i. They wish to maximize their expected payoff, which is their expec- tation of the outcome (x = xh or x = : x1) minus the payment of the campaign contribution (h or 1). Write the expected payoff of the voters π(h,l, e). ii. Voters cannot contribute negative amounts to the campaign. What does this mean to h and 1? Write the mathematical condition. iii. Voters want to make sure that the politician wants to join the arrange- ment. Therefore, the expected payoffs to the politician less the cost of effort need to be at least as much as his outside option. Write the mathematical condition. iv. Voters need to keep in mind that the politician will choose an effort level that maximizes his own expected payoff. Write the mathematical constraints. v. Using the results of iv., write the maximization problem that the voter faces subject to the two relevant constraints: that 1 has to be non- negative and that the politician finds it worthwhile to participate. vi. Assume that the politician will be indifferent between participating in the process and taking his outside option. Show that this implies that (h - 1)² 2c +1 = m vii. Show that this allows to write the voter maximization as h - (h – 1)² Мах,п(h, 1) = m с 2c viiii. Assume that > 0. Then what does h - 1 have to be? What is l? What is h? What is e(h, l)? ix. Assume that = 0. Then show that h = √√2cm. What is e(√2cm, 0)? x. In this model, if the politician has better outside options (i.e., m in- creases), what happens to effort? 3. Reflect on how commitment affects politician's incentives by comparing 1. and 2. - 1 with Imagine that voters and politicians play the following game. A politician who is in office has to decide how much effort e to apply to affect the outcome of some policy. The policy output x has two states, indicated by xh = 1 or xi = 0 re- spectively. In particular, when the politician applies effort e, we have x = probability e and x = 0 with probability 1 e. Moreover, the politician dislikes applying effort, and faces a cost ce²/2. Before the the politician takes a decision, voters tell the politician that they will contribute h to the re-election campaign if x = 1 and if x 0. Finally, if the politician does not like the contract setup, he can walk away and just get some payoff m. = The timing of the game is as follows: 1) Voters offer a take-it-or-leave-it contract {h,1} and politicians choose to accept or not. If not, her outside option is m 2) politicians apply hidden effort e = [0,1] 3) x is realized 4) voters pays w(x) = h,1 and keep x - w(x) 5) payoffs are uv(x) = x = w(x) and up (w) Answer the following questions: = w- - ce²/2. 1. First, assume that voters cannot commit to paying the declared w(x). This is because, after politicians apply their effort and the outcome x is realized, voters can change their mind. Then what is the equilibrium level of effort by the politician? 2. Now assume that voters can commit to paying the declared w(x). They thus solve the following problem: