Adam and his friends Brigit, Cheryl, David, Emily, Frank, Gail, Henry, Ivan, and Juliet have two choices for weekend activities. They can either go to the local park or get together in Adams hot tub.

The local park isnt much fun, which means that the benefits from being there are low on the friends common utility scale. In fact, each of the friends receives a benefit equal to 3 utils from being at the park.

This benefit doesnt depend on how many of the friends go to the park. Adams hot tub, on the other hand, can be fun, but the benefits of using it depend on how many of the friends are present. When the tub isnt too crowded, its quite enjoyable.

When lots of people show up, however, the tub is decidedly less pleasant. The relationship between benefit per person (measured in utils) and the number of people in the hot tub (denoted by T) is AB =9+6TT2, where AB denotes average benefit.

(a) Using the above formula, compute AB for T = 1, 2, 3, . . . ,10. Next compute total benefit from use of the hot tub for the above T values as well as T = 0. Total benefit is just T times AB. Finally, compute marginal benefit (MB), which equals the change in total benefit from adding a person to the hot tub. To do so, adopt the following convention: define MB at T = T to be the change in total benefit when T is increased from T 1 to T (in other words, MB gives the change in total benefits from entry of the last person). Deviation from this convention will lead to inappropriate answers. For example, computation of MB using calculus will lead you astray, since we are dealing with a discrete problem rather than a continuous one.

(b) Recalling that the park yields 3 utils in benefits to each person, find the equilibrium size of the group using the hot tub. Show that (aside from the owner Adam), we cant be sure of the identities of the other hot tub users. (Hint: In contrast to the freeway case, the relevant benefit number will not exactly equal 3 at the equilibrium, with a similar outcome occurring in the other cases considered below.)

(c) Find the optimal size of the hot tub group, and give an explanation of why it differs from the equilibrium size. Next compute the grand total of benefits for all the friends, which is the sum of total benefits for the hot tub group and total benefits for those using the park. Perform this computation for both the equilibrium and the optimal group sizes. What do your results show?

(d)Draw the average benefit, marginal benefit and marginal cost curves on the same graph. Now suppose that a new video game arcade opens in the friends town. While all the friends prefer playing video games to going to the park, some friends like video games more than others. The utils received from playing video games are as follows for the friends:

A 3

B 7

C 11

D 15

E 22

F 30

G 31

H 34

I 36

J 37

(d) Using the above tub users. Then identify the optimal group of hot tub users. Why do the groups differ? Compute the grand total of benefits for hot tub users and video game players in both situations.

(e) Compute the toll/subsidy schedule required to support the optimum. Recall that in the freeway case, the toll at a given T is equal to the difference between MC and AC at that T. In the present case, the toll (or subsidy) is given by AB minus MB. Show that when this toll/ subsidy schedule is used, the equilibrium coincides with the optimum. Find the toll charged (or subsidy paid) in the new equilibrium in both the park case and the video games case.

(f) Draw the average benefit, marginal benefit and marginal cost curves on the same graph.