Economics Letters 41 (1993) 41-45 0165-1765/93/$06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved 41 On the role of generating functions when preferences are recursive Hiroaki Hayakawa * zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Department of Economics, Ritsumeikan University, 56-l Kita-machi, Toji-in, Kita-ku, Kyoto 603, Japan Suezo Ishizawa Department of Economics, Nagoy a Gakuin University, 1350 Kamishinano-cho, Seto-Shi, Aichi-Ken 480- 12, Japan Received 7 December 1992 Accepted 25 January 1993 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Abstract The directional derivative of the Uzawa-Epstein utility functional gives rise to Epstein’s generating function. The function has two important attributes: its partial derivatives, when discounted, yield directly the marginal utilities, while it also serves as a generator of preferences. 1. Introduction Applications of the Uzawa-Epstein class of recursive preferences in continuous time [Uzawa (1968) and Epstein (1987)] b a ound in the literature. Recently, Epstein (1987) introduced the notion of a generating function as a way of characterizing such preferences. Given a path of decision variables, the current value of subsequent future lifetime utility from a given future point in time forward is represented by the utility of its right-hand-tail path. The derivative of this utility with respect to time then measures how fast the current valuation of future lifetime utility changes instantaneously. The generating function is defined by the negative of this derivative, its arguments being the values of the decision variables at the moment as well as their future paths. We show in this paper that this function arises naturally from the definition of the directional derivative of the utility functional, and that the function is capable of carrying all relevant information regarding preferences in the sense that it serves as a generator of preferences and that its partial derivatives discounted to the time of decision-making yield directly the marginal utilities in the Volterra derivative sense [see Volterra (1959) and Wan (1970)]. To the extent that the derivation of the generating function is straightforward, such attributes deserve attention. 2. Analysis Consider the following framework. There are n goods as decision variables. The consumption space C consists of n-tuples of their time paths x 5 (x1, x2,. . . , xn), where each x1 is a piecewise * Corresponding author.