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 = 23.121 g ,- is then the size of the library. Each family incurs an addition cost of y[g,]2/2 for contributing g,, equivalent to the cost of transporting the books to the collection point. Therefore, the utility function of family 1' is U-[gugF] G- 7Zg, (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 = 23.121 U ,- [g ,- , g_J,-]. Maximise V with respect to g,- for i = 1, - -- , 11. Using the rst order condi- tions, determine the socially optimal level of contribution per family. 830,; for i = 1,-- ,n and the socially optimal size of the libraryzG so (b) [5 marks.] How do you know that the solutions for 8'50,- are a maximum? (Hint: what conditions are needed for the solution of an optimisation to be a maximum)