In this chapter, we modeled growth in an economy by a growing population. We could also achieve a growing economy by having an endowment that increases over time. To see this, consider the following economy: Let the number of young people born in each period be constant at N. There is a constant stock of fiat money, M. Each young person born in period t is endowed with y_t units of the consumption good when young and nothing when old. The person’s endowment grows over time so that y_t = αy_t-1 where α > 1. For simplicity, assume that in each period t, people desire to hold real money balances equal to one-half of their endowment, so that v_tm_t = y_t/2.
a. Write down equations that represent the constraints on first- and second-period consumption for a typical person. Combine these constraints into a lifetime budget constraint.
b. Write down the condition that 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 in a monetary equilibrium. Explain the path over time of the value of fiat money.