2. Public Good and Contribution. [32 marks.] Consider the following scheme for contributing to the provision of a public good, say the opening of a community library for a community of n families. Each family 1' decides a contribution g,- towards a public good, which represents a number of books. The total quantity of public good is equal to G = 2;; g}- is then the size of the library. Each family incurs an addition cost of y[g,]2/2 for contributing gf, equivalent to the cost of transporting the books to the collection point. Therefore, the utility function of family 1' is (a) (b) (C) [d] '2 n _ Uilgiugz']= Gy[g;) =Zgj_7,[g'] [5 marks.] Consider the socially optimal solution. Dene the social wel- fare V as the sum of the all the families' utility functions V = 23.121 U ,- [g j, g_ j]. Maximise V with respect to g,- for i = 1, - -- , n. Using the rst order condi tions, determine the socially optimal level of contribution per family. ggm for i = 1, - -- ,n and the socially optimal size of the library: Gg'o [5 marks.] How do you know that the solutions for go i are a maximum? (Hint: what conditions are needed for the solution of an optimisation to be a maximum) [4 marks.] Consider now the simultaneous contribution to the public good by the families in the community. Find the Nash equilibrium in the con- tribution level, by having each family maximise their utility holding the contribution of others gji constant. What is the quantity contributed g\"? [4 marks.] Compare the contributions in the Nash equilibrium to the so- cial optimum quantity ggo'i. What intuition can you use to explain the difference? In particular, compute the ratio gf/ggolf. How does this vary with the number of families n? In which communities is the problem of underprovision of public goods more serious