1. Consider a 3period environment similar to the Diamond and Dybvig model presented in class. There is a mass of size 1 of households, with E type A agents and 1 H type 13 agents. Types are not revealed until t=1. Households can invest their endowment, y=1 in a rm who pays gross return R:- 1 at t=2. However, unlike the model presented in class, if households try to withdraw from the rm at t=1, the rm cannot give them anything. Therefore, households must decide in period I] how much of their endowment they want to keep as reserves. {a} Let the utility of type A agents, uA = in[c1} and the utility of type B agents, ii}; = pin[c1 +cg}. Recall that ear-ante [at period B], there is a 6 chance of being type A and a 1 H of being type B. Write down an individual household's period D expected utility. [b] If an individual household splits their y=1 endowment between reserves and rm investment [goes it alone], how much 'will they keep as reserves? HINT: let E3 = R{1 c1} and nd the Cl that maximizes their expected utility. Call this cf for \"Autarky\". {c} Now consider the social planner's problem. Remember they are subject to the same technology as the households, but can split the economy's resources however they want. Let of, be the resources set aside in period D for each individual type A agent to consume in period 1. Should the social planner set any additional reserves aside for type H's to consume in the rst period? Why or why not? Assuming they don't, how much is left over for individual type H's to consume in the second period? [d] Set up and solve the planner's problem. Is c1?\" {household's autarl-ry solu tion) different than the planner's? Assume that 19R 3- R 1. Is cf\" greater than or less than a}? {e} Do rI'ype H households consume more in period 1 than the planner would want them to? {f} Give intuition for parts iii} and {e}