Exercise $5\mathrm{.}1$

Suppose there are three potential users of a freeway: Mr$.1,$ Mr$.2,$ and Mr$.3.$ The cost of the best alternative route for each commuter is as follows:

$256$

Exercises

Commuter

Mr$.1$ Mr$.2$ Mr$.3$

Alternate cost

$$7$ $$5$ $$3$

The average cost AC of using the freeway $($i$.$e$.,$ the cost per car$)$ as a function of traffic volume T is as follows:

T AC

$1$ $$22$ $$53$ $$9$

Using this information, answer the following questions:

$($a$)$ Find the equilibrium allocation of traffic between the freeway and alternate routes.

$($b$)$ Compute the total commuting cost for all commuters for the fol$-$ lowing four allocations of traffic. Total cost is the cost incurred by freeway users plus the cost incurred by commuters who use their alter$-$ nate routes.

On freeway

No one

Mr$.1$

Mr$.1,$ Mr$.2$

Mr$.1,$ Mr$.2,$ Mr$.3$

On alternate routes

Mr$.1,$ Mr$.2,$ Mr$.3$ Mr$.2,$ Mr$.3$

Mr$.3$

No one

$($Hint: Remember the definition of AC$.)$

$($c$)$ Remember that the socially optimal allocation of traffic between the freeway and alternate routes is the one that minimizes total commuting cost for all commuters. On the basis of your answer to $($b$),$ which alloca$-$ tion is socially optimal? How does total cost at the optimum compare with the total cost at the equilibrium? $($Note that you don$$t have to use an MC curve to get the answer to this question.$)$

Exercises $257$

Exercise $5\mathrm{.}2$

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 Adam$$s hot tub. The local park isn$$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$$t depend on how many of the friends go to the park. Adam$$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 T$)$ is AB $=2+8$T $$ T$2,$ where AB denotes $$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 $\text{'}$ to be the change in total benefit when T is increased from T $\text{'}1$ to T $\text{'}($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 calcu$-$ lus 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 can$$t 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?