## The level of a public good chosen under majority voting

In this problem, you will compare the level of a public good chosen under majority voting with the socially optimal level under three different sets of circumstances. Suppose first that individual i’s demand curve for z is given by αi/z, where αi is a positive parameter. Instead of being linear, this demand curve is a hyperbola. Suppose further that z costs $1 per unit to produce (c = 1) and that this cost is shared equally among consumers. Therefore, cost per person is 1/n per unit of z. Then consider the three sets of circumstances listed below. Each situation has a different number of consumers in the economy and different collections of α values for the consumers. The number of consumers is denoted by n and the vector of α values by A = (α1, α2, . . . , αn – 1, αn).

Case 1: n = 7, A = (4, 2, 12, 4, 5, 13, 8).

Case 2: n = 5, A = (10, 6, 11, 14, 8).

Case 3: n = 9, A = (6, 9, 10, 4.5, 12, 7, 13.5, 8, 11).

Using this information, do the following:

(a) For each case, compute the preferred z level for each voter. Identify the median voter, and indicate the z level chosen under majority voting.

(b) For each case, compute the D∑ curve, and find the socially optimal level of z.

(c) For each case, compare the z level under majority voting with the socially optimal z. Explain the difference (if any) between the two z values.