Interdependent Utility Functions in an Intergenerational Context

Since Ray (1987) posed a question of representing nonpaternalistic functions (namely aggregators) in paternalistic form, Bergstrom (1999) identified sufficient conditions for a given list of linear aggregators to possess a unique list of utility functions (over consumption allocations) as the representations of the aggregators. Hori (2001) considered the representation problem for the case of a finite number of agents with possibly nonlinear aggregators. The model in this paper is a natural extension of Hori's (2001) to the case of countably many generations. As in Hori (2001), the aggregators in this paper may differ across generations and depend possibly on the utility levels of all other generations. We discuss two approaches to deal with infinite horizon. The first one explores monotonicity structures inherent in nonpaternalistic altruism. By means of lattice-theoretic arguments alone, we establish the existence of representations of nonpaternalistic functions in paternalistic form. A somewhat surprising feature of the lattice-theoretic approach is that the existence result is obtained without requiring that the degree of altruism is "small". The second approach uses the requirement of small degree of altruism in terms of "uniformly small Fréche derivative" (with respect to the utility level of other generations). We regard this approach as a natural extension of Hori's (2001) as we see later. Along the way, we discuss the case of linear aggregators. Our treatment in this paper is a little different from Bergstrom (1999) in that we view the infinite matrix as a representation of a continuous linear operator on 1∞.