.

• Typically a 3-step game of incomplete info
Step 1: Principal designs mechanism/contract
Step 2: Agents accept/reject the mechanism
Step 3: Agents that have accepted, play the game specified by mechanism
• Constant theme: Incomplete information and binding individual rationality constraints prevent efficient outcomes

• A monopolist produces good at marginal cost c and sells quantity q
• Consumer transfers T to seller and has utility u
1
( q , T , θ)= θ V ( q )T , V (0)=0, V / >0, V // <0
• θ is private knowledge for buyer
• The game:
1. Seller offers tariff T ( q ): specifies a price for qty q
2. Consumer accepts/rejects
• If seller knows θ, she will charge T = θ V ( q ), her profit, θ V ( q )cq. This is maximized at some q given by θ V / ( q )= c


• Seller’s expected profit:
Eu
0

• Seller faces two constraints: p ( T
 c q )
 p ( T
 c q )
1. Individual Rationality ( IR ): Consumer should be willing to purchase
2. Incentive Compatibility ( IC ): Consumer should consume the bundle intended for his type
• IR
1
: 
V ( q )
• IC
1
: 
V ( q )

T

T

0 ; and IR
2
:
 
V ( q )

T

V ( q )

T

0
; and IC
2
: 
V ( q )

T
 
V ( q )

T
• First step: To show that only IR
1 and IC
2 are binding

• First note: IR
1 and IC
2 imply IR
2
• IR
2 can’t be binding unless =0
• However, IR
1 must bind. Else seller can increase
T & T by same amount and increase revenue
• Also, IC
2 must be binding, else seller can increase
T , satisfy all constraints and increase revenue
• The high-type’s indifference curve is always steeper than the low type’s for any allocation
• This implies that high type consumes more than low type: q
 q

• Eliminating transfers, principal’s objective function is: max {([ q , q p
  p (
  
)] V ( q )
 p c q )
 p (

V ( q )
 c q )}
• FOC wrt q :

V
/
( q )
 c /( 1
 p (
 p
 

)
)
• FOC wrt q :

V
/
( q )
 c
• Check that IC
1 is satisfied
• Note: Quantity purchased by high-type is optimal
Quantity purchased by low-type is sub-optimal
• Seller sacrifices efficiency for rent-extraction!

• Seller has unit of good and there are two bidders
   p
• Buyer’s expected probability of getting the good are
X & X and payments are T & T
• The constraints are:
IR
1
:

X

T

0 ; IR
2
:
IC
1
: 
X

T
 
X

T
; IC
2
:

X

T

0

X

T
 
X

T
• What is seller’s optimal contract?

• Seller’s expected profit is: p T
 p T
• Again, IR
1 and IC
2 are binding. The seller’s profit:
Eu
0

(
  p

) X
 p

X
• Also, ex-ante prob of a player getting good, p X
 p X
• Moreover,
X
 p
 p
2

1
2
• Case 1:   p

. The seller sets and X
 p
 p
2
Optimal mechanism: Not to sell if both announce low-type; sell to hightype if they announce different types; sell wp ½ to each if both announce high type
  p

X
 p
 p
2
Optimal mechanism: Sell to high-type if bidders announce different types, and sell wp ½ to each if they both announce high-type or low-type

•Consider a Principal and an agent who can exert costly effort, e
•Agent receives transfer, t, and has utility;
U=u(t)ψ(e), with u / >0, u // <0.
•Production is stochastic, and production level, q

{ q , q } , q
 q
• Stochastic influence of effort on production:
Pr{ q
 q e

0 }
 
0
; Pr{ q
 q e

1 }
 
1
,

1
 
0

• Principal can offer a contract, {t( )}, that depends on observed, random output t q
• Let Principal’s profit with qty q be S(q)
• His profit when agent expends effort e=0 is:
V
0
 
0
[ S ( q )
 t ]

( 1
 
0
)[ S ( q )
 t ]
• His profit when agent expends effort e=1 is:
V
1
 
1
[ S ( q )
 t ]

( 1
 
1
)[ S ( q )
 t ]

• Induce positive effort and ensure participation
• Incentive constraint :

1 u ( t )

( 1
 
1
) u ( t )
   
0 u ( t )

( 1
 
0
) u ( t )
• Participation constraint :

1 u ( t )

( 1
 
1
) u ( t )
  
0

• Complete info or First-Best: Principal observes effort
• Principal’s problem is: max
{( t , t )}

1
( S
 t )

( 1
 
1
)( S
 t ) subject to:

1 u ( t )

( 1
 
1
) u ( t )
  
0
• Using Lagrangian, μ, and from FOCs we have,
  u
/
1
( t
*
)

1
( t
*
)

0 u
/
• From the above equations, we have that: t
*  t
*  t
*
• Thus, Agent obtains full insurance!
• The optimal transfer is: t * = u -1 ( ψ)=h(ψ), where h=u -1

• When there is complete information
• Principal’s profit from inducing effort e=1:
V
1
=

1
S

( 1
 
1
) S
 h (

)
• If agent exerted 0 effort, principal would earn:
V
0
=

0
S

( 1
 
0
) S
• Inducing effort is optimal for principal if:
  
S
 h (

)
, where
   
1
 
0
;

S

S

S
• Principal’s First-Best cost of inducing effort is: h(ψ)

• Agent is risk-averse
• Principal’s problem, P, is:
• (P): max
{( t , t )}

1
( S
 t )

( 1
 
1
)( S
 t ) subject to:

1 u ( t )

( 1
 
1
) u ( t )
  
0 , and

1 u ( t )

( 1
 
1
) u ( t )
   
0 u ( t )

( 1
 
0
) u ( t )
• First ensure concavity of (P): Let u
 u ( t ); u
 u ( t )

• The Principal’s program can be rewritten in terms of utilities
• (P / ): max
{( u , u )}

1
( S
 h ( u ))

( 1
 
1
)( S
 h ( u )) subject to :

1 u

( 1
 
1
) u
 

1 u

( 1
 
1
) u
 
 
0 u

( 1
 
0
) u

0
• Principal’s objective function is concave in ( u , u ) because h(.) is convex, and the constraints are linear
• The KKT conditions are necessary and sufficient

• Let λ & μ be Lagrange multipliers for IC & IR
• The FOCs, upon rearranging terms, are:
1 u
/
( t
SB
)
   
 

1
1
; u
/
( t
SB
)
   
 
1
 
1 where, t
SB
, t
SB are second-best optimal transfers
• From these,  
1
 
1 u
/
( t
SB
)


1 u
/
( t
SB
)

0 , so IR is binding
• Also,  

1
( 1

 

1
) 1
( u
/
( t
SB
)

1 u
/
( t
SB
)
)

0
, so IC is binding

• The variables (
SB t , t
SB
, λ, μ ) are solved simultaneously from two FOCs, IC and IR
•
• The second-best optimal transfers are: t
SB  h (
 
( 1
 
1
)

 
); t
SB  h (
   t
SB  t
SB

1 t
1

 
)
: contract does not provide full insurance
• 2 nd Best cost of inducing effort: C SB =
SB 
( 1
 
1
) t
SB
• Clearly, for the Principal, C SB > C FB . So Principal induces high effort (e=1) less often than in first-best
• There is under-provision of effort in the second-best

• Agent’s type  
[

,

] with distribution/density P (

) / p (

)
• Type-contingent allocation is fn.
  y (

)

( x (

), t (

• Defn: A decision function x :
  X if
)) there exists a transfer t (.) such that allocation y (.) is incentive-compatible, i.e.
u
1
( y (

),

)
 u
1
( y (
 ˆ
),

),

(

,
 ˆ
)

[

,

]

[

,

]
• Theorem: A piecewise C 1 decision fn x (.) is implementable only if k n 

1

 
(
 u
1
 u
1
/
/
 x k
 t
) dx k d


0 whenever x
 x (

), t
 t (

) and x is differentiable at θ

• Sketch of proof: Type
 ˆ

(
 ˆ
,

)
 u
1
( x (
 ˆ
), t (
 ˆ
),

)
The FOC and SOC are
 
(

  ˆ
,

)

0 ,
 2 
(

  ˆ
2
,

)

0
Totally differentiating the first equation,
 2 
(

  ˆ
2
,

)

 2 
(

  ˆ  
,

)

0
The (local) SOC becomes k n 

1

 


 u
1
 x k

 dx k d



 
 u
1
 t dt d


0
 2 
(

 
,
  ˆ

)

0
Rewrite the FOC we get, k n 

1


 u
1
 x k

 dx k d

Eliminating, dt / d θ, k n 
{[

1

 


 u
1
 x k


 u
1
 t


 
 u
1
 t
 or,
 u
1
 t dt d


0
 u
1
 x k
] /
 u
1
 t
} dx d
 k

0

• The sorting / single crossing / constant sign (CS) condition is:

 


 u
1
 u
1
/
/
 x k
 t



0
• Note that
 u
1
 u
1
/
/
 x
 t k is agent’s marginal rate of substitution between decision k and transfer t
• Consider x to be output supplied by agent, i.e.,
 u
1
 x

0
• Then sorting condition means that the agent’s indifference curve in ( x , t ) space,
 u
1
 u
1
/
/
 x
 t
, is decreasing in θ
• If θ
2
> θ
1
, y ( θ
1
)=( x ( θ
1
), t ( θ
1
)), y ( θ
2
)=( x ( θ
2
), t ( θ
2
)), then y ( θ
2
)> y ( θ
1
)
• Theorem: If decision space is 1-dim and CS holds, then a necessary condition for x (.) to be implementable is that it is monotonic.
• What about sufficiency?

• The assumptions:
A1: u
A2: Quasi-linear utilities:
Principal: u
0
( x, t, θ )= V
0
( x, θ )t ; Agent: u
1
( x, t, θ )= V
1
( x, θ )+ t
A3: n=1, i.e., decision is 1-dim and CS holds.
 2
V
1
 x
 

0
A4:

V
1
 

0
A5:
 2
V
0
 x
 

0
A6:

3 V
1
 x
 
2

0 , &

 x 2
3 V
1
 

0

• The problem: Principal maximizes his expected utility
{ x max
(

), t (

)}
E
 u
0
( x (

), t (

),

) subject to: (IR) u
1
( x ( θ ), t ( θ ), θ u θ
(IC) u
1
( x ( θ ), t ( θ ), θ )≥ u
1
( x (
 ˆ
), t (
 ˆ
),

)

(

,
 ˆ
)
• Let
U
1
(

)

  max
 ˆ
  u
1
( x (

), t (

),

) u
1
( x (
 ˆ
), t (
 ˆ
),

)

 u u
1
( x (

), t (

),

)

0
• From Envelope theorem, dU
1 d


 u
1
 


V
1
 
• This implies that,
U
1
(

)
 u
 



V
1
( x

(


~
~
,

~
) d

~

• Further, u
0
= V
0
+ V
1
- U
1
≡ Social surplus-
• Principal’s objective function:
Agent’s utility



[ V
0
( x (

),

)

V
1
( x (

),

)
 



V
1
( x
 

~
(
~
),

~
) d

~
] p (

) d
 



[ V
0
( x (

),

)

V
1
( x (

),

)

1

P (
 p (

)
)

V
1
( x (

 
),

)
] p (

) d

• Since monotonicity is necessary and sufficient for implementability, Principal’s optimization program becomes max
{x(.)}



[ V
0
( x ,

)

V
1
( x ,

)

1

P (
 p (

)
)

V
1
( x ,
 

)
] p (

) d
 s.t. x (.) is monotonic

• We solve the principal’s program ignoring monotonicity
• The solution to the relaxed program is

V
0
 x


V
1
 x

1

P (
 p (

)
)
 2
V
1
 x
 
• The principal faces a trade-off between maximizing total surplus ( V
0
+ V
1
) and appropriating the agent’s info rent ( U
1
)
• When is it legit to focus on relaxed program?
When solution x * ( θ) to above eq is monotonic. Differentiating,
(
 2
V
0
 x
2

 2
V
1
 x
2

1

P (
 p (

)
)

 x
2
3
V
1
 
) dx
* d


 2
V
1
 x
  d
[ d

(
1

P p (

(

)
)
)

1 ]

 2
V
0
 x
 

1

P ( p (

)

)
 3
V
1
 x
 
2
When Hazard rate is monotone : d d



1
 p (
P

)
(

)



0