1. Consider an overlapping generations model where agents live for two periods, with population, Nt, evolving according to Nt = (1 + n)t N0. Assume logarithmic utility: Ut = ln c1,t +ln c2,t+1, where c1,t is the consumption of a person when young in period t and c2,t+1 is the consumption of the same agent when old, in period t + 1.

Each individual born at time t is endowed with 1 unit of the economy’s single good, which can be consumed or stored. Each unit stored yields (1 + r) > 0 units of the good in the following period (if the good is perishable, r = 1).

There are 1 1+nN0 agents who are alive only in period 0 and they are endowed with some amount Z of the good as well as with M units of a storable, completely divisible pieces of paper that we shall call money. The initial old’s utility is just c2,0. Suppose that the old and every subsequent generation believe that they will be able to exchange money for goods.

The price of the good in units of money in period t is Pt so that the rate of return on money is (1 + gt) ⌘ Pt/Pt+1.

In the absence of storage, the agent’s two budget constraints are given by

Pt(1 c1t) = Md t , Pt+1c2t+1 = Md

where Md denotes the demand for money by the young. In contrast, with (only) storage they are

(1 c1t) = x c2t+1 = (1 + r)x

where x denotes the amount that is stored.

(a) Discuss why the existence of a monetary equilibrium in this model is dependent on the values of n and r. Briefly explain whether the monetary equilibrium, if it exists, is a Pareto optimum. 

(b) Assuming that the conditions are such that the monetary equilibrium exists, solve for the steady state value of inflation.

(c) Suppose that the money stock M grows at the rate . Briefly discuss conditions that are necessary to ensure that a monetary equilibrium exists. Obtain the value of the steady state inflation and comment on its determinants.

Assume that the new money is introduced as a nominal transfer, Mt = Mt Mt1, made to each old person at the beginning of period t + 1 so that

Mt+1 = Mt + Mt = (1 + )Mt

and the budget constraints for an agent born in period t (assuming that a monetary equilibrium exists) are given by

Pt(1 c1t) = Md t Pt+1c2t+1 = Mt + Md

where Mt is taken as exogenous by the agent.