Download Solution to In-Text Exercise 20.3: First we compute the marginal

Solution to In-Text Exercise 20.3:
First we compute the marginal social benefit of security:
The marginal social benefit equals marginal cost at the value of S that satisfies:
Solving for S, we obtain S = 45. Therefore, the socially efficient level of security is 45 personhours of patrols per week.
If one store provided all of the security by itself, it would set its marginal benefit equal to the
marginal cost:
Solving for S, we obtain S = 35. A store’s marginal benefit is greater than marginal cost when S
< 35 and less than marginal cost when S > 35.
If security is left to the independent decisions of the stores, they will provide 35 person-hours of
security patrols in total. If the total were less than 35 person-hours, marginal benefit would
exceed marginal cost for all of the stores, and they would all have an incentive to spend more on
security. If the total were greater than 35 hours, marginal cost would exceed marginal benefit for
all of the stores, and any store that was providing security would have an incentive to reduce the
amount provided.
Now let’s repeat the calculation with N firms (which we will take to be 5, 10, or 100). The
marginal social benefit of security is:
The marginal social benefit equals marginal cost at the value of S that satisfies:
1
Solving for S, we obtain
. Therefore, the socially efficient level of security is
person-hours of patrols per week. With 5 stores, it is 47 person-hours; with 10 stores, it
is 48.5 person-hours; and with 100 stores, it is 49.85 person-hours.
In contrast, if security is left to the independent decisions of the stores, the amount provided will
be 35 person-hours regardless of the number of firms. To see why, notice that the argument
given above with three firms didn’t depend on the number of firms.