Question

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, D3 =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 DΣ. In other words, don’t assume that the curves become horizontal once they hit the axis.

(a) The height of the DΣ 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 DΣ 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 welfare at the optimal z. This is just the area of the surplus triangle between DΣ and the cost line.

(d) Suppose there are two other jurisdictions, each with three consum- ers, just like the given jurisdiction. Compute total social welfare 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 DΣ curve, the optimal number of park acres, and social welfare in each jurisdiction.

(f) Compute total social welfare by summing the social welfare results from (e) across jurisdictions. How does the answer compare with social welfare from (d)? On the basis of your answer, are homogeneous juris- dictions superior to the original mixed jurisdictions?

Answer 1

Consider a city having three types of consumers and their demand for park is also given in the question. Now, here because “z” be the public good which is equally available to all, => here the summation of these demand curves are the vertical summation. So, the “total demand” curve is given below.

=> D? = D1 + D2 + D3 = 40 – z + 30 – z + 20 – z = 90 – 3*z, => D? = 90 – 3*z, z ? 20.

Now, for 20 ? z ? 30, => D? = D1 + D2 = 40 – z + 30 – z = 70 – 2*z, => D? = 70 – 2*z, 20 ? z ? 30.

Now, for 30 ? z ? 40, => D? = D1 = 40 – z, 30 ? z ? 40.

So, the total demand is given below.

b).

Now, let’s assume that the cost of park is “c=9” < 20. Now, at “z=30”, => the “D?” is “40 – z = 40 – 30 = 10 > 9”.

So, the “c=9” intersect the “D?” in “30 ? z ? 40” range. So, the socially optimum level of “z” will be determined by the intersection of “D?” and “c=9”, => 40 – z = 9, => z = 31. So, the socially optimum level of “z” is “31”.

Consider the following fig.

c).

So, here the social welfare is given below.

=> ?ABE + ?EJF + ?FHG + BEID + JFHI = (90-30)*20*0.5 + (30-10)*(30-20)*0.5 + (10-9)*(31-30)*0.5 + (30-9)*20 + (10-9)*(30*20) = 600 + 100 + 0.5 + 420 + 10 = 1,130.5.

So, here the total welfare is “1,130.5”.

d).

Now, suppose that there are two other jurisdiction each with the three consumers just like the given jurisdiction, => under this situation the “total welfare” of these three jurisdiction is the “3*1130.5 = 3391.5”.

Since here also the optimum value of “z” will not change given the cost “c=9”, => in each cases the welfare will be same, => the total welfare here is the “3*from the 1^st one”, => “3*1130.5 = 3391.5”.