exercise 8 3 consider a city with three consumers 1 2 and 3 the city p

Question

Exercise 8.3
Consider a city with three consumers: 1, 2, and 3. The city provides
park land for the enjoyment of its residents. Parks are a public good,
and the amount of park land (which is measured in acres) is denoted
by z. The demands for park land for the three consumers are as follows:
D₁ = 40 - z,
D₂ = 30-z,
D₂ = 20 - z.
These formulas give the height of each consumer's demand curve at a
given level of z. Note that each demand curve cuts the horizontal axis,
eventually becoming negative. For the problem to work out right, you
must use this feature of the curves in deriving D₂. In other words, don't
assume that the curves become horizontal once they hit the axis.
(a) The height of the De curve at a given z is just the sum of the heights
of the individual demands at that z. Using this fact, compute the expres-
sion that gives the height up to the De curve at each z.
(b) The cost of park land per acre, denoted by c, is 9 (like the demand
intercepts, you can think of this cost as measured in thousands of
dollars). Given the cost of park land, compute the socially optimal
number of acres of park land in the city.
(c) Compute the level of social surplus at the optimal z. This is just the
area of the surplus triangle between Dy and the cost line.
(d) Suppose there are two other jurisdictions, each with three consum-
ers, just like the given jurisdiction. Compute total social surplus in the
three jurisdictions, assuming each chooses the same amount of park
acres as the first jurisdiction.
(e) Now suppose the population is reorganized into three homoge-
neous jurisdictions. The first has three type-1 consumers (i.e., high
demanders). The second has three type-2 consumers (medium demand-
ers), and the third has three type-3 consumers (low demanders). Repeat
(a), (b), and (c) for each jurisdiction, finding the De curve, the optimal
number of park acres, and social surplus in each jurisdiction.
(f) Compute total social surplus by summing the social surplus results
from (e) across jurisdictions. How does the answer compare with social
surplus from (d)? On the basis of your answer, are homogeneous juris-
dictions superior to the original mixed jurisdictions?