# Discussion of Rui Zhao \Repeated Two-Sided Moral Hazard&#34; Dirk Krueger

Discussion of Rui Zhao

\Repeated Two-Sided Moral

Hazard"

Dirk Krueger

Stanford University

January 2002

Econometric Society Winter Meetings in

Atlanta

MAIN RESULTS

1. Under fairly general conditions Pareto optimal contracts under repeated two-sided moral hazard are recursive

2. Partial characterization of optimal contracts

3. Obvious next step: explore further properties of optimal contracts using numerical \examples"

MAIN ASSUMPTIONS

1. For all µ i

2

£ i

; all a i

2

A i

¼ ( µ i j a i

) > 0

2.

u i

: ( c;

1

)

! < is strictly concave and lim c !

c

( c ) =

¡1

3. There exist a i

; a

0 i

2 A i such that g i

( a i

) = g i

( a

0 i

)

4. (In¯nite horizon)

RECURSIVE FORMULATION

² State variable

U : promise of expected discounted lifetime utility for agent 1

²

Control Variables:

® i

: probability distribution over e®ort a i

2 A i c i

(

µ

) : consumption, conditional on public signal

µ

2 £

1

£ £

2

U

(

µ

) : continuation utility of agent 1, conditional on µ

²

Timing

Pick efforts

?

i

Draw

?

=(

?

,

?

)

1 2

Output x(

?

) realized

Pick cons. c (

?

), U(

?

)

i

Come in with U with

?

~

?

(

?

|

?

)

i i i

Come in with U’=U(

?

)

Today Tomorrow Time

Figure 1:

²

Bellman equation

=

V ( U )

® i

;c i max

( µ ) ;U ( µ ) f u

2

( c

2

( µ )) +

X

µ

1

2 £

1

±V

X

µ

2

2 £

2

( U ( µ ))

¼

1

( µ

1 j

®

1

) ¼

2

( µ

2 j

®

2

)

¡ g

2

( ®

2

) g

¤ subject to

{ Promise Keeping

U =

X X

¼

1

( µ

1 j

®

1

) ¼

2

( µ

2 j

®

2

)

µ

1

2

£

1

µ

2

2

£

2 f u

1

( c

1

( µ )) + ±U ( µ )

¡ g

1

( ®

1

) g

¤

{ IC, agent 1: for all

¸ a

1

2

A

1

X X

¼

1

(

µ

1 j

®

1

)

¼

2

(

µ

2 j

®

2

)

µ

1

2

£

1

µ

2

2

£

2 f u

1

( c

1

(

µ

)) +

X

¼

±U

(

µ

)

1

( µ

1 j a

1

¡

) ¼ g

2

1

(

(

®

1

µ

2

) g j

®

2

)

µ

1

2

£

1

µ

2

2

£

2

¤

¤ f u

1

( c

1

( µ )) + ±U ( µ )

¡ g

1

( a

1

) g

{ IC for agent 2: for all

¸ a

2

2 A

2

X X

¼

1

(

µ

1 j

®

1

)

¼

2

(

µ

2 j

®

2

) ¤

µ

1

2 £

1

µ

2

2 £

2 f u

2

( c

2

(

µ

)) +

X

¼

±V

(

1

(

µ

1

U j

®

(

1

µ

)

))

¼

2

¡ g

(

µ

2

2 j

( a

®

2

)

2

)

¤ g

µ

1

2

£

1

µ

2

2

£

2 f u

2

( c

2

(

µ

)) +

±V

(

U

(

µ

)) g ¡ g

2

( a

2

)

{ Resource Feasibility: for all µ

2

£

1

£

£

2 c

1

( µ ) + c

2

( µ ) = x ( µ )

MAIN CHARACTERIZATION

²

Two-Sided Moral Hazard: Sub-martingale result u u

0

2

0

1

( c

( c

1

2

(

(

µ

µ

))

))

=

·

X

µ

0

2

X

µ

0

1

¼ ( µ

0

2 j

®

0

2

)

X

µ

0

2

2

X

¼ ( µ

0

1 j

®

0

1

) u u

0

1

0

2

( c

2

( c

1

( µ

0

))

(

µ

0

))

3

¡

1

µ

0

1

¼

(

µ

0

2 j

®

0

2

)

¼

(

µ

0

1 j

®

0

1

) u

0 u

0

1

2

( c

1

( µ

0

( c

2

( µ

0

))

)) where c i

(

µ

) = c i

®

0 i

(

µ

0

=

) = c i

(

U

;

µ

)

® c i

0 i

(

(

U

U

0

) =

®

0 i

0

;

µ

0

) =

(

U

(

µ

)) c i

(

U

(

µ

);

µ

0

)

² One-Sided Moral Hazard (Rogerson 1985)): Agent

1 is risk neutral principal; agent 2 has moral hazard problem; equal discount factors: Martingale result or

1 u

0

2

( c

2

1 u

0

2

( c

2

(

µ

))

( µ ))

=

=

=

X

¼ ( µ

0 j

®

0

2

)

"

µ

0

X

µ

0

E

µ

¼ ( µ

0 j

®

0

2

) u

0

2

(

1

0 j

®

0

2 u

0

2

( c

2 u

(

0

2 c

(

1

2 c

2

1

( µ

( µ

0

))

(

µ

0

)

0

))

))

#

¡ 1

MAIN CHARACTERIZATION

² Two-Sided Moral Hazard u

0

1 u

0

2

( c

1

( c

2

(

(

µ

µ

))

))

=

X

µ

0

2

¼ ( µ

0

2 j

®

0

2

)

2

X

µ

0

1

¼ ( µ

0

1 j

®

0

1

) u

0

2 u

0

1

( c

2

(

µ

( c

1

( µ

0

))

3

¡ 1

0

)) where c i

( µ ) = c i

®

0 i

( µ

0

=

) = c i

( U ; µ )

® c i

0 i

(

( U

U

0

) = ®

0 i

0

; µ

0

) =

( U ( µ )) c i

( U ( µ ); µ

0

)

² One-Sided Moral Hazard (Rogerson 1985)): Agent

1 is risk neutral principal; agent 2 has moral hazard problem; equal discount factors or

1 u

0

2

( c

2

1 u

0

2

( c

2

(

µ

))

( µ ))

=

=

X

¼ ( µ

0 j

®

0

2

)

µ

0

(

" u

0

2

( c

2

(

µ

0

))

#

¡

1

1

)

1

E

µ

0 j

®

0

2 u

0

2

( c

2

( µ

0

))

SMALL REMARKS

² What restrictions does the use of public actions impose?

²

Implementation of optimal contracts?

²

Connection with observables? Characterization of consumption (compensation) process? Note that the results about long-run behavior still involve endogenous choices.

NUMERICAL EXAMPLES

² Suppose

A i

= f a l

; a h g and £ i

= f

µ

1

; µ

2 g

²

Only one (continuous) state variable U

² 14 control variables

²

Potential problem: unbounded domain of

U

(

µ

)

U ,

²

Potential resolution: lower bounds on continuation utilities (because of limited commitment)

U ( µ )

V ( U ( µ ))

¸

¸ u

1 u

2

²

But: this may change implications of the model drastically: see Atkeson and Lucas (1992) vs.

(1995)