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Time Consistent Policies and Quasi-Hyperbolic Discounting  

Łukasz Balbus, Kevin Reffett, and Łukasz Woźny

In dynamic choice models, dynamic inconsistency of preferences is a situation in which a decision-maker’s preferences change over time. Optimal plans under such preferences are time inconsistent if a decision-maker has no incentive to follow in the future the (previously chosen) optimal plan. A typical example of dynamic inconsistency is the case of present bias preferences, where there is a repeated preference toward smaller present rewards versus larger future rewards. The study of dynamic choice of decision-makers who possess dynamically inconsistent preferences has long been the focal point of much work in behavioral economics. Experimental and empirical literatures both point to the importance of various forms of present-bias. The canonical model of dynamically inconsistent preferences exhibiting present-bias is a model of quasi-hyperbolic discounting. A quasi-hyperbolic discounting model is a dynamic choice model, in which the standard exponential discounting is modified by adding an impatience parameter that additionally discounts the immediately succeeding period. A central problem with the analytical study of decision-makers who possess dynamically inconsistent preferences is how to model their choices in sequential decision problems. One general answer to this problem is to characterize and compute (if they exist) constrained optimal plans that are optimal among the set of time consistent sequential plans. Time consistent plans are those among the set of feasible plans that will actually be followed, or not reoptimized, by agents whose preferences change over time. These are called time consistent plans or policies (TCPs). Many results of the existence, uniqueness, and characterization of stationary, or time invariant, TCPs in a class of consumption-savings problems with quasi-hyperbolic discounting, as well as provide some discussion of how to compute TCPs in some extensions of the model are presented, and the role of the generalized Bellman equation operator approach is central. This approach provides sufficient conditions for the existence of time consistent solutions and facilitates their computation. Importantly, the generalized Bellman approach can also be related to a common first-order approach in the literature known as the generalized Euler equation approach. By constructing sufficient conditions for continuously differentiable TCPs on the primitives of the model, sufficient conditions under which a generalized Euler equation approach is valid can be provided. There are other important facets of TCP, including sufficient conditions for the existence of monotone comparative statics in interesting parameters of the decision environment, as well as generalizations of the generalized Bellman approach to allow for unbounded returns and general certainty equivalents. In addition, the case of multidimensional state space, as well as a general self generation method for characterizing nonstationary TCPs must be considered as well.