Now suppose that a new miniature golf course opens in the friends' town. While all the friends prefer playing golf to going to the park, some friends like golf more than others. The utils received from playing golf are as follows for the friends: Aaron Britney Carlos Diana Emily Frank Gwen Hugo Isabela Juliana 8 13 18 20 27 30 31 34 36 37 d) Using the above information, identify the equilibrium group of hot 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 golf players in both situations. Equilibrium group: Optimal group: Why do the groups differ? Grand total benefits for hot tub users_ Grand total benefits for golf players_ e) Compute the toll/subsidy schedule required to support the optimum and add it to the right-hand side of the table in part a). 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. Find the toll charged (or subsidy paid) to achieve the optimal number of hot tub users in both the park case and the golf case. Toll/subsidy in the park case_ Toll/subsidy in the golf case 2. Aaron and his friends Britney, Carlos, Diana, Emily, Frank, Gwen, Hugo, Isabela, and Juliana have two choices for weekend activities. They can either go to the neighborhood park or get together in Aaron's hot tub. The park isn't 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 doesn't depend on how many of the friends go to the park. Aaron's 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 isn't too crowded, it's 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 7) is 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 7 values as well as 7 = 0. Total benefit is just 7 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. 2 3 4 5 6 7 8 9 AB = 2 +8T-T, 10 AB TB MB b) Recalling that the park yields 3 utils in benefits to each person, find the equilibrium size of the group using the hot tub. (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.) Equilibrium size c) Find the optimal size of the hot tub group and explain 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? Optimal size Explanation: Grand total benefits at equilibrium_ What do your results show? Grand total benefits at optimum