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2. [24' marks} Consider a Lucas type overlapping generations model1 where individuals live on three spatially separated islands. The total population N is constant over time. From the old, %N live on each of the three islands. The number of young on an island Ni is a random variable with the three possible outcomes for Ni being EEN, gN, gN}. The monetary policy captured by the growth rate of the money supply at is also a random variable with the three possible outcomes for at being {1, 2, 3}. Individuals only observe the price on their island pi. Except for the number of possible outcomes the structure of the model and the assumptions are the same as in the standard Lucas model. In particular1 an individual faces the following budget constraints over their lifetime, Ei1t+ fl 2 Ci; + Him: 2 y .. . _ _ pr _ Balt+l = L+1ml+t+1=(::1)ft+t+l = (p?! )Ei+~+1 t t+1 where prL is the real rate of return to work. t+1 {a} {5 marks} Using the market clearing condition for money on an island solve for the prices under each possible combination [N 1', at}. (Hint: construct a matrix with the possible combinations of {Zn Ni}, and denote these combinations by a, h, c, d, ....) [b] {5 marks} Order the possible prices from lowest to highest for a given E. How many prices are not unique?I What will a young individual choose to do1 in terms of output, if they observe a nonunique price and why? {c} {5 marks} Compute aggregate output for this economy. when the monetary policy is at = 11 when it is at = 21 and when it is at = 3. 1With which policy is aggregate output the highest and with which is it lowest? 4 [d] {ti marks} Does this model give rise to a Phillips Curve and why? Explain what would happen if policymakers tried to systematically exploit the policy that gener ates the highest aggregate output and why