2012-1 Final Solutions
A1. 4, 5. Type A values each of the first 4 units at more than the cost, so they should consume
at least 4. After that, the benefit is nil, while the cost is positive, so the quantity should be
exactly 4. Similarly, type B’s quantity should be exactly 5.
A2. In the first best, the monopoly chooses the efficient allocation to maximize the surplus
because it can take all the surplus. Thus it makes 4*(4-1) = 12 from type A and 5*(5-1) =
20 from type B.
A3. subject to
IR means “individual rationality”, and IC means “incentive compatibility.”
A4. Because , and together imply , which can be
eliminated. This means that must be binding, because otherwise the firm can raise .
Therefore, , which implies as long as
because increases with . This in turn implies that
must bind because otherwise the firm can raise .
A5. The firm’s maximization problem becomes:
subject to .
By inspection (or economic intuition), the firm must pick and , so the
problem becomes: ,
or , subject to , , and .
It’s easy to see that must be as high as possible, so . The same goes for
when is positive, so when . Otherwise, must be as low as
possible, so when . (If , the firm is indifferent between all
from 0 to 4.)
Thus when , and . From , we get , and then from ,
we get .
When , and . Then , and from , we get .
A6. In the model from class, the low type’s second-best quantity is always lower than its first-
best quantity. That is not the case here when . The reason is that the marginal
value of the good is not continuous (it drops from 4 to 0), so it is not equal to the
marginal cost (1) at the first-best quantity. This means that when the firm decreases
from 4, the loss in profit is first-order rather than second-order. Thus, when there are
enough customers of type A, the firm is unwilling to do so. A7. Type A always gets 0 surplus ( always binds), so it has no reason to perform the task.
Type B would like to pool with type A, because if the fraction of type A in the pool is
above ¼, type B gets positive surplus. Since type A never performs the task, type B also
won’t – if it does, it would incur a cost and not gain any surplus. Thus the only
equilibrium is a pooling equilibrium where no one performs the task. Obviously, k>c
or c>k doesn’t matter!
B1. . Certainty equivalent is
$11.4 , or $129.96. Thus you need to offer your friend at least $29.96.
B2. In the first best, Dean is employed (his value to the firm is $200,000 and he only costs
$19,600), but exerts no effort (effort is useless, but the firm would have to pay him more
if it requires effort). The first best is achievable: the firm pays Dean a fixed $19,600.
B3. The efficient allocation is for all the good cars to go to the buyers, and for all the bad
cars to stay with the sellers. This can never happen regardless of p: the price must be
above $6,000 for the good cars to sell, but then the owners of bad cars would sell as well.
B4. FALSE (or UNCERTAIN): While it’s true that the appropriate size of a carbon tax is
hard to determine, if the right size were known, then the efficient allocation of emissions
will be achieved. With regulation, even if the r