ECON601
Spring, 2015
UBC
Li, Hao
Lecture 9. Moral Hazard
Motivation
• Origin of “moral hazard:” insurance on babies
• General characteristics of moral hazard problems
– Unobserved actions of one party, called agent, exert externality on the payoff
of the other, called principal.
– Principal can write an enforceable contract on verifiable outcomes that depend
on agent’s action.
– Agent enters the contract on a voluntary basis.
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• Hidden action problems vs hidden information problems
– Solution to hidden information problems is self-selection menus; solution to
hidden action problems is contracts.
– Incentive condition is truthful reporting of hidden information; and obeying
with recommended action.
General setup of moral hazard model in employer-employee relationship
• Employee chooses effort e, not observed by employer.
– Employee’s payoff is U (w, e) = u(w) − c(e), where w is wage paid by employer,
with u0 > 0 and u00 < 0 (employee is risk averse), and c0 > 0 and c00 ≥ 0 (effort
cost function is convex).
– Employee’s reservation utility is U.
• Employer’s payoff is V (y).
– y is employer’s profit, with V 0 > 0 and V 00 ≤ 0 (employer may be risk neutral).
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• Wage contract takes form of sharing of output x.
– w( x ) is wage, and x − w( x ) is profit when output is x.
• Modeling stochastic dependence of output x on employee’s effort e.
– Direct approach: postulate a function x (e, θ ), where θ is a random variable that
also affects output x, with some known distribution function—for example,
x (e, θ ) = e + θ.
– Parameterized approach: suppose output x ∈ R takes value from some set X,
which is either an interval [ x, x ] or finite (common support); write as f ( x |e)
density function (or probability function) of output when effort is e, and as
F ( x |e) the corresponding distribution function.
• Timing of game
– Employer makes a take-it-or-leave-it offer of a contract to employee.
– Game ends if employee rejects the offer; otherwise, employee chooses effort e,
output x is realized, and wage is paid according to contract and game ends.
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• Implicit assumptions:
– Symmetric information at contracting stage.
– Effort e is not observable to employer.
– Output is verifiable and contractible.
– No renegotiation of contract—both employee and employer commit to contract.
Optimal contract design problem
• Formulate contract design problem as a constrained optimization problem.
– Employer chooses wage function w( x ) and effort e to maximize payoff given
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by x∈X V ( x − w( x )) f ( x |e)dx, subject to
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– (IC) x∈X u(w( x )) f ( x |e)dx − c(e) ≥ x∈X u(w( x )) f ( x |e0 )dx − c(e0 ) for all e0 .
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– (IR) x∈X u(w( x )) f ( x |e)dx − c(e) ≥ U.
– We may think of the choice e in the maximization problem as “recommended
effort;” (IC) is then interpreted as “obedience.”
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• A two-step procedure for solving for optimal contract
– Step 1: For each recommended effort e, choose a wage function w( x ) in order
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to maximize x∈X V ( x − w( x )) f ( x |e)dx subject to (IC) and (IR), yielding the
expected profit as a function of e.
– Step 2: maximize the expected profit function by choosing e.
Benchmark: Observed actions
• Suppose that efforts are observable and contractible, so contract can be written on
actions directly.
– Still need to condition wage on output—employee is risk averse and values
insurance.
– But there is no (IC) constraint.
– Optimal contract problem is then to choose wage function w( x ) and effort e to
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maximize x∈X V ( x − w( x )) f ( x |e)dx subject to (IR) only.
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• Use the two-step procedure to solve for the benchmark.
– For any fixed e, in first step, (IR) binds at solution: if not could reduce w( x ) for
each x and increase profit.
– Let λ be the Lagrangian multiplier, and take derivative of objective with respect
to w( x ) for each x ∈ X: first order condition is V 0 ( x − w( x ))/u0 (w( x )) = λ.
– Can solve for each w( x ) in terms of λ and substitute in (IR) to get λ, and then
each w( x ), as functions of e.
– Optimal e can be found in second step.
• Optimal risk-sharing: ratio of marginal utilities is independent across output.
– In special case with V 0 constant, optimal risk-sharing implies full insurance for
employee: w( x ) is constant.
– (IR) becomes u(w) − c(e) = U, implying w = u−1 (c(e) + U ).
– In second step, optimal effort e∗ , called “first best,” maximizes total “surplus”
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−1
x ∈ X x f ( x | e )dx − u ( c ( e ) + U ).
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Hidden actions: When both employer and employee are risk neutral
• First best can be achieved even though employee’s effort is unobservable.
– Contract cannot be written on effort.
• Without loss of generality, assume u(y) = V (y) = y.
– Key observation: when employee is risk neutral, the first best e∗ is incentive
compatible if he is residual claimant to output—employee pays a constant fee
to employer (franchise fee, tenancy).
– After paying the fee, employee chooses e to maximize
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x∈X
x f ( x |e)dx − c(e),
leading to e∗ (for the case of risk neutral employee).
• Optimal contract for employer: choose a fee equal to
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x∈X
f ( x |e∗ )dx − (c(e∗ ) + U )
to bind (IR).
– Interpretation: employer “sells the firm” to employee.
– Employee is willing to bear all output risk, because of risk-neutrality.
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Two-effort model
• Suppose set of effort choices of employee is {e H , e L }, with e L < e H .
– A representative model demonstrating how to apply the two-step procedure.
– The central idea of moral hazard models: incentives, which require wage to
increase when higher output is realized, versus insurance, which requires riskneutral employer to insure risk-averse employee against inherent output risks.
• Assumption on output distributions: F ( x |e H ) first-order stochastically dominates
F ( x |e L ), that is, F ( x |e H ) ≤ F ( x |e L ) for all x ∈ X.
– e H leads to a higher output than e L , but only stochastically.
• Step 1: if e L is recommended effort.
– Choose w to bind (IR): u(w) − c(e L ) = U.
– (IC) is satisfied under constant wage, because c(e H ) > c(e L ): no need to provide
incentives through wage contract if employer wants to implement e L .
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• Step 1, continued: if e H is recommended effort.
– Consider the constrained maximization problem of choosing a wage contract
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w( x ) to maximize the expected profit x∈X ( x − w( x )) f ( x |e H )dx, subject to
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(IC) x∈X u(w( x )) f ( x |e H )dx − c(e H ) ≥ x∈X u(w( x )) f ( x |e L )dx − c(e L ), and (IR)
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x ∈ X u ( w ( x )) f ( x | e H )dx − c ( e H ) ≥ U.
– Let µ be multiplier for (IC) and λ multiplier for (IR), and write first order
condition with respect to w( x ) as: 1/u0 (w( x )) = λ + µ(1 − f ( x |e L )/ f ( x |e H )).
– µ > 0 at any solution w( x ) so that (IC) binds: otherwise, first-order condition
implies full insurance and constant wage, but then (IC) cannot hold because
c ( e H ) > c ( e L ).
– λ > 0 at any solution w( x ) so that (IR) binds: otherwise, for small positive e,
can define new contract w˜ ( x ) such that u(w˜ ( x )) = u(w( x )) − e for all x ∈ X and
(IR) remains satisfied; then (IC) is unaffected but value of objective is increased;
alternatively, if λ = 0, then first-order condition implies that f ( x |e H ) > f ( x |e L )
for all x ∈ X, an impossibility since they are density functions.
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• Step 2: does employer want to implement e H ?
– The answer may be no, if incentive cost of implementing e H is higher than the
benefit relative to first best solution to e L .
• Interpreting first order condition: tradeoff between incentive and insurance.
– Full insurance is not compatible with effort incentives: to provide incentives for
employee to choose e H instead of shirking by choosing e L , wage w must vary
with output x.
– Incomplete insurance is costly to employer: concavity of u implies that certainty
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equivalent of w( x ) to employee, wˆ such that u(wˆ ) = x∈X u(w( x ))dF ( x |e H ), is
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less than expected cost of the wage contract to employer, x∈X w( x )dF ( x |e H )
(Jensen’s inequality).
• Under what condition optimal contract exhibits wage monotonicity?
– Answer is: if f ( x |e H )/ f ( x |e L ) is increasing in x, from first order condition with
respect to w( x ) in step 1 when recommended effort is e H .
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• Monotone likelihood ratio property (MLRP): statistics
– f ( x |e H )/ f ( x |e L ) is likelihood ratio: for any given x, the ratio gives likelihood
of e H is chosen relative to likelihood e L is chosen, so under MLRP, the higher
the output x, the more likely that e H was chosen instead of e L .
– MLRP implies first order stochastic dominance: take any x ∈ [ x, x ]; by MLRP
f ( x1 |e H ) f ( x2 |e L ) ≤ f ( x1 |e L ) f ( x2 |e H ) for any x1 , x2 ∈ X and x1 ≤ x ≤ x2 ;
integrate both sides first from x1 = x to x1 = x, and then from x2 = x to x2 = x
to get F ( x |e H )(1 − F ( x |e L )) ≤ F ( x |e L )(1 − F ( x |e H )), so F ( x |e H ) ≤ F ( x |e L ).
• Monotone likelihood ratio property (MLRP): economics
– Wage monotonicity requires MLRP; first order stochastic dominance is not
enough.
– Wage monotonicity at optimal wage contract arises from incentive provision,
not from inference about effort; indeed, at the optimal wage contract, employer
knows employee has chosen e H .
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• Value of information
– Suppose contractible information is richer: besides output x, wage can also
depend on another consequence y of e.
– Same first order condition: 1/u0 (w( x, y)) = λ + µ(1 − f ( x, y|e L )/ f ( x, y|e H )).
– Optimal wage contract w( x, y) depends also on y if and only if the likelihood
ratio f ( x, y|e H )/ f ( x, y|e L ) depends on y.
– Statistics: x is “sufficient statistic” for ( x, y) in inference problem with regards
to e H versus e L , if f ( x, y|e H )/ f ( x, y|e L ) does not depend on y.
– Economics: contractible information should be used in the optimal contract
together with output only if it affects likelihood ratio; otherwise, dependence
would reduce employee’s certainty equivalent and hence profit.
– Application: is relative performance evaluation optimal when the outputs are
separately produced and observed? Answer is “no” if output distributions are
independent—“competition” is not useful per se.
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• Renegotiation?
– Suppose original wage contract is w( x ), to implement e H .
– After e H is chosen and before output x is realized, employee’s cost of effort
is sunk but there is output risk due to incomplete insurance provided by the
employer in order to provide incentives.
– Employer would have incentive to renegotiate the wage contract at this stage,
by offering the certainty equivalent w˜ of original contract w( x ) under e H , that
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is, u(w˜ ) = x∈X u(w( x )) f ( x |e H )dx.
– Employee is happy to accept new offer (is indifferent), but employer is better off
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by Jensen’s inequality: u(w˜ ) = x∈X u(w( x )) f ( x |e H )dx < u x∈X w( x ) f ( x |e H )dx ,
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implying that w˜ < x∈X w( x ) f ( x |e H )dx.
– Of course, if such renegotiation is anticipated, employer would not be able to
implement e H in the first place: inability to commit to not to renegotiate hurts
principal.
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Multiple levels of effort
• Grossman-Hart (ECMA83) setup.
– Finitely many effort levels, and finitely many outputs, x1 < x2 < . . . < xn , with
probability f ( xi |e) > 0 of output level xi given effort level e for each i.
• Two-step procedure again: first find the least costly way to implement each effort
level, and then find the effort that leads to maximum profit.
– For given e to be implemented, principal chooses wi , i = 1, . . . , n, to minimize
∑in=1 f ( xi |e)wi , subject to (IR) ∑in=1 f ( xi |e)u(wi ) − c(e) ≥ u, and (IC) that e solves
maxe˜ ∑in=1 f ( xi |e˜)u(wi ) − c(e˜).
– Consider change of variables: choose ui , i = 1, . . . , n.
– Objective is convex in choice variables, as wi = u−1 (ui ); (IR) is linear (with
multiplier λ); (IC) is linear constraints (with multiplier µ j for each e j 6= e); and
first order conditions 1/u0 (ui ) = λ + ∑ j:e j 6=e µ j (1 − f ( xi |e j )/ f ( xi |e)) are both
necessary and sufficient.
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Continuous effort
• Mirrlees’ setup
– Effort e is chosen from some compact interval, and f ( x |e) has common support
[ x, x ] for all feasible e.
• Main issue: use the same two-step procedure but in first step, there is a continuum
of IC for recommended effort.
• Rogerson (ECMA85)’s first order approach.
– Replace (IC) by agent’s first order condition with respect to e, solve resulting
relaxed problem, and impose conditions such that solution satisfies original
(IC) constraint.
– In last step, ensure agent’s effort choice problem is concave.
– For fixed e, if principal chooses w( x ), agent’s first order condition in effort
Rx
choice is x u(w( x )) f e ( x |e)dx = c0 (e), where f e ( x |e) ≡ ∂ f ( x |e)/∂e.
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• The relaxed problem: principal chooses w( x ) to maximize
subject to (IR)
Rx
x
Rx
x
( x − w( x )) f ( x |e)dx,
u(w( x )) f ( x |e)dx − c(e) ≥ u, and above first order condition of
agent.
– First order condition with respect to w( x ) from point-wise maximization is:
1/u0 (w( x )) = λ + µ f e ( x |e)/ f ( x |e), where λ is the multiplier for (IR) and µ the
multiplier for agent’s first order condition.
– Assume the continuous version of MLRP that f e ( x |e)/ f ( x |e) is nondecreasing.
– Solution to relaxed problem satisfies w0 ( x ) ≥ 0.
– Assume the convexity of distribution function condition (CDFC) on F ( x |e) that
F ( x |λe1 + (1 − λ)e2 ) ≤ λF ( x |e1 ) + (1 − λ) F ( x |e2 ) for all x ∈ [ x, x ], λ ∈ [0, 1],
and feasible e1 and e2 .
– With c00 ≥ 0, agent’s effort choice problem is concave in effort choice because
Rx
Rx 0
0
x u ( w ( x )) f ( x | e ) dx = u ( w ( x )) − x u ( w ( x )) w ( x ) F ( x | e ) dx is concave in e due
to CDFC and w0 ( x ) ≥ 0.
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Moral hazard in teams
• When multiple agents are engaged in joint production, there is also the free-rider
problem when individual efforts are not observed, in addition to trade-off between
insurance and incentives.
• Setup of a joint production problem:
– There are n agents; no principal.
– Each i, i = 1, . . . , n, chooses an effort ei ≥ 0, unobserved to any other agent or
the principal.
– No uncertainty: output x is deterministic function of effort profile: x = g(e1 , . . . , en ),
with g continuous, positively-valued, and increasing in each argument.
– Agents are all risk-neutral: payoff to i is si − ei , where si is i’s share of output
(effort cost function is linear).
– Agents’ non-participation payoff is 0.
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• A contract in this context is an output sharing agreement.
– Suppose that agents agree beforehand on sharing contract: (s1 ( x ), . . . , sn ( x ))
such that si ( x ) ≥ 0 for each i = 1, . . . , n, and ∑in=1 si ( x ) = x.
• Efficient (first best) efforts
– Suppose efforts are observed.
– Efficient effort profile (e1∗ , . . . , en∗ ) maximizes g(e1 , . . . , en ) − ∑in=1 ei : for any
other effort profile (eˆ1 , . . . , eˆn ), and for any sharing (s1 ( xˆ ), . . . , sn ( xˆ )) where
xˆ = g(eˆ1 , . . . , eˆn ), if each i switches to ei∗ and receives si ( xˆ ) − eˆi + ei∗ he would
be indifferent, but then we would have ∑in=1 (si ( xˆ ) − eˆi + ei∗ ) is smaller than
∑in=1 si ( g(e1∗ , . . . , en∗ )) by construction, implying the gain can be distributed to
make everyone better off.
– First order condition with respect to ei : ∂g(e1∗ , . . . , en∗ )/∂ei = 1.
– Suppose that there is a sharing arrangement (s1∗ , . . . , s∗n ) such that each agent is
willing to particapte: si∗ − ei∗ ≥ 0 for each i, and ∑in=1 si∗ = g(e1∗ , . . . , en∗ ).
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• Can efficient efforts be implemented under some sharing contract?
– Answer is “no:” simple proof restricts to differentiable sharing contracts.
– Suppose (e1∗ , . . . , en∗ ) is a Nash equilibrium under some (s1 ( x ), . . . , sn ( x )).
– Each i chooses ei to maximize si ( g(e1∗ , . . . , ei , . . . , en∗ ) − ei —first order condition
is given by si0 ( g(e1∗ , . . . , en∗ ))(∂g(e1∗ , . . . , en∗ )/∂ei ) = 1.
– Using the first order conditions for efficient profile (e1∗ , . . . , en∗ ), and summing
first order conditions over i, we have ∑in=1 si0 ( g(e1∗ , . . . , en∗ )) = n, contradicting
budget balance ∑in=1 si ( x ) = x.
• Intuition: output sharing means that each agent gets less than 100% of gain in
output as result of increase in his effort.
– Efficiency requires ∂g(e1∗ , . . . , en∗ )/∂ei = 1: each i exerts effort up to marginal
cost equal to marginal output.
– This means si0 ( g(e1∗ , . . . , en∗ )) = 1: to implement efficient effort, each i must
receive all the gain in output as his share, impossible under output sharing.
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• Budget balance and budget breaking
– A simple contract achieves the first best effort profile as a Nash equilibrium:
si ( x ) = si∗ if x = g( x1∗ , . . . , xn∗ ) and 0 otherwise.
– This contract violates budget-balance off the path.
– The budget-breaking theory of the firm: principal is someone who has no effort
decision to make and who is a participant to the sharing agreement to balance
the budget off the equilibrium path in order to achieve the first best.
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