379
Principal–Agent Contracts under the Threat
of Insurance
by
M ARIANO T OMMASI
AND
F EDERICO W EINSCHELBAUM∗
We show that standard principal–agent results are not robust to the introduction
of additional contracting opportunities for the agent. We analyze extended games
including additional players who might trade risk away from the agent. For some
settings and parameter values, the principal is worse off, and total welfare is lower.
In some cases lower effort is implemented. The principal’s contract, when high
effort is implemented, is steeper than in the standard model. In some settings,
the agent unwinds part of those incentives through additional trades. These findings may call for a revision of some previous theoretical and applied conclusions.
(JEL: C 72, D 82, J 33)
1 Introduction
Most of agency theory proceeds under the implicit assumption that the relationship between principal and agent can be isolated from other trading opportunities.
This includes the assumption that the consumption vector of the agent across states
of nature is the one stipulated in the contract with the principal. In the standard
principal–agent contract the principal can give incentives to a risk-averse agent
only at the cost of exposing her to some risk – indeed, the minimum amount of
risk compatible with the incentive to provide high levels of effort. (The standard
reference for moral hazard is HOLMSTROM [1979].) If the agent were able to reduce her risk exposure just marginally, the standard incentive contract would be in
trouble.
There are different ways in which the agent could alter the consumption lottery
induced by the contract with the principal, and hence alter her willingness to exert
high levels of effort. Additional income would suffice to affect her marginal risk
∗ A previous version of this paper has circulated with the title “The Threat of Insurance: On the Robustness of Principal–Agent Models.” We received valuable comments from two anonymous referees, as well as from Federico Echenique, Bryan Ellickson, Hugo Hopenhayn, Hernán Ianello, David Levine, Alejandro Manelli, David
Perez-Castrillo, Jorge Streb, and Bill Zame, and from participants at various seminars.
We received very valuable suggestions and excellent research assistance from Ignacio
Esponda.
Journal of Institutional and Theoretical Economics
JITE 163 (2007), 379–393 © 2007 Mohr Siebeck – ISSN 0932-4569
380
Mariano Tommasi and Federico Weinschelbaum
JITE 163
aversion (except for some specific preferences); saving and borrowing would have
the same effect in a dynamic principal–agent model (C HIAPPORI et al. [1994]).
More to the point of the modeling in this paper, agents might have explicit trading
opportunities allowing them to get rid of some of the risk provided by the principal’s
contract.
This paper investigates how the solution to the standard moral-hazard principal–
agent problem is affected if the agent has the opportunity to engage in side contracts
with third parties who can provide some insurance. The analysis is related to, but
different from, the literature of nonexclusive contracts in insurance markets.1 That
literature deals with homogeneous “principals,” none of whom has a productive
technology, and concentrates on what we call the aggregate contract. In our paper
there is a principal who has a productive technology and employs the services of
the agent, while the other players can only provide insurance. (In our setup there
is nothing to insure unless the agent has signed a contract with the principal.) We
focus on the shape and characteristics of the contract signed by the principal and
the agent, and on how the productive relationship (the level of effort provided in
equilibrium, the distribution of the surplus) changes with the presence of these
outside opportunities. Substantively, our paper is better suited to study applications
of incentive contracts to employment relationships, franchising, sharecropping, and
the like; while other papers are better suited to study insurance markets, financial
markets, and the like.
We work out the case with two levels of effort and a continuum of outcomes, under
noncontractible effort. We examine what happens if there are additional players with
whom the agent can contract. The standard second-best contract is not robust to the
introduction of further trading opportunities, since the risk born by the agent in
the second-best contract creates a surplus for an agent–insurer relationship. After
showing that initial nonrobustness result, we characterize equilibria of different
extended games. These settings, with varying number of insurers and sequence of
play, allow us to study incentive contracting across environments with different
outside trading opportunities. The existence of these trading opportunities increases
the cost of inducing high effort in some cases. Due to this higher cost, for some
model specifications low effort is implemented in equilibrium for some parameter
values in which the standard principal–agent model would implement high effort.
When high effort is indeed implemented, the contract offered by the principal is
steeper than the second-best contract.
Other papers have explored situations of nonexclusive contracting in which the
agent has additional trading opportunities. These further moves in the extensiveform game come under different shapes.
In FUDENBERG AND TIROLE [1990] the new opportunity is a renegotiation with
the principal after choosing effort. In that case the traditional contract is not robust,
1
The literature on moral hazard with nonexclusive contracts starts with a sequence of papers by Arnott and Stiglitz (collected in A RNOTT AND STIGLITZ [1993])
and HELLWIG [1983], and has been recently advanced by K AHN AND MOOKHERJEE
[1998], BISIN AND GUAITOLI [2004], and BISIN AND RAMPINI [2006].
(2007)
Principal–Agent Contracts
381
since after effort the principal would rather offer full insurance to the agent. Foreseeing that, the agent would provide low effort; this forces the principal to act
otherwise than in the standard case. One key difference with our setup is their
timing assumption, in which some contracts are signed after effort, transforming
the original moral-hazard problem into an adverse-selection one. Other aspects of
the setup are equivalent: the second incarnation of the principal, “the renegotiator,”
is analogous to our insurer; and their renegotiation-proofness requirement plays
a similar role to our noninsurability constraint.
In CHIAPPORI et al. [1994] the agent has the possibility of making intertemporal
trades via saving and borrowing. Some of our results based on intratemporal trade
across states (insurance) resemble some of their results based on intertemporal
financial opportunities.
KAHN AND MOOKHERJEE [1998] study the effects of nonexclusive credit or insurance contracts in moral-hazard economies with hidden actions. They find that
competition between firms can induce a reduction in customers’ effort, and that the
lack of coordination among insurers may affect the cost of implementation even
without affecting effort levels. They analyze a case of sequential contracting in
which the agent has all the bargaining power. Unlike them, we give the principal the
bargaining power in the productive relationship, as standard in the principal–agent
literature – and as empirically more likely in most employment and franchising
relationships.
BISIN AND GUAITOLI [2004] analyze a case where “intermediaries” design and
offer contracts simultaneously. They show, as we do, that the optimal action is not
implemented in equilibrium for some economies. They also show that whenever the
equilibrium contracts implement the optimal action, intermediaries make positive
profits and equilibrium allocations are inefficient.
Those previous papers study the case of free entry; one of the main contributions of this paper is to consider cases with a finite number of insurers. We show,
among other things, that with a finite number of principals, the extensive-form
game has an effect on the region of parameters in which high effort is implemented.
The remainder of the paper is organized as follows. In section 2 we show nonrobustness of the traditional second-best equilibrium. In section 3 we present
a generic game with the explicit presence of other players and some general properties of equilibria in such a game, and we introduce a third constraint into the
design problem, a noninsurability condition under which a contract that implements
high effort is robust to the presence of additional players. That third-best contract
is more costly to the principal than the traditional second best. We also present the
game with free entry of insurers and show that in that case the principal pays the
third-best cost. In section 4 we consider settings with a finite number of insurers, in
which the principal manages to pay less than the third-best cost, and in some cases
even the second-best cost. That section shows that the details of the extensive form
of the game do affect the properties of the resulting equilibria. Section 5 provides
concluding remarks.
382
Mariano Tommasi and Federico Weinschelbaum
JITE 163
2 Nonrobustness of the Standard Results
We consider a traditional principal–agent problem where a principal hires an agent
who bears a cost to provide (noncontractible) effort. The possible outcomes, given
the agent’s effort, are distributed in the set Π ⊂ IR with density f(π|e) > 0 for all
π ∈ Π. There are two levels of effort, e ∈ {el , eh }, low and high effort respectively.
benefits under high effort are greater than under low effort:
The principal’s gross
π f(π|eh ) dπ > π f(π|el ) dπ. The sequence of the game is the following. The
principal offers a contract wp : Π → IR to the agent. The agent accepts or rejects the
contract. Rejection gives the agent reservation utility u . Upon acceptance, the agent
must decide on the level of effort. Given the level of effort, payoffs are realized
according to the density f(π|e).
The agent maximizes expected utility2
Ua (w(π), e) = v(w(π)) f(π|e) dπ − g(e) ,
where the cost of high effort is strictly greater than the cost of low effort, i.e.,
g(eh ) > g(el ). Additionally, v > 0 and v < 0, so that the agent is strictly riskaverse, and v(0) − g(el ) < u , so that the agent will not work for free.
The solution concept is subgame-perfect equilibrium (SPE). Following GROSSMAN AND HART [1983], the problem of the principal can be decomposed in two
steps. First, he finds the optimal incentive scheme for each level of effort; then he
selects the profit-maximizing level of effort. Once step 1 is performed, step 2 is
trivial. In the rest of the paper, unless otherwise noted, we concentrate on step 1
and assume that the result of step 2 is implementation of high effort. The optimal
incentive scheme for implementing e must solve
min w (π) f (π|e) dπ
w(π)
subject to
(PC)
(IC)
v (w (π)) f (π|e) dπ − g (e) u ,
e ) dπ − g (
e) ,
e solves max v(w (π)) f (π|
e
where PC is the participation constraint and IC is the incentive (compatibility)
constraint.
Let φ ≡ v−1 (·). The minimum cost of implementing el is achieved by offering
wp = φ [g(el ) + u], a flat wage that makes the agent just indifferent between accepting
and rejecting the contract.
The cost-minimizing contract that implements eh is the solution to the two binding
constraints PC and IC, and to the first-order conditions. This is known as the
second-best solution, w2nd , and it is unique given our assumptions (see, for example,
MAS-COLELL , WHINSTON, AND GREEN [1995]).
2
Her utility function is u a (w, e) = v(w) − g(e).
(2007)
Principal–Agent Contracts
383
Given g(eh ) > g(el ) and the fact that the agent is risk-averse, it is straightforward
to show that
(1)
w2nd (π) f(π|eh ) dπ > φ [g(el ) + u] > 0 ;
that is, the cost of implementing eh is strictly greater than the cost of implementing el .
Denote these minimum costs by c2nd (eh ) and c2nd (el ), respectively.3
The traditional principal–agent model implicitly assumes that either the agent
has an “exclusivity” contract or that there is no other player who, with the same
information as the principal (i.e., observing outcomes but not effort), can provide
the agent with insurance. We say a contract is robust to the presence of a third (riskneutral) player if there are no profitable opportunities for third players to contract
with the agent in order to take her from high to low effort. We now show our
nonrobustness result.
T HEOREM 1 The second-best contract is not robust to the presence of a third
(risk-neutral) player.
P ROOF Using the fact that the second best contract satisfies PC and IC with equality
and Jensen’s inequality, we get
2nd
2nd
w (π) f (π|el ) dπ
u + g (el ) = v w (π) f (π|el ) dπ < v
Then there exists a ρ > 0 such that
v
w2nd (π) f(π|el ) dπ − ρ = u + g (el ) .
If a risk-neutral player pays w2nd (π) f (π|el ) dπ − ρ − w2nd (π o ) when the profits
are π o , then the risk-neutral player will get an expected profit of ρ (the risk premium),
and the agent will accept the contract.
Q.E.D.
Thus, under the threat of insurance the principal will not offer the second-best
contract. In the rest of the paper, we study an explicit model of the principal–
agent–insurers problem under different settings. We study the characteristics of the
principal’s contract, and pay special attention to the conditions under which the
implementation of high effort is indeed costlier for the principal when we allow for
the existence of these risk-neutral players, whom we call, from now on, the insurers.
3 The Principal–Agent–Insurers Model
3.1 Generalities
Let I = {1, 2, ..., N} denote the set of insurers (where N is infinity in the free-entry
case). The set of players is { p} ∪ I ∪ {a} , where p is the principal and a the agent. In
3
Given the optimal incentive scheme for each level of effort
(step 1), the principal now chooses the level of effort that maximizes his profits, π f(π|e) dπ − c2nd (e)
(step 2).
384
Mariano Tommasi and Federico Weinschelbaum
JITE 163
all the settings we will consider, the players move sequentially, the principal moves
first, and the insurers and the principal have the same information (they can observe
and contract on outcomes, but not efforts).
We denote by w j the contract offered by party j ; the amount that the agent
receives when the output is π is w j (π). We say a player (other than the agent) is
active if he signs a contract w = 0 with the agent. Denote the set of active players
Πi (e), i ∈ I, stand for insurer i ’s benefits from implementing e,4 and let
by A. Let
w A (π) = i∈A wi (π) represent the aggregate contract.
L EMMA 1 Any subgame-perfect equilibria of this game must satisfy the following
properties: (1) The aggregate cost of implementing effort e is at least as great as the
cost in the traditional principal–agent problem. (2) The principal is never better off
than in the traditional problem. (3) In any case where the agent chooses el or eh ,
the principal is always an active player.
P ROOF (1): Suppose the statement is not true. Then the principal could mimic such
an aggregate contract in the traditional principal–agent problem and pay less to
implement e than the minimum found in section
2. We know that is not true.
w A (π) f(π|e) dπ = wp (π) f(π|e) dπ
(2):
From
the
previous
point
we
know
that
− i∈I Πi (e) ≥ c2nd (e). If the principal were better off, at least one insurer would
be making negative profits. This is not possible in equilibrium.
(3): Notice that, ex ante, insurers are not interested in the agents’s effort. This
implies that, by themselves, they can never offer a profitable contract that gives
an agent incentives to choose e. The principal must be active in every such case.
e is greater
Formally, we know from (1) that the aggregate
cost of implementing
dπ
=
w
(π)
f(π|e)
dπ
−
Π
(e)
> 0. If the
than zero, so that w A (π) f(π|e)
p
i
i∈I
principal is not active, then wp (π) f(π|e) dπ = 0, and at least one insurer makes
negative profits, which is not possible in equilibrium.
Q.E.D.
Lemma 1 says that the introduction of these new players can add nothing to our
previous economy. If our insurers had an informational advantage over the principal,
such that they could observe the agent’s level of effort, then the principal could be
better off, as in ITOH [1993] and ARNOTT AND STIGLITZ [1991]. The fact that the
principal is always an active player in our setup is quite relevant, given our focus
on situations in which the first relationship is a productive one. This focus is one of
the features that distinguishes our contribution from those of other recent papers on
nonexclusive contracting.
Having established these preliminary results, we now turn to consider the type of
contracts that might implement high effort in the model with insurers.
4 We say a principal or an insurer implements e if he makes an offer (maximizing profits taking into account both past offers and the best responses of players moving next) such that the agent accepts his contract (and possibly others) and chooses
effort e.
(2007)
Principal–Agent Contracts
385
3.2 Contracts Robust to the Presence of Insurers
We know from section 2 that the second-best contract is not robust in our game. In
this subsection we consider whether there are contracts that implement high effort
in the presence of insurers. Such (robust) contracts would satisfy the condition that
the profits for an insurer taking the agent from high to low effort are not positive.
Thus, a contract that intends to implement high effort is robust to a third risk-neutral
player if and only if the following condition holds:5
w(π) f(π|el ) dπ − g(el ) ,
(2) max u, v (w(π)) f(π|eh ) dπ − g(eh ) ≥ v
When insurers are
present, the agent’s utility under low effort is no longer given
by the expression v(w(π)) f(π|el ) dπ − g(el ), but now takes the form of the righthand side of (2). Then (2) requires that choosing low effort be not the (only) best
alternative. Notice that we have allowed the possibility that the principal leaves the
agent above her reservation utility. This could be a possibility when insurers are
present. When the contract for high effort does not offer the agent more than her
reservation utility, then the condition (2) is replaced by
(NI)
u≥v
w(π) f(π|el ) dπ − g(el ) ,
where NI stands for “noninsurability constraint.”6 NI implies equation (2). Thus, NI
is a sufficient condition for robustness. To aid the intuition, it is helpful to rewrite
NI as follows:
φ (u + g(el )) ≥ w(π) f(π|el ) dπ .
This equation means that the contract pays for low effort no more than what the
agent requires to provide low effort. In such a case, the potential profits for an insurer
taking the agent to low effort could not be positive.
For brevity, in the rest of the paper we concentrate on cases in which PC is
binding. Thus, NI is the relevant restriction.
5
A formal proof of this statement is provided in the working-paper version.
A sufficient condition for the agent being offered just her reservation utility is that
she has nondecreasing absolute risk aversion. In some of the settings analyzed below,
in which the principal manages to overcome the presence of insurers at no additional
cost, the agent will be at her reservation utility, and NI will be necessary and sufficient
for any specification of the utility function.
6
386
Mariano Tommasi and Federico Weinschelbaum
JITE 163
We now turn to the analysis of the optimal contract when NI is an additional
constraint. The optimal incentive scheme for implementing eh must solve
min w(π) f(π|eh ) dπ
w(π)
subject to
(PC)
(IC)
(NI)
v(w(π)) f(π|eh ) dπ − g(eh ) ≥ u ,
v(w(π)) f(π|eh ) dπ − g(eh ) ≥ v(w(π)) f(π|el ) dπ − g(el ) ,
u ≥v
w(π) f(π|el ) dπ − g(el ) .
Notice that this is the traditional problem for implementing high effort, with an
additional constraint, NI. The solution to this problem will be different from that of
the traditional one, since we know from Theorem 1 that the traditional result is not
robust to the presence of insurers.
Given strict risk aversion, NI and PC imply that IC is not binding.
Next we provide a characterization of the optimal incentive scheme under those
constraints, which we call the third-best contract.
P ROPOSITION 1 The optimal contract w3rd (π) is fully characterized by the solution
to constraints PC and NI binding, and to the following first-order condition:
λ
1
=
(FC)
,
f(π|el )
v (w3rd (π))
1+δ
f(π|eh )
where λ > 0 and δ > 0 are the usual Lagrange multipliers.
P ROOF The proof is in the working-paper version.
It is easy to verify that a property valid in the traditional problem holds for the
third-best contract as well: the optimal third-best rewarding scheme is increasing
in π under the assumption that the monotone likelihood-ratio property holds. Additionally, implementing high effort is costlier in the third best than in the second
best:
P ROPOSITION 2 The cost of implementing the third best is strictly greater than the
cost of implementing the second best; that is,
w3rd (π) f(π|eh ) dπ > w2nd (π) f(π|eh ) dπ .
P ROOF Relative
to the second-best
problem, there is an additional constraint, implying that w3rd (π) f(π|eh ) dπ w2nd (π) f(π|eh ) dπ . The strict inequality comes
from the fact that the solution to the second-best problem does not satisfy the new
restriction, NI.
Q.E.D.
(2007)
Principal–Agent Contracts
387
Notice also that the third-best contract is riskier (steeper) than the second-best
contract. That is the case because the (risk-averse) agent is at her reservation utility
in both cases, and it is costlier to implement high effort.
The principal ends up paying indeed the third-best cost in some settings of the
extended game – in particular, in those with free entry of insurers. In a free-entry
game, the principal moves first, and a potentially infinite number of insurers move
sequentially after that. It is easy to show that in such cases, implementing high
effort requires paying the third-best cost.7 If the principal’s gross benefits are large
enough, he will pay that cost and implement high effort. There will be parameter
values for which low effort is implemented in equilibrium while high effort would
have been implemented in the standard case without insurers.
The principal can always implement high effort by offering the third-best contract,
which is accepted by the agent and leaves no room for active insurers. Depending
on other details of the sequence of play, there might be other equilibria in which
the principal offers a contract riskier than the third best (but with the same expected
cost), leaving the agent under her reservation utility, and then the agent trades that
extra risk away with insurers.
The following proposition summarizes the main result of the free-entry case.
P ROPOSITION 3 In a setting with free entry, the principal’s cost of implementing
eh is equal to the third-best cost.
So, in the free-entry case, the cost of implementing high effort is the third-best cost.
This case is not only more costly for the principal than the second best, but also
more inefficient, as the risk-averse agent is exposed to greater risk.
In the next section we analyze environments (with a finite number of insurers) in
which the implementation of high effort costs the principal less than the third-best
cost. The section also shows that the sequence of play can affect the characteristics
of equilibrium outcomes.
4 Finite Number of Insurers
In this section we consider settings in which there are a finite number of potential insurers that appear sequentially after the principal. In section 4.1 we focus
on a case in which the agent receives all the offers (including that of the principal) before deciding which contracts to accept – we call that setting sequential
offering. In section 4.2 we consider the case in which after signing the contract
with the principal, the agent can engage in contracts with insurers – we call that
setting sequential contracting. In both cases we present in more detail the analysis when there is only one insurer, and then we extend to the case with more
insurers.
7
All the claims stated here are proven in the working-paper version.
388
Mariano Tommasi and Federico Weinschelbaum
JITE 163
4.1 Sequential Offering
Consider the implementation of high effort in a setting in which (1) the principal
offers a nonexclusive contract wp ; (2) an insurer observes wp and offers a contract w1 ;
(3) the agent considers which contracts (if any) to accept, and chooses the level of
effort; (4) payoffs are realized.
From Lemma 1 we know that the cost of the principal of implementing high effort
is at least c2nd (eh ). If that is the case, then the insurer can either stay out or implement
high effort. If the insurer were to stay out, the combination of the noninsurability
constraint with the participation constraint would lead to a cost of high effort that
was higher than the second best, leaving the principal worse off.
The other possibility (the one that will be chosen in equilibrium) is for the
principal to offer a contract such that the insurer is willing to participate and take
the agent to high effort. In order to implement eh the insurer will offer the contract
such that the aggregate contract is the second-best contract:
w1 (π) = w2nd (π) − wp (π) .
The expected benefits to the insurer from implementing eh are therefore
Π1 (eh ) = − w1 (π) f(π|eh ) dπ = wp (π) f(π|eh ) dπ − w2nd (π) f(π|eh ) dπ .
Whenever Π1 (eh ) ≥ 0, the insurer will have incentives to take the agent to high
effort – he will not have incentives to take her to low effort, given that the contract
satisfies NI. The principal’s least costly way of implementing eh is by setting wp
such that Π1 (eh ) = 0. We show next that this is possible. The zero-profit condition
implies that the contract must satisfy
wp (π) f(π|eh ) dπ = w2nd (π) f(π|eh ) dπ ;
(SC)
that is, the principal is paying the same expected cost as in the traditional problem.
(SC stands for second-best expected cost.)
We are therefore looking for a contract that satisfies both SC and NI:
wp (π) f(π|eh ) dπ = w2nd (π) f(π|eh ) dπ
(3)
> φ [g(el ) + u ] ≥
wp (π) f(π|el )dπ ,
where the equality is SC, the first inequality follows from (1), and the second
inequality follows from a version of NI where φ is applied to both sides. (It is easy
to see that existence is guaranteed.)
Equation (3) has a nice interpretation. Compared to the second-best contract, the
principal offers higher payments in the “good” states and lower payments in the
“bad” ones in such a way that average compensation is the same when the agent
makes high effort, but lower when she makes low effort. Therefore, the principal is
offering a riskier contract to the agent but still paying the cost of the second-best
contract. Since this alternative is less costly than preempting the insurer’s entry, this
is the option chosen in equilibrium.
(2007)
Principal–Agent Contracts
389
We therefore have the following result:
P ROPOSITION 4 In the sequential offering case with one insurer: (1) The principal’s cost of implementing high effort is identical to the traditional second-best cost.
(2) Equilibria where high effort is implemented are characterized by the following
properties: (i) The principal offers a contract wp that satisfies both SC and NI. The
contract is riskier than the second-best contract. (ii) The insurer takes the agent to
the second-best contract, w A = w2nd , and makes zero profits. (iii) The agent accepts
both contracts, chooses eh , and makes her reservation level of utility u .
Greater risk to the agent implies that the principal’s contract, by itself, leaves the
agent under her reservation utility u . (Call this the agent’s interim utility.)8 The
insurer is given the incentives necessary to insure the agent and take her to the
second-best aggregate contract and, consequently, to her reservation utility u.9
The intuition is simple. Knowing that an insurer has incentives to reduce any risk
the agent might take and eventually give her incentives to choose el , the principal
offers a riskier contract such that the insurer is just willing to participate and insure
the agent up to the second-best contract. As a result, the principal manages to pay
the traditional second-best cost, even in the presence of an insurer.
Given that the results also remain the same for the case of el , the resulting
equilibria of this extended game share some of the properties of the traditional
equilibria. The principal still pays the second-best cost to implement high effort,
the agent makes her reservation level of utility, and the insurer makes zero profits.
What differs is the principal’s optimal contract, which is now steeper. However, the
aggregate contract is still the second-best contract.
These results generalize to a finite number of insurers, in which N insurers
offer contracts to the agent, and at the end the agent decides which of the N + 1
contracts to accept and which level of effort to exert. Equilibria where high effort is
implemented fulfill the following properties: (1) The principal offers a contract wp
that satisfies SC and NI. The contract is riskier than the second-best contract. (2) The
last insurer (insurer N ) is always active, makes zero profits, and takes the agent to
the second-best contract, w A = w2nd . (3) The agent makes her reservation level of
utility u . (These results are proven in the working-paper version of this article.)
4.2 Sequential Contracting
In this section we focus on the case in which after signing the contract with the
principal (and before exerting effort), the agent can engage in additional contracts
8 It turns out that in the cases when the principal cannot leave the agent below her
reservation utility, he is worse off (see section 4.2).
9 There are multiple equilibria because many contracts satisfy (i) in Proposition
4. This multiplicity appears because the principal can expose the agent to different
amounts of risk. That is, while there is an upper bound to the agent’s interim utility
(which is SC and NI with equality), there is no lower bound. Uniqueness of the contract can be achieved by assuming that the insurer has a slight degree of risk aversion.
390
Mariano Tommasi and Federico Weinschelbaum
JITE 163
– we call this case sequential contracting. It turns out that in this case, the results
are intermediate between those of the second and third best.10
We analyze the case of one insurer, in which: (1) the principal offers a nonexclusive
contract wp ; (2) the agent either accepts or rejects the contract; (3) an insurer
observes wp and the agent’s decision; (4) the insurer offers the agent the contract w1 ;
(5) the agent either accepts or rejects the contract; (6) the agent chooses the level of
effort; (7) payoffs are realized.
The principal would like, once again, to offer SC and NI, but that would give
the agent less than u . This would be the agent reservation utility in her relationship
with the insurer, since the agent has already signed the contract with the principal.
Then the insurer, who has all the bargaining power, would leave the agent at this
lower utility. The agent will never accept the principal’s contract in the first place.
Thus, the principal faces three constraints: (1) give the agent the reservation level of
utility; (2) give the agent the incentives to exert high effort; and (3) give the insurer
the incentives to implement high effort. Hence the principal’s cost of implementing
high effort is higher than the second-best cost.
The third-best contract, wp (π) = w3rd (π), is feasible, since it fulfills the three
constraints. If the principal were to offer the third-best contract, the insurer would
offer a contract that takes the agent to the second-best contract and get a profit
equal to the difference between the second- and the third-best cost. This means
that the insurer strictly prefers the implementation of high effort. The principal,
instead of the third-best contract, will offer a less costly contract that makes
the insurer indifferent between implementing high and low effort. Therefore, in
the optimal contract the principal pays less than the third-best cost. (Since the
agent is always at her reservation utility, higher costs are associated with riskier
contracts.)
The next proposition summarizes the results in this case (proven in the workingpaper version).
P ROPOSITION 5 In the case of sequential contracting when the insurer has the
bargaining power, the principal’s cost of implementing high effort is greater than
the second-best cost but smaller than the third-best cost. Furthermore, equilibria
where high effort is implemented are characterized by the following properties:
(i) The principal offers a contract wp that is riskier than the second-best contract
but less risky than the third best. (ii) The insurer takes the agent to the secondbest contract, w A = w2nd , and makes positive profits. (iii) The agent accepts both
contracts, chooses eh , and makes her reservation level of utility u .
In this case, the insurer appropriates some rents from being able to control the
agent’s consumption vector and having the bargaining power in his relationship
with the agent. This bargaining power of the insurer does hurt the principal. But
the principal is still able to capture more than in the free-entry case, because his
10 We assume that insurers offer contracts to the agent. In the alternative bargaining
assumption in which it is the agent who makes offers to insurers, the results turn out to
be almost identical to those of sequential offering.
(2007)
Principal–Agent Contracts
391
first-mover advantage allows him to capture some of the surplus generated for the
reason that somebody is able to take the agent to the second best. In this case there
are parameter values for which low effort obtains in equilibrium, while high effort
would be obtained in the absence of outside trading opportunities. (The range of
parameters over which this is the case is smaller than in the free-entry case.) These
results stress the point that the extensive-form game has an effect on the region of
parameters in which high effort is implemented.
When there are more than one insurer (but still a finite number), the last insurer
takes her to the second-best aggregate contract, and all insurers are active and make
positive profits. In this case the cost of the principal is increasing in the number of
insurers, but never higher than the third-best cost.
5 Concluding Remarks
5.1 Comparison across Cases
In all the sequences with a finite number of insurers, the last insurer is a key player,
who is responsible for taking the agent to the second-best aggregate contract. He
is able to do so because he knows there is no next insurer willing to take the agent
to an aggregate flat wage. In the traditional principal–agent problem the principal
possesses the technology that makes the agent’s effort valuable, as well as the power
to control her consumption vector. With a finite number of insurers the agent’s
consumption vector is under the control of the last insurer, but the timing of the
game gives the principal the ability to appropriate rents generated by this power.
With free entry of insurers, there is no last insurer, and consequently no one willing
to take the agent to the second best, since nobody has the power to control the agent’s
consumption vector. In that case the threat of insurance does hurt the principal, and
does affect the equilibrium contract(s) more substantively, leading to inefficient
extra risk for the risk-averse agent.
While in the finite cases the aggregate contract is the second-best contract (as
in the standard case), in the free-entry case the aggregate contract is a third-best
contract. In all the cases with insurers, the principal’s contract, when implementing
high effort, is steeper than in the standard cost. Under free entry and in the sequential
contracting case, the contract’s variance is amplified by increasing the reward to high
effort. (This, as we have noticed, involves a greater cost.) On the other hand, in the
sequential offering case the principal’s optimal contract pays the same to high effort
but decreases the expected payoff of undertaking low effort. The variance is also
increased. Incentives are enhanced in both cases, the difference being that in one
case the reward to high effort increases, while in the other the reward to low effort
decreases.
The threat of insurance does not change equilibrium effort in the sequential offering case; it reduces the range of parameters of high effort in the sequential contracting
case, and even more so in the free-entry case. That is, for some specifications and
parameter values, the threat of insurance makes incentives vanish.
392
Mariano Tommasi and Federico Weinschelbaum
JITE 163
5.2 Implications
In this paper we have studied how the presence of outside trading opportunities
(“the threat of insurance”) affects contracting in moral-hazard agency relationships.
The type of outside trading opportunities studied in this paper might be of more
relevance in some environments (industries, job types, legal environments) than
in others, providing potentially interesting empirical implications. The threat of
insurance might not be an issue in some environments, either because the risk in
question is not insurable (technologically, or because of the absence of the relevant
markets) or because, being insurable, further contracting can be forbidden in a legally
enforceable manner. This in turn might relate to the overall institutional and legal
capability of a given country, or to the observability and verifiability of the additional
contracts, which might vary across markets or across activities.
This article predicts: (i) that in cases in which it is easier to trade risk away,
contracts for performance will be less common, and effort (and hence output) will
be lower on average; (ii) that when performance contracts are indeed used, they will
tend to be steeper; and (iii) that we might observe agents facing (steep) contracts
for performance from their principals, and then partially unwinding those incentives
through further trades.11
Some additional predictions might be obtained, although they would require
further modeling, extending our analysis – for instance by endogenizing some
contractual restrictions, or enriching the set of variables over which there might be
observation, monitoring, etc. We conjecture that:
(1) We might observe a tendency to utilize compensation instruments that are
more difficult to trade away (this might relate to the tendency to use discretionary
promotions or other rewards for performance that are not directly monetary).12
(2) The tendency to observe exclusivity clauses (or other trading restrictions)
will depend on the ex ante incentives of the principal and the agent to use such
commitment technology if available. It will be more useful in situations in which
the lost surplus due to insurance is higher.
(3) The tendency to use different indicators in performance contracts will depend
not only on their observability by the principal (positively), as predicted by the
informativeness principle (HOLMSTROM [1979], SHAVELL [1979], GROSSMAN AND
HART [1983]), but also on their observability by outside parties (negatively).13
(4) This might also influence the way in which effort is monitored inside organizations. If costly and imperfect monitoring is possible, firms may find it more
advantageous to focus on indicators of effort that are less easy to observe for outsiders.
11 This seems consistent with the findings in O FEK AND Y ERMACK [2000] for
managerial contracts.
12 See for instance BARON AND KREPS [1999], MURPHY [1999], and GIBBONS
AND WALDMAN [1999].
13 Proving this intuition would require a multisignal extension of our work.
(2007)
Principal–Agent Contracts
393
References
ASHENFELTER, O., AND D. CARD (eds.) [1999], Handbook of Labor Economics, Vol. 3,
Elsevier: Amsterdam.
ARNOTT, R., AND J. STIGLITZ [1991], “Moral Hazard and Nonmarket Institutions: Dysfunctional Crowding Out or Peer Monitoring?” American Economic Review, 81, 179–190.
– – AND – – [1993], “Equilibrium in Competitive Insurance Markets with Moral Hazard,”
Mimeo, Boston College.
BARON, J., AND D. KREPS [1999], Strategic Human Resources: Frameworks for General
Managers, John Wiley & Sons: New York.
BISIN, A., AND D. GUAITOLI [2004], “Moral Hazard and Non-Exclusive Contracts,” RAND
Journal of Economics, 35, 306–328.
– – AND A. RAMPINI [2006], “Exclusive Contracts and the Institution of Bankruptcy,” Economic Theory, 27, 277–304.
CHIAPPORI, P. A., I. MACHO, P. REY, AND B. SALANIÉ [1994], “Repeated Moral Hazard:
The Role of Memory and Commitment, and the Access to Credit Markets,” European
Economic Review, 38, 1527–1553.
FUDENBERG, D., AND J. TIROLE [1990], “Moral Hazard and Renegotiation in Agency Contracts,” Econometrica, 58, 1279–1319.
GIBBONS, R., AND M. WALDMAN [1999], “Careers in Organizations: Theory and Evidence,”
chapter 36 in: ASHENFELTER AND STIGLITZ (eds.) [1999].
GROSSMAN, S., AND O. HART [1983], “An Analysis of the Principal–Agent Problem,” Econometrica, 51, 7–45.
HELLWIG, M. [1983], “On Moral Hazard and Non-Price Equilibria in Competitive Market
Insurance,” Mimeo, University of Bonn.
HOLMSTROM, B. [1979], “Moral Hazard and Observability,” Bell Journal of Economics, 10,
74–79.
ITOH, H. [1993], “Coalitions, Incentives, and Risk Sharing,” Journal of Economic Theory,
60, 410–427.
KAHN, C., AND D. MOOKHERJEE [1998], “Competition and Incentives with Nonexclusive
Contracts,” RAND Journal of Economics, 29, 443–465.
MAS-COLELL, A., M. D. WHINSTON, AND J. R. GREEN [1995], Microeconomic Theory, Oxford University Press: New York.
MURPHY, K. J. [1999], “Executive Compensation” chapter 38 in: ASHENFELTER AND STIGLITZ (eds.) [1999].
OFEK, E., AND D. YERMACK [2000], “Taking Stock: Equity-Based Compensation and the
Evolution of Managerial Ownership,” The Journal of Finance, LV, 1367–1384.
SHAVELL, S. [1979], “Risk Sharing and Incentives in the Principal and Agent Relationship,”
Bell Journal of Economics, 10, 55–73.
Mariano Tommasi
Federico Weinschelbaum
Department of Economics
University of San Andres
Vito Dumas 284
(1644) Victoria, Provincia de Buenos Aires
Argentina
E-mail:
[email protected]
[email protected]