research article sequential divestiture and firm...

Hindawi Publishing CorporationEconomics Research InternationalVolume 2013 Article ID 352847 11 pageshttpdxdoiorg1011552013352847

Research ArticleSequential Divestiture and Firm Asymmetry

Wen Zhou

School of Business The University of Hong Kong Hong Kong

Correspondence should be addressed to Wen Zhou wzhoubusinesshkuhk

Received 13 November 2012 Revised 22 January 2013 Accepted 23 January 2013

Academic Editor Jean Paul Chavas

Copyright copy 2013 Wen Zhou This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Simple Cournot models of divestiture tend to generate incentives to divest which are too strong predicting that firms will break upinto an infinite number of divisions resulting in perfect competitionThis paper shows that if the order of divestitures is endogenizedfirms will always choose sequential and hence very limited divestitures Divestitures favor the larger firm and the follower ina sequential game Divestitures in which the larger firm is the follower generate greater industry profit and social welfare but asmaller consumer surplus

1 Introduction

Firms often spin off divisions that compete directly with theparent business For example Siemens planned in early 2009to divest its 34 stake in AREVA NP a Franco-German jointventure in nuclear reactors and develop its own nuclear capa-bilities (The Economist 12909) Fast-food chain Wendyrsquosspun off fast-growing Tim Hortons in 2006 after realizingthat ldquoTimrsquos was beginning to compete directly with Wendyrsquosrdquo(The Economist 92306) Note that such spinoffs promiseto increase competition rather than reduce it On August 32010 FordMotor Company completed the sale of Volvo Carsto Chinese carmaker Geely for $18 billion Ford CEO AlanMulally said that the divestiture ldquowill allow us to sharpen ourfocus on the Ford brand around the world and continue todeliver on our One Ford plan serving our customers with thevery best cars and trucks in the worldrdquo (AutoWeek 8310)Geelyrsquos president Li Shufu remarked that Volvo would nowhave the freedom to ldquoenter market segments that were previ-ously closed to it because they were occupied bymodels fromJaguar Land Rover or Ford itself rdquo (The Economist 33110)

Separated from Ford Volvo will surely compete with itsold stablemate in the premium car market Obviously Fordhas created a competitor through the divestiture and onewonders why it wanted to do so Would not it have beenbetter for Ford to keep Volvo under its roof and thus containthe competition between the two brands1 As these examplesshow divestitures are very common in the business worldandmany divestitures create direct competitors for the parent

business2 This seems contrary to the common understand-ing that a company should seek to minimize competitionWhy would a company ever want to spin off a businesswhich will compete directly with itself The answer lies inthe responses from competitors Although the newly createdcompetition will eat into the parent companyrsquos business italso applies pressure on rival companies If the separationsucceeds in squeezing existing rivalsrsquo market shares the jointprofit of the parent and the divested unit may increase andthe parent gains by extracting surplus through the unitrsquos saleprice In other words a company uses internal competitionto gain competitive advantage over its external rivals It istherefore not surprising that firms have an incentive to divestWhat is surprising is that the incentive can be very strongStandard Cournot models would predict that when firmsproduce homogeneous products at constant marginal costand divestitures cost nothing firms will divest into an infinitenumber of offsprings leading to perfect competition [1]

This seemingly unreasonable prediction that divestitureshould always be profitable can be called the divestitureparadox Some solutions to the paradox have been offereddivestitures will be limited if they are costly [2] or productsare differentiated [3 4] Those solutions assume that firmsdivest simultaneously In real life however divestitures taketime to formulate and the decisions are often reported longbefore completion It is therefore more appropriate to modeldivestitures as sequential choices rather than simultaneousones At the very least the order of divesting should be endog-enously determined

2 Economics Research International

This study investigated the incentive to divest when theorder of divestiture is decided endogenously In an industrywith two firms producing homogeneous products firms werefound always to divest sequentially and doing so greatlylimits the number of divisions The divestiture paradox cantherefore be solved by endogenizing the order of divestituresSimultaneous divestiture is no longer an equilibrium becausea firm is always better off playing a sequential game either as aleader or as a follower A leader can always choose its equilib-rium strategy in the simultaneous game and achieve its payoffthere so it must be no worse off when it chooses differently inthe sequential game Because divestitures are strategic com-plements and the leader will be hurt by the followerrsquos divesti-ture the leader limits its own divestiture in order to constrainthe followerrsquos As a result a follower is also better off

This research reflects the simple idea that the orderof business choices should be endogenized whenever it isappropriate especially when the results depend greatly onthe order Some choices such as prices or output levelsmay be regarded as being made simultaneously becausethey can easily be changed or are unlikely to be known torivals beforehand but divestitures take time and are usuallyreported by themedia Endogenizing the order of divestituresis particularly appropriate and easy because both firms benefitbymoving from a simultaneous game to a sequential one andsuch a move does not require any coordination Even if orig-inally the decisions were supposed to be simultaneous a firmcan easily turn them into a sequential one by say committingto a choice before other firms have made their decisions

So the major novelty of this study is the endogenizationof the order of divestitures Another innovation is its way ofmodeling divestitures Capital was assumed to be requiredin the production which implies increasing marginal costs(in the short run) Compared with the commonly usedformulation assuming constant marginal costs this coststructure leads to a more reasonable modeling of divestiturea divestiture decomposes the parentrsquos capital so that eachoffspring is smaller than the parent It also admits constantmarginal cost as a special case where endogenizing the orderof divestitures generates the most striking difference Withconstant marginal costs simultaneous divestitures induceeach firm to divest into an infinite number of divisions lead-ing to perfect competition When the order is endogenizedby contrast the leader will not divest at all while the followerdivests into only two divisions

The presence of capital also enables the study of firmasymmetry Divestitures can be shown to favor the larger firmand the follower Divestitures increase social welfare at theexpense of the industryrsquos total profitThe leader is always hurtbut the followermay benefit if it is sufficiently large A sequen-tial divestiture in which the larger firm is the follower gener-ates greater industry profit and social welfare but a smallerconsumer surplus than the alternative sequence This isbecause the smaller firm tends to divest more than the largerone and the follower tends to divest more than the leader Ifthe larger firm is the follower the industryrsquos overall divestitureis more limited and the resulting divisions are more balanced

in size The first effect helps industry profit and hurts con-sumer surplus while the second effect improves productionefficiency and hence helps industry profit and social welfare

There is a small body of literature on divestiture (alsoknown as strategic divisionalization) all assuming symmetricfirms constant marginal costs and simultaneous divestitureCorchon [1] and Polasky [5] have demonstrated how a firmmay gain competitive advantage by breaking itself up intoautonomous units3 and found that the incentive to divest canbe too strong to the detriment of the divesting firms Baye etal [2] have suggested that divestitures will be limited if theyare costly All three of these studies assumed homogeneousproducts By contrast Ziss [3] and Yuan [4] showed thatthe incentive to divest can also be reduced by productdifferentiation that is the number of divisions decreaseswhen products are more differentiated4 As a result thedivestiture paradox can be solved by introducing productdifferentiation each firm will divest into a finite number ofdivisions when products are sufficiently differentiated andperfect competition will not result5

This paper assumes homogeneous products with increas-ing marginal costs Such a setting is mathematically equiv-alent to assuming differentiated products with constantmarginal costs [6] It is therefore no wonder that bothapproaches can solve the divestiture paradox Neverthelessthere are some subtle differences between the two formu-lations as will be explained More importantly this paperrsquosconclusions are based on the endogenized order of divesti-tures while Ziss [3] and Yuan [4] both assumed simultaneousdivestitures

The incentive to divest echoes that for creating com-peting divisions within a company6 Creane and David-son [7] have provided many examples of multidivisionalfirms that encourage internal competition especially in thehotel brewery fast-food automobile and tobacco industriesThey pointed out (p 954) that ldquomany firms offer multipleproducts that appear to be either identical or extremelyclose substitutes for one anotherrdquo Conlin [8] found evidenceof strong competition between different brands within ahotel chain Kalnins and Lafontaine [9] discovered empiricalregularities in the ownership of newly opened franchisedunits in the hotel industry and speculated that the regularitiesare best explained by an optimal balance between internal andexternal competition

The divestiture paradoxmirrors the better-knownmergerparadox which says that firms tend to have too weak anincentive to merge [10] The two paradoxes are actuallytwo sides of the same coinmdashboth are caused by a toostrong response from competitors7 This is hardly surprisingas divestitures are simply reverse mergers Endogenizingmerger decisions however may or may not solve the mergerparadox8 A recent study by Qiu and Zhou [11] treateddivestitures and mergers in a single model which solvesboth paradoxes simultaneously as the interaction betweenthe two restructuring activities weakens the incentive todivest and strengthens the incentive to merge Reflecting ina merger context the logic of using internal competition togain competitive advantage Mialon [12] demonstrated that

Economics Research International 3

two merging firms may choose to remain as independentand competing divisions after proper reallocation of capitalbetween them9

A number of studies have addressed the order of movesin two-player games Dowrick [13] endogenized the order byallowing duopolists to simultaneously choose a role as eitherthe leader or the follower before they engage in competitionin the product market Hamilton and Slutsky [14] havesuggested that two perfectly symmetric firms may chooseto play a sequential game and the order of moves can thenbe endogenized through either observable delay or actioncommitment Finally Henkel [15] has suggested that a playermay choose a role somewhere between a leader and a followerbymaking a commitment to a role which can be revoked laterat some cost

2 The Model

Assume that two firms indexed by 119909 and 119910 produce ahomogenous good The production cost of firm 119894 (119894 isin 119909 119910)is assumed to be 119862 (119905

119894 119902119894) = 119902

2

1198942119905119894 where 119905

119894is 119894rsquos capital

stock and 119902119894is its output This cost function has been used

previously in merger studies [16ndash18] and can be viewed asa short-run cost derived from a Cobb-Douglas productionfunction10

The two firms play a three-stage game In stage one(the timing stage) the two firms determine the sequence oftheir future divestitures by simultaneously choosing a role119903119894from 119871 119865 where 119871 means leader and 119865 means follower

If 119903119909

= 119903119910the two firms will divest sequentially in stage two

with the one choosing 119871 divesting first and the other divestingafterward If 119903

119909= 119903119910 they will divest simultaneously In stage

two (the divestiture stage) the two firms divest according tothe sequence determined in the previous stage11 To divest isto break up a firmrsquos capital into several smaller units termeddivisions which will compete independently in the productmarket Each parent firmrsquos payoff will be the sum of all itsdivisionsrsquo future profits Since it is optimal for a parent todistribute its capital equally among its divisions12 the divesti-ture decision comes down to a single choice of the number ofdivisions which for simplicity will be treated as a continuousvariable In stage three (the competition stage) all the divi-sions fromboth firms compete independently and simultane-ously a la CournotThe product demand is 119901 = 119886minus119887119876 where119886 119887 gt 0 and 119876 is the total output of all competing divisions

3 Analysis

31 Stage Three Cournot Competition The appendix showshow to derive the Cournot outcome Denote the set of allcompeting divisions by 119863 For division 119896 isin 119863 let 119892

119896equiv

119887119905119896(1 + 119887119905

119896) and 119866 equiv sum

119889isin119863119892119889 Division 119896rsquos Cournot profit

is then 120587119896= (1198862119892119896(1 + 119892

119896))(2119887(1 + 119866)

2) Without loss of

generality13 normalize 119886 equiv radic2 and 119887 equiv 1 Then

120587119896=

119892119896(1 + 119892

119896)

(1 + 119866)2

(1)

where

119892119896=

119905119896

1 + 119905119896

119866 = sum

119889isin119863

119892119889 (2)

with 119902119896= (119886119887) (119892

119896(1 + 119866)) 119876 = (119886119887) (119866(1 + 119866)) and

119901 = 119886(1 + 119866) Division 119896rsquos marginal cost at equilibrium istherefore 119898119888

119896= 119902119896119905119896= (119886(119887(1 + 119866))) (1(1 + 119905

119896)) and its

market share is 119904119896equiv 119902119896119876 = 119892

119896119866

32 Stage Two Divestitures Let 119894 isin 119909 119910 and 119895 = 119909 119910 119894 Iffirm 119894 chooses to form 119899

119894divisions each divisionrsquos capital will

be 119905119894119899119894 Let 119892

119894= (119905119894119899119894)(1 + (119905

119894119899119894)) = 119905

119894(119899119894+ 119905119894) Firm 119894rsquos

payoff is then

120587119894(119899119894 119899119895) =

119899119894119892119894(1 + 119892

119894)

(1 + 119899119894119892119894+ 119899119895119892119895)

2 (3)

Given 119899119895 119894rsquos optimal choice is derived as 119899

lowast

119894equiv

argmax119899119894gt0

120587119894(119899119894 119899119895) = 1 + 119899

119895119892119895 Define a firmrsquos best

response to its rivalrsquos choice of 119899119895as

119899lowast(119899119895) equiv 1 + 119899

119895119892119895= 1 +

119899119895119905119895

119899119895+ 119905119895

(4)

If the two firms divest simultaneously the equilibrium isdetermined by solving the two best response equations 119899119872

119894=

119899lowast(119899119872

119895) and 119899119872

119895= 119899lowast(119899119872

119894) where the superscript119872 indicates

simultaneous divestiture As a result

119899119872

119894=

1 + 2119905119895+ radic(1 + 2119905

119894) (1 + 2119905

119895) (1 + 2119905

119894+ 2119905119895)

2 (1 + 119905119894+ 119905119895)

(5)

If the two firms divest sequentially the follower will adoptthe best response 119899119865 = 119899

lowast(119899119871) Anticipating this the leaderrsquos

optimal choice is 119899119871 = argmax119899119894gt0

120587119894(119899119894 119899lowast(119899119894)) which has a

unique solution on (0infin) defined by

120573 (119899119871) = 0 (6)

where

120573 (119899119871) equiv minus [(1 + 119905

119871+ 119905119865)2+ 1199052

119865] (119899119871)

3

+ [(1 + 119905119865) (1 + 2119905

119865minus 1199052

119871) minus 4119905

1198711199052

119865] (119899119871)

2

+ 119905119871[(1 + 2119905

119865) (2 + 2119905

119865+ 119905119871) minus 21199051198711199052

119865] 119899119871

+ 1199052

119871(1 + 119905119865) (1 + 2119905

119865)

(7)

33 Stage One The Order of Moving Denote firm 119894rsquos equilib-rium choice of the number of divisions by 119899

119871

119894when 119894 is the

leader in a sequential divestiture (and accordingly 119895 is thefollower) by 119899119865

119894when 119894 is the follower (and 119895 is the leader) and

by 119899119872119894when 119894 and 119895 divest simultaneouslyThe corresponding

equilibrium payoff for 119894 is denoted by 120587119877119894 where 119877 = 119871 119865119872

The appendix proves the following ranking which is centralto the major conclusions of this analysis


Lemma 1 119899119871119894lt 119899119865

119894lt 119899119872

119894and 120587

119865

119894gt 120587119871

119894gt 120587119872

119894

Because 120587119865119894gt 120587119871

119894gt 120587119872

119894and 120587

119865

119895gt 120587119871

119895gt 120587119872

119895 it is never an

equilibrium for the two firms to choose 119903119894= 119903119895in stage one

and it is always an equilibrium for 119903119894= 119871 and 119903

119895= 119865 or for

119903119894= 119865 and 119903

119895= 119871 Therefore consider the following

Proposition 2 (i) Simultaneous divestiture is never an equi-librium

(ii) Sequential divestiture (with either firm serving as theleader) is always an equilibrium

(iii) In equilibrium 119899119894is finite and 120587

119894gt 0 even though the

products are homogeneous and divestiture is costless

Propositions 2(i) and (ii) are corollaries of Lemma 1Proposition 2(iii) is true because 119899119871

119894lt 119899119865

119894lt 119899119872

119894(Lemma 1)

and 119899119872

119894is finite

4 Discussion

Proposition 2 delivers the key message of the analysis whichis that divestitures will be limited in scope if firms can choosethe order of their divestitures The following discussionpresents the intuition suggesting this result the uniquefeatures of the model and the properties of the equilibriumTo understand the discussion it is useful to first look at theincentives for and effects of a unilateral divestiture

41 The Incentive to Divest In what follows the numberof divisions will be called the scope of divestiture or sim-ply divestiture Increasing a firmrsquos divestiture generates thefollowing tradeoff14 it increases the competition betweenthe firmrsquos own divisions hurting the firm but the increasedcompetition also forces rival firms to retreat benefitingthe firm When divisions from the same parent competeindependently they create a negative externality for oneanother From the viewpoint of the parent firm fixing itsCournot rivalsrsquo outputs these divisions produce too muchRivalsrsquo outputs however will not be fixed Precisely becausethe divested divisions overproduce rivals will produce less (asoutputs are strategic substitutes) which benefits the divestingfirm The tradeoff between the two effects determines theoptimal scope of a divestiture

Now consider the effects of increasing one firmrsquos divesti-ture while holding the otherrsquos constant Firm 119894rsquos profit hasbeen given by (3) 120587

119894= (119899119894119892119894(1 + 119892

119894))(1 + 119899

119894119892119894+ 119899119895119892119895)2 The

joint output of all its divisions is 119902119894= (119886119887) (119899

119894119892119894(1 + 119899

119894119892119894+

119899119895119892119895)) and the total market share of those divisions which

for simplicity is called 119894rsquos market share is

119904119894equiv

119902119894

119902119894+ 119902119895

=

119899119894119892119894

119899119894119892119894+ 119899119895119892119895

(8)

All these variables depend on the comparison between 119899119894119892119894

and 119899119895119892119895 Let us call 119899

119894119892119894equiv 119899119894119905119894(119899119894+ 119905119894) firm 119894rsquos dispersion

which obviously increaseswith the scope of its divestiture Fora fixed 119899

119895 therefore increasing 119899

119894increases 119894rsquos dispersion and

thus the industryrsquos aggregate dispersion while 119895rsquos dispersionremains unchanged As a result increasing a firmrsquos divestiture

will expand the total output and market share of its divisionsand depress the joint output market share and profit of itsrivalrsquos divisions

Direct observation of the best response (4) reveals thefollowing properties

120597119899lowast

120597119905119895

gt 0

120597119899lowast

120597119899119895

gt 0 (9)

That is a firm will divest more if its rival is larger or divestsmore The second property indicates that divestitures arestrategic complements Both properties arise for the samereason When the rival increases its capital or divestiture itsdispersion increases expanding its market share and reduc-ing the market share of the firm contemplating divestitureAs has been discussed earlier the optimal divestiture isdetermined by the tradeoff between the competition amongthe divesting firmrsquos own divisions and the retreat of the rivalfirmrsquos divisions When the rivalrsquos market share is larger theadvantage of divestiture increases because there are nowmore rival divisions to respond to the divestiture while thedisadvantage of the divestiture will be smaller because thedivesting firmrsquos divisions are not earning much anyway Asa result the firm will divest more15

42 Endogenized Order and the Scope of Divestiture Theprevious discussion concerns the incentives for and effectsof a unilateral divestiture When both firms divest the equi-librium is governed by two forces a firmrsquos divestiture hurtsrival firms and divestitures are strategic complements Thetwo forces imply that the two firms overdivest to each otherrsquosdetriment In fact the incentive to divest can be so strong that(assuming a homogeneous product linear demand constantmarginal costs and costless divestitures) firmswill divest intoan infinite number of divisions even when the industry hasonly two firms leading to perfect competition [1]

This seemingly unreasonable prediction that breaking upa firm is always profitable is the ldquodivestiture paradoxrdquo Boththis and the merger paradox are driven by the same forcerival firms are too responsive to other firmsrsquo restructuringbe it a divestiture or a merger One solution to the divestitureparadox is to make divestitures costly [2] A second solutionis to let firms produce differentiated products so that theresponse from rival products is weakened [3 4]

Surprisingly all previous studies have assumed that firmsdivest simultaneously According to Lemma 1 however bothfirms will be better off playing a sequential game rather thanthe simultaneous one The reason why sequential moves arebetter for both players can be understood from the two forcesat workThe leader in a sequential game will choose a smallerscope of divestiture than in the simultaneous game becausedoing so will constrain the followerrsquos divestiture and henceits damage to the leader Since the followerrsquos payoff increaseswhen its rival divests less the follower is better off Theleader must also be better off because it can guarantee itselfthe simultaneous-move payoff by choosing its simultaneous-game strategy in the sequential game and can only do betterif it chooses differently


Once it has been established that both firms are better offin a sequential game it is hard to see why they would playthe simultaneous one which gives them an inferior outcomeScholars have never explained why they have assumed firmsdivest simultaneously but as has been shown once the orderof moves is endogenized divestitures will be carried outsequentially In real life the order of business decisions isusually determined endogenously Endogenization is partic-ularly easy in the case of divestiture because both firms benefitby moving from the simultaneous game to a sequentialgame and the move does not require any coordinationwhich is not the case in many other situations (such as theprisonersrsquo dilemma) where the players are trapped in aninferior outcome

To summarize even if products are homogeneous anddivestitures are costless firms will not divest into an infinitenumber of divisions and therefore will not face perfectcompetition Proposition 2 demonstrates that after all firmswill find a way to constrain mutually damaging divestituresThe divestiture paradox disappears as soon as the order ofdivestitures is endogenized

43 Cost Structure Having established the major conclusionof the paper we are now ready to discuss the features of themodel and the properties of the equilibrium In addition to anendogenized order of divestiture this model has assumed theinvolvement of capital which leads to a marginal cost (mc)that increases with output By contrast all previous modelshave assumed constant mc Increasing marginal cost yieldsimplications for divestiture which are more reasonable Ifmarginal costs are constant each extra division will be anexact replica of the parent firm to divest amounts to createsomething out of nothing Divestitures modelled in this waycontradict the general understanding that a division shouldbe somewhat smaller than the parent firm By contrast whenmarginal costs are increasing each division gets a portion ofthe parentrsquos capital and is therefore smaller (in the sense ofhaving higher production costs) than the parent16

Note that modeling divestitures when breaking up a firmis reasonable and meaningful only when mc is increasingTo understand this it is helpful to consider divestitures asreverse mergers In merger the merged entity takes overthe production facilities of the merging firms and optimizesits production using those facilities It can be easily shownthat such within-firm optimization will give rise to a costfunction of the form 119902

2(2(119905119894+119905119895)) for themerged firm where

119905119894and 119905

119895are the capital stocks of the two merging firms

Because themerged firmrsquos cost function is that of a single firmwith capital 119905

119894+ 119905119895 we may regard the merger as a process

of pooling the merging firmsrsquo capital As a reverse mergerthen a divestiture can be viewed as decomposing the parentrsquoscapitalThat explains themodeling of divestiture in this studyand why a divisionrsquos cost is higher than the parentrsquos

To summarize the cost structure assumed in this studydiffers from that assumed in previous studies in two respectsmarginal cost is increasing rather than being constant anda division is smaller than the parent The second aspect isto generate a reasonable modeling of divestitures To justifysuch modeling however we need to introduce capital stock

which naturally gives rise to the first aspect of increasingmcSo increasingmc is a necessary element of themodel to justifywhy and how a division is smaller than the parent and istherefore essential for modeling divestitures

Once capital is introduced into the model the formula-tion with constant mc becomes a special case where both 119905

119894

and 119905119895approach infinity so that the two firmsrsquo marginal costs

approach zero In that case a division is indeed identical tothe parent

At this point it is helpful to discuss the relationshipbetween increasing marginal cost and product differentia-tion As Vives [6] has pointed out assuming homogeneousproducts with increasing mc is mathematically equivalent toassuming differentiated products with constantmc Econom-ically the two situations produce the same effect of weakeningthe interaction between Cournot competitors That is whyeither setting can solve the divestiture paradox even withconstant mc the scope of divestitures will be limited ifproducts are differentiated [3 4] Although mathematicallyequivalent assuming increasing mc has the advantage ofavoiding the complication of dealing with different degrees ofsubstitution for products within and across groups17 Finallyincreasingmc and product differentiation can be combined inthemodeling but that would not change any of the qualitativeresults18

44 The Role of Capital This model differs from thosepreviously proposed both in its cost structure and in itsendogenized order of moves It must be pointed out thatthe assumption of increasing mc is not driving the resultsof Proposition 2 Even when marginal costs are constantthe scope of divestiture is still limited as long as the orderof divestiture is determined endogenously Inspecting thesimultaneous and sequential divestiture choices depicted in(5) and (6) we have the following lemma

Lemma 3 If 119905119894rarr infin and 119905

119895rarr infin then 119899

119871= 1 and 119899119865 = 2

but 119899119872119894

= infin

The lemma says that even when marginal costs areconstant divestiture is limited if the firms divest sequentiallyIn particular the leader will not divest at all while the followerdivests into two divisions This is in sharp contrast with theresult when firms have to divest simultaneously In that caseeach firm will divest into an infinite number of divisionsleading to perfect competition as previous studies haveconcluded [1]The reason for such contrast is that as has beenexplained in Section 42 when divestitures are sequentialthe leader has a chance to influence the followerrsquos choiceand it will limit its own divestiture in order to constrain thefollowerrsquos Such reasoning is independent of the cost structureand is therefore still valid when marginal costs are constantProposition 2 still holds so the equilibriumwill be sequentialdivestitures of limited scope

To further investigate the difference between sequen-tial and simultaneous divestitures and how the differencedepends on capital consider the overall degree of divestiturein the industry and the two firmsrsquo profits Because thefirms are asymmetric in terms of both their capital stocks


and the order of divestiture adding up the number of thetwo firmsrsquo divisions is meaningless The scope of divestitureindustrywide is better measured by the Cournot equilibriumprice119901 = 119886(1+119866) To visualize the effects assume symmetricfirms (119905

119894= 119905119895

= 119905) and use the no-divestiture situation(119899119894= 119899119895= 1) as the benchmark19 Superscripts 119876 119872 and 0

indicate sequential simultaneous and no divestiture Figure 1shows the price and profits as functions of 119905 Since these twovariables depend on 119905 even without divestiture the effect ofdivestiture is captured in the price ratios (the left panel ofFigure 1) and profit ratios (the right panel)

A firmrsquos capital stock represents its size more specificallyits capacity relative to the demand Capital matters fordivestiture because it determines how responsive a firm is toother firmsrsquo output changes When both firms are very smalleach operates with a very steep marginal cost curve Sincethere is little interaction between the two firms there is noneed to divest When 119905 rarr 0 119899119871 = 119899

119865= 119899119872

= 1 so theorder of divestiture does not matter As both firmsrsquo capitalincreases the interaction between them becomes strongerand so does the incentive to divest Equilibrium divestitureincreases regardless of the order of moves as 119901

1198761199010 and

1199011198721199010 both decline with 119905 However the overall divestiture is

always larger with simultaneous moves than with sequentialmoves (119901119872 lt 119901

119876) and capital matters more in simultaneousdivestiture than in sequential divestiture At the limit when 119905

approaches infinity 1199011198721199010 = 0 but 1199011198761199010 = 34 meaningthat simultaneous divestiture leads to perfect competitionbut sequential divestiture will lead to competition that isgreatly constrained (the market price is 34 of the level whenneither firm divests) In terms of profits whenmc is constant(119905 is infinity) simultaneous divestiture leads to zero profit forboth firms but sequential divestiture gives substantial profitsto both firms with the follower earning twice as much as theleader 120587119871

1198941205870

119894= 916 while 120587

119865

1198941205870

119894= 98 Note that the

followerrsquos profit is larger than without divestiture (1205871198651198941205870

119894gt 1)

when 119905 is sufficiently largeTo summarize for any given stock of capital there is a dif-

ference between simultaneous moves and sequential movesand the difference increases with the amount of capitalbecoming most dramatic as capital approaches infinity thatis when marginal costs are constant

45 Firm Asymmetry All previous models have assumedsymmetric firms Because the major innovation of this studyis to endogenize the order of moving it has for simplicityfocused on an industry with only two firms This simpli-fication allows the firms to be asymmetric which offersan opportunity to study how divestitures change the firmsrsquorelative strengths and market shares Since the equilibriumchoice depicted in (6) does not admit any closed-form solu-tion numerical calculation is sometimes required Doing soinvolves little loss of rigor as there are only two parameters (119905

119871

and 119905119865) and all the results can be inspected using 3D graphs20

Proposition 4 In equilibrium (ie when firms divest sequen-tially) (i) 119904119871

119894lt 119904119865

119894and (ii) 120587119871

119894+ 120587119865

119895gt 120587119871

119895+ 120587119865

119894if and only if

119905119894lt 119905119895

Proposition 4(i) says that a firmrsquos market share as thefollower is always greater than its share as the leader Thisis consistent with Lemma 1 which says that a firmrsquos profit asthe follower is greater than its profit as the leader and for thesame reasonThe leader has an extra incentive to constrain itsexpansion in order to restrict the followerrsquos expansion whilethe follower does not face such a constraint

Proposition 4(ii) says that the industryrsquos total profit isgreater when the larger firm is the follower There are tworeasons for this result First the smaller firm tends to divestmore than the larger one21 and the follower tends to divestmore than the leader (Lemma 1) If the larger firm is thefollower the industrywide divestiture is more limited (seeProposition 6) The industryrsquos output will be smaller and theindustryrsquos total profit will be greater Second even with thesame industry output production cost is lower when thelarger firm is the follower Recall that a firm divests morewhen it is the follower (Lemma 1) so when the larger firmis the follower the divided capital will be more balancedtherefore the industryrsquos total production cost will be lower

As Proposition 2 shows both sequences in sequentialdivestitures constitute equilibria To arrive at a unique equi-librium suppose that the two firms bid to be the followerin some formal gamemdasha kind of war of attrition In thatcase the equilibrium sequence will be the one that yieldsthe greater industry profit According to Proposition 4(ii)then the smaller firm will be the leader in such a uniqueequilibrium while the larger firm will be the follower Sincethe follower tends to divest more this would make the sizedistribution among the divisions more balanced

46 Welfare The equilibrium price for the industryrsquos homo-geneous product has been shown to be 119901 = 119886(1 + 119866)When the industryrsquos aggregate dispersion 119866 increases theprice drops so the consumer surplus increases Since 119866 equiv

119899119894119892119894+ 119899119895119892119895and firm 119894rsquos dispersion 119899

119894119892119894equiv 119899119894119905119894(119899119894+ 119905119894)

increases with 119899119894 consumer surplus is improved if either

firmrsquos divestiture increasesSocial welfare differs from consumer surplus because pro-

duction efficiency nowmatters Efficient production requiresall Cournot competitorsrsquo marginal costs to equate in equilib-rium Section 31 has shown that each division from firm 119894 hasan equilibrium marginal cost proportional to 1(1 + (119905

119894119899119894))

where 119905119894119899119894is an 119894-divisionrsquos capital level The industryrsquos

overall production efficiency will then be improved if 119905119894119899119894is

closer to 119905119895119899119895 that is if divestitures generate divisions that

are more balanced in size Conversely production efficiencyand hence social welfare may be damaged if divestituresincrease the size discrepancy between the two firmsrsquo divi-sions This would be the case if the smaller firmrsquos divesti-ture increased while the larger firmrsquos divestiture remainedconstant22 To see an example note that social welfare is

119882 equiv CS + 120587119894+ 120587119895

=

119866

1 + 119866

+ (

119866

1 + 119866

)

2

ℎ

(10)

where CS is consumer surplus and ℎ equiv sum119889isin119863

1199042

119889= (1198991198941198922

119894+

1198991198951198922

119895)1198662 is the Hirfindahl index Let 119905

119894= 10 and 119905

119895= 1


1 1

34

119905119905

119901119876

1199010

119901119872

1199010

120587119865

119894

1205870

119894

120587119871

119894

1205870

119894

120587119872

119894

1205870

119894

916

00

00

98

Pric

e rat

io

Profi

trat

io

Figure 1 The comparison between sequential and simultaneous divestitures when the two firms are symmetric

Holding 119899119894= 1 social welfare declines monotonically as 119899

119895

increases Holding 119899119895= 1 welfare increases with 119899

119894up to 119899

119894=

50 and then declines If 119899119894and 119899

119895both increase while main-

taining the balance between divisions for example by keep-ing 119905119894119899119894= 119905119895119899119895 then social welfare always increases This

discussion of welfare would apply where the number of divi-sions is chosen by a social planner or an antitrust authority

Turning to equilibrium divestitures we must first estab-lish the following result

Lemma 5 119899119871 gt 1 when 119905119865gt 0 and 119899119871 = 1 when 119905

119865= 0

The lemma says that the leader in a sequential game willalways divest to some extent (119899119871 gt 1) unless it faces nocompetition (ie the follower has zero capital) Lemmas 1and 5 then imply that 119899119872 gt 119899

119865gt 119899119871

gt 1 As a result119866 (119899119872

119894 119899119872

119895) gt 119866 (119899

119871

119894 119899119865

119895) gt 119866 (1 1) That is the consumer

surplus will be increased by sequential divestitures and willbe increased even more by simultaneous divestitures

Between the two sequences of sequential divestitures theone in which the larger firm is the follower generates thegreater social welfare (and the greater total industry profitas established earlier) but a smaller consumer surplus thanthe alternative sequence The reason is that when the largerfirm is the follower the size of the divisions is more balancedmaking production more efficient This gain in productionefficiency outweighs the loss of consumer surplus

The above results can be summarized as follows

Proposition 6 (i) Unilaterally increasing the extent of a firmrsquosdivestiture always increases the resulting consumer surplus butit may damage social welfare Increasing the extent of bothfirmsrsquo divestitures while maintaining the balance between theirdivisions always improves social welfare

(ii) CS119872 gt CS119876 gt CS0 119882119872 gt 1198820 119882119876 gt 119882

0 It mayhappen that119882119872 lt 119882

119876 if the smaller firm serves as the leader(iii) CS119876 (119871 = 119894 119865 = 119895) gt CS119876(119871 = 119895 119865 = 119894) and

119882119876(119871 = 119894 119865 = 119895) lt 119882

119876(119871 = 119895 119865 = 119894) if and only if 119905

119894gt 119905119895

5 Concluding Remarks

This study has two distinctive features first the endogenizedorder of divestitures which solves the divestiture paradoxSecond capital is introduced into the production functionso that marginal costs are increasing The second featurealthough not essential for resolving the divestiture paradoxis nonetheless desirable because it enables more reasonablemodeling of divestitures and admits the commonly assumedconstant marginal costs as a special case A firmrsquos capitalstock is also a natural way to model its size and hence firmasymmetry Combining the two features generates some newinsights For example it reveals that the leader may divestless when the follower becomes bigger and that the followermay benefit from divestituresThese conclusions are differentfrom those applicable to simultaneous divestitures where afirm always divests more when the rival is bigger and bothfirms are hurt by divestitures

The model assumes linear demand and two firms butlinearity of demand is not driving the divestiture paradoxCorchon and Gonzalez-Maestre [19] have shown that Cor-chonrsquos [1] conclusion still holds under mild conditions evenif the demand is nonlinear The insight of this paper shouldthen still apply The number of firms does not matter eitherWhen there are three or four firms the overall divestiture isstill much smaller with sequential rather than simultaneousmoves The reason is the same as in the main model becausedivestitures are strategic complements and because a firmsuffers from other firmsrsquo divestitures an early mover willreduce its own divestiture in order to constrain later moversrsquodivestitures

Appendix

Derivation of Cournot equilibrium For division 119896 isin 119863 itsprofit is 120587

119896= 119901119902119896minus119862(119905119896 119902119896) = (119886 minus 119887119876

minus119896minus 119887119902119896)119902119896minus (1199022

1198962119905119896)

in which 119876minus119896

= sum119889isin119863119896

119902119889 The first-order condition (FOC)

is 119886 minus 119887119876minus119896

minus 2119887119902119896minus (1119905

119896)119902119896

= 0 or equivalently 119887119902119896

=

119892119896(119886 minus 119887119876) Summing up the FOC for all divisions in 119863


yields 119876 = (119886119887)(119866(1 + 119866)) Plug 119876 into the FOC to obtain119902119896= (119886119887) (119892

119896(1+119866))Then120587

119896= (1198862119892119896(1+119892119896))(2119887(1+119866)

2)

Proof of Lemma 1 119899119871119894= argmax

119899119894120587119894(119899119894 119899119895)where 119899

119895= 119899lowast(119899119894)

At 119899119894= 119899119871

119894119889120587119894119889119899119894= (120597120587

119894120597119899119894)+(120597120587

119894120597119899119895) (119889119899119895119889119899119894) = 0 But

120597120587119894120597119899119895lt 0 while 119889119899

119895119889119899119894= 119899lowast1015840(sdot) gt 0 so 120597120587

119894(119899119871

119894)120597119899119894gt 0

Meanwhile 119899119872119894

= argmax119899119894120587119894(119899119894 119899119895) so 120597120587

119894(119899119872

119894)120597119899119894= 0

Since 120597120587119894(119899119871

119894)120597119899119894gt 120597120587

119894(119899119872

119894)120597119899119894and 120597

21205871198941205971198992

119894lt 0 we

conclude that 119899119871119894lt 119899119872

119894

By symmetry 119899119871119895

lt 119899119872

119895 But 119899119865

119894= 119899lowast(119899119871

119895) and 119899

119872

119894=

119899lowast(119899119872

119895) Because 119899lowast1015840(sdot) gt 0 we have 119899119865

119894lt 119899119872

119894 Now we must

show that 119899119871119894lt 119899119865

119894 Suppose it is not (ie 119899119871

119894ge 119899119865

119894) Then by

symmetry 119899119871119895ge 119899119865

119895

119899119865

119894= 119899lowast(119899119871

119895)

ge 119899lowast(119899119865

119895) (because 119899

119871

119895ge 119899119865

119895and 119899

lowast1015840(sdot) gt 0)

= argmax119899119894

120587119894(119899119894 119899119865

119895)

gt argmax119899119894

120587119894(119899119894 119899lowast(119899119894))

(because120597120587119894

120597119899119895

lt 0 119899lowast1015840(sdot) gt 0 and 119899

119865

119895= 119899lowast(119899119871

119894))

= 119899119871

119894

(A1)

which is a clear contradictionFor profits

120587119865

119894= max119899119894

120587119894(119899119894 119899119871

119895)

gt max119899119894

120587119894(119899119894 119899119865

119895)

(by the envelope theorem

120597120587119894

120597119899119895

lt0 and 119899119871119895lt119899119865

119895)

gt 120587119894(119899119871

119894 119899119865

119895)

= 120587119871

119894

= max119899119894

120587119894(119899119894 119899lowast(119899119894))

gt 120587119894(119899119872

119894 119899lowast(119899119872

119894)) (because 119899

119871

119894= 119899119872

119894)

= 120587119872

119894

(A2)

Proof of Lemma 3 119899119871 is solved from 120573(119899119871) = 0 or equiva-

lently 120573(119899119871)1199052

1198711199052

119865= 0 When 119905

119894rarr infin and 119905

119895rarr infin

120573(119899119871)1199052

1198711199052

119865= minus2119899

119871+ 2 so 119899

119871= 1 Then 119899119865 = 119899

lowast(119899119871) =

1 + (119899119871119905119871(119899119871+ 119905119871)) = 2

Proof of Proposition 4(i) Suppose that 119904119871119894gt 119904119865

119894Then by sym-

metry 119904119871119895gt 119904119865

119895 Because 119904119871

119894= 119899119871

119894119892119871

119894(119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895) we have

120587119871

119894=

119899119871

119894119892119871

119894(1 + 119892

119871

119894)

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2

= 119904119871

119894(1 + 119892

119871

119894)

119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2

(A3)

Similarly

120587119865

119895= 119904119865

119895(1 + 119892

119865

119895)

119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2 (A4)

Then120587119871119894120587119865

119895= 119904119871

119894(1+119892119871

119894)119904119865

119895(1+119892119865

119895) Similarly120587119865

119894120587119871

119895= 119904119865

119894(1+

119892119865

119894)119904119871

119895(1 + 119892

119871

119895) Now because 119904119871

119894gt 119904119865

119894and 119904119871

119895gt 119904119865

119895 we have

119904119871

119894119904119865

119895gt 119904119865

119894119904119871

119895 Because 119899119871

119894lt 119899119865

119894and 119899

119871

119895lt 119899119865

119895(Lemma 1) we

have (1+119892119871119894)(1+119892

119865

119895) = (1+(119905

119894(119899119871

119894+119905119894)))(1+(119905

119895(119899119865

119895+119905119895))) gt

(1 + (119905119894(119899119865

119894+ 119905119894)))(1 + (119905

119895(119899119871

119895+ 119905119895))) = (1 + 119892

119865

119894)(1 + 119892

119871

119895)

As a result we conclude that 120587119871119894120587119865

119895gt 120587119865

119894120587119871

119895or equivalently

120587119871

119894120587119865

119894gt 120587119865

119895120587119871

119895 By Lemma 1 120587119871

119894lt 120587119865

119894 Therefore 120587119865

119895120587119871

119895lt

120587119871

119894120587119865

119894lt 1 or 120587119865

119895lt 120587119871

119895 But this contradicts Lemma 1

Proof of Lemma 5 119899119871 is implicitly defined by 120573(119899119871) = 0

Because 120573(1) = 119905119865(1 + 2119905

119871)2ge 0 and 120573(infin) lt 0 120573(119899119871) = 0 has

at least one root on [1infin) Furthermore the root is greaterthan 1 if and only if 119905

119865gt 0 Now it must be shown that this

root is unique on (0infin)Suppose the contrary Then because 120573(119899119871) is cubic in 119899

119871

with 120573(0) gt 0 and 120573(infin) lt 0 120573(119899119871) = 0 must have threeroots on (0infin) Further it must be true that 1205731015840(0) lt 0 and1205731015840(119899119871) gt 0 at one of the roots

Now 1205731015840(0) = 119905119871(6119905119865+ 119905119871+ 2 + 2119905

119871119905119865+ 41199052

119865minus 21199051198711199052

119865) and

[1205731015840(119899119871)]119899119871minus 120573(119899

119871) = 2120573

3(119899119871)3+ 1205732(119899119871)2minus 1205730 where 120573

3=

minus[(1 + 119905119871+ 119905119865)2+ 1199052

119865] lt 0 120573

2= [(1 + 119905

119865)(1 + 2119905

119865minus 1199052

119871) minus 41199051198711199052

119865]

and 1205730= 1199052

119871(1 + 119905

119865)(1 + 2119905

119865) gt 0 It is straightforward to

verify that 1205731015840(0)119905119871gt 1205732 Then if 120573

2gt 0 it must be true that

1205731015840(0) gt 0 If120573

2lt 0 then [1205731015840(119899119871)]119899119871minus120573(119899119871) lt 0 meaning that

at any root (so 120573(119899119871) = 0) 1205731015840(119899119871) lt 0 This is a contradictionbecause either case will violate the requirement that 1205731015840(0) lt 0

and 1205731015840(119899119871) gt 0 at one of the roots

Acknowledgments

The author would like to thank Jiahua Che Keith Head IvanPng Larry Qiu Zhiyong Yao Eden Yin and participants atthe 38th Annual Conference of the European Association forResearch in Industrial Economics (EARIE) the 9th AnnualInternational Industrial Organization Conference (Boston)and seminars at the University of Hong Kong Fudan Univer-sity and Shanghai University of Economics and Finance forhelpful comments Financial support from General ResearchFund of Hong Kongrsquos Research Grants Council (Grant HKU757910H) is greatly appreciated


Endnotes

1 This paper emphasizes the competition effects of divesti-turemdashthe intensified competition between divestedunits gives them a competitive advantage over rivalfirms Real-life examples such as the Ford-Volvo divesti-ture inevitably involve multiple reasons including costeffects (diseconomies of scope) and product differentia-tion But all divestitures regardless of and on top of firm-specific and market-specific details and complicationsinvariably involve the competition effect which is thefocus of this study

2 Divestiture is generally defined as the sale of a subsidiarya division or a minority share to a new owner In thispaper divestiture means breaking a firm up into autono-mous units that compete independently in the productmarket A common explanation for divestiture is dis-economies of scope associated with managing multipleproduct lines In large organizations it becomes moreand more difficult to manage complex tasks transmitinformation and motivate employees When the costsof producing multiple products outweigh its benefitsdivestiture becomes optimal Bresnahan et al [20] haverecently suggested that diseconomies of scope may arisefrom the need for multiple products to share some com-mon assets such as a credit rating brand recognitionor the reputation of the management team Since thediseconomies of scope motivation are straightforwardand well understood this paper abstracts away from thisforce by assuming away any cost advantage associatedwith divestiture Instead the driving force is assumed toarise through competition effects

3 Lewis [21] and Schwartz andThompson [22] have shownhow setting up competing divisionsmay deter entry Tanand Yuan [23] provided an alternative theory of divesti-tures competing conglomeratesmay divest complemen-tary product lines in order to mitigate product marketcompetition

4 Put another way firms will divest more when theirproducts are closer substitutes Since firms hurt oneanother through their divestitures a stronger incentiveto divest would induce firms to seek ways to constraintheir divestitures Qiu and Zhou [11] showed that ifmergers and divestitures are both possible in an industryfirms will avoid divestitures by merging with each otherwhen their products are close substitutes In that casea moderate degree of product differentiation will beconducive to divestitures while very strong or very weakdifferentiation will be conducive to mergers

5 Product differentiation may thus explain divestituresobserved in real life such as the one cleaving Ford andVolvo The two papers also explored other issues relatedto divestitures when products are differentiated Ziss [3]showed that product differentiation will lead to a num-ber of comparative statics results that are different fromthe ones obtained in previous studies assuming homoge-neous products while Yuan [4] demonstrated that entrywill induce vigorous competition when incumbent firms

have the option of setting up autonomous divisions sodivestiture can serve as a natural barrier to entry

6 The same logic is also behind product proliferation[25] Britainrsquos Tesco stocks 91 different shampoos 93varieties of toothpaste and 115 of household cleanermany of which are produced by the same company Forexample Tropicana offers more than 20 different freshlypulped juices (ldquoThe tyranny of choicerdquo The Economist12162010)

7 Away out of themerger paradox is to alter or weaken theresponsiveness firms may compete on price rather thanquantity [26] marginal costs may be increasing [16] orproducts may be differentiated [24] Alternatively themerger paradox disappears if mergers generate substan-tial benefits such as information pooling [24] or moreefficient production in the face of cost uncertainty [27]

8 It is natural to assume that all firms can divest so theendogenization is about the order of moves A mergerby contrast is inherently exogenous if the identityof the merging firms is exogenously determined soendogenization can be in terms of many dimensionsIf a firm can choose whether to join a single merger orstay out the merger incentive will be further weakened[28] so the merger paradox cannot be solved by endo-genization If multiple mergers are allowed the mergerincentive can be strengthened [29] Qiu and Zhou haveshown that mergers are strategic complements and thatsequential mergers expand the scope of equilibriummergers This study has found that divestitures are alsostrategic complements and sequential divestitures limitthe scope of equilibrium divestitures

9 This can be seen in real life [12 p863] ldquoAfter Volvo andFord merged Fordrsquos luxury brand Jaguar continuedto compete with Volvo A similar pattern was observedwhen Daimler and Chrysler merged When KimberlyClark and Scott Paper merged Kleenex the leadingbrand of Kimberly Clark in the facial tissue marketremained in competition with Scottie the leading brandof Scott Paperrdquo Note that even though Ford and Volvomaintained some competition after their merger thecompetition was most credible and pushed to the largestextent only after Volvo had been spun off and operatedas a truly independent entity

10 Both firms possess the same constant-return-to-scaletechnology 119902 = (119905119897)

12 in which 119905 is capital and 119897 is laborIn the short run when its capital is fixed at 119905

119894 firm 119894rsquos

variable cost is 119862119894equiv min

119897119894119908119897119894subject to 119902

119894= (119905119894119897119894)12 in

which 119908 is the wage rate The optimization leads to acost of (119908119905

119894) 1199022

119894 and the formulation assumed in the

model results when the wage rate 119908 = 1211 Commitment may seem to be an issue here Suppose

that 119910 divests after 119909 does What does then prevent 119909rsquosdivisions from further divestiture after 119910rsquos divestiture Infact this is a challenge faced by all studies of divestituresregardless of the order of moves One way out is toassume costly divestitures in particular a fixed cost for


divestiture that is independent of the divesting firmrsquoscapital A divisionwill then be less likely to divest than itsparent because the division is smaller and therefore maynot generate enough profit to cover the fixed divestiturecost Note that this explanation requires divisions tobe smaller than their parents which is true only in theformulation assumed in this model Veendorp [30] hasshown in an entry-deterrence model that it is indeedoptimal for the parent firm to allow divisions freedom intheir operations but not in investment which presum-ably would include setting up their own subdivisions

12 Itmust be emphasized that equal division is optimal onlywhen the number of divisions is optimally chosen Itmayhappen that an unequal distribution between two divi-sions generates greater total profits than an equal distri-bution but the firmrsquos optimal choice is then not to divestthat is it will not have two divisions in equilibrium

13 The demand intercept 119886 enters the equilibrium profitsonly through the common coefficient 1198862 and thereforedoes not affect a firmrsquos divestiture choice as long asdivestiture is costless As for the demand slope 119887 inaddition to being a coefficient of the profit functionit also appears in the expression for 119892

119896and therefore

will affect the divestiture incentives However since119887 is always associated with 119905

119896in the same manner its

normalization is only a rescaling of the capital stockwhich will be the focus of the discussion

14 Not divesting is a choice if 119899119894= 1 Later it will be shown

that 119899119894ge 1 in the equilibrium

15 In this model divestitures are costless so a firmrsquos optimaldivestiture depends only on the two firmsrsquo relativemarket shares This leads to strategic complementarityIf divestitures were costly the strategic interdependencewould be more complicated Because a firmrsquos profits arereduced by its rivalrsquos divestiture if the rival divests morethe firmrsquos divisions may not generate enough profit tocover the divestiture cost In that case the firm mayrespond by choosing to divest less that is divestituresmay become strategic substitutes That is why Bayeet al [2] found that a firmrsquos optimal divestiture isinversely 119880-shaped in relation to its rivalsrsquo divestituresdivestitures are strategic complements when the rivaldivests little but become strategic substitutes when therival divests more extensively Note that the nature of thestrategic interdependence depends crucially on whetherdivestitures are costly In all models where divestituresare assumed to be costless [1 3ndash5] divestitures arestrategic complements

16 Although each individual division is smaller than theparent all the divisions together ldquoequalrdquo the parentBecause the parent divides its capital equally amongall its divisions a divestiture does not impose any costadvantage or disadvantage For any given total outputthe divisionsrsquo joint production cost is exactly the sameas the parentrsquos was previously Such a formulation facili-tates focusing on the competition effects of divestitures

without the complication of any cost effect due to saydiseconomies of scope

17 Both Ziss [3] and Yuan [4] assumed that products withina group are perfect substitutes Onewonderswhether thewithin-firm substitutability can take other values or evenbe endogenized and whether their conclusions still holdunder alternative formulations about substitutabilityAs is shown in the appendix firm 119896rsquos profit is 120587

119896= (119886 minus

119887119876minus119896minus119887119902119896) 119902119896minus(1199022

1198962119905119896) which can be rewritten as 120587

119896=

[119886 minus 119887119876minus119896

minus (119887 + (12119905119896)) 119902119896] 119902119896 The latter expression

can be viewed as the profit function of a firm with zeromc and differentiated products as the own-elasticitycaptured by 119887+(12119905

119896) is greater than the cross-elasticity

captured by 119887 Firm asymmetry is then reflected in dif-ferent own-elasticities for the two firms 119887+(12119905

119894) = 119887+

(12119905119895) Unlike the product differentiation formulation

the increasing mc approach does not require anyadditional ad hoc assumptions about the substitutabilityof newly created productswhenfirmsdivest By breakingup the parentrsquos capital a divestiture means that alldivisions from the same firm produce symmetricallydifferentiated products Divisions from firm 119894 all havethe same own-elasticity captured by 119887 + 1 (2119905

119894119899119894)

which is greater than the cross-elasticity captured by119887 Substitutability across firms remains asymmetric119887 + 1 (2119905

119894119899119894) = 119887 + 1 (2119905

119895119899119895) Finally demand for

a divisionrsquos product becomes more elastic than thedemand for the parentrsquos 119887 + 1 (2119905

119894119899119894) gt 119887 + (12119905

119894)

18 Firm 119896rsquos profit is 120587119896= (119886 minus 119887119876

minus119896minus 119887119902119896) 119902119896minus (1199022

1198962119905119896)

when products are homogeneous If products are differ-entiated the profit becomes 120587

119896= (119886 minus 119889119876

minus119896minus 119887119902119896)119902119896minus

(1199022

1198962119905119896) with 119887 ge 119889 gt 0 and 119911 = 119889119887 represents the

degree of product substitutability (using the notationsof Yuan [4]) This profit expression can be rewrittenas 120587119896

= [119886 minus 119889119876minus119896

minus (119887 + (12119905119896)) 119902119896] 119902119896 which is as

if marginal costs were zero (and therefore constant)while the degree of product substitutability drops to119889 (119887 + (12119905

119896)) Product differentiation and increasing

mc can therefore be regarded as two special cases ofthe more general formulation 119905

119896= infin if products are

differentiated but 119898119888 = 0 while 119889 = 119887 if products arehomogeneous but mc is increasing Given increasingmc introducing product differentiation will changeonly the degree of substitutability but the qualitativeresults of the model remain valid In real life productsare mostly differentiated (eg as are Volvo versus Fordbrands) and one can reasonably argue that marginalcosts are increasing in the relevant range Thereforeboth elements may play a role in constraining the extentof divestiture in real life In terms of modeling sincethe role of product differentiation is well understoodthrough the work of Ziss [3] and Yuan [4] this paperfocused on two new features endogenized order ofdivestitures and increasing marginal costs

19 The result will not change when 119905119894and 119905119895differ and they

change independently


20 These graphs were generated using Maple 16 softwareProposition 4(i) is proved analytically in the appendixand the validity of (ii) can be demonstrated by numericalcalculation

21 The two firms differ in both their capital stocks and theirroles in sequential divestiture To isolate the effects of thetwo asymmetries it will be useful to look at simultaneousdivestiture even though it is off the equilibrium path Itcan be shown that when the two firms divest simulta-neously (i) the larger firm divests into fewer divisionswhich would enlarge the size discrepancy between thetwo firmsrsquo divisions (ii) the smaller firmrsquos market shareis reduced and as a result firm asymmetry is amplifiedand (iii) the smaller firm is hurtmore than the larger one

22 The flip side of this result is that a merger between twosmall firms may improve social welfare [18]

References

[1] L C Corchon ldquoOligopolistic competition among groupsrdquoEconomics Letters vol 36 pp 1ndash3 1991

[2] M R Baye K J Crocker and J Ju ldquoDivisionalization franchis-ing and divestiture incentives in oligopolyrdquoAmerican EconomicReview vol 86 no 1 pp 223ndash236 1996

[3] S Ziss ldquoDivisionalization and product differentiationrdquo Eco-nomics Letters vol 59 pp 133ndash138 1998

[4] L Yuan ldquoProduct differentiation strategic divisionalizationand persistence of monopolyrdquo Journal of Economics and Man-agement Strategy vol 8 no 4 pp 581ndash602 1999

[5] S Polasky ldquoDivide and conquerrdquo Economics Letters vol 40 pp365ndash371 1992

[6] X Vives Oligopoly Pricing The MIT Press 1999[7] A Creane and C Davidson ldquoMultidivisional firms internal

competition and themerger paradoxrdquoCanadian Journal of Eco-nomics vol 37 no 4 pp 951ndash977 2004

[8] M ConlinThe Effect of Franchising on Competition An Empir-ical Analysis Mimeo 2004

[9] A Kalnins and F Lafontaine ldquoMulti-unit ownership in fran-chising evidence from the fast-food industry in Texasrdquo RANDJournal of Economics vol 35 no 4 pp 747ndash761 2004

[10] S W Salant S Switzer and R J Reynolds ldquoLosses from hori-zontal merger the effects of an exogenous change in industrystructure on Cournot-Nash equilibriumrdquo Quarterly Journal ofEconomics vol 98 pp 185ndash199 1983

[11] D L Qiu and W Zhou ldquoMergers divestitures and industryreorganizationrdquoWorking PaperUniversity ofHongKong 2010

[12] S H Mialon ldquoEfficient horizontal mergers the effects of inter-nal capital reallocation and organizational formrdquo InternationalJournal of Industrial Organization vol 26 no 4 pp 861ndash8772008

[13] S Dowrick ldquovon Stackelberg and Cournot duopoly choosingrolesrdquo RAND Journal of Economics vol 17 pp 251ndash260 1986

[14] J H Hamilton and S M Slutsky ldquoEndogenous timing induopoly games stackelberg or cournot equilibriardquo Games andEconomic Behavior vol 2 no 1 pp 29ndash46 1990

[15] J Henkel ldquoThe 15th mover advantagerdquo RAND Journal ofEconomics vol 33 no 1 pp 156ndash170 2002

[16] M K Perry and R H Porter ldquoOligopoly and the incentive forhorizontalmergerrdquoAmerican Economic Review vol 75 pp 219ndash227 1985

[17] J Farrell and C Shapiro ldquoAsset ownership andmarket structurein oligopolyrdquo RAND Journal of Economics vol 21 pp 275ndash2921990

[18] R P McAfee and M A Williams ldquoHorizontal mergers andantitrust policyrdquo Journal of Industrial Economics vol 40 pp181ndash187 1992

[19] L C Corchon and M Gonzalez-Maestre ldquoOn the competitiveeffects of divisionalizationrdquo Mathematical Social Sciences vol39 pp 71ndash79 2000

[20] T Bresnahan S Greenstein and R Henderson ldquoSchumpete-rian competition and diseconomies of scope illustrations fromthe histories of Microsoft and IBMrdquo in The Rate and Directionof Inventive Activity Revisited J Lerner and S Stern EdsUniversity of Chicago Press 2012

[21] T R Lewis ldquoPreemption divestiture and forward contractingin a market dominated by a single firmrdquo American EconomicReview vol 73 pp 1092ndash1101 1983

[22] M Schwartz and E AThompson ldquoDivisionalization and entrydeterrencerdquo Quarterly Journal of Economics vol 101 pp 307ndash322 1986

[23] G Tan and L Yuan ldquoStrategic incentives of divestitures ofcompeting conglomeratesrdquo International Journal of IndustrialOrganization vol 21 no 5 pp 673ndash697 2003

[24] L D Qiu and W Zhou ldquoInternational mergers incentives andwelfarerdquo Journal of International Economics vol 68 no 1 pp38ndash58 2006

[25] R S Raubitschek ldquoA model of product proliferation withmultiproduct firmsrdquo Journal of Industrial Economics vol 35 pp269ndash279 1987

[26] R Deneckere and C Davidson ldquoIncentive to form coalitionswith Bertrand competitionrdquo RAND Journal of Economics vol16 pp 473ndash486 1985

[27] W Zhou ldquoLarge is beautiful horizontal mergers for betterexploitation of production shocksrdquo Journal of Industrial Eco-nomics vol 56 no 1 pp 68ndash93 2008

[28] M I Kamien and I Zang ldquoThe limits of monopolizationthrough acquisitionrdquo Quarterly Journal of Economics vol 105pp 465ndash499 1990

[29] L D Qiu andW Zhou ldquoMerger waves a model of endogenousmergersrdquo RAND Journal of Economics vol 38 no 1 pp 214ndash226 2007

[30] E C H Veendorp ldquoEntry deterrence divisionalization andinvestment decisionsrdquo Quarterly Journal of Economics vol 106pp 297ndash307 1991

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Research and TreatmentAutism


Economics Research International


This study investigated the incentive to divest when theorder of divestiture is decided endogenously In an industrywith two firms producing homogeneous products firms werefound always to divest sequentially and doing so greatlylimits the number of divisions The divestiture paradox cantherefore be solved by endogenizing the order of divestituresSimultaneous divestiture is no longer an equilibrium becausea firm is always better off playing a sequential game either as aleader or as a follower A leader can always choose its equilib-rium strategy in the simultaneous game and achieve its payoffthere so it must be no worse off when it chooses differently inthe sequential game Because divestitures are strategic com-plements and the leader will be hurt by the followerrsquos divesti-ture the leader limits its own divestiture in order to constrainthe followerrsquos As a result a follower is also better off

This research reflects the simple idea that the orderof business choices should be endogenized whenever it isappropriate especially when the results depend greatly onthe order Some choices such as prices or output levelsmay be regarded as being made simultaneously becausethey can easily be changed or are unlikely to be known torivals beforehand but divestitures take time and are usuallyreported by themedia Endogenizing the order of divestituresis particularly appropriate and easy because both firms benefitbymoving from a simultaneous game to a sequential one andsuch a move does not require any coordination Even if orig-inally the decisions were supposed to be simultaneous a firmcan easily turn them into a sequential one by say committingto a choice before other firms have made their decisions

So the major novelty of this study is the endogenizationof the order of divestitures Another innovation is its way ofmodeling divestitures Capital was assumed to be requiredin the production which implies increasing marginal costs(in the short run) Compared with the commonly usedformulation assuming constant marginal costs this coststructure leads to a more reasonable modeling of divestiturea divestiture decomposes the parentrsquos capital so that eachoffspring is smaller than the parent It also admits constantmarginal cost as a special case where endogenizing the orderof divestitures generates the most striking difference Withconstant marginal costs simultaneous divestitures induceeach firm to divest into an infinite number of divisions lead-ing to perfect competition When the order is endogenizedby contrast the leader will not divest at all while the followerdivests into only two divisions

The presence of capital also enables the study of firmasymmetry Divestitures can be shown to favor the larger firmand the follower Divestitures increase social welfare at theexpense of the industryrsquos total profitThe leader is always hurtbut the followermay benefit if it is sufficiently large A sequen-tial divestiture in which the larger firm is the follower gener-ates greater industry profit and social welfare but a smallerconsumer surplus than the alternative sequence This isbecause the smaller firm tends to divest more than the largerone and the follower tends to divest more than the leader Ifthe larger firm is the follower the industryrsquos overall divestitureis more limited and the resulting divisions are more balanced

in size The first effect helps industry profit and hurts con-sumer surplus while the second effect improves productionefficiency and hence helps industry profit and social welfare

There is a small body of literature on divestiture (alsoknown as strategic divisionalization) all assuming symmetricfirms constant marginal costs and simultaneous divestitureCorchon [1] and Polasky [5] have demonstrated how a firmmay gain competitive advantage by breaking itself up intoautonomous units3 and found that the incentive to divest canbe too strong to the detriment of the divesting firms Baye etal [2] have suggested that divestitures will be limited if theyare costly All three of these studies assumed homogeneousproducts By contrast Ziss [3] and Yuan [4] showed thatthe incentive to divest can also be reduced by productdifferentiation that is the number of divisions decreaseswhen products are more differentiated4 As a result thedivestiture paradox can be solved by introducing productdifferentiation each firm will divest into a finite number ofdivisions when products are sufficiently differentiated andperfect competition will not result5

This paper assumes homogeneous products with increas-ing marginal costs Such a setting is mathematically equiv-alent to assuming differentiated products with constantmarginal costs [6] It is therefore no wonder that bothapproaches can solve the divestiture paradox Neverthelessthere are some subtle differences between the two formu-lations as will be explained More importantly this paperrsquosconclusions are based on the endogenized order of divesti-tures while Ziss [3] and Yuan [4] both assumed simultaneousdivestitures

The incentive to divest echoes that for creating com-peting divisions within a company6 Creane and David-son [7] have provided many examples of multidivisionalfirms that encourage internal competition especially in thehotel brewery fast-food automobile and tobacco industriesThey pointed out (p 954) that ldquomany firms offer multipleproducts that appear to be either identical or extremelyclose substitutes for one anotherrdquo Conlin [8] found evidenceof strong competition between different brands within ahotel chain Kalnins and Lafontaine [9] discovered empiricalregularities in the ownership of newly opened franchisedunits in the hotel industry and speculated that the regularitiesare best explained by an optimal balance between internal andexternal competition

The divestiture paradoxmirrors the better-knownmergerparadox which says that firms tend to have too weak anincentive to merge [10] The two paradoxes are actuallytwo sides of the same coinmdashboth are caused by a toostrong response from competitors7 This is hardly surprisingas divestitures are simply reverse mergers Endogenizingmerger decisions however may or may not solve the mergerparadox8 A recent study by Qiu and Zhou [11] treateddivestitures and mergers in a single model which solvesboth paradoxes simultaneously as the interaction betweenthe two restructuring activities weakens the incentive todivest and strengthens the incentive to merge Reflecting ina merger context the logic of using internal competition togain competitive advantage Mialon [12] demonstrated that




2 The Model


119894 119902119894) = 119902

2

1198942119905119894 where 119905





If 119903119909





3 Analysis


119896equiv

119887119905119896(1 + 119887119905

119896) and 119866 equiv sum


is then 120587119896= (1198862119892119896(1 + 119892

119896))(2119887(1 + 119866)

2) Without loss of


120587119896=

119892119896(1 + 119892

119896)

(1 + 119866)2

(1)

where

119892119896=

119905119896

1 + 119905119896

119866 = sum

119889isin119863

119892119889 (2)

with 119902119896= (119886119887) (119892

119896(1 + 119866)) 119876 = (119886119887) (119866(1 + 119866)) and


119896= 119902119896119905119896= (119886(119887(1 + 119866))) (1(1 + 119905

119896)) and its


119896119866



be 119905119894119899119894 Let 119892

119894= (119905119894119899119894)(1 + (119905

119894119899119894)) = 119905

119894(119899119894+ 119905119894) Firm 119894rsquos

payoff is then

120587119894(119899119894 119899119895) =

119899119894119892119894(1 + 119892

119894)

(1 + 119899119894119892119894+ 119899119895119892119895)

2 (3)


lowast

119894equiv

argmax119899119894gt0

120587119894(119899119894 119899119895) = 1 + 119899



119899lowast(119899119895) equiv 1 + 119899

119895119892119895= 1 +

119899119895119905119895

119899119895+ 119905119895

(4)


119894=

119899lowast(119899119872

119895) and 119899119872

119895= 119899lowast(119899119872



119899119872

119894=

1 + 2119905119895+ radic(1 + 2119905

119894) (1 + 2119905

119895) (1 + 2119905

119894+ 2119905119895)

2 (1 + 119905119894+ 119905119895)

(5)






120573 (119899119871) = 0 (6)

where

120573 (119899119871) equiv minus [(1 + 119905

119871+ 119905119865)2+ 1199052

119865] (119899119871)

3

+ [(1 + 119905119865) (1 + 2119905

119865minus 1199052

119871) minus 4119905

1198711199052

119865] (119899119871)

2

+ 119905119871[(1 + 2119905

119865) (2 + 2119905

119865+ 119905119871) minus 21199051198711199052

119865] 119899119871

+ 1199052

119871(1 + 119905119865) (1 + 2119905

119865)

(7)


119871

119894when 119894 is the







Lemma 1 119899119871119894lt 119899119865

119894lt 119899119872

119894and 120587

119865

119894gt 120587119871

119894gt 120587119872

119894

Because 120587119865119894gt 120587119871

119894gt 120587119872

119894and 120587

119865

119895gt 120587119871

119895gt 120587119872




119895= 119865 or for

119903119894= 119865 and 119903








119894lt 119899119865

119894lt 119899119872

119894(Lemma 1)

and 119899119872

119894is finite

4 Discussion




119894= (119899119894119892119894(1 + 119892

119894))(1 + 119899

119894119892119894+ 119899119895119892119895)2 The


119894119892119894(1 + 119899

119894119892119894+



119904119894equiv

119902119894

119902119894+ 119902119895

=

119899119894119892119894

119899119894119892119894+ 119899119895119892119895

(8)


and 119899119895119892119895 Let us call 119899








120597119899lowast

120597119905119895

gt 0

120597119899lowast

120597119899119895

gt 0 (9)











119905119894and 119905








119894







119871= 1 and 119899119865 = 2

but 119899119872119894

= infin





119894= 119905119895




119865= 119899119872


1198761199010 and





1198941205870

119894= 916 while 120587

119865

1198941205870



119894gt 1)




119871



119894lt 119904119865

119894and (ii) 120587119871

119894+ 120587119865

119895gt 120587119871

119895+ 120587119865


119905119894lt 119905119895






119894119892119894equiv 119899119894119905119894(119899119894+ 119905119894)




119894119899119894))





119882 equiv CS + 120587119894+ 120587119895

=

119866

1 + 119866

+ (

119866

1 + 119866

)

2

ℎ

(10)


1199042

119889= (1198991198941198922

119894+

1198991198951198922


119894= 10 and 119905

119895= 1


1 1

34

119905119905

119901119876

1199010

119901119872

1199010

120587119865

119894

1205870

119894

120587119871

119894

1205870

119894

120587119872

119894

1205870

119894

916

00

00

98

Pric

e rat

io

Profi

trat

io



119895


119894up to 119899

119894=







119865= 0


119865gt 119899119871

gt 1 As a result119866 (119899119872

119894 119899119872

119895) gt 119866 (119899

119871

119894 119899119865









119882119876(119871 = 119894 119865 = 119895) lt 119882

119876(119871 = 119895 119865 = 119894) if and only if 119905

119894gt 119905119895




Appendix


119896= 119901119902119896minus119862(119905119896 119902119896) = (119886 minus 119887119876


1198962119905119896)


= sum119889isin119863119896


is 119886 minus 119887119876minus119896

minus 2119887119902119896minus (1119905

119896)119902119896

= 0 or equivalently 119887119902119896

=




119896(1+119866))Then120587

119896= (1198862119892119896(1+119892119896))(2119887(1+119866)

2)


119899119894120587119894(119899119894 119899119895)where 119899

119895= 119899lowast(119899119894)

At 119899119894= 119899119871

119894119889120587119894119889119899119894= (120597120587

119894120597119899119894)+(120597120587

119894120597119899119895) (119889119899119895119889119899119894) = 0 But

120597120587119894120597119899119895lt 0 while 119889119899

119895119889119899119894= 119899lowast1015840(sdot) gt 0 so 120597120587

119894(119899119871

119894)120597119899119894gt 0

Meanwhile 119899119872119894

= argmax119899119894120587119894(119899119894 119899119895) so 120597120587

119894(119899119872

119894)120597119899119894= 0

Since 120597120587119894(119899119871

119894)120597119899119894gt 120597120587

119894(119899119872

119894)120597119899119894and 120597

21205871198941205971198992

119894lt 0 we

conclude that 119899119871119894lt 119899119872

119894

By symmetry 119899119871119895

lt 119899119872

119895 But 119899119865

119894= 119899lowast(119899119871

119895) and 119899

119872

119894=

119899lowast(119899119872


119894lt 119899119872

119894 Now we must

show that 119899119871119894lt 119899119865


119894ge 119899119865

119894) Then by

symmetry 119899119871119895ge 119899119865

119895

119899119865

119894= 119899lowast(119899119871

119895)

ge 119899lowast(119899119865

119895) (because 119899

119871

119895ge 119899119865

119895and 119899


= argmax119899119894

120587119894(119899119894 119899119865

119895)

gt argmax119899119894

120587119894(119899119894 119899lowast(119899119894))

(because120597120587119894

120597119899119895


119865

119895= 119899lowast(119899119871

119894))

= 119899119871

119894

(A1)


120587119865

119894= max119899119894

120587119894(119899119894 119899119871

119895)

gt max119899119894

120587119894(119899119894 119899119865

119895)


120597120587119894

120597119899119895

lt0 and 119899119871119895lt119899119865

119895)

gt 120587119894(119899119871

119894 119899119865

119895)

= 120587119871

119894

= max119899119894

120587119894(119899119894 119899lowast(119899119894))

gt 120587119894(119899119872

119894 119899lowast(119899119872

119894)) (because 119899

119871

119894= 119899119872

119894)

= 120587119872

119894

(A2)


lently 120573(119899119871)1199052

1198711199052

119865= 0 When 119905


119895rarr infin

120573(119899119871)1199052

1198711199052

119865= minus2119899

119871+ 2 so 119899

119871= 1 Then 119899119865 = 119899

lowast(119899119871) =

1 + (119899119871119905119871(119899119871+ 119905119871)) = 2


119894Then by sym-

metry 119904119871119895gt 119904119865

119895 Because 119904119871

119894= 119899119871

119894119892119871

119894(119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895) we have

120587119871

119894=

119899119871

119894119892119871

119894(1 + 119892

119871

119894)

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2

= 119904119871

119894(1 + 119892

119871

119894)

119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2

(A3)

Similarly

120587119865

119895= 119904119865

119895(1 + 119892

119865

119895)

119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2 (A4)

Then120587119871119894120587119865

119895= 119904119871

119894(1+119892119871

119894)119904119865

119895(1+119892119865

119895) Similarly120587119865

119894120587119871

119895= 119904119865

119894(1+

119892119865

119894)119904119871

119895(1 + 119892

119871

119895) Now because 119904119871

119894gt 119904119865

119894and 119904119871

119895gt 119904119865

119895 we have

119904119871

119894119904119865

119895gt 119904119865

119894119904119871

119895 Because 119899119871

119894lt 119899119865

119894and 119899

119871

119895lt 119899119865

119895(Lemma 1) we

have (1+119892119871119894)(1+119892

119865

119895) = (1+(119905

119894(119899119871

119894+119905119894)))(1+(119905

119895(119899119865

119895+119905119895))) gt

(1 + (119905119894(119899119865

119894+ 119905119894)))(1 + (119905

119895(119899119871

119895+ 119905119895))) = (1 + 119892

119865

119894)(1 + 119892

119871

119895)


119895gt 120587119865

119894120587119871


120587119871

119894120587119865

119894gt 120587119865

119895120587119871

119895 By Lemma 1 120587119871

119894lt 120587119865

119894 Therefore 120587119865

119895120587119871

119895lt

120587119871

119894120587119865

119894lt 1 or 120587119865

119895lt 120587119871



Because 120573(1) = 119905119865(1 + 2119905





119871


Now 1205731015840(0) = 119905119871(6119905119865+ 119905119871+ 2 + 2119905

119871119905119865+ 41199052

119865minus 21199051198711199052

119865) and

[1205731015840(119899119871)]119899119871minus 120573(119899

119871) = 2120573

3(119899119871)3+ 1205732(119899119871)2minus 1205730 where 120573

3=

minus[(1 + 119905119871+ 119905119865)2+ 1199052

119865] lt 0 120573

2= [(1 + 119905

119865)(1 + 2119905

119865minus 1199052

119871) minus 41199051198711199052

119865]

and 1205730= 1199052

119871(1 + 119905

119865)(1 + 2119905




1205731015840(0) gt 0 If120573




Acknowledgments



Endnotes















119897119894119908119897119894subject to 119902

119894= (119905119894119897119894)12 in


119894) 1199022








119896and therefore










119896= (119886 minus



119896=

[119886 minus 119887119876minus119896





119894) = 119887+



119894119899119894)


119894119899119894) = 119887 + 1 (2119905



119894119899119894) gt 119887 + (12119905

119894)



1198962119905119896)


119896= (119886 minus 119889119876


(1199022



= [119886 minus 119889119876minus119896

minus (119887 + (12119905119896)) 119902119896] 119902119896 which is as












References







































Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal








2 The Model


119894 119902119894) = 119902

2

1198942119905119894 where 119905





If 119903119909





3 Analysis


119896equiv

119887119905119896(1 + 119887119905

119896) and 119866 equiv sum


is then 120587119896= (1198862119892119896(1 + 119892

119896))(2119887(1 + 119866)

2) Without loss of


120587119896=

119892119896(1 + 119892

119896)

(1 + 119866)2

(1)

where

119892119896=

119905119896

1 + 119905119896

119866 = sum

119889isin119863

119892119889 (2)

with 119902119896= (119886119887) (119892

119896(1 + 119866)) 119876 = (119886119887) (119866(1 + 119866)) and


119896= 119902119896119905119896= (119886(119887(1 + 119866))) (1(1 + 119905

119896)) and its


119896119866



be 119905119894119899119894 Let 119892

119894= (119905119894119899119894)(1 + (119905

119894119899119894)) = 119905

119894(119899119894+ 119905119894) Firm 119894rsquos

payoff is then

120587119894(119899119894 119899119895) =

119899119894119892119894(1 + 119892

119894)

(1 + 119899119894119892119894+ 119899119895119892119895)

2 (3)


lowast

119894equiv

argmax119899119894gt0

120587119894(119899119894 119899119895) = 1 + 119899



119899lowast(119899119895) equiv 1 + 119899

119895119892119895= 1 +

119899119895119905119895

119899119895+ 119905119895

(4)


119894=

119899lowast(119899119872

119895) and 119899119872

119895= 119899lowast(119899119872



119899119872

119894=

1 + 2119905119895+ radic(1 + 2119905

119894) (1 + 2119905

119895) (1 + 2119905

119894+ 2119905119895)

2 (1 + 119905119894+ 119905119895)

(5)






120573 (119899119871) = 0 (6)

where

120573 (119899119871) equiv minus [(1 + 119905

119871+ 119905119865)2+ 1199052

119865] (119899119871)

3

+ [(1 + 119905119865) (1 + 2119905

119865minus 1199052

119871) minus 4119905

1198711199052

119865] (119899119871)

2

+ 119905119871[(1 + 2119905

119865) (2 + 2119905

119865+ 119905119871) minus 21199051198711199052

119865] 119899119871

+ 1199052

119871(1 + 119905119865) (1 + 2119905

119865)

(7)


119871

119894when 119894 is the







Lemma 1 119899119871119894lt 119899119865

119894lt 119899119872

119894and 120587

119865

119894gt 120587119871

119894gt 120587119872

119894

Because 120587119865119894gt 120587119871

119894gt 120587119872

119894and 120587

119865

119895gt 120587119871

119895gt 120587119872




119895= 119865 or for

119903119894= 119865 and 119903








119894lt 119899119865

119894lt 119899119872

119894(Lemma 1)

and 119899119872

119894is finite

4 Discussion




119894= (119899119894119892119894(1 + 119892

119894))(1 + 119899

119894119892119894+ 119899119895119892119895)2 The


119894119892119894(1 + 119899

119894119892119894+



119904119894equiv

119902119894

119902119894+ 119902119895

=

119899119894119892119894

119899119894119892119894+ 119899119895119892119895

(8)


and 119899119895119892119895 Let us call 119899








120597119899lowast

120597119905119895

gt 0

120597119899lowast

120597119899119895

gt 0 (9)











119905119894and 119905








119894







119871= 1 and 119899119865 = 2

but 119899119872119894

= infin





119894= 119905119895




119865= 119899119872


1198761199010 and





1198941205870

119894= 916 while 120587

119865

1198941205870



119894gt 1)




119871



119894lt 119904119865

119894and (ii) 120587119871

119894+ 120587119865

119895gt 120587119871

119895+ 120587119865


119905119894lt 119905119895






119894119892119894equiv 119899119894119905119894(119899119894+ 119905119894)




119894119899119894))





119882 equiv CS + 120587119894+ 120587119895

=

119866

1 + 119866

+ (

119866

1 + 119866

)

2

ℎ

(10)


1199042

119889= (1198991198941198922

119894+

1198991198951198922


119894= 10 and 119905

119895= 1


1 1

34

119905119905

119901119876

1199010

119901119872

1199010

120587119865

119894

1205870

119894

120587119871

119894

1205870

119894

120587119872

119894

1205870

119894

916

00

00

98

Pric

e rat

io

Profi

trat

io



119895


119894up to 119899

119894=







119865= 0


119865gt 119899119871

gt 1 As a result119866 (119899119872

119894 119899119872

119895) gt 119866 (119899

119871

119894 119899119865









119882119876(119871 = 119894 119865 = 119895) lt 119882

119876(119871 = 119895 119865 = 119894) if and only if 119905

119894gt 119905119895




Appendix


119896= 119901119902119896minus119862(119905119896 119902119896) = (119886 minus 119887119876


1198962119905119896)


= sum119889isin119863119896


is 119886 minus 119887119876minus119896

minus 2119887119902119896minus (1119905

119896)119902119896

= 0 or equivalently 119887119902119896

=




119896(1+119866))Then120587

119896= (1198862119892119896(1+119892119896))(2119887(1+119866)

2)


119899119894120587119894(119899119894 119899119895)where 119899

119895= 119899lowast(119899119894)

At 119899119894= 119899119871

119894119889120587119894119889119899119894= (120597120587

119894120597119899119894)+(120597120587

119894120597119899119895) (119889119899119895119889119899119894) = 0 But

120597120587119894120597119899119895lt 0 while 119889119899

119895119889119899119894= 119899lowast1015840(sdot) gt 0 so 120597120587

119894(119899119871

119894)120597119899119894gt 0

Meanwhile 119899119872119894

= argmax119899119894120587119894(119899119894 119899119895) so 120597120587

119894(119899119872

119894)120597119899119894= 0

Since 120597120587119894(119899119871

119894)120597119899119894gt 120597120587

119894(119899119872

119894)120597119899119894and 120597

21205871198941205971198992

119894lt 0 we

conclude that 119899119871119894lt 119899119872

119894

By symmetry 119899119871119895

lt 119899119872

119895 But 119899119865

119894= 119899lowast(119899119871

119895) and 119899

119872

119894=

119899lowast(119899119872


119894lt 119899119872

119894 Now we must

show that 119899119871119894lt 119899119865


119894ge 119899119865

119894) Then by

symmetry 119899119871119895ge 119899119865

119895

119899119865

119894= 119899lowast(119899119871

119895)

ge 119899lowast(119899119865

119895) (because 119899

119871

119895ge 119899119865

119895and 119899


= argmax119899119894

120587119894(119899119894 119899119865

119895)

gt argmax119899119894

120587119894(119899119894 119899lowast(119899119894))

(because120597120587119894

120597119899119895


119865

119895= 119899lowast(119899119871

119894))

= 119899119871

119894

(A1)


120587119865

119894= max119899119894

120587119894(119899119894 119899119871

119895)

gt max119899119894

120587119894(119899119894 119899119865

119895)


120597120587119894

120597119899119895

lt0 and 119899119871119895lt119899119865

119895)

gt 120587119894(119899119871

119894 119899119865

119895)

= 120587119871

119894

= max119899119894

120587119894(119899119894 119899lowast(119899119894))

gt 120587119894(119899119872

119894 119899lowast(119899119872

119894)) (because 119899

119871

119894= 119899119872

119894)

= 120587119872

119894

(A2)


lently 120573(119899119871)1199052

1198711199052

119865= 0 When 119905


119895rarr infin

120573(119899119871)1199052

1198711199052

119865= minus2119899

119871+ 2 so 119899

119871= 1 Then 119899119865 = 119899

lowast(119899119871) =

1 + (119899119871119905119871(119899119871+ 119905119871)) = 2


119894Then by sym-

metry 119904119871119895gt 119904119865

119895 Because 119904119871

119894= 119899119871

119894119892119871

119894(119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895) we have

120587119871

119894=

119899119871

119894119892119871

119894(1 + 119892

119871

119894)

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2

= 119904119871

119894(1 + 119892

119871

119894)

119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2

(A3)

Similarly

120587119865

119895= 119904119865

119895(1 + 119892

119865

119895)

119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2 (A4)

Then120587119871119894120587119865

119895= 119904119871

119894(1+119892119871

119894)119904119865

119895(1+119892119865

119895) Similarly120587119865

119894120587119871

119895= 119904119865

119894(1+

119892119865

119894)119904119871

119895(1 + 119892

119871

119895) Now because 119904119871

119894gt 119904119865

119894and 119904119871

119895gt 119904119865

119895 we have

119904119871

119894119904119865

119895gt 119904119865

119894119904119871

119895 Because 119899119871

119894lt 119899119865

119894and 119899

119871

119895lt 119899119865

119895(Lemma 1) we

have (1+119892119871119894)(1+119892

119865

119895) = (1+(119905

119894(119899119871

119894+119905119894)))(1+(119905

119895(119899119865

119895+119905119895))) gt

(1 + (119905119894(119899119865

119894+ 119905119894)))(1 + (119905

119895(119899119871

119895+ 119905119895))) = (1 + 119892

119865

119894)(1 + 119892

119871

119895)


119895gt 120587119865

119894120587119871


120587119871

119894120587119865

119894gt 120587119865

119895120587119871

119895 By Lemma 1 120587119871

119894lt 120587119865

119894 Therefore 120587119865

119895120587119871

119895lt

120587119871

119894120587119865

119894lt 1 or 120587119865

119895lt 120587119871



Because 120573(1) = 119905119865(1 + 2119905





119871


Now 1205731015840(0) = 119905119871(6119905119865+ 119905119871+ 2 + 2119905

119871119905119865+ 41199052

119865minus 21199051198711199052

119865) and

[1205731015840(119899119871)]119899119871minus 120573(119899

119871) = 2120573

3(119899119871)3+ 1205732(119899119871)2minus 1205730 where 120573

3=

minus[(1 + 119905119871+ 119905119865)2+ 1199052

119865] lt 0 120573

2= [(1 + 119905

119865)(1 + 2119905

119865minus 1199052

119871) minus 41199051198711199052

119865]

and 1205730= 1199052

119871(1 + 119905

119865)(1 + 2119905




1205731015840(0) gt 0 If120573




Acknowledgments



Endnotes















119897119894119908119897119894subject to 119902

119894= (119905119894119897119894)12 in


119894) 1199022








119896and therefore










119896= (119886 minus



119896=

[119886 minus 119887119876minus119896





119894) = 119887+



119894119899119894)


119894119899119894) = 119887 + 1 (2119905



119894119899119894) gt 119887 + (12119905

119894)



1198962119905119896)


119896= (119886 minus 119889119876


(1199022



= [119886 minus 119889119876minus119896

minus (119887 + (12119905119896)) 119902119896] 119902119896 which is as












References







































Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal






Lemma 1 119899119871119894lt 119899119865

119894lt 119899119872

119894and 120587

119865

119894gt 120587119871

119894gt 120587119872

119894

Because 120587119865119894gt 120587119871

119894gt 120587119872

119894and 120587

119865

119895gt 120587119871

119895gt 120587119872




119895= 119865 or for

119903119894= 119865 and 119903








119894lt 119899119865

119894lt 119899119872

119894(Lemma 1)

and 119899119872

119894is finite

4 Discussion




119894= (119899119894119892119894(1 + 119892

119894))(1 + 119899

119894119892119894+ 119899119895119892119895)2 The


119894119892119894(1 + 119899

119894119892119894+



119904119894equiv

119902119894

119902119894+ 119902119895

=

119899119894119892119894

119899119894119892119894+ 119899119895119892119895

(8)


and 119899119895119892119895 Let us call 119899








120597119899lowast

120597119905119895

gt 0

120597119899lowast

120597119899119895

gt 0 (9)











119905119894and 119905








119894







119871= 1 and 119899119865 = 2

but 119899119872119894

= infin





119894= 119905119895




119865= 119899119872


1198761199010 and





1198941205870

119894= 916 while 120587

119865

1198941205870



119894gt 1)




119871



119894lt 119904119865

119894and (ii) 120587119871

119894+ 120587119865

119895gt 120587119871

119895+ 120587119865


119905119894lt 119905119895






119894119892119894equiv 119899119894119905119894(119899119894+ 119905119894)




119894119899119894))





119882 equiv CS + 120587119894+ 120587119895

=

119866

1 + 119866

+ (

119866

1 + 119866

)

2

ℎ

(10)


1199042

119889= (1198991198941198922

119894+

1198991198951198922


119894= 10 and 119905

119895= 1


1 1

34

119905119905

119901119876

1199010

119901119872

1199010

120587119865

119894

1205870

119894

120587119871

119894

1205870

119894

120587119872

119894

1205870

119894

916

00

00

98

Pric

e rat

io

Profi

trat

io



119895


119894up to 119899

119894=







119865= 0


119865gt 119899119871

gt 1 As a result119866 (119899119872

119894 119899119872

119895) gt 119866 (119899

119871

119894 119899119865









119882119876(119871 = 119894 119865 = 119895) lt 119882

119876(119871 = 119895 119865 = 119894) if and only if 119905

119894gt 119905119895




Appendix


119896= 119901119902119896minus119862(119905119896 119902119896) = (119886 minus 119887119876


1198962119905119896)


= sum119889isin119863119896


is 119886 minus 119887119876minus119896

minus 2119887119902119896minus (1119905

119896)119902119896

= 0 or equivalently 119887119902119896

=




119896(1+119866))Then120587

119896= (1198862119892119896(1+119892119896))(2119887(1+119866)

2)


119899119894120587119894(119899119894 119899119895)where 119899

119895= 119899lowast(119899119894)

At 119899119894= 119899119871

119894119889120587119894119889119899119894= (120597120587

119894120597119899119894)+(120597120587

119894120597119899119895) (119889119899119895119889119899119894) = 0 But

120597120587119894120597119899119895lt 0 while 119889119899

119895119889119899119894= 119899lowast1015840(sdot) gt 0 so 120597120587

119894(119899119871

119894)120597119899119894gt 0

Meanwhile 119899119872119894

= argmax119899119894120587119894(119899119894 119899119895) so 120597120587

119894(119899119872

119894)120597119899119894= 0

Since 120597120587119894(119899119871

119894)120597119899119894gt 120597120587

119894(119899119872

119894)120597119899119894and 120597

21205871198941205971198992

119894lt 0 we

conclude that 119899119871119894lt 119899119872

119894

By symmetry 119899119871119895

lt 119899119872

119895 But 119899119865

119894= 119899lowast(119899119871

119895) and 119899

119872

119894=

119899lowast(119899119872


119894lt 119899119872

119894 Now we must

show that 119899119871119894lt 119899119865


119894ge 119899119865

119894) Then by

symmetry 119899119871119895ge 119899119865

119895

119899119865

119894= 119899lowast(119899119871

119895)

ge 119899lowast(119899119865

119895) (because 119899

119871

119895ge 119899119865

119895and 119899


= argmax119899119894

120587119894(119899119894 119899119865

119895)

gt argmax119899119894

120587119894(119899119894 119899lowast(119899119894))

(because120597120587119894

120597119899119895


119865

119895= 119899lowast(119899119871

119894))

= 119899119871

119894

(A1)


120587119865

119894= max119899119894

120587119894(119899119894 119899119871

119895)

gt max119899119894

120587119894(119899119894 119899119865

119895)


120597120587119894

120597119899119895

lt0 and 119899119871119895lt119899119865

119895)

gt 120587119894(119899119871

119894 119899119865

119895)

= 120587119871

119894

= max119899119894

120587119894(119899119894 119899lowast(119899119894))

gt 120587119894(119899119872

119894 119899lowast(119899119872

119894)) (because 119899

119871

119894= 119899119872

119894)

= 120587119872

119894

(A2)


lently 120573(119899119871)1199052

1198711199052

119865= 0 When 119905


119895rarr infin

120573(119899119871)1199052

1198711199052

119865= minus2119899

119871+ 2 so 119899

119871= 1 Then 119899119865 = 119899

lowast(119899119871) =

1 + (119899119871119905119871(119899119871+ 119905119871)) = 2


119894Then by sym-

metry 119904119871119895gt 119904119865

119895 Because 119904119871

119894= 119899119871

119894119892119871

119894(119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895) we have

120587119871

119894=

119899119871

119894119892119871

119894(1 + 119892

119871

119894)

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2

= 119904119871

119894(1 + 119892

119871

119894)

119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2

(A3)

Similarly

120587119865

119895= 119904119865

119895(1 + 119892

119865

119895)

119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2 (A4)

Then120587119871119894120587119865

119895= 119904119871

119894(1+119892119871

119894)119904119865

119895(1+119892119865

119895) Similarly120587119865

119894120587119871

119895= 119904119865

119894(1+

119892119865

119894)119904119871

119895(1 + 119892

119871

119895) Now because 119904119871

119894gt 119904119865

119894and 119904119871

119895gt 119904119865

119895 we have

119904119871

119894119904119865

119895gt 119904119865

119894119904119871

119895 Because 119899119871

119894lt 119899119865

119894and 119899

119871

119895lt 119899119865

119895(Lemma 1) we

have (1+119892119871119894)(1+119892

119865

119895) = (1+(119905

119894(119899119871

119894+119905119894)))(1+(119905

119895(119899119865

119895+119905119895))) gt

(1 + (119905119894(119899119865

119894+ 119905119894)))(1 + (119905

119895(119899119871

119895+ 119905119895))) = (1 + 119892

119865

119894)(1 + 119892

119871

119895)


119895gt 120587119865

119894120587119871


120587119871

119894120587119865

119894gt 120587119865

119895120587119871

119895 By Lemma 1 120587119871

119894lt 120587119865

119894 Therefore 120587119865

119895120587119871

119895lt

120587119871

119894120587119865

119894lt 1 or 120587119865

119895lt 120587119871



Because 120573(1) = 119905119865(1 + 2119905





119871


Now 1205731015840(0) = 119905119871(6119905119865+ 119905119871+ 2 + 2119905

119871119905119865+ 41199052

119865minus 21199051198711199052

119865) and

[1205731015840(119899119871)]119899119871minus 120573(119899

119871) = 2120573

3(119899119871)3+ 1205732(119899119871)2minus 1205730 where 120573

3=

minus[(1 + 119905119871+ 119905119865)2+ 1199052

119865] lt 0 120573

2= [(1 + 119905

119865)(1 + 2119905

119865minus 1199052

119871) minus 41199051198711199052

119865]

and 1205730= 1199052

119871(1 + 119905

119865)(1 + 2119905




1205731015840(0) gt 0 If120573




Acknowledgments



Endnotes















119897119894119908119897119894subject to 119902

119894= (119905119894119897119894)12 in


119894) 1199022








119896and therefore










119896= (119886 minus



119896=

[119886 minus 119887119876minus119896





119894) = 119887+



119894119899119894)


119894119899119894) = 119887 + 1 (2119905



119894119899119894) gt 119887 + (12119905

119894)



1198962119905119896)


119896= (119886 minus 119889119876


(1199022



= [119886 minus 119889119876minus119896

minus (119887 + (12119905119896)) 119902119896] 119902119896 which is as












References







































Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal











119905119894and 119905








119894







119871= 1 and 119899119865 = 2

but 119899119872119894

= infin





119894= 119905119895




119865= 119899119872


1198761199010 and





1198941205870

119894= 916 while 120587

119865

1198941205870



119894gt 1)




119871



119894lt 119904119865

119894and (ii) 120587119871

119894+ 120587119865

119895gt 120587119871

119895+ 120587119865


119905119894lt 119905119895






119894119892119894equiv 119899119894119905119894(119899119894+ 119905119894)




119894119899119894))





119882 equiv CS + 120587119894+ 120587119895

=

119866

1 + 119866

+ (

119866

1 + 119866

)

2

ℎ

(10)


1199042

119889= (1198991198941198922

119894+

1198991198951198922


119894= 10 and 119905

119895= 1


1 1

34

119905119905

119901119876

1199010

119901119872

1199010

120587119865

119894

1205870

119894

120587119871

119894

1205870

119894

120587119872

119894

1205870

119894

916

00

00

98

Pric

e rat

io

Profi

trat

io



119895


119894up to 119899

119894=







119865= 0


119865gt 119899119871

gt 1 As a result119866 (119899119872

119894 119899119872

119895) gt 119866 (119899

119871

119894 119899119865









119882119876(119871 = 119894 119865 = 119895) lt 119882

119876(119871 = 119895 119865 = 119894) if and only if 119905

119894gt 119905119895




Appendix


119896= 119901119902119896minus119862(119905119896 119902119896) = (119886 minus 119887119876


1198962119905119896)


= sum119889isin119863119896


is 119886 minus 119887119876minus119896

minus 2119887119902119896minus (1119905

119896)119902119896

= 0 or equivalently 119887119902119896

=




119896(1+119866))Then120587

119896= (1198862119892119896(1+119892119896))(2119887(1+119866)

2)


119899119894120587119894(119899119894 119899119895)where 119899

119895= 119899lowast(119899119894)

At 119899119894= 119899119871

119894119889120587119894119889119899119894= (120597120587

119894120597119899119894)+(120597120587

119894120597119899119895) (119889119899119895119889119899119894) = 0 But

120597120587119894120597119899119895lt 0 while 119889119899

119895119889119899119894= 119899lowast1015840(sdot) gt 0 so 120597120587

119894(119899119871

119894)120597119899119894gt 0

Meanwhile 119899119872119894

= argmax119899119894120587119894(119899119894 119899119895) so 120597120587

119894(119899119872

119894)120597119899119894= 0

Since 120597120587119894(119899119871

119894)120597119899119894gt 120597120587

119894(119899119872

119894)120597119899119894and 120597

21205871198941205971198992

119894lt 0 we

conclude that 119899119871119894lt 119899119872

119894

By symmetry 119899119871119895

lt 119899119872

119895 But 119899119865

119894= 119899lowast(119899119871

119895) and 119899

119872

119894=

119899lowast(119899119872


119894lt 119899119872

119894 Now we must

show that 119899119871119894lt 119899119865


119894ge 119899119865

119894) Then by

symmetry 119899119871119895ge 119899119865

119895

119899119865

119894= 119899lowast(119899119871

119895)

ge 119899lowast(119899119865

119895) (because 119899

119871

119895ge 119899119865

119895and 119899


= argmax119899119894

120587119894(119899119894 119899119865

119895)

gt argmax119899119894

120587119894(119899119894 119899lowast(119899119894))

(because120597120587119894

120597119899119895


119865

119895= 119899lowast(119899119871

119894))

= 119899119871

119894

(A1)


120587119865

119894= max119899119894

120587119894(119899119894 119899119871

119895)

gt max119899119894

120587119894(119899119894 119899119865

119895)


120597120587119894

120597119899119895

lt0 and 119899119871119895lt119899119865

119895)

gt 120587119894(119899119871

119894 119899119865

119895)

= 120587119871

119894

= max119899119894

120587119894(119899119894 119899lowast(119899119894))

gt 120587119894(119899119872

119894 119899lowast(119899119872

119894)) (because 119899

119871

119894= 119899119872

119894)

= 120587119872

119894

(A2)


lently 120573(119899119871)1199052

1198711199052

119865= 0 When 119905


119895rarr infin

120573(119899119871)1199052

1198711199052

119865= minus2119899

119871+ 2 so 119899

119871= 1 Then 119899119865 = 119899

lowast(119899119871) =

1 + (119899119871119905119871(119899119871+ 119905119871)) = 2


119894Then by sym-

metry 119904119871119895gt 119904119865

119895 Because 119904119871

119894= 119899119871

119894119892119871

119894(119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895) we have

120587119871

119894=

119899119871

119894119892119871

119894(1 + 119892

119871

119894)

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2

= 119904119871

119894(1 + 119892

119871

119894)

119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2

(A3)

Similarly

120587119865

119895= 119904119865

119895(1 + 119892

119865

119895)

119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2 (A4)

Then120587119871119894120587119865

119895= 119904119871

119894(1+119892119871

119894)119904119865

119895(1+119892119865

119895) Similarly120587119865

119894120587119871

119895= 119904119865

119894(1+

119892119865

119894)119904119871

119895(1 + 119892

119871

119895) Now because 119904119871

119894gt 119904119865

119894and 119904119871

119895gt 119904119865

119895 we have

119904119871

119894119904119865

119895gt 119904119865

119894119904119871

119895 Because 119899119871

119894lt 119899119865

119894and 119899

119871

119895lt 119899119865

119895(Lemma 1) we

have (1+119892119871119894)(1+119892

119865

119895) = (1+(119905

119894(119899119871

119894+119905119894)))(1+(119905

119895(119899119865

119895+119905119895))) gt

(1 + (119905119894(119899119865

119894+ 119905119894)))(1 + (119905

119895(119899119871

119895+ 119905119895))) = (1 + 119892

119865

119894)(1 + 119892

119871

119895)


119895gt 120587119865

119894120587119871


120587119871

119894120587119865

119894gt 120587119865

119895120587119871

119895 By Lemma 1 120587119871

119894lt 120587119865

119894 Therefore 120587119865

119895120587119871

119895lt

120587119871

119894120587119865

119894lt 1 or 120587119865

119895lt 120587119871



Because 120573(1) = 119905119865(1 + 2119905





119871


Now 1205731015840(0) = 119905119871(6119905119865+ 119905119871+ 2 + 2119905

119871119905119865+ 41199052

119865minus 21199051198711199052

119865) and

[1205731015840(119899119871)]119899119871minus 120573(119899

119871) = 2120573

3(119899119871)3+ 1205732(119899119871)2minus 1205730 where 120573

3=

minus[(1 + 119905119871+ 119905119865)2+ 1199052

119865] lt 0 120573

2= [(1 + 119905

119865)(1 + 2119905

119865minus 1199052

119871) minus 41199051198711199052

119865]

and 1205730= 1199052

119871(1 + 119905

119865)(1 + 2119905




1205731015840(0) gt 0 If120573




Acknowledgments



Endnotes















119897119894119908119897119894subject to 119902

119894= (119905119894119897119894)12 in


119894) 1199022








119896and therefore










119896= (119886 minus



119896=

[119886 minus 119887119876minus119896





119894) = 119887+



119894119899119894)


119894119899119894) = 119887 + 1 (2119905



119894119899119894) gt 119887 + (12119905

119894)



1198962119905119896)


119896= (119886 minus 119889119876


(1199022



= [119886 minus 119889119876minus119896

minus (119887 + (12119905119896)) 119902119896] 119902119896 which is as












References







































Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal







119894= 119905119895




119865= 119899119872


1198761199010 and





1198941205870

119894= 916 while 120587

119865

1198941205870



119894gt 1)




119871



119894lt 119904119865

119894and (ii) 120587119871

119894+ 120587119865

119895gt 120587119871

119895+ 120587119865


119905119894lt 119905119895






119894119892119894equiv 119899119894119905119894(119899119894+ 119905119894)




119894119899119894))





119882 equiv CS + 120587119894+ 120587119895

=

119866

1 + 119866

+ (

119866

1 + 119866

)

2

ℎ

(10)


1199042

119889= (1198991198941198922

119894+

1198991198951198922


119894= 10 and 119905

119895= 1


1 1

34

119905119905

119901119876

1199010

119901119872

1199010

120587119865

119894

1205870

119894

120587119871

119894

1205870

119894

120587119872

119894

1205870

119894

916

00

00

98

Pric

e rat

io

Profi

trat

io



119895


119894up to 119899

119894=







119865= 0


119865gt 119899119871

gt 1 As a result119866 (119899119872

119894 119899119872

119895) gt 119866 (119899

119871

119894 119899119865









119882119876(119871 = 119894 119865 = 119895) lt 119882

119876(119871 = 119895 119865 = 119894) if and only if 119905

119894gt 119905119895




Appendix


119896= 119901119902119896minus119862(119905119896 119902119896) = (119886 minus 119887119876


1198962119905119896)


= sum119889isin119863119896


is 119886 minus 119887119876minus119896

minus 2119887119902119896minus (1119905

119896)119902119896

= 0 or equivalently 119887119902119896

=




119896(1+119866))Then120587

119896= (1198862119892119896(1+119892119896))(2119887(1+119866)

2)


119899119894120587119894(119899119894 119899119895)where 119899

119895= 119899lowast(119899119894)

At 119899119894= 119899119871

119894119889120587119894119889119899119894= (120597120587

119894120597119899119894)+(120597120587

119894120597119899119895) (119889119899119895119889119899119894) = 0 But

120597120587119894120597119899119895lt 0 while 119889119899

119895119889119899119894= 119899lowast1015840(sdot) gt 0 so 120597120587

119894(119899119871

119894)120597119899119894gt 0

Meanwhile 119899119872119894

= argmax119899119894120587119894(119899119894 119899119895) so 120597120587

119894(119899119872

119894)120597119899119894= 0

Since 120597120587119894(119899119871

119894)120597119899119894gt 120597120587

119894(119899119872

119894)120597119899119894and 120597

21205871198941205971198992

119894lt 0 we

conclude that 119899119871119894lt 119899119872

119894

By symmetry 119899119871119895

lt 119899119872

119895 But 119899119865

119894= 119899lowast(119899119871

119895) and 119899

119872

119894=

119899lowast(119899119872


119894lt 119899119872

119894 Now we must

show that 119899119871119894lt 119899119865


119894ge 119899119865

119894) Then by

symmetry 119899119871119895ge 119899119865

119895

119899119865

119894= 119899lowast(119899119871

119895)

ge 119899lowast(119899119865

119895) (because 119899

119871

119895ge 119899119865

119895and 119899


= argmax119899119894

120587119894(119899119894 119899119865

119895)

gt argmax119899119894

120587119894(119899119894 119899lowast(119899119894))

(because120597120587119894

120597119899119895


119865

119895= 119899lowast(119899119871

119894))

= 119899119871

119894

(A1)


120587119865

119894= max119899119894

120587119894(119899119894 119899119871

119895)

gt max119899119894

120587119894(119899119894 119899119865

119895)


120597120587119894

120597119899119895

lt0 and 119899119871119895lt119899119865

119895)

gt 120587119894(119899119871

119894 119899119865

119895)

= 120587119871

119894

= max119899119894

120587119894(119899119894 119899lowast(119899119894))

gt 120587119894(119899119872

119894 119899lowast(119899119872

119894)) (because 119899

119871

119894= 119899119872

119894)

= 120587119872

119894

(A2)


lently 120573(119899119871)1199052

1198711199052

119865= 0 When 119905


119895rarr infin

120573(119899119871)1199052

1198711199052

119865= minus2119899

119871+ 2 so 119899

119871= 1 Then 119899119865 = 119899

lowast(119899119871) =

1 + (119899119871119905119871(119899119871+ 119905119871)) = 2


119894Then by sym-

metry 119904119871119895gt 119904119865

119895 Because 119904119871

119894= 119899119871

119894119892119871

119894(119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895) we have

120587119871

119894=

119899119871

119894119892119871

119894(1 + 119892

119871

119894)

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2

= 119904119871

119894(1 + 119892

119871

119894)

119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2

(A3)

Similarly

120587119865

119895= 119904119865

119895(1 + 119892

119865

119895)

119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2 (A4)

Then120587119871119894120587119865

119895= 119904119871

119894(1+119892119871

119894)119904119865

119895(1+119892119865

119895) Similarly120587119865

119894120587119871

119895= 119904119865

119894(1+

119892119865

119894)119904119871

119895(1 + 119892

119871

119895) Now because 119904119871

119894gt 119904119865

119894and 119904119871

119895gt 119904119865

119895 we have

119904119871

119894119904119865

119895gt 119904119865

119894119904119871

119895 Because 119899119871

119894lt 119899119865

119894and 119899

119871

119895lt 119899119865

119895(Lemma 1) we

have (1+119892119871119894)(1+119892

119865

119895) = (1+(119905

119894(119899119871

119894+119905119894)))(1+(119905

119895(119899119865

119895+119905119895))) gt

(1 + (119905119894(119899119865

119894+ 119905119894)))(1 + (119905

119895(119899119871

119895+ 119905119895))) = (1 + 119892

119865

119894)(1 + 119892

119871

119895)


119895gt 120587119865

119894120587119871


120587119871

119894120587119865

119894gt 120587119865

119895120587119871

119895 By Lemma 1 120587119871

119894lt 120587119865

119894 Therefore 120587119865

119895120587119871

119895lt

120587119871

119894120587119865

119894lt 1 or 120587119865

119895lt 120587119871



Because 120573(1) = 119905119865(1 + 2119905





119871


Now 1205731015840(0) = 119905119871(6119905119865+ 119905119871+ 2 + 2119905

119871119905119865+ 41199052

119865minus 21199051198711199052

119865) and

[1205731015840(119899119871)]119899119871minus 120573(119899

119871) = 2120573

3(119899119871)3+ 1205732(119899119871)2minus 1205730 where 120573

3=

minus[(1 + 119905119871+ 119905119865)2+ 1199052

119865] lt 0 120573

2= [(1 + 119905

119865)(1 + 2119905

119865minus 1199052

119871) minus 41199051198711199052

119865]

and 1205730= 1199052

119871(1 + 119905

119865)(1 + 2119905




1205731015840(0) gt 0 If120573




Acknowledgments



Endnotes















119897119894119908119897119894subject to 119902

119894= (119905119894119897119894)12 in


119894) 1199022








119896and therefore










119896= (119886 minus



119896=

[119886 minus 119887119876minus119896





119894) = 119887+



119894119899119894)


119894119899119894) = 119887 + 1 (2119905



119894119899119894) gt 119887 + (12119905

119894)



1198962119905119896)


119896= (119886 minus 119889119876


(1199022



= [119886 minus 119889119876minus119896

minus (119887 + (12119905119896)) 119902119896] 119902119896 which is as












References







































Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal






1 1

34

119905119905

119901119876

1199010

119901119872

1199010

120587119865

119894

1205870

119894

120587119871

119894

1205870

119894

120587119872

119894

1205870

119894

916

00

00

98

Pric

e rat

io

Profi

trat

io



119895


119894up to 119899

119894=







119865= 0


119865gt 119899119871

gt 1 As a result119866 (119899119872

119894 119899119872

119895) gt 119866 (119899

119871

119894 119899119865









119882119876(119871 = 119894 119865 = 119895) lt 119882

119876(119871 = 119895 119865 = 119894) if and only if 119905

119894gt 119905119895




Appendix


119896= 119901119902119896minus119862(119905119896 119902119896) = (119886 minus 119887119876


1198962119905119896)


= sum119889isin119863119896


is 119886 minus 119887119876minus119896

minus 2119887119902119896minus (1119905

119896)119902119896

= 0 or equivalently 119887119902119896

=




119896(1+119866))Then120587

119896= (1198862119892119896(1+119892119896))(2119887(1+119866)

2)


119899119894120587119894(119899119894 119899119895)where 119899

119895= 119899lowast(119899119894)

At 119899119894= 119899119871

119894119889120587119894119889119899119894= (120597120587

119894120597119899119894)+(120597120587

119894120597119899119895) (119889119899119895119889119899119894) = 0 But

120597120587119894120597119899119895lt 0 while 119889119899

119895119889119899119894= 119899lowast1015840(sdot) gt 0 so 120597120587

119894(119899119871

119894)120597119899119894gt 0

Meanwhile 119899119872119894

= argmax119899119894120587119894(119899119894 119899119895) so 120597120587

119894(119899119872

119894)120597119899119894= 0

Since 120597120587119894(119899119871

119894)120597119899119894gt 120597120587

119894(119899119872

119894)120597119899119894and 120597

21205871198941205971198992

119894lt 0 we

conclude that 119899119871119894lt 119899119872

119894

By symmetry 119899119871119895

lt 119899119872

119895 But 119899119865

119894= 119899lowast(119899119871

119895) and 119899

119872

119894=

119899lowast(119899119872


119894lt 119899119872

119894 Now we must

show that 119899119871119894lt 119899119865


119894ge 119899119865

119894) Then by

symmetry 119899119871119895ge 119899119865

119895

119899119865

119894= 119899lowast(119899119871

119895)

ge 119899lowast(119899119865

119895) (because 119899

119871

119895ge 119899119865

119895and 119899


= argmax119899119894

120587119894(119899119894 119899119865

119895)

gt argmax119899119894

120587119894(119899119894 119899lowast(119899119894))

(because120597120587119894

120597119899119895


119865

119895= 119899lowast(119899119871

119894))

= 119899119871

119894

(A1)


120587119865

119894= max119899119894

120587119894(119899119894 119899119871

119895)

gt max119899119894

120587119894(119899119894 119899119865

119895)


120597120587119894

120597119899119895

lt0 and 119899119871119895lt119899119865

119895)

gt 120587119894(119899119871

119894 119899119865

119895)

= 120587119871

119894

= max119899119894

120587119894(119899119894 119899lowast(119899119894))

gt 120587119894(119899119872

119894 119899lowast(119899119872

119894)) (because 119899

119871

119894= 119899119872

119894)

= 120587119872

119894

(A2)


lently 120573(119899119871)1199052

1198711199052

119865= 0 When 119905


119895rarr infin

120573(119899119871)1199052

1198711199052

119865= minus2119899

119871+ 2 so 119899

119871= 1 Then 119899119865 = 119899

lowast(119899119871) =

1 + (119899119871119905119871(119899119871+ 119905119871)) = 2


119894Then by sym-

metry 119904119871119895gt 119904119865

119895 Because 119904119871

119894= 119899119871

119894119892119871

119894(119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895) we have

120587119871

119894=

119899119871

119894119892119871

119894(1 + 119892

119871

119894)

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2

= 119904119871

119894(1 + 119892

119871

119894)

119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2

(A3)

Similarly

120587119865

119895= 119904119865

119895(1 + 119892

119865

119895)

119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2 (A4)

Then120587119871119894120587119865

119895= 119904119871

119894(1+119892119871

119894)119904119865

119895(1+119892119865

119895) Similarly120587119865

119894120587119871

119895= 119904119865

119894(1+

119892119865

119894)119904119871

119895(1 + 119892

119871

119895) Now because 119904119871

119894gt 119904119865

119894and 119904119871

119895gt 119904119865

119895 we have

119904119871

119894119904119865

119895gt 119904119865

119894119904119871

119895 Because 119899119871

119894lt 119899119865

119894and 119899

119871

119895lt 119899119865

119895(Lemma 1) we

have (1+119892119871119894)(1+119892

119865

119895) = (1+(119905

119894(119899119871

119894+119905119894)))(1+(119905

119895(119899119865

119895+119905119895))) gt

(1 + (119905119894(119899119865

119894+ 119905119894)))(1 + (119905

119895(119899119871

119895+ 119905119895))) = (1 + 119892

119865

119894)(1 + 119892

119871

119895)


119895gt 120587119865

119894120587119871


120587119871

119894120587119865

119894gt 120587119865

119895120587119871

119895 By Lemma 1 120587119871

119894lt 120587119865

119894 Therefore 120587119865

119895120587119871

119895lt

120587119871

119894120587119865

119894lt 1 or 120587119865

119895lt 120587119871



Because 120573(1) = 119905119865(1 + 2119905





119871


Now 1205731015840(0) = 119905119871(6119905119865+ 119905119871+ 2 + 2119905

119871119905119865+ 41199052

119865minus 21199051198711199052

119865) and

[1205731015840(119899119871)]119899119871minus 120573(119899

119871) = 2120573

3(119899119871)3+ 1205732(119899119871)2minus 1205730 where 120573

3=

minus[(1 + 119905119871+ 119905119865)2+ 1199052

119865] lt 0 120573

2= [(1 + 119905

119865)(1 + 2119905

119865minus 1199052

119871) minus 41199051198711199052

119865]

and 1205730= 1199052

119871(1 + 119905

119865)(1 + 2119905




1205731015840(0) gt 0 If120573




Acknowledgments



Endnotes















119897119894119908119897119894subject to 119902

119894= (119905119894119897119894)12 in


119894) 1199022








119896and therefore










119896= (119886 minus



119896=

[119886 minus 119887119876minus119896





119894) = 119887+



119894119899119894)


119894119899119894) = 119887 + 1 (2119905



119894119899119894) gt 119887 + (12119905

119894)



1198962119905119896)


119896= (119886 minus 119889119876


(1199022



= [119886 minus 119889119876minus119896

minus (119887 + (12119905119896)) 119902119896] 119902119896 which is as












References







































Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal







119896(1+119866))Then120587

119896= (1198862119892119896(1+119892119896))(2119887(1+119866)

2)


119899119894120587119894(119899119894 119899119895)where 119899

119895= 119899lowast(119899119894)

At 119899119894= 119899119871

119894119889120587119894119889119899119894= (120597120587

119894120597119899119894)+(120597120587

119894120597119899119895) (119889119899119895119889119899119894) = 0 But

120597120587119894120597119899119895lt 0 while 119889119899

119895119889119899119894= 119899lowast1015840(sdot) gt 0 so 120597120587

119894(119899119871

119894)120597119899119894gt 0

Meanwhile 119899119872119894

= argmax119899119894120587119894(119899119894 119899119895) so 120597120587

119894(119899119872

119894)120597119899119894= 0

Since 120597120587119894(119899119871

119894)120597119899119894gt 120597120587

119894(119899119872

119894)120597119899119894and 120597

21205871198941205971198992

119894lt 0 we

conclude that 119899119871119894lt 119899119872

119894

By symmetry 119899119871119895

lt 119899119872

119895 But 119899119865

119894= 119899lowast(119899119871

119895) and 119899

119872

119894=

119899lowast(119899119872


119894lt 119899119872

119894 Now we must

show that 119899119871119894lt 119899119865


119894ge 119899119865

119894) Then by

symmetry 119899119871119895ge 119899119865

119895

119899119865

119894= 119899lowast(119899119871

119895)

ge 119899lowast(119899119865

119895) (because 119899

119871

119895ge 119899119865

119895and 119899


= argmax119899119894

120587119894(119899119894 119899119865

119895)

gt argmax119899119894

120587119894(119899119894 119899lowast(119899119894))

(because120597120587119894

120597119899119895


119865

119895= 119899lowast(119899119871

119894))

= 119899119871

119894

(A1)


120587119865

119894= max119899119894

120587119894(119899119894 119899119871

119895)

gt max119899119894

120587119894(119899119894 119899119865

119895)


120597120587119894

120597119899119895

lt0 and 119899119871119895lt119899119865

119895)

gt 120587119894(119899119871

119894 119899119865

119895)

= 120587119871

119894

= max119899119894

120587119894(119899119894 119899lowast(119899119894))

gt 120587119894(119899119872

119894 119899lowast(119899119872

119894)) (because 119899

119871

119894= 119899119872

119894)

= 120587119872

119894

(A2)


lently 120573(119899119871)1199052

1198711199052

119865= 0 When 119905


119895rarr infin

120573(119899119871)1199052

1198711199052

119865= minus2119899

119871+ 2 so 119899

119871= 1 Then 119899119865 = 119899

lowast(119899119871) =

1 + (119899119871119905119871(119899119871+ 119905119871)) = 2


119894Then by sym-

metry 119904119871119895gt 119904119865

119895 Because 119904119871

119894= 119899119871

119894119892119871

119894(119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895) we have

120587119871

119894=

119899119871

119894119892119871

119894(1 + 119892

119871

119894)

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2

= 119904119871

119894(1 + 119892

119871

119894)

119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2

(A3)

Similarly

120587119865

119895= 119904119865

119895(1 + 119892

119865

119895)

119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895

(1 + 119899119871

119894119892119871

119894+ 119899119865

119895119892119865

119895)

2 (A4)

Then120587119871119894120587119865

119895= 119904119871

119894(1+119892119871

119894)119904119865

119895(1+119892119865

119895) Similarly120587119865

119894120587119871

119895= 119904119865

119894(1+

119892119865

119894)119904119871

119895(1 + 119892

119871

119895) Now because 119904119871

119894gt 119904119865

119894and 119904119871

119895gt 119904119865

119895 we have

119904119871

119894119904119865

119895gt 119904119865

119894119904119871

119895 Because 119899119871

119894lt 119899119865

119894and 119899

119871

119895lt 119899119865

119895(Lemma 1) we

have (1+119892119871119894)(1+119892

119865

119895) = (1+(119905

119894(119899119871

119894+119905119894)))(1+(119905

119895(119899119865

119895+119905119895))) gt

(1 + (119905119894(119899119865

119894+ 119905119894)))(1 + (119905

119895(119899119871

119895+ 119905119895))) = (1 + 119892

119865

119894)(1 + 119892

119871

119895)


119895gt 120587119865

119894120587119871


120587119871

119894120587119865

119894gt 120587119865

119895120587119871

119895 By Lemma 1 120587119871

119894lt 120587119865

119894 Therefore 120587119865

119895120587119871

119895lt

120587119871

119894120587119865

119894lt 1 or 120587119865

119895lt 120587119871



Because 120573(1) = 119905119865(1 + 2119905





119871


Now 1205731015840(0) = 119905119871(6119905119865+ 119905119871+ 2 + 2119905

119871119905119865+ 41199052

119865minus 21199051198711199052

119865) and

[1205731015840(119899119871)]119899119871minus 120573(119899

119871) = 2120573

3(119899119871)3+ 1205732(119899119871)2minus 1205730 where 120573

3=

minus[(1 + 119905119871+ 119905119865)2+ 1199052

119865] lt 0 120573

2= [(1 + 119905

119865)(1 + 2119905

119865minus 1199052

119871) minus 41199051198711199052

119865]

and 1205730= 1199052

119871(1 + 119905

119865)(1 + 2119905




1205731015840(0) gt 0 If120573




Acknowledgments



Endnotes















119897119894119908119897119894subject to 119902

119894= (119905119894119897119894)12 in


119894) 1199022








119896and therefore










119896= (119886 minus



119896=

[119886 minus 119887119876minus119896





119894) = 119887+



119894119899119894)


119894119899119894) = 119887 + 1 (2119905



119894119899119894) gt 119887 + (12119905

119894)



1198962119905119896)


119896= (119886 minus 119889119876


(1199022



= [119886 minus 119889119876minus119896

minus (119887 + (12119905119896)) 119902119896] 119902119896 which is as












References







































Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal






Endnotes















119897119894119908119897119894subject to 119902

119894= (119905119894119897119894)12 in


119894) 1199022








119896and therefore










119896= (119886 minus



119896=

[119886 minus 119887119876minus119896





119894) = 119887+



119894119899119894)


119894119899119894) = 119887 + 1 (2119905



119894119899119894) gt 119887 + (12119905

119894)



1198962119905119896)


119896= (119886 minus 119889119876


(1199022



= [119886 minus 119889119876minus119896

minus (119887 + (12119905119896)) 119902119896] 119902119896 which is as












References







































Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal









119896and therefore










119896= (119886 minus



119896=

[119886 minus 119887119876minus119896





119894) = 119887+



119894119899119894)


119894119899119894) = 119887 + 1 (2119905



119894119899119894) gt 119887 + (12119905

119894)



1198962119905119896)


119896= (119886 minus 119889119876


(1199022



= [119886 minus 119889119876minus119896

minus (119887 + (12119905119896)) 119902119896] 119902119896 which is as












References







































Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal









References







































Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal













Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal





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