Economics Question

Consider an OLG economy with an infinite horizon. Individuals live for three periods. You can call these stages young, middle-age, and old. Individuals do not want to consume when young, but want to consume in both middle-age and when old. That is, an individual born in period t receives utility that depends on their consumption in middle and old age given by u(c2,t+1,c3,t+2). Each young individual is endowed with y goods when young and nothing in middle age and when old. Here we consider a stationary equilibrium so that c1,t = c1 = 0, c2,t+1 = c2, and c3,t+2 = c3 for all t.

The number of young born in each period is growing at rate n > 1 so that Nt = nNt1 and the money stock is constant at M so that z = 1. One unit of the consumption good can be converted into one unit of capital in any period. Capital takes two periods to mature and each unit of capital produces X > n2 units of the consumption good when it matures. Unmatured capital cannot be bought or sold. Money can be acquired and traded in any period.

(a) (10 marks) Consider an individual born in period t. Suppose there are no banks so individuals must choose how to allocate their endowment between capital and fiat money when they are young.

Carefully explain the liquidity mismatch problem faced by the individual. In your answer please include a figure, make specific reference to the one and two-period returns on fiat money and capital, and include the young, middle-age and old budget constraints.

(b) (10 marks) Now suppose there is an infinitely-lived bank that is able to issue inside money (i.e. take deposits and commit to one period returns when the depositor withdraws). The bank is risk-neutral and simply chooses how much money to hold as reserves and how much capital to create to maximize the utility of depositors. For this part assume there is no reserve ratio.

1. Carefully explain how the existence of the bank solves the liquidity mismatch problem faced by individuals. In your answer please include a figure, make specific reference to the one and two-period returns on fiat money, deposits and capital, and include the young, middle-age and old budget constraints.
2. What is the maximum one-period return the bank can offer on deposits? When offered this return, does a young individual create more or less capital compared to part (a)?
3. If the bank offers the maximum return on deposits above, what is the demand for fiat money in the economy? Explain.

(c) (10 marks) Now suppose there is a reserve ratio equal to (0, 1).

1. What is the maximum one-period return the bank can promise on a deposit? When offered this return, does a young individual create more or less capital compared to part (b)?
2. Using a figure, carefully show that the bank is able to provide this return while maintaining the required reserve ratio. Note: you need to consider the total amount of goods being deposited in each period, not just the goods deposited by one individual.
3. Compared to part (b), carefully explain the trade-off faced by the central bank when introducing a reserve ratio.