Bayes Rule applied to Political Business Cycle Model

Hello All,

I have answers for the problem below and am now overanalyzing the problem and doubting the answers. Bayes Rule is made easier with backward induction; however, this is my first course with this material. Any help would be greatly appreciated.

Consider a simple political business cycle model. Suppose that outcomes can be classified as either Good or Bad: x {Good, Bad}. Voters receive a payoff of one for a good outcome and zero for a bad outcome. Governments differ in ability a, and this affects the likelihood of good outcomes. Specifically Pr(Good|a) = a. To keep the question simple suppose that goverments can be either high ability (h, a= 3/4) or low quality (l , a=1/4). Suppose the voters' initial belief is that the government is high quality is lamda, (i.e. lamda=Pr(a=3/4). If you find the general formulas difficult, then focus on the case of lamda=1/2.

When an election occurs, the opposition launches a campaign. From this campaign, the voters learn about the quality of the opposition. In particular, for this question, the voters' expected belief that the opposition is of high quality is uniformly distributed on the unit interval. (What does this mean? the probability that after the opposition campaign, the voters believe the expected probability that the opposition is high quality is less than 1/4 is 1/4; with probability 1/2 the voters believe that the expected probability of the opposition being high ability is less than 1/2.) When the voters vote, they consider their expected payoff in the next period only. That is, they elect the government they believe to be the most able.

a) Use Bayes Rule to calculate the probability that the government is of a high type given a good outcome and a bad outcome: Pr(h|Good) and Pr(h|Bad).

b) Given the beliefs about the government’s type given good or bad outcomes, what is the government’s probability of reelection given each outcome. What is the likelihood of reelection for each type of government? [Hint: for a high ability government, the probability of reelection is Pr(Elect|h) and for a low ability government, the probability of reelection is Pr(Elect|l).]

Above are the first two questions. The greatest hurdle is the set up of the expressions to calculate the results. Please help.

(Hi)