Public procurement under moral hazard and
hidden information with a risk-averse agent
Philippe BONTEMS,
Alban THOMAS,
********
04. 13. 150
LERNA
LABORATOIRE D'ECONOMIE DES RESSOURCES NATURELLES
Unité Mixte de Recherche du Commissariat de l'Energie Atomique (CEA),
de l'Institut National de la Recherche Agronomique (INRA)
et de l'Université des Sciences Sociales de Toulouse (UT1)
LERNA
CEA-INRA-UT1
21 Allée de Brienne - 31000 TOULOUSE - FRANCE
Tél. : 33 (0)5 61 12 86 21 - Fax : 33 (0)5 61 12 85 20
E-Mail : [email protected]
Public procurement under moral hazard and hidden
information with a risk-averse agent
Philippe Bontemsy
Alban Thomasz
Abstract
We consider a model of regulation of public procurement with a risk-averse agent who
becomes privately informed on production conditions after the contract is signed. The
agent also provides unobservable eort under uncertainty on her private information
parameter. We characterize the optimal solution to the regulator problem in the presence
of moral hazard, asymmetric information and risk sharing, given general assumptions on
net expected social surplus and the agent utility function.
We show that usual results such as \no distortion at the top" no longer hold: depending on the cost function properties, overproduction or underproduction may occur.
Indeed, compared to the perfect information production level, distortions due to moral
hazard and asymmetric information can be additive or not. Moreover, risk aversion for
the agent will often generate bunching for the less eÆcient types.
JEL
: D82, Q19.
Key-words
: Mechanism Design - Asymmetric Information - Risk aversion - Moral
Hazard.
Corresponding author : Philippe Bontems, INRA, Universite des Sciences Sociales de Toulouse, 21 Allee
de Brienne, 31000 Toulouse, France. Email: [email protected].
y University of Toulouse (INRA, IDEI and LEERNA).
z LEERNA-INRA, Toulouse.
1
1
Introduction
Optimal contract schemes when the relationship between a regulator (the principal) and an
agent is subject to asymmetric information (in the form of a private-information parameter
to the agent) are now widely documented in the literature (see Laont and Tirole [1993] for
a synthesis). In these models, private information to the agent may be used for strategic
purposes in the relationship with the principal, but it is most of the time assumed that this
parameter is observed to the agent when the contract is signed. Consequently, the regulator
has to design a contract mechanism consistent with incentive-compatibility and individualrationality constraints (following the Revelation Principle, the contract must produce truthful
revelation of the agent's type, and the agent must nd protable to accept the contract).
In a number of situations however, it may be the case that the agent does not observe
her private-information parameter before engaging in a contract-based relationship with the
regulator. A rst example of this situation is the relationship between a producer and a
retailer, where nal demand for commodities sold by the retailer are still unknown. Another
example is the regulation of a producer emissions by an environmental agency, where some
production conditions are still unknown (for instance climatic ones), and may aect nal
pollution induced by production. A nal example is the employer-employee relationship
where the diÆculty of the task for which the agent is recruited is still unknown. In these
situations, when signing a contract, the agent faces uncertainty which is not resolved yet.
Informational asymmetries of this type would be of no consequence for the principal if the
agent were risk neutral or if there were no limited liability (or ex post individual rationality)
constraints. Indeed, it would then be possible for the principal to design a contract inducing
the rst best outcome where the agent supports all the risk and thus takes all appropriate
decisions that maximize principal's gain. This contract would also provide the agent with
exactly her reservation utility in expectation.
However, when the agent is risk-averse, the task of the regulator is more diÆcult, as
2
inducing the appropriate risk-sharing is required in addition to the informational constraints
mentioned above. The situation is even more complex when some decisions taken by the
agent are non contractible and yield to moral hazard problems. Consider for instance the
employer-employee relationship described above when the eort of the agent is also non
contractible. For instance, once hired, the employee must attend to a training course and
accumulate human capital (learning eort) before entering the production period and learning
the diÆculty of her task (type). An alternative view would amount to consider that the
learning eort of the agent subsequently determines stochastically her ability in undertaking
the production task.
The purpose of this paper is to explore such situations by considering a principal - riskaverse agent model and to illustrate the complex trade-o between the problems of hidden
information, moral hazard, and risk sharing. More precisely, the task of the principal is now
to nd an incentive scheme designed for a risk-averse agent whose non contractible actions are
taken after the date of the contract, but before the private-information parameter is observed
by the agent.
We develop our model in the context of regulation where the principal maximizes welfare,
but we also briey consider the case of a private principal. Our setting is closely related to
existing models of regulation under hidden information and with risk-averse agents. More
precisely, Salanie (1990) and Laont-Rochet (1998) provide analysis of related cases, in the
context of producer-retailer contract for the rst paper and in the context of public regulation
for the second. There, it is shown that a nite risk aversion implies that optimal mechanisms
are less eÆcient than under risk neutrality. In this case, distortions of agent actions are larger
and agent information rents are socially more costly and thus diminished by the principal.
Our analysis diers from these two papers in that the regulated agent takes here an action
(in the rst stage of the production process) when uncertainty is still unresolved. As this
decision is not observable and thus non contractible, the optimal contract has to induce the
agent to take the optimal decision on the rst action (moral hazard) as well as inducing
3
truthful revelation of her type. The contract also has to achieve an optimal risk sharing
between the principal and the agent: the principal being risk neutral, he has to absorb as
much risk as possible through the design of contract.
An earlier contribution related to the present paper is the analysis of Baron and Besanko
(1987). They examine an optimal procurement contract for a private principal who contracts
with a risk-averse supplier with private information about her costs and who contributes to
an unobservable eort. Their contribution is mainly to show that randomness in cost or noisy
monitoring of supplier's deterministic cost are no longer equivalent as one considers a riskaverse supplier, contrary to the case of a risk-neutral agent as in Laont-Tirole (1986). The
principal observes only production and a noisy monitoring of cost which remains unobservable
for the principal. The dierence between our model and theirs is that, in our analysis, the
private information parameter is revealed to the agent after the signature of contract. While
in their analysis private information does not entail uncertainty (which in fact originates from
an independent noise), it does here.1
Our main results are that usual results such as \no distortion at the top" no longer hold:
depending on the cost function properties, overproduction or underproduction at the top
may occur. Indeed, compared to the perfect information production level, distortions due to
moral hazard and asymmetric information can be additive or not. Moreover, there can be
over or under-provision of eort at the optimum. Finally, risk aversion for the agent may
generate bunching for the less eÆcient types as emphasized by the analysis of a particular
specication which is widely considered in the literature (Laont and Tirole, 1986). It is also
important to note that even if the shadow cost of public funds is nil, optimal production
levels and eort are still distorted compared to rst best levels, because of the presence of
risk aversion.
The paper outline is as follows. In section 2, we present the model of production under
Also, a minor dierence is that preferences with regard to prot and the (non contractible) eort of the
risk-averse supplier are additively separable in Baron and Besanko's paper, while in ours they are not.
1
4
risk conditions and describe the agency problem. We characterize in section 3 the solution to
the agency problem under perfect information and imperfect information. We then analyze
in section 4 special cases such as risk neutrality for the agent, the situation without moral
hazard and a CARA specication for the agent utility function. An illustrative example is
developed in section 5. Section 6 is devoted to the analysis of an alternative model where the
eort inuences the distribution of private information. Concluding remarks are in the last
section. Most of proofs are relegated into an appendix.
2 The model
2.1
Production decisions under uncertainty
Consider an agent engaged in a production process under the supervision of an utilitarian
regulator. In the rst stage, the agent makes a decision on an initial eort denoted e 2 R+,
which is going to aect the nal output through the cost function. This action can be
thought of, for example, as an investment or the use of a production input. When deciding
upon the level of e, the agent has no information on the value of parameter 2 = ; ,
which enters the cost function. We also assume in the following that is the only random
component in the prot function, its randomness being understood as associated with the
fact that decision on e is taken before the true value of the private information parameter
is observed. In the second stage, information on becomes available to the agent. In
the third stage, a production decision q is taken. The associated total cost of production
is C (q; e; ) + (e) where C (:) is the cost function, which is increasing and convex in the
production level (Cq > 0, Cqq 0) and (e) is the monetary cost of eort.2 We assume that
C is increasing in and that Cq = @ 2 C=@@q is strictly positive (single-crossing condition).
We also assume that the eort e decreases the production cost (Ce < 0) and that the cost of
eort is increasing and convex ( 0 (e) > 0; 00 (e) 0).
An important assumption in this basic framework is the independence of the probability
2
x.
In the following, we denote by Gx the partial derivative of any function G(:) with respect to the variable
5
distribution function from e: P r( je) = P r( ) 8 . With this assumption, the
riskiness in the production process cannot be controlled by adjusting the level of e in the rst
stage. This enables us to focus on the consequences of risk aversion in the optimal design
of the contract. Of course, this diers from more standard models of production under
risk conditions, where risk-decreasing inputs or investments may be used to hedge against
production risk, by using more of these inputs than in the risk-neutrality case. This case is
subsequently analyzed in section 6. Of course, in the rst model considered here, the choice
of e will nevertheless aect the wealth level, while parameter would aect both eÆciency
and the level of production risk independently from agent's actions.
Finally, we assume implicitely that eort cannot be exerted after the private revelation of
. Actually, if possible, it would be interesting for the agent to wait until she learns her type
before making the cost reducing eort. This is essentially the timing analyzed by Laont
and Rochet (1998). Such a strategy would obviously eliminate all the production risk borne
by the agent and allows her to choose the eort level which is optimal ex post from her
viewpoint. In fact, moral hazard does not play any role as the agent fully bears the cost of
production (Laont and Martimort (2002) coined the term \false moral hazard" for such a
setting).
2.2
The agency problem
Consider now the problem of a risk-neutral utilitarian regulatory agency engaged in a principalagent relationship with the agent. The goal of the principal is to maximize expected net social
surplus associated with production, where the policy instrument is a contract to be signed
between both parties. This contract species a production level q and the level of a lumpsum
transfer t from the principal to the agent. A crucial assumption in this case is that the
contract is signed before any action the agent might take, and importantly, before parameter
is observed by the agent.
At the time of contracting, both the risk-neutral agency and the risk-averse agent have
6
a common prior on parameter , which is represented by a cumulative density F (:) and f (:)
a density function on the compact set ; . The agent privately learns the true value of
before production q is undertaken but after the decision on e. The agency can only get
information on through communication with the agent, and observes only production q.
More formally, a contract is a mapping t : R+ ! R, where t(q) is the transfer paid by the
agency to the agent if the latter commits to a production level q. Firm's prot is given by:
(q; t; e; ) = t(q )
C (q; e; )
(e):
Alternatively, one may view the optimal contract as a pair of functions ht(); q()i from
; to R R+ , depending on the report of her type by the agent. The timing of the game
is as follows (see Figure 1):
at t = 0; the contract ht(); q()i or equivalently the non linear transfer t(q) is oered
to the agent;
at t = 1; the agent chooses e to maximize her expected utility, given the prior F (:) on
;
at t = 2, nature chooses which is private information to the agent and the latter
announces ^ to the agency or equivalently picks a production level q(^) in the contract;
at t = 3, the transfer t is paid.
Figure 1. The timing of the game
7
Contract is signed
Action e is taken
Prot is obtained
#
#
is observed
#
#
t=0
t=1
t=2
t=3
"
^ is announced
The rm's problem is to maximize expected prot if it is risk-neutral, or to maximize the
expected utility of prot if it is risk-averse. In the latter case, the rm's problem would be
max
E max U ( ) = max
e
q
e
Z max
U ( )
q
dF ( );
(1)
where U (:) is a Von Neuman-Morgenstern utility function (with U 0 > 0; U 00 0) and F (:)
is the probability distribution function for private information parameter which is common
knowledge in the economy.
A major diÆculty for the agency is that the level of e is not veriable. Moreover, as
parameter is not observed to the agent in the rst stage of the game, it is not observed
to the principal either. The objective of the principal is to design a mechanism that would
be accepted by the agent, given that the latter has no information on when signing the
contract. The principal has therefore to account for the sequential nature of the agent's
decision strategy in designing the optimal contract.
From the revelation principle, there is no loss of generality in paying attention to direct
revelation mechanisms only. As the communication between the agency and the rm must
take place after the uncertainty upon is resolved for the agent, a direct revelation mechanism
can be described by a pair of functions (q; t) such that
8 ; 8 ^
( ) (; ) (; ^ )
8
(IC)
where (; ^) = t(^) C (q(^); e; ) (e).
We restrict our attention to the case of non stochastic, continuous and piecewise dierentiable contracts.3
We also assume that the principal is exogenously constrained to oer a minimum prot
whatever the realized value of , because for instance of innite risk aversion below zero (or
institutional or political constraints): We normalize the reservation utility of the agent to
zero and ex-post participation constraints thus read:
( ) 0; 8 :
(IR)
For simplicity, we follow Laont and Rochet (1998) by assuming that the ex post participation
constraints imply the ex ante participation constraint which we neglect hereafter.
The agency seeks to maximize the expected consumer surplus net of the transfer t paid to
the rm, plus the certainty equivalent of rm's prot. Let S (q) be the gross surplus that the
agency derives from the procurement of the good, with S 0 (q) > 0 and S 00 (q) 0. The social
cost of public transfers between the regulator and the agent is assumed strictly positive, with
an opportunity marginal cost of public funds equal to .
Let CE () = U 1[E U ()] denote the certainty equivalent of , where E is the expectation operator with respect to the distribution of . The agency problem can now be written
as
Z [S (q()) (1 + )t()] dF ()
(P)
max
CE ( ) +
e; q (:);t(:)
subject to
( ) = t( )
C (q ( ); e; )
( ) (; ^);
e 2 arg max
e
Z
(e);
8; 8 ^;
U ( ( ))dF ( );
( ) 0; 8 :
(IC)
(MH)
(IR)
3
Stochastic contracts could be better in this framework as it could help reducing the incentive cost with
the drawback of insurance costs.
9
where (MH) is the moral hazard constraint stipulating that the optimal eort has to maximize
the expected utility of agent.
Before solving this program, we analyze in the following subsection the situation of perfect
information as a benchmark case.
2.3
Regulation under perfect information
Assume here that the principal can observe both the eort e and the type of the agent. We
thus can ignore the moral hazard (MH) and incentive compatibility (IC) constraints in the
program above. Eliminating the transfer t(:) in (P), we rewrite the agency problem as:
max CE () +
e;q (:); (:)
Z
W(; e; q(); ())dF ()
s.t. (IR)
with W(; e; q(); ()) = S (q()) (1 + ) (() + C (q(); e; ) + (e)) : Maximizing the
expected welfare over e and q(:) gives the following rst order conditions:
0 (eP I ) =
Z
Ce (q P I ( ); eP I ; )dF ( );
(2)
= Cq (qP I (); eP I ; ):
(3)
S 0 (q P I ( ))
1+
where P I stands for perfect information. The rst condition stipulates that the perfect
information eort level should equalize the marginal cost of eort and the expected marginal
benet in terms of production cost savings. The second one indicates that social marginal
benet of production weighted by the social cost of public funds is equal to its marginal cost.
Our assumptions on the cost and surplus functions imply that production optimally decreases
with the type .
Finally, the agent has no rent at the optimum whatever her type, or equivalently, () =
0; 8 : To see that (IR) is binding at the optimum, assume rst that it is not. Then, the
R
derivation of the principal's objective CE ()+ W(:)dF () with regards to () yields the
following rst order condition (for an interior solution):
U 0 ( ( ))
U 0 (CE ( ))
(1 + ) f () = 0:
10
(4)
This means that the marginal utility of prot is constant for every : Intuitively, because the
principal is risk neutral, he should absorb all the risk. However this also means that () is a
constant, say , and as CE () = , we should have = 0 as long as > 0. This contradicts
the fact that (IR) is not binding. As a conclusion, in any case, the regulator puts every agent
exactly to her reservation utility level.
Finally, note that risk aversion has no impact on the optimal regulation when information is symmetric. This property no longer remains true when one consider asymmetric
information as it will be shown below.
3 Regulation under imperfect information
3.1
Analysis
Assume now that there is incomplete information on and non observability of eort. In
order to solve the agency problem we have to transform the program to be solved. First, as
it is well known (see Guesnerie and Laont, [1984]), under single-crossing conditions, (IC)
can be reduced to the following set of conditions
_ ( ) =
C (q ( ); e; )
Cq (q ( ); e; )q_( ) 0:
(5)
(6)
Moreover, given that _ is strictly negative as indicated by (5), (IR) reduces to:
( ) 0:
(7)
Also, as we have Cq > 0 (single-crossing condition), (6) reduces to q_ 0. The optimal
production is thus non-increasing in : We dene q_ = ' where ' is a control variable and
we redene q as a state variable. Then constraint (6) becomes ' 0.
Finally, we use in the program above a rst order approach for the moral hazard constraint
on e. We have to check that local second order conditions are satised, i.e.
@2
@e2
"Z
#
U ( ( ))dF ( )
11
0:
It can be proved easily that a suÆcient condition is Cee + 00 0 which we assume in the
following:4
Let us redene e as a state variable with e_ = 0 and e() 0; e() = e; e() being
unconstrained. We then rewrite (MH) as follows
@
@e
"Z
#
U ( ( ))dF ( )
=
where M (:) U 0 (()) [ Ce(q(); e(); )
dene a new state variable as follows:
K_
Z
M (e( ); q ( ); ( ); )dF ( ) = 0;
0 (e)].
To deal with this integral constraint, we
= M (e(); q(); (); )f () with K () = K () = 0:
Let W(; e(); q(); ()) denote the term equal to:
W(; e(); q(); ()) = S (q()) (1 + ) (() + C (q(); e(); ) + (e())) :
We then transform the program (P)-(IC), (MH), (IR) into the following optimal control
problem:
Z max
CE ( ) +
W(; e(); q(); ())dF ()
(8)
'(:)
subject to
e_ = 0;
_ = C (q ( ); e( ); );
q_ = '( );
K_ = M (; e( ); q ( ); ( ))f ( );
'( ) 0;
e( ) 0;
e( ) = e;
() 0;
K () = K ( ) = 0:
()
( )
()
()
(1 )
(2 )
Solving this optimal control program, we obtain the necessary conditions for an optimum,
which are gathered in the following proposition.
Proposition 1 In case of no bunching and assuming an interior solution for e and q, necessary conditions for an optimum are:
4
We follow here the so-called \rst order approach" (see Grossman and Hart (1983), Rogerson, (1985) and
Jewitt (1988) for problems associated with such a method).
12
(1 + )Cq (q(); e; ) = U 0 (())Ceq (q(); e; ) + f(()) Cq (q(); e; ),
(i) S 0 (q())
where
()
=
Z
+
1+
Z
U 0 ( (u))
U 0 (CE ( ))
f (u)du
Ce (q (u); e; u) + 0 (e) U 00 ( (u)) f (u)du:
and (dropping all arguments for clarity)
=
(ii)
R
Rn
R n
o
0 ()
(1 + ) (Ce + 0 )f + U 0 (UCE
()) (1 + ) f du Ce
n
o
R
R
0 )U 00 ()f du Ce + U 0 ()(Cee + 00 )f d
(
C
+
e
o
d
:
(Ce(q(); e; ) + 0(e))U 0 (())dF () = 0,
(iii) () = () +
R
C (q (u); e; u)du.
(iv) If U is CARA, then the less eÆcient agent has no rent (() = 0): Otherwise, the less
eÆcient agent may earn a positive rent (() > 0)
Proof: See appendix A.
Condition (ii) simply states that the optimal eort should be chosen optimally by the
agent while condition (iii) denes the ex post level of rent for any type of agent. Condition (i)
denes the optimal level of production for the -type agent. These three conditions (together
with (iv)) jointly determine the optimal eort and the optimal production and prot patterns
through a system of three integral equations.
Note that from (iv) it appears that the most ineÆcient rm may have a positive rent.
Intuitively, a marginal increase in prot at () = 0 may increase the certainty equivalent by
more than the sum of the social cost of public funds 1+ and the social cost of moral hazard.
However, when income eects are eliminated by choosing the exponential specication CARA,
then ex post prot constraint (7) is binding at the optimum. Indeed, given (iii), decreasing
() by leads to an uniform decrease of ( ): Therefore, both E ( ) and CE ( ) decrease
by : As indicated by (8), the total eect on welfare is positive because > 0:
13
Compared to the perfect information outcome (see (3)), condition (i) indicates that there
are now two incentive distortions, one due to the presence of moral hazard and the other due
to hidden information. Note that the direction of the distortions compared to the perfect
information level is ambiguous for this level of generality. Even for the most eÆcient type,
there can be underproduction or overproduction, so that the usual result of \no distortion at
the top" no longer holds. Indeed for the most eÆcient agent ( = ), we get () = 0 as usual
in a standard procurement problem so that the distortion due to the incentive compatibility
constraints disappears. But here there is still a distortion due to the moral hazard constraint
which direction is given by the sign of the multiplier and the cross-derivative Ceq . Note
that if > 0, then there is under-provision of eort at the optimum: If a third party could
observe eort, the principal would like her to subsidize e in order to increase it. On the
contrary, if < 0 there is over-provision of eort from the principal viewpoint.
Finally, when = 0, it is worth noting that there are still incentive distortions on both
the eort and the production levels compared to rst best, because of the presence of moral
hazard and risk aversion. Before investigating more precisely the properties of the optimal
contract given in Proposition 1, it is very useful to consider rst the two following benchmark
cases.
3.2
Some useful benchmark cases
3.2.1 Risk neutrality
When the agent is risk neutral in the range of non-negative rents (U () = ), we get immediately that () = 0, that is the less eÆcient agent earns no rent. And the optimal eort
for the agent is still given by condition (ii) of proposition 1:
Z
(Ce(q(); e; ) + 0(e))dF () = 0:
Thus, the optimal eort follows the rule of perfect information, that is the eort should minimize the total expected cost of production (including the cost of eort) for a given production
level. However, given that the production is distorted away from perfect information level (as
14
shown above), the optimal level of eort will not be equal to the perfect information level,
except if the production level does not inuence the marginal cost saving due to eort (that
is when Ceq = 0). In the latter case, the condition which gives the optimal eort does not
depend on production at all.
Moreover, from Proposition 1, we also have
( ) = F ( ) = BM ( )
(9)
where BM stands for Baron-Myerson. Indeed, if we would have assumed further that the
eort is contractible then we would have obtained a model similar to the one proposed by
Baron and Myerson (1982). Importantly, note that under risk neutrality there is no longer
direct interaction between the cost of hidden information and the cost of moral hazard, i.e.
( ) does not depend on . This illustrates the main impact of risk aversion.
Futhermore, we easily get:
=
R
R
F ( )Ce d
(Cee + 00 )f d
and condition (i) of Proposition 1 reduces to:
S 0 (q ( ))
(1 + )Cq (q(); e; ) = Ceq (q(); e; ) + Ff (()) Cq (q(); e; ):
(10)
Thus, social marginal benet of production is equal to its marginal cost weighted by
the social cost of public funds plus two terms, the rst one coming from the moral hazard
constraint on the choice of e and the other one coming from the IC constraint. Given that
Cq > 0, private information on yields the principal to distort downwards the production
compared to the perfect information level (except for the most eÆcient type ). This is
because of the social cost of informational rents ( > 0), as in Baron and Myerson (1982).
The sign of the distortion coming from moral hazard about e will depend on the sign of
multiplier and the sign of cross derivative Ceq : Recalling that Cee + 00 0 (second-order
condition on eort), it appears that the sign of is entirely determined by the sign of Ce.
15
Assume that the marginal cost saving due to eort decreases in then there is under-provision
of eort ( > 0). Assume further that production decreases the marginal cost saving due
to eort (Ceq < 0), then it is easy to see that this leads the principal to distort upward
the production level in order to take account of the moral hazard constraint. Thus, in this
case, the two distortions work in opposite directions around the perfect information level.
Of course, other sets of assumptions can lead to both eects working together to distort
downwards the production level. Moreover, if either eort has no inuence on the marginal
cost of production (Ceq = 0) or marginal eect of eort on production cost is independent of
type (Ce = 0), then the distortion coming from moral hazard disappears.
3.2.2 Contractible eort
Note that when eort is observable and contractible, then our model contains as a particular
case a variant of the Laont-Rochet model (1998) where the agent takes also an eort decision
under conditions of risk. To see this, take to be equal to 0 in Proposition 1. This readily
implies that the optimal eort and production levels must satisfy the following conditions:
Z
(1 + ) (Ce(q(); e; ) +
S 0 (q ( ))
0 (e)) + () Ce (q(); e; ) f ()d = 0;
f ( )
(1 + )Cq (q(); e; ) = f (()) Cq (q(); e; );
((u)) f (u)du where LR stands for Laont-Rochet.
with () = LR() = (1 + ) F () UU0(0CE
())
The optimal eort level should balance in expectations the total marginal net benet
of eort weighted by the shadow cost of public funds ((1 + ) (Ce + 0 )) and the marginal
cost in terms of rents left to the agent. Indeed, an increase of e has a marginal eect on
hidden information cost equal to ()Ce(q(); e; ) which should be taken into account when
determining the optimal eort level.
Concerning the marginal cost of incentive constraints () which determines the incentive
distortion for the optimal production level, Laont and Rochet provide some insights on the
direction of the distortion imposed on the optimal level of production compared to the case
R
16
of risk neutrality. The shadow cost of incentive constraints in the case of risk neutrality (i.e.
the Baron-Myerson setting) is given by (9). Thus, the dierence between shadow costs can
be expressed as follows:
LR ()
BM ( ) = F ( )
Z
U 0 ( (u))
f (u)du
U 0 (CE ( ))
R
Let () U 0 (CE ())F () U 0((u))f (u)du: We have () = 0 and 0 is rst positive
then negative. This means that when () < 0 (which is equivalent to assume that U is
DARA5 ) there exist a threshold ~ such that for < ~, risk aversion leads to a greater
incentive distortion compared to risk neutrality and for > ~ risk aversion leads to a smaller
distortion. On the contrary, when () 0 (U is CARA or IARA), then risk aversion always
leads to a greater incentive distortion compared to risk neutrality. In particular, when U is
CARA, we have () = 0.
3.3
Back to the general case
The distortion on q due to the moral hazard constraint comes from the fact that an increase in
q for the -type agent will impact the (ex post) social cost of the moral hazard constraint for
the -type rm (U 0 (())Ceq (q(); e; )). This eect is purely due to the presence of moral
hazard and is eective even if the rm is risk neutral. If there is under provision of eort at
the optimum ( > 0) and eort contributes to decrease marginal cost of production (Ceq < 0),
then increasing the production level also gives incentives to increase eort. Intuitively, an
increase in q decreases the (ex post) cost of moral hazard by raising the marginal benet
from eort ( Ceq > 0).
When determining the optimal eort, the principal also takes into account that increasing
the eort may have eects on the informational rents. Indeed, when Ce > 0, then increasing
the eort raises the rate of growth C of socially costly rents.
Corollary 2 Assume that Ce = 0 and Ceq has constant sign. Then, at the optimum,
5
Indeed, we have () < 0 when U 0 (U
1
(x)) is convex, which is equivalent to say that U is DARA.
17
(i) if the eort decreases the marginal cost of production (i.e. Ceq <
0), there is under-
provision of eort ( > 0). On the contrary, if the eort increases the marginal cost of
production (i.e. Ceq > 0), there is over-provision of eort ( < 0).
(ii) for a given level of eort, there is over-production at the top.
Proof: See appendix B.
Now, the distortion due to the imperfect information on comes from the fact that
increasing the production level of a particular type will also increase the informational rent
left to all more eÆcient agents and the corresponding marginal cost is f(()) Cq . We then
rewrite the shadow cost of rents () as follows:
() = LR ( ) + ( )
(11)
R
where () = f(Ce(q(u); e; u) + 0 (e))U 00 ((u))g f (u)du: Note that () = 0 and that
0() = (Ce(q; e; ) + 0 (e))U 00 ()f () which sign is ambiguous because both the signs of and Ce + 0 are unknown for this level of generality. The term () is due to the interaction
of moral hazard and risk aversion. If either the eort is contractible or the agent is risk
neutral, then it disappears.
However, if we once again assume that Ce = 0 and Ceq has constant sign, then we obtain
the following result.
Corollary 3 Assume that Ce = 0 and Ceq has constant sign. Then, at the optimum, in the
no-bunching case, we have:
(i) if U is CARA, then () LR () BM ()( 0) with equality for = and = .
(ii) if the agent is imprudent (U 000 < 0); then () LR () with equality for = Proof: Assume that Ce = 0 and Ceq has constant sign.
Part (i): assume further that U is CARA. It is not diÆcult to show that for the less
eÆcient agent, we have () = 0 (because of the moral hazard constraint) and consequently,
18
() = LR () = BM ( ) = :
Then we have
() = LR ( ) + ( )
with () = LR() = BM () = and () = LR() = BM () = 0. Moreover, LR () BM ( ) for any . Furthermore,
0() = (Ce(q(); e; ) + 0 (e))U 00 ()f ()
=
(Ce (q ( ); e; ) + 0 (e))U 0 ( )f ( )
As dd (Ce + 0 ) = Ceq q_ which sign is the same as for Ceq because q_ < 0. If Ceq < 0 then
Ce + 0 is increasing and rst negative then positive (recall that there exists a unique ~ such
that Ce + 0 = 0, otherwise, the moral hazard constraint would be violated). Moreover, > 0
and consequently 0() is rst positive then negative. Finally, () 0 with equality for
= and = .
Now, assume that Ceq > 0 then Ce + 0 is decreasing and rst positive then negative.
Moreover, < 0 and consequently 0() is rst positive then negative. Finally, () 0
with equality for = and = .
Part (ii) : For any U , we still have 0() = (Ce (q(); e; ) + 0(e))U 00 ()f () and () =
LR ( ) = BM () = 0: Moreover, whatever the sign of Ceq we have 0 ( ) rst positive then
negative (see part (i)). It remains to identify the sign of (): We have:
() =
=
Z
"
(Ce(q(u); e; u) + 0(e))U 00 ((u)) f (u)du
cov (Ce + 0 ; U 00 ) +
Z
(Ce(q(u); e; u) + 0(e))f (u)du
R
Z
#
U 00 ( (u))f (u)du
Now assume that Ceq < 0, then > 0 and (Ce(q(u); e; u)+ 0 (e))f (u)du < 0. Consequently,
if cov(Ce + 0 ; U 00 ) > 0 or equivalently U 000 < 0 (the agent is imprudent) then () > 0 and
() LR ( ) with equality for = . Finally, assume that Ceq > 0, then < 0 and
R
0 (e))f (u)du < 0. Consequently, if cov(Ce + 0 ; U 00 ) < 0 or equivalently
(Ce (q (u); e; u) +
U 000 < 0 (the agent is imprudent) then () > 0 and ( ) LR ( ) with equality for = .
19
Note that when the agent is prudent, the sign of () is ambiguous. This concludes the proof.
Hence, the downward distortion due to hidden information on is often greater than
the one under risk neutrality (Baron and Myerson case) and the one without moral hazard
(Laont and Rochet case). However, as shown above, the distortion due to the moral hazard
constraint works in the opposite direction, so that it may be the case that for a given level of
eort, the optimal production level is closer to the perfect information level than under risk
neutrality or without moral hazard.
The impact of an increase in rents left to all more eÆcient agents (than the -type agent)
can be decomposed as follows. First, the social cost of rents is increased at a rate 1 + :
Second, the certainty equivalent of prot is also increased. Both eects are gathered in the
shadow cost LR (): Finally, there is interplay between hidden information and moral hazard
due purely to risk aversion because () is also a function of only if U 00 6= 0. Indeed, the
expression (Ce(q(); e; )+ 0 (e))U 0 (()) represents the ex-post marginal net value of eort
by the -type agent from the agency viewpoint. Hence, increasing q() by dq which leads to
increase the rent () of any more eÆcient type of rm will also impact the ex-post marginal
net value of eort for the agency ( [Ce + 0 ] U 00 ()).
4 A linear example with the possibility of bunching
Up to now, we have neglected the possibility of bunching in the optimal solution by focusing
on the case of a separating contract. It is useful to derive conditions upon which there is no
bunching for the optimal solution described in Proposition 1. Dierentiating the equation
given by part (i) in Proposition 1 with respect to , we obtain:
S 00
(1 + )Cqq q q_ (1 + )Cq =
d
C Ceq U 00 + (Ceqq q_ + Ceq )U 0 + (Cqq q_ + Cq )
d
20
( )
f ( )
It is easily seen that, for this level of generality, it is quite diÆcult to nd suÆcient conditions
for an optimal separating contract.6 We pursue this task using a particular specication.
Throughout this section, we make the following assumption:
H1 :
C (q; e; ) = (
(e) = e2 =2:
e)q
The cost function C is linear with respect to the production level and the eort e induces a
decrease in the marginal cost of production. But importantly, the rate of decrease does not
depend on the eÆciency parameter (Ce = 0). Also, the marginal impact of production on
marginal cost savings is of constant (positive) sign, i.e. Ceq = 1 > 0. These specications
will also allow us to explore the optimal regulation in more details. Note also that they are
commonly used in the literature, see for instance Laont and Tirole (1986) or Laont and
Rochet (1998).
Using H 1, the rst order condition of moral hazard constraint reduces to:
Z
q ( ))U 0 ( ( ))dF ( ) = 0;
(e
(12)
R
and the level of rents is given by () = () + q(u)du.
We also easily obtain the following equation from part (i) in Proposition 1:
S 0 (q ( ))
(1 + )(
e) =
U 0 ( ( )) +
( )
f ( )
(13)
where
()
6
=
=
Z
U 0 ( (u))
U 0 (CE ( ))
1+
1+
Z
(q()
dF (u) + Z
(e
q (u))U 00 ( (u))dF (u);
e)dF ( );
Even under risk neutrality, the condition under which no bunching occurs is diÆcult to exhibit:
q_
=
S 00
Cq
1+
+
Cqq
1+
Ceq
1+
+
Ceqq
1+
F
C
1 + f qq
d(F=f )
d
Cq
+
F
Cq
f
:
If for risk neutrality, the optimal contract is separating (i.e. q_ < 0) then this will also be true for small enough
risk aversion by continuity.
21
R
with = U 0 (())dF () > 0: From corollary 2, we know that > 0. Increasing the
production level for the most eÆcient type of agent allows to increase the incentive to make
eort. Note that this does not allow to state that there is overproduction at the top () at
the optimum because the eort level diers from the perfect information level. Indeed, when
information is perfect and symmetric, then the optimal eort and production levels are given
by (2) and (3) respectively, i.e.:
e
PI
=
Z
PI
q P I ( )dF ( ) = qm
;
S 0 (q P I ( ))
1+
=
eP I :
Equation (13) describes the optimal production level in case of a separating equilibrium
with q_ 0, which is not a priori guaranteed. Assume further that the following assumption
holds:
F is uniform with f = 1=( )
H2 :
U (:) is CARA with U ( ) = (1 exp( ))=
where = U 00()=U 0 () is the (constant) degree of absolute risk aversion. This gives
R
CE ( ) = 1 log exp( ( ))dF ( ) .
Under H 1 and H 2, it is also possible to characterize more precisely the direction of the
incentive distortion due to hidden information in (13). We state the following proposition.
Proposition 4 Under H 1 and H 2,
(i) if 2(1 + )
1= < 0, then there is bunching for production for the less eÆcient
h
i
types (i.e. for 2 ~(); ),
e
(ii) in the no-bunching case, there is overproduction for the most ineÆcient type () for a
given level of eort i > f :
Proof: Part (i): Dierentiating (13) with respect to , we obtain:
d
S 00 (q ( ))q_( ) = 1 + + q ( )U 00 ( ( )) +
d
22
( )
:
f ( )
(14)
Under H 2, equation (14) becomes after straightforward computations:
S 00 (q ( ))q_( ) = 2(1 + )
(e + 1= ) exp(
( )):
(15)
Denote the right member of the equality by & () 2(1+ ) (e + 1= ) exp( ()), which
is strictly decreasing in (recall that 0 () < 0). Whenever & () = 2(1 + ) e 1= is
strictly positive, then q_() < 0 for all and the equilibrium contract is separating. On the
contrary, if & () < 0, then there is bunching over an interval (~(); ], that is for the more
ineÆcient types.
If there is bunching (q_() = 0), then (15) implies that:
e + 1=
1
~
~
2(1 + ) (e + 1= ) exp( ()) = 0 , () = log 2(1 + )
which gives ~():
Part (ii): For = , (13) reduces to:
()
U 0 (0) +
S 0 (q ( )) (1 + )( e) =
=
+
f
f
and the conclusion follows.
Figures 2 and 3 depict the multiplier () and the output level q() in the CARA case.
This indicates the increase in positive distortion due to risk aversion compared to risk neutrality. Risk aversion entails more production for eÆcient types of agent than under risk
neutrality and less for intermediate types. Figures 4 and 5 depict the multiplier and the
eort level in function of the risk aversion degree. It is shown that the eort level exterted
by the agent decreases with his risk aversion degree. In parallel, the risk premium increases
with (Figure 6).
5 Private principal
We briey consider in this section the extension of our model to the case of a private principal.
In that case, the principal's objective reduces to maximize the expected surplus of production
23
less the expected transfer given to the agent:
Z
max
e; q (:);t(:)
[S (q())
t( )] dF ( )
which amounts to observe that the certainty equivalent of prot disappears in the objective
and that the cost of transfer 1 + is 1. However, the set of constraints faced by the private
principal is the same as the one for the public regulator. Hence, it is not diÆcult to show
that the costate variable () is now:
( ) = F ( ) + Z
Ce (q (u); e; u) + 0 (e) U 00 ( (u)) f (u)du
while the shadow cost of moral hazard becomes:
R
0 )f + F ()Ce g d
f(Ce +
o :
= R nR
0
00
0
00
(
C
+
)
U
(
)
f
du
C
+
U
(
)(
C
+
)
f
d
e
ee
e
It is worth noting that when there is no moral hazard (that is = 0, i.e. the LaontRochet setting) then risk aversion for the agent plays absolutely no role in the optimal
contract, given the assumption that ex-post individual rationality constraints imply ex-ante
individual rationality. Obviously, this is no longer the case when one consider the presence
of moral hazard.
6 An alternative model with endogenous distribution of types
We analyze here an alternative situation where eort exerts an inuence on the distribution
of parameter instead of reducing the cost of production C .7 We can think of e as an
investment decision chosen by the agent. This class of models has been studied by Laont
and Tirole (Chapter 1, 1993) and more recently by Szalay (2001) for the case of a private
principal under the assumption of risk neutrality. Laont and Martimort (2001, chapter 7.3)
also analyze this model under the assumption of risk neutrality with limited liability. We
propose here to extend these analyses to the case of a risk averse agent.
We still assume that the cost of production C (q; ) is increasing in
condition).
7
24
and that
Cq >
0 (single-crossing
Consider the case where the risk-averse agent may exert a continuous level of eort e
which inuences the realization of parameter through the conditional distribution F (; e)
and an associated density function f (; e) on a non-moving support = ; for any eort e.
We assume that F (:; e) is twice dierentiable with respect to e and that all these distributions
have the same full support . We assume that a higher eort or investment makes it more
likely that the agent is eÆcient, that is we assume rst-order stochastic dominance with
Fe = @F=@e > 0 for 2 (; ). We also assume there are decreasing returns in investment:
Fee 0.
Assume for the moment that information is symmetric so that the level of eort and are both observable by the principal. The principal's problem can be written as follows:
max
e; q (:);t(:)
CE ( ) +
Z
[S (q()) (1 + )t()] f (; e)d
subject to
( ) = t( )
C (q ( ); )
(e)
( ) 0; 8 :
(IR)
First, as in the preceding model, it is easy to see that the agent has no rent ex post at the
optimum whatever her type, () = 0; 8 . Furthermore, maximizing the expected welfare
over q(:) gives the following necessary condition for an optimum:
S 0 (q ( )) = (1 + ) Cq (q ( ); );
(16)
which indicates that the marginal social benet of production is equal to its marginal cost
weighted by the social cost of public funds. Finally, maximising the expected welfare over e
gives:
Z [S (q()) (1 + ) (C (q(); ) + (e))] fe(; e)d = (1 + ) 0 (e)
Integrating by parts the left hand side of this equality and using (16) leads to the following
condition:
Z Fe (; e)
0 (e) =
C (q ( ); )
dF (; e);
(17)
f (; e)
25
which states that the perfect information eort level should equalize the marginal cost of
eort and the expected marginal benet in terms of cost savings.
Now, assume that the eort is non contractible and that is private information to the
agent. The principal's program is now:
max
e; q (:);t(:)
CE ( ) +
Z
[S (q()) (1 + )t()] f (; e)d
(P2)
subject to
( ) = t( )
C (q ( ); )
( ) (; ^);
e 2 arg max
e
Z
(e)
8; 8 ^
(IC)
U ( ( ))f (; e)d
(MH)
( ) 0; 8 :
(IR)
where (IC), (MH) and (IR) respectively are the incentive compatibility constraints, the moral
hazard constraint and the ex post individual rationality constraints. The derivation of the
optimal solution to this program is somewhat analogous to the preceding case of cost reducing
eort and the exposition is relegated into appendix C. The following proposition indicates
necessary conditions for an optimum involving separation.
Proposition 5 In case of no bunching and assuming an interior solution for e and q, necessary conditions for an optimum are:
(i) S 0 (q())
(1 + )Cq (q(); ) = f((;e)) Cq (q(); ),
where
( )
=
Z
+
1+
Z
U 0 ( (u))
U 0 (CE ( ))
f (u; e)du
0 (e)U 00 ((u))f (u; e)
26
U 0 ( (u))fe (u; e) du:
and (dropping all arguments for clarity)
=
(ii)
R
U 0 ( ( ))
h
(iii) () = () +
R
0 ( e)
R
f (1 + ) 0 f + [S (1 + ) ( + C + )] feg d
:
R
00 f + 0 fe )U 0 () feeU ()g d
f(
i
;e)
C (q ( ); ) Ffe((;e
) f (; e)d = 0,
C (q (u); u)du.
Proof: See appendix C.
The main dierence with the previous model is that all distortions on the screening
variable q due to moral hazard and risk sharing come through the distortion due to hidden
information (as indicated by (i)). Condition (ii) is the rst order condition corresponding to
the moral hazard constraint while condition (iii) states the level of ex post rents left to the
agent.
7 Conclusion
We have analyzed a model of public procurement where the agent takes an unobservable
eort before learning privately the particular conditions of production. The novelty of our
model is that we consider that the agent is risk averse. We characterize the optimal solution
to the regulator's problem in the presence of moral hazard, asymmetric information and
risk sharing, given general assumptions on net expected social surplus and the agent utility
function. We show that usual results such as \no distortion at the top" no longer holds:
depending on the cost function properties, overproduction or underproduction may occur.
Moreover, compared to the perfect information production level, distortions due to moral
hazard and asymmetric information can be additive or not. Finally, risk aversion for the
agent will often generate bunching for the less eÆcient types.
27
Appendix
A Proof of proposition 1
The Lagrangian of the program reads:
L = CE () +
Z
f [S (q()) (1 + ) (() + C (q(); e(); ) + (e()))]f ()
( )_e + ( )( C (q ( ); e( ); )
_ ( ))
( )('( ) + q_( )) + ( )(M (; e( ); q ( ); ( ))f ( )
K_ )
+1()'() + 2()e()g d:
Integrating by parts and using initial and terminal conditions gives:
Z
Z
Z
( )_e( )d
Z
=
( )q_( )d
_ ( )e( )d;
_ ( )K ( )d;
_ ( )q ( )d;
Z
( )K_ ( )d =
Z
=
because () = () = 0 (q() and q() are free) and () = () = 0 (e() = e and e() is
free).
Thus, plugging these expressions into the Lagrangian and dropping for the sake of
clarity, L becomes
Z L = CE () + H(; e; q; ;_ ; '; K )d;
where
H(; e; q; ;_ ; '; K ) = W(; e; q; )f + (
+e
_
C (q; e; )
' + q
_ + M (; e; q; )f
+1' + 2 e:
Pointwise maximization gives us the following necessary conditions:
28
_ )
+ _ K
First,
@H
@'
and + 1
condition:
<
0)
'
=
+ 1
0 everywhere
= 0: Moreover, we also have the complementary slackness
1
0; ' 0 and 1 ' = 0:
Thus, whenever '() > 0; we have () = 1() = 0.
Moreover, we have
@H
@e
= Wef
@H
@K
@H
@q
Ce + _ + Me f
+ 2 = 0;
(18)
= _ = 0 ) () = everywhere;
(19)
= Wq f
(20)
Cq + _ + Mq f
= 0;
with the following slackness condition:
2
0; e 0 and 2 e = 0;
and the transversality conditions:
( )
= () = 0;
( )
= () = 0:
(21)
With respect to we have to compute the derivative of L with respect to in the
direction of an arbitrary dierentiable function h() such that h() >0 and h()= 0: Let us
dene ~ () = ()+ th() with t a real number and I (t) = L( + th): Assume that is optimal
which means that function I is maximum at 0, then a necessary condition is I 0 (0) = 0 which
can be written as follows:
I 0 (0)
L( + th) L() = @ L h
tlim
!0
t
@
Z 0
= UE0((UCE(()h)))
(1 + )hf + h0 + (Ce + 0 )U 00()hf d = 0
29
or
Z
U 0 ( )
U 0 (CE ( ))
Z
0 )U 00 () hf d
1 + + (Ce +
h0 d = 0:
(22)
Let us denote
B ( ) =
Z
U 0 ( (u))
U 0 (CE ( ))
0 (e))U 00 ((u)) f (u)du
1 + + (Ce (q(u); e; u) +
then (22) can be rewritten as
Z
Z
B 0 ( )h( )d
( )h0 ( )d = 0:
Integrating by part the rst integral, we obtain:
Z
[B ()h()]
(B () + ()) h0()d = 0:
(23)
Recall that h is arbitrary and that h() = 0: Moreover B () = 0: Then (23) implies that
( ) =
B ( );
or
( )
Z
=
Z
+
U 0 ( (u))
U 0 (CE ( ))
(1 + ) f (u)du
(Ce (q(u); e; u) + 0 (e))U 00 ((u)) f (u)du:
(24)
Plugging (24) into (18) and assuming an interior solution for e and q (2 = 0) gives:
_
=
We f +
+
Z
Z
U 0 ( )
U 0 (CE ( ))
(Ce (q; e; u) +
(1 + )
f du
0 (e))U 00 ()f du Ce
Me f:
Integrating and using (21), we obtain:
Z
d
_
= 0=
+
Z
Z
Wef
Z
Z
(Ce +
U 0 ( )
U 0 (CE ( ))
0 )U 00 ()f du Ce
30
(1 + )
f du Ce d
Me f d:
Thus, the co-state variable is equal to:
o
o
R n
Rn
U 0 ( )
(1
+
)
f
du
C
d
W
f
+
0
e
e
U (CE ( ))
o
=
R nR 0 )U 00 ()f du Ce Me f d
(Ce +
and the co-state variable () is given by (24)8 . Plugging these values in (20) and recalling
that in the no bunching case () = 0 ) _ () = 0 gives (i). Condition (MH) gives (ii).
Finally, integrating (5) gives (iii). To obtain (iv), it suÆces to maximize the Lagrangian with
respect to () given (iii):
@L
@ ( )
(1 + )
=
R
U0 U 1
Z
(
U 0 ( ) +
hR
R
C (q (u); e; u)du dF ( )
U ( ) +
R
i
C (q (u); e; u)du dF ( )
Ce (q ( ); e; ) + 0 (e) U 00 () +
Z
!)
C (q (u); e; u)du
dF ( ) =
():
If () = 0 at the optimum, then a necessary condition is :j@ L@() 0 or equivalently
() 0. When U is CARA with being the absolute degree of risk aversion, then from
(24), we obtain that:
Z Z U 0 ( (u))
U 0 ( (u))
() =
(1
+
)
f
(
u
)
du
=
1
+
f (u)du
0
U 0 (CE ( ))
U (CE ( ))
because rst U 00 = U 0 and second the moral hazard constraint together imply that the
second term of () is zero. Furthermore, we also have U 0(CE ()) = E (U 0 (()) and
consequently, () = > 0. The conclusion follows.
Otherwise, if () > 0 then it is dened by the solution of @@L() = 0, because the
Lagrangian is not necessarily concave in () due to the presence of the certainty equivalent,
CE ( ).
( )=0
B
Proof of corollary 2
Part (i): When Ce = 0, then we obtain from proposition 1:
R
(1 + ) (Ce + 0)f d
:
= R
0
00 )f d
U
(
)(
C
+
ee
8
Note that We = (1 + ) (Ce + 0 ) and Wq = S 0
(1 + )Cq .
31
(25)
Recall that we have assumed that Cee + 00 0 for the second order conditions for the agent's
choice of eort to be fullled. Hence, the sign of is determined by the sign of the numerator
in (25). Now, from the rst order condition corresponding to the moral hazard constraint,
we get that:
Z (Ce(q(); e; ) + 0 (e))U 0 (())dF () = 0;
which can be rewritten as follows (dropping arguments for clarity):
cov (Ce +
0 ; U 0 ()) +
We also have:
"Z
(Ce +
# "Z
0 )dF ()
U 0 ( )dF ( )
#
= 0:
(26)
d
(Ce + 0 ) = Ceq q_
d
and
d 0
(U ()) = U 00 ()_ > 0
d
because _ = C < 0. Hence, because q_ < 0, the sign of cov(Ce + 0 ; U 0()) is given by
the sign of Ceq . If Ceq < 0, then cov(Ce + 0 ; U 0()) > 0 and consequently, from (26),
R
0
(Ce + )dF ( ) is necessarily non positive, which states that > 0. On the contrary, when
R
Ceq > 0, then cov (Ce + 0 ; U 0 ( )) < 0 and (Ce + 0 )dF ( ) > 0, so that < 0.
Part (ii): At the top ( = ), we have from part (i) in Proposition 1:
S 0 (q ( ))
(1 + )Cq (q(); e; ) = U 0 (())Ceq (q(); e; )
recalling that () = 0. As shown above, Ceq < 0 in any case and consequently S 0 (q()) <
(1 + )Cq (q(); e; ) which states the conclusion.
C
Proof of proposition 5
First of all, (IC) can be reduced as usual to the following set of conditions
_ ( ) =
C (q ( ); ) < 0
Cq (q ( ); )q_( ) 0:
32
(27)
(28)
Given that _ is strictly negative, constraints (IR) reduce to:
( ) 0:
Also, as we have Cq > 0 (single-crossing conditions), (28) reduces to q_ 0. The optimal
production is thus non-increasing in : Dene q_ = ' where ' is a control variable and
redene q as a state variable. Then constraint (28) reads as ' 0.
Once again, we follow a rst order approach for the moral hazard constraint on e. Redene
e as a state variable with the equation e_ = 0 and e( ) 0; e( ) = e; e( ) free. Recalling that
( ) = t( ) C (q ( ); )
(e()); the rst order condition of the program dened by (MH)
writes as follows
d
de( )
=
=
Z
U ( ( ))f (; e( ))d
N (e( ); ( ); )d = 0
Z
"Z
#
(29)
U 0 ( ( )) 0 (e( ))f (; e( )) + U ( ( ))fe (; e( )) d
where N (:) U 0(()) 0 (e())f (; e()) + U (())fe(; e()). To deal with this integral
constraint, dene a new state variable as follows:
K_
= N (e(); (); ) with K () = K () = 0:
We have to check that local second order conditions are satised, i.e
d2
de( )2
"Z
U ( ( ))f (; e( ))d
#
0:
We obtain that,
d2
de( )2
"Z
U ( ( ))f (; e( ))d
#
=
Z
( 0 )2 f U 00
U 0 00 f
2U 0
0 fe + U fee d
Given our assumptions on Fe and Fee , we have fe > 0 and fee < 0 and therefore the second
order conditions are satised.
33
Let us denote W(; e(); q(); ()) the term equal to:
W(; e(); q(); ()) = S (q()) (1 + ) (() + C (q(); ) + (e())) :
We then transform the program (P2)-(IC), (MH), (IR) into an optimal control problem:
max
CE ( ) +
'(:)
subject to
Z
W(; e(); q(); ())f (; e())d
e_ = 0;
_ = C (q ( ); );
q_ = '( );
K_ = N (; e( ); ( ));
'( ) 0;
e( ) 0;
e( ) = e;
( ) 0;
K ( ) = K () = 0:
()
( )
( )
()
(1 )
(2 )
The Lagrangian of the program reads:
L = CE () +
Z
f [S (q()) (1 + ) (() + C (q(); ) + (e()))]f (; e)
( )_e + ( )( C (q ( ); )
_ ( ))
( )('( ) + q_( )) + ( )(N (; e( ); q ( ); ( ))
K_ )
+1()'() + 2()e()g d:
Integrating by part and using initial and terminal conditions gives:
Z
Z
( )_e( )d
( )q_( )d
=
=
Z
Z
_ ( )e( )d;
Z
( )K_ ( )d =
Z
_ ( )K ( )d;
_ ( )q ( )d;
because () = () = 0 (q() and q() are free) and () = () = 0 (e() = e and e() is
free).
Plugging these expressions into the Lagrangian and dropping for the sake of clarity, L
becomes
Z L = CE () + H(; e; q; ;_ ; '; K )d;
34
where
H(; e; q; ;_ ; '; K ) = W(; e; q; )f (; e) + (
+e
_
C (q; )
_ )
' + q
_ + N (; e; ) + _ K + 1' + 2 e:
Pointwise maximization gives us the following necessary conditions:
First,
and
@H
@'
+ 1 < 0 ) ' = 0:
=
+ 1
0 everywhere
Moreover, the complementary slackness condition is:
1
0; ' 0 and 1 ' = 0:
Thus, whenever '() > 0; we have () = 1() = 0.
Moreover, we have
@H
= Wef + W fe + _ + Ne + 2 = 0;
@e
@H
= _ = 0 ) () = everywhere;
@K
@H
= Wq f Cq + _ = 0;
@q
(30)
(31)
(32)
with the following slackness condition:
2
0; e 0 and 2 e = 0;
and the transversality conditions:
( )
= () = 0;
( )
= () = 0:
From (30) and assuming an interior solution for e and q (2 = 0), we get:
_ =
where We = (1 + ) 0 and Ne = (
Wef W fe
Ne :
00 f + 0 fe)U 0 () + feeU ().
35
(33)
Integrating and using (33), we obtain:
Z
d
_ =0=
Z
f Wef W fe
Ne g d:
Thus, the co-state variable is equal to:
R
=
fWef + W feg d
R
R
Ne d
f (1 + ) 0 f + [S (1 + ) ( + C + )] feg d
:
=
R
00 f + 0 fe )U 0 () feeU ()g d
f(
Integrating by parts both the numerator and the denominator, we further obtain:
R
f (1 + ) 0 f + [S (1 + ) ( + C + )] feg d
=
:
R
00 f + 0 fe )U 0 () feeU ()g d
f(
Now, with respect to we have to compute the derivative of L with respect to in the
direction of an arbitrary dierentiable function h() such that h() >0 and h()= 0: Let us
dene ~ () = () + th() with t a real number and I (t) = L( + th): Assume that is
optimal which means that I (0) is a maximum of the function I , then a necessary condition
is I 0(0) = 0 which can be written as follows:
L( + th) L() = @ L h
tlim
!0
t
@
Z 0
(1 + )hf + h0 + = UE0((UCE(()h)))
I 0 (0)
0 f U 00 ()
fe U 0 ( ) h d = 0
or
Z
U 0 ( )
U 0 (CE ( ))
1++
fe 0
U ( )
f
0 U 00 ()
Z
hf d
h0 d = 0:
(34)
Let us denote
B ( ) =
Z
U 0 ( (u))
U 0 (CE ( ))
1++
0 U 00 ((u))
fe (u; e) 0
U ( (u))
f (u; e)
then (34) can be rewritten as
Z
B 0 ( )h( )d
Z
36
( )h0 ( )d = 0:
f (u; e)du
Integrating by part the rst integral, we obtain:
[B ()h()]
Z
(B () + ()) h0()d = 0:
(35)
Recall that h is arbitrary and that h() = 0: Moreover B () = 0: Then (23) implies that
( ) =
B ( );
or
( )
=
Z
+
Z
U 0 ( (u))
U 0 (CE ( ))
(1 + ) f (u; e)du
0 (e)U 00 ((u))
fe (u; e) 0
U ( (u)) f (u; e)du:
f (u; e)
(36)
Plugging these values in (32) and recalling that in the no bunching case () = 0 )
_ ( ) = 0 gives (i). Condition (29) gives (ii). Finally, integrating (27) gives (iii).
D
Numerical simulation procedure
We describe in this section the numerical procedure used to solve for the optimal solution,
in the linear example of section 5. The agent's utility function has the CARA specication:
U ( ) = [1 exp( )]=(1 ), and the social surplus function is logarithmic: S (q ) = " log(q ),
where scale parameter " is set to 0.5. The opportunity cost of public funds has parameter set to 0.4.
The numerical algorithm essentially solves for a xed-point problem in q(). The distribution function for the agent's type is assumed uniform in [; ]. We rst generate 200
equidistant values for in the interval [ = 1; = 3], and choose initial values for eort level
(e0 = 0:5), multipliers and () (0 = 0:5, 0 () = 0:1), and prot (0() = 0). The initial
output supply path q() is computed from initial values above, as a crude solution to part
(i) in Proposition 1:
q0 ( ) = "
(1 + )(
e0 ) + 0 U 0 (0 )
37
0 ( )
f ( )
1
:
This is a crude solution path, as the right-hand side also implicitly depends on q(), but as
multiplier and prot paths are updated, convergence toward the actual solution for q() on
the right-hand side of this expression is expected. We then solve for () from the rst-order
condition to the Lagrangian problem described in Appendix A. Based on these values for
q0 ( ) and ( ), we compute the new prot path ( ), parameters and , as well as the
updated eort level e and the () multiplier path. Expressions for multiplier and parameter
values are in the text, see Proposition 2. The nal step is then to update the output level path
q ( ) using the crude approximation as above. In updating parameters and variable paths,
we use a smoothing technique allowing for soft convergence as the algorithm might otherwise
become stuck if successive values are too far apart. The smoothing parameter, weighting the
new value as opposed to the former one, is set to 0.6. The algorithm stops when convergence
criteria are met, i.e., when the relative change in prot, eort and output level is less than
1.0E-8.
38
References
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[4] Jewitt, I. (1988), \Justifying the First-Order Approach to Principal-Agent Problems",
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[5] Laont J.J. and D. Martimort (2002),
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The theory of incentives: The principal-agent
[6] Laont J.J. and J.C. Rochet (1998), \Regulation of a risk averse rm",
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Games and
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39
Figure 2: () multiplier path.
40
Figure 3: Output supply path q():
41
Figure 4: Shadow cost of moral hazard as a function of .
42
Figure 5: Eort level e:
43
Figure 6: Relative risk premium as a function of .
44