3. Suppose that the cost function for a particular public good is given by C(z,n) = (40n - 12n' + n' )z In this case, the unit cost of z is 40n - 12n' + n', a function of n. a) Using the above formula for the unit cost of z, derive the formula for the unit cost of z per capita. This is just the unit cost of z divided by n. b) The best community size is where the unit cost of z on a per capita basis is as small as possible. Using the results of part (a), compute this per capita cost for n = 1,2,3,...,8,9,10. What community size minimizes unit cost per capita? Multiply your unit cost per capita by the relevant n to get total unit cost in the optimal-size community (i.e., the c value for use with the D: curve below). c) Suppose that all consumers in the economy are identical, and each consumer's demand for z is given by D = 20 - z. Using the results of part (b), compute the Dr curve for an optimal-size community (this is a community of the size found in part (b)). This is done by adding up as many individual D's as there are people in the optimal community. d) Using the unit cost figure from part (b), find the optimal level of z in the optimal-size community. Also, compute social surplus in the optimal-size community. e) Suppose the economy contains 18 people. How many optimal-size communities can be created out of this population? Using the results of part (d), what is total social surplus in the economy? f) Now suppose that instead of being divided into optimal-size communities, the population is divided into 2 communities of size 9. Using the previous results, find the unit cost of z per capita in these communities. Then find the optimal level of z in each community, as well as social surplus in each community. g) Compute social surplus in the whole economy when there are two 9-person communities. Compare you answer to that from part (e). How big is the loss from non- optimal community sizes