Adverse selection without single crossing: monotone solutions∗ Christoph Schottmu ¨ller† March 7, 2013

Abstract The single crossing assumption simplifies the analysis of screening models as local incentive compatibility becomes sufficient for global incentive compatibility. If single crossing is violated, global incentive compatibility constraints have to be taken into account. This paper studies monotone solutions in a screening model that allows a one-time violation of single crossing. It is shown that local and non-local incentive constraints distort the solution in opposite directions. Therefore, the optimal decision might involve distortions above as well as below the first best decision. Furthermore, the well known “no distortion at the top” property does not necessarily hold. It is shown that the decision can even be distorted above first best for all types. Sufficient conditions for existence, monotonicity and continuity of the solution and an algorithm to obtain such a solution are presented. Keywords: Spence-Mirrlees condition, global incentive compatibility JEL classification: C61, D82, D86 ∗

I want to thank Jan Boone, Bruno Jullien, Peter Norman Sørensen and Bert Willems for interesting

discussions and numerous suggestions. I have also benefited from comments by C´edric Argenton, Johan Lagerl¨of, Matthias Lang, Humberto Moreira, Jens Pr¨ ufer, Fran¸cois Salani´e, Florian Sch¨ utt, Eric van Damme and seminar participants at the Toulouse School of Economics, University of Southern Denmark Odense, Aalto University Helsinki, University of Copenhagen, Lund University, UT Sydney, Monash University Melbourne, Adelaide University and Tilburg University as well as conference participants at EARIE, the ENTER Jamboree and the European Winter Meeting of the Econometric Society. † Department of Economics, University of Copenhagen, Øster Farimagsgade 5, building 26, 1353 Copenhagen K, Denmark; email: christoph.schottmuller [at] econ.ku.dk

1. Introduction Screening models are among the most commonly used tools in microeconomics. In these models, a principal offers a menu of contracts from which an agent with private information about his “type” chooses his preferred option. Depending on the application, the type represents the production technology of a firm in models of regulation (Baron and Myerson, 1982; Laffont and Tirole, 1987), the productivity of a worker in an employment relationship (Guasch and Weiss, 1981) as well as in optimal taxation models (Mirrlees, 1971), the probability of an accident in insurance models (Stiglitz, 1977) or the willingness to pay for a product in models of monopoly pricing (Mussa and Rosen, 1978) and auctions (Myerson, 1981). In all these applications, the authors assume single crossing:1 Types can be ordered according to their marginal rate of substitution between money and the decision, e.g. the quantity purchased in a monopoly pricing problem. With the common quasilinear preferences, single crossing is equivalent to a type ordering according to marginal utilities. Put differently, the private information is how eager an agent is to consume more (or how marginally efficient a firm is). More technically, single crossing says that a higher type has a higher marginal utility at every consumption level. While single crossing is a technically convenient assumption, we can think of many unobserved differences other than eagerness/efficiency. Some people are very eager at first but quickly saturated while others have more steady preferences. Compare, for example, a single person household with a family. At low quantities, the single will have the higher marginal willingness to pay for standard groceries because of a higher income per household member. At high quantities, however, the family will have the higher marginal willingness to pay as the single does not want to consume more. This violates single crossing: Types, i.e. family or single, cannot be ordered according to their marginal willingness to pay. As a second example, think of firms that have private information about their production technology. A capital intensive, fully automated production facility will at normal output levels have lower marginal costs than a labor intensive firm. But as soon as quantity approaches the capacity level of the capital intensive firm, the labor intensive 1

Other names for this assumption include “Spence-Mirrlees” or “constant sign” condition.

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production technology should result in lower marginal costs. Another motivation for studying models where single crossing is violated is that in some strands of the screening literature, e.g. the common agency literature, violations of single crossing emerge naturally. Authors have, so far, restricted themselves to analyze models with specific functional forms, e.g. quadratic utility functions and uniformly distributed types, for which they could show that non local incentive constraints do not bind, see Martimort and Stole (2009), Calzolari (2004), Jullien et al. (2007) or Hoffmann and Inderst (2011). In the signaling literature, single crossing fails in models where agents’ signaling is motivated by social image concerns, see Bernheim (1994); Bagwell and Bernheim (1996) and Andreoni and Bernheim (2009). Consequently, violations of single crossing is likely to matter in non-linear pricing problems with status goods. This paper analyzes a screening model in which single crossing is violated. Agents have quasilinear preferences and a one-dimensional type. The setting allows for a one time violation of single crossing; i.e. for a given quantity, marginal utility is first increasing and then decreasing in type. Without single crossing, local incentive compatibility does no longer guarantee global incentive compatibility. Therefore, non-local incentive compatibility constraints must be taken into account. The paper analyzes monotone solutions in this setup, that is, situations in which higher types consume higher quantities under the optimal contract.2 Sufficient conditions for the existence of a monotone solution and an algorithm to calculate such a solution are presented. With single crossing, there is no distortion at the top and the distortion for all types goes in the same direction, e.g. all types consume/produce a quantity which is weakly below their first best quantity. When single crossing is violated, neither result has to hold. The reason is that binding non-local incentive constraints will counteract the normal distortion stemming from local incentive compatibility and rent extraction motives. A rough intuition for this result is the following: With single crossing, distortions occur because the principal wants to lower the agent’s informational rent. If a non-local incentive constraint is violated, a certain type’s rent at “his contract” is too low compared with the rent he can get from misrepresenting as another type. To satisfy the non-local incentive constraint of this type, the principal must leave a higher rent to him. Reducing the 2

Araujo and Moreira (2010) analyze U-shaped solutions in such a setting; see section 1.2.

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normal distortion (or even distorting the decision in the opposite direction) will result in less rent extraction and therefore relax the non-local incentive constraint. As an illustration, consider the production technology example above. If the principal offered only contracts with quantities at or above the capacity level of capital intensive firms, labor intensive firms would have lower marginal costs in the relevant range of quantities. Hence, single crossing would be satisfied in this range and non-local incentive constraints would not bind. Such an extreme upwards distortion will not be optimal, i.e. optimal quantities will be lower such that single crossing is still violated and non-local incentive constraints bind, but moving into this direction will intuitively help to relax the non-local incentive constraints. The effect of non-local incentive constraints can be so strong that, in contrast to the previous literature, all types’ decisions are distorted. This result can, for example, explain flat rate tariffs where a zero marginal price incentivizes overconsumption. The following subsection gives two examples of preferences that violate single crossing. The related literature is reviewed in section 1.2 and the formal model is introduced in section 2. Section 3 analyzes the solution: Subsection 3.1 introduces necessary conditions which have to hold at types where non-local incentive constraints are binding. Subsection 3.2 characterizes monotone solutions while subsection 3.3 focuses on novel results concerning the distortions in the optimal contract. Section 4 gives more technical results, i.e. sufficient conditions for existence of a monotone solution, continuity and determinism of the solution as well as an algorithm which allows to compute such a solution. Section 5 discusses how the results depend on the assumptions as well as possible generalizations. Most proofs are relegated to the appendix. A webappendix includes solved numerical examples and those proofs that are straightforward extensions of results in the literature. 1.1. Example settings where single crossing is violated This section introduces two numerical examples which are used for illustration later on. Example setting 1: incentive regulation.

Consider a setting of monopoly

regulation. The monopolist produces the good with two input factors which he uses in fixed proportions. The costs of the first input factor are proportional to output, e.g. unqualified labor can be hired in any quantity on a competitive labor market. The

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costs of the second input factor are convexly increasing in output, e.g. skilled labor is increasingly difficult to hire. The monopolist’s private information is whether his production technology is more intense in the first or second factor. A cost function representing this is q2 + γ(θ) θ where γ(θ) are (possibly type dependent) fixed costs, q is quantity and θ is the type. The c(q, θ) = θq +

cross partial derivative cqθ (q, θ) = 1 − 2q/θ2 can change sign and therefore single crossing is violated: Marginal costs can be in- or decreasing in type (depending on q). For low quantities, the linear part of the cost function dominates marginal costs and therefore high types have higher marginal costs. For high quantities, the convex part of the cost function is more relevant and therefore high types have lower marginal costs. The cost function in this example is a simplified version of the flexible fixed cost quadratic cost function suggested by Baumol et al. (1982). Beard et al. (1991) estimate such a cost function for savings and loans associations. Interestingly, they allow for two unobservable types (“mixtures” in their language) of production technology in their estimation. It follows from their estimates in table 5 that mixture 1 has lower marginal costs at low output levels but higher marginal costs at high output levels. Hence, that cost functions in the savings and loan sector violate single crossing. The example above illustrates a more general point: To avoid the difficulties of multidimensional screening, some papers collapse a multidimensional type into a onedimensional type; e.g. we could have used the cost function θ1 q +θ2 q 2 instead and assumed that θ1 and θ2 are negatively correlated. When there is a tradeoff between being marginally efficient in either one or the other dimension, this collapsing will lead to a setting where single crossing is violated. I want to point out that non-local incentive constraints will also bind in the corresponding multidimensional model. The multidimensional screening literature mainly sidesteps this complication, for example by using linear utility/cost functions, see section 1.2. Example setting 2: non-linear pricing. There are many reasons why single crossing can be violated in a setting where a monopolist sets a nonlinear price schedule. First, let consumers differ in their demand elasticity. More specifically, their utility from consuming q units for a price p is given by v(q, θ) − p = q θ − p 4

where θ is distributed on a subset of (0, 1). If the relevant quantities are in (0, 1), higher types have a lower willingness to pay and a higher price elasticity of demand.3 The cross-derivative vqθ = q θ−1 (1 + θln(q)) changes sign at q = e−1/θ . Hence, marginal utility is increasing in type for quantities q > e−1/θ but decreasing in type for q < e−1/θ , i.e. single crossing is violated. 1.2. Literature Closest to this paper is Araujo and Moreira (2010). Their paper characterizes (inversely) U-shaped solutions in a setup with a one-time violation of single crossing. In these solutions, some contracts are chosen by two types (“discrete pooling”). It turns out that in (inversely) U-shaped solutions non-local incentive constraints are only binding between types which are pooled under the optimal menu, that is, between types choosing the same contract. My paper complements their work by characterizing monotone solutions in the same model. Whether the optimal contract is monotone or (inversely) U-shaped depends on the utility functions of the agents. Roughly speaking, the shape of the solution is determined by the shape of the first best decision; see section 4.1 for details. The main technical difference is that in my paper non-local incentive constraints can bind between types choosing different contracts from the menu. Qualitatively, the solution in Araujo and Moreira (2010) features either a discontinuity or a bunching interval. Furthermore, there is a no distortion at the top result, i.e. the type with the highest first best decision4 will be assigned this first best decision under the optimal contract. My paper shows that monotone solutions can be strictly monotone and continuous and therefore bunching and discontinuities are not a necessary implication of a violation of single crossing. Furthermore, I show that distortion at the top and even distortion of all types’ decision is possible in monotone solutions. The (inverse) U-shape solution and its critical condition (see section 3.1 and the webappendix) has been applied in an insurance model (Araujo and Moreira, 2003), in non-linear monopoly pricing (Araujo et al., 2010) 3

The price elasticity of demand is here defined as the relative demand change caused by a 1% increase

in the marginal price. Using the first order condition p′ (q) = θq θ−1 , the elasticity can be derived as |1/(θ − 1)|. 4 In Araujo and Moreira (2010), the function s(θ) is downward sloping and the analyzed solution is actually U-shaped (not inversely U-shaped as it would be when applied to my setting). In this setting, the undistorted top type is then the type with the lowest decision.

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and signaling games (Araujo et al., 2007). Several authors have analyzed perfect competition insurance models with discrete types where single crossing is violated. In their papers, private information has two dimensions and can take either a high or a low value in each dimension, i.e there are 2×2 types. In Smart (2000), the two dimensions are risk and risk aversion while in Wambach (2000) they are wealth and risk. Netzer and Scheuer (2010) model an additional labor supply decision and the two dimensions are productivity and risk. All three papers share a pooling result, i.e. if single crossing is violated two of the four types can be pooled. In contrast to my paper, these discrete type models have no distortion at the top and decisions of all other types are downward distorted. Violations of single crossing are also related to the literature on multidimensional screening, see Armstrong (1996) and Rochet and Chon´e (1998) for seminal contributions, Rochet (2009) for a recent related paper and Rochet and Stole (2003) for a survey. As pointed out in the survey, “the problems arise not because of multiple dimensionality itself, but because of a commonly associated lack of exogenous type-ordering in multipledimensional environments.” A violation of single crossing also conveys a lack of typeordering. To make the relationship clear, think of a multidimensional, discrete type model. Clearly, one can reassign types to a one-dimensional parameter but with those reassigned types single crossing will normally be violated. Generally speaking, multidimensionality leads to two new challenges concerning incentive compatibility: First, it is ex ante unclear which local incentive constraints are binding. Second, non-local incentive constraints can be binding. The literature has mainly focused on the first challenge5 and provided sufficient conditions ensuring that non-local incentive constraints are not binding, see Carroll (2012) for the latter. My paper analyzes a model where non-local incentive constraints are binding. Although the model is onedimensional, this can be viewed as a natural starting point for solving more general setups. The paper also relates to work relaxing the basic assumptions of the textbook screening model (Fudenberg and Tirole, 1991; Bolton and Dewatripont, 2005). Jullien (2000) allows for type dependent participation constraints while Hellwig (2010) analyzes the 5

See Moore (1984) and Matthews and Moore (1987) for early exceptions to this in the discrete type

case.

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case of irregular type distributions, i.e. distributions with mass points and zero densities. Hellwig (2010) shows that this leads to discontinuities as well as bunching in the optimal decision schedule. Contrary to my paper, there is no distortion above first best and no distortion at the top. In Jullien (2000), distortion can be above as well as below first best. The reason is that incentive constraints can bind upward as well as downward. If a participation constraint binds in the interior, it is relaxed by increasing the decision of lower types: This increases the slope of the rent function and leads to higher rents for the interior type. I show that binding non-local incentive constraints can also lead to distortion above first best although incentive constraints are only downward binding. ˆ one can relax The intuition is that if a type θ wants to misrepresent as a lower type θ, this non-local incentive constraint by increasing the decisions of types between θˆ and θ: This will increase the slope of the rent function and lead to higher rents for θ at his own contract. Hence, misrepresentation is less attractive. Furthermore, I show that all types can have distorted decisions which is–to the best of my knowledge–a new result in the screening literature.

2. Model There is a one-dimensional decision in a principal agent relationship which is denoted by q ∈ R+ . Furthermore, there is a monetary transfer t ∈ R. The agent’s utility is ¯ ⊂ R is the type of the agent which is his private π = t − c(q, θ) where θ ∈ Θ ≡ [θ, θ] information. The function c(q, θ) is assumed to be three times continuously differentiable with cq > 0, cθ < 0. The assumption cθ < 0 ensures that the participation constraint can only bind at the lowest type. Hence, any deviation from the standard solution will not be due to participation constraints binding in the interior, see Jullien (2000) for this, but to the violation of single crossing. The principal’s utility is u(q, θ) − t and is two times continuously differentiable with uq > 0. The principal has the prior distribution F (θ) with continuous density f (θ) > 0 ¯ To simplify the exposition, I assume full participation, i.e. the surplus for all θ ∈ [θ, θ]. from trade is so high that it is not beneficial to exclude some types.6 6

This is less restrictive than it might seem. By cθ < 0, only types at the low end could be excluded.

If exclusion is optimal, the characterization in this paper applies to the set of not excluded types. Using the methods of this paper, one can calculate the solution for any given cutoff type and then maximize

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The following assumptions are standard in the screening literature. They would ensure that the problem is well behaved if single crossing was satisfied. Section 5 discusses how the results of the paper extend when these assumptions are violated.7 Assumption 1. The principal’s utility satisfies uqθ ≥ 0 and uqq ≤ 0. The agent’s cost function satisfies cqq ≥ 0, cqqθ < 0 and cqθθ > 0. The monotone hazard rate property is satisfied by F , i.e. f /(1 − F ) is non-decreasing. Together with the single crossing assumption cqθ < 0, assumption 1 ensures then that the optimization problem is concave and the solution is strictly monotone. By making assumptions on the third derivatives of the cost function, I allow for a one time violation of single crossing, i.e. for a given type cqθ can change sign once when varying the decision q (see figure 1). Hence, the type decision plane is divided into two regions: One where cqθ < 0 and one where cqθ > 0. These two regions are separated by a strictly increasing function s defined by cqθ (s(θ), θ) = 0. By the revelation principle, any mechanism can also be implemented by a direct revelation mechanism in which the agent truthfully reports his type (Myerson, 1979). The task is to design a menu q(θ), implemented by transfers t(θ), which is individually rational (ir) and incentive compatible (ic) for the agent and maximizes the principal’s objective under these two constraints. Define π(θ) as the rents (or “profits”) a type θ gets under an implementable menu ˆ − c(q(θ), ˆ θ) (q(θ), t(θ)). Faced with a menu (q(θ), t(θ)), a type θ agent will maximize t(θ) ˆ The envelope theorem and truthful revelation require over his type announcement θ. πθ (θ) = −cθ (q(θ), θ). Incentive compatibility of a decision q(θ) requires in general for any θ, θˆ ∈ Θ ˆ θ)] ≥ 0. ˆ ≡ π(θ) − [π(θ) ˆ + c(q(θ), ˆ θ) ˆ −c(q(θ), Φ(θ, θ) {z } |

(IC)

ˆ t(θ)

Using the envelope condition above, Φ can be rewritten as Z θˆ Z θ Z q(t) ˆ ˆ Φ(θ, θ) = cθ (q(t), t) − cθ (q(θ), t) dt = − cqθ (s, t) ds dt ≥ 0. θˆ

θ

ˆ q(θ)

(IC’)

the principal’s payoff over the cutoff type. 7 While the assumptions on cqqθ and cqθθ are vital, the other assumptions could be substituted by the weaker requirement that the relaxed program is concave and yields a strictly monotone solution. Hence, my assumptions are equivalent to those in Araujo and Moreira (2010).

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q

q cqθ < 0

q(θ) cqθ < 0 s(θ) cqθ > 0

s(θ)

cqθ > 0 q(θ) θ

θˆ

θ (b) monotone solution

(a) inverse U-shape solution

Figure 1: possible solution shapes

With single crossing incentive compatibility in (IC’) boils down to a simple monotonicity condition on q(θ) (plus the envelope condition): If cqθ < 0, then inequality (IC’) will hold whenever q(θ) is monotonically increasing. When cqθ can change sign, this is no longer true. It remains true that q(θ) has to be increasing (decreasing) at θ if cqθ (q(θ), θ) < (>)0. Otherwise, (IC’) would be violated for types close enough to θ. But ˆ this no longer implies global incentive compatibility for two arbitrary types θ and θ. The intuition is the following: With single crossing, if a type θ prefers a high decision/high transfer contract over a low decision/low transfer contract, then all higher types will do so as well. The reason is that–by single crossing–they have lower marginal costs. Without single crossing, the latter does not necessarily hold and therefore also the first statement can be wrong. As incentive compatibility requires sgn(−cqθ (q(θ), θ)) = sgn(qθ (θ)), the optimal decision schedule cannot have arbitrary shapes. Araujo and Moreira (2010) analyze inverse U-shape decisions, see figure 1a. In such a solution, one decision can be assigned to two types (“discrete pooling”) and therefore these two types have the same contract. Clearly, a non-local incentive constraint is binding between them. It turns out that these are the only binding non-local incentive constraints. I will analyze monotone solutions in this paper, see figure 1b.8 Although cqθ < 0 at the decision q(θ) for all types, the violation of single crossing still plays a role in monotone solutions. It follows from (IC’) that one can represent incentive 8

I will only look at monotonically increasing solutions. It is easy to show that in solutions that

are below s(θ) for all types (and therefore are decreasing) non-local incentive constraints do not bind. Hence, standard methods are sufficient for such problems.

9

θ

compatibility as an integral over the shaded area in figure 1b: If the integral of cqθ over ˆ Hence, the this shaded area is negative, incentive compatibility is satisfied for θ and θ. region where cqθ > 0 is relevant for incentive compatibility although the solution does not pass through it. Before turning to the analysis of the solution, some definitions and one assumption is needed. I define the first best solution denoted by q f b (θ) as argmaxq u(q, θ) − c(q, θ) which would be the optimal decision if the principal observed the agent’s type. As a second reference point, it is useful to look at the relaxed program. This is the program taking only local incentive compatibility into account: Z θ¯ {u(q(θ), θ) − c(q(θ), θ) − π(θ)}f (θ) dθ max q(θ)

(RP)

θ

s.t. : πθ (θ) = −cθ (q(θ), θ) qθ (θ)cqθ (q(θ), θ) ≤ 0 π(θ) ≥ 0 The first and second constraint are the local incentive compatibility constraints. More specifically, the first constraint is the envelope condition. It corresponds to a first order condition of the problem in which the agent maximizes his utility over his type announcement. The second constraint is the so called monotonicity constraint which corresponds to the second order condition of the same problem.9 The third constraint is the participation constraint which will bind only for θ by the assumption cθ < 0. I will call the solution of (RP) the relaxed solution and denote it by q r (θ). The inverse U-shape solutions in Araujo and Moreira (2010) can only occur when q f b is inversely U-shaped. The monotone solutions analyzed in this paper require the first best decision to be monotone. Assumption 2. The first best decision is monotonically increasing and strictly above s, i.e. qθf b ≥ 0 and q f b (θ) > s(θ). Under assumption 1 and 2, the following lemma is standard: Lemma 1. The relaxed program is concave.10 The relaxed decision is strictly monotonically increasing and strictly above s(θ), i.e. qθr > 0 and q r (θ) > s(θ). 9 10

For a brief proof of this, see lemma 1 in Araujo and Moreira (2010). With this I mean that the usual virtual valuation is concave in q, i.e (uqq − cqq )f + (1 − F )cqqθ < 0.

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This lemma implies that q r is characterized by the first order condition {uq (q(θ), θ) − cq (q(θ), θ)}f (θ) + (1 − F (θ))cqθ (q(θ), θ) = 0.

(1)

If single crossing is satisfied, the relaxed decision is the solution of the principal’s problem. Without single crossing, non-local incentive constraints can be violated under the relaxed decision. The analysis in this paper will show (i) how to deal with these non-local incentive constraints and (ii) how the solution differs from q r qualitatively as a result of binding non-local incentive constraints.

3. Optimal contract 3.1. Necessary conditions This subsection presents necessary conditions which have to be met whenever a non-local incentive constraint is binding. Since these conditions are only a slight generalization of those presented in Araujo and Moreira (2010), the presentation will be brief and more intuitive than formal. Take an optimal decision schedule q(θ) and let transfers be determined by local incentive compatibility, i.e. such that πθ (θ) = −cθ (q(θ), θ) and π(θ) = 0.11 Furthermore, ˆ i.e. Φ(θ, θ) ˆ = 0. By incentive compatsuppose that IC is binding for two types θ and θ, ˆ ∈ argmin(s,t) Φ(s, t) as ibility, Φ has to be non-negative for all types. Therefore, (θ, θ) ˆ = 0. Hence, the first order condition for a minimum have to hold:1213 Φ(θ, θ) Z q(θ) ˆ ∂Φ(θ, θ) = −cqθ (q, θ) dq ≤ 0 with “=” if θ < θ¯ ∂θ ˆ q(θ) Z θ ˆ ∂Φ(θ, θ) ˆ ˆ t) dt ≥ 0 = qθ (θ) cqθ (q(θ), with “=” if θˆ > θ ˆ ∂ θˆ θ

(C1) (C2)

ˆ = 0 meant that the integral of cqθ over the shaded area in 1b is Graphically, Φ(θ, θ) zero. (C1) and (C2) imply that the integral of cqθ over the respective boundaries of 11

In the remainder of the paper transfers are mainly neglected and it is understood that they are

determined by these two equations. 12 It turns out that non-local incentive compatibility constraints are only downward binding, see lemma ¯ = 0 and Φ(θ, θ) ˆ =0 2. For this reason as well as notational convenience, I ignore the possibilities Φ(θ, θ) already here. 13 Differentiability of q(·) at θˆ is not essential for condition (C2); see theorem 1A in Araujo and Moreira (2010) and the graphical interpretation below.

11

q(θ) cqθ < 0 s(θ) cqθ > 0

θˆ1 θˆ2

θ

θ

Figure 2: necessary conditions at discontinuity

the shaded area in figure 1b is zero. If, for example, the integral in (C1) was positive ˆ = 0, then incentive compatibility would be violated for θ + ε and θˆ as and Φ(θ, θ) R ˆ ≈ Φ(θ, θ) ˆ − ε q(θ) cqθ (q, θ) dq, i.e. the “shaded area” for θ + ε would be the Φ(θ + ε, θ) ˆ q(θ)

same plus some area having the “wrong” sign.

The graphical interpretation also allows to quickly generalize these conditions at points of discontinuity and bunching. This situation is depicted in figure 2. Assume Φ(θ, θˆi ) = 0 for i = 1, 2. To keep incentive compatibility for types close to θ, θˆ1 and θˆ2 the following conditions have to hold:14 ❼ ❼ ❼ ❼

R q− (θ) q(θˆi )

R q+ (θ) q(θˆi )

Rθ

θˆ1

Rθ

θˆ2

cqθ (q, θ) dq ≥ 0 as otherwise Φ(θ − ε, θˆi ) < 0 cqθ (q, θ) dq ≤ 0 as otherwise Φ(θ + ε, θˆi ) < 0

ˆ t) dt ≤ 0 as otherwise Φ(θ, θˆ1 − ε) < 0 cqθ (q(θ), ˆ t) dt ≥ 0 as otherwise Φ(θ, θˆ1 + ε) < 0 cqθ (q(θ),

Given (C1) and (C2), one can use variational calculus to derive a third necessary condition for types at which the incentive constraint binds. While (C1) and (C2) are purely driven by incentive compatibility, this third condition will be derived from the principal’s optimization. The idea is to to have a variation of the optimal decision around θ and θˆ such that the two necessary conditions (C1) and (C2) are still satisfied. The derivation is contained in Araujo and Moreira (2001) and sketched in the webappendix. The following variational condition results: ˆ θ) ˆ − cq (q(θ), ˆ θ)]f ˆ (θ) ˆ [uq (q(θ), θ) − cq (q(θ), θ)]f (θ) [uq (q(θ), ˆ (C3) +1−F (θ) = +1−F (θ) ˆ θ) ˆ cqθ (q(θ), θ) cqθ (q(θ), 14

The superscript “–” (“+”) indicates limits from below (above).

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q(θ)

q(θ)

cqθ < 0 s(θ) cqθ > 0

θ

cqθ < 0 s(θ) cqθ > 0

θ

θˆ

θ4 θ3

(a) no upwards binding

θ2 θ1

θ

(b) no overlap

Figure 3: non-binding constraints

Section 3.2 will give a shadow value interpretation to the terms on both sides of (C3) and thereby provide some intuition for this condition. 3.2. Monotone solution This section gives a characterization of monotone solutions. For now, it is simply assumed that a solution exists and that this solution is monotone. However, continuity of the solution is not assumed. The following section will give sufficient conditions which ensure that the optimal decision is indeed monotone. As pointed out before, the main difficulties are non-locally binding incentive constraints. The following two lemmas show that only a certain subset of non-local incentive constraints can be binding. Lemma 2 implies that incentive constraints cannot be upward binding in monotone solutions. Put differently, no type will be indifferent between the contract designated for him and the contract of a higher type (with the exception of bunched types). The intuition for lemma 2 is the same as in models with single crossing. A higher decision increases the costs for higher types less than for lower types. For a low type, this holds true for all decisions above his own. Lemma 2. No type wants to (non-locally) misrepresent upwards if q(θ) ≥ s(θ); i.e. if ˆ > 0 if θ < θˆ and the two types are not bunched. q(θ) ≥ s(θ), then Φ(θ, θ) The following lemma puts more structure on the ways incentive compatibility constraints can bind. It states that binding non-local incentive constraints cannot overlap; ˆ then no other incentive i.e. if a non-local incentive constraint binds between θ and θ, ˆ θ] and a type outside this interval constraint can bind between a type in the interval [θ, 13

(abstracting from the trivial exception where θˆ is bunched). I will use the following phrase to describe binding non-local incentive constraints: A non-local incentive conˆ = 0. straint binds from θ to θˆ if Φ(θ, θ) Lemma 3. Assume the solution is monotone and a non-local incentive constraint binds ˆ Then non-local incentive constraints cannot bind from any θ′ ∈ [θ, ˆ θ) to from θ to θ. ˆ > q(θˆ′ ). Neither can they bind for any θˆ′ ∈ (θ, ˆ θ] and θ′ > θ with any θˆ′ < θˆ with q(θ) q(θ′ ) > q(θ). Proof. The proof is by contradiction. Suppose, contrary to the lemma, there are types θ1 > θ2 ≥ θ3 > θ4 with Φ(θ1 , θ3 ) = 0 and Φ(θ2 , θ4 ) = 0. Then the incentive constraint between θ1 and θ4 will be violated, i.e. Φ(θ1 , θ4 ) < 0: Z θ1 Z q(t) Φ(θ1 , θ4 ) = − cqθ (s, t) ds dt q(θ4 )

θ4

= − = − +

Z

θ2

θ4 Z θ2

θ4 Z θ2 θ3

Z

q(t)

q(θ4 ) Z q(t)

q(θ4 ) Z q(t)

cqθ (s, t) ds dt − cqθ (s, t) ds dt − cqθ (s, t) ds dt −

θ1

θ2 Z θ1

θ2 Z θ1 θ3

q(θ3 )

= −Φ(θ2 , θ3 ) −

Z

Z

θ1 θ2

Z

q(θ3 )

Z

q(θ3 )

cqθ (s, t) ds dt −

q(θ4 ) Z q(θ3 )

q(θ4 ) Z q(t)

Z

θ1 θ2

Z

q(t)

cqθ (s, t) ds dt q(θ3 )

cqθ (s, t) ds dt

cqθ (s, t) ds dt

q(θ3 )

cqθ (s, t) ds dt < 0 q(θ4 )

The second and third equality are simple splitting up the integral steps (and are easy to visualize in figure 3b), the fourth uses the fact that Φ(θ1 , θ3 ) = Φ(θ2 , θ4 ) = 0 and the last inequality follows from the incentive compatibility between θ2 and θ3 as well as the following idea: By the binding constraint between θ2 and θ4 and the fact that θ2 is R q− (θ ) interior, q(θ4 ) 2 cqθ (s, θ2 ) ds ≥ 0 holds by C1 (with equality if q(θ) is continuous at θ2 ). R q(θ ) By the monotonicity of q, q(θ3 ) ≤ q − (θ2 ) and therefore q(θ43) cqθ (s, θ2 ) ds ≥ 0 (see figure 3b). The inequality above follows then from cqθθ > 0.

As a special case, i.e. with θˆ = θ′ , the preceding lemma includes the following: If ˆ contract, i.e. Φ(θ, θ) ˆ = 0, then no other type θ′ is θ is indifferent between his and θ’s indifferent between his contract and θ’s contract, i.e. Φ(θ′ , θ) > 0 for all θ′ ∈ Θ \ θ. Put differently, incentive compatibility can bind non-locally from a type or to a type but not both. Figure 4 summarizes the two previous lemmas by showing how non-local incentive compatibility constraints can bind in a monotone solution. 14

q(θ) cqθ < 0

s(θ)

cqθ > 0

θˆ1 θˆ0

θ0 θ1θˆ3 θˆ2

θ2 θ3

θ

Figure 4: how incentive constraints can bind One of the contributions of this paper is that binding non-local incentive constraints can affect the solution without leading to irregularities, i.e. discontinuities or bunching. The following lemma shows that some irregularities can be ruled out on the grounds of incentive compatibility alone. ˆ i.e. Φ(θ, θ) ˆ = 0. Lemma 4. Assume a non local incentive constraint binds from θ to θ, The decision is continuous at θˆ if θˆ is not the boundary type of a bunching interval. ¯ Furthermore, θ cannot be bunched if the decision is continuous at θ and θ < θ. After these technical results, it is possible to obtain a qualitative result. If the solution is monotone, non-local incentive compatibility might require “distortions” that are unusual: With single crossing, local incentive constraints are downward binding. This explains why the relaxed solution is below the first best decision: A high type has lower marginal costs than a low type. By distorting the low type’s decision downward, the cost advantage of the high type is reduced, i.e. the low type’s contract becomes less attractive for the high type. Without single crossing, it is no longer clear that a high type has lower marginal costs than a low type at the low type’s decision. Figure 1b, Rθ ˆ t) dt = cq (q(θ), ˆ θ) − cq (q(θ), ˆ θ) ˆ could be positive. for example, illustrates that θˆ cqθ (q(θ),

Therefore, making the low type’s contract unattractive might require increasing the low type’s decision. Informational distortion from local and non-local incentive constraints will then go in opposite directions. In short, binding non-local incentive constraints reduce the usual downward distortion. Proposition 1. A monotone solution is above the relaxed solution, i.e. q(θ) ≥ q r (θ) for all θ. Proof. Suppose q(θ) < q r (θ) for some types. Since local incentive compatibility does 15

not allow downward jumps, q(θ) has to be strictly below q r (θ) for a mass of types. Consider changing this ‘optimal’ decision to q ∗ (θ) where q ∗ (θ) = max{q(θ), q r (θ)}. Transfers t∗ (θ) are determined such that π(θ) = 0 and πθ (θ) = −cθ (q ∗ (θ), θ). As q r (θ) is the optimal relaxed decision, this change will increase the principal’s expected payoff. It remains to check incentive compatibility, i.e Z θ Z q∗ (t) ∗ ˆ =− Φ (θ, θ) cqθ (q, t) dq dt ≥ 0 θˆ

ˆ q ∗ (θ)

ˆ = q(θ), ˆ incentive compatibility follows from for arbitrary types θ and θˆ < θ. If q ∗ (θ) q ∗ (t) ≥ q(t) and as q(t) ≥ s(t) the corresponding ‘additional’ cqθ (q, t) are negative. ˆ > q(θ) ˆ (and therefore q ∗ (θ) ˆ = q r (θ)), ˆ there are three possibilities: (i) There If q ∗ (θ) ˆ θ) with q(θ′ ) = q ∗ (θ), ˆ (ii) all types θ′ ∈ (θ, ˆ θ) have q(θ′ ) < q ∗ (θ) ˆ exists a type θ′ ∈ (θ, ˆ θ) with q(θ′ ) > q ∗ (θ) ˆ but no type θ′ with q(θ′ ) = q ∗ (θ), ˆ and (iii) there are types θ′ ∈ (θ, hence q(·) is discontinuous. ˆ > Φ(θ, θ′ ). In case If (i), then Φ(θ, θ′ ) ≥ 0 implies incentive compatibility as Φ∗ (θ, θ) ˆ has to be above q(θ′ ) for all θ′ ∈ (θ, ˆ θ). But since q(θ′ ) > s(θ′ ) for all these types (ii) q ∗ (θ) ˆ > s(θ) and therefore incentive compatibility is trivially satisfied. it follows that q ∗ (θ) ˆ θ) : q(t) < q ∗ (θ)} ˆ that is θ′ is the jump point. In case (iii) define θ′ = sup{t ∈ (θ, R θ R q(t) Incentive compatibility between θ and θ′ implies θ′ q− (θ′ ) cqθ (q, t) dq dt ≤ 0 as well as R θ R q(t) c (q, t) dq dt ≤ 0 where q − (θ′ ) denotes the limit of q(t) as t → θ′ from below. θ ′ q + (θ ′ ) qθ R R ˆ < q + (θ′ ), it follows that θ′ q(t) From cqqθ < 0 and q − (θ′ ) < q ∗ (θ) ˆ cqθ (q, t) dq dt ≤ 0. θ q ∗ (θ) R R θ q(t) ˆ > − ′ ∗ cqθ (q, t) dq dt ≥ 0 incentive compatibility is satisfied. But as Φ∗ (θ, θ) ˆ θ q (θ) The previous proposition highlights how violations of non-local ic are dealt with in

monotone solutions. This can also be illustrated with figure 1b. Incentive compatibility is violated if the gray area weighted by cqθ is positive. To satisfy incentive compatibility q is raised for all types between θˆ and θ. The additional gray area features cqθ < 0 and therefore the incentive problem is mitigated. One noteworthy point is that the incentive constraint is mainly relaxed by increasing q for types at which the incentive constraint is non-binding; i.e. if (IC) is binding from θ′ to θˆ′ , it is less q(θ′ ) and q(θˆ′ ) that have to be increased but q for the types between θˆ′ and θ′ . To see the intuition, recall that πθ (θ) = −cθ (q(θ), θ) and that cqθ (q(θ), θ) < 0. Therefore, increasing q will raise the slope of the rent function π. Increasing q for types 16

in (θˆ′ , θ′ ) will therefore increase the rent of θ′ at his own contract which relaxes the non-local incentive constraint. The last paragraph illustrates that non-local incentive constraints are potentially difficult to handle: The decision of a type is not only influenced by the incentive constraints binding for him but also by binding incentive constraints of other types. The following theorem structures this intuition and characterizes the solution. Theorem 1. A monotone solution is characterized by the equation [uq (q(θ), θ) − cq (q(θ), θ)]f (θ) + (1 − F (θ))cqθ (q(θ), θ) = η(θ)cqθ (q(θ), θ)

(2)

where η(θ) is a non-negative function with the following properties: ❼ η(θ) is constant on each interval of types for which non-local incentive constraints

are not binding and the decision is strictly increasing. ❼ η(θ) is non-decreasing at types θˆ to which non-local incentive constraints are binding

whenever θˆ is not bunched. ❼ η(θ) is non-increasing at types from which non-local incentive constraints are bind-

ing. ¯ is zero if no non-local incentive constraint is binding from θ. ¯ ❼ η(θ) ❼ η(θ) is zero if no non-local incentive constraint is binding to θ.

Before giving an intuitive interpretation of η, let me briefly sketch the idea behind the proof of the theorem. Given the solution q(θ), one can simply define η(θ) by (2). The properties of η are derived by showing that q could be changed in a way that (i) is incentive compatible and (ii) increases the principal’s payoff if these properties were not satisfied. Figure 5 shows feasible changes when a non-local incentive constraint is binding from θ′ to θˆ′ . Increasing the decision for types slightly below θ′ will relax (or not affect) binding non-local incentive constraints. Since this change relaxes the incentive constraints from types above θ′ to types below θˆ′ , it is then feasible to assign types slightly above θ′ a lower decision, see figure 5. Note that lemma 3 is essential for feasibility as it assures that no non-local incentive constraint is binding to types slightly above θ′ . It can then be shown that such a feasible change would increase the principal’s payoff if 17

q(θ)

q c (θ)

cqθ < 0 s(θ) ˆ q c (θ)

cqθ > 0

θˆ′

θ′

θ

Figure 5: feasible changes

ˆ a different change in the decision is feasible, see figure 5, η(θ) was increasing at θ′ . At θ, ˆ At types where non-local which can be used to show that η(θ) cannot be decreasing at θ. incentive constraints are lax, both kind of changes are feasible and consequently η has to be constant. The properties of η have an intuitive interpretation. The left hand side of (2) is well known from models with single crossing: Increasing q(θ) affects the surplus from type θ but also the rents of all types above θ. Marginally increasing q(θ) will also relax all non-local incentive constraints binding from types θ′ > θ to types θˆ′ < θ, see figure 1b. As these incentive constraints can be expressed as integrals over cqθ (see equation (IC’)), the “amount” by which those non-local incentive constraints are relaxed is given by cqθ (q(θ), θ) which can be found on the right hand side of (2). Consequently, η(θ) could be interpreted as the shadow value of all non-local incentive constraints binding from types θ′ > θ to types θˆ′ < θ. These binding constraints are the same for all types in an interval of types for which non-local incentive constraints are lax, see figure 4. This explains the first property of η. The other properties can also be explained by the shadow value interpretation of η. ˆ then there are more non-local If a non-local incentive constraint is binding to a type θ, incentive constraints binding “over” θˆ + ε than “over” θˆ − ε.15 Consequently, the shadow value of non-local incentive constraints binding over a type has to be higher for θˆ+ε than for θˆ − ε. Put differently, increasing q(θˆ + ε) relaxes more non-local incentive constraints than increasing q(θˆ − ε). Also the last two properties are straightforward: Increasing the decision of the bound15

With binding “over” θ I mean binding from a type θ′ > θ to a type θˆ < θ.

18

ary types does not affect non-local incentive constraints of other types. Furthermore, the interpretation as shadow value provides some intuition for the necˆ when a non-local incentive essary condition (C3) which basically says that η(θ) = η(θ) ˆ This makes sense in light of lemma 3. Because there constraint is binding from θ to θ. is no overlap in binding incentive constraints, the non-local incentive constraints bindˆ Consequently, the shadow value of ing over θ are the same as the ones binding over θ. relaxing those constraints is the same for the two types. Theorem 1 establishes what happens at types where non-local incentive constraints are binding (or slack). Here I want to argue that non-local incentive constraints are ¯ θ) or from and to intervals of types. More binding either at the boundary types (θ, specifically, there are intervals [θ0 , θ1 ] and [θˆ1 , θˆ0 ] such that a non-local incentive constraint is binding from each θ′ ∈ [θ0 , θ1 ] to some θˆ′ ∈ [θˆ1 , θˆ0 ]. Furthermore, a higher θ′ corresponds to a lower θˆ′ (this follows from lemma 3). From theorem 1, it follows that η(θ′ ) = η(θˆ′ ) and η(θ) is increasing (decreasing) on [θˆ1 , θˆ0 ] (on [θ0 , θ1 ]). Lemma 5. If the optimal solution is monotone and the relaxed solution is not implementable16 , the set of types from (to) which non-local incentive constraints bind cannot only consist of isolated interior types.17 At the highest type from which non-local incentive constraints bind, i.e. at θ1 = sup{θ : η(θ) > 0}, η is continuous. The lemma implies that non-local incentive constraints are binding at more than just a finite number of interior types. If θ1 is interior, its definition and the continuity of η imply that η has to be decreasing for types slightly below θ1 . Then theorem 1 implies that non-local incentive constraints bind from these types.18 This leads to the idea that there has to be an interval [θ0 , θ1 ] from which non-local incentive constraints bind.19 Some of the properties of η in theorem 1 hold only at types where the decision is strictly increasing. The reason is that, the way (2) is written, η captures not only the 16 17

The relaxed decision is said to be not implementable if it violates non-local incentive constraints. Isolated means here that for each type θ from (to) which a non-local incentive constraint binds,

there exists a neighborhood of θ in which non-local incentive constraints are slack for all types but θ. 18 Using proposition 4 and the monotonicity constraint, it is easy to show η has to be continuous in a neighborhood of θ1 in this case. 19 Strictly speaking, the is the option that non-local incentive constraints are binding from a Cantor set of interior types. As the following results do not depend on this artificial looking case, I will ignore this possibility and speak of intervals in the remainder of the paper.

19

effect of non-local incentive constraints but also the effect of the monotonicity constraint. If one wants to avoid this cluttering of effects, it is straightforward to introduce a monotonicity parameter ν which captures the effect of the monotonicity constraint. In this case it is easy to see that the properties of η described in theorem 1 extend also to bunched types. Instead of (2) the solution would then be characterized by νθ (θ) = (uq (q(θ), θ) − cq (q(θ), θ))f (θ) + (1 − F (θ) − η(θ))cqθ (q(θ), θ)

(3)

where ν(θ)qθ (θ) = 0 for all θ ∈ Θ, i.e. ν(θ) is the Lagrange parameter of the monotonicity constraint. If the start and ending type of a bunching interval are denoted by θsb and θeb , R θb then obviously θbe νθ (θ) dθ = 0. As described in the existing literature on ironing, see s

Guesnerie and Laffont (1984) or the exposition in Fudenberg and Tirole (1991, ch. 7), the bunching interval is characterized by this last condition and the endpoint conditions ν(θsb ) = ν(θeb ) = 0. Lemma WA3 in the webappendix formally shows that η in (3) has

the properties of theorem 1 also for bunched types. 3.3. Distortion for all types “No distortion at the top” is probably the most famous result of the screening literature. The idea underlying distortion is that distortion at a certain type allows to reduce the ¯ there are less and less rent of all higher types. As one moves across types towards θ, higher types and therefore the distortion vanishes. Eventually, there is no distortion for ¯ θ. ¯ It is Recall that the necessary condition (C1) might hold with inequality at θ = θ. therefore possible that non-local incentive constraints bind from θ¯ to several non-bunched θˆ even if the solution is continuous. Note that this is impossible for interior types: For ˆ a given q(θ), (C1) and (C2) will uniquely determine θˆ and q(θ). Now consider the case where the non-local incentive constraint binds not only to several but to a mass of types θˆ (or to θ). Then the shadow value η(θ) will be strictly ¯ Hence, these types have a positive and bounded away from 0 for types slightly below θ. decision q(θ) which is at least ε away from their relaxed decision q r (θ) for some ε > 0. The same has then to apply for θ¯ because of the monotonicity constraint. Put differently, ¯ > 0 and therefore q(θ) ¯ is distorted: There is distortion at the top. η(θ) The main difference to the “no distortion at the top” intuition is the following: The effect of distortion on the binding non-local incentive constraint does not vanish as we 20

¯ Increasing the decision of any type below θ¯ will–through move across types towards θ. the slope of the rent function–increase the rent of θ¯ and therefore relax the binding non-local incentive constraints. The discussion in this subsection points to an even more striking result. The optimal contract can assign distorted decisions to all types, i.e. no type will produce at his first best decision, even if the optimal decision schedule is continuous. This will happen if a non-local incentive constraint is binding from θ¯ to θ and η(θ) > 1. Intuitively (see figure 1b), this will be the case whenever the first best decision is very close to the sign switch function s. Section 4.2 contains a numerical example where optimally all types’ decisions are strictly above first best which implies the following theorem. Theorem 2. The optimal contract can have upward distorted decisions for all types, i.e. q(θ) > q f b (θ) ∀θ. This result differs from results in the literature on type dependent participation constraints, see for example Jullien (2000). In this literature, the decision can also be distorted above as well as below first best but there is always at least one type with an undistorted decision. Also in the inverse U-shape solutions of Araujo and Moreira (2010) there is always one type with an undistorted decision.

4. Sufficient conditions for monotone contracts 4.1. Existence and monotonictiy As already indicated, solutions can be monotone or inversely U-shaped (or even jumping over s(θ) discontinuously). It is therefore useful to have a sufficient condition under which the solution is monotone. To get such a sufficient condition, a technical condition has to be added to the assumptions in section 2. This technical assumption is sufficient to rule out solutions jumping discontinuously over s(θ). Given this, assumption 2 ensures that the solution is not inversely U-shaped but monotone. To state this technical condition two associated decisions have to be defined for each decision q below s(θ): First, q s (q, θ) is defined by cθ (q, θ) = cθ (q s (q, θ), θ); i.e. q s (q, θ) is the decision that would lead to the same slope of the rent function π(θ) as q at type θ. Second, q v (q, θ) is the decision which gives the same virtual valuation as q, i.e.

21

[u(q, θ) − c(q, θ)]f (θ) + (1 − F (θ))cθ (q(θ), θ) is the same for q and q v (q, θ). Since cθ (q, θ) and (RP) are concave in q, the two associated decisions are well defined. ¯ then an Proposition 2. If q v (q, θ) ≥ q s (q, θ) for all q ∈ [0, s(θ)] and all θ ∈ [θ, θ], optimal contract exists and the optimal decision q is above s and increasing in type. Note that the imposed condition is automatically satisfied for q close to s(θ) as q r (θ) > s(θ) by lemma 1. Hence, the condition roughly states that q s (q, θ) is not much steeper in q than q v (q, θ). This holds, for example, true if {u(q, θ) − c(q, θ)}f (θ) + (1 − F (θ))cθ (q, θ) and cθ (q, θ) are both symmetric in q as then qqv (q, θ) = qqs (q, θ) = −1. To illustrate, take example 1 from section 1.1 and assume that u(q, θ) = Sq with S > 0. Straightforward calculation shows that in this case q v (q, θ) = 2q r (θ) − q and q s = θ2 − q = 2s(θ) − q. Hence, qqv (q, θ) = −1 = qqs (q, θ) and–regardless of the type distribution –the conditions of the proposition are satisfied. 4.2. Continuity This subsection has two goals: First, to provide sufficient conditions under which a monotone solution is continuous and, second, to introduce an algorithm for determining such a continuous solution. The first sufficient condition for continuity is loosely based on the idea of having a one-to-one relationship between η and q for a given type θ; i.e. the idea that for a given type θ and η(θ) > 0, equation (2) yields a unique solution for q. The condition in the proposition ensures this and also ascertains that this relationship is monotonic, i.e. a higher η(θ) results in a higher q. Furthermore, the same condition ensures that the principal’s objective is concave when the optimization problem is written as an optimization over the rent profile π(θ) instead as an optimization over q(θ), see the proof of proposition 6 or Jullien (2000). Proposition 3. A monotone solution is continuous if uqq (q, θ) − cqq (q, θ) uq (q, θ) − cq (q, θ) > cqqθ (q, θ) cqθ (q, θ)

(CVR)

holds for all types and all q ≥ q f b (θ).20 20

Obviously, it is enough if the condition holds for all q ∈ [q f b (θ), q¯] where q¯ is defined as in the

webappendix (section B.1).

22

To illustrate, take the cost function in example 1 in section 1.1 and assume that u(q, θ) = Sq. It turns out that (CVR) is equivalent to the condition for q f b (θ) > s(θ), ¯ 21 i.e. S > 2θ. The following proposition gives an alternative condition under which the optimal solution is below the first best decision. Having a solution below first best turns out to be sufficient for continuity and strict monotonicity of the solution. This is in itself remarkable. As the relaxed solution is below first best, one should expect the solution to be below first best whenever non-local incentive constraints are not violated “too much” by the relaxed solution. Hence, there is a broad class of problems in which the solution will be strictly monotone and continuous. Furthermore, the proof of the following proposition shows that the property holds also locally. That is, if the decision is below q f b on some interval (θ1 , θ2 ), then the decision will be strictly increasing and continuous on (θ1 , θ2 ). Before stating the proposition some additional notation is needed. Define q m (θ) such that cθ (q f b (θ), θ) = cθ (q m (θ), θ), i.e. q f b (θ) = q s (q m (θ), θ). Hence, q m is the decision schedule below the sign switch decision s which would lead to the same slope of the rent function as q f b . Proposition 4. The optimal solution is strictly increasing and continuous at all types where it is below first best. Assume that q m (θ) is non-decreasing and that there is no distortion at the top.22 Then the optimal solution is below first best and therefore continuous and strictly increasing. One example for a class of function where q m is increasing are cost functions of the form c(q, θ) = θq + φ(q − αθ) + γ(θ) where φ(·) is a function of which the first three derivatives are positive.23 With such a cost function, any increasing and concave benefit function u(q, θ) with uqθ = 0 and q f b (θ) > s(θ) yields an increasing q m . Now I want to turn to the issue of calculating a solution. In principle, the solution is already described by (2), the properties of η and the necessary conditions C1, C2 and C3. 21

In fact, this also holds true if q 2 in the cost function is replaced by any increasing and convex

function. 22 See the following section for a simple sufficient condition for no distortion at the top. 23 The interpretation of this cost function is that there is a “normal scale” of αθ. Producing above this normal scale is increasingly costly. Type reflects a tradeoff between the size of the normal scale and marginal cost when producing within the normal scale.

23

If a non-local incentive constraint binds from a type θ, the three necessary conditions ˆ q(θ) and q(θ) ˆ (assuming that there is a unique solution). could be used to determine θ, If non-local incentive constraints are slack at a type θ, (2) can be used to calculate q(θ) where η(θ) equals η(θˆ′ ) with θˆ′ being defined as the next lower type to which a nonlocal incentive constraint is binding. However, a more structured alternative to obtain a continuous solution might be helpful. This alternative will also give some additional insights into the logic behind the solution. The algorithm is based on proposition 5. The rough idea behind this proposition is the following. Take the optimal contract q as given and say a non-local incentive constraint binds between θ′ and θˆ′ , i.e. Φ(θ′ , θˆ′ ) = 0. Then both types have the same η, call it η ′ , according to (C3). Furthermore, (θ′ , θˆ′ ) ˆ I will now argue that (θ′ , θˆ′ ) still minimizes Φ if the optimal decision minimizes Φ(θ, θ). q is manipulated in the following way: According to theorem 1, η is decreasing at θ. Now let us manipulate q such that η defined by (2) is constant (and equal to η ′ ) in some neighborhood of θ′ . This implies that the decision of types slightly below θ′ is decreased and the decision of types slightly above θ′ is increased. This manipulation makes Φθ (θ, θˆ′ ) lower for θ = θ′ − ε, higher for types θ = θ′ + ε and leaves it at zero for θ = θ′ . Consequently, (θ′ , θˆ′ ) are minimizers of this manipulated Φ. The following proposition shows that this holds still true if we manipulate q not only in a neighborhood of θ but for all types, namely if all types are assigned η ′ . The proposition, gives, for this case, even some non-local minimizer properties for (θ′ , θˆ′ ). ˆ as Φ(θ, θ) ˆ under q(θ, η) where q(θ, η) is defined as the Proposition 5. Define Φη (θ, θ) q solving {uq (q, θ) − cq (q, θ)}f (θ) + (1 − F (θ) − η)cqθ (q, θ) = 0. If the incentive constraint binds between θ′ and θˆ′ in a continuous solution q(θ), then ˆ on [θˆ′ , θ′ ] where η = η(θ′ ) = η(θˆ′ ). Furthermore, Φη (θ′ , θˆ′ ) < (θ′ , θˆ′ ) minimize Φη (θ, θ) Φη (θ′′ , θˆ′′ ) for any θ′′ > θ′ and θˆ′′ < θˆ′ . To get a feeling for this proposition, take η = 0. Then q(θ, 0) = q r (θ). Denote ˆ by (θr , θˆr ). Although a little extra work is needed, the the global minimizer of Φ0 (θ, θ) following result follows almost directly from proposition 5: Corollary 1. If the relaxed solution is not implementable, the non-local incentive constraint from θr to θˆr will bind in the optimal decision. If one of the two types (both) is 24

interior, his (their) optimal decision is the relaxed decision; i.e. q(θ) = q r (θ) or (and) ˆ = q r (θ) ˆ respectively. q(θ) The proposition then says that a similar logic applies for all pairs (θ′ , θˆ′ ) at which non-local incentive compatibility is binding: One only has to replace q r (θ) in the corollary by the corresponding q(θ, η). This q(θ, η) is the decision that would result from (2) if all types had the same η(θ) and this η(θ) would equal η(θ′ ) in the optimal decision. The last proposition in connection with theorem 1 gives a method for determining q(θ). One obtains the types for which non-local incentive constraints bind (as well as their η and their decisions) from the minimization of Φη (·). For all remaining types, the decision is determined by (2) and the fact that η is constant: 1. Solve (2) for q(θ, η) (assuming η ≥ 0). ˆ = 2. Calculate Φη (θ, θ)

R θˆ θ

ˆ η), t) dt = cθ (q(t, η), t) − cθ (q(θ,

R θ R q(t,η) θˆ

ˆ q(θ,η)

−cqθ (q, t) dq dt.

ˆ over θ and θ. ˆ Denote by θ(η) and θ(η) ˆ 3. Minimize Φη (θ, θ) the minimizers that ˆ ˆ θ] and (ii) there satisfy: (i) (θ(η), θ(η)) globally minimize Φη (·) on the interval [θ, ˆ For exists no Φη (·) minimizer (θ′ , θˆ′ ) with θ′ > θ, θˆ′ < θˆ and Φη (θ′ , θˆ′ ) < Φη (θ, θ). now, assume there is only one such pair for every η. ˆ ≥ 0. 4. Define η¯ as the smallest η such that Φη (θ(η), θ(η) 5. The solution is then the following: ❼ types from/to which non-local incentive constraints are binding: For all η ∈

ˆ ˆ [0, η¯], q(θ(η)) = q(θ(η), η) and q(θ(η)) = q(θ(η), η). ˆ ❼ types with η = 0: For types above θ(0) and below θ(0), the optimal decision is the relaxed decision. ❼ types where non-local incentive constraints are slack but η > 0: All remaining

types have the η of the next highest type from which a non local incentive ˆ ˆ constraint binds, i.e. η(θ) = η(inf {θ(η) : 0 ≤ η ≤ η¯ and θ(η) < θ < θ(θ)}). The optimal decision for these types is q(θ, η(θ)). ˆ One remark on the possibility that several relevant pairs (θ(η), θ(η)) exist. For example, say there exist the pairs (θ1 (η), θˆ1 (η)) and (θ2 (η), θˆ2 (η)) both satisfying (i) and 25

(ii) above. The non-local incentive constraint could in this case bind from an interval [θ0 , θ1 ] to the interval [θˆ1 , θˆ0 ] as well as from the interval [θ2 , θ3 ] to the interval [θˆ3 , θˆ2 ] where θˆ1 < θˆ0 < θ0 < θ1 < θˆ3 < θˆ2 < θ2 < θ3 ; see figure 4 for an illustration. Indeed one has to be a bit more precise in this case: There will be different η¯ for the two “brackets” of binding incentive constraints. In this case, η(θ) will not be single peaked. Hence, the algorithm will then be applied to the two brackets separately and nothing else changes. A second remark has to be made with regard to bunching. Some types might have an ironed out solution. This solution is then not q(θ, η(θ)) as described above but an ˆ ironed out version of it. The condition for determining η¯, i.e. Φη (θ(η), θ(η)) = 0 has to hold for the ironed out decision whenever ironing is relevant; see the webappendix for an example. One does not have to worry about ironing as long as η ≤ 1 − F (θ(η)): This implies q(θ) ≤ q f b (θ) for all types for which bunching could have been possible and the decision will be strictly increasing (see proposition 4). I want to illustrate the sufficient conditions and the algorithm with one numerical example. Amend example 1 in the following way: c(q, θ) = qθ + u(q, θ) = f (θ) =

θ 1 q2 − 2θ 10

21 qθ 10   4, if 1/4 ≤ θ ≤ 1/2  0,

else .

Hence, s(θ) = θ2 and the first best decision q f b (θ) =

11 2 θ . 10

The relaxed solution can be

easily calculated as 1 q r (θ) = θ2 + θ3 . 5 The example is illustrated in figure 6. As the first best decision is close to s for all types, one would expect distortion above first best and a non-local incentive constraints to bind between the boundary types. Indeed this is shown below. To check the sufficient condition for monotonicity from proposition 2 the functions q s and q v have to be calculated: 2 q v (q, θ) = −q + 2θ2 + θ3 5 s 2 q (q, θ) = −q + 2θ

26

Clearly, q v (q, θ) > q s (q, θ) and therefore the solution has to be monotone. Also (CVR) is satisfied as it boils down to −1 θ −1 θ2

>

⇔0 >

− θq 1 − θq2

11 θ 10

1 2 θ 10 θ2 − q

which holds for all q > s(θ) = θ2 . Hence, the solution is continuous.24 Next use the algorithm to compute a solution: 1. Solving (2) for q yields q(θ, η) = θ2 +

2θ3 . 5(2 − η)

2. Computing Φη is slightly cumbersome but possible: Z θ η ˆ η), t) − cθ (q(t, η), t) dt ˆ cθ (q(θ, Φ (θ, θ) = θˆ θ  4θˆ6 4θˆ5 4 ˆ 3 5 3 ˆ + θ + t 2t 5(2−η) 25(2−η)2 ˆ + 2θ  = θt t+ − + 5(2 − η) 2t 6 125(2 − η)2

θˆ

= θˆ2



 ˆ θˆ4 + 4ˆ 2θˆ3 (θ − 2θ) θ− θ + + 3 5(2 − η)

4θˆ5 5(2−η)

+

4θˆ6 25(2−η)2

2θ

θ3 2θ5 − 12θˆ5 − + 6 125(2 − η)2

3. The first order conditions for minimization, i.e. the equivalents of (C1) and (C2), are 1 2θˆ3 − 2 θˆ2 + 5(2 − η) 2θ

4θˆ6 4θˆ5 + θˆ4 + 5(2 − η) 25(2 − η)2 ≤

1 θ+ θ

2θˆ3 θˆ2 + 5(2 − η)

!

≥ 2θˆ +

!

θ2 2θ4 − 2 25(2 − η)2 2θˆ2 5(2 − η)

with “=” if θ < θ¯ (4)

with “=” if θˆ > θ.

(5)

It is easy to verify that the first order conditions are satisfied for the boundary types θ¯ = 1/2 and θ = 1/4 for all relevant η. It is a bit more tedious to verify that there is no interior solution, i.e. that there are no other (local) minimizers. This is done in the webappendix. Hence, the only binding non-local incentive constraint will be the one between the boundary types. 24

As pointed out in the following subsection condition (6) is satisfied as well and therefore the optimal

contract cannot be stochastic.

27

q(θ)

0.3

q f b (θ) s(θ)

0.1 q r (θ) 0.25

0.5

θ

Figure 6: numerical example: all types are distorted

¯ θ) = 0. 4. For η¯ ≈ 1.67, Φη¯(θ, 5. For all types, η(θ) = 1.67. Hence, 2θ3 . q(θ) = θ + 5 ∗ 0.33 2

Since η(θ) > 1 for all types, all types have decisions above first best. 4.3. Stochastic contracts So far, this paper concentrated on deterministic contracts. Although hardly observed in practice, one could think of stochastic contracts. In the framework of this paper, this would mean that a type θ is assigned a probability distribution over the decision q instead of one deterministic decision q(θ). The idea behind a stochastic contract is to relax (non-local) incentive constraints. Intuitively, this could work if different types have different degrees of risk aversion. See Rochet (2009) for an example where random contracts are optimal. The following proposition gives a sufficient condition under which deterministic decisions are optimal. Proposition 6. The optimal decision is deterministic if the assumptions of proposition 2, (CVR) and ∂

cqθθ cqθ

∂q 28

≥0

(6)

hold. Condition (CVR) and (6) differ from the conditions for non-stochastic contracts in Maskin and Riley (1984). In Maskin and Riley (1984), only local incentive constraints bind and they bind “downward”. It is then shown that assigning the expected decision increases the principal’s payoff and relaxes local incentive constraints if risk aversion is decreasing in type. Decreasing risk aversion is therefore a sufficient condition for the optimality of deterministic contracts. This reasoning is flawed in case non-local incentive constraints are binding: Assigning the expected decision decreases the slope of the rent function π(θ) because −cθ is convex in q. Hence, profit differences between θ and θˆ are smaller under the expected decision compared to the stochastic contract, i.e. non-local incentive constraints are harder to satisfy. Proposition 6 takes therefore another way known from Jullien (2000). When rewriting the principal’s optimization problem as an optimization over rent profiles π (instead of over decision q) condition (CVR) ensures that the resulting program is concave. Condition (6) ensures that the set of implementable utility profiles is convex.25 These two properties imply that a stochastic decision is worse for the principal than a deterministic decision implementing the same utility profile. The conditions of proposition 2 allow to focus on decisions above s(θ), i.e. monotone solutions. 4.4. No distortion at the top There are sufficient conditions for having no distortion at the top. Corollary 1 allows to ˆ where Φ0 (·) is formulate such a condition. If θ¯ is not the global minimizer of Φ0 (θ, θ) Φ(·) under the relaxed solution q r (·), then non local incentive constraints cannot bind ¯ Therefore, the relaxed decision is optimal for θ¯ implying that q(θ) ¯ = q f b (θ). ¯ from θ. Another sufficient condition for no distortion at the top can be formulated using (C1): R qf b (θ) ¯ ¯ dq ≤ 0 is sufficient since (C1) cannot hold with inequality in this case. cqθ (q, θ) 0

5. Discussion This section discusses relaxing the assumptions and possible generalizations. First, note that for the results in section 3 the assumptions on uqθ , uqq , cqq and the monotone hazard 25

To illustrate: In example 1,

cqθθ cqθ

=

−2 θ

and condition (6) holds.

29

rate assumption in 1 can be relaxed as long as the relaxed program remains concave and the relaxed decision remains strictly increasing, i.e. as long as the properties in lemma 1 hold. These properties of the relaxed program are used for showing proposition 1, i.e. that non-local incentive constraints lead to upwards distortion. The assumption that uqq − cqq ≤ 0 and the monotone hazard rate assumption are only used in the proof of proposition 4. Theorem 1 holds more generally. The only requirement for this theorem is that the variations indicated in figure 5 are feasible. Feasibility follows from lemma 2 and 3 which only require that the third derivatives cqqθ and cqθθ do not change sign. Hence, theorem 1 holds independent of assumptions on u or F . This leads to the question how important the assumptions on the third derivatives of c are. It is not important which signs these third derivatives have as long as they do not switch sign. This is illustrated with an example in the webappendix. However, the assumption that they do not switch sign is necessary to make the analysis tractable. For illustration, suppose that the third derivatives of c can switch sign in the relevant range of q and θ. Then we could have a second function s2 (θ) where cqθ changes sign. Consider figure 3a and imagine such an s2 function somewhere above q and going through the shaded area. It is clear that in this case upwards binding incentive constraints can no longer be ruled out, i.e. lemma 2 no longer holds (also lemma 3 is no longer valid). Nevertheless, theorem 1 will not become irrelevant in such a framework: As long as the variations in figure 5 are feasible, theorem 1 holds. In particular, η has to be constant on an interval of types to and from which no non-local incentive constraints are binding. Similarly, η has to be decreasing at a type θ satisfying (i) a non-local incentive constraint is binding from θ to θˆ < θ and (ii) no non-local constraint is binding to θ.

6. Conclusion This paper characterizes monotone solutions in a screening environment where single crossing is violated. Although the model restricts itself to a one time violation of single crossing, the main effects of a violation of single crossing can be illustrated. Non-local incentive constraints can be binding. The distortion caused by non-locally binding incentive constraints can counteract the normal rent extraction distortion. The solution

30

can therefore be partly above as well as below the first best decision. Distortion at the top occurs if non local incentive constraints are binding from the top type to a mass of types or to the lowest type. In the latter case, the decision of all types can be upward distorted. Possible applications can be found in various fields of economics. While the paper uses the notation of a regulation or procurement setting, the same model is applicable, for example, in models of labor, insurance or monopoly pricing. It is hard to judge how common distortions above first best are in these applications. However, there is some evidence of it in non-linear pricing environments. Telecommunication could be an example for this:26 Consumers often buy packages where an additional unit of calling (or internet use) is for free. If the marginal costs of the provider are only ε above zero, such a price scheme will lead to consumption above the socially optimal consumption. Overconsumption in terms of quality–even at the top–is already described in Dupuit (1849) who describes non-linear pricing in train travel. After explaining why price discrimination motives lead to a downward distortion of quality in the third and second class, he continues: And it is again for the same reason [i.e. incentive compatibility] that the companies, having proved almost cruel to third-class passengers and mean to second-class ones, have become lavish in dealing with first class passengers. Having refused the poor what is necessary, they give the rich what is superfluous.

26

Restaurants with “all you can eat” offers or free refills on drinks could be another example.

31

Appendix: Proofs Proof of lemma 1: see webappendix Proof of lemma 2: Recall that local incentive compatibility requires monotonicity of q(θ), i.e. q(θ) has to be monotonically increasing as q(θ) ≥ s(θ). Now take θˆ > θ. By (IC’), incentive compatibility requires ˆ =− Φ(θ, θ)

Z

θ

ˆ θˆ Z q(θ)

cqθ (s, t) ds dt ≥ 0.

q(t)

But the last inequality holds automatically since q(θ) ≥ s(θ) and qθ (θ) ≥ 0. This implies that the integrand is non-positive for all (s, t) in question. Figure 3a gives a graphical representation of this. ˆ Take a Proof of lemma 4: First, it is shown that there cannot be a discontinuity at θ. type θˆ to which non-local incentive constraint is binding from some type θ. Suppose that ˆ i.e. q − (θ) ˆ < q + (θ) ˆ by local incentive compatibility (monotonicq(·) is discontinuous at θ, R θ R q(t) ity). Binding incentive constraint means that either (i) θˆ q− (θ) ˆ cqθ (q, t) dq dt = 0 or (ii) R R R θ R q(t) θ q(t) 27 − ˆ + ˆ ˆ ˆ cqθ (q, t) dq dt = 0. ˆ cqθ (q, t) dq dt = 0 or (iii) q (θ) < q(θ) < q (θ) and θˆ q(θ) θˆ q + (θ) Rθ ˆ t) dt ≤ 0 which is just (C2) adapted to In case (i) it must hold that θˆ cqθ (q − (θ),

apply for a right hand side discontinuity, i.e. if this did not hold incentive compatibility R θ R q+ (θ) ˆ would be violated for θ and θˆ − ε. But then θˆ q− (θ) ˆ cqθ (q, t) dq dt < 0 from cqqθ < 0. R θ R q+ (θ) ˆ ˆ− Hence, Φ(θ, θˆ+ ) = Φ(θ, θˆ− ) + θˆ q− (θ) ˆ cqθ (q, t) dq dt < 0 as Φ(θ, θ ) = 0 by assumption. ˆ This is the Hence, incentive compatibility is violated from θ to types slightly above θ.

desired contradiction. R R +ˆ ˆ t) dt ≥ 0. But then ˆθ q− (θ) cqθ (q + (θ), ˆ cqθ (q, t) dq dt > θ q (θ) R R ˆ θ q + (θ) < 0. Consequently, Φ(θ, θˆ− ) = Φ(θ, θˆ+ ) − θˆ q− (θ) ˆ cqθ (q, t) dq dt < 0 and

In case (ii) it must hold that 0 from cqqθ

Rθ θˆ

ˆ therefore incentive compatibility is violated from θ to types slightly below θ. Rθ ˆ t) dt ≤ 0 while the In case (iii) the same arguments as in case (i) apply if θˆ cqθ (q(θ), Rθ ˆ t) dt > 0. same arguments as in case (ii) apply if θˆ cqθ (q(θ), Second, it is shown that θ < θ¯ cannot be bunched with some type θ′ if q(·) is

continuous at θ. Suppose θ and θ′ were bunched on q b (and by monotonicity all types in 27

ˆ and q + (θ) ˆ are offered. Only one Case (iii) does not imply that all/several decisions between q − (θ)

decision is offered for each type and at a discontinuity this decision might be strictly between the two limits (case (iii)).

32

ˆ < 0 and ic is between them are as well) and suppose for now θ < θ′ . But then Φ(θ′ , θ) violated as ˆ = − Φ(θ , θ) ′

Z

θ′ θˆ

Z

q(t)

cqθ (s, t) ds dt = −

ˆ q(θ)

ˆ − = Φ(θ, θ)

Z

θ′ θ

Z

qb ˆ q(θ)

Z

θ θˆ

Z

q(t) ˆ q(θ)

cqθ (s, t) ds dt −

Z

θ θ

Z

q(t) ˆ q(θ)

cqθ (s, t) ds dt

cqθ (s, t) ds dt < 0

where the last inequality follows from (C1) and cqθθ > 0. Now suppose θ > θ′ and both types are bunched. From condition (C1) for θ < θ¯ R q(t) and cqθθ > 0 it follows that q(θ) ˆ cqθ (q, t) dq < 0 for every t ∈ (θ − ε, θ). But then R θ R q(t) ˆ = Φ(θ, θ) ˆ + Φ(θ − ε, θ) ˆ cqθ (q, t) dq dt < 0, so incentive compatibility would be θ−ε q(θ)

violated.

Proof of theorem 1: Note that even if the theorem was not true one could still define a function η(θ) by rearranging (2). What one has to show are the properties of this function.28 η(θ) ≥ 0 follows immediately from proposition 1 and the fact that the left hand side of (2) is decreasing in q. Next I show that η(θ) is constant on an interval of types on which non local incentive constraints are lax. Suppose to the contrary that η(θ) is not constant. In particular, suppose that η is increasing at some type θ1 , i.e. either ηθ (θ1 ) > 0 or η makes an upward jump at θ1 .29 Now define for types θ above θ1 a corresponding type θ′ (θ) = θ1 − (θ − θ1 ). I will show that one can change such a decision on (θ1 − ε, θ1 + ε) in a way which increases the principal’s payoff (while keeping incentive compatibility). This contradicts the optimality of q(θ). Consider a changed decision q c (·) such that (i) q c (θ) > q(θ) on (θ1 − ε, θ1 ), (ii) q c (θ) ≤ R qc (θ′ ) q(θ) on (θ1 , θ2 +ε), (iii) for corresponding types θ and θ′ it holds that q(θ′ ) cqθ (q, θ′ ) dq = R qc (θ) − q(θ) cqθ (q, θ) dq and (iv) qθc (θ) ≥ 0 on (θ1 − ε, θ1 + ε). The changed decision will

therefore display upwards jumps at θ1 − ε and θ1 + ε. For small changes in q, (iii) can be written as δ(θ′ )cqθ (q(θ′ ), θ′ ) = −δ(θ)cqθ (q(θ), θ) where δ(θ) = q c (θ) − q(θ). This in turn can be written as δ(θ′ ) = −δ(θ)k(θ) where k(θ) is defined as 28

cqθ (q(θ),θ) . cqθ (q(θ ′ (θ)),θ ′ (θ))

As q is monotonic, it is differentiable almost everywhere and continuous almost everywhere. These

two properties are then also inherited by η. 29 If η is not constant, there has to exist a type where either ηθ exists and is not zero or a type where η has a discontinuity. This is true because η is differentiable almost everywhere. The case of a decreasing η is dealt with later.

33

Before proceeding, let me show that a function q c (θ) satisfying (i)-(iv) exists. Note that k(θ1 ) = 1 and that–due to the differentiability and continuity assumptions on c(·) and the monotonicity of q(θ)–the function k(θ) is continuously differentiable almost everywhere.30 First, consider the case where kθ+ (θ1 ) < 0. Then it is feasible to set q c (θ) = q(θ1 ) for types θ ∈ [θ1 , θ1 + ε) if ε > 0 is chosen small enough. Feasibility means that determining q(θ′ ) by δ(θ′ ) = −δ(θ)k(θ) will satisfy all conditions especially (iv). Feasibility of q c (θ) = q(θ1 ) for θ ∈ [θ1 , θ1 + ε] and monotonicity of q(θ) imply that q c∗ = αq c (θ) + (1 − α)q(θ) is also feasible. The effect of a marginal change of q is the effect changing q(·) to q c∗ (·) as α → 0. Second, consider kθ (θ1 )+ > 0. By the same argument, it is feasible to bunch types θ ∈ (θ1 −ε, θ1 ] on q(θ1 ) and the remaining argument goes through analogously. Obviously, the third case kθ+ (θ2 ) = 0 is analogous to either the first or the second case (depending on the second derivative). The effect of a marginal change on the principal’s objective is Z

θ1 +ε

{(uq (q(θ), θ) − cq (q(θ), θ))f (θ) + (1 − F (θ))cqθ (q(θ), θ)} δ(θ) dθ θ1 −ε

=

Z

θ1 +ε

η(θ)cqθ (q(θ), θ)δ(θ) dθ = θ1 −ε

Z

θ1 +ε

δ(θ)cqθ (q(θ), θ)[η(θ) − η(θ′ (θ))] > 0 θ1

where the last inequality follows (for ε > 0 small enough) from δ(θ) ≤ 0 for θ ∈ [θ1 , θ1 +ε) and the assumption that η is increasing at θ1 . Hence, the principal’s objective increases. Due to (iii) incentive compatibility is still satisfied. This contradicts the optimality of q(θ). A similar argument can be made when η is decreasing at some type in an interval where non-local ic is lax. The only difference is that (i) and (ii) are substituted by (i) q c (θ) < q(θ) on (θ1 − ε, θ1 ), (ii) q c (θ) ≥ q(θ) on (θ1 , θ1 + ε). The argument for existence is then that for kθ (θ1 ) < 0 one can choose a θ1 + ε such that setting q c (θ) = q(θ1 + ε) for all θ ∈ [θ1 , θ1 + ε] is feasible. Everything else goes through accordingly. Hence, η(θ) is constant on all intervals on which non-local incentive constraints do not bind. To see that η(θ) is non-decreasing at types θˆ to which a non-local incentive constraint 30

Note that a feasible q c (θ) exists even around types θ1 where q(θ) is discontinuous: Whether bunching

types (θ1 − ε, θ1 ) on q − (θ1 ) or bunching types (θ1 , θ1 + ε] on q + (θ1 ) is feasible is then decided by kθ+ (θ1 ) just as in the text.

34

is binding one can use the same steps as above for types where non-local incentive constraints were lax. The key insight is that such a change is feasible due to the structure given by lemma 2 and lemma 3 (see also figure 4): Increasing q for slightly higher types ˆ will relax (or not affect) binding than θˆ (and reducing for slightly lower types than θ) non-local incentive constraints because these constraints are downward binding and not overlapping. The argument why η(θ) is non-increasing at types θ from which non-local incentive constraints bind is also equivalent to the one above. The key with respect to feasibility is now that reducing q for types slightly below θ (and increasing for types slightly above θ) will again relax (or not affect) binding non-local incentive constraints because these constraints are downward binding. ¯ = 0 (and therefore q(θ) ¯ = q f b (θ)) ¯ whenever no non-local incentive Now turn to η(θ) ¯ Clearly, q(θ) ¯ does not affect non-local incentive constraints constraint is binding from θ. of other types, see figure 1b for an illustration. Consequently, the principal’s payoff is ¯ = q r (θ). ¯ The only thing to show is that the monotonicity maximized by setting q(θ) ¯ Suppose to the contrary that types [θ′ , θ] ¯ were bunched on constraint is not binding at θ. ¯ By lemma 4, non-local incentive constraints cannot be binding for types in q b > q f b (θ). ¯ First, note that q(θ) has to be continuous at θ′ as otherwise the principal’s payoff (θ′ , θ]. could be increased by reducing q b . Therefore–by the same argument as in the proof of lemma 4–non-local incentive constraints cannot bind from types [θ′ − ε, θ′ ] for some small ¯ > q r (θ) for all θ ∈ [θ′ − ε, θ), ¯ the principal’s payoff could ε > 0. Given that q(θ) > q f b (θ) ¯ This contradicts the be increased by changing q(θ) to q(θ′ − ε) for all θ ∈ [θ′ − ε, θ]. optimality of q(θ). The part that η(θ) = 0 if no non-local incentive constraint is binding to θ is even simpler: Reducing q(θ) to q r (θ) cannot violate the monotonicity constraint as q(θ) ≥ q r (θ) ≥ q r (θ) by proposition 1. Proof of lemma 5:

The proof is by contradiction. Suppose, non-local incentive

constraints bound only from isolated interior types. Denote by θ1 the supremum of all types with η(θ) > 0, i.e. θ1 = sup{θ : η(θ) > 0}. By theorem 1, a non-local incentive constraint is binding from θ1 and η(θ) = 0 for all θ > θ1 .31 As the set of types from which non-local incentive constraints bind consists only of isolated types, there exists an ε > 0 31

Note that θ1 cannot be bunched because of proposition 4 and q − (θ1 ) = q r (θ1 ).

35

such that non-local incentive constraints are slack for all θ ∈ (θ1 − ε, θ1 ). By theorem 1, η(θ) is constant on (θ1 − ε, θ1 ) and by the definition of θ1 there has to be a discontinuity in η(θ) at θ1 , i.e. η − (θ1 ) > η + (θ1 ) = 0. The definition of η(θ) in (2) implies then that q − (θ1 ) > q + (θ1 ) = q r (θ1 ). But this violates the monotonicity constraint. Hence, θ1 cannot be isolated in the set of types from which non-local incentive constraints bind. The previous argument also implies that η has to be continuous at θ1 . Similarly, take θˆ1 = inf(θˆ :

ˆ > 0). It holds that η(θ) = 0 for all θ < θˆ1 . η(θ)

Therefore, by proposition 1, θˆ1 cannot be bunched. Consequently, a non-local incentive constraint has to bind to θˆ1 . If θˆ1 is isolated in the set of types to which non-local incentive constraints are binding, η(θ) has to be discontinuous at θˆ1 by the definition of θˆ1 . Then also q(θ) is discontinuous at θˆ1 . But this is impossible by lemma 4. Hence, θˆ1 cannot be isolated in the set of types to which non-local incentive constraints bind. Proof of theorem 2: see the example in section A.3 of the webappendix Proof of proposition 2: Existence is shown in the webappendix. A decision which is above s has to be monotone because of local incentive compatibility (“monotonicity constraint”). Here I show that any incentive compatible decision function q(θ) which imposes decisions below s(θ) for some type is dominated by the following changed decision   q(θ) if q(θ) ≥ s(θ) c q (θ) = (7)  q s (q(θ), θ) if q(θ) < s(θ) combined with transfers such that πθc = −cθ (q c (θ), θ). This changed decision is shown to be above s(θ), monotonically increasing and incentive compatible. First, it is shown that the principal’s payoff is higher under q c (θ) than under q(θ): The principal maximizes expectation of u(q, θ) − c(q, θ) + [(1 − F (θ))/f (θ)]cθ (q, θ). If q s (q(θ), θ) ≤ q r (θ), the principal’s objective increases due to the change because of the concavity of (RP) and q r (θ) > s(θ). If q s (q(θ), θ) > q r (θ), then the same conclusion follows from q v (q(θ), θ) ≥ q s (q(θ), θ) > q r (θ) and the concavity of (RP). Second, the changed decision q c (θ) is monotonically increasing: From local incentive compatibility q(θ) was already increasing wherever it was above s(θ). At types with q(θ) < s(θ) the decision q(θ) had to be decreasing because of local incentive compatibility. But then q s (q(θ), θ) is clearly increasing in θ for these types because of cqθθ > 0. This leaves types at which q(θ) jumped discontinuously over s(θ). But at these jump types 36

local incentive compatibility required cθ (q − (θ), θ) − cθ (q + (θ), θ) ≥ 0 at downwards jumps (and the converse inequality at upwards jumps) across s(θ). This implies that also at jump points of q(θ) monotonicity of q c (θ) is guaranteed. Third, the changed decision q c (θ) is incentive compatible: Since q c (θ) is monotonically increasing, only downward misrepresentation has to be considered (see lemma 2). Note that the profit function π(θ) was not affected by the change from q(θ) to q c (θ) because of the definition of q s (θ) and πθ (θ) = −cθ (q(θ), θ) by local incentive compatibility. Therefore, one has only to check whether any type wants to misrepresent as a lower type ˆ < s(θ). ˆ Since π(θ) is unchanged, one can write incentive compatibility θˆ at which q(θ) under the changed decision as ˆ = − Φ (θ, θ) c

=

Z

Z θ

θˆ

θ θˆ

Z

Z

q(t)

ˆ q c (θ) c ˆ q (θ)

ˆ q(θ)

cqθ (q, t) dq dt = −

θ θˆ

Z

ˆ q(θ) ˆ q c (θ)

cqθ (q, t) dq dt −

Z

θ θˆ

Z

q(t) ˆ q(θ)

cqθ (q, t) dq dt

ˆ >0 cqθ (q, t) dq dt + Φ(θ, θ)

where the inequality follows from cqθθ > 0.

Z

R qc (θ) ˆ ˆ q(θ)

ˆ dq = 0 by the definition of q s (·) and cqθ (q, θ)

Proof of proposition 3: By lemma 4, q(θ) cannot be discontinuous at a type to which a non-local incentive constraint binds (with the exception of boundary types of bunching intervals). Therefore, theorem 1 implies that a solution could only be discontinuous at types where η(θ) is non-increasing or at the boundary types of a bunching interval to which a non-local incentive constraint is binding. First, it is shown that η(θ) is also non-increasing at such boundary types of a bunching interval. To see this take a bunching interval [θˆ1 , θˆ2 ] to which non-local incentive ˆ i.e. q − (θˆ2 ) < q + (θˆ2 ). constraints bind and suppose the solution was discontinuous at θ, Rθ By the arguments in the proof of lemma 4, θˆ2 cqθ (q − (θˆ2 ), t) dt > 0 for any θ such that Φ(θ, θˆ2 ) = 0. But then an argument as in the proof of theorem 1 applies: There is an

ˆ for θˆ ∈ [θˆ2 − ε, θˆ2 ] and decrease the decision incentive compatible way to increase q(θ) for types in [θˆ2 , θˆ + ε]. Incentive compatible means that binding non-local incentive constraints are not violated and the decision remains monotone (details in the proof of theorem 1). If η(·) was strictly increasing at θˆ2 , such a change would increase the principal’s payoff. Therefore, η(·) has to be decreasing at θˆ2 . A similar argument applies Rθ at θˆ1 . A discontinuity is only possible at θˆ1 if θˆ1 cqθ (q(θˆ1 ), t) dt < 0 for all θ such that 37

Φ(θ, θˆ1 ) = 0. Therefore, decreasing the decision on [θˆ1 , θˆ1 + ε] and increasing the decision on [θˆ1 − ε, θˆ1 ) can be done in an incentive compatible way. If η(·) was strictly increasing, such a change would increase the principal’s payoff. Hence, q(θ) can only be discontinuous at types where η(θ) is non-increasing. By local incentive compatibility, q(θ) can only jump upwards, i.e. q − (θ′ ) < q + (θ′ ) at a hypothetical discontinuity type θ′ . Using the definition of η(θ) in (2) one can calculate the change in η(θ′ ) at the discontinuity type +

′

−

′

η (θ ) − η (θ ) = =

Z

q + (θ ′ ) q − (θ ′ )

Z

q + (θ ′ ) q − (θ ′ )

d η(θ′ ) dq d q(θ′ )

(uqq − cqq )f cqθ + (1 − F )cqqθ cqθ − (uq − cq )f cqqθ − (1 − F )cqθ cqqθ dq c2qθ

where all functions are evaluated at (q, θ′ ). Note that the integrand is positive whenever q ≤ q f b (θ′ ). If q > q f b (θ′ ), the integrand can be written as   f (uq − cq ) uqq − cqq cqqθ − cqθ uq − cq cqθ which is also positive due to the condition of the proposition. Hence, η(θ) would jump up at θ′ but this contradicts that q(θ) can only be discontinuous at types where η(θ) is non-increasing. Proof of proposition 4: The optimal decision is continuous at types where it is below q f b : q(θ) ≤ q f b (θ) implies that 1 − F (θ) − η(θ) ≥ 0. Therefore, the left hand side of the first order condition uq − cq + (1 − F − η)cqθ = 0 is strictly decreasing in q. The same arguments as in the proof of proposition 3 show that q(θ) has to be continuous. Next it has to be shown that the decision is strictly monotone when it is below first best . This will be done in two steps. The first step is to show that q(θ) is strictly increasing if ηθ (θ) ≥ 0. The second step is to show that in a hypothetical bunching interval there are types θ at which ηθ (θ) ≥ 0 which by the first step contradicts that these types are bunched. First, the decision q(θ) has to satisfy the first order condition [uq − cq ] + 1−Ff −η cqθ = 0 by theorem 1. From the implicit function theorem, the sign of qθ (θ) can be determined. Note that q(θ) ≤ q f b (θ) implies 1 − F (θ) − η(θ) ≥ 0. This in turn implies that the derivative of the left hand side of the first order condition with respect to q is negative. 38

Hence, the sign of qθ (θ) is the sign of the partial derivative of the equation above with respect to θ. Denoting (1 − F (θ) − η(θ)) by λ(θ) this derivative is uqθ (q(θ), θ) − cqθ (q(θ), θ) +

∂ λ(θ)/f (θ) λ(θ) cqθθ + cqθ (q(θ), θ). f (θ) ∂θ

The first three terms are clearly positive as q(θ) ≤ q f b (θ) implies λ(θ) ≥ 0. The fourth term is positive if ηθ (θ) ≥ 0 as then −f 2 (θ) − fθ (θ)(1 − F (θ)) ηθ (θ) fθ (θ)η(θ) ∂ λ(θ)/f (θ) = − + <0 ∂θ f 2 (θ) f (θ) f 2 (θ) where the inequality comes from the monotone hazard rate assumption if fθ (θ) ≤ 0. If fθ (θ) > 0, then q f b (θ) ≥ q(θ) implies λ(θ) ≥ 0 which ensures the inequality above. Now turn to the second step. Suppose contrary to the proposition that an interval (θ1 , θ2 ) exists in which types are bunched and non-local incentive constraints are either binding to these types or are lax.32 Using the same argument as in the proof of theorem 1, it becomes evident that η(θ) as defined by (2) cannot be decreasing on the whole interval (θ1 , θ2 ). From the definition of η(θ) and the differentiability of q on the bunching interval, it follows that η(θ) is continuous and differentiable on this interval. Consequently, there has to be some type in the interior of the bunching interval where ηθ (θ) ≥ 0. But then the first step shows that this type cannot be bunched. The last part: q is below q f b if q m increases and there is no distortion at the top. The proof is by contradiction. Suppose the optimal decision q(θ) was above the first best decision for some types. Since there is no distortion at the top by assumption and since the optimal decision cannot drop discontinuously downward (local incentive compatibility), there has to be a type θ′ at which the optimal decision intersects q f b (θ) from above. The proof works now in two steps. First, I show that a non local incentive constraint must bind from θ′ and second that then non local incentive compatibility is violated for some type close to θ′ . Note that q(θ) > q f b (θ) if and only if η(θ) > 1 − F (θ). Since 1 − F (θ) is decreasing and q(θ) > (<)q f b (θ) slightly above (below) θ′ , it follows that ηθ (θ′ ) is negative. But then, by theorem 1, a non local incentive constraint has to be binding from θ′ to some R q(θ′ ) θˆ′ . Furthermore, the necessary condition q(θˆ′ ) cqθ (q, θ′ ) dq = 0 has to hold. Next consider a type θ′′ = θ′ − ǫ with ǫ > 0 very small. Since q m (θ) is increasing R qf b (θ′′ ) R q(θ) ′ cqθ (q, θ′′ ) dq < 0. Since q(θ′′ ) > q f b (θ′′ ), it has and q(θ) ˆ cqθ (q, θ ) dq = 0, clearly q(θˆ′ ) 32

By lemma 4, types from which non-local incentive constraints bind cannot be bunched.

39

R q(θ′′ )

cqθ (q, θ′′ ) dq < 0 as well. The same inequality holds for all θ ∈ (θ′′ , θ′ ). R θ′ R q(t) But then Φ(θ′′ , θˆ′ ) = Φ(θ′ , θˆ′ ) + θ′′ q(θˆ′ ) cqθ (q, t) dq dt < 0, i.e. incentive compatibility to hold that

q(θˆ′ )

from θ′′ to θˆ′ is violated. Hence, the optimal decision cannot be above the first best decision. Proof of proposition 5: Take two types θ′ and θˆ′ such that a non-local incentive constraint is binding from θ to θˆ under the optimal decision q(θ). By (C3), η(θ′ ) = η(θˆ′ ) and for this proof η (in Φη ()) simply denotes this common value η(θ′ ) = η(θˆ′ ). ˆ on [θˆ′ , θ] and call the minimizer First, suppose that (θ′ , θˆ′ ) does not minimize Φη (θ, θ) (θ′′ , θˆ′′ ). Then incentive compatibility under the optimal decision requires Φ(θ′′ , θˆ′′ ) ≥ 0. If q(θ) was q(θ, η) for all types in [θˆ′ , θˆ′′ ] ∪ [θ′′ , θ′ ], then Φ(θ′ , θˆ′ ) = Φη (θ′ , θˆ′ ) + Φ(θ′′ , θˆ′′ ) − Φη (θ′′ , θˆ′′ ) > 0 where the inequality stems from the definition of (θ′′ , θˆ′′ ) as global miniˆ Therefore ic would not be binding between θ′ and θˆ′ . mizer of Φη (θ, θ). If q(θ) 6= q˜(θ) for some types in [θˆ′ , θˆ′′ ] ∪ [θ′′ , θ′ ], then (IC) must be binding for some of these types.33 But this will only relax (IC), i.e. q(θ) ≥ q(θ, η) in a monotone solution. Therefore Φ(θ′ , θˆ′ ) will be even higher than when q(θ) = q(θ, η) and therefore (IC) cannot bind between θ′ and θˆ′ . This is the desired contradiction. Consequently, (θ′ , θˆ′ ) has to ˆ on [θˆ′ , θ′ ]. minimize Φη (θ, θ) Second, suppose that (θ′′ , θˆ′′ ) with θˆ′′ < θˆ′ < θ′ < θ′′ has Φη (θ′ , θˆ′ ) > Φ′ (θ′′ , θˆ′′ ). In ˆ under the constraint fact, choose θ′′ and θˆ′′ such that it is the global minimizer of Φη (θ, θ) θˆ < θˆ′ < θ′ < θ. Now suppose for the moment that all types in [θˆ′′ , θˆ′ ] ∪ [θ′ , θ′′ ] had q(θ) = q(θ, η). ˆ ic would be violated for θ′′ and Then since Φ(θ′ , θˆ′ ) = 0 but (θ′′ , θˆ′′ ) minimizes Φη (θ, θ), θˆ′′ . If q(θ) 6= q(θ, η) for some types in [θˆ′′ , θˆ′ ] ∪ [θ′ , θ′′ ], then (IC) was binding for some types in those intervals. In a monotone solution, this implies that q(θ) < q(θ, η) for these types. Put differently, (IC) is stricter under q(θ) than under q(θ, η).34 But then (IC) 33

Because of lemma 3 ic cannot bind from outside [θˆ′ , θ] into the interval (neither the other way

round). 34 Strictly speaking one also has to show that (IC) did not bind from outside [θˆ′′ , θ′′ ] into this interval (or the other way round), thereby increasing q(θ) for some types in say (θ′ , θ′′ ). If however this was the case and the increase in q(θ) was such that (IC) between θ′′ and θˆ′′ was relaxed by it, then there has to exist a type θˆ′′′ ∈ (θ′ , θ′′ ) and a type θ′′′ > θ′′ with Φ(θ′′′ , θˆ′′′ ) = 0 and q(θˆ′′′ ) = q(θˆ′′′ , η). But this ˆ (analogously to the proof of lemma 3), would contradict that (θ′′ , θˆ′′ ) is a global minimum of Φη (θ, θ)

40

will be even more violated for θ′′ and θˆ′′ under q(θ) than under q(θ, η). Therefore, there cannot be a global minimizer (θ′′ , θˆ′′ ) with θˆ′′ < θˆ′ < θ′ < θ′′ . Proof of corollary 1: Note first that the highest type θ from which a non-local incentive constraint is binding must have q(θ) = q r (θ) if θ is interior. This follows from the reasoning in the proof of lemma 5. The same holds for the lowest type θˆ to which a non-local incentive constraint binds. Therefore, there is a type pair such that (i) q(θ′ ) = q r (θ′ ), (ii) q(θˆ′ ) = q r (θˆ′ ) and (iii) Φ(θ′ , θˆ′ ) = 0. Since (θ′ , θˆ′ ) satisfy (C2) and (C1) with q r and given the results of proposition 5, ˆ Proposition 5 rules out that θˆr < θˆ′ < θ′ < θr and (θ′ , θˆ′ ) locally minimize Φr (θ, θ). also θˆ′ < θˆr < θr < θ′ . Hence, it still has to be shown that there cannot be an overlap between the two type pairs, i.e. θˆ′ < θˆr < θ′ < θr or θˆr < θˆ′ < θr < θ′ . To get a contradiction suppose θˆ′ < θˆr < θ′ < θr . In a similar way as in lemma 3, one can now show that in this case Φr (θr , θˆ′ ) < Φr (θr , θˆr ) thereby contradicting that (θr , θˆr ) is the ˆ global minimizer of Φr (θ, θ): r

r

ˆ′

r

r

ˆr

r

′

ˆ′

Φ (θ , θ ) = Φ (θ , θ ) + Φ (θ , θ ) +

Z

θ′ θˆr

Z

q r (t) q r (θˆr )

cqθ (q, t) dq dt −

= Φ (θ , θˆr ) + Φr (θ′ , θˆ′ ) − Φr (θ′ , θˆr ) − r

r

Z

θr θ′

Z

q r (θˆr ) q r (θˆ′ )

Z

θr θ′

Z

q r (θˆr ) q r (θˆ′ )

cqθ (q, t) dq dt

cqθ (q, t) dq dt

R qr (θ′ ) By proposition 5, Φr (θ′ , θˆ′ ) − Φr (θ′ , θˆr ) ≤ 0. Furthermore, qr (θˆ′ ) cqθ dq = 0 since (θ′ , θˆ′ ) R r ˆ Therefore, q(r θˆ ′) cqθ dq > 0 as q r (θ′ ) > q r (θˆr ) and cqqθ < 0. locally minimize Φr (θ, θ). q (θˆ ) R θr R qr (θˆr ) From cqθθ > 0 it follows that θ′ qr (θˆ′ ) cqθ (q, t) dq dt > 0 which shows that Φr (θr , θˆ′ ) < Φr (θr , θˆr ). This is the desired contradiction.

A similar argument can be made for the case θˆr < θˆ′ < θr < θ′ . Consequently, the only possibility is that (θ′ , θˆ′ ) = (θr , θˆr ) which had to be shown. If the highest/lowest type from/to which a non-local incentive constraint is binding is a boundary type, this type’s decision is not necessarily the relaxed decision. However, the minimization argument does not change which concludes the proof. Proof of proposition 6: see section B.3 in the webappendix or Jullien (2000) for a similar proof

i.e. Φ′ (θ′′′ , θˆ′′ ) < Φ′ (θ′′ , θˆ′′ ).

41

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