Antton is the head of the Economics department at Hasparren university. The university is about to vote on how to allocate its budget across departments. The game is sequential: Antton is good friends with the president of the university, so he can propose to allocate an amount $31 to the department, to which the president can either accept or refuse. If he refuses, the amount allocated to the department is the same as the previous year, $so. Hence, the decision of the president, 52, will either be so (rejection of 31} or 31 (acceptance of 31). The president's utility function is up(32) = (32 (1)2. (a) What is the amount .5; that maximizes the president's utility? What is the maximized utility 1: 12(33) equal to? {3 points) (b) Consider the last subgame, where the president either accepts 31 or rejects it, in which case so will be allocated to the department. What values of 31 will the president accept to allocate? (your answer in (a) will be useful) (5 points) (c) Assume Antton wants to see the highest possible amount voted, i.e. his utility is \"A(32) = 32- If so 2 a, what amount 31 will Antton propose? Deduce the subgame perfect Nash equilibrium and write it down. (6 points) (d) Is there a subgame perfect Nash equilibrium where Antton proposes some value 51 > so and the president rejects? Explain. (6 points) (e) If so