An entrepreneur with initial wealth 1 – I dollars has a new project in which to invest. To com« plete the project the entrepreneur needs additional i dollars that they can borrow from a bank. If they borrow l dollars, they invest the 1 euro in the project. With probability p the project succeeds and brings it > 1 dollars and with probability 1 – p it fails and brings 0.11″ the entrepreneur does not borrow the money they keep the l — l dollars (outside option). There are two types of entrepreneurs : those who have good projects with a high probability of success p = pH and those who have bad projects with a lower probability of success p = pt < pH. The prior probability that an entrepreneur has a good project is If = 0.5. The game is as follows. At time t = 0 the bank quotes an interest rate i on loans. Then. "nature” decides if the project is good or bad. the entrepreneur learn their type and chooses whether to borrow at rate 1′ and to invest in the project or not. If the project is successful the bank gets (1 + illr and the entrepreneur 8— it] + i). If the project fails they both get 0. (a) Throughout the exercise. we assume that RpL > I. How can we interpret this assumption? (b) What is the expected payoff ofan entrepreneur oi tupe j = L. H if they borrow at rate 1′ ? Show that a type L will borrow if and only if i < 11 and type H if and only if i < in for some i L. i n that you will characterize. Show that in > i t.- (c) What are the bank’s updated beliefs about the entrepreneur’s type if a loan is taken at i > EL? At i < 51,? Compute the bank’s expected profit in both cases and find the bank’s optimal interest rate offer given the model’s parameters. (d) Carefully state the weak perfect bayesian equilibria of this game.