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Equity and Adverse Selection

RAMARAO DESIRAJU

Department of MarketingCollege of Business Administration

University of Central FloridaOrlando, FL 32816

[email protected]

DAVID E. M. SAPPINGTON

Department of EconomicsUniversity of Florida

PO Box 117140, Gainesville, FL [email protected]

We introduce concerns with inequity into the canonical adverse selection model.We find that aversion to ex post inequity is not constraining for the principalif the two agents are identical ex ante, but generally is constraining when theagents differ ex ante. Constraining equity concerns can lead to output levelsthat are either above or below standard levels, and can result in only one agentexperiencing systematic inequity in equilibrium.

1. Introduction

There is substantial evidence that individuals typically care about boththeir absolute and their relative well-being. For example, workers gen-erally are less satisfied with the wages they receive if coworkers withsimilar skills and job assignments are paid higher wages. Consistentfindings of this sort in the management, psychology, and economicsliteratures suggest the importance of incorporating concerns with rel-ative payoffs—or, more generally, with “fairness” or “equity”—intoformal models of economic activity.1 Yet surprisingly few models inthe economics literature in fact do so.

The standard adverse selection model is a case in point. In thesimplest of these models, a principal (e.g., an employer) contracts with

The authors thank two anonymous referees and the coeditor for very helpful commentsand suggestions.

1. See Adams (1963, 1965), Pritchard et al. (1972), Akerlof and Yellen (1988), Loewen-stein et al. (1989), Pinder (1998), Isaac (2001), Bolton and Ockenfels (2005), and Luttmer(2005), for example. In practice, individuals may be averse both to inequitable outcomesand to inequitable procedures. See Greenberg (1990) and Colquitt et al. (2006), for example.

C© 2007, The Author(s)Journal Compilation C© 2007 Blackwell PublishingJournal of Economics & Management Strategy, Volume 16, Number 2, Summer 2007, 285–318

286 Journal of Economics & Management Strategy

a risk-neutral agent (e.g., a worker) who is privately informed aboutwhether his production cost is high or low. It is well known that theoptimal contract for the principal in this standard setting eliminates allrent for the high-cost agent and affords the low-cost agent exactly the(positive) rent he could secure by exaggerating his cost. Furthermore,the output of the high-cost agent is reduced below the efficient level inorder to reduce the rent secured by the low-cost agent.

Corresponding conclusions emerge from standard models of ad-verse selection in which a principal contracts with multiple agentswhose costs are uncorrelated. An important feature of these standardmodels is that each agent’s well being is assumed to be affected onlyby his own net compensation. Thus, given a specified level of netcompensation, an agent’s utility is presumed to be the same whetheranother agent receives the same, a much higher, or a much lower netcompensation. In particular, although a high-cost agent would preferto secure positive rent, his utility is not reduced by the fact that a low-cost agent secures positive rent in equilibrium. In this sense, agents arenot troubled by reward structures that produce outcomes that mightbe construed as “inequitable” or “unfair” in the standard models ofadverse selection.

The purpose of this research is to begin to assess whether the well-known conclusions in the adverse selection literature require modifi-cation when agents are averse to inequity. In particular, we examinewhether aversion to inequity on the part of agents is constraining for theprincipal in the standard adverse selection model with two risk-neutralagents. We find that aversion to inequity (i.e., aversion to unequalequilibrium net compensation) is not constraining for the principalwhen the agents are identical ex ante (i.e., when their cost realizations areindependent draws from the same distribution. Even when agents areidentical ex ante, though, their costs can differ ex post). We demonstratethat when the agents are identical ex ante, the principal can structurepayments so as to eliminate inequity without ceding additional rent tothe agents or altering their outputs.

In contrast, when the agents differ ex ante, aversion to inequitygenerally is constraining for the principal. When agents differ ex ante,the principal typically is unable to structure payments to avoid ex postinequity without ceding additional rent to the agents. Consequently,the principal may optimally admit some inequity in equilibrium. Whenshe does so, one agent systematically experiences inequity for all costrealizations.2 To limit the extent of this inequity, the principal reduces

2. Depending upon parameter values, the agent that experiences inequity in equilib-rium can be either the agent with lower expected cost or the agent with higher expectedcost.

Equity and Adverse Selection 287

below the standard output levels (i.e., the output levels in the standardadverse selection model) the output of the agent that does not experienceinequity in equilibrium.3

When the agents are sufficiently averse to inequity, the principaloptimally eliminates all ex post inequity. When she does so, the principalreduces the output of one high-cost agent below the standard outputlevel and increases the output of the other high-cost agent above thestandard level. The pronounced output reduction is optimal becauseit serves both to limit one agent’s rent and to reduce the inequityexperienced by the other agent. The output increase is the most efficientmeans to provide the additional rent an agent requires to reduce theinequity he faces.

These findings are stated more precisely and explained in detailin section 3. First, though, the key elements of our formal model arepresented in section 2. The implications of our findings and directionsfor future research are discussed in section 4. The proofs of our mainfindings are provided in the Appendix.

Before proceeding, we explain how our analysis relates to otherstudies in the literature. Our formal treatment of inequity aversionparallels Fehr and Schmidt (FS)’s (1999) treatment in that both analysesassume an agent’s utility is the difference between his direct payoff andthe disutility he suffers due to disparities in payoffs. An agent’s payoffin our model is his net compensation, which is the difference betweenthe payment he receives from the principal and his production costs. InFS’ model, the agent’s payoff is simply the payment he receives.4 LikeFS, we focus on inequity in outcomes rather than inequity in processesor procedures. Consequently, we abstract from the possibility that anagent’s evaluation of payoff disparities might vary with the intent ormotivation of other agents in generating observed outcomes (Rabin,1993; Fehr et al., 1997; Fehr and Gachter, 2000; Charness and Rabin,2002; Bolton and Ockenfels, 2005). We also abstract from endogenousperceptions of what constitutes a fair outcome (Konow, 1996, 2000).

Our focus on whether inequity aversion complicates adverseselection problems complements the analyses of Demougin and Fluet(2003), Itoh (2004), and Grund and Sliwka (2005), which examine

3. The outputs of the agent that experiences inequity for all cost realizations coincidewith outputs in the canonical adverse selection problem.

4. In assuming that the disutility from inequity reflects differences in realized netcompensation, we depart from models in the psychology literature (e.g., Adams, 1963,1965) that presume disutility from inequity increases with differences in the ratios ofoutputs (e.g., payment) to inputs (e.g., effort or contribution to total output). However,as shown in Desiraju and Sappington (2005), many of the qualitative conclusions drawnbelow persist in this alternative setting.


whether inequity aversion complicates moral hazard problems.5 Theirfindings that concerns with inequity generally are constraining in moralhazard models parallel our conclusion that inequity aversion typicallyis constraining in adverse selection models when agents differ ex ante.

2. The Model

A principal contracts with two risk-neutral agents, called agent A andagent B, to produce output, x. The principal’s valuation of the outputof agent i (denoted xi, for i = A, B) is given by the strictly increasingand strictly concave function V(xi).6 The principal seeks to maximizethe difference between the expected value of the agents’ outputs andthe payments she delivers to the agents.

Agent i‘s cost of producing output x is C(x, ci) for i = A, B. Theparameter ci can take on one of n ≥ 2 possible values, c1 < c2 < · · · < cn.Higher realizations of c are associated with higher total and marginalcosts of production.7 Thus, c might represent the agent’s constantmarginal cost of production, for example. Each agent is privately in-formed about his cost parameter from the outset of his interaction withthe principal. The principal believes that agent i‘s cost parameter is cj

with probability φij ∈ (0, 1). To emphasize the changes introduced by

concerns with inequity, we assume the distributions of the agents’ costsare independent.8 Recall that when agents are not averse to inequity, nostrict gains arise from linking an agent’s payment to the performance(or the cost report) of another agent when their costs are independent.In contrast, as demonstrated below, such linking is a central feature ofoptimal compensation structures when agents are averse to inequity(even when the agents’ costs are independent).9

5. Grund and Sliwka (2005), who find that it is more costly to induce agents toparticipate in tournaments when the agents are averse to inequity, provide additionalreferences to studies of inequity concerns in moral hazard settings.

6. The assumption that the principal values the output of each agent independentlyhelps to draw useful parallels to the canonical adverse selection model in which theprincipal interacts with a single agent. The assumption that the principal values the outputof each agent symmetrically facilitates the exposition by allowing all relevant differencesbetween the agents to be reflected in their expected costs.

7. Formally, for any ck > cj and for any x > 0, C(x, ck) > C(x, cj) and Cx(x, ck) >

Cx(x, cj), where Cx(·) denotes the partial derivative of C(·) with respect to x. C(x, ci) isa convex function of x for all ci.

8. Thus, the random cost parameter is best viewed as reflecting the residual, id-iosyncratic component of cost after all common elements of the agents’ costs have beenaccounted for.

9. As we explain further below, our primary conclusion extends to settings in whichthe agents’ costs are correlated.


An agent will agree to work for the principal if his expectedutility exceeds his reservation utility level, which is normalized to zero.An agent’s utility (U(·)) is the difference between the payment (T) hereceives from the principal and the sum of his physical productioncosts, C(·), and the disutility D(·) ≥ 0 he incurs due to inequity (soU(·) = T − C(·) −D(·)). An agent experiences inequity when his netcompensation (N = T − C(·)) differs from the net compensation of hiscounterpart. We will denote by Di(Ni, N−i) the disutility that agent iincurs when his net compensation is Ni and the net compensationof the other agent is N−i. The agents experience no disutility frominequity when they receive the same net compensation (so DA(·) =DB(·) = 0 when NA = NB). However, at least one agent experiencesstrictly positive disutility from inequity when their net compensationsdiffer (so DA(·) > 0 and/or DB(·) > 0 when NA �= NB). No additionalstructure is imposed on D(·) at this point.10

To provide a formal statement of the principal’s problem in thissetting, the following additional notation is helpful. Let Ti

jk denotethe payment from the principal to agent i when agent i reports hiscost parameter to be cj and the other agent (agent −i) reports his costparameter to be ck (for i, −i = A, B; j, k = 1, . . . , n). Also, let xi

jk denotethe corresponding output of agent i.11

The timing in the model is as follows. First, each agent privatelyobserves his cost parameter. Second, the principal designs the contractthat will govern her future interaction with the agents. A contractspecifies payments and output levels for the two agents, based on theirreported costs. Third, after observing the terms of the contract, the agentsannounce their cost parameters simultaneously and independently. (Theagents are unable to coordinate their actions or share their private costinformation in any way.) Finally, outputs are produced and paymentsare made as per the terms of the contract. The interaction between theprincipal and the agents is not repeated.

The principal’s problem [P] is:

maxTi

jk,xijk

n∑j=1

n∑k=1

φAj φB

k

[V

(xA

jk

) − T Ajk + V

(xB

kj

) − T Bkj

](1)

10. We will assume later that an agent incurs disutility from inequity only when his netcompensation is lower than the net compensation of the other agent. For now, though, wealso allow for the possibility that an agent might incur disutility (perhaps due to feelingsof guilt associated with being relatively overcompensated) when his net compensationexceeds that of his counterpart. See, for example, Fehr and Schmidt (1999), Bolton andOckenfels (2000), and Charness and Rabin (2002).

11. For simplicity, we assume V′(0) is sufficiently large for i = A,B and Cx(0, cn) issufficiently small that the principal always secures strictly positive output from bothagents.


s.t. for all i = A, B and j, k = 1, . . . , n :

EUi (c j | c j ) ≥ 0, and (2)

EUi (c j |c j ) ≥ EUi (ck |c j ), for k �= j, (3)

where EUi (ck | c j ) ≡n∑

m=1

φ−im

[Ti

km − C(xi

km, c j) − Di(Ni

kmj, N−imk

)], (4)

where Nikmj = Ti

km − C(xi

km, c j)

and

N−imk = T−i

mk − C(x−i

mk, cm)

for all i �= −i, i, −i = A, B.

Expression (1) reflects the principal’s desire to maximize the differencebetween the expected value of the agents’ outputs and the paymentsmade to the agents. Inequality (2) ensures the agents receive at leasttheir reservation expected utility, and so will agree to work for theprincipal. Equation (4) defines agent i‘s expected utility when his costparameter is cj and he reports this parameter to be ck, given that agent − ireports his cost parameter truthfully. Inequality (3) ensures that eachagent reports his cost parameter truthfully as a Nash response to truthfulcost reporting by the other agent.

3. Findings

We seek first to determine when the agents’ concern with inequityis constraining for the principal. To do so, consider two variants ofproblem [P]. The first variant, [P − N], is problem [P] in the special casewhere Di(Ni

jk, N−ikj ) = 0 for all Ni

jk, N−ikj . In words, [P − N] denotes the

principal’s problem in the simple (canonical) setting where inequity isof no concern to the agents. The second variant of [P], problem [P − E],appends constraint (5) to problem [P]:

Tijk − C

(xi

jk, c j) = T−i

kj − C(x−i

kj , ck)

for all

j, k = 1, . . . , n and i �= −i, i, −i = A, B. (5)

Constraint (5) eliminates inequity by ensuring identical net compensa-tion for the two agents for every possible pair of cost realizations.

We can conclude that the agents’ concern with inequity is notconstraining for the principal if there is a solution to [P − N] that is alsoa solution to [P − E]. In contrast, inequity concerns will be constrainingfor the principal if no solution to [P − N] is a solution to [P − E].


Proposition 1: If the agents are symmetric ex ante (so φAi = φB

i for alli = 1, . . . , n), there is a solution to [P − N] that is also a solution to [P − E],and so concerns with inequity are not constraining for the principal.

Proposition 1 implies that if the agents are identical ex ante, theprincipal can eliminate all inequity ex post while ensuring the standardoutput levels and expected payments (i.e., the outputs and expectedpayments that she optimally secures in the absence of concerns aboutinequity).12 Consequently, regardless of the exact nature of the agents’aversion to unequal net compensation, the aversion is not constrainingfor the principal when the agents are identical ex ante.

To provide some understanding of this finding, recall the standardsolution to the binary adverse selection problem (i.e., the solution to[P − N] with n = 2 in which the payment to an agent depends only onhis own cost report, not the report of the other agent). At this standardsolution, the high-cost agent receives no rent and the low-cost agentearns the (positive) rent he could secure by exaggerating his cost.13 Thus,this solution produces ex post inequity for an agent that is the only one tohave high costs. This inequity for agent A and for agent B, respectively,can be expressed formally as:

NA21 ≡ T A

21 − C(xA

21, c2)

< T B12 − C

(xB

12, c1) ≡ NB

12; and (6)

NB21 ≡ T B

21 − C(xB

21, c2)

< T A12 − C

(xA

12, c1) ≡ NA

12. (7)

When the agents are symmetric ex ante, payments can be struc-tured to eliminate the inequities in (6) and (7) without ceding anyadditional rent to the agents (relative to the standard solution), withoutreducing total surplus, and without introducing any additional formsof inequity. The inequities in (6) and (7) can be eliminated, for example,by increasing TA

21 and TB21 while reducing TA

22 and TB22 at the rate φA

1 /

φA2 = φB

1 /φB2 to the point where NA

21 = NB12 and NB

21 = NA12. Notice that

these changes do not alter either agent’s expected net compensationwhen they have high costs (ENi

2) because, holding output constant:14

−dTi22

dTi21

∣∣∣∣∣dENi

2=0

= φ−i1

φ−i2

for i, −i = A, B. (8)

12. Formally, the standard outputs and expected payments are those induced at thesolution to [P − N].

13. The “high-cost agent” is the agent with the high cost parameter c2. The “low-costagent” is the agent with the low cost parameter c1.

14. ENji ≡ φ1

−i[Tj1i − C(xj1

i, cj)] + φ2−i[Tj2

i − C(xj2i, cj)].


Notice also that when the agents are symmetric ex ante at the standardsolution, they are induced to produce the same output (x2) and areafforded the same payment (T2 = C(x2, c2)) when they have high costs.The agents also secure the same rent (C(x2, c2) − C(x2, c1)) when theyhave low costs. Consequently, the payment variations identified aboveeliminate inequity without ceding any additional rent to the agents (inexpectation).

The same is not true when the agents differ ex ante. In this case,the agents will be induced to produce different output levels and sowill receive different payments when they have the same high cost(c2) in the standard solution. The agents also will be afforded differentlevels of rent when they have the same low cost (c1). Furthermore,as equation (8) indicates, the variations in T21 and T22 that leave onehigh-cost agent’s expected net compensation unchanged will alter thecorresponding net compensation of the other high-cost agent. Thesecomplications generally make it impossible for the principal to securethe same expected outcomes that she secures in the standard solutionwhen the agents differ ex ante. This conclusion is recorded formally inProposition 2. In the statement of the proposition and throughout theensuing discussion, agent A is presumed to have the lowest expectedcosts (φA

1 > φB1 ) when the agents have binary costs and differ ex ante.

Proposition 2 also refers to condition (9), which states that the two agentsreceive the same expected rent at the solution to [P − N] when costs arebinary:

φA1

[C

(xA

2 , c2) − C

(xA

2 , c1)] = φB

1

[C

(xB

2 , c2) − C

(xB

2 , c1)]

, (9)

where xij is the output of agent i at the solution to [P − N] when his cost

realization is cj.

Proposition 2: Suppose n = 2 and the agents are asymmetric (so φA1 >

φB1 ). Then aversion to inequity is constraining for the principal (because there

is no solution to [P − N] that is a solution to [P − E]) whenever condition (9)does not hold.

To provide additional understanding of why concerns with in-equity generally are constraining for the principal in the setting ofProposition 2, return to the solution to the standard binary adverseselection problem. Recall that at this standard solution, the outputinduced from each high-cost agent is reduced below the efficient outputlevel (i.e., xi

2 < x∗2 ) in order to reduce the rent that accrues to the low-

cost agents.15 This output reduction is most pronounced for agent A (soxA

2 < xB2 ) because he is more likely than agent B to have the low cost

15. xi∗ is defined by V′(xi

∗) = Cx (xi∗, ci) for i = 1,2.


realization. It is readily shown that when: (i) the standard outputs areinduced and ex post equity is ensured for all cost realizations; and (ii)each high-cost agent is held to zero rent, agent A will truthfully revealhis low cost realization if and only if inequality (10) holds. Similarly,agent B will truthfully reveal the c1 realization under these conditionsif and only if inequality (11) holds.

T A11 ≥ C

(x∗

1 , c1) + 1

φB1

[C

(xA

2 , c2) − C

(xA

2 , c1)]

+ φA2 φB

2

φA1 φB

1

[T A

22 − C(xA

2 , c2)]

. (10)

T B11 = TA

11 ≥ C(x∗

1 , c1) + 1

φA1

[C

(xB

2 , c2) − C

(xB

2 , c1)]

+ φA2 φB

2

φA1 φB

1

[T A

22 − C(xA

2 , c2)]

. (11)

It is apparent from inequalities (10) and (11) that the principalwill be able to secure the same expected payoff that she secures in thestandard solution while eliminating all ex post inequity only if condition(9) holds. Suppose instead that condition (12) holds:

φA1

[C

(xA

2 , c2) − C

(xA

2 , c1)]

> φB1

[C

(xB

2 , c2) − C

(xB

2 , c1)]

. (12)

Then in order to induce agent A to truthfully reveal his low costrealization, TA

11 has to be increased above the minimum value of TB11 that

will ensure agent B truthfully reveals his low cost realization (holdingxA

2 , xB2 , and TA

22 constant). This increase in TA11 will cede more rent to

agent A than he receives in the standard solution. In contrast, supposecondition (13) holds:

φA1

[C

(xA

2 , c2) − C

(xA

2 , c1)]

< φB1

[C

(xB

2 , c2) − C

(xB

2 , c1)]

. (13)

Then in order to induce agent B to truthfully reveal his low costrealization, TB

11 has to be increased above the minimum value of TA11

that ensures agent A truthfully reveals his low cost realization. Thisincrease in TB

11 will deliver more rent to agent B than he receives in thestandard solution.

When condition (9) does not hold, the principal faces a choice. Shecan either eliminate all ex post inequity and cede additional rent to at leastone agent or she can tolerate some ex post inequity in order to reduce the


rent afforded the agents.16 The principal will pursue the former strategywhen the agents are sufficiently averse to inequity. This will be the case,for example, when assumption 1 holds and d is sufficiently large.

Assumption 1: Di(Ni, N−i) = d[max{N−i − Ni, 0}], where d is a strictlypositive constant.

When assumption 1 holds, an agent experiences inequity if andonly if he receives less net compensation than the other agent. Further-more, the disutility the agent suffers from inequity increases linearly (atrate d) with the extent to which the agent’s net compensation is less thanthe net compensation of his counterpart.

The precise form of the agents’ aversion to inequity is largelyinconsequential when this aversion is so pronounced that the principaloptimally avoids all inequity. Proposition 3 describes the key featuresof the principal’s optimal strategy in such settings.

Proposition 3: Suppose n = 2, the agents are asymmetric (so φA1 > φB

1 ),and ex post inequity is eliminated at the solution to [P] (because, for example,assumption 1 holds and d is sufficiently large). Then at this solution:

(i) each agent produces the efficient output when he has the low cost (i.e.,xi

11 = xi12 = x∗

1 for i = A, B);(ii) when condition (12) holds, agent A produces less than the standard output

(xA21 = xA

22 ∈ (0, xA2 )) and receives no rent when he has the high cost, while

agent B produces more than the standard output (xB21 = xB

22 ∈ (xB2 , x∗

2 ])when he has the high cost;

(iii) when condition (13) holds, agent B produces less than the standard output(xB

21 = xB22 ∈ (0, xB

2 )) and receives no rent when he has the high cost, whileagent A produces more than the standard output (xA

21 = xA22 ∈ (xA

2 , x∗2 ])

when he has the high cost.

The discussion that follows Proposition 2 explains that whenagent A receives greater expected rent than agent B at the standardsolution (so condition (12) holds) and the principal is compelled toeliminate ex post inequity, she cannot avoid delivering more rent toagent B than he receives in the standard solution unless induced output

16. The fact that concerns with inequity typically are constraining for the principalwhen the costs of asymmetric agents are uncorrelated does not imply that these concernswill be constraining when the costs of symmetric agents are correlated. To the contrary,it can be shown that when n = 2 and the costs of symmetric agents are correlated, theprincipal can ensure the first-best (full-information) outcome while avoiding equilibriuminequity. (See Desiraju and Sappington (2007). The analyses of Demski and Sappington(1984) and Cremer and McLean (1985) explain why the relevant benchmark is the first-bestoutcome when the agents’ costs are correlated.)


levels are changed.17 The extra rent that must be delivered to agentB increases with the equilibrium rent that agent A secures. In typicalfashion, this rent increases as agent A’s output when he has high costs(xA

2 ) increases. Therefore, as property (ii) of Proposition 3 reveals, theprincipal optimally reduces xA

2 below its standard level (xA2 ) in order to

reduce the rent that agent A, and thus agent B, commands when lowcosts are realized. The best way to provide the extra rent that must bedelivered to agent B to avoid ex post inequity is to increase agent B’soutput when he has high costs (xB

2 ) above the standard level (xB2 ). This

increase in xB2 toward the efficient output (x∗

2) increases total surplus,and thereby provides direct benefits to the principal.

Analogous logic explains property (iii) of Proposition 3. Whenagent B receives greater expected rent than agent A at the standardsolution (so condition (13) holds), the value of TA

11 = TB11 that just induces

agent B to truthfully reveal his low costs exceeds the correspondingvalue of TA

11 that would induce agent A to truthfully reveal his low costs.Therefore, any increase in agent B’s rent in this setting is particularlycostly for the principal because it necessitates a corresponding increasein agent A’s rent. Therefore, the principal optimally limits agent B’s rentbelow the level he secures in the standard solution. This rent reductionis achieved by reducing xB

2 below xB2 . Furthermore, the additional rent

that must be provided to agent A is best delivered in a manner thatincreases total surplus, that is, by increasing xA

2 toward x∗2.

When the agents’ aversion to inequity is less pronounced, theprincipal can implement some variation in the agents’ net compensationlevels if she chooses to do so. Proposition 4 reveals how the principaloptimally employs this additional flexibility.

Proposition 4: Suppose n = 2, the agents are asymmetric (so φA1 > φB

1 ),condition (9) does not hold, assumption 1 holds, and d is sufficiently small thatsome inequity is optimally tolerated at the solution to [P]. Then at this solution:

(i) each agent earns no rent when he has the high cost (c2) and is indifferentbetween exaggerating and truthfully revealing his low cost (c1);

(ii) one agent (i) experiences inequity for all cost realizations (so Nijk < N−i

kj );(iii) the agent (i) that systematically experiences inequity produces the stan-

dard outputs for all cost realizations (so xijk = x j );

(iv) the agent that does not experience inequity (agent − i) produces thestandard output only when both agents have low costs (so x−i

11 = x1)and otherwise produces less than the standard output (so x−i

jk < x j forjk �= 11); and

(v) either agent can experience systematic inequity.

17. In particular, when TA11 = TB

11 is set to ensure (10) holds as an equality, (11) holdsas a strict inequality when condition (12) holds if outputs are at their standard levels.


Property (i) of Proposition 4 reflects the familiar conclusion thatthe principal best limits the agents’ rents by holding a high-cost agentto his reservation level of expected utility and ceding to a low-costagent only the rent he could secure by exaggerating his productioncost. When condition (9) does not hold, the principal cannot securethis preferred rent pattern while eliminating all ex post inequity andholding the agents’ outputs at their standard levels. As Lemma A2 inthe Appendix reveals and property (ii) in Proposition 4 indicates, thepayments that maximize the principal’s expected return for any inducedoutput levels while respecting the agents’ participation and incentivecompatibility constraints ((2) and (3)) induce systematic inequity forone of the agents, say agent i, when condition (9) does not hold. Thisinequity is costly for the principal because she must compensate agenti for the inequity he suffers. To reduce the requisite compensation, theprincipal reduces the inequity incurred by agent i below the level thatwould prevail if standard outputs were induced. As property (iv) ofProposition 4 reveals, she does so by reducing the output induced fromagent −i when he has the high cost (c2). In typical fashion, the reductionsin x−i

21 and x−i22 reduce the rent that accrues to agent −i when he has the low

cost (c1). This rent reduction, in turn, reduces the extent of the inequityexperienced by agent i whenever his counterpart has the low cost.

The principal also reduces x−i12 below its standard level. This

reduced output permits a reduction in T−i12 without affecting agent −i‘s

incentive to report his cost realization truthfully. The reduction in T−i12 , in

turn, helps to reduce the critical inequity experienced by agent i whenhe alone has the high cost realization.

As property (iii) of Proposition 4 reports, the agent that suffersinequity (agent i) optimally is induced to produce the standard outputs.Because agent −i does not experience inequity at the solution to [P],marginal deviations in the outputs of agent i from their standard levelsin order to alter the net compensation of agent i do not serve to limit theequilibrium extent of inequity. Consequently, the outputs of the agentthat experiences inequity optimally reflect the familiar trade-off betweenrent and efficiency, and so are set at their standard levels.

As property (v) of Proposition 4 reveals, either agent can experi-ence inequity in equilibrium. To identify conditions under which eachagent optimally experiences inequity, consider the following setting.Suppose the principal derives value V(xi) = 100

√xi from the output

of each agent. Also suppose production costs are linear in output (i.e.,C(x, c) = cx) and there are two possible cost realizations: c1 = 20 andc2 = 25. In addition, let the probability that agent B has the low costbe 0.3, and let ε ∈ (0, 0.7) denote the incremental probability that agentA has the low cost (so φB

1 = 0.3 and φA1 = 0.3 + ε). Further suppose


Table I.

Standard (Second-Best) Outcomes in the Example

ε = 0.1 ε = 0.5

Agent A Agent B Agent A Agent B(i = A) (i = B) (i = A) (i = B)

xi1 6.25 6.25 6.25 6.25

xi2 3.11 3.39 1.23 3.39

Expected payment (T) 102.94 101.97 111.11 101.97Value of principal’s objective 204.92 213.09

assumption 1 holds, so d is the rate at which an agent’s disutilityincreases as his net compensation (T − C(x, c)) declines below the netcompensation of his counterpart.

Table I reports output levels (x), expected transfer payments (T),and the value of the principal’s objective function (as stated in expression(1)) at the standard (second-best) outcome for this example, where theagents are not averse to inequity (so d = 0). Two settings are considered:one where agent A’s cost advantage is relatively limited (ε = 0.1) andone where this advantage is relatively pronounced (ε = 0.5).

Table II reports the corresponding data for the example in a settingwhere the agents’ aversion to inequity is relatively limited (d = 0.2). Thetable reports both the outcomes that would arise at the solution to [P]if agent A were optimally induced to experience systematic inequityand the corresponding outcomes if agent B were induced to experienceinequity. Table III reports the analogous data for the example in a settingwhere the agents’ aversion to inequity is more pronounced (d = 1.0).

The tables reveal that the basic patterns that arise when agent A’scost advantage is limited persist and become more pronounced whenagent A’s cost advantage is larger. In both cases, the principal prefers tohave agent B experience inequity when the agents’ aversion to inequityis relatively limited and to have agent A experience inequity whenthis aversion is more pronounced. As the agents’ aversion to inequitybecomes more pronounced, the principal sacrifices more surplus (byreducing induced output levels further below standard levels) in orderto reduce inequity.18 Tables II and III indicate that when she is compelledto sacrifice considerable surplus in order to limit inequity (because dis large), the principal optimally arranges payments so that agent Aexperiences inequity and then reduces the magnitude of the equilibrium

18. Notice that if the output of an agent were reduced to zero whenever he had highcosts, the principal could hold the agents to zero rent for all cost realizations, and therebyeliminate all inequity.


Table II.

Outcomes in the Example with Relatively LimitedAversion to Inequity (d = 0.2)

Agent A Experiences Inequity Agent B Experiences Inequity

ε = 0.1 ε = 0.5 ε = 0.1 ε = 0.5

Agent A Agent B Agent A Agent B Agent A Agent B Agent A Agent B(i = A) (i = B) (i = A) (i = B) (i = A) (i = B) (i = A) (i = B)

xi11 6.25 6.25 6.25 6.25 6.25 6.25 6.25 6.25

xi12 6.25 3.83 6.25 1.86 4.08 6.25 4.08 6.25

xi21 3.11 3.11 1.23 3.31 3.00 3.39 1.07 3.39

xi22 3.11 2.11 1.23 1.07 2.06 3.39 0.85 3.39

Expected 103.59 77.93 111.83 86.55 77.62 102.75 83.90 102.58payment(T)

Value of 201.46 208.33 201.54 209.64principal’sobjective

Table III.

Outcomes in the Example with RelativelyPronounced Aversion to Inequity (d = 1.0)

Agent A Experiences Inequity Agent B Experiences Inequity

ε = 0.1 ε = 0.5 ε = 0.1 ε = 0.5

Agent A Agent B Agent A Agent B Agent A Agent B Agent A Agent B(i = A) (i = B) (i = A) (i = B) (i = A) (i = B) (i = A) (i = B)

xi11 6.25 6.25 6.25 6.25 6.25 6.25 6.25 6.25

xi12 6.25 1.86 6.25 0.51 2.13 6.25 6.25 6.25

xi21 3.11 3.14 1.23 3.14 2.78 3.39 0.83 3.39

xi22 3.11 1.04 1.23 0.30 1.09 3.39 0.47 3.39

Expected 104.35 57.41 113.04 79.50 54.07 103.57 59.02 103.13payment(T)

Value of 191.10 200.92 191.02 198.53principal’sobjective

inequity by reducing the output induced from agent B.19 In contrast,when relatively little surplus is sacrificed to limit inequity (because d issmall), the principal optimally arranges payments so that agent B incurs

19. Table I illustrates the more general conclusion that agent B’s standard outputexceeds agent A’s standard output when the high cost (c2) is realized.


inequity and then reduces the magnitude of the equilibrium inequity byreducing the output induced from agent A.20

4. Conclusions

We have found that concerns with inequity are not constraining fora principal when the risk-neutral agents that are privately informedabout their capabilities are identical ex ante (but may differ ex post).However, concerns with inequity generally are constraining when theagents differ ex ante. When agent aversion to inequity is sufficientlypronounced, all ex post inequity will be eliminated, typically by cedingadditional rent to one of the agents. In this case, the output of one high-cost agent is reduced below the standard level and the output of theother high-cost agent is increased above the standard level. When agentaversion to inequity is less pronounced, some inequity typically will betolerated in order to reduce the agents’ rent. In this case, (only) one agentsystematically experiences inequity in equilibrium. This agent producesthe standard outputs. The outputs of the agent that does not experienceinequity are reduced below standard levels in order to reduce the extentof the inequity experienced by his counterpart.

Our analysis suggests that conclusions drawn from the canonicalmodels of adverse selection may require modification when concernswith inequity are relevant. In particular, inequity concerns can ensurestrict gains for the principal from basing an agent’s compensation onboth his relative and his absolute performance even when the agents’costs are uncorrelated. For example, to avoid inequity for an agent withhigh costs when his counterpart has low costs, the low-cost agent mayoptimally receive: (i) a high payment for producing a large output whenhis counterpart also produces a large output; and (ii) a small paymentfor producing the same large output when his counterpart produces asmall output. Consequently, a low-cost agent may only enjoy substantialreturns to his pronounced skills when his counterpart is also highlyskilled. When his counterpart suffers from more limited ability, the low-cost agent optimally bears some of the associated financial burden whenthe agents are averse to inequity.

20. It is readily shown that agent A’s expected rent exceeds agent B’s expected rent inthe standard outcome when ε = 0.1. In contrast, agent B’s expected rent exceeds agent A’sexpected rent in the standard outcome when ε = 0.5. Recall also that agent B optimallyexperiences inequity when d = 0.2 while agent A optimally experiences inequity when d =1.0 for both values of ε. These facts imply that although conditions (12) and (13) determinethe nature of the optimal output distortions when inequity is optimally eliminated, theyare not determinative when some inequity optimally is tolerated in equilibrium.


Our analysis also suggests that a principal might rationally chooseto hire an agent with higher expected costs than an agent with lowerexpected costs if the former is more similar to the other agents alreadyin the principal’s employ. Although the high-cost agent will be lessproductive on average, his similarity to the other agents could enable theprincipal to avoid the need to cede rent to the agents in order to limit theirdisutility from inequity. If this latter advantage outweighs the formerdisadvantage, the principal will prefer to hire the less capable agent.

Many extensions of our simple model remain to be analyzed,including more general cost distributions, private information aboutaversion to inequity, and risk aversion. When more than two costrealizations are possible, there will be more than four possible statesof the world (corresponding to distinct pairs of cost realizations) inwhich inequity can arise. The additional states of the world likely willcomplicate the principal’s task of managing or eliminating inequityunder plausible conditions. In particular, the distortions from standardoutcomes that are implemented in response to concerns with inequitymay interact in complex ways, perhaps introducing regions of outputpooling in optimal contracts.21

Von Siemens (2005) demonstrates that a different form of poolingcan arise when agents are privately informed about both their ability(e.g., their cost parameter, c) and the extent of their aversion to inequity.22

In such a setting, the principal may be unable to induce different levelsof performance from agents with the same ability but different levels ofaversion to inequity. Risk aversion also can complicate the principal’stask of motivating agents who are averse to inequity. The stochasticpayments that eliminate all ex post inequity in the setting of Proposition1, for example, will be costly for the principal to introduce when agentsare averse to risk. More generally, aversion to inequity likely will bemore constraining for the principal when agents are risk averse.

Future research also should consider alternative characterizationsof inequity. For example, agents may be more concerned with variationsin the ratio of payment to output (T/x) than with variations in net com-pensation.23 Alternatively, experimental research suggests individualsmay be less averse to unequal payoffs when the agents with the higher

21. Output pooling arises when an agent produces the same output for multiple costrealizations. Output pooling may be particularly likely if agents experience no inequitywhenever they produce the same output and receive the same payment.

22. Lopomo and Ok (2001) analyze a related bargaining model with asymmetricinformation about the extent to which agents value relative payoffs.

23. See Adams (1963, 1965), for example. Desiraju and Sappington (2005) show thatsome of the key qualitative conclusions drawn above persist under such conditions. Anagent’s disutility also might arise from variations in ratios of (rather than differences in)net compensation. See Lopomo and Ok (2001), for example.


payoffs are judged to have earned their more pronounced rewards.24

In this regard, a principal generally will be less constrained by inequityconcerns if an agent only experiences inequity when he receives lessnet compensation than another agent who has the same or lower innateability (i.e., the same or higher costs). Inequity concerns also will be lessconstraining if agents’ costs vary over time and agents are only averseto persistent inequality in net payoffs.

Appendix

Proof of Proposition 1. The standard (second-best) outputs (xh1 > xh

2 ≥xh

3 ≥ · · · ≥ xhn ≥ 0 for h = A, B) and expected payoffs can be achieved

while eliminating all ex post inequity if and only if equations (A1)–(A5)hold:25

n∑j=1

φBj T A

n, j − C(xA

n , cn) = 0; (A1)

n∑i=1

φAi T B

n,i − C(xB

n , cn) = 0; (A2)

n∑j=1

φBj T A

i, j − C(xA

i , ci) =

n∑j=1

φBj T A

i+1, j − C(xA

i+1, ci)

for i = 1, . . . , n − 1;

(A3)

n∑i=1

φAi T B

j,i − C(xB

j , c j) =

n∑i=1

φAi T B

j+1,i − C(xB

j+1, c j)

for

j = 1, . . . , n − 1; and (A4)

T Ai, j − C

(xA

i , ci) = T B

j,i − C(xB

j , c j)

for all i, j = 1, . . . , n. (A5)

Equations (A2) and (A4) imply:

n∑i=1

φAi T B

n−1,i =n∑

k=n−1

[C B

k,k − C Bk+1,k

], (A6)

where Chk, j ≡ C(xh

k , c j ) and Chn+1,n ≡ 0 for h = A, B and k, j = 1, . . . , n.

24. See Hoffman et al. (1994), for example.25. For expositional ease, commas are inserted between the elements of subscripts

throughout the proof of Proposition 1.


Now consider the following induction argument. Supposen∑

i=1

φAi T B

j,i =n∑

s= j

[C B

s,s − C Bs+1,s

]for some j < n. (A7)

Then it follows from equation (A4) thatn∑

i=1

φAi T B

j−1,i =n∑

i=1

φAi T B

j,i + C Bj−1, j−1 − C B

j, j−1. (A8)

Substituting from (A7) into (A8) providesn∑

i=1

φAi T B

j−1,i =n∑

s= j−1

[C B

s,s − C Bs+1,s

]. (A9)

This induction argument ensures equalities (A2) and (A4) will be satis-fied if and only if

n∑i=1

φAi T B

j,i =n∑

s= j

[C B

s,s − C Bs+1,s

]for each j = 1, . . . , n. (A10)

Analogous arguments reveal that equalities (A1) and (A3) will besatisfied if and only if

n∑j=1

φBj T A

i, j =n∑

s=i

[C A

s,s − C As+1,s

]for each i = 1, . . . , n. (A11)

Therefore, the proof is complete if there exist TAi,j and TB

j,i payments suchthat equations (A5), (A10), and (A11) hold whenever the agents aresymmetric ex ante. To construct one such set of payments, let

T Ai, j =

n∑s=i

[C A

s,s − C As+1,s

] + �Ai, j , and

T Bj,i =

n∑s= j

[C B

s,s − C Bs+1,s

] + �Bj,i for all i, j = 1, . . . , n. (A12)

To ensure that equations (A10) and (A11) hold when the agents aresymmetric ex ante, it must be the case that

n∑j=1

φBj �A

i, j = 0 for each i = 1, . . . , n; and (A13)

n∑i=1

φAi �B

j,i = 0 for each j = 1, . . . , n. (A14)


To ensure that equation (A5) holds when the agents are symmetric exante, equation (A12) implies

�Ai, j = �B

j,i + �Cj − �C

i , (A15)

where �Cj ≡ ∑n

s= j [Cs,s − Cs+1,s] − C(x j , c j ) and where the superscriptson the x‘s and C‘s are omitted because xA

i = xBi and CA

s,h = CBs,h when the

agents are symmetric ex ante.To satisfy equations (A13) and (A14), it must be the case that

�Ai,i = − 1

φAi

n∑k=1k �=i

φAk �A

i,k for all i = 1, . . . , n; and (A16)

�Bi,i = − 1

φBi

n∑k=1k �=i

φBk �B

i,k for all i = 1, . . . , n. (A17)

For those i, j realizations (i �= j) for which �Cj ≥ �C

i , let

�Bj,i = 0 and �A

i, j = �Cj − �C

i ≥ 0. (A18)

Similarly, for those i, j realizations (i �= j) for which �Cj < �C

i , let

�Ai, j = 0 and �B

j,i = �Ci − �C

j > 0. (A19)

Finally, let

�Ai,i = �B

i,i = − 1φA

i

n∑k=1k �=i

φAk max

{0, �c

k − �ci

}for all i = 1, . . . , n. (A20)

It is apparent that when the �Ai,j and �B

j,i variables are as specified inequations (A18) – (A20) and when the agents are symmetric ex ante

n∑j=1

φBj �A

i, j = φAi �A

i,i +n∑

k=1k �=i

φAk �A

i,k = 0

= φAi �B

i,i +n∑

k=1k �=i

φBk �B

i,k =n∑

i=1

φAi �B

j,i for all i = 1, . . . , n.

(A21)

Consequently, equations (A13) – (A15) are satisfied, and so equations(A5), (A10), and (A11) will be satisfied when the agents are symmetric


ex ante and when payments are as specified in equations (A12) and(A18)–(A20). �

Lemma A1 facilitates the proof of Proposition 2. Lemma A1, likePropositions 2 and 4, holds even if the principal does not value theoutputs of agents A and B symmetrically. To demonstrate this fact,these formal conclusions and/or their proofs refer to Vi(xi), which isthe principal’s valuation of agent i‘s output, for i = A, B. VA(·) and VB(·)are strictly increasing, strictly concave functions.

Lemma A1: Suppose n = 2. Then, the solution to [P − N] has the followingproperties for all i, j, −i ∈ {1, 2}, where i �= −i:

(i) Vi ′(xi1 j ) = Cx(xi

1 j , c1);

(ii) Vi ′(xi2 j ) = Cx(xi

2 j , c2) + φi1

φi2[Cx(xi

2 j , c2) − Cx(xi2 j , c1)];

(iii) φ−i2 Ti

22 + φi1Ti

21 = C(xi2, c2); and

(iv) φ−i2 Ti

12 + φ−i1 Ti

11 = C(xi2, c2) + C(xi

1, c1) − C(xi2, c1).

The proof of Lemma A1 is standard, and so is omitted.

Proof of Proposition 2. Let Tijk denote payments at the solution to [P − N],

and consider a new payment structure: T ijk = Ti

jk + �ijk. To ensure that

properties (iii) and (iv) in Lemma A1 hold, (A22) must be satisfied forall i, −i ∈ {1, 2}, where i �= −i:

φ−i2 T i

j2 + φ−i1 T i

j1 = φ−i2 Ti

j2 + φ−i1 Ti

j1. (A22)

(A22) implies:

φ−i2 �i

j2 + φ−i1 �i

j1 = 0. (A23)

If these new (T) payments are to eliminate all ex post inequity given theoutput levels identified in Lemma A1, (A24) – (A27) must hold:

T A11 − C

(xA

1 , c1) = T B

11 − C(xB

1 , c1); (A24)

T A12 − C

(xA

1 , c1) = T B

21 − C(xB

2 , c2); (A25)

T A21 − C

(xA

2 , c2) = T B

12 − C(xB

1 , c1); and (A26)

T A22 − C

(xA

2 , c2) = T B

22 − C(xB

2 , c2). (A27)

Since T i11 = T i

11 + �i11, (A24) implies:

�A11 = �B

11 + T B11 − T A

11 + C(xA

1 , c1) − C

(xB

1 , c1). (A28)


(A23) and (A28) imply

�A12 = −

[φB

1

φB2

] [�B

11 + T B11 − T A

11 + C(xA

1 , c1) − C

(xB

1 , c1)]

. (A29)

Similarly, (A25) implies

�B21 = �A

12 + T A12 − T B

21 − C(xA

1 , c1) + C

(xB

2 , c2). (A30)

Substituting (A29) into (A30) provides

�B12 = −

[φB

1

φB2

] [�B

11 + T B11 − T A

11 + C(xA

1 , c1) − C

(xB

1 , c1)]

+ T A12 − T B

21 − C(xA

1 , c1) + C

(xB

2 , c2). (A31)

From (A23):

�B22 = −

[φA

1

φA2

]�B

21. (A32)


�B22 =

[φA

1

φA2

] { [φB

1

φB2

] [�B

11 + T B11 − T A

11 + C(xA

1 , c1) − C

(xB

1 , c1)]

− T A12 + T B

21 + C(xA

1 , c1) − C

(xB

2 , c2)}

. (A33)

(A27) implies

�A22 = �B

22 + T B22 − T A

22 + C(xA

2 , c2) − C

(xB

2 , c2). (A34)

Substituting from (A33) into (A34) provides

�A22 =

[φA

1

φA2

] { [φB

1

φB2

] [�B

11 + T B11 − T A

11 + C(xA

1 , c1) − C

(xB

1 , c1)]

− T A12 + T B

21 + C(xA

1 , c1) − C

(xB

2 , c2)} + T B

22 − T A22

+ C(xA

2 , c2) − C

(xB

2 , c2). (A35)


Also, (A23) and (A35) imply

�A21 = −

[φB

2 φA1

φB1 φA

2

] {φB

1

φB2

[�B

11 + T B11 − T A

11 + C(xA

1 , c1) − C

(xB

1 , c1)]

− T A12 + T B

21 + C(xA

1 , c1) − C

(xB

2 , c2)}

−[φB

2

φB1

] [T B

22 − T A22 + C

(xA

2 , c2) − C

(xB

2 , c2)]

. (A36)

(A26) implies

�A21 = �B

12 + T B12 − T A

21 + C(xA

2 , c2) − C

(xB

1 , c1). (A37)

(A23) implies

�B12 = −

[φA

1

φA2

]�B

11. (A38)


�A21 = −

[φA

1

φA2

]�B

11 + T B12 − T A

21 + C(xA

2 , c2) − C

(xB

1 , c1). (A39)

(A36) and (A39) require

φA1 �B

11

φA2

+ T A21 − T B

12 − C(xA

2 , c2) + C

(xB

1 , c1)

= φA1 �B

11

φA2

+ φA1 T B

11

φA2

− φA1 T A

11

φA2

+ φA1 C

(xA

1 , c1)

φA2

− φA1 C

(xB

1 , c1)

φA2

+ φA1 φB

2 T B21

φA2 φB

1− φA

1 φB2 T A

12

φA2 φB

1+ φA

1 φB2 C

(xA

1 , c1)

φA2 φB

1− φA

1 φB2 C

(xB

2 , c2)

φA2 φB

1

+ φB2 T B

22

φB1

− φB2 T A

22

φB1

+ φB2 C

(xA

2 , c2)

φB1

− φB2 C

(xB

2 , c2)

φB1

. (A40)

Simplifying (A40) and substituting Nijk ≡ Ti

jk − C(xijk, c j ) provides

NA21 − NB

12 = φA1

φA2

[NB

11 − NA11

] + φA1 φB

2

φA2 φB

1

[NB

21 − NA12

] + φB2

φB1

[NB

22 − NA22

].

(A41)


Multiplying both sides of (A41) by φA2 φB

1 provides

φA2 φB

1

[NA

21 − NB12

] = φA1 φB

1

[NB

11 − NA11

]+ φA

1 φB2

[NB

21 − NA12

] + φB2 φA

2

[NB

22 − NA22

]. (A42)

Rearranging terms in (A42) provides

φA1

[φB

1 NA11 + φB

2 NA12

] + φA2

[φB

1 NA21 + φB

2 NA22

]= φB

1

[φA

1 NB11 + φA

2 NB12

] + φB2

[φA

1 NB21 + φA

2 NB22

]. (A43)

Notice that

φ−i1 Ni

j1 + φ−i2 Ni

j2 = [φ−i

1 Tij1 + φ−i

2 Tij2

] − [φ−i

1 C(xi

j , c j) + φ−i

2 C(xi

j , c j)]

.

= [φ−i

1 Tij1 + φ−i

2 Tij2

] − C(xi

j , c j). (A44)

(A43)–(A44) and properties (iii) and (iv) of Lemma A1 imply that thereis a solution to [P − N] that is also a solution to [P − E] if and only if

φA1

[C

(xA

2 , c2) + C

(xA

1 , c1) − C(xA

2 , c1) − C(xA

1 , c1)]

+ φA2

[C

(xA

2 , c2) − C

(xA

2 , c2)] = φB

1

[C

(xB

2 , c2) + C

(xB

1 , c1)

− C(xB

2 , c1) − C

(xB

1 , c1)]

+ φB2

[C

(xB

2 , c2) − C

(xB

2 , c2)]

. (A45)

(A45) simplifies to

φA1

[C

(xA

2 , c2) − C(xA

2 , c1)] = φB

1

[C

(xB

2 , c2) − C

(xB

2 , c1)]

. (A46)

Proof of Proposition 3. The principal’s problem, [P − E2], in this settingis

maxxi

jk,Tijk

2∑j=1

2∑k=1

φAj φB

k

{V

(xA

jk

) + V(xB

kj

) − T Ajk − T B

kj

}(A47)

s.t. for i �= −i, i, −i ∈ {A, B}:2∑

k=1

φ−ik

[Ti

jk − C(xi

jk, c j)] ≥ 0 for j = 1, 2; (A48)

2∑k=1

φ−ik

[Ti

jk − C(xi

jk, c j)] ≥

2∑k=1

φ−ik

[Ti

mk − C(xi

mk, c j)]

for m �= j, j, m ∈ {1, 2}; and (A49)


Tijk − C

(xi

jk, c j) = T−i

kj − C(x−i

kj , ck)

for j, k ∈ {1, 2}. (A50)

Let λij , µ

ij , and γ i

jk denote the Lagrange multipliers associated with con-straints (A48), (A49), and (A50), respectively. Using standard techniques,it is readily shown that at the solution to [P − E2] for i = {A, B} andk = 1, 2:

V′(xi1k

) = Cx(xi

1k , c1), and (A51)

V′(xi2k

) = Cx(xi

2k , c2) + µi

1

φi2

[Cx

(xi

2k , c2) − Cx

(xi

2k , c1)]

. (A52)

(A51) implies xi11 = xi

12 = x∗1 for i = A, B, which proves property (i) of

the proposition. (A52) implies

xi2k � xi

2 as µi1 � φi

1. (A53)

It is also readily shown that the solution to [P − E2] is characterized byone of the following three cases:

Case 1. λA2 = λB

2 = 1, µA1 = φA

1 , and µB1 = φB

1 . (A54)

Case 2. λA2 > 1, λB

2 < 1, µA1 > φA

1 , and µB1 < φB

1 . (A55)

Case 3. λB2 > 1, λA

2 < 1, µB1 > φB

1 , and µA1 < φA

1 . (A56)

If the solution to [P − E2] is characterized by Case 1, then (A53) implies

xA2 = xA

2 < xB2 = xB

2 . (A57)

Because λA2 > 0 and λB

2 > 0 in Case 1, (10) and (11) in the text must holdas equalities when µA

1 > 0 and µB1 > 0. Therefore, condition (9) must

hold.(A53) implies that in Case 2

xA2 < xA

2 < xB2 < xB

2 . (A58)

Furthermore, since λA2 > 0, µA

1 > 0, and (A50) holds, it is readily shownthat in Case 2

T B11 = T A

11; (A59)

T B12 = C11 + φB

2

φB1

C A22 − φB

2

φB1

T A22; (A60)

T B21 = φB

1

φB2

C11 + C B22 + 1

φB2

[C A

22 − C A21

] − φB1

φB2

T A11; (A61)


T B22 = C B

22 − C A22 + T A

22; (A62)

T A12 = 1

φB2

[C11 + C A

22 − C A21

] − φB1

φB2

T A11; and (A63)

T A21 = 1

φB1

C A22 − φB

2

φB1

T A22, (A64)

where C11 ≡ C(x∗1, c1) and Ci

jk ≡ C(xij, ck).

Using (A59) – (A64), the participation constraint (A48) for agent B whenhis cost realization is c2 can be written as

φA2 φB

2 T A22 − φA

1 φB1 T A

11 ≥ J , (A65)

where

J ≡ φA2 φB

2 C A22 − φA

1

[C A

22 − C A21

] − φA1 φB

1 C11. (A66)

Similarly, the incentive compatibility constraint (A49) for agent B whenhis cost realization is c1 can be written as

φA2 φB

2 T A22 − φA

1 φB1 T A

11 ≤ J + K , (A67)

where

K ≡ φB2

{φA

1

[C A

22 − C A21

] − φB1

[C B

22 − C B21

] − φA1 (φB

1 )2

φA2 φB

2

[φA

1 − φB1

]C11

}.

(A68)

(A65) and (A67) imply that K must be non-negative. If condition (13)holds, then (A58) implies

φA1

[C

(xA

2 , c2) − C

(xA

2 , c1)]

< φB1

[C

(xB

2 , c2) − C

(xB

2 , c1)]

(A69)

in Case 2. If (A69) holds, then K must be strictly negative. Consequently,the solution to [P − E2] must be characterized by Case 3 when condition(13) holds.

Analogous logic reveals that in Case 3, agent A’s participationconstraint (A48) when he has cost realization c2 and his incentivecompatibility constraint (A49) when he has cost realization c1 will besatisfied if and only if

R − Q ≤ φA1 φB

1 T A11 − φA

2 φB2 T A

22 ≤ R, (A70)

where

R ≡ φA1 φB

1 C11 + φB1

[C B

22 − C B21

] − φA2 φB

2 C A22, and (A71)


Q ≡ φA2

[φB

1

(C B

22 − C B21

) − φA1

(C A

22 − C A21

)]. (A72)

(A70) implies that Q must be non-negative in Case 3. If condition (12)holds, then (A53) implies that in Case 3

φA1

[C

(xA

2 , c2) − C

(xA

2 , c1)]

> φB1

[C

(xB

2 , c2) − C

(xB

2 , c1)]

. (A73)

If (A73) holds, then Q < 0. Therefore, when condition (12) holds, thesolution to [P − E2] must be characterized by Case 2. �

Lemma A2 facilitates the proof of Proposition 4. The lemma refersto the following definition of net compensation:

Nijk ≡ Ti

jk − C(xi

jk, c j). (A74)

The lemma also refers to [P]’, which is problem [P] with the participationconstraints omitted for the low-cost agents and the incentive compati-bility constraints omitted for the high-cost agents. (Formally, for i = Aand B, constraints (2) are not imposed for j = 1 and constraints (3) are notimposed for j = 2.) We will subsequently demonstrate that the omittedconstraints are satisfied at the solution to [P]’. Consequently, the solutionto [P]’ will constitute the solution to [P].

Lemma A2: Suppose n = 2 and assumption 1 holds. Then for i, −i ∈ {A, B},with i �= −i, one of the following three relationships holds for all j, k ∈ {1, 2}at the solution to [P]’: (1) Ni

jk = N−ikj ; (2) Ni

jk < N−ikj ; or (3) Ni

jk > N−ikj .

Proof of Lemma A2. Let λij and µi

j denote the Lagrange multipliersassociated with constraints (2) and (3), respectively. In addition, fori, −i ∈ {A, B} (where i �= −i) and j, k ∈ {1, 2}, let

δijk = 1 when Ni

jk < N−ikj and

δijk = 0 otherwise. (A75)

Taking partial derivatives of the Lagrangean function associated with[P]’ with respect to Ti

11, Ti12, Ti

21, Ti22, xi

11, xi12, xi

21, and xi22, respectively

(for i = A, B) provides the following necessary conditions for an interiorsolution to [P]’:

δi11µ

i1 − δ−i

11 µ−i1

[φi

1

φ−i1

]= 1

d

[φi

1 − µi1

]; (A76)

δi12µ

i1 − δ−i

21φi

1

φ−i2

[λ−i

2 − µ−i1

] = 1d

[φi

1 − µi1

]; (A77)


δi21

[λi

2 − µi1

] − δ−i12 µ−i

1φi

2

φ−i1

= 1d

[φi

2 − λi2 + µi

1

]; (A78)

δi22

[λi

2 − µi1

] − δ−i22

[φi

2

φ−i2

] [λ−i

2 − µ−i1

] = 1d

[φi

2 − λi2 + µi

1

]; (A79)

φi1

[Vi ′(

xi11

) − Cx(xi

11, c1)] + [

φi1 − µi

1

]Cx(xi

11, c1) − δi11dµi

1Cx(xi

11, c1)

+ δ−i11 dµ−i

1

[φi

1

φ−i1

]Cx(xi

11, c1) = 0; (A80)

φi1

[Vi ′(

xi12

) − Cx(xi

12, c1)] + [

φi1 − µi

1

]Cx

(xi

12, c1) − δi

12dµi1Cx

(xi

12, c1)

− δ−i21 dµ−i

1

[φi

1

φ−i2

]Cx

(xi

12, c1) = 0; (A81)

φi2

[Vi ′(

xi21

) − Cx(xi

21, c2)] + [

φi2 − λi

2

]Cx

(xi

21, c2) + µi

1Cx(xi

21, c1)

+ δi21d

[µi

1Cx(xi

21, c1) − λi

2Cx(xi

21, c2)]

+ δ−i21 dµ−i

1

[φi

2

φ−i1

]Cx

(xi

21, c2) = 0; and (A82)

φi2

[Vi ′(

xi22

) − Cx(xi

22, c2)] + [

φi2 − λi

2

]Cx

(xi

22, c2) + µi

1Cx(xi

22, c1)

+ δi22d

[µi

1Cx(xi

22, c1) − λi

2Cx(xi

22, c2)]

− δ−i22 dµ−i

1

[φi

2

φ−i2

]Cx

(xi

22, c2) = 0. (A83)

Case (a): NA11 = NB

11.(A75) implies δA

11 = δB11 = 0 in this case. Consequently, (A76) implies

µA1 = φA

1 and µB1 = φB

1 . (A84)

Case (b): NA11 < NB

11.

(A75) implies δA11 = 1 and δB

11 = 0 in this case. Consequently, (A76)implies

µA1 = φA

1

1 + d∈ (0, φA

1 ); and (A85)


µB1 = φB

1

[1 + d

1 + d

]> φB

1 . (A86)

Case (c): NA11 > NB

11.



µA1 = φA

1

[1 + d

1 + d

]> φA

1 ; and (A87)

µB1 = φB

1

1 + d∈ (0, φB

1 ). (A88)

Case (a)′: NA12 = NB

21.

(A75) implies δA12 = δB

21 = 0 in this case. Consequently, (A77) implies

µA1 = φA

1 . (A89)

(A89) is consistent with (A84), but not with (A85) or (A87). Furthermore,(A78) implies

λB2 − µB

1 = φB2 . (A90)

Case (b)′: NA12 < NB

21.



µA1 = φA

1

1 + d∈ (

0, φA1

). (A91)

(A91) is consistent with (A85), but not with (A84) or (A87). Substituting(A91) into (A78) provides:

λB2 − µB

1 = φB2

[1 + d

1 + d

]> φB

2 . (A92)

Case (c)′: NA12 > NB

21.



λB2 − µB

1 = φB2

[1

1 + d

]∈ (

0, φB2

). (A93)



µA1 = φA

1

[1 + d

1 + d

]> φA

1 . (A94)

(A94) is consistent with (A87), but not with (A84) or (A85).

Case (a)′′: NA21 = NB

12.

(A75) implies δA21 = δB

12 = 0 in this case. Consequently, (A77) implies

µB1 = φB

1 . (A95)


λA2 − µA

1 = φA2 . (A96)

Case (b)′′: NA21 < NB

12 .



λA2 − µA

1 = φA2

[1

1 + d

]∈ (

0, φA2

). (A97)


µB1 = φB

1

[1 + 1

1 + d

]> φB

1 . (A98)

(A98) is consistent with (A86), but not with (A84) or (A88).

Case (c)′′: NA21 > NB

12.



µB1 = φB

1

1 + d∈ (

0, φB1

). (A99)


λA2 − µA

1 = φA2

[1 + 1

1 + d

]> φA

2 . (A100)

Finally, consider the following three cases: Case (a)’’’: NA22 = NB

22;Case (b)’’’: NA

22 < NB22; and Case (c)’’’: NA

22 > NB22. Arguments analogous


to those employed immediately above reveal that: (1) Case (a)’’’ isconsistent only with Cases (a), (a)’, and (a)’’; (2) Case (b)’’’ is consistentonly with Cases (b), (b)’, and (b)’’; and (3) Case (c)’’’ is consistent onlywith Cases (c), (c)’, and (c)’’. These conclusions complete the proof. �

Proof of Proposition 4. Lemma A2 implies that at most one agent willexperience inequity in equilibrium at the solution to [P]’. Initially,suppose agent A experiences inequity at the solution to [P]’. Then (A75)implies

δAjk = 1 and δB

kj = 0 for all j, k ∈ {1, 2}. (A101)

Furthermore, (A85) and (A86) imply:

µA1 = φA

1

1 + dand µB

1 = φB1

[1 + d

1 + d

]. (A102)

Substituting (A101) and (A102) into (A80) and (A81) reveals that (A103)and (A104) hold for i = A:

Vi ′(xi

11

) = Cx(xi

11, c1). (A103)

Vi ′(xi

12

) = Cx(xi

12, c1). (A104)

(A101) and (A79) imply

λA2 − µA

1 = φA2

1 + d. (A105)

Substituting (A101), (A102), and (A105) into (A82) and (A83) revealsthat (A106) holds for i = A:

Vi ′(xi

2k

) = Cx(xi

2k , c2) + φi

1

φi2

[Cx

(xi

2k , c2) − Cx

(xi

2k , c1)]

for k = 1, 2.

(A106)

Corresponding substitution from (A101), (A102), and (A105) into (A80)– (A83) reveals that (A107) – (A110) hold for i = B and −i = A:

Vi ′(xi

11

) = Cx(xi

11, c1); (A107)

Vi ′(xi

12

) = Cx(xi

12, c1) +

[d

1 + d

] [1

φ−i2

]Cx

(xi

12, c1); (A108)

Vi ′(xi

21

) = Cx(xi

21, c2) +

[1 + 2d1 + d

] [φi

1

φi2

] [Cx

(xi

21, c2) − Cx

(xi

21, c1)]

; and

(A109)


Vi ′(xi

22

) = Cx(xi

22, c2) +

[1 + 2d1 + d

] [φi

1

φi2

] [Cx

(xi

22, c2) − Cx

(xi

22, c1)]

+[

d1 + d

] [1

φ−i2

]Cx

(xi

22, c2). (A110)

It is apparent from (A107) – (A110) and from properties (i) and (ii) inLemma A1 that for i = B:

xi1 = xi

11 > xi12 and xi

2 > xi21 > xi

22. (A111)

In addition, (A103), (A104), (A106), and properties (i) and (ii) in LemmaA1 reveal that for i = A:

xi1 = xi

11 = xi12 > xi

21 = xi22 = xi

2. (A112)

Arguments analogous to those in (A101) – (A112) reveal that ifagent B experiences inequity at the solution to [P]’, then: (1) (A103),(A104), and (A106) hold for i = B; and (2) (A107) – (A110) hold for i =A and −i = B. Consequently, (A111) holds for i = A and (A112) holdsfor i = B. Therefore, provided the solution to [P]’ is the solution to [P],properties (iii) and (iv) of the proposition follow from Lemma A1 andfrom (A103), (A104), and (A106) – (A110).

To conclude that the solution to [P]’ is the solution to [P], it remainsto verify that the omitted participation and incentive compatibility con-straints are satisfied at the solution to [P]’. Using (4), it is straightforwardto verify that

EUi (c2 | c1) = EUi (c2|c2) + φ−i1

[1 + dδi

21

][C

(xi

21, c2) − C

(xi

21, c1)]

+ φ−i2

[1 + dδi

22

][C

(xi

22, c2) − C

(xi

22, c1)]

. (A113)

Because C(x, c2) ≥ C(x, c1) for all x ≥ 0, (A114) implies that EUi(c2 | c1) ≥EUi(c2 | c2). Furthermore, EUi(c1 | c1) ≥ EUi(c2 | c1) from (3). Therefore,EUi(c1 | c1) ≥ EUi(c2 | c2) ≥ 0 (by (2)), and so the omitted participationconstraints do not bind at the solution to [P]’.

Notice from (A84), (A85), and (A87) that µA1 > 0. Similarly, (A84),

(A86), and (A88) imply µB1 > 0. Therefore, by complementary slackness

EUi (c1 | c1) = EUi (c2 | c1) for i = A, B. (A114)

Using (A114), it is readily verified that

EUi (c1 | c2) = EUi (c2 | c1) + φ−i1

[1 + dδi

11

][C

(xi

11, c1) − C

(xi

11, c2)]

+ φ−i2

[1 + dδi

12

][C

(xi

12, c1) − C

(xi

12, c2)]

. (A115)


Substituting from (A113) into (A115) provides

EUi (c1 | c2) = EUi (c2 | c2) + φ−i1

[1 + dδi

21

][C

(xi

21, c2) − C

(xi

21, c1)]

+ φ−i2

[1 + dδi

22

][C

(xi

22, c2) − C

(xi

22, c1)]

+ φ−i1

[1 + dδi

11

][C

(xi

11, c1) − C

(xi

11, c2)]

+ φ−i2

[1 + dδi

12

][C

(xi

12, c1) − C

(xi

12, c2)]

. (A116)

From (A75) and Lemma A2, the δijk‘s in (A116) are either all zero or all

unity. If they are all zero, then (A116) becomes

EUi (c1 | c2)

= EUi (c2 | c2) − φ−i1

(C(xi

11, c2) − C

(xi

21, c2) − [

C(xi

11, c1) − C

(xi

21, c1)])

−φ−i2

(C

(xi

12, c2) − C

(xi

22, c2) − [

C(xi

12, c1) − C

(xi

22, c1)])

. (A117)

If the δijk′s are all unity, then (A116) becomes

EUi (c1 | c2)

= EUi (c2 | c2) − φ−i1 [1 + d]

(C(xi

11, c2) − C

(xi

21, c2)

− [C

(xi

11, c1) − C

(xi

21, c1)])

− φ−i2 [1 + d]

(C

(xi

12, c2) − C

(xi

22, c2) − [

C(xi

12, c1) − C

(xi

22, c1)])

.

(A118)

C(x, c2) − C(y, c2) > C(x, c1) − C(y, c1) for all x > y, by assumption.Therefore, (A117) and (A118) imply that EUi(c1 | c2) < EUi(c2 | c2) fori = A, B. Therefore, the omitted incentive compatibility constraints donot bind at the solution to [P]’. Consequently, the solution to [P]’ is thesolution to [P].

Notice that property (i) of Proposition 4 holds because: (i) (A114)holds; (ii) λB

2 > 0 from (A90), (A92) and (A93); and (iii) λA2 > 0 from

(A96), (A97) and (A100). Property (ii) of the proposition follows fromLemma A2 and Proposition 2. Property (v) of the proposition followsfrom the simulation presented in the text. �

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