INTRODUCTION
TO
PRINCIPAL-AGENT
THEORY
UNDER MORAL HAZARD
Moral Hazard :
• Moral Hazard occurs in the Principal-Agent relationship when
some actions of the agent are not perfectly observable.
• Contractual payment cannot depend on variables that are not
observable by the Principal, the Agent, and by an outside (or
third) party : a Judge.
• This poses the problem of performance measures. They may
be more or less imprecise ”signals” of the Agent’s true activity.
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Moral Hazard and Risk
• Important property. In situations of Moral Hazard, the probabilities of outcomes, success or failure of projects, etc., depend
on hidden actions of the agents.
• Uninformed parties (i.e., Principals) must take into account
this fact.
• To create effort incentives, contracts will typically involve an
element of profit-sharing, and therefore impose risk upon the
agent.
• Risky compensation hurts the risk-averse agent. This generates
transactions costs.
2
Application of the Theory
• Hierarchical Organization in Firms. Piece Rates. Compensation
Policy.
• Delegation and Outsourcing ; Public Procurement.
• Theory of Labor Contracts.
• Theory of Financial Intermediation (Banking),
Insurance Markets,...
• Financial Theory of the Firm. Corporate Finance.
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A Model of the Principal-Agent Relationship :
• Grossman and Hart’s model (1983).
• We follow the presentation by David Kreps (1990).
• Agent chooses an action (or effort) a in a set A.
• Principal observes an outcome s in a set S.
• A = {a1, a2, ..., aN } (A is a discrete finite set).
• S = {s1, s2, ..., sM } (S is also finite).
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Formal Representation of Moral Hazard and Risk :
• Define πnm = P rob[sm | an]
•
X
πnm = 1 for all n.
m
• ASSUMPTION 1 : We assume that πnm > 0 for all n and m.
Any outcome is possible, under every action.
• The Agent is potentially risk averse. We assume,
U (w, a) = u(w) − a
where w = wage.
• ASSUMPTION 2 : Risk aversion : u : IR → IR is strictly increasing, concave and continuously differentiable.
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Definition of a Contract :
• A contract is a function w : S → IR or equivalently :
w = (w1, w2, ..., wm), where wm is compensation if outcome sm is
observed.
• Note : It is crucial here that the outcome sm is observable and
verifiable. The Principal, the Agent, and possibly a Judge, can
observe sm and check if wm = w(sm) is paid to the Agent.
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The Principal’s Objective and Information
• The Principal is risk neutral and observes the outcome (signal)
sm only.
• His objective is expected profit, net of expected wages E(w | a),
that is,
B(a) − E(w | a)
• B(an) =
X
πnmsm = E(s | an)
m
• Note E(w | an) =
X
πnmwm
m
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The Principal-Agent Problem
• Choose (a, w) ∈ A × IRM so as to maximize
B(a) − E(w | a)
subject to the Agent’s participation constraint,
E(u(w) | a) − a ≥ Uo,
and subject to the incentive constraint,
E(u(w) | a) − a = max{E(u(w) | α) − α}.
α∈A
• Note : Uo is the Agent’s best outside option.
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First-Best Effort :
• Compute the minimal certain reward for an, denoted C o(an)
C o(an) = u−1[an + Uo]
• C o(an) is the cost of action an if action a is observable.
• The First-Best effort is a solution of the problem
M ax[B(a) − C o(a)]
a∈A
• Let a∗ be the first-best effort.
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Transformation of the Principal-Agent Problem :
• Define, xm = u(wm).
• Denote, v = u−1.
• We have wm = v(xm), and v is convex.
• Then,
E(wm | an)
=
X
πnmv(xm)
m
E(u(wm) | an) =
X
πnmxm.
m
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Cost of Incentives :
• The cost minimization problem : for a given action an,
Minimize
X
πnmv(xm)
m
subject to the participation constraint,
X
πnmxm ≥ Uo + an
(IR)
m
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and s.t. the incentive constraints,
X
πnmxm − an ≥
m
X
πνmxm − aν
(ICν )
m
for all ν = 1...N .
• Minimization is with respect to
x = (x1, x2, ..., xM ).
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• The solution of the above problem is a cost function
C : A → IR+
• C(an) = M in{E(w | an) | (IR) and (ICν )},
w
the smallest expected wage required to obtain an effort level an
from the Agent.
• Second-Best Effort : a solution of
M ax[B(a) − C(a)]
a∈A
• We will show that the second-best solution may be different
from a∗ since C(a) > C o(a).
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Efficient Risk-Sharing
• Assume that an is observable, then, the optimal contract problem is just
M in
x
X
πnmv(xm)
m
s.t.,
X
πnmxm ≥ Uo + an
m
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• Write the first-order conditions (λ is a Lagrange multiplier) :
v 0(xm) = λ for all m
(Borch’s equation).
• This implies x1 = x2 = ... = xM = x∗
(Full Insurance) or equivalently, w1 = w2 = ... = wM = w∗.
•
x∗ = Uo + an
w∗ = v(Uo + an).
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Proposition 1 : If the agent is strictly risk averse (i.e., u00 < 0),
then, C o(an) < C(an) for any action an that is more costly than
some other action, i.e., such that an > M in(aν ).
A
Proof : Simple. If agent is risk averse, the unique efficient risksharing implies Full Insurance (as shown above). A constant wage
implies that chosen effort is M in(a). Thus, ICν is binding for
A
some ν...
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Proposition 2 : The IR constraint is always binding at the optimum.
Proof : If not
X
m
πnmxm > Uo + an choose ε > 0 and x0m = xm − ε
for all m. If ε small enough, IR and ICν are still satisfied. But
X
expected wage
πnm v(x0m) is smaller : contradiction.
m
• Remark : u(w) is not bounded below here.
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Taking ICν constraints into account
• Lagrangian :
L(x, λ, µ) = −
X
X
πnmv(xm) + λ
m
m
+
X
ν6=n
µν
πnmxm − an − Uo
X
(πnm − πνm)xm − an + aν
m
• First-Order Conditions : for all m,
v 0(xm) = λ +
X
ν6=n
µν
πνm 1−
πnm
(Mirrlees’ equation)
• Interpretation : variable wage ; wm = base wage + bonuses or
penalties...
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Special Case : Risk-Neutral Agent.
• Assume u(w) ≡ w (risk-neutrality)
Proposition 3 : If the Agent is risk neutral, then, the first-best
effort can be implemented,
M ax(B(a) − C(a)) = M ax(B(a) − C o(a))
a∈A
a∈A
and
wm = sm − B(a∗) + C o(a∗)
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• Proof : Under the proposed contract, the Agent chooses an so
as to maximize,
E[U (w, an)]
=
X
πnmsm − (B(a∗) − C o(a∗)) − an
m
= B(an) − an − constant,
and, C o(an) = an + Uo, thus,
E[U (w, an)] = B(an) − C o(an) + Uo − constant.
This proves that the agent will choose an = a∗.
Q.E.D.
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We have proved that
C(a∗) = B(a∗) − B(a∗) + C o(a∗)
= C o(a∗).
Since : B(a) − C(a) ≤ B(a) − C o(a)
and : B(a∗) − C o(a∗) = max(B(a) − C o(a)).
A
We proved that :
max(B(a) − C(a)) = max(B(a) − C o(a)).
A
A
Q.E.D.
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Further Properties of the Optimal Contract
• The optimal contract trades off insurance and incentives : If
the Agent is better insured, the power of incentives is reduced.
• But the contract also exploits the information conveyed by the
signal sm on effort an.
• Is it true that the optimal contract is a profit-sharing contract
in the sense that :
w1 < w2 < ... < wM
if
s1 < s2 < ... < sM ?
• Assume now that signals are ranked : sk < sk+1.
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• Assume that effort levels are ranked : a1 < a2 < ... < aN .
• Define Πnm = P rob(s ≥ sm | an)
• ASSUMPTION 3 : If ν > n then Πnm ≤ Πνm for all m = 1...M ,
with a strict inequality at least for some m.
• First-Order Stochastic Dominance :
Increasing effort increases the probability of getting a higher outcome. This implies that B(an) is increasing in n.
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Example (Kreps)
• Three levels of profit :
s1 = 1, s2 = 2, s3 = 10, 000.
• Two levels of effort :
a1 = 1 and a2 = 2.
• The probabilities πnm are :
s
10,000
2
1
P r(s | a = 1)
0.2
0.3
0.5
s
10,000
2
1
P r(s | a = 2)
0.5
0.1
0.4
• Check that πnm satisfy the FOSD assumption.
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• Suppose that a2 is the optimal effort level.
• Write Mirrlees’ equations :

µ
0.5

0


=λ−
v (x1) = λ + µ 1 −



4
0.4






0.3
= λ − 2µ
v 0(x2) = λ + µ 1 −

0.1








3
0.2

0

=λ+ µ
 v (x3) = λ + µ 1 −
0.5
5
• x1 = u(w1), x2 = u(w2), etc...
• Remark that wm is non monotonic : w2 < w1 and w3 > w2 !
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• We need two additional assumptions.
• ASSUMPTION 4 : (MLRP)
If an < aν and sm < sµ, we have
πνµ
≥
πnµ
πνm
πnm
• The Monotone-Likelihood Property. Can we now prove
wk+1 > wk ? Not yet !
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• Using Mirrlees’ equations, we get,
v 0(xm) − v 0(xµ) =
X
ν
µν
π
νµ
πνm −
πnµ πnm
and µν ≥ 0 (multipliers are ≥ 0).
• We want wµ > wm or xµ > xm ⇒ v 0(xµ) > v 0(xm).
The difference above should be negative.
• We obtain the desired results if µν = 0 for all ν > n.
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Convexity of Distribution Function :
• ASSUMPTION 5 :(CDF)
The cumulative distribution function of outcomes s is convex
with respect to effort :
Πnm = P rob(s ≥ sm | an)
is a concave function of a.
• Interpretation : increases in effort have decreasing marginal
impact on the probabilities of better outcomes.
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Proposition 4 : If u is strictly concave, (πnm) satisfy FOSD,
MLRP and CDF, then the optimal wage-incentive scheme has
wages that are nondecreasing functions of the level of firm profits, i.e.,
w1 ≤ w2 ≤ ... ≤ wM .
Proof : (see Kreps (1990))
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Readings :
• David KREPS (1990), A Course in Microeconomic Theory,
Chapter 16, Harvester Wheatsheaf, pp. 577-624.
• Jean-Jacques LAFFONT and David MARTIMORT, (2002),
The Theory of Incentives : The Principal-Agent Model, Princeton University Press, Chapter 4, pp. 145-186.
• The source for the theory presented here :
Sanford GROSSMAN and Oliver HART (1983), ”An Analysis of
the Principal-Agent Problem”, Econometrica, 51, pp. 7-45.
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