which coincides with the basic textbook definition of present value. If income growth and/or returns are not constant over time, the net growth rate may nevertheless remain constant, in which case: General Equilibrium In computing equilibrium price I have treated incomes and bond returns as exogenous at each date , but in general equilibrium these are endogenous and depend on tastes and technology. Restrictions on tastes and technology can narrow the possible range of incomes and bond returns, and the range of prices as well. Consider a complete-markets representative agent economy with a single durable consumption good which is consumed in each of the periods 0,1,...,. The agent labors in each period to produce output: with the amount of labor and a number in the range . The agent's consumption in period is , and leisure time is , with time constraint . Taking consumption in period 0 as the numeraire good, let be the competitive equilibrium price of the consumption good in period , with normalization . The agent's intertemporal budget constraint is then: The agent chooses a consumption sequence and leisure sequence so as to maximize lifetime utility subject to equations 14 and 15, with a function which I suppose is log-linear and invariant with respect to each temporal reordering of consumption-leisure pairs: with a number in the range . Denote by and the utility-maximizing choices of consumption and leisure, subject to equations 14 and 15. Since is additively separable maximizes the function subject to:
which coincides with the basic textbook definition of present value. If income growth and/or returns are not constant over time, the net growth rate may nevertheless remain constant, in which case: General Equilibrium In computing equilibrium price I have treated incomes and bond returns as exogenous at each date , but in general equilibrium these are endogenous and depend on tastes and technology. Restrictions on tastes and technology can narrow the possible range of incomes and bond returns, and the range of prices as well. Consider a complete-markets representative agent economy with a single durable consumption good which is consumed in each of the periods 0,1,...,. The agent labors in each period to produce output: with the amount of labor and a number in the range . The agent's consumption in period is , and leisure time is , with time constraint . Taking consumption in period 0 as the numeraire good, let be the competitive equilibrium price of the consumption good in period , with normalization . The agent's intertemporal budget constraint is then: The agent chooses a consumption sequence and leisure sequence so as to maximize lifetime utility subject to equations 14 and 15, with a function which I suppose is log-linear and invariant with respect to each temporal reordering of consumption-leisure pairs: with a number in the range . Denote by and the utility-maximizing choices of consumption and leisure, subject to equations 14 and 15. Since is additively separable maximizes the function subject to:
JFE_Gilbert_2011May6.docx