Competing for Consumers with Self‐Control Problems

• Published date: 01 March 2016
• DOI: http://doi.org/10.1111/jems.12147
• Date: 01 March 2016
• Author: Alexei Alexandrov

Competing for Consumers with Self-Control

Problems

ALEXEI ALEXANDROV

Office of Research

U.S. Consumer Financial Protection Bureau

Washington, D.C.

Alexei.Alexandrov@cfpb.gov

I examine strategic implications of competing for consumers with self-control problems. For in-

vestment goods, like health clubs, I find that the equilibrium sign-up (lump-sum) fees decrease

when competition intensifies, similarly to prices in standard oligopoly models. However,the equi-

librium attendance (per-unit) price increases due to firms’ deteriorated ability to take advantage of

consumers’ self-control problems. Moreover, firms earn less profit due to consumers’ self-control

problems—the firms have a unilateral incentive to charge per-unit fees lower than the marginal

cost; however, they cannot make up the lost margins by increasing the lump-sum fee, due to

competition. I also show that for plausible parameter regions the market adjusts to consumers’

self-control problem in such a way that firms play the standard equilibrium strategies that they

would have engaged in with fully rational consumers, with identical market outcomes. Most of

the results are qualitatively the same for leisure goods (for example, credit cards); however, some

results are reversed: the per-unit fees are higher than marginal cost and decrease as competition

intensifies.

1. Introduction

How does consumer irrationality affect the market equilibrium? If consumers were

to behave irrationally, would this affect outcomes in a particular way that could be

distinguished from market equilibria when consumers behave rationally? My analysis

suggests that market institutions would adjust to irrational behavior, which can alter the

effects of consumer irrationality on market outcomes. In this paper, I examine the effects

of one form of consumer irrationality—self-control problems—and develop a model for

firm strategy. I derive the optimal strategies for the initial lump-sum payment and the

usage (attendance) price in a market with several strategic firms, which are aware of

consumers’ problems with self-control (time-inconsistency).

I consider the implications of consumer self-control problems with respect to in-

vestment goods and leisure goods. Investment goods have immediate costs associ-

ated with one of the actions that consumers can take, as well as future benefits if the

I thank Priscilla Medeiros for getting me interested in this problem; Simon Loertscher and Michael Raith

for their detailed suggestions; Paul Heidhues, Ted O’Donoghue, Jeremy Tobacman, and participants of the

International Industrial Organization Conference for their comments on a much earlier draft; Thomas Gresik,

Chun-Hui Miao, and Sendhil Mullainathan for their comments on a more current draft; and the co-editor and

two anonymous referees. Some of the work was completed while the author was an assistant professor of

economics and management at the University of Rochester. The views expressed arethose of the author and

do not necessarily represent those of the Director of the Consumer Financial Protection Bureau nor those of

the staff.

Published 2015. This article is a U.S. Government work and is in the public domain in the USA.

Journal of Economics & Management Strategy, Volume25, Number 1, Spring 2016, 179–194

180 Journal of Economics & Management Strategy

consumers pursue this action. One example is health clubs, in which a consumer expe-

riences private disutility and may have to pay a usage price to attend, but attendance

yields some future health benefit. Leisure goods are the opposite of investment goods,

associating a current benefit with one of the actions, and a future cost if that action is

chosen. Some examples of leisure goods are credit cards or unhealthy products, where

current consumption implies a consequence later, in either monetary or health terms.1

Firms selling investment goods or services, health clubs for example, have two

major pricing tools at their disposal: the lump-sum fee which consumers pay when they

sign up and the attendance price which consumers pay each time they attend. With fully

rational consumers, the firms compete on the lump-sum fee, with the attendance price

equal to the marginal cost of the health club. However, if consumers have self-control

problems, DellaVigna and Malmendier (2004) showed that the optimal attendance price

is lower than the health club’s marginal cost both in the monopoly and in the perfect

competition scenario. A low attendance price and a high lump-sum fee allow the firm

to exploit consumers’ self-control problems, thus making the firm more profitable since

some consumers pay the lump-sum fee, but do not attend, so that the firm does not incur

the marginal cost.

Whether consumers indeed have self-control problems or other forms of irra-

tionality in a given market is an empirical question. I simply show the firms’ strategies

assuming that consumers do have self-control problems, and that for some of the param-

eter regions it does not matter for final market outcomes whether consumers actually

have any self-control problems.

I show that when firms compete strategically, the lump-sum fee follows the fa-

miliar (Hotelling) comparative statics: the more competitive (less differentiated) firms

become, the lower the fee is. Moreover, if the market is covered in the spatial differen-

tiation model, the lump-sum fee is exactly what it would have been with fully rational

consumers: the market adjusts in such a way that the outcome and comparative statics

with respect to lump-sum price are indistinguishable from the outcome with fully ra-

tional consumers.2On the other hand, the attendance price has different comparative

statics: lower differentiation does not allow firms to exploit consumers’ biases, pushing

the equilibrium attendance price higher, closer to the firms’ marginal cost.3As a mo-

nopolist, as in DellaVigna and Malmendier (2004), a firm has an incentive to charge an

attendance price lower than its marginal cost, while more than making up for it on the

front end. This incentive continues to exist as firms become more competitive; however,

the firms’ ability to make up for it on the front end diminishes. Thus, we see the resulting

comparative static of firms pricing their products closer to marginal cost as they become

more competitive, but closer to marginal cost happens to mean a higher attendance price

for markets with investment products.

1. The term leisure good is chosen to correspond to the earlier literature. Arguably, a more proper term

is a good with delayed cost, and it should not be confused with the more colloquial definition of leisure

good. Vacations would be colloquially characterized as leisure goods, but arenot leisure goods according to

the definition above (unless one pays for them with their credit card, but on the other hand, for many of us

it could be an investment good from the perspective of not being productive during the vacation, with the

delayed benefit of being much more productive after). Similarly, while cigarettes would not be colloquially

characterized as leisure goods, they would be a leisure good according to the definition above since this

product has a delayed cost.

2. The market is covered when consumers have no outside option. This assumption does not impose any

restrictions on cross-elasticities between the firms.

3. Chen and Riordan (2008) show how competition can increase prices in a fully rational model for a given

set of parameters, when the marginal consumer is less elastic in duopoly.Thomadsen et al. (2014) show that a

low-quality entrant might result in increased prices and profits for the incumbents.