The paper analyses, within a moral hazard scenario, a contract between an agent with anticipatory emotions and a principal who responds strategically to those emotions. The agent receives a private signal on the profitability of the task he was hired for. If the signal is informative about the return from effort, the agent would benefit from knowing accurate news. However, if the agent derives utility from the anticipation of his final payoff, the suppression of a bad signal may induce a positive interim emotional effect. We show that it may be impossible to achieve the first-best, even though the risk-neutral parties are symmetrically informed at the contracting stage and complete contracts can be written.Hidden action, anticipatory utility.

Annamaria Menichini

Giovanni Immordino

Maria Grazia Romano

English

Research Papers in Economics

 
 
 
W WO OR RK KI IN NG G   P PA AP PE ER R   N NO O. .   2 26 62 2 
 
A Simple Impossibility Result  
in Behavioral Contract Theory 
 
 
 
Giovanni Immordino, Annamaria C. Menichini and Maria Grazia Romano 
 
  
October  2010  
 
   
 
 
 
 
 
University of Naples Federico II 
 
University of Salerno 
 
Bocconi University, Milan 
CSEF - Centre for Studies in Economics and Finance  
DEPARTMENT OF ECONOMICS – UNIVERSITY OF NAPLES 
80126  NAPLES - ITALY 
Tel. and fax +39 081 675372 – e-mail: csef@unisa.it  
 
 
W WO OR RK KI IN NG G   P PA AP PE ER R   N NO O. .   2 26 62 2 
 
 
 
A Simple Impossibility Result  
in Behavioral Contract Theory 
 
 
 
Giovanni Immordino
*, Anna Maria C. Menichini
*, Maria Grazia Romano
* 
 
 
 
Abstract 
The paper analyses, within a moral hazard scenario, a contract between an agent with anticipatory emotions and a 
principal who responds strategically to those emotions. The agent receives a private signal on the profitability of the task 
he was hired for. If the signal is informative about the return from effort, the agent would benefit from knowing accurate 
news. However, if the agent derives utility from the anticipation of his final payoff, the suppression of a bad signal may 
induce a positive interim emotional effect. We show that it may be impossible to achieve the first-best, even though the 
risk-neutral parties are symmetrically informed at the contracting stage and complete contracts can be written. 
  
JEL Classification: D86. 
 
Keywords: Hidden action, anticipatory utility. 
 
 
 
*   Università di Salerno and CSEF Table of contents 
 
 
 
1.  Introduction 
2.   Model 
3.  Effort choice and memory strategy 
4.  A simple impossibility result 
References 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 1 Introduction
Forward-looking agents care about expected utility ￿ ows, and enjoy anticipatory utility if they are
optimistic about the future. Due to imperfect memory, they may then choose their beliefs so as
to enjoy the greatest comfort or happiness (cognitive dissonance).1 Building on the motivated-
beliefs and optimal awareness framework, this paper studies a contract between an (endogenously)
optimistic agent and a realistic principal. Upon receiving a private signal about the pro￿tability
of the task he is hired for, the agent has to choose the level of e⁄ort to exert that a⁄ects the
probability of success of the project. Although the risk-neutral parties are symmetrically informed
at the contracting stage and complete contracts can be written, we show that it may be impossible
to achieve the ￿rst-best, unless the weight on emotions is su¢ ciently low.
Thus, as several other works in the contract-theoretic literature, this paper also derives an
impossibility result (Myerson and Satterthwaite, 1983; Akerlof, 1970, among others). The novelty
of the paper is that this result is derived within a behavioral model that in recent years has been
applied in a variety of contexts, among which asset-pricing (Caplin and Lehay, 2001; Kuznitz et al.,
2008), health economics (Caplin and Lehay, 2004; Koszegi, 2006), policy design (Caplin and Eliaz,
2003), managerial compensation (Immordino et al., 2010) and theories of moral behavior (BØnabou
and Tirole, 2010).
2 Model
Consider a setting in which a risk neutral principal hires a risk neutral agent for a project that has
three possible outcomes, ~ v 2 f0;vL;vHg, with 0 < vL < vH. In carrying out the task he is hired for,
the agent chooses a level of e⁄ort a a⁄ecting the probability of success, with a 2 [0;1]. The e⁄ort
has disutility c(a), with c(0) = 0, c0(a) > 0 and c00(a) > 0. In order to ensure interior solutions,
we also assume that c0(0) = 0 and c0(1) ￿ vH. After signing the contract but before choosing the
1See for instance Akerlof and Dickens (1982), Loewenstein (1987), Caplin and Leahy (2001), BØnabou and Tirole
(2002, 2003, 2010), Koszegi (2006), among others.
2e⁄ort level, the agent receives a private signal ￿ 2 fL;Hg correlated with the project￿ s return ~ v.
The probability of a good signal is q; with q 2 [0;1]: In our setting, good (bad) news means that
the outcome is vH (vL) or v0 with probability a and 1 ￿ a; respectively, i.e.,
Pr(~ v = vLj￿ = H) = Pr(~ v = vHj￿ = L) = 0:
Then, ex-ante v0;vL and vH can each arise with probabilities 1 ￿ a;(1 ￿ q)a and qa, respectively.
Given that the signal is informative about the return from e⁄ort, the agent would bene￿t from
knowing accurate news when choosing a. However, if the agent derives utility from the anticipation
of his ￿nal payo⁄, the suppression of a bad signal may induce a positive emotional e⁄ect. This is
modeled assuming that total utility is a convex combination of actual (as of time 3) and anticipated
physical outcomes (as of time 2); with weights 1 ￿ s and s; respectively, with s 2 [0;1].
At the time of the e⁄ort decision, the recollection of a good signal is always accurate, whilst
a bad signal can be forgotten due to voluntary repression. Denote by b ￿ 2 fb L; b Hg the recollection
of the news ￿ and by ￿ 2 [0;1] the probability that bad news will be remembered accurately
(￿ ￿ Pr(b LjL)). Finally, denote by Agent 1 the agent￿ s self as of time 1 and by Agent 2 the agent￿ s
self as of time 2.
The agent￿ s action is not directly observable by the principal, who can o⁄er a contract
C ￿ fw0;wL;wHg with rewards contingent on the observable and veri￿able project revenues.
Finally, we maintain the standard assumption of individuals as rational Bayesian information
processors.
The time-line is the following:
t=0: The principal o⁄ers a contract to the agent.
t=1: If Agent 1 refuses, the game ends. If he accepts, he observes a private signal ￿ and, when
bad (￿ = L), chooses the probability ￿ that news will be remembered accurately.
t=2: Agent 2 observes b ￿, updates his beliefs on the outcome v, selects the e⁄ort level a and
enjoys the anticipatory utility.
3t=3: The project payo⁄ is realized and the payment is executed.
First-best benchmark. When both the action and the signal are observable, the e¢ cient memory
strategy and e⁄ort levels are obtained by maximizing the ex-ante total surplus
￿
aFB
L^ L ; aFB
L ^ H; aFB
H ^ H; ￿FB
￿
2 argmaxE0 [~ v ￿ c(a)ja;￿] ￿ (1)
￿ q
￿
aH ^ HvH ￿ c
￿
aH ^ H
￿￿
+ (1 ￿ q)
￿
(1 ￿ ￿)
￿
aL ^ HvL ￿ c
￿
aL ^ H
￿￿
+ ￿
￿
aL^ LvL ￿ c
￿
aL^ L
￿￿￿
;
where a￿^ ￿ is the e⁄ort exerted by the agent when observing ￿ and recollecting ^ ￿. Solving problem
(1) gives three sets of ￿rst-best outcomes
￿FB 2 [0;1]; aFB
L^ L = aFB
L ^ H = aFB
L ; aFB
H ^ H = aFB
H ; (2)
￿FB = 0; aFB
L^ L 2 [0;1]; aFB
L ^ H = aFB
L ; aFB
H ^ H = aFB
H ; (3)
￿FB = 1; aFB
L^ L = aFB
L ; aFB
L ^ H 2 [0;1]; aFB
H ^ H = aFB
H ; (4)
where aFB
i is such that c0 ￿
aFB
i
￿
= vi, for i = L; H.
In the following, the e⁄ort choice is a hidden action and Agent 1 observes a private signal about
the project pro￿tability which he may choose to forget at the time of the e⁄ort decision. When
e⁄ort is chosen by Agent 2 without observing ￿, (2) and (3) can never be implemented. Indeed,
when the signal is not observable to all parties, e⁄ort can only be contingent on signal recollection
(and not on the actual signal). Therefore, the ￿rst-best could be implemented only if the contract
gives Agent 1 the incentive to perfectly recollect the signal as in (4).
Next, we characterize the optimal incentive scheme for each e⁄ort level and memory strategy
that the principal may want Agent 2 and Agent 1 to select. To this aim we analyze the optimal
e⁄ort choice a, given Agent 2￿ s beliefs about ￿ and describe the Perfect Bayesian Equilibrium of
the memory game. Finally, we use the agent￿ s optimal e⁄ort choice rule and the memory game
equilibrium to show our impossibility result.
43 E⁄ort choice and memory strategy
We start by describing the incentive problem faced by the principal to induce Agent 2 to choose the
desired level of e⁄ort. Denoting by E2 the expectation at t = 2; the intertemporal utility perceived
by Agent 2, given the memory b ￿ and the contract C, is
E2 [U3] = ￿c(a) + E2 [u(C;a)jb ￿]; (5)
where E2 [u(C;a)jb ￿] is the sum of the agent￿ s material payo⁄, (1 ￿ s)E2 [u(C;a)jb ￿], and his
anticipatory utility (experienced by savoring the future material payo⁄), sE2 [u(C;a)jb ￿]. This
is equal to
E2[u(C;a)jb L] = awL + (1 ￿ a)w0;
when b ￿ = b L, and to
E2[u(C;a)j b H] = a(rwH + (1 ￿ r)wL) + (1 ￿ a)w0;
when b ￿ = b H; where r is the posterior probability attached to state H.
Agent 2 chooses the level of e⁄ort that maximizes his intertemporal expected utility. Thus, to
induce e⁄orts ab L and a b H from an agent who recalls b L and b H respectively, the contract has to be
such that
wL ￿ w0 = c0 ￿
ab L
￿
; (6)
and
rwH + (1 ￿ r)wL ￿ w0 = c0 ￿
a b H
￿
: (7)
We now consider the incentive problem faced by the principal to induce Agent 1 to correctly
recall the signal. To this aim, we describe the Perfect Bayesian Equilibrium of the memory game
for given contract C. In the PBE: i) for any realized ￿, Agent 1 chooses his message b ￿ to maximize
his expected utility, correctly anticipating what inferences he will draw from b ￿, and what action he
will choose; ii) Agent 2 forms his beliefs using Bayes rule to infer the meaning of Agent 1￿ s message,
knowing his strategy.
5Agent 2 is aware that there are incentives in manipulating memory when the true state is L.
Hence, faced with a memory b H, he assesses its credibility. If he thinks that the bad signal is recalled
with probability ￿, using Bayes￿rule he computes the likelihood of an accurate signal recollection
as
r(￿) ￿ Pr(Hj b H;￿) =
q
q + (1 ￿ q)(1 ￿ ￿)
2 [q;1]: (8)
If Agent 1 observes ￿ = L; he chooses the probability of remembering the signal so as to
maximize his expected utility
max
￿2[0;1]
E1 [U(C;a(r;C);￿)] ￿ E1[￿c(a(r;C)) + sE2 [u(C;a(r;C))]] + (1 ￿ s)E1 [u(C;a(r;C))]; (9)
where E1 denotes expectations at t = 1 and a(r;C) is the optimal strategy of Agent 2 for given r
and C 2 R3
+.
Bayesian rationality implies that Agent 2 knows that Agent 1 is choosing the recalling strategy
according to (9), and uses this optimal ￿ in his inference problem. A PBE of the memory game is
a pair (￿?; r?) 2 [0; 1] ￿ [q; 1] that solves (8) and (9).2
In order to induce perfect recall (￿ = 1) the contract must be such that
E1 [U(C;a(r;C);1)jL] ￿ E1 [U(C;a(r;C);￿)jL]: (10)
At equilibrium r = r(￿) and, from (7), the optimal e⁄ort when recalling b H depends on ￿ through
r(￿): Thus, at equilibrium a(r;C) = a(￿;C) ￿
￿
ab L;a b H(￿)
￿
and condition (10) simpli￿es to
c
￿
a b H(￿)
￿
￿ c
￿
ab L
￿
extra cost of e⁄ort
￿ sa b H(￿)r(￿)(wH ￿ wL)
emotional gain from forgetting
+
￿
a b H(￿) ￿ ab L
￿
(wL ￿ w0)
indirect gain from higher e⁄ort
: (11)
The agent has an incentive to remember when, for any ￿ < 1, the extra-cost he incurs to exert e⁄ort
a b H(￿) rather than ab L exceeds the emotional gain from forgetting due to the uncertainty about the
payment in case of success, plus the gain due to obtaining wL rather than w0 with an increased
probability
￿
a b H(￿) ￿ ab L
￿
. Notice that the incentive to forget is positively correlated with s.
2To simplify notation, we will omit the star superscript.
64 A simple impossibility result
The principal￿ s problem boils down to the choice of e⁄ort levels a b H;ab L, recall probability ￿,
and payments wH;wL and w0 that maximize his expected pro￿t subject to Agent 2￿ s incentive
constraints (6) and (7), and Agent 1￿ s recalling constraint (11). Moreover, when the principal
makes his o⁄er, the agent does not know ￿. To induce him to accept it, the contract has to satisfy
the following ex-ante participation constraint
E0 [U (C;a;￿)] = qE1 [U(C;a;￿)j￿ = H] + (1 ￿ q)E1 [U(C;a;￿)j￿ = L] =
= [w0 + qa b H (wH ￿ w0) + (1 ￿ q)
￿
(1 ￿ ￿)a b H + ￿ab L
￿
(wL ￿ w0)]+ (12)
￿[(1 ￿ ￿(1 ￿ q))c(a b H) + ￿(1 ￿ q)c
￿
ab L
￿
] ￿ 0;
where E0 denotes the expectation at t = 0, qaH = Pr0 (vHja; ￿), (1 ￿ q)((1 ￿ ￿)aH + ￿aL) =
Pr0 (vLja; ￿), and ￿(1 ￿ q) = Pr0(b Lj￿).
It should be emphasized that if the agent recalls the signal, the ￿rst-best is still attainable even
when e⁄ort is unveri￿able, as standard in principal-agent models with risk-neutrality and unlimited
liability. The same is true if e⁄ort is veri￿able but the agent may forget a bad signal. For instance,
from (11) it follows that a ￿ at contract, which satis￿es the participation constraint at the ￿rst-best
levels of e⁄ort (wH = wL = w0 = qc(aFB
H ) + (1 ￿ q)c
￿
aFB
L
￿
), removes all incentives to forget bad
news. Then,
Proposition 1 With hidden action only or with a forgetful agent only the parties would always
write a contract that implements the ￿rst-best.
In contrast, although parties are symmetrically informed at the contracting stage and can write
complete contracts specifying state-contingent rewards, e⁄ort levels contingent on recollected signals
and the memory strategy, there is no contract implementing the ￿rst-best. The next proposition
states that the simultaneous presence of hidden action and forgetful agent leads to a simple and
novel impossibility result.
7Proposition 2 If the weight on anticipatory utility s is su¢ ciently high, there is no contract that
implements the ￿rst-best.
Proof: Substituting a b H = aFB
H , ab L = aFB
L and ￿ = 1 in incentive constraints (6) and (7) we
work out the premiums for success ￿wFB
H = (wH ￿ w0) and ￿wFB
L = (wL ￿ w0) that implement the
￿rst-best levels of e⁄ort given perfect recall. Substituting ￿wFB
H and ￿wFB
L in (11), the recalling
constraint becomes
c
￿
a b H (￿)
￿
￿ (caFB
L ) ￿ s
￿
a b H(￿)c0 ￿
a b H(￿)
￿
￿ aFB
L c0(aFB
L )
￿
+ (1 ￿ s)(a b H(￿) ￿ aFB
L )c0(aFB
L ); (13)
where a b H(￿) is the level of e⁄ort chosen by a forgetful agent recalling b H and is such that
c0(a b H (￿)) = r(￿)￿wFB
H + (1 ￿ r(￿))￿wFB
L :
Then, if s = 1, (13) becomes
aFB
L c0 ￿
aFB
L
￿
￿ c
￿
aFB
L
￿
￿ a b H (￿)c0 ￿
a b H (￿)
￿
￿ c
￿
a b H (￿)
￿
;
and this is never true since a b H (￿) > aFB
L and the function h(a) = ac0 (a) ￿ c(a) is increasing in a
(h0(a) = ac00(a) > 0 by assumption). By continuity, Agent 1 also prefers to forget bad news for s
close to 1. ￿
Acknowledgements: We are indebted to Alberto Bennardo for comments and suggestions.
Usual disclaimers apply.
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A Simple Impossibility Result in Behavioral Contract Theory

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