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Essential Microeconomics -1 Sections 7-3, 7-4 © John Riley CHAPTER 7: UNCERTAINTY First and second order stochastic dominance 2 Mean preserving spread 8 Conditional Stochastic Dominance 10 Monotone Likelihood Ratio Property 12 Continuous distributions 15 Principal-Agent problem 21

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Page 1: CHAPTER 7: UNCERTAINTY - Essential Microeconomicsessentialmicroeconomics.com/ChapterY7/SlideChapter7B.pdf · Essential Microeconomics-1 Sections 7-3, 7-4 © John Riley CHAPTER 7:

Essential Microeconomics -1 Sections 7-3, 7-4

© John Riley

CHAPTER 7: UNCERTAINTY

First and second order stochastic dominance 2

Mean preserving spread 8

Conditional Stochastic Dominance 10

Monotone Likelihood Ratio Property 12

Continuous distributions 15

Principal-Agent problem 21

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Stochastic dominance

Three state case with 1 2 3x x x≺ ≺ . Preferences over prospects 1 2 3( , , )π π π π= are illustrated below.

Consider two prospects π and π̂ . The difference in expected utility is 3 3 3

1 1 1ˆ

s s s s s ss s s

U u u uπ π π= = =

Δ = − = Δ∑ ∑ ∑

Also 3 3

1 1ˆ 1s s

s sπ π

= == =∑ ∑ and so 2 1 3( ) 0π π πΔ + Δ + Δ = .

Then

1 1 1 3 2 3 3( )U u u uπ π π πΔ = Δ − Δ + Δ + Δ 1 2 1 3 3 2( ) ( )u u u uπ π= −Δ − + Δ −

Thus any change in probability that shifts weight from the worst

to the best outcome raises expected utility.

Also note that since 3 1 2( )π π πΔ = − Δ + Δ

Expected utility rises if both 1 0πΔ < and 1 2 0π πΔ + Δ < .

That is, the probability mass in each left tail declines.

1

1

First order stochastic dominance

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First order stochastic dominance

Definition: First Order Stochastic Dominance

Let Ax and Bx be two random variables with realizations in nX ⊂ . If for each x X∈ ,

Pr{ } Pr{ }B Ax x x x≤ ≤ ≤ , then Bx exhibits (first-order) stochastic dominance over Ax .

Proposition 7.3.1:

If ( )u ⋅ is increasing, 1 ... Sx x< < and the prospect ( ; )Bx π exhibits first-order stochastic dominance over

the prospect ( ; )Ax π , then prospect B is preferred to prospect A.

Exercise: Prove this result for the 4 state case. Hence sketch an argument for the S state case.

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Second Order Stochastic Dominance

Definition: Second-Order Stochastic Dominance

Let 1( ,..., )B B BSπ π π= and 1( ,..., )A A A

Sπ π π= be two probability distributions. Let BSΠ and A

SΠ be the

associated cumulative probabilities or left tail probabilities. Distribution B exhibits second order

stochastic dominance over A if the sums of left tail probabilities are lower for B than for A.

1 1

, 1,...,s s

B A s Sτ ττ τ= =

Π ≤ Π =∑ ∑ .

We now assume that 1s sx x δ+ − =

Three state example.

1 2 3( , , ) ( , , 2 )x x x w w wδ δ= + + .

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Risk-neutral individual

Three state example.

1 2 3( , , ) ( , , 2 )x x x w w wδ δ= + + .

2 1 3 2x x x x− = − therefore the slope of an

indifference curve 0 2 1

3 2

1x xmx x−

= =−

.

An equal increase in the probability of states 1 and 3 does not change the expected payoff.

Slope

Acceptable gambles for a risk-neutral individual

1

1

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Risk averse individual

Appealing to concavity 2 1 3 2( ) ( ) ( ) ( )u x u x u x u x− > − .

Then the slope of an indifference curve

2 1

3 2

1u u umu u−

= >−

Then any prospect in the shaded region is preferred

over Aπ , regardless of the degree of risk aversion.

Note that in the shaded region 1 0πΔ ≤ and 3 1π πΔ ≥ Δ .

Equivalently, 1 0πΔ < and 1 2 1( )π π π− Δ + Δ ≥ Δ .

That is , 1( ) 0πΔ ≤ and 1 1 2( ) ( ) 0π π πΔ + Δ + Δ ≤ .

Each term in parentheses is a change in a left tail probability. Thus the new prospect is preferred if the

sums of all the left tail probabilities are all negative.

Define 1 1 2 1 2,π π πΠ = Π = + . Then the sufficient condition is

1 1Π̂ ≤ Π and 1 2 1 2ˆ ˆΠ +Π ≤ Π +Π

indifference line for a risk-averse individual

Slope =

Figure 7.3-1: Preferred gambles for a risk-averse individual

1

1

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Proposition 7.3.2:

Suppose that ( )u c is increasing and concave and that 1s sc c δ+ − = . If Bπ exhibits second-order

stochastic dominance over Aπ then the prospect ( , )Bc π is preferred over ( , )Ac π .

We will prove this in the continuous case. See EM for the discrete case.

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Mean Preserving Spread

A limiting special case of SOSD is when the two

Distributions have the same mean as depicted

opposite.

Note that 1 0πΔ < and 3 1π πΔ = Δ . (*)

Thus, starting with distribution A,

an equal probability mass is taken from

each tail. To create distribution B.

Hence A is a mean preserving spread of B.

We can rewrite (*) as follows.

1 0πΔ < and 1 2 1( )π π π− Δ + Δ = Δ .

that is, 1( ) 0πΔ < and 1 1 2( ) ( ) 0π π πΔ + Δ + Δ = .

that is, 1 1Π̂ ≤ Π and 1 2 1 2ˆ ˆΠ +Π = Π +Π

indifference line for a risk-averse individual

Slope =

Figure 7.3-1: Preferred gambles for a risk-averse individual

1

1

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Other Dominance Relations

For many applications, the assumption of first order stochastic dominance needs to be strengthened to

achieve unambiguous analytical results. We now consider two stronger assumptions.

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Conditional Stochastic Dominance

For any pair of probability distributions 1( ,..., )A A ASπ π π= and 1( ,..., )B B B

Sπ π π= , suppose we truncate the

distributions at t and consider the left tail distributions. The c.d.f.’s of these left tail distributions are

As

At

ΠΠ

and Bs

Bt

ΠΠ

s t≤ .

The probability distribution Bπ exhibits conditional stochastic dominance over Aπ if the first- order

stochastic dominance property holds for every left tail distribution.

Definition: Conditional Stochastic Dominance

The probability distribution Bπ exhibits conditional stochastic dominance (CSD) over Aπ if, for every

left tail distribution the first-order stochastic dominance relation holds. That is,

, andB As s

B At t

t s tΠ Π≤ ∀ ∀ <

Π Π.

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Conditional Stochastic Dominance

, andB As s

B At t

t s tΠ Π≤ ∀ ∀ <

Π Π

Note that an implication of CSD is that

1 11 1B B A At t t t

B B A At t t t

π π− −Π Π= − ≥ − =

Π Π Π Π.

Repeating, 1 11 1B B A At t t t

B B A At t t t

π π− −Π Π= − ≥ − =

Π Π Π Π

Thus the CSD relation implies that each right truncated distribution has more weight in the extreme

right tail. We now show that the converse is also true.

Proposition 7.3-3: Conditional Stochastic Dominance1

The probability distribution Bπ exhibits CSD over Aπ if and only if

,B At t

B At t

sπ π≥ ∀

Π Π.

1 The direct proof for the continuous case is somewhat different. See Exercise 7.3-4.

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Proof:

We begin by rewriting the condition asB A A At t t t

B A B Bt t t t

π π ππ

Π≥ ≥

Π Π Π.

Define At

Bt

k Π=Π

. Then A Bt tkΠ = Π and A B

t tkπ π≤ and so ( )A A B Bt t t tkπ πΠ − ≥ Π − . Therefore

1

1

A A A At t t t

B B B Bt t t t

kππ

Π Π − Π= ≥ =

Π Π − Π.

Because this argument holds for all t it follows that

andA As t

B Bs t

t s tΠ Π≥ ∀ ∀ <

Π Π.

Rearranging this inequality,

andA Bs s

A Bt t

t s tΠ Π≥ ∀ ∀ <

Π Π.

Q.E.D.

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Monotone Likelihood Ratio Property

Two probability distributions are said to satisfy the monotone likelihood property if for some ordering

of states

,A As t

B Bs t

s tπ ππ π

≥ < .

Intuitively if the ratio of the probabilities (the “likelihood ratio”) is decreasing across states, then there

must be more weight in the left tails of the distribution of Aπ . In fact the decreasing likelihood property

is a sufficient condition for conditional stochastic dominance.

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Let AsΠ and B

sΠ be the cumulative probabilities, that is 1

tA At s

=Π = ∑ and

1

tB Bt s

=Π = ∑ .

Proposition 7.3-4:

The monotone likelihood ratio property implies conditional stochastic dominance.

Proof: Appealing to the monotone likelihood ratio property,

( ) , andB

B Ats sA

t

t s tππ ππ

≤ ∀ ∀ ≤ so that

1 1( ) ( ) ,

B Bt tB B A At tt s s st t

s sA A

tπ ππ ππ π= =

Π = ≤ = Π ∀∑ ∑ .

Rearranging this inequality,

,B At t

B At t

tπ π≥ ∀

Π Π.

Appealing to Proposition 7.3-3. The conditional stochastic dominance relation holds.

Q.E.D.

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Continuous Distributions

Ac and Bc have supports (possibly different) within the interval [ , ]α β , p.d.f.s ( ), { , }jf c j A B∈ .

c.d.f.s ( )jF c and integrals of the c.d.f.s ( ) ( )c

j jT c F dα

θ θ= ∫ .

(These integrals are the continuous equivalents of the sums of left tails.)

{ ( )} { ( )} ( ) ( ) ( ) ( ) .B A B AE v c E v c v c f c dc v c f c dcβ β

α α

− = −∫ ∫

Integrating by parts and noting that ( ) ( ) 0A BF Fα α= = and ( ) ( ) 1A BF Fβ β= = ,

{ ( )} { ( )} ( ) ( ) ( ) ( )B A B AE v c E v c v c F c dc v c F c dcβ β

α α

′ ′− = − − −∫ ∫ .

Hence,

{ ( )} { ( )} ( )[ ( ) ( )]B A A BE v c E v c v c F c F c dcβ

α

′− = −∫ . (7.3-1)

Thus, if v is increasing and Bc exhibits first-order stochastic dominance over Ac , then

{ ( )} { ( )} 0B AE v c E v c− ≥ . This is Proposition 7.3-1.

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Second order stochastic dominance

{ ( )} { ( )} ( )[ ( ) ( )]B A A BE v c E v c v c F c F c dcβ

α

′− = −∫ . (7.3-2)

Integrating again,

{ ( )} { ( )} ( )[ ( ) ( )] ( )[ ( ) ( )]B A A B A BE v c E v c v T T v c T c T c dcβ

α

β β β′ ′′− = − − −∫ .

Thus, if ( )v ⋅ is increasing and concave and the left tail sums are smaller for distribution B than A, then

{ ( )} { ( )}B AE v c E v c≥ . This is Proposition 7.3-2.

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We next seek conditions under which two distributions have the same mean. Setting ( )v c c= in

equation { ( )} { ( )} ( )[ ( ) ( )]B A A BE v c E v c v c F c F c dcβ

α

′− = −∫ (7.3-3)

[ ( ) ( )] ( ) ( )A B A BF c F c dc T Tβ

α

β β= − = −∫ .

Hence, we have the following proposition.

Proposition 7.3-5

The distributions andA Bc c with supports in [ , ]α β have the same mean if ( ) ( )A BF c dc F c dcβ β

α α

=∫ ∫ , that

is, the areas under each c.d.f. is the same.

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Consider the first of the two diagrams. If the

two shaded areas are equal, then the areas under

each c.d.f. are equal. Note that the slope of ( )AF c

is decreasing to the left of γ while the slope of

( )BF c is increasing. That is, the density of Ac is

decreasing and the density of Bc is increasing.

The opposite is true to the right of γ .

The probability density functions must therefore

be as depicted. Note also that for any c β< ,

( ) ( ) ( ) ( )c c

A A B BT c F x dx F x dx T cα α

= > =∫ ∫ .

Because the two distributions have the same mean,

it is natural to describe Ac , with its greater weight in the tails, as a mean-preserving spread of Bc .

Figure 7.3-2: Mean preserving spread

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Definition: Mean-preserving spread Suppose that Ac and Bc are random variables with supports in [ , ]α β . The random variable Ac is a

mean-preserving spread of Bc if for all [ , ]c α β∈

( ) ( ) ( ) ( )c c

B B A AT c F x dx F x dx T cα α

= ≤ =∫ ∫ and ( ) ( )B AT Tβ β= .

From (7.3-1)

{ ( )} { ( )} ( )[ ( ) ( )]B A A BE v c E v c v c F c F c dcβ

α

′− = −∫ .

Integrating again by parts,

{ ( )} { ( )} ( )[ ( ) ( )] ( )[ ( ) ( )] .A B A BB AE v c E v c v c T c T c v c T c T c dc

ββ

αα

′ ′′− = − − −∫

Appealing to the definition of a mean preserving spread, we have the following proposition.

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Proposition 7.3-6:

If Ac is a mean preserving spread of ,Bc and ( )v ⋅ is concave, then { ( )} { ( )}B AE v c E v c≥ . If ( )v ⋅ is convex

then { ( )} { ( )}B AE v c E v c≤ .

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PRINCIPAL AGENT PROBLEM

Key ideas: contracting with hidden actions, incentive constraints, insurance with moral hazard

Consider the following 2 person economy. Alex chooses an action 1{ ,.. }nx X x x∈ = , where 1 ... nx x< < .

Penny owns the single firm. The set of possible outputs is 1{ ,..., }SY y y= where 1 ... Sy y< < . For any

action x the firm’s output is a prospect ( , ( ))y xπ . Higher actions shift probability mass to higher

outputs.

If x x′ > then ( )xπ ′ exhibits FOSD over ( )xπ .

Later we will need to assume that the monotone likelihood ratio property holds. That is, for any x x′ > ,

( ) ( ) ,( ) ( )

s t

s t

x x s tx x

π ππ π

′ ′< ∀ <

Let 1( ,..., )Sw w w= be the state contingent allocation to Alex and let 1( ,..., )Sr r r= be the state

contingent allocation to Penny. Then s s sw r y+ ≤ . Finally let ( )C x be the utility cost to Alex of taking

action x.

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Then expected utilities are 1

( , ) ( ) ( ) ( )S

As s

s

U x w x v w C xπ=

= −∑ and 1

( ) ( )S

Ps s

s

U x v rπ=

=∑ where

s s sr w y+ =

Substituting for r,

1

( , ) ( ) ( )S

P s s ss

U w x x v y wπ=

= −∑

Efficient allocations

For any fixed action x we can in principle solve for

The efficient allocations by solving one of the following

problems.

(i) Maximize Penny’s expected utility given that Alex must have an expected utility of at least AU

(ii) Maximize Alex’s expected utility given that Penny must have an expected utility of at least PU

Pareto efficient outcomes with 3 actions

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Efficient Allocations

Consider problem (i).

Fix AU and x and solve for the efficient state

contingent payments

( ) { ( , ) | ( , ) }P A Aww x Max U w x U w x U= ≥ .

Then Penny’s expected utility is

1

( ( ), ) ( ) ( ( ))S

P s s ss

U w x x x u y w xπ=

= −∑ .

Repeat for each x and thus solve for * arg { ( ( ), )}Pxx Max U w x x=

Pareto efficient outcomes with 3 actions

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Asymmetric information (hidden action)

The Principal Agent problem

Thus far we have assumed that all aspects of the contract are costlessly verifiable and hence

enforceable. But what if is prohibitively costly to verify the action chosen by Alex (the agent)? We

assume that output is costlessly verifiable by a third party. Since output is higher in higher numbered

states, it follows that the state is verifiable.

Suppose then that the efficient contract is signed. Alex is paid *( )sw x in state s and so his expected

utility if he takes action x is

* *

1( ( ), ) ( ) ( ( )) ( )

S

A s ss

U w x x x v w x C xπ=

= −∑ .

As we shall see, except in special cases * *arg { ( ( ), )}A xx x Max U w x x≠ ≡ . Thus the efficient contract is

not feasible.

Then Penny (the principal) must design a contract in such a way that the agent she hires will choose

the action in the contract even without verification.

We consider two special cases

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Risk-Neutral Agent

Suppose that the principal is risk averse and the agent is risk neutral.

An efficient contract solves

, ,{ ( , ) | ( , ) }A P Px r w

Max U w x U r x U≥ . (*)

For any action x, efficiency requires that all the risk be borne by the agent. That is, the principal

receives a fixed rent r and the agent receives the residual s sw y r= − .

The constraint is then 1

( , ) ( ) ( ) ( )S

P s Ps

U r x x u r u r Uπ=

= = ≥∑ .

Then the efficient contract is a fixed rent contract with rent *r satisfying *( ) Pu r U=

To satisfy the constraint in (*), this rent must be chosen so that

1

( ) ( ) ( )S

s s s Ps

x u y w u r Uπ=

− = =∑ .

We can rewrite (*) as follows:

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* *

1{ ( , ) ( ) ( )

S

A s sx sMax U y r x x y C x rπ

=

− = − −∑ .

Hence * *

1arg { ( ) ( )}

S

s sx sx Max x y C x rπ

=

= − −∑

Asymmetric Information

Suppose that the agent’s action is not verifiable. The principal’s fixed rent contract is verifiable since it

is independent of the outcome.

The risk neutral agent chooses

* *

1arg { ( , ( )) arg { ( ) ( )}

S

A A s sx x sx Max U x w x Max x y C x rπ

=

= = − −∑

Then *Ax x=

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Risk-Neutral Principal

Suppose that the principal is risk neutral and the agent is risk averse.

An efficient contract solves

, ,{ ( , ) | , ( , ) }P A Ax r w

Max U w x r w y U r x U+ = ≥ . (*)

For any action x, it is efficient for the principal to bear all the risk. Therefore 0( )sw x w= , 1,...,s S= . At

the optimum the constraint must be binding therefore 0w satisfies 0 0( , ) ( ) ( ) AU w x v w C x U= − = .

Let 1( )v− ⋅ be the inverse function, that is 0w must satisfy 0 1( ( ))Aw v U C x−= + . Then the maximization

problem of the principal can be rewritten as follows:

0

0 1

, 1 1

{ ( ) ( ) ( ( ))S S

s s s s Ax w s s

Max x y w x y v U C xπ π −

= =

− = − +∑ ∑

Then

* 1

1

{ ( ) ( )}, where ( ) ( ( ))S

s s P P Ax s

x Max x y C x C x v U C xπ −

=

= − ≡ +∑ .

( )PC x is the virtual cost function for the principal.

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The efficient contract thus maximizes expected revenue

less the virtual cost. Consider two actions x and x x′ > .

Define

( ) ( )P P PC C x C x′Δ ≡ −

This is the marginal cost of paying the agent to take

the more costly action x′ . It is the increase in the wage

needed to compensate the agent so that the payoff remains at AU .

Next suppose that the agent’s payoff is increased to AU U+ Δ . Because v is concave, the marginal

utility of the agent is lower at the higher utility level. Thus the dollar compensation needed to pay for

the more costly action is higher. That is, the marginal cost of providing a big enough incentive for the

agent to take a more costly action rises with the utility level of the agent. Therefore, the higher the

expected utility of the agent, the lower will be the efficient action. This is the case depicted above. As

the expected utility of the agent rises, the efficient action changes first from the highest cost action 3x

to 2x and then from 2x to the lowest cost action 1x .

Pareto efficient outcomes with 3 actions

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With a fixed wage the agent’s optimal choice is the low cost action. Therefore unless *1x x= the

efficient contract is not enforceable.

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Solving the Principal-Agent problem

Continue with the special case in which the principal is risk neutral and the agent is risk averse.

Three step approach

Step 1: For any action x , and expected utility for the agent AU characterize the state contingent

payments which induce the agent to take action x rather than any other x X∈

Step 2:

Of all these payments schemes choose the one that is best for the principal. Since she is risk neutral

this is the scheme with the lowest expected cost. Write her expected payoff as **( )PU x

Step 3:

For each x X∈ solve for **( )pU x and finally choose ** **arg { ( )}Px Max U x= .

Actually we will not take the third step but leasrn what we can by examining the first two steps.

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Step 1:

Incentive constraints

For the action x , the state contingent payment vector w is incentive compatible if the agent has an

incentive to take the action x rather than any other action, that is,

( , ) ( , ),A AU x w U x w x X≤ ∈ . Incentive Constraints

Let AU be the contracted expected utility of the agent.

Then ( , )x w must also satisfy the following constraint

( , )A AU x w U≥ Participation constraint

The best contract for the principal, given the selection of action x , thus solves the following

optimization problem:

** arg { | ( , ) , ( , ) ( , ), }P A A A Aww Max U U x w U U x w U x w x X= ≥ ≥ ∈ . (**)

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Step 2:

Intuitively, the problem is one of designing a contract to deter the agent from taking ales costly action.

We will assume this so that (**) can be rewritten as follows.

** arg { | ( , ) , ( , ) ( , ), }P A A A Aw

w Max U U x w U U x w U x w x x= ≥ ≥ < .

Suppose that in fact there is a single binding constraint.2 The optimization problem can then be

rewritten more simply as follows.

* * *{ | ( , ) , ( , ) ( , ), for some }P A A A AwMax U U x w U U x w U x w x x≥ ≥ < .

The Lagrangian of this optimization problem is

* *( ( , ) ) ( ( , ) ( , ))P A A A AU U x w U U x w U x wλ μ= + − + −L

*( )( ( , ) ( , ))P A AU U x w U x wλ μ μ= + + − + a constant

* *

1 1 1

( )( ) ( ) ( ) ( ) ( ) ( ) a constantS S S

s s s s s s ss s s

x y w x v w x v wπ λ μ π μ π= = =

= − + + − +∑ ∑ ∑ .

2 It is not important that there be one binding constraint, only that the binding constraints are all associated with lower cost actions. General sufficient conditions to ensure this are quite stringent. However for numerical examples it is typically the “downward” constraints that are binding.

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* *

1 1 1

( )( ) ( ) ( ) ( ) ( ) ( ) a constantS S S

s s s s s s ss s s

x y w x v w x v wπ λ μ π μ π= = =

= − + + − +∑ ∑ ∑L .

The first-order conditions are therefore

* * * *( ) ( ) ( ) ( ) ( ) ( ) 0s s s s ss

x x v w x v ww

π λ μ π μπ∂ ′ ′= − + + − =∂L , 1,...,s S= .

Hence

* *

( )1( ) ( )

s

s s

xv w x

πλ μ μπ

= + −′

where *x x< .

By hypothesis, the likelihood ratio *( )

( )s

s

xx

ππ

is increasing in the output state s. Thus the right hand side

of this expression increases with s. Therefore, because ( )v ⋅ is concave, sw is increasing in s. Thus for

efficiency, the higher the output, the higher is the payment to the agent.

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Application: Insurance with Moral Hazard

An individual’s house burns down with probability 1( )xπ , where x is the effort the owner makes to

avoid such a catastrophe. The house can be rebuilt at a cost of L.

If the insurance company does not observe the homeowners action but the home owner is

perfectly moral, he tells the truth and the efficient contract is unaffected. However if a homeowner is

not so trustworthy, the insurance company faces a “moral hazard problem.” With full coverage the

incentive to take care is eliminated so the homeowner’s best choice is the cheapest action 1x . To give

the homeowner the incentive to take appropriate care the homeowner must be offered a contract in

which he is sufficiently penalized in the loss state.