Question

Suppose there is a society with two people who have the following valuations of level y of a public good: v1(y) = 1210√y and v2(y) = 410√y so their marginal benefits are given by 605/√y and 205/√y, respectively. The cost of providing the public good is given by c(y) = 10y. So we have MC(y) = 10.

Suppose that their utility is given by u1 = x1 + v1(y) and u2 = x2 + v2(y) , where xi denotes person i’s money (that is not spent on the public good). Further, suppose their wealth/income levels are given by w1 = 500 and w2 = 800.

(a) What is the socially optimal level of the public good?

(b) Suppose that it is proposed that y = 4 be implemented, and that this or y = 0 are the only two options. The two people vote on this proposal, and approval requires a majority vote. Suppose that, if it is approved, each will be required to pay a tax equal to half of the cost. Will this proposal be approved.

Answer #1

A).

Now, the marginal benefit of the two people are “MB1=605/y^0.5” and “MB2=205/y^0.5”, => the aggregate benefit is the vertical summation of the individual MB curves.

=> MB = MB1+MB2 = 605/Y^0.5 + 205/Y^0.5 = 810/Y^0.5. Now, the socially optimum production will be determined by the intersection of “MB” and “MC”.

=> MB = MC, => 810/Y^0.5 = 10, => 81 = Y^0.5, => Y = 81^2 = 6,561, => Y=6,561. So, the socially optimum level of public good is “Y=6,561”.

B).

Let’s assume “Y=0”, => v1 = v2= 0. So, the utility of both people are given below.

=> “U1 = (500-0)+0 = 500” and “U2 = (300-0)+0=300”.

Now, let’s assume that “Y=4”, => total cost to produce “4 units” of public good is “C=10*Y = 10*4 = 40”. Now, here both the people have to pay equally, => each of them have to pay “$20”.

So, given the information the utility of both the people are “U1 = (500-20)+1210*4^0.5 = 480+2,420 = 2,900 > 500” and “U2 = (300-20)+410*4^0.5 = 280+820 = 1,100 > 300”.

So, here both the individuals are getting more utility for “Y=4”, => both of them will vote for “Y=4”.

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