Consider the following version of the Lucas model. The number of young individuals born on island i in period t, N, is random according to the following specification: N= (1/4)N with probability 0.5, N= (3/4)N with probability 0.5, where N denotes the total number (i.e. across the two islands) of young people born in every period. Assume that the fiat money stock Mt grows at the fixed rate z in all periods. a. Set up the budget constraints of an individual when young and when old in terms of (where denotes an individual's labour supply). Find the lifetime budget constraint of an individual. b. Set up the money market-clearing condition. c. On which island would you prefer to be born? Explain with reference to the rate of return to labor. d. Show how the rate of return to labor and the individual's labor supply depend on the value of z. For the remaining questions, assume now that the growth rate of the fiat money stock z, is random according to: z = 1 with probability e, where 0 01; Z 3 with probability 1-0. The realization of z is kept secret from the young until all puchases of goods have occured (i.e. individuals do not learn M, until period t is over). Given these changes in assumption, answer the following questions: e. How many states of the world would agents be able to observe if information about every variable were perfectly available? Describe those possible states. f. How many states of the world are the agents able to distinguish when there is limited information (i.e. they do not know the value of z+)? g. Draw a graph of labour supply and the growth rate of the fiat money stock in each possible state of the world when there is limited information. What is the correlation observed between money creation and output? h. Suppose the government wanted to take advantage of the relation between money creation and output. If it always inflates, will the graph you derived in part f remain the same? Explain fully.