Honesty and Adverse Selection
Michael T. Rauh
Indiana University
[email protected]
Giulio Seccia
University of Southampton
[email protected]
May 20, 2015
Abstract
There is substantial evidence that parents value honest children and that trust has important
implications for economic performance and outcomes. We consider the problem of a parent who
chooses from a set of technologies that can make her child honest with some probability. In the
second stage of the game the child becomes the agent in a standard screening context or the
seller in a game-theoretic version of the market for lemons. We provide conditions under which
the parent makes positive investments in honesty and derive the effects on the optimal screening
contract and equilibrium prices and communication.
Keywords: deception, exchange, honesty, institutions, religion, trade, trust.
JEL Codes: D02, D03, D82, Z1.
This paper presents a struggling attempt to give structure to the statement: “Business in
underdeveloped countries is difficult”; in particular, a structure is given for determining the
economic costs of dishonesty.
Akerlof (1970, Introduction)
1
Introduction
There is substantial evidence that parents attempt and often succeed in instilling certain values
and cultural traits in their children. According to the literature on cultural transmission surveyed
in Bisin and Verdier (2011), direct socialization (i.e., parental upbringing) plays an important role
in the development of preferences for discounting and risk; generalized trust; attitudes towards
work, welfare, and individual responsibility; and generosity. In addition to the family itself, other
technologies for cultural transmission include educational, occupational, and religious institutions.
Indeed, Bisin and Verdier present evidence from a variety of sources that parents often exhibit a
strong preference for children with their own religious beliefs and are willing to make the appropriate
costly investments.
Which values and traits do parents emphasize the most? Successive versions of the General
Social Survey (GSS) have included questions such as “Which three qualities listed on this card
would you say are the most desirable for a child to have?” and “Which one of these three is the
most desirable of all?” The number one response by a substantial margin is “honesty” which is rated
more highly than “tries hard to succeed,” “good sense and sound judgement,” “self-control,” “gets
along with other children,” and “good student.” This is consistent with the experimental results in
Gneezy (2005) and Hurkens and Kartik (2009) where subjects displayed an aversion to lying even
in anonymous one-shot settings where reputation concerns are minimal. Following Kartik (2009),
we say that an agent is honest when he incurs a cost from being deceptive and dishonest otherwise.
Note that this definition refers to preferences rather than behavior (an honest agent can still be
deceptive) and intrinsic rather than extrinsic motivation (honesty is an acquired preference rather
than behavior enforced by punishments).
There is also substantial evidence that honesty and other values related to trust have important
implications for economic and organizational performance. Fukuyama (1995) stresses the role
of social capital and trust in the formation of large-scale business enterprises in countries such
as Japan, the United States, and West Germany. Landes (1999) presents historical evidence that
certain cultural traits promote national economic success. In support of this thesis, Tabellini (2010)
finds that generalized trust, respect for others, and confidence in the virtues of individualism have
positive and causal effects on economic development. Likewise, Algan and Cahuc (2010) show that
1
trust has a significant positive impact on income per capita using the inherited trust of descendants
of US immigrants as an instrument for trust in the country of origin. Guiso, Sapienza, and Zingales
(2009) find that religious similarity and somatic distance (i.e., similarity of physical appearance) are
important determinants of trust, which in turn is an important determinant of trade flows, foreign
direct investment, and portfolio choice. This is especially true for trade in goods characterized
by asymmetric information. Using data from the European Social Survey, Butler, Giuliano, and
Guiso (2014) show that trust is important not only at the aggregate but also at the individual
level. In particular, they find an inverted-U relationship between individual income and individual
trust beliefs so that individuals who exhibit the “right amount of trust” are more prosperous.
Butler, Giuliano, and Guiso (2013) obtain similar results in experiments using a version of the
trust game. In this context they find that trustworthiness is positively correlated with parental
efforts at instilling closely related values. For a survey of this literature see Algan and Cahuc (2013)
and for another that emphasizes the connections between culture and institutions see Alesina and
Giuliano (2013).
Another literature studies the relationships between trust, economic outcomes, and religion.
Weber’s classic treatise The Protestant Ethic and the Spirit of Capitalism emphasized the role of
Protestantism and specifically Calvinism in the early development of capitalism. Guiso, Sapienza,
and Zingales (2003) find that Christianity has a stronger association with attitudes conducive
to economic growth than other religions. In particular, Christianity fosters trust but more so
for Protestants than Catholics. McCleary and Barro (2006) find that religious belief has a large
and positive impact on economic growth. Algan and Cahuc (2010) find no relationship between
religion and trust (see their Table 2) but the percentage of nonreligious persons and religious
fractionalization have negative effects on income per capita which are much larger than inherited
trust, lagged income per capita, political institutions, and historical preschool enrollment (Table
12). Norenzayan and Shariff (2008) survey the literature on religious pro-sociality, costly signaling,
and trust and conclude that: (i) religious individuals are perceived to be more cooperative and
trustworthy, (ii) religious groups with more costly requirements have more committed members,
and (iii) in experiments religion is associated with trusting behavior. For example, in the trust game
proposers forward more money to responders perceived to be religious and religious responders are
more likely to reciprocate.1 Although this evidence points to specifically religious institutions, a
variety of secular institutions may serve a similar function such as schools; medieval guilds and their
1
Religion also has a darker side: Guiso, Sapienza, and Zingales (2003) find that religious individuals tend to be
more racist and Bénabou, Ticchi, and Vindigni (2013) that religion can suppress innovation both theoretically and
empirically.
2
modern equivalent, trade unions; as well as the family itself. E.g., Ariely, Garcia-Rada, Hornuf,
and Mann (2014) find that cheating declined with education in their experiments.
The existence of trust in environments where credible punishments are available and the players
have high discount factors, a high probability of continuation, and readily observed histories can
be explained by standard reputation mechanisms. But many settings such as the private used
car market do not satisfy these requirements because of the low probability of continuation and
unobservable histories between strangers. This may undermine parental investments in honesty
because private information is valuable and honesty entails divulging this information freely. In
this paper we seek to show how honesty can arise endogenously even in such challenging settings.
We consider the problem of a parent who chooses from a set of technologies that can make her
child honest with some probability. Each technology is characterized by the probability that it
makes the child honest and the corresponding deception cost which the child incurs when he lies.
After implementing the chosen technology, two random variables are realized. The first is whether
the technology was successful and the child is honest or not. In the second stage of the game
the child becomes the seller in an exchange context with asymmetric information and the second
random variable, independent from the first, is whether the child is a high or low type. We consider
the two main benchmark adverse selection models in the literature: the basic two-type screening
model where the child is the agent with high or low production cost and a game-theoretic version
of Akerlof’s (1970) market for lemons similar to that in Streb and Torrens (2012) where the child
is the seller of a high or low quality car. In the first model the uninformed party (the principal)
moves first and designs a direct mechanism which conditions the contract on the agent’s statement
about his type. In the second the informed party (the seller) moves first, sends the buyer a message
about the quality of the car, and makes a take-it-or-leave-it offer with respect to price. We assume
the informed party does not regard an abstract message or a choice from a menu of contracts to
be a lie – only a direct false statement about their type. Since the uninformed party benefits from
honesty our exclusive focus on direct mechanisms is therefore warranted.
The crucial assumptions of the paper concern the nature of the technologies available to the
parent and what the uninformed player can observe in the game with asymmetric information. As
Frank (1987) notes, the fact that a seller is honest cannot influence trade unless honesty can be
credibly signaled.2 In many cases institutional membership is easily verified as when a politician
2
Frank (1987) discusses several physical symptoms of an involuntary nature which signal trustworthiness. E.g.,
Cogsdill et al. (2014) find that the predisposition to make character inferences (including trustworthiness) from faces
fully develops in early childhood which seems consistent with the cultural transmission hypothesis. Frank, Gilovich,
and Regan (1993) find that subjects can predict with “considerable accuracy” how other players will choose in a
3
cites their military service or religious affiliation as evidence of their trustworthiness. In other
cases the signal is strongly but not perfectly correlated with the technology. According to Murdock
(2000) Weber’s Calvinists were distinguishable from other denominations due to their strict dress
codes. More generally, it may be that long-standing members of certain kinds of institutions are
able to send signals that non-members find hard to fake. To keep the model tractable we assume
the uninformed party can observe the technology (or a perfectly correlated signal) but it is clear
that perfect correlation is not necessary either in theory or in practice. We emphasize that in our
model institutional membership is not a guarantee of behavior but under certain conditions it may
serve as an endogenous and informative signal of honesty. Indeed, we will show that some members
do in fact lie in equilibrium.
The second assumption is that the parent can commit to the technology. This assumption is
crucial because in our model there is an incentive to fake the application of the technology. E.g.,
the parent could undermine the technology at home or enroll the child in the institution but make
arrangements for the technology to be withheld. Such commitment issues may be overcome by
outsourcing the application of the technology to a specific class (e.g., educators or clergy) who have
a vested interest in ensuring that it is applied and is successful. This vested interest may derive
from moral or religious conviction or a concern for the reputation of the institution. In this context
boarding arrangements where the child is sent to reside at the institution at an impressionable
young age acquire special significance.
When the second or exchange stage of the game is the screening model we obtain the following
results. When the parent chooses a relatively low deception cost the principal’s optimal screening
contract is similar to the standard one except that the efficient honest type receives less information
rent than the efficient dishonest type because the deception cost softens the former’s incentive
compatibility constraints. As usual the principal distorts the output of the inefficient types below
the first best level to reduce the amount of information rent to the efficient types and when the
deception cost is low the level of distortion is exactly the same as in the standard model. When
the parent chooses a high deception cost the optimal screening contract entails zero information
rent for the efficient honest type. Furthermore, the distortion in output depends on the probability
that the agent is dishonest. In particular, output ranges between its first and second best levels as
this probability ranges between zero and one, respectively. We show that the optimal technology
entails a high deception cost and the probability that the agent is dishonest is chosen to balance the
one-shot prisoner’s dilemma when the players can interact for 30 minutes prior to play. In contrast, Ockenfels and
Selten (2000) find little evidence of such involuntary signaling in their bargaining experiments.
4
following tradeoff: the greater the probability that the agent is dishonest the greater the probability
the agent collects rent (only the efficient dishonest type does so) but the smaller the amount of
rent collected because the corresponding distortion in output is greater.
A different set of tradeoffs emerges when the second stage is the pricing game with asymmetric
information and communication. If the high-quality seller’s reservation value is low then even with
dishonest sellers an efficient pooling equilibrium exists where high and low quality cars charge
the same price. Since the market is efficient the parent has little incentive to invest in honesty
and will not do so if such investments are costly. If the high-quality seller’s reservation value is
medium the parent must invest in honesty for high quality cars to trade. In particular, the optimal
technology entails a deception cost high enough to support an efficient semi-separating equilibrium
where the honest low quality type separates, informs the buyer that quality is low, and charges
a low price whereas all other types claim to be high quality and pool on a high price. Note that
the dishonest low quality type lies in equilibrium. When the high-quality seller’s reservation value
is high efficiency can only be achieved by making the child honest with probability one with a
deception cost high enough to support a separating equilibrium. This assumes that making the
child honest with probability one is possible and not prohibitively costly which may be unwarranted.
If so the only feasible outcome is the lemons equilibrium where the market completely unravels.
The model therefore predicts there may be less adverse selection in the used car market than
standard theory predicts.3 The empirical evidence is indeed mixed. Peterson and Schneider (2013)
do find evidence of adverse selection which they characterize as “moderate”; e.g., recently purchased
used cars have 12% more engine repairs than all used cars, traded and non-traded. They also find
no evidence of moral hazard (another dimension of trustworthiness) in the sense of cutting back
on maintenance prior to sale. The empirical results in Lacko (1986) suggest relatively high levels
of honesty even between strangers. In particular, there is no significant difference in the quality
of used cars traded between friends and relatives as compared with trade initiated by newspaper
ads for cars between 1-7 years old. For cars between 8-15 years old the difference is significant
but again not very large: average repair expenditures are only $166 dollars less for trade between
friends and relatives as compared with trade based on newspaper ads.
Our paper contributes to the literature on cultural transmission pioneered by Bisin and Verdier
(2000, 2001), including Doepke and Zilibotti (2008), Guiso, Sapienza, and Zingales (2008), and
Tabellini (2010), who shows how generalized trust can emerge in the context of the Prisoner’s
3
A complementary explanation is provided by Hendel and Lizzeri (1999) who show that adverse selection is reduced
when the interaction between the new and used car markets is explicitly modeled which makes the valuations of used
car sellers endogenous.
5
Dilemma through a dynamic process governed by direct socialization. For a comprehensive survey
see Bisin and Verdier (2011). Our paper also contributes to a growing theoretical literature on
honesty including Alger and Renault (2006, 2007), who show how the optimal second best screening
contract is modified when there is some probability that the agent is honest and Kartik (2009),
who shows how deception costs affect communication in a general class of sender-receiver games.4
The literature on honesty and mechanism design includes Kartik, Tercieux, and Holden (2014),
who show that when there are at least two agents and a separable punishment condition holds,
any social choice function can be implemented by a simple mechanism in two rounds of iterated
elimination of strictly dominated strategies.
The plan for the rest of the paper is as follows. In the next section we consider the endogenous
development of honesty in the context of screening mechanisms and in section 3 in the pricing game
with asymmetric information and communication. Section 4 concludes. All proofs not in the text
are in the appendix.
2
Honesty and Screening
There are three players: a parent (she), a child (he), and a principal (she). We say the child is
honest if he incurs a deception cost ψ > 0 when he makes a false statement about his type. The
child only incurs this cost when he lies to a direct mechanism and not when he chooses from a menu
of contracts or sends an abstract message in a general mechanism because he does not consider
these to be lies. The child is dishonest when ψ = 0.
In the first stage of the game the parent chooses a technology (ψ, ν) which instills the deception
cost ψ with probability 1 − ν, where ψ ≥ 0 and 0 ≤ ν ≤ 1. With probability ν the technology
fails and the child remains dishonest. For simplicity we assume that all technologies are costless.
Note that the parent can choose (0, ν) for any 0 ≤ ν ≤ 1 or (ψ, 1) for any ψ ≥ 0 in which case the
child is dishonest with probability one. The parent can also choose (ψ, 0) with ψ > 0 so the child
is honest with probability one. The parent is altruistic and chooses the technology that maximizes
the expected payoff of the child. We assume the parent can commit to this choice.
After the technology has been applied two independent random variables are realized. The first
is the realization of ψ̃ ∈ {0, ψ} which determines whether the child is honest or not. The second θ̃
4
Alger and Renault (2006) consider three different definitions of honesty which have different implications for the
optimal second best screening contract as compared to our definition of honesty in terms of deception costs. E.g.,
Alger and Renault show that in the case of conditional honesty any optimal truth-telling contract entails rent for
honest agents (except in the worst state) and allocations which are equivalent to the standard second best allocations.
This is not the case in our model. But the main difference is that here honesty is endogenous.
6
is the production cost of the child which is low θ ≥ 0 with probability 0 < γ < 1 and high θ with
probability 1 − γ, where ∆θ = θ − θ > 0. The realization of (ψ̃, θ̃) is the private information of the
child.
After receiving his private information the child becomes the agent in the contracting stage.
The principal observes the technology (ψ, ν) but not the realization of (ψ̃, θ̃) and offers the agent
a direct mechanism (q(ψ̂, θ̂), t(ψ̂, θ̂)), where output q and the transfer t depend on the agent’s
announcement (ψ̂, θ̂) about his type. If the agent rejects the contract the payoff of all three parties
is zero. If the agent accepts then he announces his type, the corresponding allocation (q, t) is
executed, and the parties receive their payoffs. We assume that when the agent is indifferent he is
willing to choose as the principal directs.
The payoff of the principal is V = S(q) − t, where S(q) is the expected benefit of output q. The
payoff of the agent is U = t − θq when he is dishonest or is honest and does not lie. His payoff is
U = t − θq − ψ when he is honest and lies. Utilitarian welfare independent of deception costs is
given by
W (q, θ) = U + V = S(q) − θq.
(1)
The following assumptions ensure the screening problem has a unique interior solution. An example
is S(q) = (1/α)q α for all 0 < α < 1.
Assumption 1 S : [0, ∞) → [0, ∞) is twice differentiable on (0, ∞) with S(0) = 0, S 0 > 0,
limq→0 S 0 (q) = ∞, limq→∞ S 0 (q) = 0, and S 00 < 0.
We use backward induction to solve the model starting with the contracting stage. If the agent
is dishonest with probability one we obtain the standard solution (see Lemma 1 below). In what
follows an underlined variable refers to an efficient agent with cost θ and an overlined variable to
an inefficient agent with cost θ. Let q ∗ and q ∗ be the corresponding first best output levels which
maximize utilitarian welfare
S 0 (q ∗ ) = θ
and S 0 (q ∗ ) = θ.
(2)
Lemma 1 If the agent is dishonest with probability one then the direct mechanism that maximizes
the expected profit of the principal is given by q = q ∗ ,
S 0 (q s ) = θ +
q = q s , U = ∆θq s , and U = 0.
7
γ∆θ
,
1−γ
(3)
Although the result is well-known (e.g., see Laffont and Martimort (2002, Chapter 2)), a brief
review will ease the exposition. When the agent is dishonest with probability one he effectively has
two types: efficient and inefficient. The principal chooses transfers and outputs (t, q) and (t, q) to
maximize expected profit
γ S(q) − t + (1 − γ) S(q) − t
(4)
t − θq ≥ t − θq
(5)
t − θq ≥ t − θq
(6)
t − θq ≥ 0
(7)
t − θq ≥ 0.
(8)
subject to the constraints
The first two constraints are the incentive compatibility (truth-telling) constraints while the last
two are the participation constraints. Let
U = t − θq
(9)
U = t − θq
(10)
denote the rents of the efficient and inefficient agents, respectively. We can re-write the problem as:
choose rents and outputs (U , q) and (U , q) to maximize expected utilitarian welfare less expected
rent
γW (q, θ) + (1 − γ)W (q, θ) − γU − (1 − γ)U
(11)
U ≥ U + ∆θq
(12)
U ≥ U − ∆θq
(13)
U ≥0
(14)
U ≥ 0.
(15)
subject to
Adding (12) and (13) we obtain the implementability condition q ≥ q. In the standard model
the result that the efficient agent produces more than the inefficient one follows directly from the
incentive compatibility constraints rather than the optimizing behavior of the principal. Given
8
outputs that satisfy q ≥ q the principal first chooses the rents (U , U ) to minimize the expected cost
γU + (1 − γ)U subject to (12)-(15). Intuitively, from (12) we observe that rents to the inefficient
type only increase the rents that must be offered to the efficient type. The principal therefore sets
U = 0 and the participation constraint of the inefficient type binds. The principal then offers the
minimum rent U = ∆θq necessary to induce the efficient agent to truthfully reveal his type. The
corresponding incentive compatibility constraint therefore binds.
Substituting the optimal rents into (11), the principal chooses outputs (q, q) to maximize
γW (q, θ) + (1 − γ)W (q, θ) − γ∆θq
(16)
subject to the implementability condition. At the optimum the output of the efficient agent is first
best q = q ∗ but the output of the inefficient agent is subject to a rent-extraction/efficiency tradeoff
because an increase in q towards the first best q ∗ increases utilitarian welfare but also increases the
rent U = ∆θq of the efficient type. Since S 00 < 0 the solution (3) is to distort q downwards below
the first best level to balance this tradeoff.
We now consider the opposite extreme where the agent is honest with probability one. Let
qψ =
ψ
∆θ .
Lemma 2 Assume the agent is honest with probability one. (i) If qψ < q s then U = ∆θq s − ψ and
q = q s . (ii) If q s ≤ qψ < q ∗ then U = 0 and q = qψ . (iii) If q ∗ ≤ qψ then U = 0 and q = q ∗ . In all
three cases U = 0 and q = q ∗ .
If the agent is honest with probability one the only change with respect to the statement of the
previous problem is that the incentive compatibility constraints now take the form
U ≥ U + ∆θq − ψ
(17)
U ≥ U − ∆θq − ψ
(18)
because an honest agent incurs the deception cost ψ when he lies. Adding these constraints we
obtain the new implementability condition
q ≥ q − 2qψ .
(19)
In this case the result that the efficient agent produces more than the inefficient one follows from
the optimizing behavior of the principal and not directly from the constraints.
9
We follow the same procedure as before. Given implementable outputs (q, q) we find the rents
(U , U ) that minimize the expected cost γU + (1 − γ)U subject to (14), (15), (17), and (18). As
before (17) implies that rent to the inefficient type increases the rent that must be offered to the
efficient one so U = 0 in all three cases (i)-(iii) of Lemma 2 above. If the deception cost is low in the
sense that ψ < ∆θq the principal offers the minimum rent U = ∆θq − ψ > 0 necessary for incentive
compatibility (17) as in case (i). The only difference between (i) and the standard case where the
agent is dishonest with probability one is that the principal can induce the efficient honest type to
tell the truth with less rent because the deception cost softens his incentive compatibility constraint.
If ψ ≥ ∆θq the deception cost is so high that the efficient honest agent does not require rent to
truthfully reveal his type. In cases (ii) and (iii) the solution is therefore U = 0. The latter result
adds an important new dimension to the principal’s problem not present in the standard model:
the principal now has the option of choosing q sufficiently low (specifically q ≤ qψ ) to shut off all
rent to the efficient agent.
Substituting the optimal rents into (11) the principal chooses outputs (q, q) to maximize
where
γW (q, θ) + (1 − γ)W (q, θ) − γU ,
(20)

 ∆θq − ψ if q > q ,
ψ
U=

0
if q ≤ qψ .
(21)
In all cases (i)-(iii) the output of the efficient type is first best q = q ∗ which follows from inspection.
The output q of the inefficient type is determined by the modified rent-extraction/efficiency tradeoff
in (21) where q ≤ qψ shuts off rent to the efficient honest agent. In case (i) the condition q ∗ ≤ qψ
implies that the first best output level q ∗ which maximizes utilitarian welfare also shuts off rent
to the efficient type and is therefore clearly optimal. In case (ii) the condition q s ≤ qψ < q ∗
implies that the output level q = qψ which shuts off rent to the efficient type involves a downward
distortion in utilitarian welfare. The principal would never choose q < qψ because it would only
increase that distortion and the rent of the efficient type is already zero. On the other hand, output
levels q > qψ would reduce the distortion but also necessitate rent to the efficient type. Since q s
optimally balances the rent-extraction/efficiency tradeoff and q s ≤ qψ < q ∗ , output levels q > qψ
reduce expected profit and the solution is therefore q = qψ . Case (iii) is similar except that qψ < q s ,
output levels qψ < q < q s increase expected profit, and the solution is q = q s . In all three cases
q ≥ q at the optimum.
10
We now consider intermediate technologies (ψ, ν) such that 0 < ν < 1 and ψ > 0. In this
case the agent has four possible types: the efficient honest type (ψ, θ), the inefficient honest type
(ψ, θ), the efficient dishonest type (0, θ), and the inefficient dishonest type (0, θ), where the first
component indicates the deception cost and the second the production cost. In what follows a
subscript ψ refers to an honest agent and a subscript 0 to a dishonest one. Let ~q = (q ψ , q 0 , q ψ , q 0 )
~ = (U ψ , U 0 , U ψ , U 0 ) the vector of rents for each type. The principal
be the vector of outputs and U
~ to maximize
now chooses ~q and U
γνW (q 0 , θ) + γ(1 − ν)W (q ψ , θ) + (1 − γ)νW (q 0 , θ) + (1 − γ)(1 − ν)W (q ψ , θ)
− γνU 0 − γ(1 − ν)U ψ − (1 − γ)νU 0 − (1 − γ)(1 − ν)U ψ
(22)
subject to
Uψ ≥ U0 − ψ
(23)
U ψ ≥ U ψ + ∆θq ψ − ψ
(24)
U ψ ≥ U 0 + ∆θq 0 − ψ
(25)
Uψ ≥ 0
(26)
and the incentive compatibility and participation constraints for the other three types.5 Given an
~ of rents that minimize the
implementable vector ~q of outputs we now characterize the vector U
expected cost
γνU 0 + γ(1 − ν)U ψ + (1 − γ)νU 0 + (1 − γ)(1 − ν)U ψ
(27)
subject to the incentive compatibility and participation constraints.
Proposition 1 Let ~q be an implementable vector of outputs and q = max{q ψ , q 0 }.
(i) The principal always chooses
0 < qψ , q0 ≤ q∗ < q∗ ≤ qψ , q0.
(28)
~ of rents that minimizes expected cost (27) subject to the incentive compatibility
(ii) The vector U
5
For a complete statement of the problem including the implementability conditions see the appendix.
11
and participation constraints is given by U ψ = U 0 = 0, U 0 = ∆θq, and
Uψ =


0
if ψ ≥ ∆θq
(29)
 ∆θq − ψ if ψ < ∆θq.
The intuition builds on the previous two lemmas. We first consider (i). The constraints for the
efficient dishonest type are given by
U0 ≥ Uψ
(30)
U 0 ≥ U ψ + ∆θq ψ
(31)
U 0 ≥ U 0 + ∆θq 0
(32)
U 0 ≥ 0.
(33)
If the principal distorts q ψ or q 0 away from the first best q ∗ there will be a loss in utilitarian
welfare. If the distortion is upward, above the first best, then from (24), (25), (31), and (32) this
will also increase rents. It follows that any distortion will be downwards. Since we have assumed
the expected marginal benefit of output S 0 is arbitrarily large close to zero, q ψ and q 0 will be
positive. Similarly, any distortion in q ψ or q 0 will be upwards and the inequalities in (28) follow.
We now discuss the optimal rents in (ii). From (24), (25), (31), and (32) we again observe
that rents to the inefficient types increase the rents that must be offered to the efficient ones so
U ψ = U 0 = 0. The principal then offers the efficient dishonest type the minimum necessary rent
to induce him to truthfully reveal his type. From (31) and (32) this is ∆θq ψ or ∆θq 0 , whichever
is higher. The solution is therefore U 0 = ∆θq, where q = max{q ψ , q 0 }. From (24) and (25) the
minimum necessary rent to induce the efficient honest agent to reveal his type is U ψ = ∆θq − ψ
unless this is negative. In the latter case the agent is sufficiently honest that he does not require
rent to truthfully reveal his type and his participation constraint binds U ψ = 0. For a formal proof
of these statements see the appendix.
We now find the outputs ~q that maximize expected profit (22) given the optimal rents in
Proposition 1.
Proposition 2 Let
S 0 (q h ) = θ +
12
γν∆θ
1−γ
(34)
where q s < q h < q ∗ . Define
IL = [0, ∆θq s ]
(35)
IM = [∆θq s , ∆θq h ]
(36)
IH = [∆θq h , ∞).
(37)
The optimal output levels are q ψ = q 0 = q ∗ and q ψ = q 0 = q, where



q s if ψ ∈ IL ,


q=
qψ if ψ ∈ IM ,



 q h if ψ ∈ I .
H
(38)
The outputs q ψ and q 0 of the efficient types are easily determined. Since the optimal rents in
Proposition 1 do not depend on them, q ψ and q 0 are chosen to maximize utilitarian welfare and
will therefore be first best q ψ = q 0 = q ∗ . It is also clear that the outputs q ψ and q 0 of the inefficient
types must be the same. E.g., if q ψ < q 0 then a small increase in q ψ would improve utilitarian
welfare for that type with no effect on the rent of the efficient types.
We make two preparatory remarks to elucidate the solution in (38). From Proposition 1 for
output levels q ∈ [0, qψ ] only the efficient dishonest type receives rent and the expected profit of
the principal is given by
γνW (q 0 , θ) + γ(1 − ν)W (q ψ , θ) + (1 − γ)νW (q 0 , θ) + (1 − γ)(1 − ν)W (q ψ , θ)
− γν∆θq.
(39)
The output level q that maximizes (39) ignoring the constraint 0 ≤ q ≤ qψ is given by q h in (34)
which does not necessarily belong to the interval [0, qψ ]. For output levels q ∈ [qψ , ∞) both efficient
types receive rent and expected profit is given by
γνW (q 0 , θ) + γ(1 − ν)W (q ψ , θ) + (1 − γ)νW (q 0 , θ) + (1 − γ)(1 − ν)W (q ψ , θ)
− γ∆θq + γ(1 − ν)ψ.
(40)
The output level that maximizes (40) ignoring the constraint q ≥ qψ is the standard second best
level q = q s in (3).
The solution q in (38) follows directly from these observations. E.g., when the deception cost is
13
high ψ ∈ IH then q s < q h ≤ qψ . For output levels q ∈ [0, qψ ] expected profit is given by (39). Since
q h maximizes (39) and 0 < q h ≤ qψ , expected profit on the interval [0, qψ ] increases up to q h and
declines thereafter. For output levels q ∈ [qψ , ∞) expected profit is (40). Since q s maximizes (40)
and q s < qψ , expected profit declines on [qψ , ∞) and the principal therefore chooses q = q h . The
cases where ψ ∈ IL and ψ ∈ IM are similar.
For future reference we summarize Propositions 1 and 2 as follows.
Corollary 1 For intermediate technologies (ψ, ν) such that ψ > 0 and 0 < ν < 1: (i) If ψ ∈ IL
then q = q s , U ψ = ∆θq s − ψ, and U 0 = ∆θq s . (ii) If ψ ∈ IM then q = qψ , U ψ = 0, and U 0 = ψ.
(iii) If ψ ∈ IH then q = q h , U ψ = 0, and U 0 = ∆θq h . In all cases q = q ∗ and U ψ = U 0 = 0.
When the deception cost is low ψ ∈ IL or high ψ ∈ IH the optimal screening contract is almost
the same as the standard one in Lemma 1. The only differences are that when ψ ∈ IL the efficient
honest type receives less rent than in the standard optimal screening contract and when ψ ∈ IH
the distortion entailed by q h depends on the probability ν that the agent is dishonest. From (34),
∂q h
1
γ
= 00 h
∆θ < 0
∂ν
S (q ) 1 − γ
(41)
so q h declines from the first q ∗ to the second best q s as ν increases from zero to one.
In the first stage of the game the parent chooses the technology (ψ, ν) to maximize the expected
~ , ~q) chosen by the principal. If
rent less deception costs of the child given the direct mechanism (U
ψ ∈ IL then from Corollary 1 the expected payoff of the child is
PL = γ∆θq s − γ(1 − ν)ψ.
(42)
On this region IL honesty reduces the expected payoff of the child by reducing the information rent
of the efficient honest type. A small amount of honesty is therefore worse than none at all and the
optimal choice of the parent on the interval IL is to make the child dishonest with probability one.
At the optimum PL = γ∆θq s as in the standard model.
If ψ ∈ IM the principal chooses q = qψ =
ψ
∆θ
and the expected payoff of the child is
PM = γν∆θq = γνψ.
(43)
On this region IM an increase in ψ induces the principal to increase qψ to reduce the distortion
in output and this increases the rent U 0 = ψ of the efficient dishonest type. The optimal choice
14
of ψ from the interval IM is therefore the maximum possible ψ = ∆θq h . Since this choice is also
available in IH we may henceforth restrict attention to IL and IH .
If ψ ∈ IH the expected payoff of the child is PH = γν∆θq h . Since q h and therefore PH do not
depend on the deception cost ψ the parent is indifferent over all ψ ∈ IH . From (34) we observe
that q h = q s when ν = 1 in which case PH = γ∆θq s . Since this is the same as PL when ψ is chosen
optimally from IL we can restrict attention to IH . In particular, the optimal technology (ψ, ν) is
any ψ ≥ ∆θq h and then the ν which is the solution to the following problem
max γν∆θq h .
0≤ν≤1
(44)
The parent therefore chooses a high deception cost ψ ∈ IH which secures a rent U 0 = ∆θq h which
is at least as high as the standard model because q h ≥ q s . Since
∂PH
∂q h
h
= γ∆θ ν
+q
∂ν
∂ν
(45)
the parent faces the following tradeoff with respect to the optimal choice of ν: for a given output
level q h an increase in ν implies a greater probability that the child will collect rent U 0 = ∆θq h
but for a given ν it also implies less expected rent because it induces the principal to reduce q h .
Proposition 3 (i) The parent never chooses ν = 0. (ii) If
S 0 (q) + qS 00 (q) → ∞ as q → 0
(46)
then ν = 1 is suboptimal for γ and/or ∆θ sufficiently large.
The parent never chooses ν = 0 because only the efficient dishonest type receives rent. As
the principal increases ν she faces the tradeoff in (45). In particular, given q h an increase in the
probability ν that the agent is dishonest raises the probability γν the agent will collect rent ∆θq h .
From (34) we observe that q h → 0 as γ → 1 and/or ∆θ → ∞. Intuitively, the downward distortion
in q h is large when the probability γ of an efficient agent (only the efficient dishonest type receives
rent) and/or the temptation ∆θ to mimic an inefficient type is high. The positive effect of increasing
ν therefore becomes vanishingly small when γ → 1 and/or ∆θ → ∞. At the same time, an increase
in ν reduces the rent ∆θq h of the efficient dishonest type because q h is decreasing in ν. From (41)
we observe that the size of this negative effect depends on the curvature S 00 (q h ) of the principal’s
expected benefit. The negative effect will therefore dominate the vanishingly small positive effect
15
unless S 00 → −∞ sufficiently fast as q h → 0. The condition (46) is a simple and easily applied
sufficient condition which rules this out.
We illustrate this result in the specific case where S(q) = (1/α)q α and 0 < α < 1. The condition
(46) is satisfied because
S 0 (q) + qS 00 (q) =
α
.
q 1−α
(47)
The optimal ν is given by
ν = min
1
−1
α
θ
1
−1
,1
γ
∆θ
(48)
which is less than one when α and γ are sufficiently close to one, ∆θ is sufficiently large, and θ is
sufficiently small.
3
Honesty and the Market for Lemons
We now consider a game-theoretic version of Akerlof (1970) similar to that in Streb and Torrens
(2012). There are three players: the parent (she), the child S (he), and the buyer B (she). As
before in the first stage of the game the parent chooses the technology (ψ, ν) to maximize the
expected profit less deception costs of the child. After the technology is applied two independent
random variables are realized. The first is the deception cost ψ̃ ∈ {0, ψ} and the second is the
quality of the car which is low L with probability 0 < ρ < 1 and high H with probability 1 − ρ.
The child therefore has four types (j, k), where j ∈ {0, ψ} and k ∈ {H, L}. While the technology
(ψ, ν) is observable the child’s type is private information. We continue to assume that the choice of
technology is costless but make comments on what would happen otherwise because in the present
model there are multiple equilibria and unlike the previous section the existence of such costs can
have non-trivial implications.
After learning his type the child decides whether or not to sell his car. We assume he withholds
it when indifferent. If he decides to sell the child S meets with the buyer B to discuss trade. Let
L < vH < vH
vik be the value of a car of type k to player i, where i ∈ {B, S}. We assume vSL < vB
S
B
to allow potential gains from trade and ensure that car H has a higher price under complete
information. A type (j, k) seller makes a take-it-or-leave-it offer consisting of a price pjk and a
statement mjk ∈ {H, L} about the quality of the car. The buyer updates her prior belief and
decides whether or not to buy. We assume she buys when indifferent. In this context deception
consists of making a false statement mjk 6= k about the quality of the car. If the car is sold the
16
k − pjk . The payoff to the seller is
payoff to the buyer is uk = vB
π 0k = p0k − vSk
(49)
π ψk = pψk − vSk
(50)
π ψk = pψk − vSk − ψ
(51)
when the seller is dishonest, honest and not deceptive, and honest and deceptive, respectively. If
the car is not sold the payoff of all three parties is zero.
H
Under complete information and take-it-or-leave-it offers the price of an H car would be vB
L , with expected total surplus
and the price of an L car would be vB
L
H
RF = ρ(vB
− vSL ) + (1 − ρ)(vB
− vSH ).
(52)
We refer to the above as the first best expected surplus.
We first characterize the set of perfect Bayesian equilibria when the seller is dishonest with
probability one.
L + (1 − ρ)v H . If the seller is dishonest with probability one the set of
Lemma 3 Let pP ≡ ρvB
B
equilibria is as follows:
(i) the lemons equilibrium where the H car is withheld and the L car is offered at the price
L and either message is sent.
p0L = vB
(ii) If pP > vSH there is a set of pooling equilibria where H and L cars charge p such that vSH <
p ≤ pP and either message is sent.
When the seller is dishonest with probability one a separating equilibrium where H and L are
both on the market but with different prices and/or messages cannot exist because L will deviate
and mimic H. In contrast the lemons strategy where H is withheld and L charges the buyer’s
L always exists. In this case the market is marred by an adverse selection and
reservation level vB
almost completely unravels. When H and L pool and charge the same price p and send the same
message (which could be either message) the buyer maintains her prior that the car is L with
probability ρ. If pP > vSH there is a range vSH < p ≤ pP of prices which H will accept (recall that
an indifferent seller withholds) and do not exceed the buyer’s willingness to pay pP . Note that in
all these equilibria one type lies. In the particular pooling equilibrium where p = pP the expected
17
total surplus is first best RF . If pP ≤ vSH the only equilibrium is the lemons equilibrium with
L − v L ).
expected total surplus RL = ρ(vB
S
We now consider the case where the seller is honest with probability one. In the following result
we only report those equilibria which survive the Cho-Kreps (1987) intuitive criterion.6 In the
appendix we report the full set of equilibria.
Lemma 4 If the seller is honest with probability one the set of equilibria that survive the intuitive
criterion are as follows:
L and message mψL = L.
(i) car H is withheld and L sets the price pψL = vB
L there is a set of separating equilibria where H sets the price
(ii) If ψ > vSH − vB
H L
vSH < pψH ≤ min{vB
, vB + ψ}
(53)
L and mψL = L.
and message mψH = H while L sets pψL = vB
L there is a set of pooling equilibria where H and L send the
(iii) If pP > vSH and ψ ≤ pP − vB
L + ψ.
message mψH = mψL = H and charge p such that vSH < p ≤ pP and p ≥ vB
The lemons strategy is again an equilibrium. The only difference with the previous lemma is
that now the message is truthful because the seller is honest but this is not an important distinction
L already reveals the quality. There is also a set of pooling equilibria
because the price pψL = vB
L because
similar to the previous lemma except that the seller cannot be too honest ψ ≤ pP − vB
the L type lies. But the main difference is that now there exists a class of separating equilibria
L to keep L from mimicking H.
provided the deception cost is sufficiently high ψ > vSH − vB
Our next result considers the intermediate case where ψ > 0 and 0 < ν < 1.
Proposition 4 Let
ρ
ρ+(1−ρ)ν
(1−ν)ρ
(1−ν)ρ+ν(1−ρ)
qA =
qC =
ρν
ρν+(1−ρ)(1−ν)
ρν
q S = ρν+(1−ρ)
qB =
(54)
and
L
H
p l = q l vB
+ (1 − q l )vB
(55)
for l = A, B, C, S. If ψ > 0 and 0 < ν < 1 the following equilibria survive the intuitive criterion:
6
In our model an off-the-equilibrium-path pair (p, m) is equilibrium-dominated for type (j, k) if (j, k) receives a
higher payoff in equilibrium than from (p, m) assuming the buyer buys when she observes (p, m). According to the
intuitive criterion the posterior belief of a buyer who observes (p, m) should assign probability zero to type (j, k)
when possible (i.e., when there is at least one other type for whom (p, m) is not equilibrium-dominated).
18
(i) the lemons equilibrium where the honest H and dishonest H are withheld, the honest L and
L , the honest L tells the truth mψL = L, and the dishonest
dishonest L charge pψL = p0L = vB
L can send either message.
(ii) If pA > vSH it is an equilibrium for all four types to charge p such that vSH < p ≤ pA , the honest
H to tell the truth mψH = H, and all other types to send the message m0H = mψL = m0L = L.
(iii) If pB > vSH and pC > vSH it is an equilibrium for all four types to charge p such that vSH <
p ≤ min{pB , pC } and send messages mψH = m0L = H and m0H = mψL = L.
L + ψ it is an equilibrium for all four types to send the message H
(iv) If pP > vSH and pP ≥ vB
L + ψ.
and charge p such that vSH < p ≤ pP and p ≥ vB
L it is an equilibrium for the honest L to set pψL = v L and tell
(v) If pS > vSH and ψ > vSH − vB
B
the truth mψL = L and for all the other types to charge p such that
L
vSH < p ≤ min pS , vB
+ψ
(56)
and send the message H.
We need to make an assumption about which set of equilibria the parent expects to prevail for
each choice of technology. We assume the parent is optimistic in the sense that for each (ψ, ν)
she believes that one of the class of equilibria which generates the highest expected payoff for the
seller will be realized with probability one.7 We refer to this as the class of optimistic equilibria
corresponding to (ψ, ν).
H − v L so the seller is honest with probability one and Lemma
E.g., consider ν = 0 and ψ ≥ vB
S
4 applies. Since ψ > pP − vSL the set of equilibria consists of the lemons equilibrium (i) and
H . In this case there is one optimistic equilibrium:
separating equilibria (ii) where vSH < pψH ≤ vB
H . Since this achieves the first best expected profit RF
the separating equilibrium with pψH = vB
with zero deception costs in what follows we can restrict attention to other optimistic equilibria
which also achieve RF with zero deception costs. In particular, at any optimal technology the
parent does not expect the market to unravel in the sense of the lemons equilibrium in Lemmas
3(i) and 4(i) and Proposition 4(i) nor does she expect honest types to lie as in Lemma 4(iii) and
Proposition 4(iv).
7
A pessimistic parent would always choose ν = 1 or ψ = 0 because the lemons strategy is always an equilibrium.
19
If pS > vSH the class of semi-separating equilibria in Proposition 4(v) exists for technologies
L and 0 < ν < 1. This class of equilibria is similar to that in Proposition
such that ψ > vSH − vB
L the optimistic equilibrium is where the honest
3.1 in Streb and Torrens (2012). If ψ ≥ pS − vB
L while the other types pool on the price
L type separates, tells the truth, and charges pψL = vB
p = pS and send the message H. When the buyer observes (pS , H) the posterior probability q S
that the car is L is less than the prior ρ so the willingness to pay pS is greater than pP . As in the
optimistic separating equilibrium discussed above communication does not convey any information
not already contained in prices. An investment in honesty is nevertheless necessary for the optimistic
semi-separating equilibrium to exist because it separates the honest L type from the others who
then form a higher quality pool with a correspondingly greater willingness to pay pS > vSH which
supports entry by the H types. Since expected profit is first best RF and no deception costs are
L and 0 < ν < 1 are also optimal in addition
incurred the set of technologies such that ψ ≥ pS − vB
to the set of technologies that support the optimistic separating equilibrium.
If pS > pP > vSH and the seller is dishonest with probability one there exist the pooling
equilibria in Lemma 3(ii) where all four types charge p and send the same message which can be
either message. At the optimistic pooling equilibrium p = pP expected profit is first best RF .
Since the parent can achieve the first best with ν = 1 and ψ = 0 this is the unique optimum when
pP > vSH and investments in honesty are costly.8 If pP ≤ vSH these equilibria do not exist and it
will be optimal to invest in honesty provided the cost is not so prohibitive that the parent prefers
the lemons equilibrium.
If pP > vSH then for all intermediate technologies such that ψ > 0 and 0 < ν < 1 there exist
the semi-separating equilibria in Proposition 4(iii) where all four types charge the same price p and
the honest H and dishonest L types send the message H while the honest L and the dishonest
H types send the message L. When the buyer observes (p, H) she believes that quality is L with
probability q B and has the corresponding willingness to pay pB . When she observes (p, L) she
believes that quality is L with probability q C with willingness to pay pC . This is an equilibrium
provided vSH < p ≤ min{pB , pC }. The optimistic equilibrium p = min{pB , pC } is inefficient unless
ν = 1/2 in which case p = pB = pC = pP and expected profit is first best RF . Note that when
ν = 1/2 communication does not convey any information not already contained in prices. Indeed,
this is the case for all optimistic equilibria for all optimal technologies. In summary,
Proposition 5
8
E.g., if the cost of technology (ψ, ν) is C(ψ, ν) which is increasing in ψ and decreasing in ν and C(0, 1) = 0.
20
H − v L and ν = 0 and the corresponding
(i) If vSH ≥ pS the optimal choice of the parent is ψ ≥ vB
B
H , mψH = H, pψL = v L , and mψL = L.
optimistic separating equilibrium is pψH = vB
B
L and
(ii) If pS > vSH ≥ pP then in addition to the above it is also optimal to choose ψ ≥ pS − vB
L , mψL = L,
any 0 < ν < 1 with corresponding optimistic semi-separating equilibrium pψL = vB
pψH = p0H = p0L = pS , and mψH = m0H = m0L = H.
(iii) If pP > vSH then in addition to the above it is also optimal to
(a) make the child dishonest with probability one with the corresponding optimistic pooling
equilibrium where all four types set p = pP and send the same message which could be
either message.
(b) choose any ψ > 0 and ν = 1/2 with the corresponding optimistic equilibrium where all
four types set p = pP , mψH = m0L = H, and mψL = m0H = L.
When the reservation level vSH of the H seller is low vSH < pP there exists an efficient optimistic
pooling equilibrium 5(iii)(a) where the child is dishonest with probability one and the parent has
little incentive to invest in honesty. If vSH is medium pP ≤ vSH < pS the parent must invest in
honesty if H types are to trade. An optimal investment in honesty reduces adverse selection by
making communication and prices together more informative because in equilibrium both have the
same informational content. When vSH is high vSH > pS the parent has no alternative but to make
the child honest with probability one if the lemons equilibrium is to be avoided.
4
Conclusion
There is abundant evidence that generalized trust improves economic performance and outcomes.
But what is the specific mechanism? And what is the role of institutions in that process? In this
paper we focus on one specific mechanism: honest communication. It is an act of trust when trade
occurs or a contract is agreed because one party accepts the word of the other regarding the latter’s
private information. But why should the informed party tell the truth? One possibility is a concern
for reputation but this is unlikely to be effective in settings like the private used car market where
the probability of continuation is low and knowledge of prior histories is sketchy between strangers.
Another possibility is that the seller actually is honest because he has been socialized within an
appropriate institution. This is the first role of institutions in our paper — as potentially imperfect
socialization technologies.
21
But the mere existence of institutions that socialize their members to behave in a trustworthy
fashion is insufficient unless parents can commit to their institutional choices and membership is
observable or can be credibly signaled. Otherwise everyone with private information would claim
to be members but all such claims would be non-credible. Of course, membership is no guarantee of
behavior because the technology may be imperfect and some members do in fact lie in equilibrium
but it can still serve as a credible signal of honesty.
But this is still insufficient for positive investments in honesty to occur because honesty is a
double-edged sword as when the agent in the screening model announces that he is high cost or the
seller confesses that his car is low quality. For such investments to occur market institutions must
provide the appropriate incentives. In the screening model this is the additional information rent for
the efficient dishonest type. In the pricing game with asymmetric information and communication
the incentive is that high types who would otherwise be barred by adverse selection can trade
(and the dishonest low type can sometimes trade at a higher price). In this context investments
in honesty occur when the market would otherwise be inefficient. In the screening model economic
performance is improved by the corresponding increase in allocative efficiency.
5
Appendix
Proof of Lemma 2.
Given outputs (q, q) that satisfy the implementability condition (19) we
consider the problem of choosing rents (U , U ) to minimize the expected cost γU +(1−γ)U subject to
(14), (15), and (17). Note that we omit the incentive compatibility constraint (18) of the inefficient
type. Having done so the problem is trivial with solution U = 0 and U = ∆θq − ψ if ∆θq ≥ ψ
and U = 0 otherwise. We then check that the omitted constraint (18) is satisfied using (19). The
statements about the optimal outputs are proven in the text.
~ to maximize
Proof of Proposition 1. The principal’s problem is to choose outputs ~q and rents U
(22) subject to (23)-(26), (30)-(33),
Uψ ≥ U0 − ψ
(A1)
U ψ ≥ U ψ − ∆θq ψ − ψ
(A2)
U ψ ≥ U 0 − ∆θq 0 − ψ
(A3)
U ψ ≥ 0,
(A4)
22
and
U0 ≥ Uψ
(A5)
U 0 ≥ U ψ − ∆θq ψ
(A6)
U 0 ≥ U 0 − ∆θq 0
(A7)
U 0 ≥ 0.
(A8)
Inspection of (24), (25), (31), and (32) reveals that distortions in q ψ or q 0 away from the first best
q ∗ must be downward because upward distortions also increase rents. Likewise, (A2), (A3), (A6),
and (A7) show that distortions in q ψ or q 0 must be upward. Since S 0 (0) = ∞ the inequalities in
(28) follow. We now derive the implementability conditions. From (24) and (A2)
2ψ ≥ ∆θ(q ψ − q ψ ),
(A9)
ψ ≥ ∆θ(q 0 − q ψ ),
(A10)
ψ ≥ ∆θ(q ψ − q 0 ),
(A11)
q0 ≥ q0.
(A12)
from (25) and (A6)
from (31) and (A3)
and (32) and (A7)
~ which minimize the expected cost (27) subject
Given implementable outputs ~q we find the rents U
to the constraints (23), (26), (31), (32), (A4), and (A8). This problem can be solved by inspection
with solution given in the statement. We then check that the remaining constraints are satisfied
using (28).
Proof of Proposition 3. Evaluating (45) at ν = 0,
∂PH
= γ∆θq ∗ > 0
∂ν
(A13)
because S 0 (0) = ∞ and therefore q ∗ > 0. It follows that ν = 0 is never optimal. From (34) q h = q s
at ν = 1. Evaluating (45) at ν = 1,
∂PH
1
γ
s
= γ∆θ
∆θ + q .
∂ν
S 00 (q s ) 1 − γ
23
(A14)
If the above expression is negative then ν = 1 is suboptimal. Substituting (3),
S 0 (q s ) − θ
+ q s < 0 ⇐⇒ S 0 (q s ) + q s S 00 (q s ) > θ.
S 00 (q s )
(A15)
From (3), q s ↓ 0 as γ → 1 and/or ∆θ → ∞. The assumption S 0 + qS 00 → ∞ as q → 0 ensures that
(A15) holds for γ and/or ∆θ sufficiently high.
Proof of Lemma 3. We first note that the L car is never withheld because it can be sold at the
L no matter what the buyer believes. For the rest of the proof we assume the buyer believes
price vB
the car is L with probability one off the equilibrium path. With this off-equilibrium path belief (i)
follows. If neither car is withheld it is clear that a separating strategy cannot be an equilibrium
because the L car would deviate and mimic the H car. Consider a strategy where H and L both
charge p. In equilibrium they must both send the same message which could be either message.
On the equilibrium path the buyer maintains her prior and has maximum willingness to pay pP .
Since the buyer believes that deviators are L with probability one and the H car is withheld when
indifferent this is an equilibrium provided vSH < p ≤ pP .
Proof of Lemma 4.
As before the L car is never withheld and the lemons strategy in (i) is
an equilibrium. For the rest of the proof we assume the buyer believes that deviators are L with
probability one. We now prove (ii). Say neither car is withheld and pψH 6= pψL . In equilibrium we
H and mψH = H because if mψH = L then L could mimic H with zero
must have vSH < pψH ≤ vB
L and send the message mψL = L iff
deception cost. The L car will charge pψL = vB
L
L
pψH − vSL − ψ ≤ vB
− vSL ⇐⇒ pψH ≤ vB
+ψ
(A16)
and (ii) follows. We now prove (iii). Say H and L pool on the price p and message m. On the
equilibrium path the buyer maintains her prior and has maximum willingness to pay pP so we must
have vSH < p ≤ pP . If m = L this is a class of pooling equilibria provided that p − vSH − ψ ≥ 0 or
H will withhold his car. Now consider a deviation by H where he charges p and sends the message
mψH = H. This deviation is equilibrium-dominated for L so the buyer should conclude that the
deviator is H with probability one. This class of pooling equilibria therefore does not survive the
intuitive criterion. If m = H we have a class of pooling equilibria provided that
L
L
p − vSL − ψ ≥ vB
− vSL ⇐⇒ p ≥ vB
+ ψ.
24
(A17)
This completes the proof.
Proof of Proposition 4. As in previous proofs (0, L) and (ψ, L) never withhold their cars and we
assume the buyer believes the car is L with probability one off the equilibrium path. We organize
the proof around the number of H cars present.
(I) Consider a strategy for the seller where (0, H) and (ψ, H) withhold their cars. The buyer
therefore believes the car is L on and off the equilibrium path. The equilibrium prices must
L . There is no reason to lie so mψL = L to avoid the deception
therefore be pψL = p0L = vB
cost but (0, L) can lie or tell the truth in equilibrium. This is the equilibrium in (i).
(II) Consider a strategy where exactly one H type withholds. This cannot be (0, H) because he
could enter with the same strategy as (ψ, H). Furthermore, we cannot have m0H = H because
(ψ, H) could then enter with the same strategy. In equilibrium we must have p0H > vSH ,
m0H = L, and p0H − vSH − ψ ≤ 0 so (ψ, H) withholds. In that case (0, L) and (ψ, L) will
pool with (0, H): p0H = p0L = pψL = p and m0H = m0L = mψL = L. On the equilibrium
path the buyer believes the car is L with probability q A so vSH < p ≤ pA . This class of
equilibria does not satisfy the intuitive criterion because (ψ, H) could enter with the message
mψH = H and price slightly below p. This deviation is equilibrium-dominated for all the
other types.
(III) Assume neither (0, H) or (ψ, H) withhold. In equilibrium we cannot have p0H < pψH because
(0, H) can mimic (ψ, H). Furthermore, we cannot have pψH < p0H and m0H = H because
(ψ, H) could then mimic (0, H). There are therefore two possibilities: (A) pψH < p0H and
m0H = L or (B) pψH = p0H . We consider each case in turn.
(A) In this case (0, L) and (ψ, L) will pool with (0, H): p0H = p0L = pψL = p and m0H =
m0L = mψL = L. In equilibrium we cannot have mψH = L because (ψ, H) would deviate
and pool with the others. To be an equilibrium we must therefore have mψH = H,
vSH < pψH < p ≤ pA ,
(A18)
pψH − vSH ≥ p − vSH − ψ.
(A19)
and
This class of equilibria does not survive the intuitive criterion because (ψ, H) could
increase his price slightly and continue telling the truth.
25
(B) Finally, we consider the case pψH = p0H = p.
(1) If mψH = m0H = L then all types pool: pjk = p and mjk = L for all j ∈ {ψ, 0}
and k ∈ {H, L}. This is an equilibrium if p ≤ pP and (ψ, H) prefers not to exit
p − vSH − ψ > 0. This does not survive the intuitive criterion because (ψ, H) could
reduce his price slightly and tell the truth.
(2) If mψH = H and m0H = L then (ψ, L) pools with (0, H): pψL = p and mψL = L.
(a) If (0, L) also pools with (0, H) then the seller’s strategy is for all four types
to charge p, (ψ, H) tells the truth mψH = H, while the rest send the message
m0H = mψL = m0L = L. When the buyer observes p and L she believes the car
is L with probability q A . When she observes p and H she believes the car is H
with probability one. This is the class of equilibria in (ii) if vSH < p ≤ pA .
(b) If (0, L) pools with (ψ, H) then all four types charge p and mψH = m0L = H
and m0H = mψL = L. On the equilibrium path the buyer believes the car is L
with probability q B when she observes p and H and with probability q C when
she observes p and L so this is an equilibrium when vSH < p ≤ min{pB , pC }.
This is the class of equilibria in (iii).
(3) The case mψH = L and m0H = H cannot be an equilibrium.
(4) If mψH = m0H = H then (0, L) will pool with the H types.
(a) If (ψ, L) pools with the others then the seller’s strategy is for all four types to
charge p and send the message H. On the equilibrium path the buyer maintains
her prior with maximum willingness to pay pP . Unless
L
p − vSL − ψ ≥ vB
− vSL
(A20)
L and tell the truth mψL = L. We
(ψ, L) will deviate and charge pψL = vB
L + ψ. This is the
therefore have an equilibrium when vSH < p ≤ pP and p ≥ vB
class of equilibria in (iv).
L and mψL = L and the other types charge p and
(b) If (ψ, L) separates with pψL = vB
send the message H then the buyer believes that the car is L with probability
one when she observes the former and q S when she observes the latter. This is
L + ψ. This is the class of
an equilibrium provided that vSH < p ≤ pS and p ≤ vB
equilibria in (v).
This completes the proof.
26
Proof of Proposition 5. We first observe that pA < pP < pS because q S < ρ < q A . Furthermore,
q B R ρ and pB Q pP iff ν R 1/2 and q C R ρ and pC Q pP iff ν Q 1/2. It follows that min{pB , pC } =
pP iff ν = 1/2 and min{pB , pC } < pP otherwise. Next, we note that the optimistic separating
H achieves the first best expected profit RF when ψ ≥
equilibrium in Lemma 4(ii) with pψH = vB
H − v L and ν = 0. Any equilibrium which does not achieve RF is therefore suboptimal. This
vB
B
includes the lemons equilibria in Lemmas 3(i) and 4(i) and Proposition 4(i) as well as the equilibria
in Lemma 4(iii) and Proposition 4(iv) where deception costs are incurred. Since pA < pP the
optimistic equilibrium in Proposition 4(ii) with p = pA does not achieve RF . If pP > vSH then the
optimistic pooling equilibrium in Lemma 3(ii) with p = pP achieves RF . Similarly, if pP > vSH
the optimistic equilibrium in Proposition 4(iii) with p = pP and ν = 1/2 and ψ > 0 also achieves
RF . Finally, if pS > vSH the optimistic semi-separating equilibrium in Proposition 4(v) with p = pS
L.
achieves RF when ψ ≥ pS − vB
6
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