Answers to Final Examination
Economics 703
Spring 2015
√
√
1. (a) The consumer’s expected utility if he is
√cautious is (1/2) 100 + (1/2) 36 − 1 = 7.
His expected utility if he is not cautious is 36 = 6. So the consumer will be cautious
and have expected utility of 7.
(b) First, note that if the contract √
(ts , tL ) induces the consumer to not be cautious,
then the consumer would accept it iff 36 + tL ≥ 7 or tL ≥ 13. Clearly, the insurance
company’s profit from this contract is −13, so they’d rather not offer any contract than
offer this one.
Hence the optimal contract must satisfy
√
1√
1√
100 + ts +
36 + tL − 1 ≥ 36 + tL ,
2
2
so that the consumer will choose to be cautious. It must also satisfy the individual
rationality constraint
1√
1√
100 + ts +
36 + tL − 1 ≥ 7.
2
2
So the consumer chooses (ts , tL ) to maximize (1/2)(−ts ) + (1/2)(−tL ) subject to these
constraints. It is not hard to see that both are binding. To see that the first one must
bind, note that if it didn’t, the contract would have to have the insurance company
bearing all the risk, in which case the consumer would never incur a cost to be cautious.
If the first is binding but the second isn’t, then the company could lower tL and be better
off.
Since both constraints are binding, we have
√
√
100 + ts = 36 + tL + 2
√
√
100 + t2 + 36 + tL = 16.
Substituting from the first into the second gives
√
2 36 + tL + 2 = 16
1
so
√
36 + tL = 7 or tL = 13. Hence
√
100 + ts = 7 + 2 = 9,
so ts = −19. The insurance company’s expected profit is (1/2)(19 − 13) = 3. The
consumer’s expected utility is 7 since the individual rationality constraint is binding.
2. We know that we must have
Z
θi
Ui (θi ) = Ui (0) +
ȳi (s) ds.
0
Since the good is always allocated to the bidder with the highest type, ȳi (s) is the
probability that the other I − 1 bidders have types below s. Hence ȳi (s) = sI−1 . So
Z θi
θI
Ui (θi ) = Ui (0) +
sI−1 ds = ,
I
0
where I also use the assumption that Ui (0) = 0. We also know that
−t̄i (θi ) = θi ȳi (θi ) − Ui (θ) = θiI −
θI
,
I
so the seller’s expected revenue is
Z 1
1 I
I −1
I
1−
θ dθ =
.
I
I +1
0
This is exactly what we computed for the first and second price auctions in class using
the equilibrium strategies in these auctions.
3. As usual, the individual rationality constraint is binding for the low type and the
incentive constraint is binding for the high type. Thus
wL = αeL
wH = eH + αeL − eL .
So the principal chooses the efforts, eH and eL , to maximize
1 √
1 √
[2 eH − eH + αeL − eL ] + [2 eL − αeL ] .
2
2
The first–order condition for eH is
1
= 1,
√
eH
so eH = 1. The first–order condition for eL is
1
√ − α = α − 1.
eL
2
So
eL =
1
.
(2α − 1)2
wL =
α
(2α − 1)2
Hence
wH = 1 +
α−1
.
(2α − 1)2
(b) The low type gets his outside option regardless of α, so he doesn’t care what α
is. The high type’s payoff is
α−1
.
wH − e H =
(2α − 1)2
The derivative of his payoff with respect to α is
4(α − 1)
1
−
.
2
(2α − 1)
(2α − 1)3
This is greater than or equal to zero iff
(2α − 1) − 4(α − 1) ≥ 0
or 2α ≤ 3 or α ≤ 3/2. Hence the derivative is positive for α ≤ 3/2 and negative for
α ≥ 3/2. Hence high type’s preferred α is α = 3/2.
Finally, the principal’s expected payoff is
" s
#
1 √
1
1
[2 1 − wH ] +
2
− wL
2
2
(2α − 1)2
1
2
α
α−1
=
+
−
2−1−
2
(2α − 1)2 2α − 1 (2α − 1)2
2
1
2α − 1
=
1−
+
2
(2α − 1)2 2α − 1
1
1
=
1+
2
2α − 1
This is obviously decreasing in α, so the best α for the principal, given the constraint
α ≥ 1, is α = 1.
3