A note on Markov perfect equilibria in a class of non-stationary stochastic bequest games

In this note, we prove the existence of a Markov perfect equilibrium in a non-stationary version of a paternalistic bequest game. The method we advocate is general and allows to study models with unbounded state space and unbounded utility functions. We cover both, the stochastic and deterministic cases. We provide a characterization of the set of all Markov perfect equilibria by means of a set-valued recursive equation involving the best response operator. In the stationary case, we show that there exists a set of strategies that is invariant under the best response mapping.