Self‐Control, Effort Procrastination, and Competitive Equilibrium in Insurance Markets

• Author: Wei Zhu,Lin Zhao,Jing Ai
• Published date: 01 September 2020
• Date: 01 September 2020
• DOI: http://doi.org/10.1111/jori.12275

©2019 The Journal of Risk and Insurance. Vol.XX, No. XX, 1–32 (2019).

DOI: 10.1111/jori.12275

Self-Control, Effort Procrastination, and

Competitive Equilibrium in Insurance Markets

Jing Ai

Lin Zhao

Wei Zhu

Abstract

This article studies consumers’ self-control problems in precautionary ac-

tivities, their contract choices, and the welfare implications in a competitive

insurance market. Present bias and consumer naivete both induce consumers

to procrastinate or eventually give up precautionary efforts. In consequence,

self-control problems disrupt the monotonicity of consumers’ indifference

curve on contract choices, leading to a pooling equilibrium or an absence

of risk–coverage correlation, in addition to the classic result of adverse se-

lection. Compulsory insurance raises all consumers’ welfare only in adverse

selection, but not in other equilibrium patterns.

Introduction

Consumers often have multiple opportunities to undertake precautionary activities

that are beneficial to them. Given the flexibility in whether and when to exert precau-

tionary effort, however, they have the propensity to procrastinate or eventually give

up effort, exhibiting a self-control problem. For example, in the health insurance mar-

ket, Gallup reports that 74 percent of smokers would like to give up smoking, but only

4–7 percent of the attempts are successful; 59 percentof Americans said they wanted to

lose weight, but at least 15 percent of them werestill trying 7 years later (Bryan, Karlan,

and Nelson, 2010). In the automobile insurance market, Brockett and Golden (2007)

suggest that impulsive behavior due to lack of self-control is present in both driving

Jing Ai is at the Shidler College of Business, University of Hawai’i at M¯

anoa, 2404 Maile Way

Honolulu, HI 96822. Ai can be contacted via e-mail: jing.ai@hawaii.edu. Lin Zhao is at the

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190,

PRC. Zhao can be contacted vi e-mail: zhaolin@iss.ac.cn. Wei Zhu is at the School of Insurance

and Economics, University of International Business and Economics, Beijing, 100029, PRC.

Zhu can be contacted via e-mail: zhuwei@uibe.edu.cn. Zhao acknowledges financial support

from the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy

of Sciences, and the National Natural Science Foundation of China (Grant Nos. 71871212 and

71532013). Zhu acknowledges financial support from the Humanity and Social Science Founda-

tion of the Ministry of Education in China (Grant No. 16YJA790072), Beijing Municipal Social

Science Foundation (Grant No. 17YJC039), the Fundamental Research Funds for the Central

Universities in UIBE (Grant No. CXTD8-03), and the Program for Young Excellent Talents,

UIBE (Grant No. 17YQ13). 1

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2The Journal of Risk and Insurance

and financial risk taking, evident by a strong negative correlation between insurance

losses and personal credit scores. Although precautionary effort is an important issue

in insurance economics and has been studied extensively, implications of the self-

control problem on market equilibrium and welfare have yet to be well understood.

A precautionary activity is typically a task requiring immediate effort but providing

delayed benefit. When implementing such tasks, empirical and experimental stud-

ies suggest that individuals can easily lose self-control when they are present bi-

ased: they value the present significantly more than the future and yet place similar

weights on different future events (Ainslie, 1992; O’Donoghue and Rabin, 1999, 2001;

DellaVigna and Malmendier, 2004). Present bias introduces an inconsistency in in-

dividuals’ preferences over time, leading consumers to make time-inconsistent de-

cisions, such as procrastinating or giving up the planned precautionary activities.

Consumers are also found to differ in their recognition of self-control problems. Em-

pirical evidence of such differences is provided by Madrian and Shea (2001) on 401(k)

investment, Ariely and Wertenbroch (2002) on homework completion, DellaVigna

and Malmendier (2006) on health club attendance, and Wong (2008) on examination

preparation. Theoretically, two polar types of consumers are labeled by economists as

“sophisticates” and “naifs”: sophisticates fully foresee the self-control problem and

account for it in the decision process, while naifs do not foresee the problem and

make decisions in a myopic way (O’Donoghue and Rabin, 2001; DellaVigna and Mal-

mendier, 2004). Compared with time-consistent consumers, sophisticates suffer only

from present bias, whereas naifs suffer from both present bias and naivete.

Allowing for flexible timing in individuals’ effort choices, the objective of this article

is to understand how present bias and consumer naivete affect consumers’ precau-

tionary effort making and insurance contract choices, and consequently equilibrium

and welfare in a competitive insurance market.

The “Self-Control Problemsin Precautionary Activities” section lays out a multiperiod

setting to describe the effort making by time-consistent consumers, sophisticates, and

naifs, respectively. In our setting, the cost of precautionary effort is lower when it is

completed earlier.Time-dependent cost is an important feature of real-life precaution-

ary activities. For example, performing maintenance on a car sooner rather than later

can efficiently lower the cost of keeping the car in good running condition. Similarly,

receiving a smoking cessation treatment sooner makes it easier to see results,reducing

the medical expenditure. Under this setting, we show how present bias and consumer

naivete induce the self-control problem and lead consumers to procrastinate. Com-

pared to time-consistent consumers, present-biased consumers weigh the current ef-

fort costs higher than the delayed benefits, and thus are less likely to make effort.Naive

consumers, unaware of the self-control problem, have even weaker effort incentives

than sophisticates. They always procrastinate effort and can eventually give up effort.

The “Equilibrium When Consumer Type Is Observable” section characterizes con-

sumers’ preferences over insurance and studies the equilibrium when consumer

types are observable. We reproduce a standard result that with moral hazard, time-

consistent consumers can no longer obtain full insurance at low premium rate. Present

bias exacerbates this negative impact by reducing the coverage insurance companies

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Self-Control, Effort Procrastination, and Competitive Equilibrium in Insurance Markets 3

are willing to provide at the low premium rate, driving down consumers’ welfare.

Consumer naivete further lowers the effort incentive, and as a result, naifs are pro-

vided with an even smaller coverage than sophisticates and have the lowest welfare

among all three types of consumers.

The “Equilibrium When Consumer Type Is Unobservable” section investigates the

equilibrium and welfare implications in a competitive market when neither con-

sumers’ precautionary efforts nor their types are observable to insurance companies.

An important finding is that along sophisticates’ indifference curve, the insurance

premium is no longer monotonically increasing in the insurance coverage. Rather, it

exhibits a downward jump when the insurance coverage exceeds a certain threshold.

When the coverage is below the threshold, sophisticates plan to make effort when

purchasing insurance and will be able to keep this plan later on. However, when the

coverage becomes larger, sophisticates prefer to make effort originally but will be

tempted to switch to no effort later on due to their present bias. Foreseeing that the

increased insurance coverage triggers the self-control problem and eventually makes

them worse off, sophisticates actually prefer to pay less for the higher coverage con-

tract, resulting in a downward jump in their indifference curve. This downwardjump

in sophisticates’ indifference curve leads to various possible market equilibria, de-

pending on the mix of consumers in the market.

rWhen the market consists of time-consistent consumers and sophisticates, it ex-

hibits either adverse selection or an absence of risk–coverage correlation. Adverse

selection arises when the effort cost is moderately high such that sophisticates prefer

to make no effort (and become the high-risk type) while time-consistent consumers

choose to exert effort (and become the low-risk type). Insurance companies, as a re-

sponse, offer full insurance to sophisticates and partial insurance to time-consistent

consumers. The absence of risk–coverage correlation arises when the effort cost is

low such that both time-consistent consumers and sophisticates prefer to exert effort

(and become the same low-risk type). However, insurance companies provide less

insurance to sophisticates than to time-consistent consumers in order to incentivize

effort.

rWhen the market consists of sophisticates and naifs, it can exhibit adverse selection,

an absence of risk–coverage correlation, or a pooling equilibrium. The adverse selec-

tion and the absence of risk–coverage correlation are driven by the same screening

mechanism as previously described. The existence of pooling equilibrium is remark-

able, given Rothschild and Stiglitz’s (1976) well-known result that a pooling equi-

librium does not exist in the presence of the single-crossing property of consumers’

indifference curves. In our setting, the single-crossing property no longer holds as

the indifference curves of sophisticates always lie (weakly) below those of naifs at

the same utility level due to naifs’ misbelief, allowing for a pooling equilibrium.

rWhen the market consists of all three types of consumers, insurance companies

cannot screen naifs from time-consistent consumers because naifs’ (misperceived)

preference is the same as that of time-consistent consumers. They thus screen the

pool of naifs and time-consistent consumers from sophisticates. The equilibrium re-

sembles either one of the two markets discussed above, depending on the fractions

of naifs and time-consistent consumers relative to the sophisticates in the market.

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