Bryant-Keynes-Wallace Consider an economy consisting of overlapping generations of two-period-lived agents. There is a constant population of N young agents born at each date t ≥ 1. There is a single consumption good that is not storable. Each agent born in t ≥ 1 is endowed with w1 units of the consumption good when young and with w2 units when old, where 0 2 1. Each agent born at t ≥ 1 has identical preferences ln cht (t) + ln cht (t + 1), where cht (s) is time-s consumption of agent h born at time t. In addition, at time 1, there are alive N old people who are endowed with H(0) units of unbacked paper currency and who want to maximize their consumption of the time-1 good. A government attempts to finance a constant level of government purchases G(t) = G > 0 for t ≥ 1 by printing new base money. The government’s budget constraint is

Where p(t) is the price level at t, and H(t) is the stock of currency carried over from t to (t + 1) by agents born in t. Let g = G/N be government purchases per young person. Assume that purchases G(t) yield no utility to private agents.

a. Define a stationary equilibrium with valued fiat currency.

b. Prove that, for g sufficiently small, there exists a stationary equilibrium with valued fiat currency.

c. Prove that, in general, if there exists one stationary equilibrium with valued fiat currency, with rate of return on currency 1 + r(t) = 1+ r1 , then there exists at least one other stationary equilibrium with valued currency with 1 + r(t) = 1 + r2 =1+ r1 .

d. Tell whether the equilibria described in parts b and c are Pareto optimal, among allocations among private agents of what is left after the government takes G(t) = G each period. (A proof is not required here: an informal argument will suffice.) Now let the government institute a forced saving program of the following form. At time 1, the government redeems the outstanding stock of currency H(0), exchanging it for government bonds. For t ≥ 1, the government offers each young consumer the option of saving at least F worth of time t goods in the form of bonds bearing a constant rate of return (1 + r2). A legal prohibition against private intermediation is instituted that prevents two or more private agents from sharing one of these bonds. The government’s budget constraint for t ≥ 2 is

Where B(t) ≥ F . Here B(t) is the saving of a young agent at t. At time t = 1, the government’s budget constraint is

Where p(1) is the price level at which the initial currency stock is redeemed at t = 1. The government sets F and r2. Consider stationary equilibria with B(t) = B for t ≥ 1 and r2 and F constant.

e. Prove that if g is small enough for an equilibrium of the type described in part a to exist, then a stationary equilibrium with forced saving exists. (Either a graphical argument or an algebraic argument is sufficient.)

f. Given g, find the values of F and r2 that maximize the utility of a representative young agent for t ≥ 1. g. Is the equilibrium allocation associated with the values of F and (1 + r2) found in part f optimal among those allocations that give G(t) = G to the government for all t ≥ 1? (Here an informal argument will suffice.)