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 i decides a contribution g,- towards a public good, which represents a number of books. The total quantity of public good is equal to G = 23.; g j is then the size of the library. Each family incurs an addition cost of gl- )2 / 2 for contributing gt, equivalent to the cost of transporting the books to the collection point. Therefore, the utility function of family i is n 2 M(giigi): G_ TgTJZZIgj' (35? (a) [5 marks.] Consider the socially optimal solution. Dene the social wel- fare V as the sum of the all the families' utility functions V 2 22:1 UJ-(gj, 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. g0,i for i = 1, - -- , n and the socially optimal size of the library: Gg'o (b) [5 marks.] How do you know that the solutions for 3:0,: are a maximum? (Hint: what conditions are needed for the solution of an optimisation to be a maximum)