2. Aaron and his friends Britney, Carlos, Diana, Emily, Frank, Gwen, Hugo, Isabela, and Juliana have...

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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 6 "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 AB 3+7 T-T = 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 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. T AB TB MB 1 2 3 5 6 7 8 9 10 b) Recalling that the park yields 6 utils in benefits to each person, find the equilibrium size of the group using the hot tub. 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. c) Equilibrium size Grand total benefits at equilibrium Find the optimal size of the hot tub group. 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. Optimal size Grand total benefits at optimum 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: 8 Aaron Britney 10 Carlos 14 20 Diana Emily 27 Frank 30 Gwen 31 Hugo 34 Isabela 36 Juliana 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 (names): Optimal group (names): Grand total benefits for hot tub users Grand total benefits for golf players_ e) 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 6 "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 AB 3+7 T-T = 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 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. T AB TB MB 1 2 3 5 6 7 8 9 10 b) Recalling that the park yields 6 utils in benefits to each person, find the equilibrium size of the group using the hot tub. 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. c) Equilibrium size Grand total benefits at equilibrium Find the optimal size of the hot tub group. 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. Optimal size Grand total benefits at optimum 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: 8 Aaron Britney 10 Carlos 14 20 Diana Emily 27 Frank 30 Gwen 31 Hugo 34 Isabela 36 Juliana 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 (names): Optimal group (names): Grand total benefits for hot tub users Grand total benefits for golf players_ e) 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