Exercise 1: Suppose that landowners have the power to restrict x¯, the distance to the edge ofthe city, in order to increase the land rent they earn. Suppose that, with no restriction, the urban land function is given by r = 100 - x, where x is distance in blocks to the CBD. Suppose that agricultural land rent rA is equal to 20.
a) Compute x¯ in the absence of any restriction by landowners, and show your result in a diagram.
Now suppose that landowners can restrict x¯ to a value of 65 blocks. When they impose this restriction, the urban land rent curve shifts up, with the new rent function given by r = 105 - x.
b) Show the new land-rent curve in your diagram, and indicate the area corresponding to the land-rent loss from the restriction, as well as the additional area showing the land-rent gain.
c) Compute the sizes of these areas, and compute the net gain or loss in land rent from imposing the x¯ restriction. Is the restriction beneficial to the landlords? Will it be imposed? (Hint: the area corresponding to the land rent gain is a parallelogram, but instead of using the area formula for that type of shape, the area can be computed more easily by multiplying the horizontal length of the parallelogram (65 blocks) by its height.)
Now suppose that the landowners can impose a further restriction, with x¯ set equal to 50 blocks. When this restriction is imposed, the urban land-rent curve shifts up more, with the new rent function given by r = 110 - x.
d) Repeat parts (b) and (c). Relative to the original x¯ restriction of 65, is this further restriction beneficial to the landlords? Will it be imposed?
e) If your answer is different from before, explain intuitively why a difference emerges.
Exercise 2: Suppose there are 3 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:
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Commuter
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Alternate Cost (ga)
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|
Mr. 1
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$7
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|
Mr. 2
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$5
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|
Mr. 3
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$3
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The average cost AC of using the freeway (i.e., the cost per car) as a function of traffic volume T is as follows:
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T
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Average Cost (AC)
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1
|
$2
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2
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$5
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3
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$9
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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 following four allocations of traffic. Total cost is the cost incurred by freeway users plus the cost incurred by commuters who use their alternate routes. (Hint: Remember the definition of AC, while carrying out your calculations.)
| On freeway |
On alternate routes |
| Nobody |
Mr. 1, Mr. 2, Mr. 3 |
| Mr. 1 |
Mr. 2, Mr. 3 |
| Mr. 1, Mr. 2 |
Mr. 3 |
| Mr. 1, Mr. 2, Mr. 3 |
nobody |
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. Based on your answer to (b), which allocation is socially optimal? How does total cost at the optimum compare to total cost at the equilibrium? (Note that you don't need to use an MC curve to get the answer to this question.)
Exercise 3: 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 T) is
AB = 2 + 8T - T 2, where AB denotes "average benefit".
a) Using the above formula, compute AB for T = 1, 2, 3... 8, 9, 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 this, adopt the following convention:
define MB at T = T' to be the change in total benefit when T changes 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 given that we're dealing with a discrete rather than continuous problem.
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 won't 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) 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:
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Adam
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8
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Brigit
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13
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Cheryl
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18
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David
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20
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Emily
|
27
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|
Frank
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30
|
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Gail
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31
|
|
Henry
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34
|
|
Ivan
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36
|
|
Juliet
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37
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Using the above information, identify the equilibrium group of hot tub users. Then identify the optimal group of hot tub users. Explain why 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 and video games cases.