4. Consider the model of cleaning an apartment in Appendix A.2.3. Suppose the government offers a subsidy, o, for each unit of effort by each person. So 0x Si. if person i chooses efforts, she receives a subsidy of The subsidy is funded through taxation. Each player pays the tax for the other person's subsidy. Moreover, taxation is inefficient. Each dollar of subsidy paid costs the tax payer T> 1 dollars. So if person 1 receives a subsidy of ox s, then player 2's tax burden is O X S x T. (a) Write down each player's utility given a strategy profile subsidy rate o. (S, S) and a (b) Solve for each player's best response as a function of the other player's strategy and the subsidy. (c) Identify the Nash equilibrium level of effort by the players, given a subsidy o. (d) Write down each player's payoff in equilibrium as a function of the subsidy. (e) Use your answer to part (d) to write down the utilitarian social welfare as a function of 0. (f) Use your answer to part (e) to solve for the second-best subsidy and effort. (g) The first-best effort in this model is 2. Show, given the inefficiency of taxes, that the second-best subsidy leads to higher payoffs than either no subsidy or a subsidy that leads players to choose the first-best effort.