Consider a three-period OLG model. There are Nt people born in time t and gross rate of population growth is n. A member of generation t consumes c1,t when young, c2,t+1 when middle age and c3,t+2 when old. An Individual is endowed with y1 and y2 units of the consumption goods in the first and second period of his life and nothing in the third period. There is one single physical asset: capital, kt . A unit of capital is created from a unit of the consumption good in any period t. Two periods after its creation, a unit of capital produces X units of the consumption good and then disintegrates. Let X > n2 : Suppose it is impossible to observe the capital created by others. Hence, there is no credit market. Suppose sum of savings in the first period of life (in the form of fiat money) and y2 finances the expenses for consumption good in the second period of life.
Assume that an individual faces a lump-sum taxes of r1 and r2 goods in the first and second period of his life, respectively. These taxes finances the government debt. Moreover, suppose Mt = Mt-1.
a) Write down the feasible constraint. Rewrite it for stationary case
b) Find the individuals budget constraints when young, middle age and old. Combine them and find the individuals lifetime budget constraint
c) Find the rate of return on money. Compare the rate of return on capital, X, and that of fiat money and discuss whether an individual will choose to provide for
c3,t+1 by holding capital
d) Discuss the substitutability of fiat money and capital
e) Suppose Mt = zMt1 where z > 1. Find the rate of return on money in this case. Find an assumption that you need to impose so that an individual will choose to provide for c3,t+1 by holding capital
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