Consider the following version of the model of this chapter. The number of young people born on island i in period , t, Nⁱ_t is random according to the following specification:

Nⁱ_t = 4/5 N with probability .5
      = 1/5 N with probability .5

Assume that the fiat money stock grows at the fixed rate z_t = z in all periods.

a. Set up the budget constraints that a person faces when young and when old in terms of lⁱ_t . Also set up the government budget constraint and money market-clearing condition. Find the lifetime budget constraint (combine the budget constraints of the young and old by substituting for lⁱ_t).

b. On which island would you prefer to be born? Explain with reference to the rate of return to labor.

c. Show how the rate of return to labor and the person’s labor supply depend on the value of z. For the following parts, assume that the growth rate of the fiat money stock z_t is random according to

z_t = 1 with probability θ
     = 4 with probability 1 – θ

The realization of z_t is kept secret from the young until all purchases of goods have occurred (people do not learn M_t until period t is over). Given these changes in assumption, answer the following questions.
d. How many states of the world would agents be able to observe if information about every variable were perfectly available? Describe those possible states.

e. How many states of the world are the agents able to distinguish when there is limited information (they do not know the value of z_t)?

f. Draw a graph of labor 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?

g. Suppose the government wanted to take advantage of the relation between money creation and output. If it always inflates, θ – 1 will the graph you derived in part f remain the same? Explain fully.