Consider a Lucas-type overlapping generations model, where individuals live on two spatially separated islands. The total population N is constant over time. From the old, 1 2N live on each of the three islands. The number of young on an island Ni is a random variable with the two possible outcomes for Ni being 1 4N, 3 4N . The monetary policy captured by the growth rate of the money supply zt is also a random variable with the three possible outcomes for zt being {1, 3}. Individuals only observe the price on their island p i t . The structure of the model and the assumptions are the same as in the standard Lucas model.

(a)  Using the market clearing condition for money on an island solve for the prices under each possible combination (Ni , zt). (Hint: construct a matrix with the possible combinations of (zt , Ni ), and denote these combinations by a, b, c, d).

(b)  Order the possible prices from lowest to highest for a given `. How many prices are not unique? What will a young individual choose to do, in terms of output, if they observe a non-unique price and why?

(c) ( Compute aggregate output for this economy, when the monetary policy is zt = 1, and when it is zt = 3. With which policy is aggregate output the highest and with which is it lowest?

(d) Does this model give rise to a Phillips Curve and why? Explain what would happen if policymakers tried to systematically exploit the policy that generates the highest aggregate output and why.