2.18. Stationary monetary equilibria in the Samuelson overlapping-generations model. (Again this follows Samuelson, 1958.) Consider the setup described in Problem 2.17. Assume that x < 1+n. Suppose that the old individuals in period 0, in addition to being endowed with Z units of the good, are each endowed with M units of a storable, divisible commodity, which we will call money. Money is not a source of utility.

(a) Consider an individual born at t. Suppose the price of the good in units of money is P, in t and P in t + 1. Thus the individual can sell units of endowment for P, units of money and then use that money to buy Pr/P+1 units of the next generation's endowment the following period. What is the individual's behavior as a function of P/P+1? =

(b) Show that there is an equilibrium with P+1 P+/(1+ n) for all t 0 and no storage, and thus that the presence of "money" allows the economy to reach the golden-rule level of storage.

(c) Show that there are also equilibria with P+1 P/x for all t 0.

(d) Finally, explain why P = 0 for all t (that is, money is worthless) is also an equilibrium. Explain why this is the only equilibrium if the economy ends at some date, as in Problem 2.19(b), below. (Hint: reason backward from the last period.)