Synopses & Reviews
Synopsis
Excerpt from On Intertemporal Preferences in Continuous Time: The Case of Certainty
A continuous function can be nowhere differentiable take, for example, a Brownian motion sample path. In fact, most continuous functions are nowhere differentiable. This is perhaps not so counter-intuitive when time is not the characteristic of the commodity in question. When the commodities are indexed by time, our intuition certainly suggests that prices for consumption should be continuous functions of time. We also think that more should be true. Prices should not ﬂuctuate over time in a nowhere differentiable fashion.2 In particular, we think that interest rates should exist in a reasonable economic intertemporal model. This condition is equivalent to the requirement that prices for consumptions be absolutely continuous functions of time. This additional requirement is what makes the results of the commodity differentiation literature unsatisfactory for our purposes.
We begin in Section 2 with a formulation of the consumption set and consumption space. For the set of feasible consumption patterns, we use the space of positive, increasing, right continuous functions on denoted X].3 For given a: e X+ and t 6 x(i) denotes the cumulative consumption from time zero to time t under consumption pattern 1. In order to speak of net trades, the space of consumption bundles, or the commodity space, is the linear span of X+. Denoted X, this is the space of functions of bounded variation on 0, 1] that are right-continuous.
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