Intertemporal consumption in discrete time: two examples

The intertemporal consumption problem may be formulated as an optimal control problem, in which case maximization is made subject to a difference equation, or as a constrained optimization problem, in which case maximization is made subject to lifetime income. Two examples of discrete optimization are used to show that, under the assumption of a perfect capital market along with the usual assumptions of perfect foresight and well-behaved utility functions, both formulations yield equivalent results. Specifically, current consumption expenditure is a constant proportion of current holdings of wealth and lifetime income; in other words, the permanent income hypothesis is derived.