Based on game theory answer the question.

3 Sharing Is Weak - 20 points Let us extend the Tragedy of the Commons problem we discussed during lectures. Con sider N players, each of whom has the following utility function each period: log{1r}, where 1: is the amount consumed. Suppose there is a connnon property resource of size :1; 3: I]. \"The game has two stages. In stage one player, 1': 1,. ..N can withdraw c1, such that Z c, s: y. In stage two, each player consumes of the remaining quantity, 11 Z; c,. i=1 In the case that they attempt to consume in excess of the available amount, then each player splits the resource equally. For the sake of simplicity, let 2 c; = E. i=1 {a} Write down each player's intertemporal utility function.1 {2 points} {b} What is the Nash equilibrium in Stage Two? {3 points} {c} What is this game's SPNE? [E points) {d} What is the socially optimal level of consumption in Stage One? [4 points) {c} How does the difference between the SPNE consumption in Stage One, and the socially optimal consumption in Stage One, change as N becomes very large? What is the intuition for this, and does this explain why ecologist Garrett Hardin suggested sterilisation as a solution to this problem? [l-lint: take the limit as N approaches innity] {5 points}