Free entry under uncertainty

Munich Personal RePEc Archive

Free entry under uncertainty

Jellal, Mohamed and wolff, François charles

Al Makrîzî Institut d’Economie

2005

Online at https://mpra.ub.uni-muenchen.de/38376/

MPRA Paper No. 38376, posted 26 Apr 2012 13:04 UTC

Free entry under unertainty

∗

Mohamed Jellal

†

François-Charles Wol

‡

Seond revision, April 2004

Abstrat

When fousing on rm's risk-aversion in industry equilibrium, the number of

rms maybeeitherlargeror smallerwhen omparingmarket equilibrium withand

without prie unertainty. In this paper, we introdue risk-averse rms under ost

unertainty in a model of spatial dierentiation and show that the impat of un-

ertainty will always inrease the number of rms in an industry. This nding is

explained by the higher pries that rms harge to onsumers under unertainty.

With inreased unertainty, rms have greater inentive to enter themarket sine

they maybenetfrom higherlevels ofprot.

JEL lassiation: D43, D81,L12

Key words: Spatialdierentiation;Risk-averse rms;Cost unertainty

∗

Weareindebtedtotwoanonymousrefereesfortheirhelpfulommentsandsuggestionsonaprevious

draft. Theusualdislaimerapplies.

†

UniversitéMohammedV, Rabat,Morroo; Toulouse BusinessShool,Frane;and Conseils-Eo,10

ImpassedeMansenal,31500Toulouse,Frane. E-mail: jellalmohamedyahoo.fr

‡

Correspondingauthor. LEN-CEBS,UniversitédeNantes,hemindelaCensiveduTertreBP52231,

44322NantesCedex3,Frane;CNAVandINED, Paris,Frane. Tel: 33240141742. Fax: 33240141743.

E-mail: wols-eo.univ-nantes.fr

When explainingvariationsinthenumberof rmsaross industries,standard arguments

drawing onsale eonomies and entryonditions usually negletthe issue of unertainty.

Unfortunately, the prevalent assumption of risk-neutral rms is not really appropriate.

Severaltheoretial ontributions have reentlyonsidered asettingwherermsbehavein

a risk-averse manner (see Asplund, 2002, and the referenes therein). Among the most

frequent explanations, one an invoke the presene of liquidity onstraints, the manage-

ment by non-diversied owners ordelegation ofontroltorisk-averse supervisors, aswell

as nanial distress (Drèze, 1987). In partiular, the extent of orporate hedging ativ-

ities may be interpreted as a relutane to bear risk (Nane et alii, 1993). Clearly, the

introdutionof unertainty has strong impliations for the produt marketompetition.

The pioneeringwork dealingwith the impatof unertainty on rms'deisionsisdue

to Sandmo (1971). Within a partial equilibrium framework, greater prie unertainty

is expeted to lower the optimal quantity produed in a perfetly ompetitive market 1

.

Then,thedegreeanddistributionofprieunertaintyare signiantfatorstoexplainin-

dustrystruture. Attheequilibrium,Sandmo(1971)provesthataninreasedunertainty

about prie lowers the number of rms in the industry. A more general question is to

fousontheimpatofriskaversioninamodelinwhihthenumberofrmsisdetermined

endogenously. Appelbaum and Katz (1986) were the rst to address that issue (see also

Haruna, 1996). One a ompetitiveequilibriumis introdued, they show that the eets

of prie unertainty onthe numberof rms inan industry an nolonger be signed, even

with additionalassumptions about relative orabsoluterisk aversion.

Despite the ambiguous predition of prie unertainty on the industry equilibrium,

it seems tempting to believe that a negative relationship between unertainty and the

number of rms is more likely 2

. Intuitively, and following the disussion in Sandmo

(1971), rms that are haraterized by ahigh value for risk aversion ertainly prefer not

to operate in a market where prie unertainty prevails. Indeed, unertainty may be

seen as a natural barrier to entry, thereby leading to a derease in the number of rms

1

SeealsoLeland(1972)fortheeet ofunertaintyinamonopolysetting.

2

And suh anegativerelationshipseems rathersupported by thedata. Using aross-setion ofUS

manufaturingindustries,Ghosal(1996)showsthatgreaterunertaintyexertsanegativeimpatonthe

variability may alsooer opportunities for inreasing average prot for risk-averse rms.

Average prots of a prie taker are inreasing in the variability of the output prie and

Oi's onlusion does generalize to a onsiderable extent (Friberg and Martensen, 2000).

Suh positive eets onprots ould have abeneial inuene onthe entry of rms.

In this paper, followingSandmo (1971) and Appelbaum and Katz (1986),we further

examinetheeetsofunertaintywithinanindustryequilibriumframework. Weexamine

the problem offree entry andexit of rmsin asettingof spatial dierentiation withost

unertainty. Speially, we draw on the loation model originally proposed by Salop

(1979), who introdues dierentiation using a irular ity with onsumers uniformly

distributed on its irumferene. Our main result is to prove that the indeterminate

eet of unertainty on the number of rms in an industry does no longer hold. In a

loation model with horizontally dierentiated produts and risk-averse rms, greater

ost unertainty always inreases the numberof rms operatinginthe industry.

Theintuitionof thatresultisasfollows. In aloationmodel(eitherlinearorspatial),

it is well known that the ompetitive prie under produt dierentiation is dened as

the sum of the marginal ost and the transportation ost, whih leads to a monopoly

powerforthe dierentrmsinthe industry(see Tirole,1988). Whenone introduesost

unertainty, the optimal prie now inludesanadditionalterm orresponding to the risk

premium faed by the rms. So, when omparing market equilibrium with and without

unertainty, it turns out that rms harge higherpries toonsumers under unertainty.

This leads to higher prots for risk-averse rms, and greater unertainty inreases the

number of rms in the industry. Thus, ina ertain sense, our theoretial ontributionis

lose tothe famous Oi's variability result.

By fousing on unertainty in a loation model, our paper is related to the reent

literature onrisk-averse rms inanoligopoly. Ina ontext ofost unertainty, Wambah

(1999)provesthattheBertrandparadoxsuhthattwormsaresuientforperfetom-

petitiondoesnolongerholdwithrisk-averserms. Inanindustrywithprieompetition,

the equilibriumprie isexpetedto exeedthe ompetitiveprie and then inreasingthe

number of rms may lead to an inrease in prie 3

. Janssen and Rasmusen (2002) also

3

Speially,thenewprieisexpetedtobehigherwhenthereisaninreaseinthesizeofthemarket

industry. With an unertain number of ompetitors, there exists a unique symmetri

equilibrium in mixed strategy and again eah rms harges a prie larger than marginal

ost 4

. The questionofstrategihoiesofrisk-averse rmsisfurtheranalyzedinAsplund

(2002), who examines how the degree of risk aversion and dierent types of unertainty

aet ompetition in an oligopolisti framework. The key feature of this insightful on-

tributionistopropose ageneralompetitionmodelofrisk-averse rmsthat enompasses

prie ompetition with dierentiated produts under various forms of ost and demand

unertainty. In partiular, ompetition issofter inase of marginal ost unertainty.

Thus,our workmaybeseen asomplementarytothe analysisofAsplund(2002). Our

ontribution is twofold. First, we fous on the onsequenes of unertainty in a model

with produtdierentiationand freeentry ofrms. Seond,wepresentawelfareanalysis

whih aounts for the osts involved by rms in bearing risk. The remainder of the

paperisorganizedasfollows. Insetion2,weextendthe irularloationmodelof Salop

(1979) and assume that marginalost is unertain. In setion 3,we determine the Nash

equilibriuminpries forany numberof rmsand showthat rmshargehigher priesto

onsumers beause of unertainty. The Nash equilibrium in the entry game is analyzed

in setion 4, with a positive impat of unertainty on the number of rms. Setion 5

examinesthe prieequilibriumfromanormativeviewpoint. Conludingommentsarein

setion 6.

2 The spatial model

We onsider a model with rms produing dierentiated produts, in whih onsumers

are heterogeneous and where rms have unertain marginal osts. Thus, we relax the

prevalent assumption behind the Bertrand paradox that rms produe an homogeneous

good,a situation analyzed by Janssen and Rasmusen (2002)and Wambah (1999) inan

unertain setting. Witha loationmodel, itfollows that rmsan raise theirprie above

the marginalost withoutlosing their entire market share.

and thenumberofrmsin thesameproportion(seeWambah,1999).

4

Theperfetlyompetitiveequilibriumisthelimitasewhenthenumberofrmsbeomeslarge. As

theprobabilityofompetitioninreases,eahrmreduesitspries.

are not uniformly ranked by all onsumers. As usual in the literature, eah onsumer

has a dierent preferene for the brands sold in the market due to dierent loation.

In our setting, loation orresponds to the physial loation of a partiular onsumer.

Eah agent observes the pries harged by all the rms, and then deides to purhase

the good from the rm at whih the prie plus the transportation ost is minimized.

Another onvenient interpretationis that loation an alsorepresent a distane between

the brand harateristis viewed as ideal by the onsumer and the harateristis of the

brand atually purhased 5

. Thus, rms hoose their produts antiipating that their

loationdeisioninprodutspaeisexpetedtoaet the intensity ofprie ompetition.

Ourtheoretial analysisofthe impatofunertainty onthenumberofrmsdraws on

the spatial dierentiation model originally desribed by Salop (1979), orresponding to

the ase ofairularity. Insodoing, weare abletoexaminethe problemof rms'entry

on the market given marginalost unertainty. Speially, we study entry and loation

deisions when there existnobarriers toentry other than xed osts.

Wesupposethatonsumersareloateduniformlyonairle

C

^{,whihhas aperimeter}

equal to

L

^{. Clearly,the irumferene}

L

^{isa measure for the} heterogeneity of onsumers and itmay beseen asanindiatorfordemandintensity. Individualsareontinuously and

uniformlydistributedalongthisirumferene. Weassumewithoutlossofgeneralitythat

the density is onstant, and it is denoted by

∆

^{6. Thus, the parameter}

∆

^{expresses the}

thikness of the market. Given the loation of rms, onsumers inur a transportation

ost equal to

t

^{per unit of length,suh that this ost inludesthe value of time spent in}

travel. Eah onsumer buys exatly one unit of the brand that minimizesthe sum ofthe

prie and thetransportationost. Nevertheless, this generalizedost has toremainlower

than the gross surplusthat the onsumer an obtain fromthe good. This outside option

is denoted by

s

^{. It is assumed to be large enough, sothat the market is always overed}

in equilibrium(goods are boughtby allonsumers).

Firms are loated around the irle. Although the irular model of Salop (1979) is

a loation model, it does not expliitly explain how rms hoose their loation (see the

5

Inthatase,distaneisameasureofthedisutilityfromonsumingaless-than-idealprodut.

6

Relaxing this assumption does not modify our theoretial onlusions. See Calvo-Armengol and

Zenou(2002)fortheaseofageneraldensityinamodelofdierentiatedproduts,butunderertainty.

two-stagestruture. First,thenumberofrmsisendogenouslydetermined. Itisassumed

that rms are automatially loated at anequal distane from one another. Thus, if the

number of entering rms is denoted by

n

^{and given the irumferene}

L

^{, the distane}

betweenanytwormsisequalto

L/n

^{. Seond,rmsompeteinpriesgiven theprevious}

loations. So, a key feature of this horizontaldierentiation modelis the fous on rms'

entry, and we examinethe impatof unertainty on entry.

There are many potential rms in the loation model, whih have all the same teh-

nology. To address the issue of entry, we suppose that eah rm is haraterized by a

xed ostofentrydenotedby

f

^{. Onethermisloatedatapointontheprodutspae,}

it faes a marginalost

c

^{that is supposed tobe onstant. We depart from the model of}

Salop (1979) by assumingthat this marginalost is unertain, sothat rms fae supply-

indued ost utuationsinour setting. Toformalizethis typeofunertainty, weassume

that the marginalost is desribed by a random variable

˜ c

^{whose mean is}

E(˜ c) = c

^and

the orrespondingvarianeis

V ar(˜ c) = σ ²

^{. Asusual,greaterostunertaintyismeasured}

by aninrease inthe variane

σ ²

^{(amean preserving spread in osts).}

Itseemsimportanttonotethatourwaytoinludeunertaintyintheloationmodelis

absolutelynotrestritive. Indeed,thereare numerousexamplesintheindustryofsoures

of unertainty arising by the marginalost of prodution. Forinstane, Wambah(1999,

p. 946) mentions the ase of insurane orporationswhere the probability of aident is

imperfetlyknowntotheinsurers, rmswhihprovideguaranteesfornewproduts(given

random breakdown), or simply rms whih import brands and then fae exhange-rate

unertainty. Other explanations onern poorlimati onditions for rms that produe

or use agriultural goods orunertain wages linked toeieny wage onsiderations and

shirkingbehaviorsaswellasunertaintyoverthenumberofativeworkers(duetoillness).

Eahrmislabelledbysubsript

i

⁽

i = 1, . . . , n)

^{,andtherm'sloationisdenotedby}

x i

^{. Armisfullydesribedby thelistofprieshargedononsumers(}

p 1 , . . . , p i , . . . , p n

^).

A onsumer is loated at the distane

x ∈ C

^{. Then, the} generalized prie to buy the brand is equal to

p i + t|x − x i |

^{under linear} transportation osts

7

. Firms antiipate that

7

While werestrit ourattentionto thease oflineartransportationostsfor thesakeof simpliity,

ourtheoretialresultsremainsunhangedwithquadratitransportationosts.

In the irular model, a representative rm has only two ompetitors. Given two level

of pries

p i − 1

^and

p i+1

^{, the demand poolfor the rm}

i

^{is omposed of two} sub-segments.

The outside boundaries of the pool are given by two marginal onsumers, respetively

denoted by

x

^and

x

^{, for whom the} generalized prie is idential between two adjaent rms : respetively between

i − 1

^and

i

^for

x

^{, and between}

i

^and

i + 1

^for

x

^{. Thus, the}

marginal value

x

^{is the solutionof the followingequation :}

p i + t(x i − x) = p i− 1 + t(x − x i− 1 )

⁽¹⁾

Hene, the onsumer whih is indierent between purhasing the brand from rm

i

^and

purhasing itfrom itslosest neighbor

i − 1

^isharaterizedby :

x = (p i − p i − 1 ) + t(x i + x i − 1 )

2t

⁽²⁾

So, therm

i

^{faesademandfromalltheonsumerswhoseloationbelongtotheinterval}

[x; x]

^,sinethe generalizedprietheseonsumers obtainfromrm

i

^{islowerthanthe one}

they would obtainfromrm

i − 1

^{. In asimilarway,the marginalonsumer}

x

^{issuh that}

p i + t(x − x i ) = p i+1 + t(x i+1 − x)

^{, whihimplies :}

x = (p _i +1 − p _i ) + t(x _i + x _i +1 )

2t

⁽³⁾

Finally, the demand poolfor the rm

i

^{onsists of all onsumers whose loation is om-}

prised in the losed interval

[x; x]

^.

Now, let

Π i

^{bethe protlevelofthe rm}

i

^{. Knowingthe rm's demand, thepresene}

of a xed ost and given the unertainty on marginalost, the prot for the rm is also

a randomvariablewhih isgiven by :

Π ˜ i =

Z x

x ∆(p i − c)dx ˜ − f

⁽⁴⁾

so that the random prot

Π ˜ i

^{an be expressed as :}

Π ˜ i = ∆(p i − ˜ c)(x − x) − f

⁽⁵⁾

Given the unertainenvironment, weassumethatrms areriskaverse followingsome

reentextensions inoligopolytheory(see Asplund,2002,Haruna, 1996,Maietalii,1993,

risk-neutral has strong impliationsfor the produt market ompetition.

Thereare several reasonsthat may explainwhy rmsbehaveina risk-averse manner.

The existene of xed osts means that rms are making ostly investment before pro-

duing, so that risk aversion is driven by liquidity onstraints (see Drèze, 1987). Many

rms have an imperfet aess to the apital markets, and thus they have to bear part

of the risk assoiated with their prodution. Another reason deals with non-diversied

owners. Although owners may be tempted to maximize expeted prots, the delegation

of ontrol to managers in hierarhial struture favors the relutane to bear risk sine

the managers' inome is learly related to the rm's performane. Others arguments in

the prevalent literature are linked to ostly nanial distress and to non-linear tax sys-

tems. Somestudieshavesuggested thatthe extentof orporatehedgingativitiesmaybe

interpreted asthe result of risk-averse behavior (Nane et alii,1993,Gézi et alii,1997).

Given the unertainty on the marginal ost, the rm

i

^is haraterized by a Von

Neumann-Morgenstern utility funtion denoted by

U i

^{, sothat the objetive funtion for}

the rm may be expressed as :

max V _i = E[U _i ( ˜ Π _i )]

⁽⁶⁾

where

U i

^{is a ontinuous,} twie-dierentiable and onave utility funtion (

U _i ^′ > 0

^,

U _i ^′′ <

0

^{). From the denition of}

Π ˜ i

^{, the} representative rm

i

^{seeks to maximize the expeted}

utility funtion :

V i = E ^h U i

∆(p i − ˜ c)(x − x) − f ⁱ

⁽⁷⁾

Letusnallyremindthedenition ofthemonopolisti-ompetitionequilibriuminthe

irular ity. At the optimum, eah rm behaves as a monopoly on its brand, meaning

that the rm hooses the prie that maximizes its utility funtion given the demand for

brand

i

^{and given that allother rms harge the sameprie, and then freeentry of rms}

results in zero prot. So, we solve the model by rst determining the Nash equilibrium

in pries for any numberof rms, then by alulating the Nash equilibriumin the entry

game (see Salop,1979, Tirole, 1988).

Letusassumethat

n

^{rmshaveenteredthemarket. Sinethesedierentrmsareloated}

symmetrially around the irle, we examine an equilibrium in whih eah rm harges

the sameprie. Werestritourattentiontotheaseofaovert market,whihmeansthat

there are enough rms inthe market. This orresponds toa situationwherethe value of

the xed ost

f

^{isnot toohigh.}

Thus, the maximizationprogramfor the rm

i

^is

max _p i V _i

^{, so that the} orresponding rst-order ondition given by

∂V i /∂p i = 0

^{under marginalost unertainty is:}

E

"

U _i ^′ (.) ∆(p i − c) ˜ ∂x

∂p _i − ∂x

∂p _i

!

+ ∆(x − x)

!#

= 0

⁽⁸⁾

with

U _i ^′ (.) = U _i ^′ ∆(p _i − ˜ c)(x − x) − f

^{for the notation. We also hek that the seond-}

order ondition

∂ ² V i /∂p ² _i < 0

^{for a maximum issatised sine :}

E



 U _i ^′′ (.) ∆(p _i − ˜ c) ∂x

∂p i

− ∂x

∂p i

!

+ ∆(x − x)

! 2

+ 2∆ ∂x

∂p i

− ∂x

∂p i

!

U _i ^′ (.)



 < 0

using

U _i ^′′ (.) < 0

^and

∂x/∂p i − ∂x/∂p i < 0

^{. Sine}

Π i

^{is ontinuous in}

(p i− 1 , p i , p i +1 )

^and

sine

Π i

^{is stritly onave in}

p i

^{, we dedue that there always exists a Nash} equilibrium in pries and that this Nash equilibriumis unique.

Proposition 1 The symmetri Nash equilibrium prie denoted by

p ^∗ _i

^{is given by:}

p ^∗ _i = c + tL

n + cov[˜ c, U _i ^′ (∆(p ^∗ _i − ˜ c)L/n − f )]

E[U _i ^′ (∆(p ^∗ _i − c)L/n ˜ − f)]

⁽⁹⁾

Proof : The optimal prie is given by ondition (8). First, we know that rms are

symmetriallyloatedandthusthedistanebetweentwormsis

L/n

^{,sothatthemarket}

area foreahrm is

x − x = L/n

^{. Seond,given the denitionofthe marginalonsumers}

x

^and

x

^{, using (2)and (3)leads to}

∂x/∂p i − ∂x/∂p i = −1/t

^{. Thus, we get :}

E

U _i ^′ (.)∆

L

n − p i − ˜ c t

= 0

Given the properties of the expetany operator,it follows that :

p ^∗ _i = tL

n + E[˜ cU _i ^′ (∆(p ^∗ _i − c)L/n ˜ − f )]

E[U _i ^′ (∆(p ^∗ _i − ˜ c)L/n − f )]

Sine

˜ c

^{is an argument of}

U _i ^′ (.)

^{, we an further simplify the optimal prie using the fat}

that

E(XY ) = E(X)E(Y ) + cov (X, Y )

^{for twovariables}

X

^and

Y

^{. Sinethe meanofthe}

random marginalost is

E(˜ c) = c

^{, we nallydedue (9). QED}

Clearly,thesignoftheovariane

cov[˜ c, U _i ^′ (.)]

^{ispositivesineBaron(1971)hasshown}

that the inequality

cov[˜ p, U _i ^′ (.)] < 0

^{holds under prie unertainty and provided that the}

marginal utility

U _i ^′ (.)

^{is dereasing.} Proposition 1 gives us a rst result onerning the role of ost unertainty on the spatial monopolisti-ompetition equilibrium. A greater

ost unertainty when produing brands leads to higher generalized pries harged to

onsumers. At the equilibrium, the prie

p ^∗ _i

^{is the sum of three elements : the marginal}

ost of prodution

c

^{, the} transportation ost

tL/n

^{, whih measures the} monopsonisti behavior of rms, and the risk premiumgiven by

cov [˜ c, U _i ^′ (.)]/E[U _i ^′ (.)]

^.

As the optimalprie stands, it seems at rst sight diult tointerpret the last term

dealingwith riskaversion. Tond amoreexpliitresultand getlosedformsolutionsfor

our problem,we have to makean additionalassumption onerningthe marginalost.

Assumption 1 The marginal ost

˜ c

^{follows a Normal} distribution, with

E(˜ c) = c

^and

V ar(˜ c) = σ ²

^.

Under assumption1, we an use the Stein's lemma(Huang and Litzenberger, 1988). Let

usonsidertwovariables

X

^and

Y

^{suhthattheyarebivariatenormally}distributed. Ifthe funtion

f (Y )

^isontinuouslydierentiable,Rubinstein(1976)provethat

cov[X, f (Y )] = E[f ^′ (Y )]cov(X, Y )

^{. Now, ifwe apply the lemmaofStein toour problem,itfollowsthat :}

cov [˜ c, U _i ^′ (∆(p i − ˜ c)L/n − f )] = E[U _i ^′′ (∆(p i − ˜ c)L/n − f )]cov [˜ c, ∆(p i − c)L/n ˜ − f]

Sine we have

cov[˜ c, ∆(p i − c)L/n ˜ − f] = −∆σ ² L/n

^{, this implies :}

cov[˜ c, U ^′ (∆(p i − ˜ c)L/n − f )] = −E[U _i ^′′ (∆(p i − ˜ c)L/n − f )] ∆L n σ ²

and thusthe symmetri Nashequilibrium priemay beexpressed as 8

:

p ^∗ _i = c + tL

n − E[U _i ^′′ (∆(p ^∗ _i − c)L/n ˜ − f)]

E[U _i ^′ (∆(p ^∗ _i − ˜ c)L/n − f )]

∆L

n σ ²

⁽¹⁰⁾

8

The derivation of the rst-order ondition in the ase of normally distributed unertainty is also

Letus denethe parameter

a

^{suh that :}

a = − E[U _i ^′′ (∆(p ^∗ _i − c)L/n ˜ − f)]

E[U _i ^′ (∆(p ^∗ _i − ˜ c)L/n − f )]

In the literature,

a

^{isknown as the} Rubinstein's measure of absolute risk aversion 9

. Ru-

binstein (1973, 1976) has proved that this measure based on the expetations of

U _i ^′′ (.)

and

U _i ^′ (.)

^{remainsonstant.}

Proposition 2 Under assumption 1, the Nash symmetri prie

p ^∗ _i

^{is given by:}

p ^∗ _i = c + tL

n + ∆L

n aσ ²

⁽¹¹⁾

Assumption 1 leads to a losed-form solutionfor the positive risk premium, whih is

now equalto

∆Laσ ² /n

^{. Itis aninreasingfuntion ofthe density}

∆

^{of onsumersonthe}

irle and of the demand intensity

L

^{, but it is negatively related tothe number of rms}

n

^{. In that ase, the risk due to unertain marginal ost is spread over a larger number}

of rms. A novel result in our analysis is that rms harge higher pries for onsumers

given ost unertainty. When rms are haraterized by risk aversion (a>0), we obtain

∂p ^∗ _i /∂a = ∆Lσ ² /n > 0

^{. Also,the optimal prie ispositively relatedto the variane}

σ ²

^of

the marginalostsinethe derivative

∂p ^∗ _i /∂σ ² = ∆La/n

^{ispositive. Bothresultsindiate}

that rms share with onsumers the risk generated by ost utuations. In industries

haraterizedby greater ost unertainty, higherpries for brandsare expeted sinethe

risk premiuminreases.

Another interesting result is that the optimal prie is an inreasing funtion of the

demand intensity

L

^{and of the onsumer density}

∆

^{(onlyin an unertain ontext), with}

inreasedopportunitiesofdierentiationforrms. Otherndingsonerningthevariables

that aet the optimal prie are more standard. With risk-averse rms in the industry

(

a > 0

^{), a larger produt market exerts a positive eet on the} equilibrium prie, given the higher possibility of dierentiation for rms (the market area for eah rm is xed,

given by

L/n

^{). Eah rm faes the same degree of unertainty onits marginal ost and}

9

Asplund (2002,appendix1)alsousesthemeasure

−EU _i ^′′ ( ˜ Π i − f)/EU _i ^′ ( ˜ Π i − f )

^{. Theauthordenes}

this ratioasthe Arrow-Prattmeasure of global absolute risk aversion. However,aspointedoutby an

anonymousreferee,this expression annotbeonsideredastheArrow-Prattmeasure whihis givenby

−U i ^′′ ( ˜ Π i − f )/U i ^′ ( ˜ Π i − f )

^.

higher prie. Also, the optimal prie inreases with

t

^{sine the market power of rms is}

inreased for onsumers who are loated lose to the rms (Salop, 1979). Finally, given

the inreased ompetition,webasially observe that the priedereases with the number

of rms inthe market sine

∂p ^∗ _i /∂n = −t/n ² − ∆Laσ ² /n ² < 0

^10.

Beforendingtheequilibriumnumberofbrands(

n

^isendogenous),webrieyexamine the situationwhere rms are riskneutral. When ost utuations have noimpatonthe

utility derived by the rms (

a = 0

^{),the optimal prie is:}

p ^∗ _i = c + tL

n

whih is the result obtained by Salop (1979) in a spatial model under ertainty 11

. In

the ase of risk neutrality, we note that the onsumer density does not inuene the

equilibriumprie. Thisonlusion doesnot longerhold whenrmsshare with onsumers

part of the riskgenerated by ost volatility, asshown below.

So, at this rst-stage of the loation model, our main onlusion is that pries are

higher with ost unertainty. The ost of an inrease inunertainty is supported by on-

sumers withdierentiatedproduts. Asaonsequene,greaterost unertaintyinreases

average prots for rms, and this positive eet of variability on rms' prot should be

linked to the inuentialontribution of Oi (1961), who evidenes a positive relationship

between the variability of the output prie and average protsof a prie taker.

4 Free entry of rms

We nowturn to the determination of the endogenous number of rms

n ^∗

^{,assuming that}

thereareenoughpotentialentrantstooverthemarket. Letusbrieydetailtheondition

forthemarkettobeovert 12

. Weknowthattheequilibriumpriehastobelowerthanthe

gross surplus

s

^{. Sine the maximum distane for aonsumer is}

L/2n

^,the orresponding 10

The ompetitive outome anbe regardedas alimit ase of our model when the numberof rms

beomesverylarge.

11

IntheoriginalpresentationofSalop(1979),thelengthoftheirle isset toone.

12

On this issueofovertmarketin spatial model,see thefurther disussion ofJellal et alli(1998)in

theontextofalabormarket.

p ^∗ + tL

2n ≤ s

⁽¹²⁾

Using the denition of

p ^∗

^{, itan also be expressed as :}

aσ ² ∆L

n ≤ (s − c) ² − 3 2

tL

n

⁽¹³⁾

so that the ondition ensuring that the marketis overed atthe prie equilibriumis:

0 < σ ² < 2n(s − c) ² − 3tL

2a∆L

⁽¹⁴⁾

Thus, the variane

σ ²

^{has totake}intermediatevaluesforeahonsumertobuythe brand at the equilibrium. The interpretationof this result is asfollows. When the variane

σ ²

is small, the equilibrium prie is above the prie under unertainty, but the inrease in

prie remains limited sine rms harge a low risk premium to the onsumers. Hene,

the market is overt. Conversely, when the risk premium beomes important, the rms

are expeted toset priesthat are exessivelyhigh. Then,some onsumerswillnolonger

purhase anything.

Bydenition, the equilibrium numberof rms

n ^∗

^{is given by :}

E[U _i ( ˜ Π _i )] = 0

⁽¹⁵⁾

Ignoring assumption 1, let us suppose more generally that the unertain ost

˜ c

^{is dis-}

tributed aording to a density funtion

g(˜ c)

^{dened over the support}

Ω = [c; c]

^{. Thus,}

the previous ondition may be expressed as

R

Ω U i [Π(˜ c)]dg(˜ c) = 0

^{, the} reservation prot beingnormalized to0. Again, thediulty forour problemistond anexpliitsolution

for the optimal number of rms

n ^∗

^{, whih involves additional} restritions either on the distribution of

c ˜

^{or onthe funtionalformfor}

U

^.

Reallthattoderivetheoptimalprie

p ^∗ _i

^{,wehaveusedtheStein'slemmabyassuming}

that the marginal ost is normally distributed. It is well known that the mean and the

variane provide a omplete haraterization of a random variable whih is normally

distributed. Thus, under assumption 1, we an rely on the mean-variane speiation

for the utility funtion

U i

^{13. Thus, the problemfor arm may beexpressed as :}

V _i = E( ˜ Π _i ) − a

2 V ar( ˜ Π _i ) − f

⁽¹⁶⁾

13

Themean-varianeapproahanbeusedifthestohasti distributionof themarginalostbelongs

to apartiularparametrizedfamily,normalorelliptialrandomvariable.

where

a

^{isthedegreeofabsoluteriskaversion(}

a ≥ 0

^{)andthe protis}

Π ˜ _i = ∆(p _i − c)(x ˜ − x) − f

^{. Itfollows that :}

V i = ∆(p i − ˜ c)(x − x) − a

2 (∆(x − x)) ² σ ² − f

⁽¹⁷⁾

One an easilyhek that with the mean-variane utility, the optimal symmetriprie is

p ^∗ _i = c + tL/n + ∆Laσ ² /n

^{as laimed in} Proposition 2. Using this optimal value for

p ^∗ _i

^,

we nallyobtain

V i

^{suh that :}

V i = t∆

L n

²

+ a 2 σ ² ∆ ²

L n

²

− f

⁽¹⁸⁾

Sine the numberof rms

n ^∗

^{isgiven by}

V i (n ^∗ ) = 0

^,weget

^L _n ² t∆ + ^a ₂ ∆ ² σ ² = f

^.

Proposition 3 Under assumption 1 andwith a mean-variane utilityfuntion, theopti-

mal number of rms

n ^∗

^{in a situation of imperfet ompetition with free entry is:}

n ^∗ =

v u u t

(t∆ + ^a ₂ ∆ ² σ ² )L ²

f

⁽¹⁹⁾

Proposition 4 Under assumption 1 andwith a mean-variane utilityfuntion, theopti-

mal prie value

p ^∗

^{under free entry isgiven by :}

p ^∗ = c +

s tf

∆

v u u t

(1 + aσ ^{2 ∆} _t ) ²

(1 + ^a ₂ σ ^{2 ∆} _t )

⁽²⁰⁾

Now, let usdene

φ(a, σ)

^{suh that :}

φ(a, σ) = 1 + aσ ^{2 ∆} _t

q 1 + ^a ₂ σ ^{2 ∆} _t

Clearly, we have

φ(a, σ) > 1

^,

φ(0, σ) = 1

^and

φ(a, 0) = 1

^{. Thus, the optimal prie under}

ertainty

p ^∗ 0

^issimply

p ^∗ 0 = c + ^{q tf} _∆

^{and we are now able toompare}

p ^∗ 0

^and

p ^∗

^.

Corollary 1 With free entry of rms, the prie ishigher under unertainy.

Inthismodelofspatialdierentiation,themainontributionofourpaperistoformally

more rms beause of unertainty and risk aversion . Clearly, both the degree of risk

aversion

a

^{andthemeasureofvariane}

σ ²

^{exertapositiveeet ontheoptimalnumberof}

rms. That unertainty positivelyaets free entry may besurprising, sine itis usually

admitted that greater unertainty is rather expeted to derease the number of rms in

anindustry. Forinstane, inthe ontextof prieunertainty,Sandmo(1971)argues that

rmsharaterizedbyalargevalueforriskaversionwillhoosenottoenter inanindustry

faing a high degree of unertainty. Only low risk-averse rms are expeted to enter in

industries with greaterunertainty, thereby reduing the numberof rms.

Then, how an we justify that greater unertainty does not at as a barrier to entry

under spatial ompetition ? In fat, we have previously shown that rms an harge a

higherprietoonsumers undermarginalostunertainty,sine theyshiftthe risktothe

onsumers. So, withgreaterunertainty,the riskpremiumbeomeslargerand risk-averse

rms have greaterinentives toenter the marketsine entering rms may benet froma

higherprie. Thispositiverelationshipbetweenentryandunertaintyundermonopolisti

ompetition is a novel result with respet to the previous literature for models in whih

the number of rms inthe market isendogenously determined 15

.

5 Welfare analysis

Wenowonsider the prieequilibriumunderunertaintyfromanormativeviewpoint. In

partiular, we examine the impat of marginal ost unertainty in a free-entry and exit

equilibrium in order to know whether unertainty produes a larger ora smaller variety

of brands than the optimal variety level 16

.

With respet to the previous literature, we have to aount for the additional ost

involvedinbearingrisksinethe rmsarerisk-averse. Fromthedenitionof

V _i

^suhthat

V _i = E( ˜ Π _i ) − ^a ₂ V ar( ˜ Π _i ) − f

^{,wenote thatthe term}

^a ₂ V ar( ˜ Π _i )

^{indiatestherisksupported}

14

Whenthedegreeofriskaversion

a

^{isset to0(or}

σ ² = 0

^{),wendthattheoptimalnumberofrms}

is

n ^∗ = q

t∆L ² /f

^{,whihistheoriginalresultofSalop(1979).}

15

Also,weobservethat aninreasein thexedostvalueausesadereasein thenumberofrmsin

themarketandthatariseinthetransportationostleadstoaninreaseintheprotmarginsinethere

is ahigherprobabilityofdierentiationforrms.

16

Under ertainty, itis wellknown that privateand soialinentivesdonot neessarilyoinide and

themarketisexpetedtogeneratetoomanyrms(see Tirole,1988).

by eah rm given the randomness of

Π ˜ _i

^{. Using the denition of the prot level}

Π ˜ _i

^{, we}

dedue that

V ar( ˜ Π i ) = ∆ ² L ² σ ² /n ²

^{. Thus, the ost of risk bearing by a rm denoted by}

B i

^{isgiven by :}

B i = a 2

∆L n

²

σ ²

⁽²¹⁾

We note that this ost inreases with the absolute degree of risk aversion

a

^{, with the}

demand intensity

L

^{and with the variane of the marginal ost}

σ ²

^. Conversely, risk bearing osts are a dereasing funtion of the number of rms

n

^{. The aggregate ost of}

risk bearingis simply

nB i

^.

In the spatial modelof Salop (1979),the aggregate transportationost

T

^is:

T = 2nt

Z _L/ 2 n

0 ∆xdx

⁽²²⁾

sine all onsumers purhasing the brand from a rm are loated between 0 and

L/2n

units of distane from that rm. So, the average onsumer has to travel

L/4n

^{units of}

distane, whih leads tothe following aggregate transportationost :

T = t∆L ²

4n

⁽²³⁾

Now, the problemforthe soialplanneristominimize thesum of xed ostspaid by

the produing rms, aggregate transportation osts and aggregate osts of risk bearing.

The soial aggregate ost

S

^{is then equal to}

S = nf + T + nB i

^{. Formally, the problem}

for the soialplannermay beexpressed as:

min n nf + t∆L ² 4n + a

2 (∆L) ²

n σ ²

⁽²⁴⁾

Proposition 5 Under ost unertainty, the optimal number of rms

n ˆ

^{hosen by an}

omnisient planner is:

ˆ n =

s L ² f

t∆

4 + a 2 σ ² ∆ ²

(25)

Proof. Sine the problem for the soial planner is

min n S

^{, we solve the} orresponding rst-order ondition

∂S/∂n = 0

^{and obtain :}

f − 1 ˆ n ²

t∆L ² 4 + a

2 σ ² (∆L) ²

!

= 0

Corollary 2 The market generates too many rms at the equilibrium, i.e.

n < n ˆ ^∗

^.

When omparingthe numberof rms hosen by the soialplanner and the deentralized

equilibrium, itfollows that :

ˆ

n < n ^∗ =

s L ² f

t∆ + a 2 σ ² ∆ ²

(26)

So, inthefree-entryloationmodel, wenotethatthe marketgeneratestoomanyrmsat

the equilibrium. Clearly,too many brandsare produedsine rms have too muhof an

inentive toenter. Of ourse,suh aresult alsoholds inthe modelof Salop (1979)under

ertainty. But with respet tospatialdierentiationunder ertainty, weobserve thatthe

soialplannerhoosesahighernumberofrmsinordertoahieveanoptimalrisk-sharing

among rms. Inreasing the numberof rmsinthe markets leadstoanimpliithedging.

Finally, whenthe transportationost is very low, we nd that

n ^∗

^is approximatelyequal to

n ˆ

^{. In that ase, the number of rms only depends on osts involved in bearing risk,}

and this fator whihis equalto

a

2 σ ² ∆ ²

^{is idential in}

n ^∗

^and

n ˆ

^17.

Sineentryofrmsissoiallyjustiedbythesavingsintransportationostsand osts

of risk bearing, we suggest that there are some poliy solutions for the soialplanner in

order to redue the exessive entry of rms in the market. In partiular, any poliy

designed toderease the levelofrisk inindustries maybeaneetiveway toregulatethe

market. Resouresdevotedtothepoolingofindustrialrisksshouldsigniantlyontribute

to the deline of pries harged by the rms, by lessening the prodution risk premium

supported by onsumers whenbuying the goodsgiven spatial dierentiation.

6 Conluding omments

In this paper, we have analyzed a loation model to examine the eets of unertainty

in an industry equilibrium. We extend the model of spatial dierentiation proposed

by Salop (1979) by introduing marginal ost unertainty and examine the free-entry

equilibrium. Aounting for horizontalprodut dierentiation strongly aets the eets

ofunertainty onthenumberofrmsinanindustry,whihisindeterminateinastandard

17

When

t → 0

^,weget

n ^∗ = ˆ n = q _a

2 σ ² ∆ ² L ²

f

^.

Our analysis is a ontribution to the reent developments on the theory of oligopolisti

rms underunertainty with dierentiated produtspresented inAsplund (2002).

In our setting, the optimal prie harged to onsumers inludes an additional term

orresponding to a measure of the risk premium faed by risk-averse rms, so that the

ost of unertainty is supported by onsumers with dierentiation. As a onsequene,

when there are no barriers to entry other than xed osts, rms have greater prots

opportunities and then inentives to enter the market are inreased. Finally, omparing

the numberof goods inamarketeonomy and a soialeonomy indiatesthattoo many

brandsare produedinafree-entryloationmodel,ost unertaintyhavinganadditional

positiveimpat onthe distortion.

A nal omment deals with empirialtesting. Our framework suggests a positive re-

lationship between ost unertainty and entry of rms in industries with dierentiated

produts. However, evideneonthe eets of unertaintyonthe industry equilibriumre-

mains sare. Usinga ross-setionofAmerianmanufaturingindustries, Ghosal(1996)

nds thatgreaterprieunertaintyhasasigniantandlargenegativeeetonthenum-

berof rmsinanindustry. Fousingonthe intertemporaldynamisofindustry struture

again for manufaturing rms in the United States, Ghosal (2002) shows that greater

unertainty does not aet large establishments, while it has a negative impat on the

numberof smallrms in anindustry (see alsoGhosal and Loungani,2000).

Nevertheless, this observed negative relationship between unertainty and industry

equilibrium should not neessarily be interpreted against our model of spatial ompeti-

tion. Forinstane, Ghosal (1996)onlyinludesaprie unertainty measure anddoesnot

aount for ost unertainty. Asplund (2002) learly shows that dierenttypes of uner-

tainty mayhave opposite eets on ompetitionfor risk-averse rms inoligopolies. Also,

the issue of dierentiated produts isnot speially addressed in the previous empirial

literature. Thus, itwouldbeusefultoinvestigatetheeetsofunertaintyonthenumber

of rms for markets with dierentiated produts and signiant ost unertainty. Suh

markets ould be identied with unertainty measures based on the standard deviations

of residuals in prie equations for most important inputs. This empirial issue, whih

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