Online Appendices for “Self Control and Commitment: Can Decreasing the Liquidity of a Savings Account Increase Deposits?” John Beshears, Harvard University and NBER

B. Preliminaries B.1. Functions of Bounded Variation. We begin by discussing the concept of bounded variation. This concept will be used to formulate our assumptions on the distribution function F in the subsection immediately following this one, namely Section B.2. More importantly, it plays an essential role in our proof of sufficiency in a much later section, namely Section P. The simplest definition of a function of bounded variation is probably that given in the main text: a function f : (0,∞) → R is of bounded variation iff it is the difference of two bounded and non-decreasing functions f, f− : (0,∞) → R. This definition forms the starting point for the definition that we shall use. However, it needs to be developed into a form that is more convenient for the Lagrangean analysis below. The first step is to collect the functions of bounded variation into equivalence classes. Intuitively speaking, two functions of bounded variation are equivalent iff they differ only at their points of discontinuity. This step is analogous to the first step in defining spaces of Lebesgue integrable functions. (In that case, one collects the Lebesgue integrable functions into equivalence classes. Two Lebesgue integrable functions are equivalent if they differ only on a set of measure 0.)