Suppose that there are two people living on a block who benefit from street lights S and their private consumption ci for i = 1, 2. Suppose that the price of street lights and private consumption is normalized to 1 and each person is identical with an income of 100 dollar to be spent on street lights and private consumption. The utility function of each i is given by ui(ci, S) = ci 2 ln S where S is the total number of street lights provided. (a) (7 points) What is the social optimum quantity S? That is, what is the quantity chosen by a utilitarian social planner who maximizes the sum of the utilities? Please write down the social planer's problem to get full credit. (b) (8 points) Private provision of public good: Suppose that each person independently buys street lights an amount that is denoted by si where S = si si for i = 1, 2. Write down the individuals problem. Then fond the total number of street lights provided by private provision, call it S