Time Preferences, Intertemporal Optimization, and the Permanent Income-Life Cycle Hypothesis

This paper analyzes how the three lines of development, namely, the structure of dynamic recursive preferences, Pontryagin's maximum principle, and the permanent income-life cycle hypothesis, are interrelated in the context of consumer choice in continuous time. First, by introducing variational changes in consumption around two different points in time that are compensating to the lifetime utility or having canceling effects on asset accumulation, we define and compute the marginal rates of substitution and transformation and express the optimality condition in terms of the instantaneous time rates of change of these rates. This condition is then related to the Böhm-Bawerkian and Fisherian rates of time preference, which are defined, calculated, and shown to give an equivalent measure whether inclusive of the first order change of consumption or not. The Keynes-Ramsey rule of consumption is related to this optimality condition and is given a new interpretation. Second, we inquire into the preference basis of the permanent income-life cycle hypothesis and show explicitly that the proportionality of optimal consumption to wealth and permanent income follows from the homotheticity of recursive time preferences and a dynamic budget constraint, thereby confirming that the hypothesis, as originally conceived by Friedman and Modigliani-Brumberg, is a dynamic version of the static demand under homothetic preferences.