The Increasing Convex Order and the Trade–off of Size for Risk

• DOI: http://doi.org/10.1111/jori.12132
• Date: 01 September 2017
• Author: Liqun Liu,Jack Meyer
• Published date: 01 September 2017

© 2015 The Journal of Risk and Insurance. Vol. 84, No. 3, 881–897 (2017).

DOI: 10.1111/jori.12132

881

THE INCREASING CONVEX ORDER AND THE TRADE-OFF OF

SIZE FOR RISK

Liqun Liu

Jack Meyer

ABSTRACT

One random variable is larger than another in the increasing convex order if

that random variable is preferred or indifferent to the other by all decision

makers with increasing and convex utility functions. Decision makers in this

set prefer larger random variables and are risk loving. When a decision

maker whose utility function is increasing and concave is indifferent

between such a pair of random variables, a trade-off of size for risk is

revealed, and this information can be used to make comparative static

predictions concerning the choices of other decision makers. Specifically, the

choices of all those who are strongly more (or less) risk averse than the

reference decision maker can be predicted. Thus, the increasing convex

order, together with Ross’s (1981) definition of strongly more risk averse, can

provide additional comparative static findings in a variety of decision

problems. The analysis here discusses the decisions to self-protect and to

purchase insurance.

INTRODUCTION

Building decision models that involve trade-offs is a distinguishing feature of

economic analysis.

1

In models with randomness, the most important of these trade-

offs involves choosing between a larger and riskier random variable and a smaller

and less risky one. Mean–variance (M–V) decision models represent this trade-off in a

very simple and direct fashion. M–V models measure size by the mean mand risk by

Liqun Liu is at the Private Enterprise Research Center, Texas A&M University, College Station,

TX 77843. Liu can be contacted via e-mail: lliu@tamu.edu. Jack Meyer is at the Department of

Economics, Michigan State University, East Lansing, MI 48824. Meyer can be contacted via

e-mail: jmeyer@msu.edu. We would like to thank Henry Chiu, Louis Eeckhoudt, Stephen Ross,

and Harris Schlesinger for helpful comments and suggestions with special thanks to Michel

Denuit for pointing out the connection to the mathematical statistics literature. All remaining

errors are our own.

1

Greg Mankiw, in his Principles of Economics textbook, lists as Principle 1: “People Face Trade-

Offs.”

882 THE JOURNAL OF RISK AND INSURANCE

the standard deviation s, and use a utility function V(s,m) to represent a decision

maker’s willingness to trade off size for risk.

In expected utility (EU) decision models, representing the size for risk trade-off is a

much more complex problem. There is no agreed-upon measure of the size of a

random variable, nor is there an agreed-upon measure of the magnitude of risk.

2

First-degree stochastic dominance (FSD) provides an agreed-upon partial order for

size, and the Rothschild and Stiglitz (1970) (R–S) definition of an increase in risk

provides a partial order for risk, but since neither provides a measure, defining the

trade-off of size for risk is still not simple.

This article shows that the increasing convex order from the mathematical statistics

literature can be used to frame such trade-offs. Random variable 

yis said to be larger

than another random variable 

xin the increasing convex order if 

yis preferred or

indifferent to 

xby all decision makers with increasing and convex utility functions.

Larger in the increasing convex order can also be characterized in a number of other

ways. First, 

yis larger in the increasing convex order than 

xif the change from 

xto 

y

can be decomposed into a change becoming larger (in the FSD sense) and a change

becoming riskier (in the R–S sense). Therefore, the choice between 

xand 

ywhen 

yis

larger in the increasing convex order than 

xinvolves trading off size for risk for a non-

satiated and risk averse decision maker. Second, larger in the increasing convex order

can also be characterized by a stochastic dominance like inequality that specifies a

necessary and sufficient condition on cumulative distribution functions (CDFs) of 

x

and 

yfor 

yto be larger than 

xin the increasing convex order. The decision models

examined here show how this CDF characterization can be a fundamental tool for

doing comparative static analysis.

This article does two main things: it reviews the literature presenting the theory, and it

provides applications of the theory. First, in the section “Literature Review: Several

Characterizations of the Increasing Convex Order,” the increasing convex order,

Ross’s strongly more risk averse order, and the theory indicating how these two

orders can be combined to facilitate comparative statics analysis are reviewed. The

increasing convex order has been used by others doing economic analysis, but much

of this work does not refer to the mathematical statistics literature and appears to be

unaware of the full range of characterizations of the increasing convex order that are

available. In particular, the CDF-based characterization of the increasing convex

order that is emphasized here has not been used before for economic analysis. The

terminology employed by mathematical statisticians is different from that used in

economics, but is easily translated into discussion that fits the more familiar stochastic

dominance framework.

The section “Trading Off Size for Risk When Making Decisions” uses four decision

models to illustrate how the increasing convex order and Ross’s strongly more risk

averse order can be used together to present new comparative static statements. The

section begins with the decision to self-protect. With self-protection, the least

2

Aumann and Serrano (2008) do provide a measure for risk, but their measure appears to

combine both the elements of size and risk.