Time Consistency and Utility Weighted Discount Rates

The discount rate is a tool used to measure the preference for immediate gratification or utility over delayed gratification. This paper considers the modified Merton problem of an economic agent maximizing his utility from consumption and final wealth when his discount rate is not constant. The question we answer is the following: if we allow the individual to update his decisions, will he stick to his original strategy or will he switch? We show that there are cases in which the agent's strategy keeps changing thus his behaviour becomes time inconsistent. We introduce two notions to solve this inconsistency problem. The agent can pre commit i.e. he does not change his original optimal strategy. He can also plan for his future changes of strategy and adopt time consistent strategies also known as subgame perfect strategies. We show that the subgame perfect strategy can be obtained from the pre commitment strategy if we replace the agent's discount rate by his utility weighted discount rate. This last quantity is shown to be the solution of a fixed point problem.