Markov Perfect Equilibria in Stochastic Bequest Games

In many real-life situations, the preferences of an economic agent change over time. Rational behaviour of such agents was studied by many authors (Strotz, Pollak, Bernheim and Ray) who considered so-called “consistent plans”. Phelps and Pollak [10] introduced the notion of “quasi-hyperbolic discounting”, which is a modification of the classical discounting proposed in 1937 by Samuelson. Within such framework an economic agent is represented by a sequence of “selves” who play a non-cooperative discrete-time dynamic game with appropriately defined payoff functions, see [6] and [7] and references cited therein. Alternatively, one can look at the model of Phelps and Pollak as an intergenerational game with altruism between generations. In such a game it is assumed that each generation lives over just one period and consumes a fixed good. The part left after consumption constitutes an investment for following generations. Therefore, each generation derives utility from its own consumption and those of its descendants. The next generation’s endowment is determined by investment and certain production function. The existence of a Markov perfect equilibrium in an intergenerational dynamic game is a fixed point problem in an appropriately defined function space. The set of possible endowments is usually an interval in the real line. This problem was successfully treated by Bernheim and Ray [3,4] and Leininger [8] for certain classes of deterministic bequest games, where utility of each generation depends only on its own consumption and that of an immediate descendant. Other, more special cases were studied by Kohlberg, Peleg and Yaari; see [8,11] for references. Stochastic intergenerational dynamic games were studied by Bernheim and Ray in [5], but their proof is incorrect. It is based on a false lemma (Lemma 6) in [11]. The existence of stationary Markov perfect equilibria in intergenerational stochastic games is proved under general assumptions on utility functions for the generations and for non-atomic transition probabilities in [1]. In particular, the case studied by Bernheim and Ray [5] is covered. The transition probabilities considered in [1] are of very natural type, e.g., they can be determined as in [6] by some difference equations and i.i.d. random shocks. A different class of intergenerational stochastic games is studied in [7] where the transition probability functions are additive in some sense. The paper [7] generalizes many earlier results by Amir, Balbus, Reffett and Woźny who also gave some constructive proofs; for references see [7]. In some applications of intergenerational dynamic games it is assumed that each generation derives utility from its own consumption and the utilities of some or all future generations (so-called non-paternalistic models). The way how these utilities are derived from consumption is not important. Results establishing existence and characterization of equilibrium in non=paternalistic models are surprisingly few and incomplete; see Ray [11]. We prove that a stochastic version of the deterministic game