Question 6. Consider a firm that has a project. There are two possible outcomes of the project in each period: good or bad. The payment of the project is for good and bad outcome, respectively. The quality of the project is unknown. If the quality of a project is , then the probability that observing a good outcome is , i.e. . Suppose there are two qualities: high ( ) and low ( ). Let . Suppose that investors are risk neutral and update their beliefs using Bayes' rule. In the beginning, they believe that the probability that the project has high quality is . Let the risk-free interest rate be and assume there are only two periods (the project ends and does not generate any outcome after period 3). Observe that a bad outcome fully reveals to quality of the project since (i.e. a high-quality project never generates a bad outcome). This implies that . Let be the history of our past observations. For example, if the outcome is good in the first period, then . If the outcome is good in the first period and bad in the second period, then . Let be the probability of a good outcome after some history . a) [10 points] Calculate and . Specifically, you have to calculate numerically: . . x =g10,x =b5 Pr(g) = H L =H1, =L0.1 Pr( ) =H0.5 r =f0.1 =H1 Pr( b) =H0 h h= g h= gb h h g b =gPr(gg) = HPr( g)+H LPr( g)L =bPr(gb) = HPr( b)+H LPr( b)

g =

b =