Adverse Selection Problems without the

Spence-Mirrlees Condition ⋆

Aloisio Araujo a,b and Humberto Moreira a

aGraduate School of Economics, Getulio Vargas Foundation, Praia de Botafogo,

190, Rio de Janeiro, Brazil

bInstituto Nacional de Matematica Pura e Aplicada, Estrada Dona Castorina,

110, Rio de Janeiro, Brazil

Abstract

This paper studies a class of one-dimensional screening problems where the agent’sutility function does not satisfy the Spence-Mirrlees condition (SMC). The strengthof the SMC for hidden information problems is to provide a full characterizationof implementable contracts using only the local incentive compatibility (IC) con-straints. These constraints are equivalent to the monotonicity of the decision vari-able with respect to the agent’s unobservable one-dimensional parameter. When theSMC is violated the local IC constraints are no longer sufficient for implementabilityand additional (global) IC constraints have to be taken into account. In particular,implementable decisions may not be monotonic and discretely pooled types musthave the same marginal utility of the decision (or equivalently, receive the samemarginal tariff). Moreover, at the optimal decision, the principal must preserve thesame trade-off between rent extraction and allocative distortion measured in theagent’s marginal rent unit. In a specific setting where non-monotone contracts maybe optimal we fully characterize the solution.

Keywords: Spence-Mirrlees condition; global incentive compatibility; U-shapedcondition; discrete pooling.JEL Codes: H41, D82.

Preprint submitted to Journal of Economic Theory 6 August 2009

1 Introduction

The Spence-Mirrlees condition (SMC) has been until now the main assumptionthat allows us to fully characterize the solution of one-dimensional adverseselection or hidden information models. Specifically, implementable contracts(and a fortiori optimal contracts) of standard principal-agent models andseparating equilibria of signaling games can be easily characterized under thiscondition. 1

We say that the SMC holds when the privately informed part has preferencesthat have increasing marginal rates of substitution between a decision variableand money with respect to a one-dimensional parameter or type (asymmet-ric information). 2 From the Revelation Principle, implementable contractsare mechanisms (i.e., maps from types to decision and money) that inducethe agent to truthfully reveal her type, i.e., they must satisfy the agent’sincentive compatibility (IC) constraints. Under the SMC, these constraintsare equivalent to their local first- and second-order conditions and hence im-plementable decisions are non-decreasing in types. For principal-agent models,this is enough to provide an algorithm for computing the optimal contract (see[10], for instance). Therefore, the SMC transforms a two-level (complex) max-imization problem into a much simpler optimal control program. The mono-tonicity of implementable (and optimal) contracts is a robust property in such

⋆ We acknowledge CNPq and FAPERJ of Brazil for financial support. Preliminaryversions of this paper were presented at the Universite de Toulouse 1, PompeuFabra (Barcelona), IMPA, PUC-Rio, EPGE/FGV, University of Chicago, Universitede Paris 1, the Nineteenth Meeting of the Brazilian Econometric Society, the 2000World Congress of the Econometric Society (Seattle), 2006 Latin American Meetingof the Econometric Society (Cidade de Mexico) and the 2006 Stony Brook GameTheory Festival. We are grateful to A. Chassagnon, P.-A. Chiappori, D. Gottlieb, J.-J. Laffont, G. Maggi, A. Mas-Colell, J.-C. Rochet, J. Scheinkman, L. Stole, J. Tirole,an associate editor, and a referee for helpful discussions and suggestions. We wouldlike to thank Caio Almeida for helping us with the numerical implementation of theexample of the previous version of the paper and Marcos Tsuchida for importantcomments. Finally, we are in debt with Sergei Vieira who did a careful revision ofthe proofs and checked some of them numerically.

Email address: humberto@fgv.br (Humberto Moreira).1 Mirrlees [18] is the pioneering reference and Guesnerie and Laffont [10] give themost complete treatment for the principal-agent problem under adverse selectionwith one-dimensional parameter. Spence [26] is the classical reference for signalinggames.2 In the literature of monotone methods for comparative statics there are somesubtle differences among the various concepts of single-crossing. Milgrom and Shan-non [16] define the order-theoretic single-crossing property for general lattice spaces.Edlin and Shannon [7] extend the analysis to strict monotone comparative statics(the strict SMC) by imposing a stronger differential restriction.

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models.

Our main goal is to study a class of one-dimensional screening problems wherethe agent’s utility function does not satisfy the SMC. For a quite generalframework we obtain necessary conditions for incentive compatibility whichare now much more complex since they involve both the traditional local andsome new global conditions. Next, we derive the necessary conditions for opti-mality. Finally, for a more special case we also obtain sufficient conditions foran optimum. Furthermore, for the reader not interested in going throughoutall the mathematics of the paper we provide a practical algorithm for com-puting solutions which has an analogy with the SMC case described in theprevious paragraph.

Assuming quasi-linearity, the SMC is equivalent to the constant sign of thecross derivative of the utility function with respect to decision and type. Ourclass of problems is characterized by the existence of a decreasing limit curvethat separates two regions in the type versus decision plane where the signof the cross derivative changes: the positive (above the limit curve) and neg-ative (below the limit curve) single-crossing regions. The necessary local ICconstraints imply that implementable decisions preserve monotonicity in eachregion, crossing or not the limit curve. In particular, when they cross, theymust have a U-shaped form and a new necessary global IC condition emerges.Two types taking the same decision must get the same marginal tariff whichleads to the same marginal utility of decision (Theorem 1). We will refer to itas the U-shaped condition. This indicates that the principal has to take care ofa wider variety of strategic behavior from the agent when the SMC is violated,i.e., the local IC constraints no longer imply global truth-telling.

In general it is very hard to characterize the set of implementable decisionseven in this class of problems. Following the relaxed approach of the literature,our strategy is to characterize the solution of a more relaxed program thattakes into account, in addition to the first- and second-order conditions ofthe IC constraint, the U-shaped condition, called the U-shaped problem. Apreliminary and important result is the necessary optimality condition for thebinding U-shaped condition (Theorem 2). It says that pooling types musthave the same trade-off between rent extraction and distortion measured inthe agent’s marginal rent. The remaining of the paper provides a specificsetup where these necessary conditions for implementability and optimalityare sufficient (Theorem 3).

In one-dimensional single-crossing literature, Jullien’s work [11] is the mostrelated to ours. He gives a complete analysis of the agent’s participation con-straint and the possibility of countervailing incentives (see also [13] and [15]).Although the distortions of the optimal contract found by him are similar toours, they come from the endogenous binding participation constraint whereas

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ours are due to binding global IC constraints. Furthermore, he keeps the single-crossing assumption and, a fortiori, the monotonicity of the optimal contractand we in a complementary way avoid countervailing incentives due to theparticipation constraints (see Section 6 for more on this).

There are few examples in the literature that violate the SMC. Two of themare on signaling games and show that breaking down the SMC may lead to in-teresting phenomena. Bernheim [4] presents a model for a theory of conformityin which individuals care about status as well as intrinsic utility. When statusis sufficiently more important than intrinsic utility, the SMC fails and manyindividuals conform to a single, homogeneous behavior (continuous pooling),despite the heterogenous underlying preferences. Bagwell and Bernheim [3]give a theory of conspicuous consumption where the Veblen effect (i.e., will-ingness to pay a higher price for a functionally equivalent good) occurs if andonly if the SMC fails. However, in both applications the signaling equilibriumis always monotonic.

The paper is organized as follows. Section 2 presents the model and Section3 relaxes the SMC. Section 4 provides new necessary conditions for imple-mentability and optimality. Section 5 shows that these new necessary condi-tions are sufficient to characterize the optimal contract in a special case. Inthat section we also provide a practical guide for the reader not interestedin going through the technical details in the Appendix. Section 6 gives theextensions and concluding remarks. Proofs and technical details are relegatedto the Appendix.

2 The Model

The principal-agent relationship involves a one-dimensional decision (con-sumption, production, etc.) ξ ∈ [x, x] ≡ X ⊂ R and a monetary transfer t ∈ R.The agent’s utility depends on a type parameter θ ∈ Θ = [θ, θ], known by herand unobservable to the principal, and is quasi-linear: v(ξ, θ) + t. The princi-pal’s utility function is also quasi-linear and may depend on the agent’s type:u(ξ, θ)− t. The total surplus is then u(ξ, θ)+v(ξ, θ), maximal at the first-bestlevel xFB(θ), for each θ. The principal has prior distribution over Θ accordingto the cumulative distribution P with continuous density p : Θ → R++. Thefunctions u and v are three times continuously differentiable on X ×Θ.

Using the revelation principle, any general communication mechanism betweenthe principal and the agent can be mimicked by a direct truthful one with nowelfare loss to the principal. A direct mechanism (or contract) is a pair offunctions (x, t) : Θ→ X ×R which can be viewed as a procedure committingthe principal to a rule relating the choice of x and t to messages sent by the

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agent about their types.

A decision function x : Θ→ X is implementable by a money transfer functiont : Θ→ R if the incentive-compatibility constraint is satisfied:

v(x(θ), θ) + t(θ) ≥ v(x(θ), θ) + t(θ) ∀θ, θ ∈ Θ. (1)

Equivalently, we say that the contract (x, t) is implementable.

We assume that the agent’s reservation utility, v(0, θ), is independent of histype and normalized to zero. A contract (x, t) satisfies the individual-rationalityconstraint if

v(x(θ), θ) + t(θ) ≥ 0, ∀θ ∈ Θ. (2)

An implementable contract that satisfies the IR constraint is called feasible.

We present the definition of the SMC that will be relaxed.

Definition 1 (SMC) The single-crossing or Spence-Mirrlees condition (SMC)is the constant sign of the cross partial derivative with respect to decision andtype: 3

vxθ > 0 on X ×Θ (CS+)

or

vxθ < 0 on X ×Θ. (CS−)

Definition 1 implies that indifference curves of two different types cross atmost once in the plane (ξ, t). Under CS+ (CS−), higher types are associatedwith higher (lower) marginal valuations of the decision. In what follows wealso refer to CS+ (CS−) as the region where the sign of the cross derivative ispositive (negative).

Below we present the standard necessary first- and second-order conditionsof the IC constraint. Both the statement and proof are simple extensions toexisting results in the literature (and do not depend on the SMC). The firstpart is the usual envelope theorem (see [17]).

Lemma 1 (Necessary conditions for local IC constraints) Suppose thatx is an implementable bounded decision.

(i) (First-order condition) The agent’s rent function of x, Vx, is given by

Vx(θ) := v(x(θ), θ) + t(θ) = Vx(θ)−∫ θ

θvθ(x(θ), θ)dθ, ∀θ ∈ Θ. (3)

3 The sub-index in the function denotes the partial derivative. The superior orderderivative is denoted by a multi-index notation.

5

(ii) (Second-order condition) If x : Θ→ X is cadlag 4 , then

x is non-decreasing (non-increasing) in CS+ (CS−). (M)

By Lemma 1 (i), for each implementable x there exists a unique t (up to theconstant Vx(θ)) that implements it:

t(θ) = Vx(θ)− v(x(θ), θ), ∀θ ∈ Θ. (4)

The global incentive compatibility (1) requires that

Vx(θ)− v(x(θ), θ)− t(θ) ≥ 0.

But, using (4), we get

Vx(θ)− v(x(θ), θ)− t(θ) =Vx(θ)− Vx(θ) + v(x(θ), θ)− v(x(θ), θ)

=∫ θ

θ[vθ(x(θ), θ)− vθ(x(θ), θ)]dθ

=∫ θ

θ

[∫ x(θ)

x(θ)vxθ(x, θ)dx

]dθ,

where the second equality results from (3) and the third is a consequence ofthe fundamental theorem of calculus.

We can then define the global incentive function (GIF) of x that satisfies (3):

Φx(θ, θ) :=∫ θ

θ

[∫ x(θ)

x(θ)vxθ(x, θ)dx

]dθ. (GIF)

Hence, x is implementable if and only if Φx is non-negative. For instance, if vxθhas a constant positive sign everywhere, then high types have higher marginalvaluations for x, i.e., their indifference curves in the space (x, t) are less steepthan those for low types. As a consequence, Φx(θ, ·) is non-negative everywhereif and only if x is non-decreasing. In other words, if the local incentive to cheatis avoidable in the report of true type, then the same is true for global reports.Therefore, the necessary monotonicity condition (M) joint with (3) are alsosufficient for implementability under the SMC.

The principal’s hidden information or second-best problem is to choose a fea-sible contract with the highest expected payoff:

maxx(·),t(·)

E[u(x(·), ·)− t(·)] (SB)

s.t. IC and IR constraints,

4 That is, right-continuous with left-hand side limit at each point of the domain.

6

where E is the expectation operator with respect to the prior distribution p.

Plugging (4) into the objective function, we get

E[u(x(·), ·)− t(·)] = E[u(x(·), ·) + v(x(·), ·)− Vx(·)]. (5)

Integrating by parts the term E[Vx(·)] and using (3), it becomes

E [f(x(·), ·)]− Vx(θ)

where

f(ξ, θ) = u(ξ, θ) + v(ξ, θ) +P (θ)

p(θ)vθ(ξ, θ)

is the virtual surplus, i.e., the social surplus (u+ v) discounted by the agent’sinformational rent (−P

pvθ).

If vθ ≤ 0, then Vx given by (3) has a minimum at θ. Then, the IR constraintneeds to be checked only at θ and, since transfer is costly to the principal, atthe optimal contract it must bind, i.e., the IR constraint can be replaced byVx(θ) = 0.

If vθ changes its sign, then Vx can assume its minimum value in the interior ofΘ (or even, at θ) depending on x. However, in order to simplify matters, weinvestigate the effect of breaking down the assumption on the cross derivative,not on vθ. From now on we assume that vθ ≤ 0 (see the last section and Jullien[11] for more details on the general case). 5

If we only consider the (local) first-order condition of the IC constraint (3),then we can define the following relaxed problem:

maxx(·)

E [f(x(·), ·)] (RP)

and its solution, denoted by x1, is called the relaxed solution. The relaxedproblem reduces to a pointwise maximization of the virtual surplus whosefirst-order condition is

fx(x1(θ), θ) = 0, (6)

for all θ ∈ Θ such that x1(θ) ∈ (x, x).

Under the assumption that follows (made throughout the paper), this point-wise first-order condition is sufficient for optimality.

A1. f(·, θ) is concave, for all θ ∈ Θ.

5 The case vθ ≥ 0 would be completely analogous. The only difference is that theinverse of the hazard rate would be (1− P (θ))/p(θ) instead.

7

If the solution of the relaxed problem (RP) is implementable, then it is thesolution of Problem (SB). Therefore, under the SMC, x1 is the solution ofProblem (SB) if and only if it is monotonic. Otherwise, the monotonicitycondition (M) is binding.

When (M) is binding, one has to perform the ironing principle 6 on x1. Usingthis principle Guesnerie and Laffont [10] managed to provide an algorithm todetermine the solution under the SMC (for a complete analysis of the standardcase see also [12]).

3 Relaxing the SMC

In this paper, we are concerned with a particular type of failure of the SMCin which there is a decreasing boundary x0 that splits the space Θ × X intotwo single-crossing regions CS+ above and CS− below. More formally: 7

A2. vxθ(ξ, θ) = 0 defines implicitly a decreasing function x0 of θ on Θ suchthat ξ ≥ x0(θ) if and only if vxθ(ξ, θ) ≥ 0 (CS+), for all θ ∈ Θ. Moreover,vx2θ > 0 holds on X ×Θ. 8

Figure 1 shows an example of such a boundary. Clearly, the SMC does not holdin Figure 1. To see how this can cause a problem, suppose that the solutionto the standard relaxed problem is the function x1 as shown. The relaxedsolution lies entirely in CS− and, since it is decreasing, it satisfies the localmonotonicity. Nevertheless, the global IC condition GIF may fail. Consider the

hatched area in Figure 1 given by∫ θ

θ

[∫ x(θ)

x(θ)dx

]dθ. If we weight by vxθ, this

area becomes the global incentive function Φx1(θ, θ) defined by (GIF). Thus,if the weighted hatched area above x0 is greater in absolute terms than the

6 Mussa and Rosen [19] is the pioneering references of this procedure for adverseselection problems. Rochet and Chone [22] provide a generalization of the “ironingprinciple” for multidimensional screening models.7 Athey [2] introduces “limited-complex” strategies which have a finite number ofpeaks in the context of screening equilibrium models like auctions (although sheonly shows the existence of pure strategy equilibria). In our case, A2 will implythat implementable decisions have at most one peak. Chassagnon and Chiappori[5] study a competitive insurance model with moral hazard and adverse selectionwhere the SMC may fail, but in a two-type model.

8 By the implicit function theorem, a sufficient condition for the first part of A2,besides vx2θ > 0, is vxθ2 > 0 on X ×Θ. Making the other possible sign changes forξ and/or θ, we obtain other three equivalent cases.

8

��������������������������������

��������������������������������

θθ

ξ

x0

x1CS+

CS−

θθ

θ

Fig. 1. Global IC constraints under A2.

weighted area below it, the global incentive compatibility condition is violated.More generally, when x0 separates the space Θ × X into two regions, thenet effect on the global incentive function (GIF) may be positive or negativedepending on the relative intensity of the local incentive effect in each region.

This indicates that other (global) IC constraints might be important. This isthe subject of the next section.

4 Necessary Conditions

4.1 Necessary conditions for implementability

For the purpose of this paper, let us consider a particular and interesting caseof global binding IC constraint. Fix an implementable decision x and supposethat it crosses continuously the limit curve x0. Then, x must cross x0 in aU-shaped form due to condition (M) and there must be θ 6= θ in (θ, θ) suchthat x(θ) = x(θ) = ξ (discrete pooling). Figure 2 illustrates this situation.

The right graph shows a U-shaped decision such that θ and θ are poolingand the left one is the correspondent contract in the plane X × R of indirectmechanisms. Hence, if the contract is implementable, the indifference curves(of θ and θ) and the non-linear tariff are tangent at ξ. Equivalently, discretelypooled types must obtain the same marginal tariff (see subsection 5.1 for moreon the indirect approach).

9

indifferencecurves

x(θ)

x(θ) = x(θ)

θ

θ

ξθ

decisionx0

ξ

θ θ

nonlineartariff

t

Fig. 2. The U-shaped condition.

We then get our new global necessary incentive compatibility condition: 9

vx(ξ, θ) = vx(ξ, θ), (UC)

i.e., the U-shaped condition. Formally, we have:

Theorem 1 (U-shaped condition) Let x be an implementable decision suchthat x is continuous and strictly monotonic at θ and θ such that ξ = x(θ) =x(θ). Then, (UC) must hold.

Under the SMC, implementability is equivalent to checking the local upstreamor downstream incentives, i.e., monotonicity. When it does not hold, the up-stream and downstream incentives are necessary but not sufficient for imple-mentability. At least an extra necessary global IC condition must be checked:the cross-stream incentives (UC).

The SMC implies the quasi-convexity of Φx(θ, ·), for all θ, and the convexityof the space of implementable decisions (since monotonicity is preserved byconvex combination). However, these are no longer true when the SMC isbroken: Φx(θ, ·) may have two distinct minima (the pooling types) and theconvex combination of implementable decisions that satisfy condition (UC)may not satisfy it. 10

9 This condition is equivalent to vx being constant on every level set of an imple-mentable decision. Rochet [21] presents the generalization of necessary and sufficientconditions for implementability. However, he does not use these conditions to treata non-single crossing case and he does not characterize the associate optimalityconditions (as we do here in a specific setup).

10 Araujo and Moreira [1] study moral hazard principal-agent problems relaxingthe standard conditions that make the agent’s problem concave. They provided ageneralization of the first-order approach. This paper follows the same route foradverse selection problems.

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Taxation Principle

The “Taxation Principle” establishes that, given an implementable directmechanism (x(·), t(·)), there is a non-linear tariff T (·) that implements x(·):

T (ξ) = t(θ),

where ξ ∈ x(θ). Reciprocally, given a continuous non-linear tariff T (·), we candefine the agent’s optimal decision such that

x(θ) ∈ arg maxξ∈X

{v(ξ, θ) + T (ξ)}

and transfer t(θ) = T (x(θ)). In this case, we say that x(·) is implemented byT (·). See [9] for a complete exposition on the topic.

Remark 1 By Lemma 1 (i), if T implements x, then T is a.e. differentiableand

vx(ξ, θ) + T ′(ξ) = 0, (7)

for all ξ ∈ x(θ), where T ′(ξ) is the marginal tariff. Hence, pooling types mustpresent identical marginal utility when choosing the same decision, which isthe content of Theorem 1 (see the left graph of Figure 2). 11

4.2 Necessary conditions for optimality

The next theorem is one of our main results since it gives the necessary op-timality condition associated to the condition (UC), i.e., it characterizes thecritical U-shaped parts. Thus, let us define the U-shaped problem as the max-imization of the expected virtual surplus subject to conditions (M) and (UC):

maxx(·)

E[f(x(·), ·)] (UP)

s.t. (M) and (UC).

Let x∗ be the optimal solution of Problem (UP).

Theorem 2 (Critical U-shaped curve) Suppose that x∗ is continuous andstrictly monotonic at θ and θ such that ξ = x∗(θ) = x∗(θ). Then,

fx(ξ, θ)

vxθ(ξ, θ)p(θ) =

fx(ξ, θ)

vxθ(ξ, θ)p(θ). (8)

11 This is also called the demand-profile approach. See [23] for an exposition of thetwo approaches to dealing with screening problems: the parametric-utility and thedemand-profile approaches.

11

To prove this theorem it is enough to make admissible perturbations h aroundθ of the optimum solution x∗ to perform the variational calculus (see [14] forthis type of approach). Let [θ1, θ2] be a small interval around θ in the U-shapedpart. Define the space of admissible perturbations by:

H = {h : [θ1, θ2]→ R; h(θ1) = h(θ2) = 0 and x = x∗+h satisfies (M) and (UC)}.

Figure 3 shows a typical perturbation h ∈ H where x0 is the vertical axis.

ξ1

ξ2

x0

x∗

θ1 θ2 θ1

θθ2

Fig. 3. U-shaped admissible pertubations.

Take x = x∗ + h. In order to be admissible, the function

U(ξ, θ, θ) = vx(ξ, θ)− vx(ξ, θ)

should be identical to zero for ξ = x(θ) = x(θ) with θ ∈ [θ1, θ2] and θ ∈ [θ2, θ1].Taking the derivative with respect to θ, we see that U

θ(ξ, θ, θ) = −vxθ(ξ, θ) <

0. By the implicit function theorem, for each (ξ, θ, θ) such that U(ξ, θ, θ) = 0,there exists a neighborhood of (ξ, θ) where the conjugate type θ can be writtenas a continuously differentiable function of (ξ, θ): θ = ϕ(ξ, θ).

To simplify the exposition, let us suppose that x∗ and h are differentiable on[θ1, θ2] and denote the derivative with respect to θ by a dot over the function.Let us consider only the part of the principal’s objective functional affectedby the admissible perturbation:

12

G(h) =∫ θ2

θ1

g(x(θ), θ)dθ +∫ ϕ(x(θ1),θ1)

ϕ(x(θ2),θ2)g(x(θ), θ)dθ

=∫ θ2

θ1

[g(x(θ), θ)− g(x(θ), ϕ(x(θ), θ))(ϕx(x(θ), θ)·x(θ) + ϕθ(x(θ), θ))]dθ

where g(ξ, θ) = f(ξ, θ)p(θ) and the last equality is a simple change of variable.

Taking the Gateaux differential at 0 in the direction h, we must have:

δhG(0) =∫ θ2

θ1

[gxh− g((ϕxx·x∗+ϕxθ)h+ϕx

·

h)−(gx+ gθϕx)(ϕx·x∗+ϕθ)h]dθ = 0,

where we are omitting the arguments of the functions and the hat means

computation at the conjugate type. Integrating by parts∫ θ2

θ1

(gϕx)·

hdθ and

plugging it into the previous equation we get

∫ θ2

θ1

[gx − gxϕθ]hdθ = 0

(see the Appendix A). Again, by the implicit function theorem ϕθ = vxθ/vxθand since h is arbitrary we have the result.

Theorem 2 has a straightforward interpretation. Notice that the marginalchange in the virtual surplus due to an increase of the slope of the type-θagent’s rent at a given decision ξ is exactly fx(ξ, θ)/vxθ(ξ, θ) (i.e., the ratio be-tween the allocative distortion and marginal rent). Hence, Theorem 2 simplysays that at the optimal U-shaped part these marginal increases weighted bythe density should be equalized across pooling types. 12 Combining this con-dition and (UC), we can generally find a critical U-shaped part. See AppendixD for a derivation of Theorem 2 using the demand-profile approach.

5 Sufficient Conditions

The aim of this section is to present a specific setup where the necessaryconditions for implementability (Lemma 1 and Theorem 1) and optimality(condition (6) and Theorem 2) are indeed sufficient to characterize the second-best solution. Let us start by the definition of the space of contracts where wewill provide this characterization.

12 In Appendix B we present another derivation of Theorem 2 through a Lagrangianapproach that does not depend on the differentiability of the optimal decision andperturbation. In particular, we will show that these ratios are equal to the La-grangian multiplier of the condition (UC) (see equation (12)).

13

From the SMC case (e.g., [19]) one would expect the optimal decision to becontinuous. However, without single-crossing this might not be the case as wewill see in what follows. Without the continuity property, a weaker requirementis the convex-valued correspondence property. By the maximum theorem, animplementable decision is a non-empty and upper semi-continuous compactvalued correspondence. In this case, we can always select a subcorrespondencethat is a cadlag function. 13 Reciprocally, given a cadlag decision x we definethe extended version of x as the correspondence whose values are the intervalsbetween the right- and left-hand limits of x at every θ, i.e., the extended x isa convex-valued correspondence.

To avoid further technical difficulties due to a richer space of possible devi-ations for the principal and the agent and, at the same time, to allow fordiscontinuities, we consider the following:

Definition 2 (Space of decisions) We call X the space of extended versionof cadlag decisions.

Notice that X is restrictive once it enlarges the strategic behavior of theagent who is now endowed with more options to mimic other types. Theprincipal’s welfare would increase with more control over the out of contractrealizations (i.e., implementable decisions without convex values). In AppendixD we provide an example with implementable decisions not in X that improvethe welfare by crossing (discontinuously) the limit curve x0 many times.

5.1 A class of U-shaped problems

From the discussion on Figure 1, the necessary conditions may not be suffi-cient in general for implementability. Indeed, we argued that there may existmonotonic decisions that do not cross the limit curve x0 (so trivially satisfyconditions (M) and (UC)) and that are not implementable. Therefore, to atthe same time deal with a situation where global IC conditions matter and tohave a tractable case we will assume two more conditions besides assumptionsA1 and A2. First, we focus our analysis on a case where the solution x1 ofProblem (RP) has a unique minimum (unique peak). 14

13 Christensen [6] shows that any such correspondence has a minimal sub-correspondence with the same properties which is single-valued and continuous ata Gδ (i.e., a countable intersection of open sets - see [25]) and dense subset of thedomain. Given that in our case this sub-correspondence has a closed graph and islocally monotonic at the points of continuity in the domain (see the proof of Lemma1 (ii)), we can assume that x is a cadlag function without loss of generality.14 Notice that for the private value case (uxθ = 0), it is easy to check that thefirst-best solution xFB is U-shaped. Moreover, under A1 and A2, xFB is inbetween

14

C1. The relaxed solution x1 has a unique minimum and increasingly crossesx0. Moreover, x1(θ) < x1(θ).

Let xu : [θ, θu] → X be the continuous decision function defined by equation(8), where [θ, θu] ⊂ Θ and xu(θ) = xu(θu). To complete our specific case, weassume that the following condition holds.

C2. The function xu is the unique solution of equation (8) that satisfies con-dition (M). 15

Since the left and right hand sides of equation (8) must have the same sign, itis easy to see that, from assumptions A1-2, conditions C1-2 and Theorem 2,xu must lie inbetween x0 and x1. In particular, x1 does not satisfy condition(UC) and the problem raised in Figure 1 occurs for x1.

Let x∗ be the optimum for the U-shaped problem (UP) in the space X . Ap-pendix B 16 provides necessary and sufficient conditions for x∗ for the casewhere C1-2 hold. Let us only describe this solution here. As we saw in Section4 standard variational calculus implies that: (i) x∗ is the relaxed solution x1

in every interval where conditions (M) and (UC) do not bind (from condition(6)); and (ii) x∗ is the critical U-shaped curve xu in every interval where onlycondition (UC) binds (from Theorem 2 and condition C2). Using this obser-vation and the fact that xu is inbetween x0 and x1, we have two natural casesto consider:

• x∗(θ) < x∗(θ). Then, x∗ must be xu in the first part of Θ and jumps “ver-tically” to be x1 in the second part of Θ. We then need to find the optimaljump point θ1 (vertical ironing procedure) to determine a solution. That is,we must find θ1 such that

x∗(θ) =

xu(θ, θ1), if θ < θ1

xu(θ), if θ ∈ [θ1, θ1)

x1(θ), if θ ≥ θ1,

where θ1 is discrete pooled with θ1 and xu(θ, θ1) is the curve discrete pooled

the curves x0 and x1 with no distortion at the bottom type θ and at the crossingpoint between x0 and x1. Therefore, condition C1 represents a natural case wherex1 crosses increasingly x0.15 In Appendix B we provide sufficient conditions on preferences and distribution oftypes to ensure this uniqueness.16 There we use an approach based on a reformulation of the space of decisionsproposed by Noldeke and Samuelson [20] and built on an insight of Goldman, Lelandand Sibley [8]. The reformulation consists in writing decisions as type assignmentfunctions (i.e., the inverse functions of the decisions) instead.

15

with the vertical line θ = θ1 such that condition (UC) holds. Notice that therequirement of being in the space X is important to define x∗ as xu(·, θ1) inthe interval [θ, θ1]. We show that θ1 must satisfy condition (VI) in AppendixB which is a generalization of Theorem 2. In the left graph of Figure 4 thethick line represents this solution candidate.• x∗(θ) ≥ x∗(θ). Then, x∗ must be x1 in the first part of Θ and jumps “hori-

zontally” to be xu in the middle of Θ. We then need to perform the optimalbunching (horizontal ironing procedure) to determine a solution. That is,we must find θ2 such that

x∗(θ) =

x1(θ), if θ < θ2

ξ, if θ ∈ [θ2, θ3) ∪ [θ3, θ]

xu(θ), if θ ∈ [θ3, θ3),

where θ3 > θ2 is discrete pooled with θ3 such that condition (UC) holdsand ξ ≥ x1(θ2) is the bunching level. We show that the optimal bunchingξ (horizontal ironing procedure) must satisfy condition (HI) in Appendix Bwhich is a generalization of the “ironing principle” of the standard litera-ture. In the right graph of Figure 4 the thick line represents this solutioncandidate. 17

θ

ξ

θ θ1θ1 θu θ

xu

x1

x0

θ

ξ

θ2θ3θ3

ξ xu

x1

x0

(9)

Fig. 4. The solutions of Problem (UP).

The solution of Problem (UP) is the candidate that gives the highest expectedpayoff for the principal. Since Problem (UP) is evidently more relaxed than

17 There are two degenerate cases of the right graph of Figure 4: (i) θ2 = θ andξ > x1(θ) such that x∗ is the composition of bunching intervals and the U-shapedpart; (ii) θ3 = θ3 such that x∗ is the composition of the relaxed solution and aunique bunching interval.

16

the second-best problem (SB) in X , to show that x∗ is the second-best solutionit is enough to show that x∗ is implementable. The following theorem, whoseproof is in Appendix B, concludes that, under conditions C1 and C2, this isthe case.

Theorem 3 (Second best decision) Assume that A1-2 and C1-2 hold. Thesolution of Problem (UP) is the solution of Problem (SB) when decisions arerestricted to the space X .

Algorithm. Figure 4 suggests also an algorithm to compute the second-bestdecision:

• First, draw the relaxed solution x1 (satisfying condition C1) and the criticalU-shaped solution xu (satisfying condition C2).• Second, find the best ironing procedure such that conditions (M) and (UC)

are satisfied. There are two possibilities: the vertical ironing principle shownin the left graph of Figure 4 or the horizontal ironing principle shown in theright graph of Figure 4.• Third, compute the principal’s payoff at each of these solutions to obtain

the second-best solution.

Summarizing, one has to adjust optimally first the level of the vertical ironing(θ1) and then the level of horizontal ironing (ξ) on the curves x1 and xu suchthat conditions (M) and (UC) hold in order to find the second-best solution.If ξ = x we get the solution presented in the left graph of Figure 4. Otherwise,we get the solution presented in the right graph of Figure 4.

Economic interpretation. The relaxed solution x1 represents the best trade-off between rent extraction and distortion when one takes into account onlythe local first-order condition of the IC constraint. When x1 is not imple-mentable, we have to characterize the least distorted implementable decisioncompared to x1. Under our assumptions, there are two extreme cases to con-sider. In the first distortions of high types are relatively more costly whichfavors no further distortion for them and new distortions for low types. Thesedistortions are related to the binding monotonicity and U-shaped conditions.Since the monotonicity condition implies that local IC constraints are up-ward (downward) binding in CS− (CS+), the critical U-shaped curve, xu, isupward (downward) distorted when compared to x1 in CS− (CS+). Theorem2 implies that the trade-off between rent extraction and distortion measuredin the agent’s marginal rent unit are equalized between pooling types. Thisprovides our first optimal decision (the left graph of Figure 4).

On the other hand, if the distortion costs for low types are relatively high, thereis no further distortion for these types. However, high types may be partiallyor completely pooled (the right graph of Figure 4). In the former case, someintermediate types may be discretely pooled, i.e., they may present partial

17

separation. The interpretation is similar to the classical “ironing principle” ifwe add the new U-shaped feature: full separation for high (low) types mayinduce costly pooling for low (high) types. 18

One of the new feature of the optimal decision is the possibility of discretepooling, i.e., isolated types taking the same decision. In the standard literaturethere are only two possibilities: separation or continuous pooling. In the formerthe agent’s type is known ex-post by the principal and in the last the principalknows a range of types where the agent belongs to. When the SMC fails,discrete pooling may arise and the principal may not know the true typebetween two types or two ranges of types (i.e., disconnected continuous poolingsets). Moreover, middle types are assigned to lowest decision levels, but enjoypositive rents. More interestingly, when there is discrete pooling, there is nodistortion for the type where x1 crosses x0.

Unlike the single-crossing case where continuous pooling implies discontinuityonly for the derivative of the optimal decision function with respect to type,it may also be discontinuous as well. The reason is that the hazard rate oftypes jumps upward at the transition point from the discrete pooling regionto the separation region (see the discussion of the demand-profile approachin Appendix D). That is, the trade-off between rent extraction and distortionchanges discontinuously from discrete pooling to separation.

6 Extension and Concluding Remarks

Before giving the final concluding remarks, let us present a quick discussionon the countervailing incentives extension. The model studied in this paperrules out exclusion of types, i.e., the agent’s reserve utility is given by

V(θ) = v(0, θ), (RU)

assumed to be constant and equal to zero. However, even when it is not con-stant, the IR constraint is equivalent to checking it only at one of the extreme

18 In Appendix C, we provide a monopolistic labor market example. In the modelworkers have two productivity parameters that are mixed through a verifiable signalsuch that the SMC is violated. We perform some numerical examples consistent withthe solutions characterized in this paper.

18

types under the SMC. 19

Jullien [11] studied a more general case where V(θ) does not necessarily satisfy(RU). His main insight is that, in the presence of countervailing incentives,upward and downward distortions compared to the first-best may happen.Under CS− (vxθ < 0), upward distortions better prevent low types to overstatetheir types, because they have higher marginal benefit from the decision. Whenthe reserve utility is large for low types, the contracts offered to them maybecome attractive for high types. If the reserve utility is large enough, thendownward distortions better prevent understatement of high types.

However, countervailing incentives can also occur when the SMC does not holdeven if the agent’s reserve utility does satisfy (RU). Like in [11], the derivativeof the difference between the marginal rent and the marginal reserve utilitywith respect to type can change sign when the SMC fails.

Under countervailing incentives, the necessary conditions of Theorem 2 remainthe same when the Lagrangian multiplier of the IR constraint, γ, is constantin θ. Otherwise, there may be a stronger conflict between rent extraction andefficiency. This is because the equalization of the marginal virtual benefit ofincreasing the rent profile between pooling types is affected by the shadowvalue associated with a reduction in the participation level (γ). [11] madeassumptions (potential separation and homogeneity) to avoid this extra dif-ficulty under single-crossing. This potential conflict is related to the local ICconstraints in his case whereas in ours is to the global IC constraints.

This paper provides a first treatment of one-dimensional screening problemswithout the SMC. Although it leads to a non-concave principal-agent prob-lem, we were able to deal with it mathematically and gave an economicallymeaningful solution. We first derive new necessary global implementabilityconditions which are translated into the U-shaped conditions joint with theirnecessary optimality conditions. These and the local IC conditions are suffi-cient for implementability and optimality in a specific set up.

There are several directions we may pursue. One is to cover other possibleshapes for x0 and x1. The first interesting case is when x1 is decreasing con-tained in CS−, but the global IC does not hold (see Figure 1). Although, thisseems to be a natural case to study, it leads to extra difficulties avoided inthis paper. Indeed, besides condition (UC), there is at least one extra class of

19 For instance, if vxθ < 0 then Lemma 1 (i) implies that

·

Vx

(θ)−·

V(θ) = vθ(x(θ), θ)− vθ(0, θ) ≤ 0

for every implementable x and the IR constraint is binding for θ = θ at the optimalcontract.

19

global IC constraints that can bind: the marginal rent identity. Theorem 1Ain the Appendix A derives this extra necessary condition. Another direction isto explore these new necessary implementability and optimality conditions formultidimensional screening problems. Finally, one may want to investigate theconsequences of the lack of the monotonicity property of implementable (andoptimal) contracts for the empirical literature on hidden information models.

20

Appendix A. Proofs (necessity)

Proof of Lemma 1. (i) The IC constraint (1) implies that, for θ > θ,

v(x(θ), θ)− v(x(θ), θ)

θ − θ≥Vx(θ)− Vx(θ)

θ − θ≥v(x(θ), θ)− v(x(θ), θ)

θ − θ.

Since v is C3 and x is bounded, the above inequality shows that Vx is aLipschitz function. Then, Vx is differentiable a.e. and

d

dθVx(θ) = vθ(x(θ), θ)

a.e. The fundamental theorem of calculus for absolutely continuous functionsgives (i).

(ii) By (GIF), we know that the global IC constraint for x is equivalent to

Φx(θ, θ) =∫ θ

θ

[∫ x(θ)

x(θ)vxθ(x, θ)dx

]dθ ≥ 0,

for all θ, θ ∈ Θ. Let I ⊂ Θ be an interval such that vxθ(x(θ), θ) > 0 for allθ ∈ I. We claim that x is non-decreasing in I. Otherwise, there exists θ ∈ Isuch that the set Iθ = {θ ∈ I; θ > θ and x(θ) < x(θ)} is a non-empty set. Bythe right continuity, Iθ contains a non-empty interval J . In this case, it is easyto see that Φx(θ, θ) < 0 for all θ ∈ J , which is a contradiction. The other caseis analogous.

Proof of Theorem 1. We will prove a more general result than Theorem1. In particular, item (ii) of Theorem 1A below is a kind of dual necessaryimplementability condition of item (i) not explored in this paper.

Theorem 1A. Let x be an implementable decision and θ, θ ∈ (θ, θ) be suchthat Φx(θ, θ) = 0.

(i) If x is strictly monotonic and continuous at θ, then

vx(x(θ), θ) = vx(x(θ), θ).

(ii) If x is continuous at θ, then

vθ(x(θ), θ) = vθ(x(θ), θ).

Proof of Theorem 1A. (i) Suppose that θ < θ and x(θ) > x(θ) = minθ∈[θ,θ]

x(θ)

21

(the other cases are analogous). Define the local inverse of x at θ by ψ. Ap-plying Fubini’s theorem, (GIF) can be written, with some abuse of notation,as

Φx(θ, ξ) =∫ x(θ)

ξ

∫ θ

ψ(ξ)vxθ(ξ, θ)dθdξ.

Since ξ = x(θ) is a minimum for Φx(θ, ·), taking the derivative with respectto ξ at x(θ) and equating to zero, we obtain the result.

(ii) Observe that if we fix θ, θ is a minimum for Φx(·, θ). Thus, the result isagain a direct consequence of the first-order condition.

Turning back to the proof of Theorem 1, suppose that x(θ) = x(θ). Inter-changing the parameters and using the definition of Φx in (GIF), we havethat Φx(θ, θ) = −Φx(θ, θ). The implementability of x implies that Φx is non-negative and hence Φx(θ, θ) = 0. Therefore, Theorem 1 follows from Theorem1A (i).

Proof of Theorem 2. The only remaining part is the integration by parts:

−∫ θ2

θ1

(gϕx)hdθ =∫ θ2

θ1

·

(gϕx)hdθ

=∫ θ2

θ1

[(gxx∗ + gθ(ϕxx

∗ + ϕθ))ϕx + g(ϕxxx∗ + ϕxθ)]hdθ.

Plugging this last equation into the first-order condition δhG(0) = 0 in thetext, for all direction h,

∫ θ2

θ1

[gx − gxϕθ]hdθ = 0.

Appendix B. Proofs (sufficiency)

As in the standard problem, we will construct a relaxed problem and iron it.First, we will introduce a reformulation of the space of decisions proposed byNoldeke and Samuelson [20]. The reformulation consists in writing decisions astype assignment functions (i.e., the inverse functions of the decisions) instead.

Under A2, decision functions in X can have more than one inverse. Figure 5shows how to define these inverse functions in this case. The left graph showsa typical x ∈ X and the right one gives their two respective inverses, ψb andψs. Figure 5 suggests that we define them as extended cadlag versions in thewhole domain [x, x] (as in Definition 2):

As we will see below, it turns out that this technique is helpful for the non-single crossing case because it avoids the difficulties of the Hamiltonian ap-proach to deal with the global IC constraints. More precisely, it transforms

22

CS+

CS−

θ

ξ

x

x0

x

θ θ

ψb( )

ψs( )

x−1

0

ψb

ψs ξ

ξ

xx

θ

ξx

θ

θ

Fig. 4. Fig. 5. A decision function in X and their type assignment functions.

the principal’s problem into a pointwise maximization problem.

The reformulation of the problem. Our goal now is to rewrite Prob-lem (UP) in terms of type assignment functions. Let x ∈ X , implementedby T (·), and ψb and ψs their inverses. By Remark 1, T ′(ξ) = −vx(ξ, ψb(ξ))when ψb(ξ) > θ and T ′(ξ) = −vx(ξ, ψs(ξ)) when ψs(ξ) < θ, a.e. (Using theterminology of [20], the pair (T, ψi) is consistent for each i = b, s).

Using Fubini’s theorem, the principal’s objective function (5) at x in terms oftype assignment function becomes

E[u(x(·), ·)− T (x(·))]=E[u(x, ·)]− T (x)− E

[∫ x

x(·)(ux(ξ, ·)− T

′(ξ))dξ

]

=E[u(x, ·)]− T (x) +∫ x

x[F (ξ, ψb(ξ))− F (ξ, ψs(ξ))]dξ,

where

F (ξ, θ) =∫ θ

θ[ux(ξ, θ) + vx(ξ, θ)]p(θ)dθ (10)

is the cumulative marginal virtual surplus.

Let us rewrite the monotonicity and U-shaped conditions in terms of typeassignment functions. The monotonicity condition (M) is trivially equivalentto:

ψb and ψs are non-increasing and non-decreasing, respectively. (M’)

The following lemma says that condition (UC) as a constraint in Problem(UP) is equivalent to (UC’) below in the type assignment approach. 20

20 Roughly speaking we are writing a version of the U-shaped condition valid whereit binds. This approach resembles the “double relaxed program” of Rogerson [24].Although studying a moral hazard problem, his idea was to transform an “equalityconstraint” into an “inequality constraint”.

23

Lemma 2 (U’-shaped condition) Assume A2. Let x ∈ X be an imple-mentable decision with type assignments ψb and ψs. If ξ ∈ X is a point ofcontinuity for both ψb and ψs, then

ψs(ξ) < θ =⇒ vx(ξ, ψb(ξ)) ≤ vx(ξ, ψs(ξ)). (UC’)

Proof. If ψs(ξ) < θ, then condition (UC’) holds with equality when ψb(ξ) > θdue to Theorem 1. Suppose that ψb(ξ) = θ. First, take ξ the infimum of ξ

such that ψb(ξ) = θ and ψs(ξ) < θ. Thus, either ψb(ξ) = ψs(ξ) = θ (i.e., thegraph of x belongs to CS+) and then (UC’) trivially holds with equality. Or ξis the point where ψb(ξ) starts to be the constant function θ (i.e., ξ = x+(θ)).Notice that condition (M’) implies that ψb(ξ) = θ, for all ξ > ξ. By continuity,condition (UC’) holds with equality for all ξ below ξ such that ψb and ψs arecontinuous at ξ. A2 and Remark 2 below imply that the function

γ(ξ) = vx(ξ, ψs(ξ))− vx(ξ, θ)

is increasing. Since limξ↑ξγ(ξ) = 0, γ(ξ) > 0 for all ξ > ξ. This is equivalent to

say that condition (UC’) holds with strict inequality for all ξ > ξ.

Observe that, since vθ ≤ 0, the IR constraint is equivalent to T (x)+v(x, θ) = 0.We can then define the following relaxed version of the U-shaped problem (withsome abuse of notation we use the same label for both problems, the originaland this relaxed version):

maxψb,ψs

∫ x

x[F (ξ, ψb(ξ))− F (ξ, ψs(ξ))]dξ

s.t. (UC’) and (M’).(UP)

Let ψb(·) and ψs(·) be any feasible pair of type assignment functions for Prob-lem (UP). Let ξ ∈ X the level that ψs(·) hits θ, i.e., the U-shaped part ends.By conditions (M’) and (UC’), we must then have

vx(ξ, ψb(ξ)) ≤ vx(ξ, ψs(ξ)), for all ξ ≤ ξ, and

ψs(ξ) = θ, for all ξ ≥ ξ.

Therefore, we can parametrize the pair of feasible type assignment functionsof Problem (UP) by ξ. Using this observation, we now proceed analogously tothe standard literature by relaxing the constraint (M’) from Problem (UP).For each ξ ∈ X we then define the problems that constitute the relaxed versionof Problem (UP), the pointwise relaxed U-shaped problems (RUP):

24

(i) for ξ ≤ ξ, (ψb, ψs) ∈ [θ, x−10 (ξ)]× [x−1

0 (ξ), θ] solves

maxF (ξ, ψb)− F (ξ, ψs)

s.t. vx(ξ, ψb) ≤ vx(ξ, ψs)(RUP)

(ii) for ξ ≥ ξ, ψs = θ and ψb ∈ [θ, x−10 (ξ)] solves

maxF (ξ, ψb)− F (ξ, θ). (RUP)

Observe that formally this relaxed version of the U-shaped problem dependson ξ. With some abuse of notation we do not make this dependence explicit inwhat follows. Notice also that compactness and continuity trivially imply that

Problem (RUP) always has a solution. Let (ψξb, ψξs) be the solution of Problem

(RUP).

The strategy to characterize the solution of Problem (UP) is based on threesteps. First, for each fixed ξ we characterize the solutions of Problem (RUP).Second, we deal with the possible lack of monotonicity of these solutions.Third, we compute the optimal ξ to find the solution of Problem (UP).

Characterization of the solution of Problem (RUP). From now onwe use the convention: x−1

0 (ξ) = θ for all ξ ≥ x0(θ) and x−10 (ξ) = θ for all

ξ ≤ x0(θ).

(1) ξ ≤ ξ. Dropping the Lagrangian multipliers of the boundary conditionsfor simplicity, we define the Lagrangian of Problem (RUP) as:

L(ξ, ψb, ψs, µ) = F (ξ, ψb)− F (ξ, ψs)− µ(vx(ξ, ψb)− vx(ξ, ψs)),

where µ ≥ 0 is the Lagrangian multiplier of condition (UC’).

Using the fundamental theorem of calculus and the definition of the cumulativemarginal virtual surplus (10) we get

L(ξ, ψb, ψs, µ) =∫ ψb

ψs

[fx(ξ, θ)p(θ)− µvxθ(ξ, θ)]dθ. (L)

Taking the derivative with respect to ψi we have the following necessary op-timality conditions:

0 ≥ fx(ξ, ψi)p(ψi)− µvxθ(ξ, ψi) (11)

with equality for ψb ∈ (θ, x−10 (ξ)) or ψs ∈ (x−1

0 (ξ), θ) and the usual comple-mentary slackness condition: 0 = µ(vx(ξ, ψb)− vx(ξ, ψs)).

In particular, we obtain the already known necessary optimality conditions:

25

• µ > 0, ψξb ∈ (θ, x−10 (ξ)) and ψξs ∈ (x−1

0 (ξ), θ) =⇒

fx(ξ, ψξb)

vxθ(ξ, ψξb)p(ψξb) = µ =

fx(ξ, ψξs)

vxθ(ξ, ψξs)p(ψξs) (12)

and (UC) by the complementary slackness condition, which are exactly thecontent of Theorem 2.• µ = 0 and ψξb ∈ (θ, x−1

0 (ξ)) =⇒ fx(ξ, ψξb) = 0, which is exactly equation (6)

(analogously for ψξs ∈ (x−10 (ξ), θ)).

• µ > 0 and ψξb = θ =⇒ ψξs = ψ(ξ), where ψ be the conjugate of the verticalline θ = θ, i.e., the curve in CS+ defined by

vx(ξ, ψ(ξ)) = vx(ξ, θ),

for all ξ ∈ X. Analogously, µ > 0 and ψξs = θ =⇒ ψξb = ψ(ξ), where ψ is

the conjugate of the vertical line θ = θ.

The following remark shows that conjugations of vertical lines are decreasingunder A2.

Remark 2 A2 implies that, for each fixed θ ∈ Θ, the implicit solution ψ(ξ) 6=θ of

vx(ξ, ψ(ξ))− vx(ξ, θ) = 0

is always decreasing. Indeed, by the implicit function theorem and vx2θ > 0,

dψ

dξ(ξ) =

vxx(ξ, θ)− vxx(ξ, ψ(ξ))

vxθ(ξ, ψ(ξ))< 0.

By Remark 2, ψ(ξ) and ψ(ξ) are decreasing functions, and eventually hit the

vertical lines θ = θ and θ = θ when ξ goes to 0 and to x, respectively. Notice

that these hitting points are the same and denoted by←→ξ ∈ X. Let us extend

ψ(ξ) (resp. ψ(ξ)) as θ (resp. θ), for all ξ ≤ (resp. ≥)←→ξ .

(2) ξ ≥ ξ. Here the analysis is more simple here. It is equivalent to makeψs = θ and µ = 0 in the above Lagrangian (L). Therefore, ψ∗

b is characterizedby (6).

The following lemma gives a complete characterization of the solution of Prob-lem (RUP). We use the following conventions for this lemma. Let us assumethat x = min

θ∈Θx1(θ) and x = max

θ∈Θx1(θ) without loss of generality. Let ψ1

b and

ψ1s be the lowest and highest inverse functions of x1 if they are in CS− and

CS+, respectively. Notice that condition C1 implies that they are, respectively,decreasing and increasing. Let ψub and ψus be the type assignment functions of

26

xu on [ξ0, ξ1], respectively, decreasing and increasing by condition C2, whereξ0 = min{xu(θ); θ ∈ [θ, θu]} and ξ1 = max{xu(θ); θ ∈ [θ, θu]} .

Lemma 3 (Solution of Problem (RUP)) Assume A1-2 and C1-2. If ξ =x, then the optimal type assignment are: 21

(i) ψξb = ψξs = x−10 (ξ), ∀ξ ∈ [x, ξ0].

(ii) ψξj = ψuj (ξ), j = b, s, ∀ξ ∈ [ξ0, ξ1].

(iii) ψξb = max{ψ1b(ξ), ψ(ξ)} and ψξs = max{ψ1

s(ξ), ψ(ξ)}, ∀ξ ∈ (ξ1, x].

If ξ ∈ X, ψξb and ψξs are characterized by (i)-(iii), ∀ξ ∈ [x, ξ], and:

(iv) ψξb = ψ1b(ξ) and ψξs = θ, ∀ξ ∈ [ξ, x].

Proof. (i) Let V (ξ) be the value function of Problem (RUP). By the envelopetheorem (see [17]) and the Lagrangian (L), we have that

V ′(ξ) =∫ ψ

ξ

b(ξ)

ψξs(ξ)

[fxx(ξ, θ)− µ(ξ)vx2θ(ξ, θ)]dθ.

By A1 and A2, V ′(ξ) ≥ 0. Notice that V (ξ0) = 0 because ψξb(ξ0) = ψξs(ξ0) (see

item (ii) below). Therefore, V (ξ) = 0 and ψξb = ψξs = x−10 (ξ), for all ξ ∈ [x, ξ0].

(ii) Let ξ ∈ [ξ0, ξ1]. Suppose that constraint (UC’) is not binding for Problem

(RUP) at the solution. Then, condition (11) implies that ψξj = ψ1j , j = b, s.

Since xu is inbetween x0 and x1, A2 implies that (UC’) is violated in this case,a contradiction. Then, constraint (UC’) must bind for Problem (RUP) at the

solution. By the uniqueness of xu, ψξb and ψξs should be the type assignment

functions of xu, for all ξ ∈ [ξ0, ξ1].

(iii) For all ξ ∈ [ξ1, x] the solution of Problem (RUP) is at the boundary. A1and condition (11) directly imply that the maximum of Problem (RUP) isattained at

ψξb =

ψ1b(ξ), if (UC’) is slack

ψ(ξ), if (UC’) is bindingand ψξs =

ψ1s(ξ), if (UC’) is slack

ψ(ξ), if (UC’) is binding,

for all ξ ∈ [ξ1, x]. Since ψ1b(ξ) ≥ ψ(ξ) if and only if (UC’) is slack, we can write

21 To be consistent with the definition of type assignment functions, on the interval[x, ξ0] they would be ψb = ψs = x−1

0 (ξ0). Notice, however, that this alternativedefinition does not modify the principal’s pointwise payoff in this interval.

27

ψξb = max{ψ1b(ξ), ψ(ξ)}. Analogously, ψξs = max{ψ1

s(ξ), ψ(ξ)}.

(iv) Now take any ξ ∈ X. For each ξ ≤ ξ, by definition of Problem (RUP), the

solution is characterized by (i), (ii) and (iii) above. For all ξ ≥ ξ, ψξs = θ bythe definition of ξ. A1 and condition (11) imply that the maximum of Problem

(RUP) is attained at ψξb = ψ1b(ξ).

From Remark 2 and Lemma 3 (iii), ψξs does not satisfy condition (M’) when

ψξb = θ. Thus, the solution of Problem (RUP) may display non-monotonicitiesand to get a solution of Problem (UP) from the solution of Problem (RUP)may involve ironing. Therefore, we determine the optimal jump (θ1) on thesolution of Problem (RUP) for each ξ and then find the optimal bunching (ξ).

Optimal jump (dealing with the lack of monotonicity). Fix the bunch-ing level ξ and consider Problem (RUP) with the constraint (M’). Let (ψ∗

b , ψ∗s)

be its optimal solution. 22 The Hamiltonian is:

H(ξ, ψb, ψs, µ, ψ′b, ψ

′s, δb, δs) = L(ξ, ψb, ψs, µ) + δbψ

′b − δsψ

′s,

where ψ′i is the derivative of ψi and δi ≥ 0 is the multiplier of (M’) for i = b, s.

The Pontryagin principle implies

·

δb = −∂ψbH and

·

δs = ∂ψsH

0 ≤ ∂ψ′bH = δb and 0 ≤ −∂ψ′

sH = δs

0 = δiψ′i, i = b, s.

For intervals where condition (M’) does not bind we have the solution ofProblem (RUP). If [ξ1, ξ2] is a maximum bunching interval for ψ∗

s then, fromthe above optimality conditions and the Lagrangian (L), we must have θ1 =ψ∗s(ξ),

vx(ξ, ψ∗b(ξ, θ1)) = vx(ξ, θ1)

and

0 ≤ δs(ξ) = −∫ ξ

ξ1

[fx(ξ, θ1)p(θ1)− µ(ξ)vxθ(ξ, θ1)]dξ, (13)

for all ξ ∈ [ξ1, ξ2], with an equality for ξ = ξ2 and θ1 < θ. By Remark2, ψ∗

b(ξ, θ1) is decreasing on [ξ1, ξ2] and condition (M’) is then slack for ψ∗b .

22 We are making some abuse of notation here because we use star for the solutionof Problem (UP) and we are not making explicit the dependence of this solution onξ. Notice, however, that when ξ is chosen to be optimal, this is exactly the solutionof Problem (UP).

28

Therefore, from condition (11),

δb(ξ) = 0 and µ(ξ) ≤fx(ξ, ψ

∗b(ξ, θ1))

vxθ(ξ, ψ∗b(ξ, θ1))

p(ψ∗b(ξ, θ1)),

for all ξ ∈ [ξ1, ξ2], with an equality when ψ∗b(ξ, θ1) > θ, and µ(ξ) = 0 otherwise.

If θu < θ, then Lemma 4 (iii) below shows that there exists 23 θ1 ∈ [θ1, θu]such that ξ1 = xu(θ1), ξ2 = x1(θ1) ∧ ξ and condition (13) at ξ = ξ2 becomes

−[f(x1(θ1) ∧ ξ, θ1)− f(xu(θ1), θ1)]p(θ1)

+∫ xu(θ,θ1)

xu(θ1)fx(ξ, ψ

∗b(ξ, θ1))

vxθ(ξ, θ1)

vxθ(ξ, ψ∗b(ξ, θ1))

p(ψ∗b(ξ, θ1))dξ = 0.

(14)

Using ψ∗b(ξ, θ1) to change variable ξ for θ in the previous integral we get the

jump condition (or vertical ironing principle): 24

∫ θ1

θ

fx(xu(θ, θ1), θ)

vxθ(xu(θ, θ1), θ)vxθ(xu(θ, θ1), θ1)p(θ)dθ = [f(x1(θ1), θ1)−f(xu(θ1), θ1)]p(θ1),

(VI)when ξ = x, for each feasible vertical bunching level θ1 ∈ [θ1, θu], let xu(·, θ1)be its associate decision function which is defined by

vx(xu(θ, θ1), θ) = vx(xu(θ, θ1), θ1),

for all θ ∈ [θ, θ1], where θ1 satisfies xu(θ1, θ1) = xu(θ1) = xu(θ1).

If θu = θ, Lemma 4 (iv) below shows that θ1 = θ is the optimal bunching ifand only if the equality in (14) becomes the inequality “greater or equal”.

Lemma 4 (Characterization of vertical ironing) For each θ1 ∈ [θ1, θu],let δs : [ξ1, ξ2] → R be the function defined by (13), where ξ1 = xu(θ1) andξ2 = x1(θ1) ∧ ξ. Then:

(i) ∃ξ∗ ∈ [ξ1, ξ2] such that δs|[ξ1,ξ∗]is positive and increasing; and δs|[ξ∗ ,ξ2]

is

decreasing and convex;

(ii) if θ1 = θ1, then ξ∗ = ξ2 and δs is always positive; if θ1 = θu < θ, thenξ∗ = ξ1 and δs is always negative;

(iii) if θu < θ, then there exists θ1 ∈ (θ1, θu) for which equation (14) holds;

23 Formally, θu is the maximal parameter where xu is defined and θ1 is parameterwhere the curves ψ1

s and ψ cross.24 Although this kind of phenomenon is not usual in the standard adverse selectionliterature, Wilson [27] mentions the possibility of “vertical ironing” (see pp. 179).Notice that condition (VI) is a natural generalization of Theorem 2.

29

(iv) if θu = θ, then either the conclusion of (iii) holds or the equation (14)becomes the inequality “greater or equal” for θ1 = θ.

Proof. (i) From the definition of δs, δ′s(ξ) = −fx(ξ, θ1)p(θ1) + µ(ξ)vxθ(ξ, θ1),

for all ξ ∈ [ξ1, ξ2]. Let ξ∗ ∈ [ξ1, ξ2] be such that µ(ξ) = 0, for all ξ ∈ [ξ∗, ξ2](which coincides with the point where the vertical line θ = θ1 crosses the graphof ψ). By A1 and A2, δ′′s(ξ) > 0 and, since δ′s(ξ2) = −fx(ξ2, θ1)p(θ1) ≤ 0,we conclude that δ′s(ξ) < 0, for all ξ ∈ (ξ∗, ξ2]. Using the uniqueness of xu(condition C2) and condition (18) below for θ = ψ∗

b(ξ, θ1) > ψ∗b(ξ), we have

that θ1 = ϕ(ξ, θ) < ψ∗s(ξ) and

µ(ξ) =fx(ξ, ψ

∗b(ξ, θ1))p(ψ

∗b(ξ, θ1))

vxθ(ξ, ψ∗b(ξ, θ1))

>fx(ξ, θ1)p(θ1)

vxθ(ξ, θ1),

for all ξ ∈ (ξ1, ξ∗]. Then, δ′s(ξ) > 0 and, since δs(ξ1) = 0, δs(ξ) > 0 for all

ξ ∈ (ξ1, ξ∗].

(ii) In particular, if θ1 = θ1 (resp. θ1 = θu < θ), then ξ∗ = ξ2 and δs(ξ) > 0(resp. ξ∗ = ξ1 and δs(ξ) < 0) for all ξ ∈ (ξ1, ξ2).

(iii) From item (ii) and the intermediate value theorem, there exists θ1 ∈(θ1, θu) such that the corresponding function δs satisfies δs(ξ2) = 0, which isexactly the equation (14).

(iv) If for θ1 = θ the function δs is such that δs(ξ2) ≥ 0, then, by item (i),δs(ξ) ≥ 0, for all ξ ∈ [ξ1, ξ2]. Otherwise, we can apply the same argumentmade in (iii) to prove the existence of θ1 ∈ (θ1, θu) for which equation (14)holds.

Optimal bunching (optimal ξ). Let [θ2, θ3] be a jump interval for ψ∗b at ξ,

i.e., θ2 and θ3 are the right- and left-hand side limits of ψ∗b at ξ, respectively.

If ξ is optimal, then

{θ2, θ3} ⊂ ψ∗b(ξ) = arg max

θ

L(ξ, θ, ψ∗s, µ)

and, in particular, we must have L(ξ, θ2, ψ∗s, 0) = L(ξ, θ3, ψ

∗s, µ(ξ)) or, using

the Lagrangian (L),

0 =L(ξ, θ3, ψ∗s, µ(ξ))− L(ξ, θ3, ψ

∗s, 0) + L(ξ, θ3, ψ

∗s, 0)− L(ξ, θ2, ψ

∗s, 0)

=∫ µ(ξ)

0∂µL(ξ, θ3, ψ

∗s, µ)dµ+

∫ θ3

θ2

∂ψbL(ξ, θ, ψ∗

s, 0)dθ

=µ(ξ)vx(ξ, θ3) +∫ θ3

θ2

fx(ξ, θ)p(θ)dθ.

30

Analogously, if [θ3, θ] is an optimal jump for ψ∗s at ξ, then L(ξ, ψ∗

b , θ3, µ(ξ)) =L(ξ, ψ∗

b , θ, 0) or

0 = −µ(ξ)vx(ξ, θ3) +∫ θ

θ3

fx(ξ, θ)p(θ)dθ.

Adding up these two last equations and using that vx(ξ, θ3) = vx(ξ, θ3), wehave the horizontal “ironing principle”

∫ θ3

θ2

fx(ξ, θ)p(θ)dθ +∫ θ

θ3

fx(ξ, θ)p(θ)dθ = 0, (HI)

where θ2, θ3 and θ3 are shown in the right graph of Figure 4. In particular, ifξ is below the crossing point of x0 and x1, then we have a degenerate case ofunique bunching interval (i.e., θ3 = θ3). Other degenerate case is when thereis no interval with the relaxed solution (i.e., θ2 = θ and x1(θ) < ξ).

Finally, we claim that both ironing principles cannot hold at the same time,i.e., the solution of Problem (UP) can present only one ironing procedure.Otherwise, from Lemma 4 (i) (in particular, its proof), for the optimal ξ theoptimal vertical ironing line must cross the curve ψ, which implies that ξ mustbe above this crossing point. Thus, the bunching interval would be completelybelow the curve x1 (i.e. θ3 < θ in the right graph of Figure 4). In this case,the integral in (HI) would be strictly positive due to A1. This implies that ξcannot be optimal unless ξ = x.

Therefore, the solution of Problem (UP) must have one of the shapes describedin the text.

The second-best solution. Next we determine whether the solution of Prob-lem (UP) is implementable. For this we need the following lemma which showsthat to get implementability, besides conditions (M’) and (UC’), we only haveto check the following condition that is complementary to (UC’): Let ψb andψs be a pair of type assignment functions of x ∈ X . Then, we that they sat-isfy the complementary relaxed U-shaped condition if ξ ∈ X is a point ofcontinuity for both ψb and ψs, then

ψb(ξ) > θ =⇒ vx(ξ, ψb(ξ)) ≥ vx(ξ, ψs(ξ)). (UC”)

Notice that conditions (UC’) and (UC”) are equivalent to condition (UC) inintervals where the decision functions have interior type assignment functions(i.e., ψb(ξ) > θ and ψs(ξ) > θ). Otherwise, condition (UC”) avoids the problemraised in Figure 1 for other global IC constraints besides (UC) that may bebinding.

31

Lemma 5 (Sufficient conditions for implementability) Assume A2. Sup-pose that ψb and ψs satisfy conditions (M’), (UC’) and (UC”). Then, x isimplementable.

Proof. We have to show that there exists a non-linear tariff T (ξ) such thatfor each θ ∈ Θ and ξ ∈ x(θ)

v(ξ, θ) + T (ξ) ≥ v(ξ, θ) + T (ξ),

for all ξ ∈ x(Θ). By the fundamental theorem of calculus, this is equivalent to

∫ ξ

ξ[vx(ξ, θ) + T ′(ξ)]dξ ≥ 0.

For such x, take the corresponding type assignments ψb and ψs. Let us considerthe following cases:

(a) x(θ) ≥ x(θ). By Remark 1, we can define T ′(ξ) = −vx(ξ, ψs(ξ)) for a.e.ξ ∈ x(Θ). There are two possibilities:

• θ = ψs(ξ). Then, implementability of x is equivalent to

∫ ξ

ξ[vx(ξ, θ) + T ′(ξ)]dξ =

∫ ξ

ξ

∫ ψs(ξ)

ψs(ξ)vxθ(ξ, θ)dθdξ ≥ 0,

for a.e. ξ ∈ x(Θ). Since x0 is a decreasing function, the area delimited by theprevious double integral is contained in CS+, for a.e. ξ ∈ x(Θ). Therefore, itis non-negative if and only if ψs is non-decreasing which is ensured by (M’).

• θ = ψb(ξ). We claim that

∫ ξ

ξvx(ξ, θ)dξ ≥

∫ ξ

ξvx(ξ, ψs(ξ))dξ, (15)

for a.e. ξ ∈ x(Θ). Indeed, defining the difference function as

a(ξ, ξ) =∫ ξ

ξ[vx(ξ, θ)− vx(ξ, ψs(ξ))]dξ,

we have that a(ξ, ξ) = ∂ξa(ξ, ξ) = 0 by (UC’) and (UC”). Moreover,

∂2

ξa(ξ, ξ) = vxx(ξ, ψs(ξ))− vxx(ξ, θ) > 0

since vx2θ > 0 and ψs(ξ) > θ. Hence, a(ξ, ξ) ≥ 0 for all ξ ∈ x(Θ).

Subtracting∫ ξ

ξvx(ξ, ψs(ξ))dξ from each side of (15), the implementability for

θ = ψb(ξ) follows from the implementability for the case θ = ψs(ξ).

32

(b) x(θ) < x(θ). Again by Remark 1, we can define T ′(ξ) = −vx(ξ, ψb(ξ)) fora.e. ξ ∈ x(Θ). Define ψb such that

vx(ξ, ψb(ξ)) = vx(ξ, θ),

for all ξ > x(θ), and identical to ψb, for all ξ ≤ x(θ). By (UC’) and (UC”),

vx(ξ, ψb(ξ)) = vx(ξ, θ) ≤ vx(ξ, ψb(ξ)),

for all ξ > x(θ). This and A2 imply that ψb ≥ ψb. From Remark 2, ψb is non-increasing (it is ψb for ξ < x(θ) and jumps downward at x(θ)). From the case(a), the decision associated to the type assignment functions ψb and ψs, x, isimplementable because x(θ) = x(θ). From this and the fact that x satisfies(M) and (UC), the only IC constraints that remain to be checked for x are theones illustrated in Figure 1 (the other IC constraints are similar to the case(a)). This is equivalent to check that

∫ ξ

ξ[vx(ξ, θ) + T ′(ξ)]dξ =

∫ ξ

ξ

∫ ψb(ξ)

ψb(ξ)vxθ(ξ, θ)dθdξ ≥ 0,

for θ = ψb(ξ) and ξ ≥ ξ. However, since ψb ≥ ψb, A2 and the implementabilityof x imply

∫ ξ

ξ

∫ θ

ψb(ξ)vxθ(ξ, θ)dθdξ ≥

∫ x(θ)

ξ

∫ θ

ψb (ξ)vxθ(ξ, θ)dθdξ ≥ 0

since ψb(x(θ)) = θ. This completes the proof of implementability of x. 25

The following condition avoids that other global IC constraints than (UC)bind for x∗. For instance, if the decreasing part of x1 is close enough to thelimit curve x0, condition C3 may fail because of the problem raised in Figure1.

C3. The type assignment functions (ψ∗b , ψ

∗s) satisfy condition (UC”).

Condition C3 holds under x1(θ) < x1(θ). Otherwise, there exists ξ ∈ Xsuch that ψ∗

b > θ and condition (UC’) is slack for Problem (RUP). Hence,ψ∗b = ψ1

b and µ(ξ) = 0. A1 and x1(θ) < x1(θ) imply

∂ψsL(ξ, ψ∗

b , θ, 0) = −fx(ξ, θ)p(θ) < 0

and then ψ∗s = ψ1

s < θ. Lemma 3 (ii) (specially its proof) shows that thiscannot be the case.

25 The intuition of this part of the proof can be explained using Figure 1. Supposenow that x = x1 is an implementable decision and x is non-increasing, and com-pletely below x. Then, x is also implementable. Indeed, the typical weighted hatchedarea of Figure 1 will be always non-negative for x once it is for x.

33

Proof of Theorem 3. We only have to check that the solutions or Problem(UP) satisfy the conditions of Lemma 5. All conditions but (UC”) are triviallysatisfied. Condition C3 implied by C2 (i.e., by x1(θ) < x1(θ)), however, ensurescondition (UC”).

Sufficient condition for the uniqueness of xu. Let ξ0 be the parameterwhere x0, x1 and xu cross. Suppose that for each ξ ≥ ξ0 and θ 6= x−1

0 (ξ): 26

∂

∂θ

(fx(ξ, θ)

vxθ(ξ, θ)p(θ)

)> 0. (16)

For each such ξ let us make the following change of variables:

Ψj = vx(ξ, ψj), for j = b, s. (17)

In the new variables Ψb and Ψs, the constraint (UC’) becomes the linearconstraint: Ψb ≤ Ψs. By the implicit function theorem, the derivative of theobjective function of Problem (RUP) with respect to Ψb and Ψs are

fx(ξ, ψb)

vxθ(ξ, ψb)p(ψb) and −

fx(ξ, ψs)

vxθ(ξ, ψs)p(ψs),

respectively. Since vx(ξ, ·) is decreasing on [θ, x−10 (ξ)] and increasing on [x−1

0 (ξ), θ],condition (16) ensures that the objective function of Problem (RUP) is strictlyconcave in Ψb and Ψs. Moreover, the cross derivative with respect to Ψb andΨs is zero, which implies that it is a strictly concave function. Therefore, inthese new variables Problem (RUP) is a strictly concave program which im-plies the uniqueness of solution. Since (17) defines a monotone transformation,the solution is unique in the original variables as well.

Jullien [11] assumes the “potential separation property” that has a similarrole of our condition C2. Although we prove below that ψ∗

b is non-increasing,we cannot ensure that ψ∗

s is always non-decreasing (as a function of ξ). In-deed, condition (M’) binds for Problem (RUP) when the solutions are at theboundary.

Monotonicity of the type assignment function ψξb. Let us consider the

26 For private value model (i.e., uxθ = 0), a simple calculation of the right and leftlimits of condition (16) at x−1

0 (ξ) shows that

limθ→(x−1

0(ξ))

±

∂

∂θ

(fx(ξ, θ)p(θ)

vxθ(ξ, θ)

)= ∓∞

for all ξ ≥ ξ0, and the reverse asymptotic limits occur for ξ < ξ0. In particular,condition (16) can only hold for ξ ≥ ξ0. Jullien [11] uses a similar condition toensure the potential separation assumption in his paper.

34

interior solution case. (The boundary solution case is an immediate conse-quence of Lemma 3.) Defining, for each ξ, D(ξ, θ) = F (ξ, θ) − F (ξ, ϕ(ξ, θ))and ϕ(ξ, θ) 6= θ implicitly by vx(ξ, θ) = vx(ξ, ϕ(ξ, θ)), we have that

ψξb(ξ) ∈ arg maxθ∈[θ,x−1

0(ξ)]

D(ξ, θ)

is a critical point of D(ξ, θ), i.e., Dθ(ξ, ψξb(ξ)) = 0. Taking the total derivative

of with respect to ξ we get

Dθθ(ξ, ψξb(ξ))

dψξbdξ

(ξ) = −Dθx(ξ, ψξb(ξ)).

Using the definition of F in (10),

Dθ(ξ, θ) = fx(ξ, θ)p(θ)− fx(ξ, ϕ)p(ϕ)ϕθ(ξ, θ)

and

Dθx(ξ, θ) = fxx(ξ, θ)p(θ)− fxx(ξ, ϕ)p(ϕ)ϕθ(ξ, θ)

−[fxθ(ξ, ϕ)p(ϕ) + fx(ξ, ϕ)p′(ϕ)]ϕθ(ξ, θ)2 − fx(ξ, ϕ)p(ϕ)ϕθx(ξ, θ),

where ϕ = ϕ(ξ, θ) and ϕθ(ξ, θ) = vxθ(ξ, θ)/vxθ(ξ, ϕ) < 0.

Since fx(ξ, ϕ)p(ϕ) = µ(ξ)vxθ(ξ, ϕ) from (12) and ϕθx(ξ, θ)vxθ(ξ, ϕ) = vx2θ(ξ, θ)−ϕθ(ξ, θ)vx2θ(ξ, ϕ)− ϕθ(ξ, θ)

2vxθ2(ξ, ϕ), we have that

Dθx(ξ, θ) = fxx(ξ, θ)p(θ)− µ(ξ)vx2θ(ξ, θ)− [fxx(ξ, ϕ)p(ϕ)− µ(ξ)vx2θ(ξ, ϕ)]ϕθ(ξ, θ)

−[fxθ(ξ, ϕ)p(ϕ) + fx(ξ, ϕ)p′(ϕ)− µ(ξ)vxθ2(ξ, ϕ)]ϕθ(ξ, θ)2.

The second-order condition of the Lagrangian (L) implies that

fxθ(ξ, ϕ)p(ϕ) + fx(ξ, ϕ)p′(ϕ)− µ(ξ)vxθ2(ξ, ϕ) ≥ 0,

where ϕ = ϕ(ξ, ψξb(ξ)). Therefore, A1 and A2 imply that Dθx(ξ, ψξb(ξ)) < 0.

By the second-order condition, Dθθ(ξ, ψb(ξ)) < 0. Then, ψξb is decreasing. 27

Using Dθ(ξ, θ) and ϕθ(ξ, θ), the uniqueness of xu implies that

fx(ξ, θ)

vxθ(ξ, θ)p(θ)−

fx(ξ, ϕ(ξ, θ))

vxθ(ξ, ϕ(ξ, θ))p(ϕ(ξ, θ)) ≶ 0⇐⇒ θ ≶ ψξb(ξ). (18)

27 We cannot have the same conclusion for ψs. The reason is that, in this case, thefirst and second lines of the last expression of Dθx have opposite signs.

35

Appendix C. Example (labor contract)

Suppose that a firm, monopolistic in the labor market, can hire a managerwith a vector of unknown abilities (θ, η) to deliver outcome x through thetechnology:

x = θe+ αη,

where e is the manager’s unknown effort, θ is her marginal productivity ofeffort and η is a fixed factor with common knowledge shadow price α.

The firm cannot contract directly on the abilities, but it contracts on anexogenous verifiable fixed signal s (schooling, interview, etc.) that mixes theseabilities:

s = θ + η.

Given s, a higher marginal return to effort (the slope θ in the productionfunction) means a lower η (and hence a lower intercept αη in the productionfunction). Therefore, the effort required by an agent with abilities (θ, s− θ)to achieve output level x,

e(x, θ) = θ−1(x− αη) = θ−1(x− αs) + α,

is not necessarily decreasing in θ.

Suppose that the manager’s cost of effort is quadratic. If t is the manager’swage, her utility function has the quasi-linear form of Section 2, V = v(x, θ)+t,with

v(x, θ) = −e(x, θ)2.

The firm is on the other hand a profit maximizer with utility function U = x−t.

Let us check the validity of the SMC:

vxθ(x, θ) = 2θ−2(2e(x, θ)− α) = 2θ−2(2θ−1(x− αs) + α).

When outputs and efforts are low (high), the intercept can be more (less)important than the slope. In this case, increasing (decreasing) the slope θ -but reducing (improving) the intercept - can make more (less) effort needed toachieve the same level of output, and hence increases (decreases) the marginalcost of output: vxθ < (>)0.

If the shadow price of η is zero (α = 0), then the marginal cost of productiondecreases with θ (vxθ > 0), which implies the SMC and that implementableoutputs are non-decreasing in θ. This is the standard case since θ is productiv-ity, η has no impact on the firm’s technology and s is an uninformative signalof the manager’s productivity.

On the other hand, if α is high enough, then the marginal cost of productionincreases with θ (vxθ < 0) because η is now relatively more valuable than θ

36

for the firm’s technology. A given signal s may convey bad news if θ is highsince it means low η. Then, the SMC holds again but outputs decrease withθ for every implementable contract.

For intermediate values of α we may have both signs: low (high) θ implies thatthe marginal cost of production increases (decreases) with θ. Indeed, we cancompute the limit curve x0 between CS− (below x0) and CS+ (above x0), i.e.,the curve implicitly defined by vxθ(x, θ) = 0 in the positive quadrant:

x0(θ) = α[s− 0.5θ]+,

where [a]+ = max{a, 0}.

Given s, suppose that θ is uniformly distributed (the conditional distributionof η is then uniform as well). Consider a particular example: Θ = [1, 1.5]. Therelaxed solution is given by:

x1(θ) = min

{αs, αs−

2α− θ2

2(2− θ)θ

}.

Observe that x1 satisfies condition C1 and crosses x0 exactly at θ = α. Hence,α ∈ (1, 1.5) if and only if the SMC does not hold.

The left and right graphs below give, besides x0 and x1, the optimal decisionscharacterized by Theorem 3 for s = 1.5 and α = 1.15 and 1.3, respectively.

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.50.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

xsb

x0

x1

Labor Contract

θ

x

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.50.9

1

1.1

1.2

1.3

1.4

1.5

xsb

x0

x1

Labor Contract

θ

x

The sign of the derivatives and the critical U-shaped decision. An easy inspec-tion gives

fxx(ξ, θ) = 2(θ − 2) < 0,

for θ ∈ [1, 1.5] which implies A1. The third derivatives of v, vx2θ(ξ, θ) = 4θ−3 >0 and vxθ2(ξ, θ) = 2θ−3[6θ−1(αs−ξ)−2α] (positive along x0), imply A2. Since

37

vx(ξ, θ) = −2θ−1e(ξ, θ), condition (UC) gives

θ = ϕ(θ, ξ) = (α(αs− ξ)−1 − θ−1)−1.

Thus, the U-shaped part characterized by Theorem 2 is a root of the followingthird degree polynomial (the monotonicity condition eliminates the other tworoots):

−α2

4+ αθ−1

(1

2+ αθ−1

)y − θ−2

(1

2+ 2αθ−1

)y2 + θ−4y3 = 0,

where y = αs− ξ and condition C2 is easily satisfied.

Appendix D. Suplements (demand profile approach and example)

Demand profile approach. For simplicity, let us assume that u does notdepend directly on θ (private value case). Define the cumulative aggregatedemand functional:

D(τ , ξ) = Prob [−vx(ξ, θ) ≤ τ ],

where −vx is the type-θ marginal cost of ξ. For each ξ, the principal’s programis equivalent to determine the marginal tariff that solves

maxτ

D(τ , ξ)[u(ξ)− τ ].

The first-order condition gives the well-known equalization of the markup tothe inverse of the demand elasticity with respect to τ :

τ − ux(ξ)

τ=

1

ǫ(τ , ξ), (19)

where ǫ(τ , ξ) = −τDτ (τ , ξ)/D(τ, ξ).

If there is only one marginal type θ choosing ξ, then D(τ , ξ) = P (θ). Usingthe implicit function theorem on vx(ξ, θ) + τ = 0, we obtain a formula for theelasticity:

ǫ(τ , ξ) =p(θ)

P (θ)

τ

vxθ(ξ, θ)which is proportional to the hazard rate p(θ)/P (θ). Plugging this into theexpression (19) gives exactly the first-order condition of the relaxed problem(6).

On the other hand, suppose that ξ belongs to a U-shaped part. That is, thereare two marginal pooling types, θ < θ, choosing ξ. Thus, D(τ , ξ) = P (θ)−P (θ)and arguing in same way as before we get

ǫ(τ , ξ) =p(θ) + λ(ξ, θ)p(θ)

P (θ)− P (θ)

τ

vxθ(ξ, θ),

38

where λ(ξ, θ) = vxθ(ξ, θ)/vxθ(ξ, θ) and vx(ξ, θ) = vx(ξ, θ) = −τ . This is exactlythe content of Theorem 2. Since λ(ξ, θ) < 0 and P (θ) ≥ 0, the hazard ratehere is lower than the corresponding in the separation case. This explains theupward jump at the transition point.

Example showing how restrictive is X . Let Θ = [0, 1] with uniformdistribution, v(x, θ) = 1

2θx(x−1) and u(x, θ) be such that the relaxed function

is f(x, θ) = x(θ − x

2

). Thus, the limit curve x0 is the horizontal line ξ = 1/2

such that CS−(+) is below (above) it and the relaxed solution is x1(θ) = θ.Therefore, the solution of Problem (UP) in X is x∗(θ) = max{x0(θ), x1(θ)}.However, if we allow for discontinuous crossings (i.e., decisions not in X ), thenone may find an incentive compatible decision that dominates x∗: 28

x∗∗(θ) =

1/4, for θ ∈ [0, 1/2)

max{3/4, x1(θ)}, for θ ∈ [1/2, 1].

Moreover, if we change the distribution of types, more crossing points mayenhance welfare. Consider this other incentive compatible decision:

x∗∗(θ) =

5/8, for θ ∈ [0, 1/4) ∪ [1/2, 5/8)

3/8, for θ ∈ [1/4, 1/2)

x1(θ), for θ ∈ [5/8, 1].

We see that, to keep incentive compatibility, the welfare gain to better ap-proximate the relaxed solution on [1/4, 1] is penalized by the loss on [0, 1/4).However, if the density weight on [0, 1/4) is much smaller than on [1/4, 1],then x∗∗ would increase the welfare gain to the principal compared with theprevious decision. One might guess that we can keep doing this process. Thequestions are, for more general distributions and problems, how to determinethe optimal decision and what would be the appropriate space of contracts.

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28 The welfare loss of x∗ w.r.t. x1 is the triangle area delimited by the vertical lineθ = 0, x0 and x1, i.e., 1/8. On the other hand, the loss of x∗∗ w.r.t. x1 is threetriangles of area of 1/32 each, i.e., 3/32 which is lower than 1/8. Moreover, one caneasily check that (GIF) is always non-negative for x∗∗, i.e., it is incentive compatible.

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