1. Consider an economy with two types of consumers labeled ?h? and ?l?. Consumers have preferences f
1. Consider an economy with two types of consumers labeled ?h? and ?l?. Consumers have preferences forcurrent and future consumption given by: uh (ch , c?h ) = lnch + ? h lnc?h ul (cl , c?l ) = lncl + ? l lnc?l Some consumers are very patient and have a high ? h , others are impatient and have low ? l (? l < ? h ) .Note that ? is the weight in future consumption, or the discount rate. Consumers are equal in theirincome and there are no taxes (that is ?t? = ?t’? = 0). The government does not have to finance any spending.There are ?N ?consumers of each type in this economy. That in total there are ?2N? consumers, N? ?of them aretype ?h? and ?N? are type ?l?.I recommend that you make a change of variable to solve this problem, instead of working with the netinterest rate (?r?) work with the gross rate ?R? = 1 + ?r?.(a) Define an equilibrium for this economy.(b) Find the equilibrium if y = y? = ych , c?h , sh .cl , c?l , sl . (i) First solve the problem of a type ?h? consumer. This is find (ii) First solve the problem of a type ?l? consumer. This is find (iii) Finally replace in the market clearing condition. Solve for ?R? as a function of ?h & ?l . (c) Describe intuitively what would have to be c, c’ and s in equilibrium if all the consumers wereidentical (if ? l = ? h ) .(d) Solve for the equilibrium R ifyour intuition from above? ? l = ? h .? ?What are c, c’ and s in this case? Does this match 2. Model on asymmetric information. The economy is populated by a large number of savers and a largenumber of borrowers, as well as a financial intermediaries that we call banks. There is no government.The banks have access to highly trained economists which inform them that a recession might hit theeconomy in the second period. If the recession hits then a fraction 1 ? ?a? of the borrowers would lose theirjobs, if this happens these borrowers will default on their debt. The remaining fraction of borrowers (?a?)will pay the debt with the bank. This means that in the future there are going to be ?a? “good borrowers†and 1 ? ?a? “bad borrowersâ€. The severity of the recession forecasted by the economists is measured by ?a?.In a stronger recession more people are laid off, and so ?a? decreases.The banks believe the economists so their profits are given by: ? = L[a(1 + r2 ) ? (1 + r1 )]Where ?L ?is the amount of loans given, ?r?2? is the lending rate and ?r?1? is the savings rate. The agents are notaware of the potential recession, and they all think that they are going to remain employed in the secondperiod. So the problem of an agent is: maxc,c?,c u(c, c?) s.t. c + s = y c? = y ? + (1 + r)s Where ?r = r?1? if ?s > 0? and ?r = r?2? if s ? 0?. (Just as in the book page 334).(a) Define an equilibrium for this economy.(b) What is the equilibrium condition for the interest rates if the economists forecast that there is notgoing to be a recession. (where a = 1). Show the problem of a borrower in a diagram.(c) What happens to the lending rate (?r?2?) when a recession is forecasted (a decreases)? Whathappens to current consumption? SHow this in a diagram.(d) Show in a diagram that if the recession is believed to be too severe then the credit markets canshut down, and there is no lending taking place in equilibrium, what happens to the lending ratein this case? We call this a credit crunch, since banks are not willing to lend to borrowers.Explain the intuition for this result.

