Two Papers on Intertemporal Choice and Self Control SS200 - Meghana Bhatt

Temptation and Self- Control Frank Gul and Wolfgang Pesendorfer

Motivations  Classical theory predicts that more choices strictly increase utility yet we often see people willingly restrict their choice sets  People will often choose alternatives that they initially deemed inferior - I.e. they show preference reversals. (Rabin ‘97)  Biological factors such as hunger can affect short term preferences and may be the source of these reversals. In order to avoid these reversal, agents may attempt to commit to the choice that they feel is ex-ante superior.

Basic Model  2 Periods  2 competing “utilities” one long term utility u, and one temptation utility v  First Period - the consumer chooses some set A of possible options.  Second Period - the consumer chooses a specific option and receives utility on both the chosen object and the set A.

First Period  The consumer has preferences over sets. In this paper the set of all lotteries over possible alternatives.  These preferences are assumed to satisfy  Completeness and transitivity  Strong Continuity (the sets {B: B ≥ A} and {A: A ≥ B} are closed  Independence  Set Betweenness: A ≥ B implies that A  A  B  B

 An agent is said to have a preference for commitment at some set A if there exists some B contained in A such that B is preferred to A.  Theorem: a preference relation satisfies the above axioms if and only if there exist von Neumann-Morgenstern utility functions u and v s.t. the utility from A is equal to: max x  A u(x) + v(x) - max y  A v(y). User:

 The agent has self-control at some set C if there exist A and B such that C = A  B and A > C > B. This implies that the chosen object from C is in A but not in B and that the most tempting object in B is in C but not A.  Note that here self-control and commitment are considered separately. Commitment is something that occurs ex-ante whereas self-control occurs at the time of consumption. Second Period Behavior

 As we allow v to get arbitrarily large as compared to u our agents will never exhibit self-control but should generally exhibit commitment - exceptions include cases where either u or v is a constant and when u is simply a positive affine transformation of v.  The representation for preferences over sets suggest that period 2 behavior should simply maximize u(x) + v(x) for the given set since the cost to temptation is fixed. So agents are making some compromise between their ex- ante preferences and the temptation preferences represented by v.

Overestimating Self- Control: Evidence from the Health Club Industry Stefano DellaVigna and Ulrike Malmendier

Model  Contracts have three components:  L’ = sign up fee (positive for flat rate memberships, 0 for pay per visit memberships)  p’ = the pay-per-visit fee (0 for flat rate memberships)  T’ = contract duration  Attending the health club at time t incurs cost c immediately and benefit b at t+1.  The cost of effort c is uncertain and drawn from some distribution G.

Intertemporal Choice in this Model  Agents are assumed to have quasi- hyperbolic preferences over time. So the utility of a payoff of u at time t, t > 0, is equal to  t u.  We also allow agents to be Partially naïve: they understand that they are hyperbolic, but underestimate the size of their immediacy factor

Implications  At time 0 a consumer evaluates his decision to go to the gym at time t according to and exponential discount - so he believes he will attend if and only if c <  b - p’, but at time t he will actually attend only if c <  b - p’. So he will always attend less often then he believes he will  Partially naïve hyperbolic individuals will also overestimate their attendence, but by a smaller margin.

Contract Choice  At time 0 the agent will maximize: In general they observed that the average price per-visit of the flat rate plans were well above the $10 price per visit of the pay as you go plan.

Possible Explanations  To the extent that agents are sophisticated they may be using the flat-rate plans as a form of commitment to going to the gym. By doing the they are insuring themselves against their future impatience  To the extent that they are naïve they are simply overestimating their future attendence.

Renewals  Sophisticated agents will have short delays before cancellation.  For the data in this paper monthly memberships are automatically renewed and must be cancelled whereas annual memberships must be renewed and will expire if no effort is expended by the agent.

 Frequent users should sort themselves into annual memberships, so we would expect annual memberships to have a higher rate of renewal than the monthly memberships. But the extra cost of cancellation for monthly memberships may reverse this relationship.