Describe the decentralized equilibrium of the given economy

Question

Problem

Stationary monetary equilibrium in the Samuelson overlapping-generations model. (Again this follows Samuelson, 1958.) Consider the setup described in Problem 2.18. 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>

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

(b) Show that there is an equilibrium with P_t+1 = P_t/(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 equilibrium with P_t+1 = P_t/x for all t ≥ 0.

(d) Finally, explain why P_t = ∞ 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 2.20(b) below. (Hint: Reason backward from the last period.)

Problem 18

The basic overlapping-generations model. (This follows Samuelson, 1958, and Allais, 1947.) Suppose, as in the Diamond model, that L_t two-period-lived individuals are born in period t and that Lt = (1 + n)L_t-1. For simplicity, let utility be logarithmic with no discounting: Ut = ln(C_1t) + ln (C_{2t +1}).

The production side of the economy is simpler than in the Diamond model. Each individual born at time t is endowed with A units of the economy's single good. The good can be either consumed or stored. Each unit stored yields x > 0 units of the good in the following period.³⁰

Finally, assume that in the initial period, period 0, in addition to the L0 young individuals each endowed with A units of the good, there are [1/(1+n)]L0 individuals who are alive only in period 0. Each of these "old" individuals is endowed with some amount Z of the good; their utility is simply their consumption in the initial period, C₂₀.

(a) Describe the decentralized equilibrium of this economy. (Hint: Given the overlapping-generations structure, will the members of any generation engage in transactions with members of another generation?)

(b) Consider paths where the fraction of agents' endowments that is stored, ft, is constant over time. What is total consumption (that is, consumption of all the young plus consumption of all the old) per person on such a path as a function of f ? If x < 1 + n, what value of f satisfying 0 ≤ f ≤ 1 maximizes consumption per person? is the decentralized equilibrium pare toe efficient in this case? if not, how can a social planner raise welfare?

The response should include a reference list. Double-space, using Times New Roman 12 pnt font, one-inch margins, and APA style of writing and citations.