EC537 Microeconomic Theory for Research Students, …econ.lse.ac.uk/staff/lfelli/teach/EC537 Slides...

EC537 Microeconomic Theory for Research Students,Part II: Lecture 2

Leonardo Felli

CLM.G.4

15 November 2011

Moral Hazard:

Consider the contractual relationship between two agents (agent 1 and 2)summarized in the following problem generated by the take-it-or-leave-itoffer that agent 1 makes to agent 2:

maxe,wi

N∑i=1

pi v(e, yi ,wi )

s.t.N∑i=1

pi u(e,wi ) ≥ U

(1)

where:

v(·, ·, ·) is agent 1’s utility function;

u(·, ·) is agent 2’s utility function;

U is the reservation utility of agent 2.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 215 November 2011 2 / 86

Moreover:

e can be interpreted as agent 2’s effort or investment;

e enhances the random variable y interpreted as expected profit orexpected outcome;

wi is a transfer contingent on yi from agent 1 to agent 2;

θ is the state of nature and pi is the probability of state θi .

Assume that:

e is chosen by agent 2 before the state of nature θ is realized;

e is only observed by agent 2. It is his private information.


Label:

agent 2, who exerts effort, as the agent A;

agent 1, who benefits from the effort, as the principal P.

Assume that:

y is verifiable information (observable to all agents involved in thecontract court included).

Moreover, it is critical for the problem to be interesting that:

y is not in a one-to-one relation with the effort e.

In other case, the contracting problem will result to a highly simplifiedversion of an optimal risk-sharing problem.


First Best:

Assume first that the effort chosen by the agent e is verifiable.

Then w can be a function of e. Let e∗ be the optimal effort from theprincipal’s view point.

The Principal’s optimal contract then will specify:

a state contingent payment {w1(e), . . . ,wN(e)} to the agent that isindividually rational if and only if e = e∗:

N∑i=1

pi u(e∗,wi (e∗)) = U

a state contingent payment {w1(e), . . . ,wN(e)} to the agent that isnot individually rational (a punishment) otherwise.


This can of course also be achieved with a y∗ that corresponds to e∗

in a one-to-one fashion.

Therefore the problem is interesting only when output is a noisysignal of effort: y = f (e, θ).

The principal is thus restricted to offer the contract w(f (e, θ)) and eis chosen by the agent so as to maximize his expected utility:

e ∈ arg maxe

N∑i=1

pi u(e,w(f (e, θi ))).

The contract w(f (e, θ)) must be such that it is in the agent’s bestinterest to choose the “right” (desired) level of effort e.


The principal’s problem is:

maxe,w(f (e,θ))

N∑i=1

pi v(e, f (e, θi ),w(f (e, θi )))

s.t.N∑i=1

pi u(e,w(f (e, θi ))) ≥ U

e ∈ arg maxe

N∑i=1

pi u(e,w(f (e, θi )))

The latter constraint is known as the agent’s incentive compatibilityconstraint.

The former constraint is the agent’s individual rationality constraint.


The solution to the principal’s problem with both constraints is ingeneral not a trivial matter.

Key tradeoff: the one between insurance and incentives.

Recall that without moral hazard the choice of e achieves ex-postallocative efficiency, while the choice of w achieves optimal risksharing.

This cannot be done in the presence of moral hazard.


Second Best: risk neutral principal

Assume that the principal is risk neutral:

v(e, f (e, θ),w) = f (e, θ)− w

Optimal risk-sharing is achieved by giving the agent full insurance:w(f (e, θi )) = τ, ∀i ∈ {1, . . . ,N}.

The optimal choice of e will then be: e ∈ arg maxe u(e, τ).

Clearly the agent will choose the same effort level independently ofthe contract (τ is independent of e).

This means that any conflict of interest between the principal and theagent will not be ameliorated by the contract.


Special case: effort as pure cost:

Assume that e is a pure cost for the agent:

u(e,w) = U(w)− c(e), u′ > 0, c ′ > 0, c ′′ > 0

Then the agent will minimize effort. If e ≥ 0 then: e = 0.

This differs from the effort level e that the principal desires:

maxe

N∑i=1

pi f (e, θi )− c(e)

orN∑i=1

pi∂f (e, θi )

∂e= c ′(e)

Optimal risk sharing implies no incentives.


Risk neutral agent:

The special case with no tradeoff between incentives and insurance iswhen the agent is risk neutral.

The solution is to sell the firm/activity to the agent. Let

w(f (e, θi )) = f (e, θi )− κ

Then e = e since:

e ∈ arg maxe

N∑i=1

pi f (e, θi )− κ− c(e)

While κ is independent of e and such that

N∑i=1

pi f (e, θi )− κ− c(e) = U


General Characterization:

We provide a more detailed characterization of the moral hazardproblem by considering its simplest general form.

The principal hires the agent to perform a task.

The agent chooses his effort intensity, e, which affects the outcome ofthe task, q.

The principal only cares about the outcome, but effort is costly forthe agent, hence the principal has to compensate the agent forincurring the cost of effort.

Effort is observable only to the agent, hence the agent’s compensationhas to be contingent on the outcome q: a noisy signal of effort.


Assume that the outcome of the task can take only two values:q ∈ {0, 1}.

We assume that when q = 1 the task is successful and when q = 0the task is a failure.

The probability of success is:

P{q = 1|e} = e.

The principal’s preferences are represented by:

V (q − w), V ′(·) > 0, V ′′(·) ≤ 0

where w is the transfer to the agent.


The agent’s preferences are represented by the utility functionseparable in income and effort:

U(w)− c(e), U ′(·) > 0, U ′′(·) ≤ 0

where c ′(·) > 0, c ′′(·) ≥ 0.

For convenience we take

c(e) = e2/2

and we normalize the agent’s outside option:

U = 0.


First Best Contract:

The first best contract can be contingent on e. It is obtained as thesolution to the problem:

maxe,wi

e V (1− w1) + (1− e) V (−w0)

s.t. e U(w1) + (1− e) U(w0) ≥ e2

2

The optimal pair of transfers w∗1 and w∗0 are such that the followingFOC (Borch optimal risk-sharing rule) are satisfied:

V ′(1− w∗1 )

U ′(w∗1 )=

V ′(−w∗0 )

U ′(w∗0 )


These transfers are paid only if the effort level coincides with e∗

satisfying:

e∗ = [U(w∗1 )− U(w∗0 )] +U ′(w∗1 )

V ′(1− w∗1 )[V (1− w∗1 )− V (−w∗0 )]

Finally the agent’s expected utility coincides with the outside option:

e∗U(w∗1 ) + (1− e∗) U(w∗0 ) = (e∗)2/2


If the principal is risk neutral:

V (x) = x

Then the conditions above become:

w∗1 = w∗0 = w∗

andU(w∗) = (e∗)2/2, e∗ = U ′(w∗)

If the agent is risk neutral:

U(x) = x

Then the optimum entails:

w∗1 − w∗0 = 1, e∗ = 1.


Second Best Contract:

If e is not verifiable then for every w1 and w0 it is determined so that:

maxe

e U(w1) + (1− e) U(w0)− e2/2 (2)

The second best contract can be contingent only on q.

It is obtained as the solution to the problem:

maxe,wi

e V (1− w1) + (1− e) V (−w0)

s.t. e U(w1) + (1− e) U(w0) ≥ (e)2/2

e ∈ arg maxe

e U(w1) + (1− e) U(w0)− e2/2


The FOC of the incentive compatibility constraint are:

e = [U(w1)− U(w0)] (3)

A first observation: from this condition it is clear that full insuranceleads to no incentives:

e = 0

Notice also that our assumptions imply that the solution to theagent’s incentive problem is unique for any pair (w0,w1).

This allow us to replace the agent’s (IC) by (3): the solution to theFOC of the agent’s incentive problem.

In general, replacing the (IC) constraint with the FOC of the agent’seffort choice problem is not a valid approach as we will see later on.


Consider now the case in which the agent is risk neutral:

U(x) = x

we have seen that first best optimality requires

e∗ = 1

In this case the FOC of the (IC) constraint becomes:

e = (w1 − w0)

Therefore settingw1 − w0 = 1

leads to the first best allocation: optimal risk sharing and optimalincentives.


The reason is that:

optimal risk sharing requires that the agent bears all the risk in theenvironment,

optimal incentives requires that the agent is residual claimant.

This is achieved by selling the activity to the agent at a fix price

−w0 > 0

so that the risk averse principal receives full insurance.

Notice that in this case we need the agent to have deep enoughpockets: when the outcome is q = 0 the agent’s payoff is w0 < 0.


In other words, the agent must be willing to incur a loss with astrictly positive probability.

It is often natural to assume that the agent has no resources to put inthe activity.

This implies a resource or limited liability constraint: wi ≥ 0.

In this case the problem becomes:

maxe,wi

e V (1− w1) + (1− e) V (−w0)

s.t. e w1 + (1− e) w0 ≥ (e)2/2

e = (w1 − w0)

wi ≥ 0 ∀i ∈ {0, 1}


In the situation in which the agent is resource constrained not all therisk can be transferred to the agent: the constraint wi ≥ 0 will bebinding for the transfer w0: w0 = 0

It is still possible to create first best incentives but for this purposethe agent’s needs to be rewarded.

If e∗ = w1 − w0 = w1 = 1 then the agent’s payoff is:

e∗w1 − (e∗)2/2 = 1/2 > 0

In other words the (IR) constraint is not binding. This is notnecessarily optimal for the principal.


In particular, if we assume that the principal is risk neutral as well:

V (x) = x

then the principal’s problem is:

maxe,wi

e (1− w1)− (1− e) w0

s.t. e w1 + (1− e) w0 ≥ (e)2/2

e = (w1 − w0)

wi ≥ 0 ∀i ∈ {0, 1}

The solution implies that

w0 = 0, w1 = e


Moreover, e solves the constrained problem:

maxe

e (1− e)

s.t. (e)2/2 ≥ 0

ore = 1/2

We then conclude:e = 1/2 < e∗ = 1.

The resource constraint implies a second best level of effort.


The principal trades off the lower effort choice by the agent againstthe higher compensation that the agents needs to provide first bestlevel of effort.

However, the agent still gets a strictly positive payoff:

ΠA = (e)2/2 = 1/8 > 0

Indeed, the principal also received a strictly positive profit:

ΠP = e(1− 2) = 1/4


Recall that social surplus associated with the first best problem i.e.the solution to the social planner problem is such that:

ΠP + ΠA = e∗ − (e∗)2/2 = 1/2

Compare it to the social surplus in equilibrium:

ΠP + ΠA = 3/8

The gap is the result of the agency problem.


Continuous Outcomes:

Consider now the more general environment in which the state-spacerepresentation of the effort’s outcome is the random variable q(e, θ),where θ ∈ [θ, θ].

First, for simplicity assume that φ(e) = e.

Second, let us consider the parameterized distribution characterizationof the same problem: q ∈ Q = [q, q] is the support of the densityf (q, e) > 0 and cdf F (q, e).

We assume that: Fe(q, e) < 0, ∀q ∈ [q, q]

In other words, the effort e produces a first-order stochastic dominantshift on Q. If e0 < e1:


-

6

F (q, e)

F (q, e1)

F (q, e0)

q qq

1


First Best:

In this setup the first best contract solves:

maxe,w(·)

∫ q

qV (q − w(q)) f (q, e)dq

s.t.

∫ q

qU(w(q)) f (q, e)dq ≥ e

That is:V ′(q − w∗(q))

U ′(w∗(q))= λ, ∀q ∈ Q

∫ q

qU(w∗(q)) f (q, e∗)dq = e∗

and ∫ q

q[V (q − w∗(q)) + λU(w∗(q))] fe(q, e∗)dq = λ


Second Best:

The second best contract solves:

maxe,w(·)

∫ q


s.t.

∫ q

qU(w(q)) f (q, e)dq ≥ e

e ∈ arg maxe

∫ q

qU(w(q)) f (q, e)dq − e

The (IC) constraint implies, assuming an interior optimum:∫ q

qU(w(q)) fe(q, e)dq = 1

and ∫ q

qU(w(q)) fee(q, e)dq ≤ 0


We proceed using the so called first order approach: substitute the(IC) constraint with the FOC of the agent’s incentive problem.

In this case the lagrangian is:

L =

∫ q


+ λ

[∫ q


]

+ µ

[∫ q

qU(w(q)) fe(q, e)dq − 1

]


There exists no constraint on the first and second derivative of w(·).

Therefore it is possible to solve the problem by pointwisemaximization:

V ′(q − w(q))

U ′(w(q))= λ+ µ

fe(q, e)

f (q, e), ∀q ∈ Q

This is, once again, the modified Borch rule.


Assume that:

the agent is strictly risk averse:

U ′′(·) < 0,

the distribution F (q, e) has fixed support Q:

Fe(q, e) = Fe(q, e) = 0,

the distribution F (q, e) satisfies first order stochastic dominance:

Fe(q, e) < 0


Result (Holmstrom, 1979)

Assume that the first order approach is valid then at the optimum: µ > 0.

Proof: Assume not: µ ≤ 0.

From the necessary condition

∂L∂e

= 0

and the modified (IC) constraint we obtain:∫ q

qV (q − w(q)) fe(q, e)dq + µ

[∫ q

qU(w(q)) fee(q, e)dq

]= 0


Using the second order condition of the agent’s incentive problem∫ q

qU(w(q)) fee(q, e)dq ≤ 0

when µ ≤ 0 it becomes:∫ q

qV (q − w(q)) fe(q, e)dq ≤ 0

Define w0(q) as the function that solves, for µ = 0:

V ′(q − w0(q))

U ′(w0(q))= λ, ∀q ∈ Q

From U ′′(·) < 0, w ′0(·) exists and is such that:

0 ≤ w ′0(q) =V ′′ U ′

V ′′ U ′ + V ′ U ′′< 1


Moreover, when µ ≤ 0 we have from

V ′(q − w(q))

U ′(w(q))= λ+ µ

fe(q, e)

f (q, e), ∀q ∈ Q

thatw(q) ≤ w0(q) ⇔ fe(q, e) ≥ 0

w(q) > w0(q) ⇔ fe(q, e) < 0

Since

∂(V ′(q − w(q)/U ′(w(q)))

∂w(q)= −V ′′ U ′ + V ′ U ′′

(U ′)2> 0

We therefore conclude that for all q ∈ Q:

V (q − w(q)) fe(q, e) ≥ V (q − w0(q)) fe(q, e)


Integrating over Q we then obtain:∫ q

qV (q − w(q)) fe(q, e) dq ≥

∫ q

qV (q − w0(q)) fe(q, e) dq

Integrating by parts and using the fixed support and the first orderstochastic dominance we conclude:∫ q

qV (q − w0(q)) fe(q, e) dq =

=[V (q − w0(q)) Fe(q, e)

]qq−

−∫ q

qV ′(q − w0(q)) (1− w ′0(q)) Fe(q, e) dq > 0

A contradiction of the necessary conditions above.


Observations:

We have assumed that e is in the interior of E , it is a much simplerproblem if the agent’s incentive problem leads to a corner solution.

The assumption that Q does not depend on e is crucial, movingsupport may lead to first best outcome.

Commitment to the contract is critical: risk will be renegotiated awayfrom the agent between choice of e and realization of q.

The fe/f is the derivative of the ln f and the gradient for a MLE of e.

We proved 0 ≤ w ′0(·) < 1 but we have not proved that w(·) ismonotonic.


MLRP:

The monotone likelihood ratio property (MLRP) is satisfied for F (·)and f (·) iff

d

dq

(fe(q, e)

f (q, e)

)> 0

Notice that MLRP implies FOSD, Fe(q, e) < 0 for every q ∈ Q.

In fact:

Fe(q, e) =

∫ q

q

fe(s, e)

f (s, e)f (s, e)ds (4)

Since ∫ q

q

fe(s, e)

f (s, e)f (s, e)ds = Fe(q, e)− Fe(q, e) = 0

Then for every q < q the fact that the likelihood ratio is increasing ins implies that (4) is strictly negative.


Result (Holmstrom, 1979, Shavell 1979)

Under the first order approach, if f (·) satisfies the MLRP, then the wagecontract w(·) is increasing in output.

The proof is a direct consequence of the definition of MLRP.

Notice that if the agent can freely dispose of output then monotonicity ofw(·) is a constraint that the solution must satisfy.

Result (Holmstrom, 1979, Shavell 1979)

Under the first order approach, if w(q) is the solution then there exists anew contract w(q, s) that strictly Pareto dominates w(q) if and only if[fe(q, s, e)/f (q, s, e)] varies with s.


First Order Approach:

The first order approach of first finding w(·) using the relaxed problemand then checking that the principal’s choice of e actually solves theagent’s incentive problem is logically invalid (additional restrictions).

Problem

if the SOC of the agent’s incentive problem are not globally satisfied, thenthe solution to the principal’s problem satisfies the agent’s FOC but notnecessarily the principal’s ones.

This is because the global maximum to the principal’s problem mightinvolve a corner solution and so the principal’s selected e may not satisfythe necessary Kuhn-Tucker conditions of the relaxed problem.


Definition

A distribution F (·) satisfies the Convexity of Distribution FunctionCondition (CDFC) if and only if for every γ ∈ [0, 1]:

F (x , γe + (1− γ)e ′) ≤ γF (x , e) + (1− γ)F (x , e ′)

orFee(x , e) ≥ 0.

Special case of CDFC is the linear distribution condition: let f (q) FOSDf (q) then f (q, e) = ef (q) + (1− e)f (q)

Result (Mirrlees 1976, Rogerson 1985)

The first order approach is valid if F (q, e) satisfies MLRP and CDFC.


“Proof:” Integrating by parts the agent’s payoff becomes, if w(·) isdifferentiable:∫ q

qU(w(q)) f (q, e)dq − e =

[U(w(q)) F (q, e)

]qq−

−∫ q

qU ′(w(q)) w ′(q) F (q, e) dq − e

Differentiating it with respect to e twice we get from CDFC for everye ∈ E :

−∫ q

qU ′(w(q)) w ′(q) Fee(q, e) dq ≤ 0


We conclude the proof with the observation that under MLRP and thefirst order approach µ > 0 and w ′(·) > 0.

This reasoning is however wrong.

We know that µ > 0 only if the first order approach is valid.

Therefore in proving that the first order approach is valid we cannot takefor granted that µ > 0.


Proof: Consider an alternative principal’s relaxed problem where the (IC)constraint is substituted by

d

de

[∫ q


]≥ 0

Inequality constraint implies that the constraint multiplier µ′ ≥ 0 thereforeunder MLRP the solution to this new problem is such that

w ′(q) ≥ 0

We still need to show that:


Lemma (Rogerson 1985)

The solution to the new principal’s problem is such that

d

de

[∫ q


]= 0

Proof: If µ′ > 0 then we are done.

Assume then that µ′ = 0.


The lagrangian is then:

L′ =

∫ q

qV (q − w(q)) f (q, e)dq +

+ λ

[∫ q


]+

+ µ′d

de

[∫ q


]Therefore from the necessary first order condition with respect to e we get:

∂L′

∂e=

∫ q

qV (q − w(q)) fe(q, e)dq +

+ λd

de

[∫ q


]= 0


Moreover if µ′ = 0 then w(q) ≡ w0(q) and we already proved that

0 ≤ w ′0(q) < 1.

Recall further that integrating by parts the principal’s expected utility weget: ∫ q

qV (q − w0(q)) fe(q, e) dq =

= −∫ q

qV ′(q − w0(q)) (1− w ′0(q)) Fe(q, e) dq > 0


Therefore the necessary first order conditions above imply that λ > 0 and:

d

de

[∫ q


]< 0

Clearly a contradiction.

Very few distribution satisfy MLRP and CDFC. An example is thegeneralization of the uniform distribution for e ∈ [0, 1):

F (q, e) =

(q − q

q − q

) 11−e

Jewitt (1988) provides a collection of alternative sufficient conditions onf (q, e) and U(w(q)) which are weaker than CDFC.


First Order Approach (Grossman and Hart 1983)

Consider now the original principal’s problem and assume that there areonly a finite number of outputs

q1 < q2 < . . . < qN .

Clearly in this casef (qi , e) = Pr{q = qi |e}

Assume that the principal is risk neutral, V (x) = x and that U(x) satisfiesInada conditions for x → 0 and E is compact.

Assume that the agent’s utility function is (K (e)U(w)− φ(e)):

U(w)− e


The second best contract is then the solution to the following problem:

maxe,w(·)

N∑i=1

f (qi , e)(qi − w(qi ))

s.t.N∑i=1

U(w(qi )) f (qi , e) ≥ e

N∑i=1

U(w(qi )) f (qi , e)− e ≥

≥N∑i=1

U(w(qi )) f (qi , e)− e ∀e ∈ E


We solve the problem in two steps:

find the least cost way to implement a given action e,

find the optimal e from the principal’s view point.


Step 1:

minwi

N∑i=1

f (qi , e) wi

s.t.N∑i=1

U(wi ) f (qi , e) ≥ e

N∑i=1

U(wi ) f (qi , e)− e ≥

≥N∑i=1

U(wi ) f (qi , e)− e ∀e ∈ E


This is not a convex programming problem, to be able to use Kuhn-Tuckertheorem we change variable. Define h(·) = U−1(·) we then use as controlvariable:

ui = U(wi )

The now convex programming problem is:

minui

N∑i=1

f (qi , e) h(ui )

s.t.N∑i=1

ui f (qi , e) ≥ e

N∑i=1

ui f (qi , e)− e ≥N∑i=1

ui f (qi , e)− e ∀e ∈ E


Result (Grossman and Hart 1983)

If either K ′(e) = 0 or φ′(e) = 0 then the (IR) constraint of this problem isbinding.

Notice that in this case the agent’s preferences over action lotteries isindependent of income.

Define:

C (e) =

inf

{∑i

f (qi , e)h(ui )

}, if e obtained;

+∞, otherwise.


Step 2: maxe

N∑i=1

f (qi , e) qi − C (e)

Result (Grossman and Hart 1983)

Step 2’s solution exists and the inf in C (e) is a min.

Assume that E = {eL, eH} and e = eH then

h′(ui ) =1

U ′(wi )= λ+ µL

(f (qi , eH)− f (qi , eL)

f (qi , eH)

)µL > 0 and wi increases with (f (qi , eL)/f (qi , eH)).


CARA Utility, Normal and Linear Contracts:

Let the principal be risk neutral V (x) = x and the agent’s preferencesbe represented (a denotes effort, r is the index of absolute riskaversion) by:

U(x , a) = −e−r(x−φ(a))

Let the outcome q be such that:

q = a + ε where ε ∼ N (0, σ2)

The cost of effort is quadratic:

φ(a) =a2

2

and we normalize U = −1.


Restrict now the principal to offer only linear contracts:

w(q) = β q + γ

Recall also that ifx ∼ N (µ, σ2)

then:

Ex

[et x]

= eµ t− 12σ2 t2


First best:

maxw(·),a

Eq [q − β q − γ]

s.t. Eq

[−e−r(β q+γ−φ(a))

]≥ −1

effort choice a∗ = 1

transfer to the agent w∗(q) = 1/2 for every q.

expected profit of the principal Π∗ = 1/2


Second best:

The principal’s problem is:

maxw(·),a

Eq [q − β q − γ]

s.t. Eq


]≥ −1

a ∈ arg maxa

Eq


]

Notice that:

Eq

[−e−r(β a+β ε+γ−φ(a))

]= − e−r(β a+γ−φ(a))+ 1

2β2 r2 σ2


Therefore the agent’s incentive problem becomes:

a ∈ arg maxa

β a + γ − a2

2− r

2β2 σ2

The unique solution to this problem is then

a = β

Then the second best problem becomes:

maxγ,a

a− a2 − γ

s.t. a2 + γ − a2

2− r

2a2 σ2 ≥ 0


The solution is then:

a = β =1

1 + r σ2, γ =

r σ2 − 1

2 (1 + r σ2)2

Moreover:

a < a∗, Π =1

2 (1 + r σ2)< Π∗

Comparative static: if either r or σ2 decreases the power of theoptimal incentive scheme increases (less distortion).


Unrestricted Contracts (Mirrlees 1999)

Question: what if the contract space is unrestricted?

Result (Mirrlees 1999)

If f (q, a) is a normal distribution with mean a and variance σ2 andunlimited punishments are possible, the first best outcome can beapproximated arbitrarily closely.

Let the principal be risk neutral V (x) = x and the agent’s preferences beU(w , a) = −e−r(w−φ(a)).


Proof:

By assumption:

f (q, a) =1√

2π σe

−(q−a)2

2σ2

Therefore:fa(q, a)

f (q, a)=

d

daln f (q, a) =

(q − a)

σ2

In other words: detection is efficient for very small q.


Recall that the first-best contract is the solution to the following problem:

maxw(q),a

Eq [q − w(q)]

s.t. Eq

[−e−r(w(q)−φ(a))

]≥ −1

The first order conditions of this problem imply:

e−r(w(q)−φ(a)) =1

λ r, ∀q ∈ R


This condition clearly implies full insurance:

w(q) = w , ∀q ∈ R

We can now re-write the principal’s problem as:

maxw ,a

[a− w ]

s.t. −e−r(w−φ(a)) ≥ −1


or equivalently:maxw ,a

[a− w ]

s.t. w − φ(a) ≥ 0

A linear programming problem where the objective function is monotonicdecreasing in w , while the constraint is monotonic increasing in w .

Therefore in equilibrium the constraint is necessarily binding:

w∗ = φ(a∗), φ′(a∗) = 1


Consider the following approximate first best contract:

for every q ≥ q0 (q0 very small) the agent is paid the constanttransfer w∗,

for every q < q0 the agent is paid the very small constant amount k .

The amount k is chosen (low enough) so that it is optimal for the agent tochoose a = a∗:∫ q0

−∞U(k) fa(q, a∗) dq +

+

∫ +∞

q0

U(w∗) fa(q, a∗) dq = φ′(a∗)


This can always be done choosing a low enough k since from the necessaryfirst order conditions of the agent’s (IC) constraint:∫ q0

−∞U(k) fa(q, a) dq +

+

∫ +∞

q0

U(w∗) fa(q, a) dq = φ′(a)

we obtain:

sign

{d a

d k

}= sign

{∫ q0

−∞U ′(k) fa(q, a) dq

}=

= sign{

U ′(k) Fa(q, a)}

Therefored a

d k< 0


We still need to show that this contract is such that the agent’s (IR)constraint can be satisfied arbitrarily closely.

The loss with respect to the first best is:

∆ =

∫ q0

−∞(U(w∗)− U(k)) f (q, a∗)dq

Define

M(q0) =fa(q0, a

∗)

f (q0, a∗)

Since the normal distribution satisfies MLRP, for all q < q0, we have thatfor a low enough q0 such that fa(q0, a

∗) < 0:

f (q, a∗) <fa(q, a∗)

M(q0)


Therefore

∆ ≤ 1

M(q0)

∫ q0

−∞(U(w∗)− U(k)) fa(q, a∗)dq

Using the necessary first order conditions of the (IC) constraint we obtainthat: ∫ q0

−∞U(k) fa(q, a∗) dq =

−∫ +∞

q0

U(w∗) fa(q, a∗) dq + φ′(a∗)


We can, therefore, rewrite this bound as:

∆ ≤ 1

M(q0)

[∫ +∞

−∞U(w∗)fa(q, a∗)dq − φ′(a∗)

]

The bound on ∆ is then a constant (with respect to k and q0) divided byM(q0).

Therefore as q0 decreases unboundedly by MLRP also M(q0) decreasesunboundedly. Therefore ∆→ 0

In other words, the (IR) is approximately satisfied for a low enough q0.


Intuition:

In the lower tail of the normal distribution q is very informative abouta.

Therefore, a harsh punishment can achieve first best incentives at acost to the principal.

The unboundedness of the support of the distribution allows theprincipal to render these costs arbitrarily small.

The result generalizes to other distributions with unbounded supportand to general utility functions.


Multitasking (Homstrom and Milgrom 1991, 1994)

Assume that the agent can carry out multiple tasks that affect output.

Consider our previous model: both the principal and the agent arerisk neutral but the agent has limited liability: wi ≥ 0.

The agent now controls two tasks: one is “standard” (S) and one is“noisy” (N).

The agent chooses two effort levels: eS and eN , eS ∈ [0, 1],eN ∈ [0, 1].

Disutility of effort: (e2S + e2N)/4.

Assume that the two tasks are perfect complements in the stochastictechnology:

P{q = 1|eS , eN} = min{eS , eN}


ΠP is expected output, minus expected wage.

ΠA equals the agent’s expected wage, minus the disutility of effort.

Assume that q is not contractible.

Task S generates the contractible binary signal σS ∈ {0, 1} that isequal to 1 with probability eS .

Task N generate the contractible binary signal σN ∈ {0, 1} that isequal to 1 with probability [eNp + (1− eN)(1− p)] with p ∈ [1/2, 1].

If p = 1/2 then σN contains no information about eS , while if p = 1,the signals σS and σN are equally informative.


A contract is now a quadruple of wages (wS0,wS1,wN0,wN1), one foreach task, and for each possible value of the corresponding signal.

Limited liability implies wS0 = wN0 = 0.

The (IC )s constraints pin down eS and eN as

eS = 2wS1, eN = 2wN1(2p − 1)

The principal’s problem of maximizing ΠP subject to the restrictionsabove gives:

eS = eN = max{0, 1/2− (1− p)/(8p − 4)}

Notice that when p = 1 we get the same outcome as in the singletask case, if instead p ≤ 3/5 then eS = eN = 0.


Notice that as p decreases (task N becomes more noisy) two changesoccur.

In equilibrium, eN decreases (the increased noise increases the cost ofN incentives).

The effort eS decreases as well: increased noise yields softerincentives on the standard task, as well as the noisy one.

Complementarity of tasks dictates that as eN becomes moreexpensive the principal chooses to induce lower eS as well.

When p ≤ 3/5, σN is not informative enough. In this caseeS = eN = wS1 = wN1 = 0 (no incentive contract is signed).


Informed Principal (Maskin and Tirole 1990, 1992)

Effort is e non-contractible and private information of A.

Output q ∈ {0, 1} is contractible and

P{q = 1|e} = e

Assume that also the principal has private information, creating apotential signaling role for the contract offer.

Consider a simple “Common values” case as in Maskin and Tirole(1992).

There are two types of principal: PH and PL.


Assume that P = PH with probability φ = 18/29 and P = PL withprobability 1− φ = 11/29.

If P is of type H, A’s outside option is U = 9/32, while if P is oftype L then A’s outside option is U = 0.

If PH and PL separate in their contract offer than in equilibrium twoIRs need to be satisfied for A.

If pooling obtains A faces a single IR and his expected outside optionis

φU = 81/464

A has limited liability as described above.


Timing:

First P learns his type.

P offers a contract to A, which may take the form of a menu (wagescontingent on output and P’s announcement).

After seeing the offer A updates his beliefs about P’s type and thendecides whether to accept or reject.

As in any signaling game, the issue of off-the-equilibrium-path beliefsarises: assume that A’s beliefs after observing an “unexpected” offerare that P is of type H with probability 1.

After a contract is signed P announces to A which part of the menuapplies in his case (if the contract is in fact a menu).

Finally, A chooses effort, output is realized and payoffs are obtained.


In a separating equilibrium PH and PL offer two distinct pairs ofoutput-contingent wages: (wH1,wH0) and (wL1,wL0).

A’s IC s dictate that after being offered (wH1, wH0) effort is eH =wH1 − wH0, while after being offered (wL1, wL0) effort is eL =wL1 − wL0.

Separation requires that neither PH nor PL has an incentive to offerthe other type’s wage pair.

Since P’s private information does not enter directly his payoff, thiscan only be true if the expected profits for the two types of principals,ΠH and ΠL are the same truth telling constraint:

ΠH = eH(1− eH) = eL(1− eL)− wL0 = ΠL


Since U = 9/32 for the H agent then his IR does bind.

Using IC this yields eH = wH1 = 3/4.

Using the principal’s truth telling constraint we obtain eL = 1/2, wL0

= 1/16 and wL1 = 9/16.

In other words, ΠH = ΠL = 3/16.


Consider now the possibility of pooling equilibria, in which thecontract is a menu.

Both PH and PL offer the same menu (wMH1, wM

H0, wML1, wM

L0), whichA has to accept or reject based on his expected IR.

After a contract is signed, P tells A which pair of output-contingentwages applies.

The truth-telling constraint still applies, since both PH and PL haveto be willing to indicate to A the appropriate wage pair.


Using IC and wMH0 = 0, the truth telling constraint reads:

ΠMH = eMH (1− eMH ) = eML (1− eML )− wM

L0 = ΠML

The single binding expected IR and the IC s yield

(18/58)(eMH )2 + (11/29)[(eML )2 + wML0] = 81/464

From the truth-telling constraint we then get: eH = wH1 = 5/8, eL =1/2, wL0 = 1/64 and wL1 = 33/64.

In other words: ΠH = ΠL = 15/64.


Both types of P enjoy strictly higher profits under pooling than underseparation.

Pooling relaxes A’s IR which binds in expectation.

PH can lower wH1 which increases ΠMH relative to the separation case.

The increased profit for PH affects PL via the principal’s truth-tellingconstraint.

PL lowers both output-contingent wages to satisfy the truth-tellingconstraint, which in turn increases ΠM

L to keep it in line with ΠMH .


EC537 Microeconomic Theory for Research Students, …econ.lse.ac.uk/staff/lfelli/teach/EC537 Slides...

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