Aubert, Cécile

English

Research Papers in Economics

Work incentives and household insurance: Sequential contracting
with altruistic individuals and moral hazard∗
C´ ecile Aubert†
Abstract
Two agents sequentially contracts with diﬀerent principals under moral hazard. If agents
care for one another, the second principal gains by insuring them over ﬁrst wages. Even with
independent tasks, the ﬁrst principal must oﬀer riskier payments to induce eﬀort.
JEL Codes: D10, D64, D82.
Keywords: Incentives, Altruism, Moral Hazard.
1 Introduction
The interplay between altruism and incentives has been the object of many studies since the
seminal papers by Becker (1974, 1976, 1991) and Buchanan (1975). Most papers focus on
incentives between two individuals, one of which (at least) is altruistic with respect to the other.
Yet the existence of altruism between individuals may also aﬀect their incentives with respect
to a third party who is not altruistic nor the object of altruism. Consider a moral hazard
setting in which two individuals are hired by diﬀerent principals; if these individuals care for
∗I am very grateful to Jean-Jacques Laﬀont† for helpful discussions on this topic. I also thank Fran¸ cois Salani´ e
for very helpful comments, and seminar participants in Cambridge, Leicester, and Toulouse.
†Eurisco, Universit´ e Paris Dauphine. Address: D´ epartement Economie Appliqu´ ee, Universit´ e Paris Dauphine,
place de Lattre de Tassigny, F - 75 016 Paris, France. Tel: (33)1 44 05 45 23. Fax: (33)1 44 05 47 33. E-mail
address: Cecile.Aubert@dauphine.fr
1one another, their ability to insure each other modiﬁes incentive costs. Household members may
be such agents, especially if one obtains a position before the other does, and employers are
aware of this fact. Such ‘small’ employment markets may especially be found in rural, possibly
developing, areas or in small communities.
We consider such a setting where two agents are altruistic with respect to each other, and
choose to share their resources. Each is engaged in a contractual relationship with a risk-
neutral principal, under moral hazard. The individuals are not altruistic with respect neither
to their principal nor to the task they undertake (contrary to the literature on non-proﬁts or on
physicians1). Outcomes are learned sequentially. Tasks are assumed to be perfectly independent.
Yet the fact that altruistic agents share resources can be exploited by the second principal – the
one for which performance is known second: this principal will indeed ﬁnd it optimal to make
her wages depend on both outcomes, as both determines her agent’s revenues. She will thus
(at least partly) insure her agent against ﬁrst-income variations. Altruism creates a situation
similar to common agency with moral hazard2, as in Tommasi and Weinschelbaum (2004). The
ﬁrst principal must take into account the best response of the second principal; she must oﬀer
steeper contracts to induce eﬀort despite indirect insurance through the second agent.
2 The model
The players Two individuals, A and B, are altruistic with respect to each other. To simplify
and avoid making assumptions on how agents solve disagreements on resource sharing, we assume
1This literature has mostly considered settings in which agents are intrinsically motivated by the task they
undertake: Chon´ e and Ma (2004) consider physicians’ private information on their altruism with respect to
patients, and on the latters’ health; Francois (2003) and Rowat and Seabright (2005) analyze eﬀort incentives for
motivated employees in NGOs. Our setting strongly diﬀers as there is no intrinsic motivation, and principals act
strategically with respect to one another, something that cannot arise in these other papers.
2See Arnott and Stiglitz (1991) on self-insurance, and Bizer and DeMarzo (1992) and Bisin and Guaitoli (2004)
on multi-contracting in banking.
2that altruism appears in an additive separable way and that both agents have the same degree
of altruism. Their utility function is thus given by a ‘von Neumann-Morgenstern’-type bi-variate
function Ui(t,Uj) = u(t) + Uj, i 6= j, i,j = 1,2, where u(.) is a strictly concave and increasing
function, with u(0) normalized to zero. The individuals will thus always prefer to share equally
total resources.3
Each individual contracts with a diﬀerent, risk-neutral, principal: principal PA hires indi-
vidual A to exerts some unobservable eﬀort ea in some task a, in exchange for some transfer tA,
and PB does the same with individual B. Project a yields a veriﬁable return, Sa > 0 in case
of success and 0 in case of failure. Eﬀort ea is discrete (0 or 1) and non observable. If agent A
exerts a high eﬀort, ea = 1, he incurs a (non observable) disutility of eﬀort ψ and the probability
that the project is successful is p1. If he shirks on the other hand, and exerts eﬀort e = 0 only,
he incurs no disutility but the probability of success falls to p0 < p1.
To simplify, we assume that principal PB does not wish to induce eﬀort in task b. She thus
does not need to make payments depend on her agent’s performance (a null transfer would be
optimal if she was facing a non-altruistic agent). Hence, performance in this task can be assumed
to be certain, ¯ Sb. The analysis is similar, but expressions become more complicated, if PB also
requires an eﬀort to increase the probability of a high performance in task b, as four diﬀerent
states of nature, and additional incentive constraints, must be considered.
The two projects are completely independent, yet due to altruism, the transfer given by one
principal aﬀects the utility level obtained by the agent of the other principal: altruism creates
an indirect externality between principals. Once inter-agent transfers are taken into account,
the setting becomes formally similar to a sequential common agency game with a single agent.
3Full insurance of one individual here implies into full insurance of the other. This is a restriction, but it allows
highlighting the interlinkage between incentive contracts for independent tasks due to altruism. It should become
clear that our qualitative results would remain valid with less altruism.
3The timing Principal PA must pay her agent ﬁrst and cannot condition transfers on B’s
wage or contract. The timing4 is the following:
1. Principal PA oﬀers a contract to agent A, who accepts or refuses it.
2. Principal PB then oﬀers a contract to B, who accepts or refuses it.
3. If agent A has accepted his contract, he privately chooses eﬀort ea.
4. The outcome for task a is observed, and tA is publicly paid by PA.
5. Agent B is paid by PB, and the total payment is shared by B and A.
A benchmark Assume that the two agents are selﬁsh, and cannot insure each other. Then
principal Pa solves a standard moral hazard game by having both the participation and the
incentive compatibility constraint of her agent binding. She oﬀers transfers t
a in case of success
and ta in case of failure, where u(t
a) =
1−p0
p1−p0ψ and u(ta) = −
p0
p1−p0ψ.
3 The second-stage contract
If agent A has refused the contract oﬀered by principal PA, the problem is standard, except
that a transfer tB yields utility 2u(tB
2 ) to the agent. Since u(.) is strictly concave, 2u(x
2) > u(x),
and PB beneﬁts from altruism. The optimal contract gives a null expected utility to agents B
and A.
Assume on the other hand that A has accepted the contract oﬀer, and suppose for the
moment that principal PA prefers to induce eﬀort from her agent. She pays him t
A in case of
success, and tA in case of failure. Principal PB can observe and contract on the outcome of task
4Since the contract oﬀered by P
A is accepted or refused before principal P
B makes an oﬀer to B, P
A cannot
take advantage of the subsequent relationship between agent B and P
B in his dealings with A.
4a. Although tasks are independent, she should oﬀer transfers tB that depend on whether agent
A has succeeded (and obtained a high transfer t
a) or not.5
One might think that making the transfer depend on an unrelated factor increases the risk
the agent bears. However, principal PB can use transfers that are diﬀerentiated according to
performance in the other task so as to insure her agent, thereby reducing expected transfers
for this agent. This allows the risk-neutral principal to oﬀer negative expected transfers to her
agent, transfers that are nevertheless accepted.
The expected utility of individual B is indeed given by
peau(
t
A + tB(t
A)
2
) + (1 − pea)u(
tA + tB(tA)
2
). (1)
Let us denote by TB(ea) the certainty equivalent that gives him the same utility as when
tB(t
A) = tB(tA) = 0 for eﬀort ea in task a. From the strict concavity of u(.), TB(ea) <
peau−1(t
A
2 )+(1−pea)u−1(
tA
2 ). Principal PB can thus make proﬁts by oﬀering agent B transfers
that compensate for the payments of the other principal so as to ensure that the agent always
obtains the certainty equivalent: tB(t
A) = TB(ea)−t
A and tB(tA) = TB(ea)−tA. Fully insuring
her agent is thus always proﬁtable for the second principal. The result holds for any given eﬀort
level ea by individual A.
Lemma 1 Due to agents’ altruism for one another, the second principal ﬁnds it optimal to
condition payments on the outcome of the ﬁrst contract even with independent tasks and no
direct externalities.
There exists a potential conﬂict of interests between the two principals since the second
5If performance in task b was also random, and depended on some eﬀort e
b, principal P
B should oﬀer four
diﬀerent transfers, t
B
ab when both tasks succeed, t
B
a when only task B succeeds, similarly t
B
b when only A succeeds,
and t
B
∅ when both tasks fail. Separating between four states of nature allows insuring agent B against income
shocks that do not depend on his eﬀort e
b, and obtaining eﬀort at a lower cost.
5principal can exploit her agent’s risk aversion to her advantage.
Given that the agents equally share their resources, when agent B is perfectly6 insured, so
is agent a. The latter no longer has incentives to exert eﬀort, and chooses ea = 0.
This analysis is not suﬃcient, though, as the second principal may ﬁnd it even more proﬁtable
not to fully insure her agent, so as to still induce eﬀort from A, by satisfying an incentive
compatibility constraint. The incentives of the second principal depend on the variability of the
contract initially oﬀered, as we discuss below.
4 The ﬁrst-stage contract
• Let us ﬁrst consider the situation in which the ﬁrst principal, PA, oﬀers a contract that is not
steep enough to make it proﬁtable for the second principal to induce eﬀort in task a. As the
agent will not exert eﬀort, principal Pa will oﬀer a ﬁxed, null, transfer to her agent. Principal
PB then oﬀers the same contract as with a selﬁsh agent7. Although the transfers received by
agent B ex post do not depend on the outcome in task a, the possibility that PB diﬀerentiates
transfers according to more states imply that it can counter the incentive scheme designed by
the ﬁrst principal and induce no eﬀort, as long as these transfers are not steep enough.
• Assume now that principal PA oﬀer a wage scheme suﬃciently risky to ensure that the second
principal prefers to still induce eﬀort in the ﬁrst task (this scheme is characterized below). When
contracting with B, the second principal will have the incentive compatibility constraint with
respect to task a bind, to limit incentive costs:
(p1 − p0)
h
u(
t
A + tB(t
A)
2
) − u(
tA + tB(tA)
2
)
i
= ψ. (2)
6In a more general model in which sharing would be incomplete, agent A might not be perfectly insured when
B is, but it may still be the case that his incentive compatibility constraint be violated due to insurance by
principal P
B.
7It would not, obviously, be an equilibrium if the game was simultaneous.
6As the participation constraint will also bind, we obtain as usual that u(
t
A+tB(t
A)
2 ) =
1−p0
p1−p0ψ
and u(
tA+tB(tA)
2 ) = −
p0
p1−p0ψ. The gain obtained by PB will be
p1[t
A − u−1(
1 − p0
p1 − p0
ψ)] + (1 − p1)[tA − u−1(−
p0
p1 − p0
ψ)]. (3)
The second principal will prefer to induce eﬀort from the agent if and only if this expected gain
is larger than the expected gain when fully insuring the agent, i.e.,
p1t
A + (1 − p1)tA − u−1

p1u(
t
A
2
) + (1 − p1)u(
tA
2
)

. (4)
If the ﬁrst principal wants to obtain eﬀort is task a, she must satisfy an incentive compatibility
constraint with respect to the second principal. This constraint is
p1u(
t
A
2
) + (1 − p1)u(
tA
2
) ≥ u

p1u−1(
1 − p0
p1 − p0
ψ) + (1 − p1)u−1(−
p0
p1 − p0
ψ)

, (5)
and replaces the incentive compatibility constraint with respect to agent A. The ﬁrst contract
must thus be riskier.
The ﬁrst principal also faces a participation constraint with respect to principal PB:
p1[t
A − u−1(
1 − p0
p1 − p0
ψ)] + (1 − p1)[tA − u−1(−
p0
p1 − p0
ψ)] ≥ 0. (6)
Because incentive constraints are changed with respect to the benchmark situation of a selﬁsh
agent, the costs of inducing eﬀort are modiﬁed. The ﬁrst principal will therefore less often prefer
to induce a high eﬀort.
Proposition 1 Assume that contracts and outcomes are public, and that principals sequentially
contract with agents who are altruistic vis-` a-vis each other. The principal who acts ﬁrst must
oﬀer riskier contracts to induce eﬀort in equilibrium, despite subsequent contracting by the second
principal. A high eﬀort level will less often be induced.
75 Conclusion
Our setting, although speciﬁc, highlights interesting insurance eﬀects of altruism. If the two
principals were acting simultaneously, the game would be formally similar (once intra-household
transfers are taken into account) to that in Bernheim and Whinston (1986) but with delegated,
common agency. In both settings, the equilibrium payment is the minimum one necessary to
obtain the eﬀort level required. And in both, equilibria can be interpreted as resulting from
the delegation by one principal of her task to the other principal, via the contract oﬀered to
the agent. Here however, there is a single equilibrium and the identity of the principal who
delegates is no longer undetermined. If outcomes were private, our results would be more
drastically aﬀected: the second principal would face adverse selection regarding the transfer
paid to agent A. As the second principal’s transfer is a decreasing function of the ﬁrst, agents
would always underestimate tA when they have CARA preferences: the second principal could
not oﬀer insurance. Our setting is more relevant for ‘small’ employment markets, in which more
information is available, as mentioned in the introduction.
The externality created between principals by altruism remains striking: Even if no insurer
is available to the ﬁrst agent, insurance by his mate, possibly via another work contract, strongly
aﬀects eﬀort incentives. Recognizing the role of the environment (including the aﬀective environ-
ment) of the agent is thus essential. Our results may apply to situations in which an individual
obtains a position after his/her spouse, and the second employer is aware of the insurance prop-
erties of the second employment – married women employment decisions, in particular, are often
partly based on insurance arguments.
In such a context, a potential solution for the ﬁrst employer to restore incentives at a lower
cost is to use non monetary beneﬁts that are more diﬃcult to share – free or subsidized lunches,
8transportation to the work place, or health beneﬁts.
References
Arnott, R., and J. E. Stiglitz, 1991, Moral Hazard and Nonmarket Institutions: Dysfunc-
tional Crowding Out of Peer Monitoring?, The American Economic Review 81: 179-190.
Becker, G., 1974, A Theory of Social Interaction, Journal of Political Economy 82: 1063-1093.
Becker, G., 1976, Altruism, Egoism, and Genetic Fitness: Economics and Sociobiology,
Journal of Economic Literature 14: 817-826.
Becker, G., 1991, A Treatise on the Family (Harvard University Press, Cambridge, MA).
Bernheim, D., and M. Whinston, 1986, Common Agency, Econometrica 54: 923942.
Bisin, A., and D. Guaitoli, 2004, Moral Hazard and Nonexclusive Contracts Production:
Theories and Evidence from Joint Ventures, The RAND Journal of Economics 35: 306-328.
Bizer, D., and P. M. DeMarzo, 1992, Sequential Banking, The Journal of Political Economy
100: 41-61.
Buchanan, J., 1975, The Samaritan’s Dilemma, in E. Phelps, ed., Altruism, Morality, and
Economic Theory (Sage Foundation, New York, NY).
Chon´ e, P., and C-T. A. Ma, 2004, Asymmetric Information from Physician Agency: Optimal
Payment and Healthcare Quantity, mimeo, Crest-Lei and Boston University.
Francois, P., 2003, Not-for-Proﬁt Provision of Public Services, Economic Journal 115: 53-61.
Rowat, C., and P. Seabright, 2005, Intermediation by Aid Agencies, Journal of Development
Economics forthcoming.
Tommasi, M., and F. Weinschelbaum, 2004, Principal-Agent Contracts under the Threat of
Insurance, mimeo, Universidad de San Andr´ es.
9

A Theory of Social Interaction,

A Treatise on the Family

Altruism, Egoism, and Genetic Fitness: Economics and Sociobiology,

Asymmetric Information from Physician Agency: Optimal Payment and Healthcare Quantity, mimeo, Crest-Lei and

Intermediation by Aid Agencies,

Moral Hazard and Nonexclusive Contracts Production: Theories and Evidence from Joint Ventures, The RAND

Moral Hazard and Nonmarket Institutions: Dysfunctional Crowding Out of Peer Monitoring?,

Not-for-Proﬁt Provision of Public Services,

Sequential Banking,

The Samaritan’s Dilemma,

Work Incentives and Household Insurance: Sequential Contracting with Altruistic Individuals and Moral Hazard.

Work Incentives and Household Insurance: Sequential Contracting with Altruistic Individuals and Moral Hazard.

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