Explosive paths in the Samuelson overlapping-generations model. (Black, 1974; Brock, 1975; Calve, 1978a.) Consider the setup described in Problem 2.19. Assume that x is zero, and assume that utility is constant-relative-risk aversion with θ < 1="" rather="" than="" logarithmic.="" finally,="" assume="" for="" simplicity="" that="" n="">

(a) What is the behavior of an individual born at t as a function of P_t/P_t +1? Show that the amount of his or her endowment that the individual sells for money is an increasing function of Pt/_Pt+1 and approaches zero as this ratio approaches zero.

(b) Suppose P₀/P₁ < 1.="" how="" much="" of="" the="" good="" are="" the="" individuals="" born="" in="" period="" 0="" planning="" to="" buy="" in="" period="" 1="" from="" the="" individuals="" born="" then?="" what="" must="" p1/p2="" be="" for="" the="" individuals="" born="" in="" period="" 1="" to="" want="" to="" supply="" this="" amount?="">

(c) Iterating this reasoning forward, what is the qualitative behavior of P_t/P_t+1 over time? Does this represent an equilibrium path for the economy?

(d) Can there be an equilibrium path with P0/P1 > 1?

Problem 19

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?