Answers to Final Examination
Economics 703
Spring 2016
1. (a) The characterization of incentive compatibility for linear environments applies
here. So we have to have
Z θi
ȳi (s) ds
Ui (θi ) = Ui (0) +
0
where ȳi (θi ) = Eθj y(θ1 , θ2 ). Also,
Ui (θi ) = θi ȳi (θi ) + t̄i (θi ),
where t̄i (θi ) = Eθj ti (θ1 , θ2 ). Note that if θi = 0, then the probability that θ1 + θ2 > 1 is 0.
By assumption, if the sum of the θ’s is less than 1, the transfers are 0. Hence Ui (0) = 0.
Rearranging the definitions and using Ui (0) = 0 gives
−t̄i (θi ) = θi ȳi (θi ) −
Z θi
0
ȳi (s) ds.
Note that y(θ1 , θ2 ) is 1 iff θ1 + θ2 ≥ 1, so ȳi (s) is the probability that s + θj ≥ 1. Given
the uniform distribution, this probability is s. So
1
1
−t̄i (θi ) = θi2 − θi2 = θi2 .
2
2
So
−Eti (θ1 , θ2 ) = −Eθi t̄i (θi ) =
Z 1
0
1 2 1
θ = .
2 i
6
(b) The answer to (a) tells us that the expected payment made by the two agents in
total is 1/3. But the public good is being provided iff θ1 + θ2 > 1. The probability of
this event is 1/2 (since this area is half the unit square). So the expected total transfers
must equal 1/2, a contradiction.
1
2. (a) The principal chooses eH , wH , eL , and wL to maximize his expected profits subject
to the individual rationality and incentive compatibility constraints:
wj −
ej
≥ 0, j = L, H
θj
eH
eL
≥ wH −
θL
θL
eH
eL
wH −
≥ wL −
.
θH
θH
As usual, the only binding constraints are the individual rationality constraint for type
θL and the incentive compatibility constraint for type θH . Hence
wL −
wL =
eL
θL
eL
eL
eH
+
−
.
θH θL θH
Substituting these results into the objective function, the principal will choose eL and eH
to maximize
1 √
eH
eL
eL
eL
1 √
2 eH −
−
+
+
2 eL −
2
θH
θL θH
2
θL
wH =
2
The first–order condition for eH is the same as in the first–best (as usual), so eH = θH
= 4.
The first–order condition for eL is
1
2
or
"
#
1 1
1
1
1
=
−
√ −
eL θL
2 θL θH
1
1
√ −1=1− .
eL
2
Rearranging gives eL = 4/9. The wages are wL = 4/9 and
wH = 2 +
4 2
20
− = .
9 9
9
(b) Suppose the principal offers only one wage–effort pair. The problem says the
principal must hire the agent, so this must satisfy the individual rationality constraint
for both types. Since the low type has higher costs, if (w, e) satisfies individual rationality
for the low type, it must satisfy it for the high type. Incentive compatibility is no longer
relevant since there’s no “lies” anyone can tell. So the
√ best version of this option for the
principal is to offer the w and e which maximize 2 e − w subject to w − (e/θL ) ≥ 0.
This is the same problem as the first–best for the low type,√so we know that e = 1 and
w = 1. If the principal follows this approach, his payoff is 2 e − w = 2 − 1 = 1.
2
If the principal offers two wage–effort pairs, then he has to choose the first–best
efforts in each case. Otherwise, the contract cannot be ex post efficient since the total
payoff to the principal and agent could be made larger. Hence we must have eL = 1 and
eH = 4. We also have to satisfy incentive compatibility and individual rationality. So
our constraints are
wL − 1 ≥ 0
wH − 2 ≥ 0
1
eL
= wL −
wH − 2 ≥ wL −
θH
2
eH
wL − 1 ≥ wH −
= wH − 4.
θL
Rewriting:
wL ≥ 1
wH ≥ 2
3
2
wL ≥ wH − 3.
wH ≥ wL +
It’s easy to see that the first and third constraints imply the second. Also, if we ignore the
fourth constraint, we get wL = 1 and wH = 5/2. Since this implies the fourth constraint,
it can’t bind and these must be the wages. The principal’s profit is (1/2)(6 − 1 − 2.5) =
5/4 > 1. Hence this is the optimal contract.
3. Let ei , i = h, `, denote the education choice by type θi . For a separating equilibrium,
we require incentive compatibility and also require that neither type wishes to deviate
to some off equilibrium education level. In this context, incentive compatibility says
√
√
eh
e`
≥ 40 + 4 e` −
100 + 4 eh −
5
5
and
√
√
2eh
2e`
≤ 40 + 4 e` −
.
100 + 4 eh −
5
5
Of course, the best way to prevent deviations to some off equilibrium education level is
to suppose that the employers’ beliefs in response to such a deviation is to believe that
the worker is type θ` . Suppose this is the belief in response to any devaition. Note that
both types have nontrivial preferences regarding education levels, so we have to ensure
that sticking to the equilibrium is better than the best deviation.
√
For type θ` , the best deviation would be to that e which maximizes 4 e − 2e/5. The
first–order condition for e is
2
2
√ − = 0,
e 5
3
so the maximizer is 25. The only way to prevent the low type from wanting to deviate
from e` to 25 is to set e` equal to 25. If we have a separating equilibrium where type
√ θ`
√
chooses any e` 6= 25, then his equilibrium payoff is 40 + 4 e` − 2e` /5 < 40 + 4 25 −
(2)(25)/5 = 50. Because we’re assuming the employers believe he is the low type if he
deviates, the right–hand side is the payoff type θ` would get if he deviated to 25. Hence
he would want to deviate. So we must have e` = 25. Plugging this into the incentive
compatibility conditions above, we can rewrite them as
√
√
eh
25
≥ 40 + 4 25 −
= 55
100 + 4 eh −
5
5
and
√
2eh
100 + 4 eh −
≤ 50.
5
√
For the high type, the best deviation would be to the e which maximizes 4 e − e/5.
The first–order condition is
1
2
√ − =0
e 5
so the √maximizing e is 100. If the high type deviates to e = 100, his payoff would be
40 + 4 100 − (100/5) = 60. So we have to add the constraint that
√
eh
100 + 4 eh −
≥ 60.
5
Clearly, this implies the first of the two incentive compatibility conditions above. Finally,
then, we see that eh must satisfy
√
eh
≥ 60
100 + 4 eh −
5
√
2eh
100 + 4 eh −
≤ 50.
5
So the set of separating equilibrium strategies for the worker consist of the pairs e` = 25
and any eh satisfying these two inequalities. You can simplify further and even prove
that the set of such eh is non–empty, but this is sufficient.
4