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Production and hedging decisions under regret aversion

Xu Guo a, Wing-Keung Wong b, Qunfang Xu c,, Xuehu Zhu d

a College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, Chinab Department of Economics, Hong Kong Baptist University, Hong Kongc Business School of Ningbo University, Ningbo, Chinad School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, China

a b s t r a c ta r t i c l e i n f o

Article history:

Accepted 8 August 2015Available online xxxx

JEL classication:

D21

D24

D81

Keywords:

Production

Hedging

Regret aversion

Risk aversion

Competitiverms

In this paper, we investigate regret-averse rms' production and hedging behaviors. We rst show that the

separation theorem is still alive under regret aversion by proving that regret aversion is independent of thelevel of optimal production. On the other hand, we nd that the full-hedging theorem does not always hold

under regret aversion as the regret-averse rms take hedged positions different from those of risk-averse rms

in some situations. With more regret aversion, regret-averse rms will hold smaller optimal hedging positions

in an unbiased futures market. Furthermore, contrary to the conventional expectations, we show that banning

rms from forward trading affects their production level in both directions.

2015 Published by Elsevier B.V.

1. Introduction

Since the seminal work ofSandmo (1971),Holthausen (1979),Katz

and Paroush (1979)and others, there has been much theoretical and

empirical research on the economic behavior of the competitive rm

under price risk. In classical economic theory under uncertainty, two

main results are derived: the separation theorem and the full-hedging

theorem. The separation theorem documents that for risk-averse

rms, their risk attitude and the distribution of prices are independent

of their optimal production decision. On the other hand, the full-

hedging theorem tells us that if the futures price is unbiased, risk-

averse rms take a fully hedged position to eliminate the risk of price

uncertainty. For reference, seeBroll and Zilcha (1992),Adam-Mller(1997),Broll and Eckwert (1998, 2000), andLien (2000).

In this paper,we extendthe theoryby examiningthe production and

hedging behavior of a competitive rm under the premise that therm

demonstrates both risk-averse behavior and regret-averse attitude. In

recent years, there has been a growing literature on rm behavior that

assumes not only risk aversion but also regret aversion. For example,

ifrms' prices turn out to be very high and sales turn out to be very

good, rms might regret not producing more. Conversely, if prices

turn out to be low and sales are poor, rms might regret over-

producing.Savage (1951)is the rst to propose that decision makers

include regret in their decision-making processes. Bell (1982),

Fishburn (1982), and Loomes and Sugden (1982, 1987) develop a

formal analysis of regret theory, which is later extended bySugden

(1993)andQuiggin (1994). Following the pioneering work ofSandmo

(1971)on the optimal output of a risk-averse rm under price uncer-

tainty, Paroush and Venezia (1979) rst assume that competitive

rms have a regret-averse utility. Nonetheless,Braun and Muermann

(2004)propose using a more specic regret-averse utility function

that is easily managed.Egozcue and Wong (2012)andWong (2014)

are therst to study the behavior of a competitive rm when price is

unknown with the regret-averse utility introduced by Braun and

Muermann.Niu et al. (2014)further examine the production decision

of a competitiverm with a regret-averse utility. Broll et al. (2015)

study the behavior of a regret-averse rm when the exchange rate is

uncertain. There are other applications of regret aversion, for example,

Economic Modelling 51 (2015) 153158

The authors thankthe Editor, Professor PareshNarayan,the Associate Editor, Professor

Niklas Wagner, and two anonymous referees for their constructive comments and

suggestions, which help us clarify more precisely the results and lead to the substantialimprovement of an early manuscript. The authors are grateful to Professors Donald Lien,

Agnar Sandmo, Robert I. Webb, and Kit Pong Wong for their valuable comments, which

have signicantly improved this manuscript. The second author would like to thank

Professors Robert B. Miller and Howard E. Thompson for their continual guidance and

encouragement. This research is partially supported by the Fundamental Research Funds

for the Central Universities (NR2015001), Hong Kong Baptist University (FRG2/14-15/

106, FRG2/14-15/040), Research Grants Council of Hong Kong (Project Number

12502814), Natural Science Foundation of Zhejiang Province (LY15A010006) Grants,

Research Project of the National Statistics (2013LY119), and Ningbo University Talent

Project (ZX2014000781).

Corresponding author at: Business School of Ningbo University, China. Tel.: + 86 186

0574 87600396.

E-mail address:[email protected](Q. Xu).

http://dx.doi.org/10.1016/j.econmod.2015.08.007

0264-9993/ 2015 Published by Elsevier B.V.

Contents lists available atScienceDirect

Economic Modelling

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e c m o d
http://dx.doi.org/10.1016/j.econmod.2015.08.007http://dx.doi.org/10.1016/j.econmod.2015.08.007http://dx.doi.org/10.1016/j.econmod.2015.08.007mailto:[email protected]://dx.doi.org/10.1016/j.econmod.2015.08.007http://www.sciencedirect.com/science/journal/02649993http://www.elsevier.com/locate/ecmodhttp://www.elsevier.com/locate/ecmodhttp://www.sciencedirect.com/science/journal/02649993http://dx.doi.org/10.1016/j.econmod.2015.08.007mailto:[email protected]://dx.doi.org/10.1016/j.econmod.2015.08.007http://crossmark.crossref.org/dialog/?doi=10.1016/j.econmod.2015.08.007&domain=pdf

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the demandfor insurance(Wong, 2012), portfolio investment (Barberis

et al., 2006; Muermann et al., 2006), auctions (Engelbrecht-Wiggans

and Katok, 2008; Feliz-Ozbay and Ozbay, 2007), newsvendor model

(Perakis and Roels, 2008), and banking issues (Tsai, 2012). To apply

the regret theory tonance,Wagner (2002)derives a Markowitz-type

model of portfolio choice with variable regret aversion. Readers may

refer toBleichrodt and Wakker (2015)for a comprehensive review of

the development of the regret theory.

As discussed above, some authors extend the production theory ofthe risk-averserm to that of the regret-averse rm to investigate the

optimal production decision without considering the presence of a

futures market. On the other hand, some authors study the production

and hedging behaviors of the risk-averserm. In addition,Michenaud

and Solnik (2008) apply regret theory to get solutions to optimal

currency hedging choices. As far as we know, there is no previous

work that studies the production and hedging behaviors of the regret-

averse rm. Nonetheless, it is of great interest to study the effect of

regret aversion on the optimal production and hedging decisions

when a futures market exists. For example, it would be interesting to

know whether the two notable resultsthe separation and full-

hedging theoremsare still valid ifrms are regret averse. Our paper

provides a natural extension ofWong (2014) by adding the opportunity

for regret-averserms to trade in futures contracts. This is in the same

spirit as Holthausen (1979), who adds this opportunity to Sandmo

(1971)for risk-averse rms.

We rst examine whether the separation theorem still holds for the

regret-averse rm. It is well known (Niu, et al., 2014; Wong, 2014) that

under certain sufcient conditions and without hedging, the optimal

output level is smaller under uncertainty than under certainty. When

a futures market exists, we nd that regret aversion has no effect on

the optimal production decision. To be precise, optimal output levels

are the same for both risk-averse and regret-averse rms that will

choose their optimal outputs when their marginal costs are equal to

the futures price. These results imply that the separation theorem is

still alive under regret aversion.

Thereafter, we examine the full-hedging theorem for the regret-

averse rm. We nd that sometimes both regret-averse and risk-

averse competitive rms behave similarly, but, in other situations,they behave differently. For example, when the futures price is smaller

than the expected price (contango market), both risk-averse and

regret-aversermswill take under-hedgedpositions. When the expect-

edprice and the futures price are the same (unbiased futures market),

the risk-averse competitive rm will take a fully hedged position,

while the regret-averse competitive rm will still take an under-

hedged position. Furthermore, when the futures price is slightly higher

than the expected price (backwardation futures market), risk-averse

rms will take an over-hedged position, but regret-averse rms

could take an under-hedged, a fully hedged, or an over-hedged

position, depending on the degree of regret aversion. These results

imply that under regret aversion, the full-hedging theorem does not

hold.

We conduct a comparative statics analysis and nd that withmore regret aversion, the optimal hedging position will become smaller

in an unbiased futures market. Moreover, wend that the regret-averse

rm optimally produces more or less in the absence than in the

presence of a futures market. This result is different from the traditional

wisdom that forward hedging always promotes production. The theory

developed in our paper is not only useful for rms in managing their

production levels and hedging positions, but it also aids rms' compet-

itors, business partners, shareholders, and stock and bond investors in

their investment decisions as well as assisting policy makers in their

policy setting processes.

InSection 2,we state the assumptions and the model setup for a

competitiverm that is not only risk averse but also regret averse. In

Section 3, we derive the theory to describe the production and hedging

behaviors for regret-averse competitive rms and compare the results

with those for risk-averse competitive rms. The nal section offers

some discussions and conclusions.

2. Assumptions and the model setup

Suppose that a competitive rm producesQunits of a product at

time 0 and will sell all units at time 1. We followBroll et al. (2006)

andothers by assuming that theproduction cost, C(Q), is strictly convex,

such thatC(0) =C

(0) = 0,C

()N

0 andC

()N

0, to re

ect the

rm'sproduction technology to have decreasing returns to scale. The price, ~P,

is random at time 1 with support on PP, such that 0 b Pb Pb . In ad-

dition, there is a corresponding futures contract that matures at time 1

with pricePf at time 0. We also assume that the producer wants to

hedge against the risk that the price of his/her produced goods may

drop so that he/she sellsX1 units of the product under the futures con-

tract and he/she will deliverXunits of the product against the futures

contract at time 1. Lettingbe the prot at time 1, we have

~ ~P QX PfXC Q : 2:1

To address the notion of regret, we followBraun and Muermann

(2004) and others by employing the following bivariate (two-attribute)

regret-averse utility functionVto get the ex-post suboptimal decision:

V ;max U gmax ; 2:2

in which the rst term is the von NeumannMorgenstern utility

function U to reect risk aversion satisfying U0 N0 and Ub 0. The

second term takes care of therm's regret prospect. The regret function

g() indicates the regret-averse attribute in which g(0) = 0, g() N 0,

andg() N 0. The parameter 0 is the weight of the regret attribute

or regret coefcient measuring the extent of regret aversion. max is

the ex-post optimal prot that the rm could have earned if there

were no price uncertainty.

In our framework, when the realized output pricePis observed,X

will be zero since there is no uncertainty. The optimal Qis calculatedby maximizing the following function

maxQ 0

U maxQ 0

U PQC Q f g

and getting C(Q) = P. When the value ofPchanges, the optimal Q

changes accordingly. Thus, max(P) appears in the form ofPQ(P)

C[Q(P)] withC[Q(P)] = P. For example, if the value of the uncertain

price ~Pis realizedto be P1, the managerwill make production and hedg-

ing decisions based on P1. In other words, the manager will make the

optimal decision based on the realized product price. However, in

practice thepriceis unknown and random. Thus, themax is also random

and depends on every possible value of the price ~P.

In order to compare the difference between regret-averse and risk-averse rms, in this paper we also investigate production and hedging

decisions when the competitive rm is risk averse. To get a risk-averse

utility function, one could simply set = 0 in (2.2).

3. The theory

We will use the term propositionto state new results obtained

in this paper and propertyto state some well-known results or the

inference drawn from the propositions obtained in this paper.

1 We note that the hedging positionXin our paper is different from the hedge ratio,

which is set to be between 0 and 1, while Xcan be negative as well as greater than one.

Readersmay refer to Michenaud and Solnik(2008) formorediscussion onthe hedge ratio.

We would like to show our appreciation to the anonymous reviewer who point out this

problem.

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To obtain the optimal production and hedging for the regret-averse

competitiverm, we maximize the expectation ofVin (2.2) such that:

maxQ 0;X

Eh

V;max

i max

Q 0;XE U ~P

h ig max ~P

~P

h in o:

3:1

The expectationE() is with respect to the cumulative distribution

function,F(P), of the random output price ~P.

Therst-order conditions are then given by:

E U0 ~P h i

g0 max ~P

~P h in o

~PC0 Q h in o

0; 3:2

E U0 ~P h i

g0 max ~P

~P h in o

Pf~Ph in o

0; 3:3

where an asterisk () indicates an optimal level.

Summing up Eqs. in(3.2) and (3.3), we getPf =C' (Q*). Thus, we

nd that the optimal output decision is solely determined by the cost

function and the forward price as stated in the following proposition:

Proposition 1. Under the assumptions described inSection 2, the optimal

production decision is independent of the regret aversion and the optimal

output level, Q*,of the regret-averse rm is the unique solution obtainedfrom the following equation:

Pf C0 Q ;

regardless of whetherPf is smaller than, equal to, or larger than E~P.

FromProposition 1, one could nd that the risk-averserm obtains

its optimal output level by setting = 0 in (2.2). We state this well-

known result in the following property:

Property 2. Under the assumptions described in Section2,the optimal

output level, Q, of the risk-averse rm is the unique solution obtained

from the following equation:

Pf C0 Q ;

regardless of whetherPf is smaller than, equal to, or larger than E~P.

In this paper, we set Q andQto be the optimal output levels of

the regret-averse and the risk-averse rms, respectively. We note

that the difference betweenProposition 1and Property 2 is that the

former states the result for regret-averserms, while the latter states

the result for risk-averse rms. FromProposition 1and Property 2, it

is clear that Q is always equal to Q, and thus, we conclude that

regret aversion has no effect on the optimal production decision, so that

the optimal output levels are the same for both risk-averse and regret-

averserms.

It is well-known (Egozcue and Wong, 2012) thatwhen a risk-averse

rm faces the certain priceP, it will choose an optimal outputQ

, suchthat its marginal revenue (=P) equals its marginal cost C(Q). Howev-

er, Wong (2014) and Niu, et al. (2014) have shown that a sufcient con-

dition is needed to obtain the traditional result in which the optimal

output level will be smaller under uncertainty than under certainty

when there is no hedging.

From Proposition 1and Property 2, we conclude thatwhen a futures

market exists, the presence of hedging can eliminate the effect of regret

aversion on the optimalproduction level, so that it is similarto the certainty

case in which the price P is known, and in this situation, Pf serves as the

known price P. This situation holds for risk-averserms also.

The intuition ofProposition 1is as follows: Given that there exists a

futures market, the rm can set the random output price, ~P, to be the

predetermined futures price,Pf. At the optimum, the rm must equate

the known marginal revenue, Pf

, from selling the last unit to the

marginal cost, C(Q*), of producing that unit, thereby yielding theoptimality condition,C0Q Pf.

In the following, we study the properties of the optimal hedging

position. It is well-known thatPf could be smaller than, equal to, or

greater thanE~P(Hirshleifer, 1988). We rst study the situation forPf

E~P. To do so, we only need to consider the rst-order condition in

(3.3).

We note that the following second-order condition:

E U ~P h i

g00 max ~P

~P h in o

Pf~Ph i2

b 0 3:4

will hold automatically given that all the assumed properties ofU and

g() are satised. This implies that the solutions obtained from the rst-

order condition in (3.3) are indeed the optimal values.

From the Eq. in(3.3), we get:

GX; Q Cov U0 ~P h i

;~P

Cov g0 max ~P

~P

h i;~P

E U0 ~P

h ig0 max ~P

~P

h in o PfE ~P

h in o 0;

3:5

whenX=XandQ=Q.

Now, we evaluate GX; Q atX= Q= Q. Notice that in this situation,

we have ~P PfQCQ, which is a xed value. This implies that

CovU0~P; ~P 0:On the other hand,

dg0 max ~P

~P h i

d~Pg max ~P

~P

h idmax ~P d~P

g max ~P

~P h i

Q ~P

N0:

As a result, we can conclude that

Cov g0 max ~P

~P h i

;~P

N0:

Theabove resultimplies that GQ; Q b 0when PfE~P. RecallthatGX; Q 0, and from the second-order condition, we can conclude

thatX b Qwhen PfE~P. We summarize thending in the following

proposition:

Proposition 3. When PfE ~P, the regret-averserm's optimal hedging

position, X, will be smaller than the rm's optimal output level, Q, in the

presence of regret aversion.

To explore the intuition for Proposition 3, we look into the following

equation by settingX=Q=QinGX; Qfrom (3.5) and it becomes

GQ; Q E U0 ~P h i

g0 max ~P

~P h in o

PfE ~P h i

Cov g0 max ~P ~P h i; ~P :3:6

Therst term on the right-hand side of (3.6) is the product of the

expected marginal utility of the regret-averse utility function V;

and the difference betweenPf and E~P . In a contango or unbiased

market, this term cannot be positive. The second term measures the

co-movement of the uncertain price ~P and marginal effect for the

regret-averse attribute. With an increase in the output price, the ex-

post loss,max~P~P, will increase atX=Q=Qand the feeling of

regret for not producing more and for selling less in the futures market

would increase too. These wouldnally lead to the nding that the

optimal production levelQis larger than the optimal hedging position

X, even in an unbiased market.

We turn to studying the behavior of the risk-averserm under the

same situation as stated inProposition 3. Similarly, we set X

andXto

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be the optimal hedging positions for the regret-averse and risk-averse

rms, respectively. We rst state the following well-known property

of the optimal hedging position for risk-averse rms (Holthausen,

1979) for comparison:

Property 4. The risk-averse rm's optimal hedging position, Xwill be

smaller than, equal to, or larger than the rm's optimal output level Qwhen Pf is smaller than, equal to, or larger than E~P, respectively.

In the following, we consider the situation with PfNE

~P . From

Eq.(3.6)to getGQ; Qto be positive, there are two situations: rst,

considering the termEfg0max~P~PgPfE~PCovg0max~P

~P; ~P. When

PfE ~P

Cov g0 max ~P

~P

h i;~P

E g0 max ~P

~P h in o ; 3:7

GQ; Q will be positive, which, in turn, implies that X N Q. In

addition, we get

Cov g0 max ~P ~P h i; ~P E g0 max ~P

~P

h in o

E g0 max ~P ~P h i~Pn oE g0 max ~P

~P

h in oE

~

P

N0:

We dene the following function:

p

ZP

p g0 max ~P

~P h i

E g0 max ~P

~P h in o dF ~P 3:8

for all pPP . Since (p) N0, P 0, and P 1, there is an

increasing function,H, of~Psuch that ~Pis a cumulative distribution

function ofH~Pand we useE~Pto denoteEH~P. Hence, the condi-

tionin (3.7) can beexpressed asP

f

E~

PN

E~

P. Thus, with regretaver-sion, different from the situation in which the managers are only risk

averse, the conditionPfNE~Pis not enough to ensure that the optimal

hedging position X is larger than the optimal production level Q.

Instead, it requires the conditionPf to be larger than a transformed

expectationE~Pof~Pwith respect to the regret functiong().

On theother hand, ifPfis not larger thanE~P, the regret parameter

is required to be small enough to ensure that the rm's optimal

hedging position is larger than the rm's optimal output level. To

achieve this, we rewriteGQ; Qin (3.6) to be:

GQ; Q E U0 ~P h in o

PfE ~P h i

E g0 max ~P ~P h i P

f~P n o: 3:9

Ifis small enough such that

bE U0 ~P

h in o PfE ~P

h i

E g0 max ~P

~P h i

~PPf n o ; 3:10

from (3.9) one could easily conclude that GQ; QN0, and thus,X* N Q*.

Now we provide some explanations for the condition in (3.10). It is easy

to see that the numerator EfU0~PgPfE~P measures the risk-

aversion effect, while the denominator Efg0max~P~P~PPfg

can explain the regret-aversion effect. Thus, when the regret parameter

is bounded by the relative effect of both risk aversion and regret

aversion, the risk-aversion effect will play a leading role and the rm

will perform similar to a rm with only risk aversion. As a result,

X N Q when PfNE~P . We summarize the results in the following

proposition.

Proposition 5. Under the assumptions described inSection2,

a. if the futures contract price Pf is larger than the transformed

expectation of~P,E~P, such thatPfE~P, or

b. if the futures contract pricePf

is larger than the expected price E~P

with the following two conditions:

E ~P

b Pfb E ~P

and

bE U0 ~P

h in o PfE ~P

h i

E g0 max ~P

~P h i

~PPf n o ; 3:11

then the regret-averse rm's optimal hedging position,X, will be

larger than the rm's optimal output level, Q, in the presence of

regret aversion.

For the optimal hedging decision, we summarize the ndings from

Propositions 3 and 5and Property 4 in the following property:

Property 6. Under the assumptions described inSection2,

a. when Pf b E~P, both regret-averse and risk-averse competitiverms

will take an under-hedged position;

b. when Pf E~P , risk-averse competitive rms will take a fully

hedged position, while regret-averse competitive rms will still

take an under-hedged position;

c. when E~P b Pf b E~P, risk-averse competitive rms will take

an over-hedged position but regret-averse rms may take either

an under-hedged, fully hedged, or over-hedged position,

depending on the size of the regret coefcient(see condition(3.11); and

d. when E~P with PfE~P, both regret-averse and risk-averse

competitive rms will take an over-hedged position.

From Property 6, it is clear that different from the purely risk-averse

rm, the optimal hedging position X is still less than the optimal

production level Q under regret aversion in an unbiased market.

Thus, the full-hedging theorem fails under regret aversion. As explained

above, regret aversion plays an important role. Another important

nding in our paper is that different from the risk-averse rm, the

regret-averse rm may still take an under-hedged position in the

backwardation futures market. Due to regret aversion, PfNE~Pis not

sufcient enough to ensure that X N Q

. Instead, it requires eitherthe condition that the futures price Pf is bigger than EH~P in the

sense that PfE~P or the condition that the regret coefcient

is bounded from above. Under either condition, the risk-aversion

effect prevails, and thus, X NQ. However, ifPf is slightly higher

than E~P in the sense that E~P b Pf b E~P and is large enough

with N EfU0 ~PgPfE~P

Efg0 max ~P~P~PPfg, the regret-aversion effect will dominate

the risk-aversion effect. In this situation,the rm will regretnot produc-

ing more and selling less in the futures market. This will lead to the

optimal production levelQ, which is larger than the optimal hedging

positionX.

It is interesting to know whether regret-averse rms will

take a higher or lower optimal hedging position when their regret-

averse attribute changes. To answer this question, we study the


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comparative statics of the optimal hedging position when the regret

coefcientvaries as shown in the following proposition:

Proposition 7. The regret-averse rm's optimal hedging position, X,

satises the followingifPf E~P, thenX will surely decrease with an

increase in the regret coefcient.

Proof. Denote~P ~PQX PfXCQ. From the rst-order

condition in (3.3), we get

f X; E U0 ~P h i

g0 max ~P

~P h in o

Pf~Ph in o

0

whenX=X. Applying the implicit function theorem and the second-

order condition, we obtain

sign dX

d

sign

f

sign E g0 max ~P

~P

h i Pf~Ph in o

:

From Proposition 3, we can obtain thatX b Qwhen Pf E~P, and

thus,CovU0~P; ~P b 0. On the other hand, from the rst-order con-

dition in (3.3), we know that

E g0 max ~P ~P h i Pf~Ph in o 1

E U0 ~P h i Pf~Ph in o

1

Cov U0 ~P

h i;~P

b 0:

Thereafter,Proposition 7holds.

We give some explanations forProposition 7here. From the proof,

we know that

sign dX

d

sign E g0 max ~P

~P

h i Pf~Ph in o

sign Cov g0 max ~P

~P h i

;~P

b 0:

We note thatCovg0max~P~P; ~Pmeasures the co-movement

of the uncertain price

~

Pand marginal effect for the regret-averse attri-bute. The above result implies that with an increase in the output

price, the ex-post loss, max~P~P, will also increase atX=Xand

Q= Q. The feeling of regret for not selling less in futures market

would increase too. In addition, when increases, rms will become

more regretaverse. This will eventually lead to a decrease in theoptimal

hedging positionX.

Last, we compare the results using hedging with the results in which

rms cannot hedge; for example, there is no futures market so that

rms cannot hedge (Niu et al., 2014; Wong, 2014). That is, X 0 in

(2.1). Under this situation, we denote the corresponding prot

and optimal output level to be0 and Q0, respectively, such that0~P ~PQ0CQ0. The corresponding rst-order condition then becomes

E U0 0 ~P h i

g0 max ~P

0 ~P h in o

~PC0 Q0 h in o

0 : 3:12

Differentiating EV;max in (3.1) with respect to Q and

evaluating the resulting derivative atQ=QandX= 0 yields

A:E U0 ~PQC Q h i

g0 max ~P

~PQC Q h in o

~PC0 Q h in o

:

3:13

If the above term is negative (positive), then, from Eq. (3.12) and the

corresponding second-order condition, we haveQ0 b(N)Q. On the

other hand, differentiatingEV;max in (3.1) with respect toX

and evaluating the resulting derivative atQ= Qand X= 0 yields

AIfX N (b) 0, it follows from the rst-order condition stated in (3.3)

and the corresponding second-order condition that A is positive

(negative), and thus, the term (3.13) is negative (positive) and Q0 b

(N)Q. We summarize the result in the following proposition:

Proposition 8. If regret-averserm's optimal hedging position X N(b) 0,

banning therm from forward trading will lead the rm to get a lower

(higher) optimal output level, i.e., Q0 b(N) Q.

From Proposition 5 and Property 6, we nd thatX NQ 0 ifthe fu-

tures price is considerably higher than the expected price. Under this

situation, from Proposition 8, we can conclude that Q0b

Q

. On theother hand, ifPfE~P, we show that X bQ. SinceXcan be positive

or negative, fromProposition 8,Q0can be smaller or larger than Q.

The result is different from that for the purely risk-averse rm.

To be precise, we set = 0 and Pf E~P, A can be rewritten

as CovU0~PQCQ; ~PN0, and thus, X N0 and Q0bQ.

2 This is

the well-known result inHolthausen (1979). That is,if Pf E~P, the

optimal output level for the purely risk-averse rm with hedging, Q,

must be larger than that without hedging, Q0. In other words, in an

unbiased forward market, forward hedging always promotes production

for the purelyrisk-averserm. However,for the regret-averserm, banning

therm from forward trading may induce therm to produce more or less,

even in an unbiased forward market.As explained above, for the regret-

averse rm, even in an unbiased forward market,X b Q, and thus,X

can be negative or positive. Thereafter, we apply Proposition 8andobtain the result thatQ0 can be smaller or larger than Q.

To see the intuition forProposition 8, we recast Eq.(3.12)as

C0 Q0

E ~P

Cov U0 0 ~P

h ig0 max ~P

0 ~P

h i;~P

n o

E U0 0 ~P h i

g0 max ~P

0 ~P h in o :

This equation states that the rm's optimal output level,Q0, is the

onethat equates themarginal cost of production, C(Q0), to the certainty

equivalent output price that takes both the rm's risk aversion and

regret aversion preferences into account. Indeed, the second term on

the right-hand side of the above equation captures the price risk

premium, which must be negative (positive) if the

rm optimally sells(purchases) its output forward, i.e., X N(b)0, thereby implying that

Q0 b(N)Q.

In the absence of regret aversion, i.e., 0, theriskpremium of price

is unambiguously negative sinceUb0. In this case,X N 0, and thus,

Q0 bQwhich is the well-known result ofHolthausen (1979). When

regret aversion prevails, the risk premium of price could be positive or

negative. This is due to the existence of the regret function g(). With

an increase in the output price, the ex-post loss, max~P0~P, will

increase. The co-movement of the uncertain price ~Pand the marginal

effect for the regret-averse attribute Covfg0max~P0~P; ~Pg will

then be positive.

4. Conclusion and discussion

In this paper, we extend previous studies on risk-averse competitive

rmsto examine the production andhedging behaviorsof regret-averse

competitive rms when there is a futures market. We rst nd that

regret aversion has no effect on the optimal production decision so

that the separation theorem is still alive under regret aversion. In addi-

tion, we show that with an unbiased futures price, the regret-averse

rm will take an under-hedged position. This implies that under regret

aversion, the full-hedging theorem does not hold. We nd that with

more regret aversion, regret-averse managers will take a smaller

optimal hedging position in an unbiased futures market. Last, we com-

pare the results using hedging with the results in which rms cannot

2 Recall that the optimal output level and hedging position for the risk-averse rm are

given byQandX, respectively.

157X. Guo et al. / Economic Modelling 51 (2015) 153158

7/23/2019 1-s2.0-S0264999315002114-main

6/6

hedge. We nd that when regret-averse rms take a positive (negative)

optimal hedging position, banning the rm from forward trading will

lead the rm to produce a lower (higher) optimal output level. This

result is different from the traditional wisdom that forward hedging

always promotes production.

In this paper, we develop some properties of production and hedg-

ing behaviors when the competitive rm is not only risk averse but

also regret averse by assuming that the regret-averserm has decided

to hedge. However, solving it is important and interesting to notewhether a regret-averserm should hedge or not. To start answering

this issue, we develop proposition 8 to compare the optimal production

levels with and without the use of futures/forward contracts. In study-

ing whether to hedge or not, one may compare the expected regret-

averse utility functions under the optimal decisions with and without

hedging. In addition, we also note that the optimal choice of the

nancial hedging instruments (futures and options) is an interesting

extension of the theory with regret aversion developed in our paper

and others.3 We leave this to further study in the future.

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3 We would like to show our appreciation to the anonymous reviewer who point out

this problem.

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