, we only assume that the surplus function s(θ,v)(here expressed in terms of a control variable v) is an upper semicontinuous function of vfor all θ, bounded from above, and that f(θ)is a positive, bounded function giving rise to an absolutely-continuous definite integral F(θ)≡θ θf(θ)dθ. Without loss of generality, we normalize fsuch that F(θ)=1, allowing us to interpret Fas a continuous probability distribution and fas its associated density for our applications. We also make a minimal technical assumption that s(·,·)is L×B-measurable, where Ldenotes the set of Lebesgue measurable subsets of and Bis the set of Borel measurable subsets of R. We define the integrand for program (P)as L(θ,u,˙ u)≡s(θ,˙ u)−uf (θ).Weshould make clear that the key restriction we have placed on (P)is that, for any θand ˙ u,the maximand is a linear function of the state variable u. As we will see below in our applications, this linearity is found in a number of economic problems, especially in contract theory where agents are risk-neutral and payoffs are linear in money. The function s(θ,˙ u)/f (θ)can there be viewed as a surplus function while uis the share of this surplus that is captured by the agent—his information rent. 2See Milgrom and Segal (2002) and Carbajal and Ely (2013). As an example, if the agent’s utility is continuously differentiable in type with a uniformly-bounded derivative, then the agent’s indirect utility function is necessarily Lipschitz continuous (and therefore absolutely continuous).
1148 Martimort and Stole Theoretical Economics 17 (2022) The linearity restriction is the primary source of many sharp results in the analysis that follows, including the ability for us to relax the continuity of s, to characterize the solution by means of a simple generalized gradient condition, and to verify that necessary conditions for optimality are also sufficient. Indeed, nonsmooth techniques are particularly useful if one can find the solutions of such control problems as pointwise optima. This is where the assumption of linearity of the maximand in uprovides purchase. Linearity allows for such a pointwise simplification—a well-known result in familiar quasilinear screening models (Myerson (1981, Lemma 3), Baron and Myerson (1982, Lemma 2), Laffont and Martimort (2002, Chapter 3)).3Complications arise if the agent’s reservation utility is type-dependent, but the basic intuition remains. Linearity allows us to separate incentive and participation concerns4from any nonsmoothness of the surplus function, the latter of which is addressed by using the super-differential of the concave envelope of s(θ,v)in place of the gradient. To better isolate the role of nonsmoothness, consider the case where the integrand is reduced to L(θ,u,v)≡s(θ,v). In the contract theory applications below, this case would correspond to a scenario of complete information in which the principal can fully extract the agent’s rent. The optimization problem so constructed can be solved pointwise by means of standard techniques for nonsmooth problems.5Any solution, v∗(θ), must satisfy the following pair of conditions: co(s)θ,v∗(θ)=sθ,v∗(θ)and 0 ∈∂co(s)θ,v∗(θ),(2) where co(s)(θ,v)is the concave majorization of the function sover vevaluated at (θ,v), and ∂vco(s)(θ,v)is the set of supported gradients of the majorized function evaluated at (θ,v). This is a nonsmooth generalization of the familiar first order necessary condition for optimality. Intuitively, the maximum of an upper semicontinuous function must also coincide with its concave majorization (the first part of (2)). If the maximand is differentiable at the maximizing point, then the derivative is necessarily zero; if it is not differentiable, it must nonetheless support a zero gradient (the second part of (2)). We briefly review this idea in an Appendix.6The solution to our original program (P) will differ from v∗in (2) as a result of the addition of the linear term −uf (θ)in the Lagrangian. Condition (5) below indicates how the solution needs to be modified. As familiar from the principal-agent literature, u(θ)will have to maximize a virtual surplus function obtained by combining the impacts of participation and incentive constraints. Heuristic Approach. Following Jullien (2000), one might address problem (P)as follows. First, one could add a Lagrange multiplier μ(dθ)to the participation constraint (1) 3When quasilinearity is not assumed, this familiar trick no longer works as, for instance, in the wellknown model of the optimal taxation due to Mirrlees (1971). 4Incentive and participation concerns are captured by the term F(θ)−γ(θ)in the optimality condition (5)below. 5See Section Abelow for details. 6Appendix Aprovides a brief discussion and survey of nonsmooth, convex analysis. Because we are focused on maximization, our tools rely on concave majorizations (i.e., the minimal concave envelope of a function) rather than convex minorizations. Likewise, we are interested in the set of gradients of a concave function (superdifferentials) rather than the set of gradients of a convex function (subdifferentials).
Theoretical Economics 17 (2022) Participation constraints 1149 with the complementarity slackness condition μ(dθ)=0ifθ∈˜ θ|u(˜ θ)>0. Second, one could then form a Lagrangian as θ θsθ,˙ u(θ)−u(θ)f(θ)dθ +θ θ u(θ)μ(dθ). Third, with a simple integration by parts, the integrand could be written as θ θsθ,˙ u(θ)+F(θ)−γ(θ)˙ u(θ)dθ where the adjoint function γ(θ)=[θ,θ)μ(d˜ θ), is right continuous, strictly increasing at points where (1) is binding, and satisfies the boundary conditions γ(0)=0, γ(1)=1. These characteristics allow us to identify γwith a distribution function. Finally, pointwise optimization implies the optimality condition ˙ u(θ)∈argmax v s(θ,v)+F(θ)−γ(θ)v. Two difficulties arise with this simplistic approach. The first one is merely technical and puts conditions on the Lagrange multiplier. Indeed, integrating by parts requires that μ(dθ)lies in the dual space of nonnegative functions in AC(,R); i.e., μ(dθ)must be the “derivative” of a function of bounded variation. The second difficulty is that, once we proceed to pointwise optimization, we may have to deal with a nonsmooth objective since s(θ,v)is only required to be upper semicontinuous. The optimality conditions have to be expressed by means of tools imported from nonsmooth, convex analysis. Main Result. We now present our main result for this class of problems. Theorem 1. uisasolutiontoprogram(P)if and only if uis admissible and there exists a probability measure μdefined over the Borel subsets of with an associated adjoint function, γ:→[0, 1],definedbyγ(θ)=0and γ(θ)=[θ,θ) μ(d˜ θ),for θ>θ, such that the following conditions are satisfied: θ θ u(˜ θ)μ(d˜ θ)=0, (3) co(s)θ,˙ u(θ)=sθ,˙ u(θ)for a.e. θ∈,(4) 0∈F(θ)−γ(θ)+∂vco(s)θ,˙ u(θ)for a.e. θ∈.(5)
1150 Martimort and Stole Theoretical Economics 17 (2022) The conditions in Theorem 1are similar to those of Theorem 1 in Jullien (2000). In both theorems, necessary and sufficient conditions are stated in terms of a probability measure, which serves to express a “complementary slackness condition” (3)anda first-order optimality condition (5). Moreover, both theorems use a similar condition to establish the continuity of ˙ u(θ)in the solution to (P). In contrast, the theorem in Jullien (2000) relies on results from Seierstad and Sydsaeter (1987, Theorems 2 and 3, Chapter 5), which use the stronger hypothesis that s(θ,v)is twice continuously differentiable in v. Our contribution is to demonstrate the force and the broader validity of these conditions for problems with integrands that are only upper semicontinuous through the use of nonsmooth analysis. As in Jullien (2000), the measure μ(dθ)stems for the shadow cost of the participation constraint (1)aroundθ. The adjoint function γ(θ)can thus be interpreted as the sum of these shadow costs for all inframarginal types. It is thus nondecreasing and constant on any open interval where the participation constraint is slack. Replacing the right-hand side of (1) uniformly by <0 for all ˜ θ≤θwould relax the optimization problem and increase its value by γ(θ). The adjoint function γso constructed is right continuous. Since the probability measure μmay have mass points where the participation constraint begins to bind, γ(θ) may have upward jumps at such points. This possibility may only arise at a countable number of points since any increasing function is almost everywhere differentiable. We now investigate under which conditions the optimal solution remains continuously differentiable. Proposition 1. If V(θ,σ)≡argmax v∈R s(θ,v)+F(θ)−σv(6) is single-valued and continuous over the domain (θ,σ)∈×[0, 1], then the solution u to (P)is continuously differentiable. That V(θ,σ)is single-valued and continuous is implied by strict concavity of s(θ,·). It is also implied by the weaker condition in Jullien (2000, Assumption 2) that s(θ,v)− (σ−F(θ))vis strictly quasiconcave in vfor any σ∈[0, 1]. Together with the stronger hypothesis that s(θ,v)is twice continuously differentiable in v, Lemma 7 in Jullien (2000, p. 32) then provides a smooth version of (5) by means of a first-order condition, namely F(θ)−γ(θ)+∂s ∂vθ,˙ u(θ)=0a.e. 7 Proposition 1is more general since it allows for the possibility that s(θ,v)fails to be continuous in vat its maximum. That uis continuously differentiable captures the fact that often in applications the optimal control, say output in a principal-agent context, is itself continuous. Examples abound in the literature where continuity is not optimal if the virtual surplus is 7Galbraith and Vinter (2004) provide also alternative conditions ensuring Lipschitz-continuity of the optimal control.
Theoretical Economics 17 (2022) Participation constraints 1151 not strictly concave. Models of bypass and regulation under the threat of entry due to Caillaud (1990), Laffont and Tirole (1990), and Curien, Jullien, and Rey (1998)), as an illustration, feature discontinuities in the optimal control because the bypass technology entails a fixed cost. In these papers, the authors generally deal with the discontinuities by using details specific to the setting to guess where the binding participation constraints lie (sometimes a complex task in itself), then constructing the agent’s profile of information rent given the conjectured constraint set, and finally constructing the principal’s nonconcave virtual surplus given the agent’s conjectured rent profile—all before proceeding to optimization and confirming that the original conjecture was correct. This approach does not provide much guidance in settings with nonconcavities, especially when the discontinuities arise exactly where the marginal type’s participation constraint binds.8In such a case, participation constraints truly interact with nonconvexities. In contrast to previous papers, our approach is more direct: (i) construct the concave envelope of sand (ii) compute the adjoint and indirect utility functions, which satisfy the first-order condition and complementary slackness. While this approach still entails jointly solving for two objects, γand u, which can be a complicated task, it is considerably more methodical. A More Primitive Statement of the Problem. Principal-agent problems with typedependent participation constraints as studied in the path-breaking works of Lewis and Sappington (1989), Maggi and Rodriguez-Clare (1995), and Jullien (2000)areoftenexpressed under the form (P)below so as to make the nature of the agent’s outside option, ˆ U(θ), and its associated participation constraint more explicit: P: Maximize U∈AC(,R),qθ θ˜ sθ,q(θ)−U(θ)f(θ)dθ subject to ˙ U(θ)=gq(θ),θa.e., and U(θ)≥ˆ U(θ)for all θ∈. The control variable q, which is assumed to be measurable, is generally interpreted as a quantity vector that belongs to a feasible set Q⊆Rk.9The primitive surplus function ˜ s(θ,q)is defined over Q.10 The differential equation that defines ˙ U(θ)immediately 8Another source of possible discontinuities comes when the types distribution has mass points; an assumption that we have ruled out for simplicity. See Lewis and Sappington (1993) and Cremer, Khalil, and Rochet (1998) for applications to information gathering where a mass point of agents remains uninformed; Hellwig (2010) provides a more general treatment. 9Incentive compatibility requires additional monotonicity conditions to hold. For instance, if gθ(q,θ)≥ 0forall(q,θ), then incentive compatibility requires q(θ)to be nondecreasing in θ. Often such monotonicity conditions are handled in the literature by adding one state variable and an associated law of motion (see Guesnerie and Laffont (1984)). To illustrate, provided that q(θ)is absolutely continuous, one can add anewcontrolp(θ)=˙ q(θ)a.e., and impose the constraint p(θ)≥0. While this introduces multiple dimensions to the state space, we note that our analysis can readily be extended to multidimensional settings because Theorem 3 from Vinter and Zheng (1998), used in Appendix Bin our one-dimensional setting, applies to multidimensional state variables. That said, this approach has limits. In a nonsmooth framework where discontinuities are pervasive, imposing that q(θ)is absolutely continuous may be excessive. Ironing techniques in the spirit of Myerson (1981) and Toikka (2011) could be powerful in this context but their development in the nonsmooth case lies outside the scope of this paper. 10Note that the domain of definition of qcan be easily included into the objective to fit with the formalism of Theorem 1if we set ˜ s(θ,q(θ)) =−∞for q/∈Q.
1152 Martimort and Stole Theoretical Economics 17 (2022) follows from the well-known envelope condition for incentive compatibility. The participation constraint U(θ)≥ˆ U(θ)allows the agent’s outside option to vary by type. The program (P)can easily be transformed into the canonical program (P)we explore if ˆ U(·)is differentiable almost everywhere. To illustrate, set u(θ)=U(θ)−ˆ U(θ) and define s(θ,v)=max q∈Q˜ s(θ,q)f(θ)s.t. v=g(q,θ)−˙ ˆ U(θ). This reduction is particularly easy when gis itself a bijection between qand ˙ ufor all θ, which implies that the control qcan be expressed as a function of (v,θ), namely q(θ)= g−1 q(v+˙ ˆ U(θ),θ). In that case, substitution yields s(θ,v)=˜ sθ,g−1 qv+˙ ˆ U(θ),θf(θ), and Preduces to P. While it is possible that the function s(θ,v), obtained as a maximum over all controls that generate the same derivative ˙ u(θ), may be smoother than ˜ s(θ,q), it will typically fall short of satisfying the twice continuous differentiability requirement in Jullien (2000). 3. Applications This section shows the broad applicability of our approach by highlighting applications that require nonsmooth analysis. Our first application (Section 3.1) deals with nonlinear pricing of a digital good under the threat of competition by a low-quality fringe. Because the provider of a high-quality good needs to build capacity for extra services in discrete bundles, the cost function is discontinuous. This model illustrates an interesting intricacy. Avoiding the fixed cost requires leaving additional rent to loyal customers since, otherwise, they would switch to a competitive fringe. The discontinuity from the fixed cost of additional capacity thus determines the subset of types for which the participation constraint is binding. Our second example, developed in Section 3.2, is another model of nonlinear pricing where a buyer may split his purchases between an incumbent firm and a competitive fringe. The fringe has limited capacity but sells a perfect substitute to the incumbent’s product. We show that this possibility introduces a discontinuity in the surplus function simply because the buyer’s rent has a different slope depending upon whether or not he purchases from the fringe. The discontinuity is endogenously derived from demand considerations. 3.1 Nonlinear pricing by a digital firm Pricing for digital products, including internet services, online trading and advertising services, is complex. The first source of complexity comes from the specific cost structure of those goods. For an infrastructure of a given size, the marginal cost of service is zero while supramarginal blocks of infrastructure must be added discretely to satisfy higher demand. The second source of complexity comes from the fact that competing
Theoretical Economics 17 (2022) Participation constraints 1159 as in Section 3.1, discontinuities come from the demand side and, more precisely, from how the buyer responds to changes in contractual terms with a dominant firm. Model. A dominant firm produces a good at constant marginal cost c≥0without any additional capacity costs. On the demand side, consumer preferences are again characterized by (7)whereθ, uniformly distributed on [θ,θ](with θ−θ=1), is private information. Importing mutatis mutandis our earlier findings of Section 3.1 with k=0 and now a nonnegative marginal cost equal to c≥0, the monopoly solution (absent competition) consists in offering qm(θ)=S−2θ+θ−c=qfb(2θ−θ). A competitive fringe sells a perfect substitute to the incumbent’s product at price p>c. We recycle our previous notation , and define =p−c>0. Taking pto be equal to the competitive fringe’s unit cost, entry is inefficient in a first-best world. We denote by ˆ qthe quantity bought from the fringe. The fringe has a capacity constraint K>0andthusˆ q∈ˆ Q=[0, K]. The key difference with Section 3.1 is that consumers can now always purchase from the fringe if the incumbent charges a price, which is too high. Formally, if a consumer with type θchooses qunits from the incumbent, his indirect utility function becomes ˆ u(θ,q)=max x∈ˆ Q (S−θ)(q+x)−(q+x)2 2−px. Define the highest level of consumption from the incumbent firm, which induces consumption of Kfrom the competitive fringe as ˇ q(θ)≡S−θ−p−k, which we assume to be positive. For q≤ˇ q(θ), the consumer purchases ˆ q(θ)=Kunits from the competitive fringe; for q≥ˇ q(θ)+K, the consumer purchases q0(θ)=0from the fringe. Straightforward computations yield ˆ u(θ,q)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (S−θ)(q+K)−(q+K)2 2−pK if q≤ˇ q(θ), ˇ q(θ)+K2 2+pq if q∈ˇ q(θ),ˇ q(θ)+K, (S−θ)q−q2 2if q≥ˇ q(θ)+K. 15 15To better understand the shape of the indirect utility function ˆ u(θ,q), we may think of the consumer as choosing between consuming 0 or Kunits from the fringe. His indirect utility function would be the (nonconcave) maximum of two concave functions because of a discontinuous jump in the corresponding choice. The possibility of consuming any arbitrary amount within the interval ˆ Qconcavifies this indirect utility function and introduces a linear segment for intermediate consumption levels from the dominant firm.
1160 Martimort and Stole Theoretical Economics 17 (2022) In particular, had the consumer not bought from the incumbent, i.e., q=0, he would consume up to capacity from the fringe and get a payoff worth ˆ U(θ)≡ˆ u(θ,0 )=ˇ q(θ)K+K2 2.16 In what follows, we will assume that qm(θ)>K, (19) which ensures that the incumbent firm still wants to serve the type with the lowest possible demand θeven when that type consumes up to the fringe’s capacity. A Discontinuous Surplus Function. When the consumer instead buys qunits from the dominant firm at a nonlinear price T(q), a consumer with type θobtains U(θ)=max q∈Qˆ u(θ,q)−T(q). To import our general formalism, we again introduce the state variable u(θ)=U(θ)− ˆ U(θ). The participation constraint takes its usual form (1) while the standard envelope condition for incentive compatibility becomes ˙ u(θ)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ −q(θ)if q(θ)<ˇ q(θ), −ˇ q(θ)if q(θ)∈ˇ q(θ),ˇ q(θ)+K, −q(θ)+Kif q(θ)>ˇ q(θ)+K. (20) This relationship actually shows that ˙ u, taken as a function of q,isnotbijective. 17 This phenomenon captures the role of competition from the fringe. If, to facilitate rent extraction, the dominant firm is willing to slightly reduce the buyer’s consumption q(θ) when it takes values in (ˇ q(θ),ˇ q(θ)+K), the buyer can instead consume more from the fringe. The dominant firm wants to maximize the bilateral surplus of contracting with the buyer, ˜ s(θ,q)=ˆ u(θ,q)−ˆ u(θ,0 )−cq, net of the rent the buyer gets when purchasing exclusively from the fringe. That is, the dominant firm wishes to implement the allocation q(·)to maximize the expectation of ˜ sθ,q(θ)−u(θ)=vθ,q(θ)−v(θ,0 )−cq(θ)−u(θ). After replacing q(θ)by its expression in terms of ˙ u(θ)wherever such inversion is possible (i.e., using (20), where ˙ u(θ)=−ˇ q(θ)), and after straightforward computations (detailed in the proof of Proposition 4), we may express the gross surplus as s(θ,v)=−ˇ q(θ)+v−v2 2+Kδv≤−ˇ q(θ). (21) 16Since v(θ)<0, this right-hand side remains positive. 17Carbajal and Ely (2016) present an interesting model of behavioral consumers having loss aversion that has similar features.
Theoretical Economics 17 (2022) Participation constraints 1161 This surplus function is upper semicontinuous, has a downward jump discontinuity at v0(θ)=−ˇ q(θ), and is maximized at v1(θ)=−ˇ q(θ)−<−ˇ q(θ).Thisdownwardjump captures the fact that, when consumption from the dominant firm is too low (which means v(θ)=−˙ u(θ)), the bilateral surplus between the dominant firm and the customer diminishes by K, reflecting the opportunity cost of purchasing Kunits from the fringe. Proposition 4. Let ˜ θ=θ+and ˜ θ0=˜ θ+√2K. Suppose that ˜ θ0≤θand (19)holds. The following optimal consumption levels and adjoint functions satisfy the necessary and sufficient conditions for optimality of Theorem 1: q(θ)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ qm(θ)if θ∈[θ,˜ θ], arbitrary ∈ˇ q(θ),ˇ q(θ)+Kif θ∈(˜ θ,˜ θ0), qm(θ)−Kif θ∈[˜ θ0,θ]. (22) γ(θ)has a mass point at θ,μ({θ})=1. The participation constraint (1)isbindingatθ only. Types who are the most eager to buy (i.e., θ∈[θ,˜ θ)) are not tempted to switch to the fringe. They consume the same downward distorted monopoly quantity qm(θ)as what the incumbent firm would offer if the fringe was absent. This quantity nevertheless remains large enough to make it unattractive for the buyer to switch. For intermediate types in [˜ θ,˜ θ0), the monopolist loses some of his ability to screen. Any attempt to reduce consumption from the incumbent is entirely compensated by the buyer consuming more from the fringe. The incumbent’s sales are indeterminate but screening distortions are reduced to avoid switching. Finally, types in [˜ θ0,θ]react to the downward screening distortion offered by the incumbent by consuming up to capacity from the fringe. Although the discontinuity of s(θ,v)bears some similarity with that analyzed in Section 3.1, the characterization of the solution is rather deferent due to the different nature of the participation constraint. Over the range (˜ θ,˜ θ0), the solution satisfies u(θ)=−ˇ q(θ)independently of the incumbent’s sales, which explains the indeterminacy in those sales. Because of the downward jump discontinuity of s(θ,v)at v=−ˇ q(θ), co(s)(θ,v)has a kink at that point and displays a flat segment for v∈[−ˇ q(˜ θ0),v1(˜ θ0)] where v1(˜ θ0)=−ˇ q(˜ θ0)+√2K.Attype ˜ θ0, the incumbent is thus indifferent between choosing ˙ u(˜ θ0)=−ˇ q(˜ θ0)and moving up to ˙ u(˜ θ0)=v1(˜ θ0), reducing his sales to qm(˜ θ0)−Kand allowing the buyer to purchase from the fringe up to capacity. Appendix A: NonSmooth optimization This Appendix briefly reminds the reader about necessary and sufficient conditions for the general problem of maximizing an upper semicontinuous function, h:R→R,over a compact set X⊂R. A generalization of the first-order condition for smooth, concave optimization programs can indeed be obtained for general upper semicontinuous programs by introducing a few concepts from nonsmooth convex analysis. The basic idea is that any solution
1162 Martimort and Stole Theoretical Economics 17 (2022) x 2 4 6 8 X Figure 1. Concavification of discontinuous, but upper-semicontinuous function on X. to the original upper semi-continuous program must lie on the minimal concave envelope or concavification of the objective. Consider, for example, the upper semicontinuous function graphed in Figure 1in bold. This function is defined over the real line, but the restricted domain of interest is X=[x,x]. The minimal-concave envelope over this domain is depicted by the dashed lines in the graph. Notice its value is negative infinity outside of [x,x]. Obviously, the maximum of this concave envelope is a solution to the original program. More generally, in the case in which there is a continuum of solutions (i.e., the maximum is achieved on a horizontal component of the majorization), there exist two solutions to the original program—the endpoints of the majorization. It is, in this sense, without loss to convert an upper semicontinuous program over a compact set into a concave (but possibly nondifferentiable) program over the same set. Formally, we will denote coX(h)to refer to the concavification of an objective function, h, over a domain, X,18 and coX(h)(x)to refer to the value of this envelope evaluated at x.19 Having reached the conclusion that we may focus on the concave envelope of the program, we can now import the generalized notion of derivative from convex analysis. Formally, we will define a set of gradients at any point to be all those vectors which 18When X=R, we simplify notation and omit the subscript. 19In the nonsmooth optimization literature, often one considers the minimal concave envelope of hover thereallineinsteadofsomedomainX, but in this case with a penalty function, X(x),whichequals0for x∈Xand −∞for x/∈X. Thus, in our notation, coX(h)=coR(h+X).
Theoretical Economics 17 (2022) Participation constraints 1163 “support” the graph at the given point, and we refer to this set-valued notion of derivative as the generalized gradient or the superdifferential, denoted ∂h(x)when applied to a concave function hat point x.20 Where his differentiable, the superdifferential is single-valued and corresponds to the gradient. If hexhibits a kink and X⊆R,thesuperdifferential is an interval of gradients with endpoints corresponding to the leftand right-side derivatives at the point. More generally, if X⊆Rn, then ∂h(x)=τ∈Rn|h(y)≤h(x)+τ,y−x∀y∈Rn. Using this generalization of gradient, we can now state the necessary and sufficient conditions for x∗to be a maximum of an upper-semicontinuous function, h,oversome given domain x∈X:ifx∗is a solution to the maximization program, then the following first-order condition must be satisfied: 0∈∂coX(h)x∗. (23) Furthermore, if x∗satisfies (23) and the envelope coincides with hat x∗, i.e., coX(h)x∗=hx∗(24) then x∗solves the maximization program. These conditions can be further tightened when a component of the objective function is affine. To this end, suppose that h=g+fwhere gis affine (and slightly abusing notation, let us write g(x)=gx). Well-known identities from convex analysis give us: coX(h)(x)=gx +coX(f)(x)and ∂coX(h)(x)=g+∂coX(f)(x). Thus, the linear linear part of the objective can be factored out and the “first order” necessary and sufficient condition for the optimality of x∗reduces to −g∈∂coX(f)x∗. This property will be repeatedly used throughout our analysis, first, to derive generalized first-order conditions for our infinite-dimensional optimal control problem, and second, to tackle applications in contract theory where such a decomposition is frequently available. 20We use the term “support” from convex analysis given it is evocative and familiar. The term subdifferential is the parallel notion of superdifferential when applied to convex functions. When we refer to the generalized gradient of a function that is understood to be convex, we will abuse notation slightly by again using the notation ∂h(x), where it is understood that when his convex, then ∂h(x)=τ∈Rn|h(y)≥h(x)+τ,y−x∀y∈Rn. See Ferrera (2014) for an introduction to nonsmooth analysis and an in-depth discussion of super and subdifferentials.
1164 Martimort and Stole Theoretical Economics 17 (2022) Appendix B: Proof of Theorem 1 Preliminaries for Nonsmooth Analysis. We draw heavily from Vinter and Zheng (1998) in the following presentation. A complete treatment can be found in the monograph of Vinter (2000). Theorem 3 from Vinter and Zheng (1998) appears as Theorem 10.2.1 in Vinter (2000). Take a closed set A⊆Rnand a point x∈A.Avectorr∈Rnis a limiting normal to Aat xif there exists a sequence (xi,ri)→(x,r)with xi∈Aand a constant M≥0such that for each iin the sequence ri·(xi−x)≤Mxi−x2,where·denotes Euclidean distance. The cone of limiting normal vectors to Aat xis denoted NA(x). Given a lower semicontinuous function g:R→R∪{+∞}and a point x∈Rsuch that g(x)<+∞,the limiting subdifferential of gat xis defined as ∂g(x)≡ξ|(ξ,−1)∈Nepi{g}x,g(x), where epi{g}is the epigraph of the function gdefined as epi{g}≡(x,α)∈R×R|α≥g(x). The asymptotic limiting subdifferential of gat x, written ∂∞g(x), is defined as ∂∞g(x)≡ξ|(ξ,0 )∈Nepi{g}x,g(x). Finally, we define ∂> xh(t,x)≡colim iξi∃ti→t,xi→xs.t. h(ti,xi)>0andξi∈∂xh(ti,xi)∀i. Two results from nonsmooth analysis (Vinter (2000, Propositions 4.3.3 and 4.3.4)) that we use are (1) ∂∞g(x)={0}if gis Lipschitz continuous and (2) for any xsuch that g(x) is finite, Nepi{g}x,g(x)=(ξd,−ξ)|ξ>0, d∈∂g(x)∪∂∞g(x)×{0}. Alocal maximizer of (x)is a feasible arc, x, which maximizes (x)over all feasible arcs x∈AC(,R+)within an εneighborhood of x,x−xac ≤εwhere we denote the norm on the space of absolutely continuous functions by xac ≡x(θ)+θ θ˙ x(θ)dθ. Alocal minimizer is defined analogously. Necessity. First, and for completeness, we reproduce here Theorem 3 of Vinter and Zheng (1998), which provides necessary conditions for solutions to the following minimization program: P: Minimize J(x)≡θ θ Lθ,x(θ),˙ x(θ)dθ subject to x∈AC(,R)and hθ,x(θ)≤0forallθ∈≡[θ,θ].21 21We specialize their theorem to our present problem in which the range of x(θ)is one-dimensional and there is no endpoint cost function.
Theoretical Economics 17 (2022) Participation constraints 1165 We will prove necessity for Theorem 1by specializing this theorem, exploiting the fact that our integrand in is a linear function of xand h(θ,x)=−x. Theorem 2(Vinter and Zheng (1998, Theorem 3)). Let xbe local minimizer for (P)in AC(,R)such that J(x)<+∞. Assume that the following hypotheses are satisfied: H1.L(·,x,·)is L×Bmeasurable for each xand L(θ,·,·)is lower semicontinuous for a.e. θ∈. H2. For every N>0,thereexistsδ>0and k∈L1such that Lθ,x,v−L(θ,x,v) ≤k(θ) x−x ,Lθ,x(θ),v≥−k(θ) for a.e. θ∈, for all x,x∈x(θ)+δB and v∈˙ x(θ)+NB,whereBis a unit Euclidean ball. H3.his upper semicontinuous near (θ,x(θ)) for all θ∈, and there exists a constant khsuch that hθ,x−h(θ,x) ≤kh x−x for all θ∈and all x,x∈x(θ)+δB. Then there exist an arc p∈AC(,R), a constant λ≥0, a nonnegative measure μon the Borel subsets of and a μ-integrable function ζ:→R,suchthat (i) λ+maxθ∈|p(θ)|+θ θμ(d˜ θ)=K>0(where Kis an arbitrary normalization constant),22 (ii) ˙ p(θ)∈coηη,p(θ)+[θ,θ) ζ(˜ θ)μ(d˜ θ),−λ ∈Nepi{L(θ,·,·)}x(θ),˙ x(θ),Lθ,x(θ),˙ x(θ)a.e., (iii) p(θ)=p(θ)−θ θ ζ(˜ θ)μ(d˜ θ)=0, (iv) p(θ)+[θ,θ) ζ(˜ θ)μ(d˜ θ)˙ x(θ)−λLθ,x(θ),˙ x(θ) ∈argmax v∈Rp(θ)+[θ,θ) ζ(˜ θ)μ(d˜ θ)v−λLθ,x(θ),v, (v) ζ(θ)∈∂> xh(θ,x(θ)) μ-a.e. and supp{μ}⊆{θ|h(θ,x(θ)) =0}. 22We choose to state the theorem using K>0 as an arbitrary normalization rather than K=1, which is the normalization chosen in Vinter and Zheng (1998). Later, by setting K=3, we will succeed in normalizing μto a probability measure, which is a more familiar object.
1166 Martimort and Stole Theoretical Economics 17 (2022) We apply this result to our setting by substituting xf (θ)−s(θ,v)in program (P)in place of L(θ,x,v)and thereby converting the maximization functional in program (P)to the minimization functional Jin program (P). We complete the transformation by requiring that h(θ,x)=−x,andthatL(θ,x,v)is a linear function of xfor any (θ,v). First, we verify that hypotheses H1–H3are satisfied for our program (P). Because s(θ,·)is upper semicontinuous and B-measurable, and because L(θ,x,v)is linear in x,H1is satisfied. H2requires that L(θ,·,v)is Lipschitz continuous, which is trivial given that Lis linear in xwith coefficient f(θ). Because the transformed program has h(θ,x)=−x,his a continuous linear function of x,andthusH3is also satisfied. Next, we specialize the conclusions of Vinter and Zheng (1998) by making use of the additional restrictions on L(·)and h(·). We present this in the following lemma. Lemma 1. Suppose that L(θ,x,v)is a linear function of xand that h(θ,x)=−x. Then the conclusions (i)–(v) of Theorem 2imply (a) λ+maxθ∈|p(θ)|+θ θμ(d˜ θ)=K, (b) ˙ p(θ)=λf (θ)a.e., (c) p(θ)=p(θ)+θ θζ(˜ θ)μ(d˜ θ)=0 (d) ˙ x(θ)∈argmaxv∈R(p(θ)+[θ,θ)ζ(˜ θ)μ(d˜ θ))v+λs(θ,v), a.e., (e) ζ(θ)=−1μ-a.e. and supp{μ}⊆{θ|u(θ)=0}. Proof of Lemma 1. Implications (i) and (a) are identical. Implication (ii) requires ˙ p(θ)∈coηη,p(θ)+[θ,θ) ζ(˜ θ)μ(d˜ θ),−λ ∈Nepi(L(θ,·,·))x(θ),˙ x(θ),Lθ,x(θ),˙ x(θ),a.e. Because L(θ,x(θ),˙ x(θ)) =f(θ)x(θ)−s(θ,˙ x(θ)) is finite, the limiting normal cone in the aboveexpressioncanbewrittenas Nepi(L(θ,·,·))x(θ),˙ x(θ),Lθ,x(θ),˙ x(θ) =(ξd1,ξd2,−ξ)|ξ>0, (d1,d2)∈∂f(θ)x(θ)−sθ,˙ x(θ) ∪∂∞f(θ)x(θ)−sθ,˙ x(θ)×{0}. Using the fact that L(·)is additively separable in xand ˙ xyields (Rockafellar and Wets (2004, Proposition 10.5)) ∂f(θ)x(θ)−sθ,˙ x(θ)=∂f(θ)x(θ)×∂−sθ,˙ x(θ) =f(θ)×∂−sθ,˙ x(θ)
Theoretical Economics 17 (2022) Participation constraints 1167 and ∂∞f(θ)x(θ)−sθ,˙ x(θ)⊆∂∞f(θ)x(θ)×∂∞−sθ,˙ x(θ) ={0}×∂∞−sθ,˙ x(θ), where the last equality uses the fact that a linear function is Lipschitz continuous, and hence ∂∞(f(θ)u(θ)) ={0}. Substituting these subdifferentials into the expression for the limiting normal cone, we have a simple inclusion: Nepi(L(θ,·,·))x(θ),˙ x(θ),Lθ,x(θ),˙ x(θ) ⊆ξf (θ),ξd2,−ξ|ξ>0, d2∈∂−sθ,˙ x(θ) ∪{0}×∂∞−sθ,˙ x(θ)×{0}. This simplifies again to the inclusion Nepi(L(θ,·,·))x(θ),˙ x(θ),Lθ,x(θ),˙ x(θ) ⊆ξf (θ),ξd2,−ξ|ξ≥0, d2∈∂−sθ,˙ x(θ)∪∂∞−sθ,˙ x(θ). The key point to note is that any vector in the limiting normal cone must point in the same direction in the (x,L)plane, regardless of d2. Returning to implication (ii), we see that any point ηin the given convex hull must satisfy (η,·,−λ)=(ξf (θ),·,−ξ)for some ξ≥0, and hence the convex hull reduces to {λf (θ)}. We conclude that implication (ii) simplifies to implication (b) given that L(·)is both additively separable and linear in x. Implication (iii) is identical to implication (c). Using the transformation L(θ,x,v)=xf (θ)−s(θ,v), implication (iv) simplifies to implication (d). Lastly, the fact that h(θ,x)=−xyields ∂xh(θ,u(θ)) =∂> xh(θ,x(θ)) = {−1}. Thus, implication (v) simplifies to ζ(θ)=−1μ-a.e. and supp{μ}⊆θ|u(θ)=0. (25) This is implication (e) and completes the proof to the lemma. Returning to the proof of Theorem 1, an immediate inspection of conditions (a)–(e) suggest further simplifications by combining these conditions. Conditions (b) and (c) jointly yield p(θ)=λF(θ). Because p(θ)=λand ζ(θ)=−1 a.e. with respect to μ, condition (c) also implies θ θ μ(d˜ θ)=λ. Becausewealsohavemaxθ∈|p(θ)|=λ, condition (a) implies λ>0 and in particular λ=K 3. Because the choice of Kis arbitrary, we choose K=3 as a normalization, yielding
1168 Martimort and Stole Theoretical Economics 17 (2022) λ=1andθ θμ(d˜ θ)=1. Thus, up to this normalization, μis a probability measure on . Defining now γ(θ)=[θ,θ)μ(d˜ θ), the implication in (d) is therefore ˙ x(θ)∈argmax v∈R s(θ,v)+F(θ)−γ(θ)v, a.e. (26) This condition can finally be expressed as (4)and(5)ofTheorem1. Lastly, implication of (e) delivers the complementary slackness condition (3). We have therefore proven the necessity of the conditions in Theorem 1. Sufficiency. Sufficiency is proven by generalizing Arrow’s sufficiency theorem to nonsmooth optimal control problems and specializing the theorem to the case in which the objective integrand is a linear function of x. We adapt the argument of Arrow’s sufficiency theorem using the approach of Seierstad and Sydsaeter (1987) but relaxing their continuity and smoothness assumptions. The regularity of the optimal solution follows from arguments involving the necessary conditions. Let xbe any admissible arc satisfying thus x∈AC(,R)and x(θ)≥0 for all θ∈. Define =θ θsθ,˙ x(θ)−x(θ)f(θ)−sθ,˙ x(θ)−x(θ)f(θ)dθ. We will demonstrate that, under conditions (25)and(26)ofTheorem1,≥0. To this end, it is useful to define the Hamiltonian for program (P) with γ(θ)being the adjoint equation, which satisfies conditions (25)and(26): H(θ,x,v)≡s(θ,v)−xf (θ)−γ(θ)−F(θ)v. Note that γ(θ)is defined for θ∈(θ,θ],andthusH(·)inherits the same domain. Nonetheless, because μis not part of expression of and Fis absolutely continuous, we can ignore the point θin the integral and conclude that =(θ,θ]Hθ,x(θ),˙ x(θ)−Hθ,x(θ),˙ x(θ)dθ +θ θF(θ)−γ(θ)˙ x(θ)−˙ x(θ)dθ. Define the optimized Hamiltonian as ˆ H(θ,x)≡sup v∈R H(θ,x,v). Because γ(θ)−F(θ)is bounded on (θ,θ]and s(θ,·)is bounded from above by assumption, ˆ H(·)must be finite. Condition (26) implies that ˆ Hθ,x(θ)=Hθ,x(θ),˙ x(θ) and for any admissible x∈AC(;R+), ˆ Hθ,x(θ)≥Hθ,x(θ),˙ x(θ).
Theoretical Economics 17 (2022) Participation constraints 1175 Figure 2. Consumption levels: Digital goods. Summarizing our previous findings in (33), (36), (40), and (41) yield the expression of qin (17). (See Figure 2.) On the other hand, (32), (35), (38)and(42) yield the expression of γin (18). Proof of Proposition 4. Standard arguments (see footnote 2) establish that U(θ)so defined is absolutely continuous, and thus a.e. differentiable with ˙ U(θ)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ −q(θ)−Kif q(θ)<ˇ q(θ), −ˇ q(θ)−Kif q(θ)∈ˇ q(θ),ˇ q(θ)+K, −q(θ)if (θ)>ˇ q(θ)+K. (44) From this and the fact that ˙ ˆ U(θ)=−K,w eget(20). We can express ˜ s(θ,q)as ˜ s(θ,q)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (S−θ−K)q−q2 2−cq if q≤ˇ q(θ), ˇ q(θ)+K2 2+q −(S−θ)K−K2 2−pKif q∈ˇ q(θ),ˇ q(θ)+K, (S−θ−c)q−q2 2−(S−θ)K−K2 2−pKif q≥ˇ q(θ)+K.
1176 Martimort and Stole Theoretical Economics 17 (2022) Observing that ˇ q(θ)+K=S−θ−pand simplifying yields ˜ s(θ,q)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˇ q(θ)+q−q2 2if q≤ˇ q(θ), ˇ q(θ)2 2+q if q∈ˇ q(θ),ˇ q(θ)+K, (S−θ−c)q−q2 2−(S−θ−p)K−K2 2if q≥ˇ q(θ)+K. Expressing q(θ)in terms of ˙ u(θ)over the different intervals yields q(θ)⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ =−˙ u(θ)if ˙ u(θ)>−ˇ q(θ), ∈ˇ q(θ),ˇ q(θ)+Kif ˙ u(θ)=−ˇ q(θ), =−˙ u(θ)+Kif ˙ u(θ)<−ˇ q(θ). (45) Inserting these expressions of q(θ)into the definition of ˜ s(θ,q)above yield s(θ,v)as (21). From there, we now compute co(s)(θ,v) =⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −ˇ q(θ)+v−v2 2if v≥v2(θ), −(√2K +)v−v2(θ)−ˇ q(θ)+v2(θ)−v2 2(θ) 2if v∈−ˇ q(θ),v2(θ), −ˇ q(θ)+v−v2 2+K if v<−ˇ q(θ) where v2(θ)=√2K −ˇ q(θ). This yields the following expression of the subdifferential for co(s)(θ,v): ∂vco(s)(θ,v)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ −ˇ q(θ)−−vif v≥v2(θ)and if v<−ˇ q(θ), −√2K −if v∈−ˇ q(θ),v2(θ), [−√2K −,−]if v=−ˇ q(θ). With a uniform distribution, the optimality condition (5) becomes γ(θ)−θ+θ∈∂vco(s)θ,˙ u(θ) or γ(θ)−θ+θ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ =−ˇ q(θ)−−˙ u(θ)if ˙ u(θ)≥v2(θ)and if ˙ u(θ)<−ˇ q(θ), =−√2K −if ˙ u(θ)∈−ˇ q(θ),v2(θ), ∈[−√2K −,−]if ˙ u(θ)=−ˇ q(θ). (46) We conjecture a solution (u(θ),γ(θ)) such that (1) binds at θonly, and thus μ({θ})> 0withγ(θ)=0on[θ,θ). Thanks to the sufficiency part of our theorem, we only check that this solution satisfies the necessary conditions for optimality.
Theoretical Economics 17 (2022) Participation constraints 1177 •On the interval [θ,˜ θ), this conjecture implies γ(θ)=0. Inserting into (46)yields −θ+θ=∂vco(s)θ,˙ u(θ)=−ˇ q(θ)−−˙ u(θ). Because θ≤˜ θ=θ+,wethushave ˙ u(θ)=−ˇ q(θ)+θ−˜ θ<−ˇ q(θ). From (45), we deduce that −q(θ)+K=˙ u(θ)=−(S−θ−p−K)+θ−˜ θ, and thus q(θ)=S−2θ+θ−c=qm(θ). •On the interval [˜ θ,˜ θ0],wehave ˙ u(θ)=−ˇ q(θ). Indeed, imposing our conjecture γ(θ)=0on(46)yields −θ+θ∈∂vco(s)θ,−ˇ q(θ)=[−√2K −,−]⇐⇒ θ∈[˜ θ,˜ θ0]. From (45), we deduce that q(θ)∈[S−θ−p−K,S−θ−p]=ˇ q(θ),ˇ q(θ)+K. •On the interval [˜ θ0,θ), our conjecture is γ(θ)=0. Inserting into (46)yields −θ+θ=∂vco(s)θ,˙ u(θ)=−ˇ q(θ)−˙ u(θ)−. Because θ≥˜ θ0>˜ θ,wehave ˙ u(θ)=−ˇ q(θ)+θ−˜ θ>−ˇ q(θ)+θ−˜ θ0≥−ˇ q(θ). From (45), we deduce that −q(θ)=˙ u(θ)=−(S−θ−p−K)+θ−˜ θ, and thus q(θ)=S−2θ+θ−c−K=qm(θ)−K. Gathering all the above findings yields (22). (See Figure 3below.) The condition (19) ensures that the incumbent firm still wants to serve the type with the lowest possible demand θeven when that type consumes up to the fringe’s capacity. It implies that u(θ) is everywhere decreasing, consistently with (1) being binding at θonly. Hence, μ({θ})= 1asrequired.
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