# 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 )

X

µ

1

2 £

1

X

µ

2

2 £

2

¼

1

( µ

1 j ®

1

) ¼ f u

2

( c

2

( µ )) + ±V ( U ( µ )) ¡ g

2

( ®

2

) g

2

( µ

2 j ®

2

) ¤ 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

1

( µ

(

1

µ ) ¡ j a

1

) ¼ g

2

1

(

( µ

®

1

) g

2 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 ( µ )) j ®

1

) ¼

2

¡

( µ g

2

2

( ®

2

) j a

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

( U ; µ ) c i

®

0 i

( µ

0

= ®

) = c i

0 i

( U

( U

0

0

) = ®

0 i

; µ

0

) =

( c i

U ( µ ))

( 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

2 u

0

1

( c

1 u

0

2

( c

2

( µ ))

( µ ))

=

X

µ

0

2

¼ ( µ

0

2 j ®

0

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

( U ; µ ) c i

®

0 i

( µ

0

= ®

) = c i

0 i

( U

( U

0

0

) = ®

0 i

; µ

0

) =

( c i

U ( µ ))

( 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 ,

U ( µ )

² Potential resolution: lower bounds on continuation utilities (because of limited commitment)

U ( µ ) ¸ u

1

V ( U ( µ )) ¸ u

2

² But: this may change implications of the model drastically: see Atkeson and Lucas (1992) vs.

(1995)