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Solutions to Chapter 13 Exercises
SOLVED EXERCISES
S1. Some examples of incentive schemes that help induce more care on the part of policy holders:
1. The insurer can provide a multiyear contract or otherwise establishes an ongoing relationship in
2. The insurer can provide discounts for observable actions that reduce the risk that the
1. Insurers can offer rewards to people who observe fraudulent behavior by neighbors or others (for
example, reckless drivers) and report it. Companies could get together and set up a Web site, for
2. Insurers could increase the consequences associated with detected fraud. A firm can, of course,
S2. (a) Here are several examples of discounts being offered for larger quantities:
(i) A daily newspaper subscription for $20 per month, versus a single copy for $1.
(ii) A mobile-phone plan has a larger number of minutes per month, at a lower cost per
minute.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(b) Those who choose a large quantity reveal themselves to be willing to purchase more of
the product, with the lower price inducing them to purchase more rather than less. For example, a large
Without offering the different package sizes, the grocery would not manage to separate the two
types of customers. See the next exercise for a numerical example showing how this type of screening
strategy may increase the firm’s profits.
S3. (a) Let’s consider the different prices P that OWL could choose for the 300-minute plan.
Looking at the customers’ values, we can see who will purchase at each price:
There’s no point charging P < $20, because that would sell to the same number of people as P =
So we can immediately narrow the optimal price down to two choices: either P = $20 or P = $25.
Let’s consider each of these in turn, to see which one is optimal.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
P = $25: Sell to 50% of potential customers
P = $20: Sell to 100% of potential customers
(b) Similarly, let’s consider the different prices P that OWL could choose for the 600-minute
plan, and which customers will purchase at each price:
By an argument similar to that given in part (a), we know that the only two choices we need to
consider are P = $30 and P = $70. Let’s consider each of these in turn, to see which one is optimal.
(c) The ICL constraint will guarantee that a Light user chooses the 300-minute plan instead of
(d) The ICR constraint will guarantee that a Regular user chooses the 600-minute plan instead
(e) To maximize profits, OWL should charge as much as it can while still making sure the
ICL and ICR constraints are satisfied. In addition, OWL should make sure the participation constraints are
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
These can be rewritten as
If all that mattered were the participation constraints, then OWL would prefer to set prices of P300
That is, the difference in price between the two plans must be at least $10, or else the Light users will buy
the larger plan instead of the smaller one. We can see that this constraint is automatically satisfied with
the prices we chose above to satisfy the participation constraints.
In other words, for Regular users to buy the 600-minute plan, the 600-minute plan must cost no more than
$45 more than the 300-minute plan:
Note that when we just satisfied the participation constraints, we wanted to charge $70 for the
The average profit per potential customer is then
(f) Let’s consider the three cases separately.
Part (a) is a pooling outcome, because when OWL charges the optimal price of $20 for
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Part (b) is a separating outcome, because when OWL changes the optimal price of $70,
Part (e) is a separating outcome, because when OWL charges the optimal prices of $20
S4. (a) To casual users, Mictel offers to sell a low-end machine for a price of x = 4. The firm
(b) If producing only low-end machines, Mictel can either set x = 4 and sell to everybody, or
(c) If producing only high-end machines, Mictel either sets y = 5 and sells to everybody, or
(d) The incentive-compatibility constraints are
(e) The participation constraints are
(f) We assume, as in part (d), that the firm wants casual users to buy the low-end machine
and intensive users to buy the high-end machine. If the firm had to worry only about the participation
To satisfy the incentive constraint for intensive users, Mictel would need to lower the price of the
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
The optimal prices are therefore x = 4 and y = 7. The company’s expected profit from this policy
is the weighted average between the profit earned from a casual user and the profit earned from an
intensive user:
Note that this expected profit is lower than what the firm could get if it could costlessly identify the type
of each user, as in part (a). Then the expected profit would be
(g) In part (b), we saw that the profit from selling only low-end machines is 3.2. In part (c),
Note that all three possibilities give lower profit than the expected profit of 5 that the firm would
get if it could identify the type of each user, as in part (a).
S5. (a) Same as the answer to Exercise S4 above. These prices don’t depend on the proportion of
casual users, only on users’ willingness to pay.
(b) If producing only low-end machines, Mictel can either set x = 4 and sell to everybody, or
(c) If producing only high-end machines, Mictel either sets y = 5 and sells to everybody, or
(d) Same as in Exercise S4.
(e) Same as in Exercise S4.
(f) As in Exercise S4, to satisfy the participation and incentive constraints with the highest
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(g) In part (b), we saw that the profit from selling only low-end machines is 3. In part (c), we
S6. (a) Same as the answer to Exercise S4 above.
(b) If producing only low-end machines, Mictel can either set x = 4 and sell to everybody, or
(c) If producing only high-end machines, Mictel either sets y = 5 and sells to everybody, or
(d) Same as in Exercise S4.
(e) Same as in Exercise S4.
(f) As in Exercise S4, to satisfy the participation and incentive constraints with the highest
Again, note that this expected profit is lower than what the firm could get if it could identify the
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(g) Summarizing the results of parts (b), (c), and (f), we know that there are three possible
cases we need to consider: c ≤ 1/4, 1/4 < c ≤ 3/5, and c > 3/5.
(i) When c ≤ 1/4:
Selling just the low-end machine earns a profit of 4(1 – c).
Which of these three expected profits is highest? We can see immediately that 5(1 – c) >
4(1 – c), so Mictel should not sell just the low-end machine. Mictel should sell just the
(ii) When 1/4 < c ≤ 3/5:
Selling just the low-end machine earns a profit of 3.
We can see readily that 4 – c > 3, because c < 1, so Mictel should not sell just the low-end
machine. Mictel should sell just the high-end machine if 5(1 – c) > 4 – c, and both
(iii) When c > 3/5:
Selling just the low-end machine earns a profit of 3.
We see that 3 > 2, so Mictel should not sell just the high-end machine. Mictel should sell
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
To summarize the three cases, we see that Mictel should do the following:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
