e 2 f0; 1g;
yk ; k 2 f0; :::; Lg; ordered so that strictly increasing.
Pr(y`) = `(e)
Monotone likelihood ratio :
` (0) is strictly decreasing in `
` (1)
u( w ) d(e); d(1) > d(0):
vNM Utility function u(:) is increasing and strictly concave
Worker's reservation utility is u
contract is (w0; ::w`; :::; wL)
Symmetric information
Firm can specify e ort level e
L
X
max ` (e)V (y` w` )
e;w0 ;w1 ;w2 ;::;wL
`=0
subject to
L
X
` (e)u(w` ) d(e) u: (IR)
`=0
Max wrt (w`) for xed e; then choose e
2 3
L
X L
X
L= 4 u5
` (e)V (y` w` ) + ` (e)u(w` ) d(e)
`=0 `=0
@L 0 (y 0
= ` (e)V ` w` ) + ` (e)u (w` ) = 0; 8`:
@w`
XL
@L
= ` (e)u(w` ) d(e) u = 0 if 6= 0:
@ `=0
> 0 ) IR binds.
u0 ( w ` ) 1
0 (y
= 8`
V ` w` )
optimal risk sharing (Borch condition)
outcome is Pareto e cient.
X
u(w`) `(e) = d(e) + u
^
This fully determines wage schedule for e ort level e; w`(e)
L
X
max ` (e)V (y` ^
w`(e)):
e2f0;1g `=0
Special cases: worker risk neutral, rm risk neutral
Asymmetric Information
Incentive constraint: if the company wants e to be chosen, this must be optimal
for worker
L
X L
X
0 d(e0):
` (e)u(w` ) d(e) ` ( e ) u( w ` ) (IC)
`=0 `=0
Max now subject to IC and IR
Suppose company wants e = 0 be be chosen
optimal policy under symmetric information
if this does not satisfy IC, rm is better
with risk neutral rm, IC will be violated
E(u(e = 1)) = E(u(e = 0)) & d(1) > d(0)
For inducing e = 1;
2 3
L
X L
X
L = 4 u5
` (e)V (y` w` ) + ` (e)u(w` ) d(e)
`=0 8 `=0 9
0)
IC must bind at optimum ( 6= 0) (otherwise contract as in rst best)
We can show that >0:
Proof: demonstrate that rst best contract satis es IC strictly if 1; wages lower than rst best, otherwise greater than
` (1)
FB
Trade o between insurance & incentives.