Consider an economy with a constant population of N = 100. Individuals areendowed with y = 20 units of the consumption good when young and nothing when old.
1. What is the equation for the feasible set of this economy? Portray the feasible set on a graph. With arbitrarily drawn indifference curves, illustrate the stationary combination of c1 and c2 that maximizes the utility of future generations
2. Now look at a monetary equilibrium. Write down equations that represent the constraints on first- and second-period consumption for a typical individual. Combine these constraints into a lifetime budget constraint
3. Suppose the initial old are endowed with a total of M = 400 units of fiat money. What condition represents the clearing of the money market in an arbitrary period t? Use this condition to find the real rate of return of fiat money
4. Assume the lifetime utility function is ln(c1,t) + ln(c2,t+1). (In this caseβ = 1.) Setup the Lagrangian function, derive the first-order conditions, and find algebraic expressions for consumption when young, when old for all t ≥ 1. Show your work.
5. How does the return of money affect consumption when young, when old,and real money holdings (vtmt)? Provide intuitive answers for each case
6. What is the value of money in period t (vt)? Use the assumption about preferences and your answer in part (3) to find an exact numerical value. What is the price of the consumption good pt?
7. If the rate of population growth increased, what would happen to the rate of return of fiat money, real money holdings, the value of a unit of fiat money in the initial period, and the utility of the initial old? Explain your answers. Hint: Answer these questions in the order asked
8. Suppose instead that the initial old were endowed with a total of 800 units of fiat money. How do your answer to part (6) change? Are the initial old better off with more units of fiat money?