Costly self-control and limited willpower

In Gul and Pesendorfer (Econometrica 69(6):1403-1435, 2001), a decision-maker, when facing a choice among menus, evaluates each menu in terms of the maximum value of its commitment utility net of self-control costs. This paper extends the model such that this maximum is constrained by the condition that the cost of self-control cannot exceed the decision-maker's stock of willpower w. Four of the five axioms of our characterization are as in their Theorem 3 except that the independence axiom is restricted to a subset of menus. We add one new axiom to regulate willpower as a limited (cognitive) resource in which the available "stock" does not vary across menus. In our characterization, choices within menus that satisfy WARP reveal a constant trade-off between commitment and temptation utilities. However, it is the discontinuity of preferences over menus (along with violations of WARP for choices within menus) that revealsw (measured in units of temptation utility), allowing for a behaviorally meaningful comparative measure of self-control across individuals.