Advanced Microeconomics

Assumptions

Assumptions

CHAPTER 2

Assumptions

Assumptions

Principal (P) - Agent (A)

Principal (P) - Agent (A)

We have a principal P (the employer, the bank) and an agent A (the worker, he has to sell

Probability distribution of Return conditional on E§

A can choose between two EFFORT levels: e can be either LOW (e )

L

A can choose between two EFFORT levels: e can be either LOW (e )

financial products). L

or HIGH (e )

Principal (P) - Agent (A) (Figure 1)

umptions H or HIGH (e )

A can choose between two effort levels: eL and eH, with eH>eL.

H

<

Disutility: e e

A can choose between two EFFORT levels: e can be either LOW (e )

L H L

<

Disutility: e e

The outcome of A’s behavior is the return (value of shares sold in one period) going to P,

L H

The outcome of A’s behavior is the RETURN going to P, which is a

or HIGH (e )

Principal (P) - Agent (A) H

Assumptions

The outcome of A’s behavior is the RETURN going to P, which is a

e

which is a random variable R~ with possible values R2>R1.

>

random variable (

R) with possible values: R R

<

Disutility: e e 2 1

L H e

A can choose between two EFFORT levels: e can be either LOW (e ) >

random variable (

R) with possible values: R R

L

The relationship between effort and returns is stochastic: 2 1

The relationship between EFFORT and RETURN is stochastic (see

The outcome of A’s behavior is the RETURN going to P, which is a

or HIGH (e )

H The relationship between EFFORT and RETURN is stochastic (see

Principal (P) - Agent (A)

e

-if effort is high

Figure 1 ): >

random variable (

R) with possible values: R R

" #

<

Disutility: e e 2 1

L H Figure 1 ):

A can choose between two EFFORT levels: e can be either LOW (e )

" #

e L

The relationship between EFFORT and RETURN is stochastic (see

= | =

If e§ort is high: P R R e p ,

The outcome of A’s behavior is the RETURN going to P, which is a

2 H H R

e R

" #

or HIGH (e ) = | =

If e§ort is high: P R R e p , 2 2

H 2 H H

Figure 1 ):

e >

" #

random variable (

R) with possible values: R R

" #

e P

= | = −

2 1

P R R e 1 p <

Disutility: e e L

1 H H P

L H

e

e

" # " # H

= | =

If e§ort is high: P R R e p ,

= | = −

P R R e 1 p

The relationship between EFFORT and RETURN is stochastic (see

2 H H

1 H H

The outcome of A’s behavior is the RETURN going to P, which is a

" # " # " #

e e

= | = = | = −

If e§ort is low: P R R e p , P R R e 1 p

Figure 1 ): 2 1

L L L L

e e

e e e e

" # L H

= | = − >

P R R e 1 p = | = = | = −

random variable (

R) with possible values: R R

If e§ort is low: P R R e p , P R R e 1 p

1 2 1

H H 2 1

L L L L

High e§ort makes higher outcome more likely than low e§ort does:

" # " #

e = | =

If e§ort is high: P R R e p ,

The relationship between EFFORT and RETURN is stochastic (see

2 H H 1-P

-if effort is low High e§ort makes higher outcome more likely than low e§ort does:

e e

" # 1-P H

= | = = | = −

If e§ort is low: P R R e p , P R R e 1 p

>

p p (stochastic dominance) L R R

2 1

L L L L

" # " #

H L 1 1

Figure 1 ):

e >

p p (stochastic dominance)

= | = − " #

P R R e 1 p " # " #

H L

e e

1 H H

High e§ort makes higher outcome more likely than low e§ort does:

| > |

Implication: E R e E R e

" # " # e

H L

e e

= | =

If e§ort is high: P R R e p ,

| > |

Implication: E R e E R e

2 H H

e e H L

>

p p (stochastic dominance)

= | = = | = −

" #

If e§ort is low: P R R e p , P R R e 1 p

" # " #

H L

2 1

L L L L P >P

Angelo Baglioni () 2018 2 / 34

e H L

e e

= | = −

P R R e 1 p

| > |

Implication: E R e E R e

1 H H

High e§ort makes higher outcome more likely than low e§ort does:

Angelo Baglioni () 2018 2 / 34

H L

" # " #

>

p p (stochastic dominance) e e

= | = = | = −

If e§ort is low: P R R e p , P R R e 1 p

" # " #

H L 2 1

L L L L

Angelo Baglioni () 2018 2 / 34

e e

| > |

Implication: E R e E R e

High effort makes higher outcome more likely than low effort does: pH>pL. Direct

High e§ort makes higher outcome more likely than low e§ort does:

H L Angelo Baglioni ()

implication: >

p p (stochastic dominance)

" # " #

H L

Angelo Baglioni () 2018 2 / 34 Preferences: Principal

e e

Preferences: Principal | > |

Implication: E R e E R e

H L

Angelo Baglioni () 2018 2 / 34

Asymmetric information: HIDDEN action A can observe both outcome R~ and action e.

P can observe R~ but he cannot observe the action e taken by A.

Principal is RISK NEUTRAL

Principal is RISK NEUTRAL

Preferences Principal: risk neutral.

Principal is Utility function is e f

( − )

Utility function v R W , where:

e f

( − )

Utility function v R W , where: e

R is his Return

e

R is his Return f

where R~ is the return, W~ is the wage it pays to A (assuming W~ is a

W is the WAGE he pays to A

Preferences: Agent

f

W is the WAGE he pays to A f e

function of R~, with W2>=W1). ≥

(assuming that W is a function of R, with W 2

f e ≥

(assuming that W is a function of R, with W W )

2 1

v() is a linear utility function v(x)=a+b(x). From the properties of the linear function

() ( ) = +

v is a LINEAR utility function: v x a bx

() ( ) = +

v is a LINEAR utility function: v x a bx ( )] = +

[ [

Property of the linear function: E v x E a

( )] = + = + ( )

[ [ ]

Property of the linear function: E v x E a bx a bE x

So P wants to maximise the Expected Value of his Net Return:

e f

Hence P wants to maximize the Expected Value of his net return E(R~-W~).

( − )

E R W Agent is RISK AVERSE

Preferences Agent: risk averse.

Agent is Utility function is f f

( ) = ( ) −

Utility function u W , e u W e

f f f

0 00

( ) ( ) > ( ) <

where u W CONCAVE: u W 0 and u W 0

where u() is concave u’(W~)>0 and u’’(W~)<0. Reservation level of utility: u

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A has a reservation level of utility of u.

Angelo Baglioni () 2018 5 / 34

Solving the P-A problem

We look for the optimal equilibrium contract, the best contract that the two parties can

design, given the constraints of the problem.

Bargaining structure is the following:

-P makes a take-it-or-leave it offer to A

-A can accept or refuse

-so all bargaining power is assigned to P, but this does not alter the properties of eq.

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Two step procedure: we find the optimal contract under SYM, the 1st best; then the

optimal with ASYM, the 2nd best.

Agency cost can be derived by compering 1st best to 2nd best.

A) SYMMETRIC INFORMATION contractible.

Givent that P can observe both R~ and e, the action e is

Symmetric info: LOW e§ort

For each action, we must find the least costly contract for P, given the reservation

constraint of A

Principal’s problem:

+ ( − )

min p W 1 p W

2 1

L L

( ) + ( − ) ( ) − ≥

s.to p u W 1 p u W e u (PC)

2 1

L L L

= + ( − ) − ( ) + ( − ) ( ) − −

l [ ]

L p W 1 p W p u W 1 p u W e u

2 1 2 1

L L L L L

FOC:

∂L 0

= − ( ) =

lp

p u W 0

2

L L

∂W 2

∂L 0

= ( − ) − ( − ) ( ) =

l

1 p 1 p u W 0 We can derive ORS principle

1

L L

∂W 1 We can derive ORS principle

=

l

Can be 0 (PC slack)?

= = ( − ) =

l

No: with 0 FOC imply p 0 and 1 p 0, which cannot

L L

be >

l

So it is 0 and PC binding FOC can be written as:

FOC can be written as:

0

= ( )

lu

1 W

FOC can be written as 2

0

= ( )

lu

1 W

0 2

= ( )

lu

1 W 1

Angelo Baglioni () 2018 9 / 34

0

= ( )

lu

1 W 1 =

which imply that W W (FIXED WAGE): Optimal Risk

1 2

=

which imply that W W (FIXED WAGE): Optimal Risk

1 2 −

P bears the whole risk: R R

Equilibrium contract (1st best)

FIXED WAGE: Optimal Risk Sharing ORS,

which imply that W1=W2 P bears the whole

2 1

−

P bears the whole risk: R R

2 1

risk R2-R1. What determines the level of W ?

Equilibrium contract (1st best)

What determines the level of W ?

∗ ( )] − =

[

PC: u W e e u

What determines the level of W? L L ∗ (

∗

The same procedure applies to HIGH e§ort, leading to W

( )] − =

[

PC: u W e e u

Equilibrium contract (1st best) L L ∗

determined by: ( )

The same procedure applies to HIGH e§ort, leading to W e H

and the same applies for HIGH eff: ∗ ( )] − =

[

PC: u W e e u

determined by: H H

∗

∗ ( )

The same procedure applies to HIGH e§ort, leading to W e ∗ ∗

( )] − =

[

PC: u W e e u > ( ) > ( )

Of course, from e e we get W e W e .

H

H H H L H L

determined by: ∗ ∗ ∗

> ( ) > ( )

Of course, from e e we get W e W e . [ (

Which contract will P propose? Either high e§ort, W e

H L H L

Of course eH>eL, so we get W*(eH)>W*(eL). H

∗ ( )] − =

[

PC: u W e e u ∗

H H ( )] ∗

e§ort, W e ? [ ( )] [

Which contract will P propose? Either high e§ort, W e or low

Which contract will P propose? [high effort, W*(eH)] or [low effort, W*(eL)]?

L H

∗ ∗ Angelo Baglioni ()

> ( ) > ( )

Of course, from e e we get W e W e . ∗

H L H L ( )]

e§ort, W e ?

P will chose the one that maximises his own NET expected

P will choose the one that maximizes his own net expected return as we’ve said:

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L ! $

∗

[ ( )] [

Which contract will P propose? Either high e§ort, W e or low

e f ∗

H − ( )

P will chose the one that maximises his own NET expected return:

E R W e

! $

∗ ( )]

e§ort, W e ? e f

L ∗

− ( )

E R W e Two alternative cases:

! $ ! $

P will chose the one that maximises his own NET expected return:

! $ e e

∗ ∗

Two alternative cases: | − ( ) > | − ( )

(A) If E R e W e E R e W e , the

! $ ! $

H H L L

e f ∗

− ( )

E R W e

Two alternative cases: e e

∗ ∗ ∗

| − ( ) > | − ( )

(A) If E R e W e E R e W e , then the 1st

( )]

[

best Equilibrium Contract is: high e§ort, W e

H H L L H

! $ ! $

Two alternative cases:

! $ ! $ ∗ ( )]

[

best Equilibrium Contract is: high e§ort, W e

e e

∗ ∗

| − ( ) < | − ( )

H

(B) If E R e W e E R e W e , the

e e ! $ ! $

∗ ∗ H H L L

| − ( ) > | − ( )

(A) If E R e W e E R e W e , then the 1st

H H L L e e

∗ ∗ ∗

| − ( ) < | − ( )

(B) If E R e W e E R e W e , then the 1st

( )]

[

best Equilibrium Contract is: low e§ort, W e

H H L L

∗ L

( )]

[

best Equilibrium Contract is: high e§ort, W e H

! $ ! $ ∗ ( )]

[

best Equilibrium Contract is: low e§ort, W e L

e e

∗ ∗

| − ( ) < | − ( )

(B) If E R e W e E R e W e , then the 1st

Angelo Baglioni ()

H H L L

∗ ( )]

[

best Equilibrium Contract is: low e§ort, W e

Angelo Baglioni () 2018 11 / 34

L

Angelo Baglioni () 2018 11 / 34

Asymmetric information

B) ASYMMETRIC INFORMATION

cannot observe

P can observe R~, but he the action e taken by A.

e

P can observe R, but he CANNOT OBSERVE the action e taken by

LOW effort.

Let us now focus on P proposes the 1st best contract, [low effort, W*(eL)].

A.

Does it work? Let us focus on LOW e§ort. P proposes the 1st best contract:

Yes, because ∗

∗ ( )]

[ low e§ort, W e . Does it work?

L

L ∗ ∗

∗ ∗

( )] − > ( )] −

[ [

Yes, because u W e e u W e e , so A will not

L L L H

L L L H

deviate.

so A will not deviate. ∗

∗ ( )

Fixed wage contract (W e : the same as under Symmetric Info)

L

L

Fixed wage contract ( the same as under symmetric info) can be used when P wants A to

can be used when P wants A to take the LEAST COSLTY action for

NO MORAL HAZARD HERE.

take the least costly action for A:

A: NO MORAL HAZARD here!

HIGH effort.

Let us now focus on P proposes the 1st best contract, [high effort, W*(eH)].

Let us focus on HIGH e§ort. P proposes the 1st best contract:

Does it work? ∗

∗ ( )]

[ high e§ort, W e . Does it work?

H

H

NO, because ∗ ∗

∗ ∗

( )] − < ( )] −

[ [

NO, becuase u W e e u W e e , so A will deviate:

H H H L

H H H L

MORAL HAZARD!

P cannot apply any penalty for deviation, because he does not

MORAL HAZARD.

so A will deviate: observe A’s e§ort (action is not contractible).

any penalty

P cannot apply for deviation, because he does not observe A’s effort

So P will not propose the 1st best contract.

(remember action e is not contractible).

Angelo Baglioni () 2018 12 / 34

Angelo Baglioni () 2018 12 / 34

So P will not propose 1st best contract with high effort.

Looking for the Incentive Compatible Contract. P looks for a contract that creates an

incentive for A to exert high effort.

Intuition: P should design a random wage W~ increasing in R~. Hence A knows that, by

taking high effort, he will make higher outcomes R~ and higher wage W~ more likely.

P will look for the wage schedule that minimizes E(W~), given the Participation Constraint

PC and the Incentive Compatibility IC constraint.

IC constraint: A gets more utility from high effort than from low effort.

Principal’s problem:

+ ( − )

min p W 1 p W

2 1

H H

s.to

Getting to the solution

( ) + ( − ) ( ) − ≥

p u W 1 p u W e u (PC)

2 1

H H H

( ) + ( − ) ( ) − ≥ ( ) + ( − ) ( ) −

p u W 1 p u W e p u W 1 p u W e

2 1 2 1

H H H L L L

(IC) > >

l µ

=

So it must be 0 and 0 (both PC and IC binding)

L + ( − ) − ( ) + ( − ) ( ) − − −

l [ ]

p W 1 p W p u W 1 p u W e u

FOC can be written as

2 1 2 1

h i

H H H H H

…

[ ( ) + ( − ) ( ) − − ( ) − ( − ) ( ) + ]

µ p u W 1 p u W e p u W 1 p u W e

p

1 = + −

l µ 1 L

2 1 2 1

H H H L L L

0 p

( )

u W

(with both lambda and mu =0 ph=0 and 1-ph=0, which cannot be.

h i

H

2

FOC: ( − )

1 p

1 L

= + −

l µ 1

with lambda=0 the second FOC becomes …, where LHS>0 and RHS<0.

∂L 0 0 0

0 ( ) ( − )

u W 1 p

= − ( ) − ( ) − ( )] =

lp µ [

p u W p u W p u W 0

1 H

2 2 2

H H H L

∂W

with mu=0 u’(W1)=u’(W2), implying W2=W1=W, which leads to a violation of IC (because

2 1

∂L > > l

First FOC: p p implies 0

= ( − ) − ( − ) ( ) −

l

1 p 1 p u W

H L 0 1

H H ( )

u W

∂W

IC becomes W-eH>=W-eL implying eL-eH>=0 which cannot be, eH>eL). )

2

1 0 0 1

− ) ( ) − ( − ) ( )] =

µ [( 1 p u W 1 p u W 0

( − ) < ( − ) < l

Second FOC: 1 p 1 p implies

1 1

H L

H L

… 0 ( )

u W 1

which can be written as:

1

0 0

So u’(W2)<u’(W1), which implies (since marginal utility u’(W1) is decreasing) W2>W1.

( ) < < ( ) >

Therefore: u W u W , which implies W W (given

2 1 2 1

l

p − − − =

lp µ [ ]

p p 0

H Equilibrium contract (2nd best)

0

H H L

( )

Bottom line: The Incentive Compatible Contract is a random wage W*~ INCREASING in

that marginal utility u W is decreasing)

0 ( )

u W 2

( − )

1 p

the outcome R~:

H − ( − ) − − ) − ( − )] =

l µ [(

1 p 1 p 1 p 0

Bottom line: the Incentive Compatible Contract is a random wage

H H L

0 ( )

u W 1

f e

∗ ∗ ∗

> >

W INCREASING in the outcome R: W W (given R R ).

2 1

2 1

Angelo Baglioni () 2018 14 / 34

Which contract will P propose? f

Which contract will P propose? W*(eL) (inducing A to take low effort) or

∗ ∗

( )

Either W e (inducing A to take low e§ort) or W (inducing A to

L

Angelo Baglioni () 2018 16 / 34

(inducing A to take high effort)? take high e§ort)?

P will chose the one that maximises his own NET expected return.

Two alternative cases:

" $ " $

e f e

∗ ∗

&minu