Question 2: Run. baby. run! (50%] An economy is populated by a large number of [for our purposes} identical individuals facing each a personal liquidity risk: they can have utility from consumption in period 1 or in period 2, but not in both. The probability of having utility in period 1 is lfi. They do not want to consume today. and have 961 of wealth to invest in one of two projects: a riskless shortterm asset, paying $1.1 at tl for each dollar invested at t. for t = 0. l denoting periods; and a long term asset. which loses all of its value if liquidated in period 1. but pays $9.9 if liquidated in period 2. In future periods individuals do not have other sources of income. but their nancial investment. An individual will learn their preference for liquidity at time t = 1. Individuals are expected utility maximizers, with utility index given by ulx] = ,1 They are concerned only about their future consumption [today's consumption has been chosen already). An allocation is an investment strategy and a state-contingent consumption plan for each individual. a) \\Nrite down the constraints that an allocation must satisfy to be fea sible in the aggregate {you can assume that the fraction of individuals in a liquidity crunch in the population is equal to the probability, and normalize the population size to one}. b) Solve for the 'Pareto optimal allocation, showing your solution steps. Draw your solution in a graph. lnterpret economically the result: is there full insurance'.' c} Organize competitive markets for investment and liquidity insurance so to achieve Pareto efciency: give the precise budget constraints and prot- functions for investment rms, and the resulting equilibrium prices and alloca tion. An indi dual will now privately learn their preference for liquidity at time t = 1. Individuals can also secretly operate the shortterm investment technol ogy at home. d] Precisely explain whether or not, and why. the equilibrium you found under (c) is still an equilibrium under the new informational situation. A bank is introduced to implement the allocation you found under {b}: it of fers deposit contracts with maximum withdrawal amount x1 and x3. and chooses an investment strategy yrs and yL, collecting deposit-s today from individuals. e} Is the bank subject to the selffullling prophecy of a bank run'? Is a central planner subject to the same problem? Why? Your answer should contain at least two fundamental ingredients. Suppose that the bank invests before finding out whether or not there is a bank run, and that the bank can now liquidate the long-term asset at 10097: recovery rate at t = l. f} Compute the largest probability of occurrence ofa bank run that will make it better than regulating the bank with a withdrawal limit of 3].] in period 1