THECHOICEOFSTOCKOWNERSHIPSTRUCTURE: M P A R ELL...THECHOICEOFSTOCKOWNERSHIPSTRUCTURE:...

THE CHOICE OF STOCK OWNERSHIP STRUCTUREAGENCY COSTS MONITORING

AND THE DECISION TO GO PUBLIC

MARCO PAGANO AND AILSA ROELL

From the viewpoint of a companyrsquos controlling shareholder the optimalownership structure generally involves some measure of dispersion to avoidexcessive monitoring by other shareholders The optimal dispersion of shareownership can be achieved by going public but this choice also entails some costs(the cost of listing and the loss of control over the shareholder register) If thecontrolling shareholder sells shares privately instead he avoids the costs of goingpublic but must tolerate large external shareholders who may monitor him tooclosely Thus the owner faces a trade-off between the cost of providing a liquidmarket and overmonitoring The incentive to go public is stronger the larger theamount of external funding required The listing decision is also affected by thestrictness of disclosure rules for public relative to private rms and the legallimits on bribes aimed at dissuading monitoring by shareholders

I INTRODUCTION

Much existing literature in corporate nance focuses onpublic companies with a large number of dispersed shareholdersand entrenched managers who effectively control the company Inthis situation typical of many large public companies in theUnited States shareholders have little incentive to monitormanagers and prevent them from putting their own personalinterest above that of the companyrsquos shareholders A potentialremedy is to have a less dispersed share ownership structureshareholders with a large stake in the company have a greaterincentive to play an active role in corporate decisions because theypartially internalize the benets from their monitoring effort

This paper is part of the research project on lsquolsquoThe decision to go public andthe stock market as a source of capitalrsquorsquo promoted by the Ente lsquolsquoLuigi Einaudirsquorsquo pergli studi monetari bancari e nanziari An earlier version was circulated under thetitle lsquolsquoThe Choice of Stock Ownership Structure Agency Costs Monitoring andLiquidityrsquorsquo We greatly beneted from the suggestions of an anonymous referee andfrom comments by Patrick Bolton Thomas Chemmanur Mikel Fishman DenisGromb Oliver Hart Stewart Myers Fausto Panunzi Rafael Repullo Michel RobeAndrei Shleifer Chester Spatt and seminar participants at institutions in Romeand the University of Bern the London School of Economics Harvard Universitythe Massachusetts Institute of Technology Tilburg University the University ofToulouse the University of Turin the 1996 WFA meetings and the 1997 AFAmeetings Carlo Croff and Luca Filippa gave us helpful advice on points of law Weacknowledge nancial support from the EC under its SPES plan from the ItalianMinistry for Universities and Scientic Research (MURST) and from the ItalianNational Research Council (CNR)

r 1998 by the President and Fellows of Harvard College and the Massachusetts Institute ofTechnologyThe Quarterly Journal of Economics February 1998

These shareholders also have a greater incentive to place atakeover bid for a badly managed company because they appropri-ate part of the appreciation of the shares due to the improvedmanagement So via better monitoring and takeover bids largeshareholders alleviate the agency problem arising from the sepa-ration between ownership and control as argued by Shleifer andVishny [1986] and Grossman and Hart [1980] Empirical evidenceon the monitoring role of large shareholders is reported by MorckShleifer and Vishny [1988] Wruck [1989] Franks Mayer andRenneboog [1997] Yafeh and Yosha [1995] Renneboog [1996] andothers

In most European countries share ownership is much moreconcentrated than in the United States1 Most companies are notlisted on stock exchanges and even when they are a single largeshareholder or a tightly knit group of shareholders retains acontrolling stake in the company Since this ownership structuremakes companies impervious to takeovers the controlling stake iscommonly retained by the founder of the company and by hisdescendants even when the company is large and publicly listedThe controlling shareholder generally takes active interest inrunning the company by choosing the management and directlytaking executive positions In this situation the main conict ofinterest is that between the controlling shareholder and theminority shareholders2 rather than between hired managers andthe generality of shareholders as in the United States Thisconict can take several formsmdashfrom the diversion of corporateearnings to the advantage of the controlling shareholder to theuse of the companyrsquos assets to favor other companies owned byhim

This conict of interest cannot be analyzed by mechanicallyapplying the models that focus on agency problems betweenmanagement and shareholders In contrast with a manager theinitial owner of a company himself determines the distribution ofshareholding sizes ie designs the companyrsquos ownership struc-ture Assuming that the initial owner retains a controlling stake

1 In Italy 88 percent of the manufacturing companies (accounting for 91percent of the employed labor force) are controlled by one person or family [Barcaet al 1994] and in the majority of publicly listed companies one shareholder ownsmore than 50 percent of the voting rights [Zingales 1994] In Germany 85 percentof the public companies have a major shareholder and in France the correspondinggure is 79 percent to be compared with only 16 percent in the United Kingdom[Franks and Mayer 1995 1997]

2 As in the classic paper by Jensen and Meckling [1976]

QUARTERLY JOURNAL OF ECONOMICS188

the degree of discretion that he maintains in running the companywill depend on how concentrated are the stakes of the outsideshareholders A large shareholder such as a venture capitalistwill want to monitor his conduct more closely than a large group ofsmall investors As one guide to raising private funding for youngcompanies puts it the question is

whether the founder of the business is willing to allow his creation to besullied with petty considerations such as cash and short-term prot Thechairman of a company which has recently received venture capital fundingremarked of its managing director lsquolsquoIn his head he understands the need forthe controls and inuences the investor wants but in his heart he cannotreconcile himself to sharing his authorityrsquorsquo [Sharp 1991 p 59]3

Thus the founder of the company may want to temper theinvolvement of outside shareholders by limiting their stakes Ourmodel captures this intuition We show that the optimal owner-ship structure chosen by the entrepreneur generally involvessome measure of dispersion (more than one external investor) Itmay also involve a certain degree of monitoring by a large externalshareholder becausemdashin order to obtain equity capital morecheaplymdashthe initial owner needs to restrain his own futuretendency to stray from value maximization Thus the ownershipstructure acts as a precommitment device to limit agency costs4

But there are other considerations involved in widening theshareholder base of a company In a private company adding moreshareholders can be very costly because each new shareholder inturn must expend time and effort to check that the company is asound investment Moreover when he wants to liquidate hisstake he must incur further costs to search for a counterpartBeyond a critical number of shareholders it becomes morecost-effective to list the stock publicly so that dissemination of

3 This point often surfaces in discussions of venture capital in the nancialpress For example The Economist [January 25 1997] mentions that lsquolsquoentrepre-neurs are sometimes suspicious of venture capitalists [because] they have forricher or poorer married a meddlesome outsider [who] will often demandmanagement changesrsquorsquo [p 21]

4 Other recent papers model the optimal design of the ownership structureby the initial owner of the rm Zingales [1995] argues that selling an initial staketo dispersed shareholders indirectly strengthens the owner rsquos bargaining power ina subsequent transfer of control and thereby the total revenue obtained Mello andParsons [1994] focus on the design of a revenue-maximizing auction method in asetting where the investorsrsquo valuations and the large investorrsquos potential contribu-tion are both private information The model of Stoughton and Zechner [forthcom-ing] is closer to ours because it focuses on the benets of monitoring by largeshareholders They consider revenue-maximizing auction techniques under theassumption that the entrepreneur is obliged to sell all shares at a common price aconstraint which we do not impose

THE CHOICE OF STOCK OWNERSHIP STRUCTURE 189

information and trading activity are more centralized In apublicly listed company expanding the shareholder base is quiteinexpensive although there is a large xed cost to listing (Weargue that in part this cost derives from the fact that upon listingthe company its initial owner is no longer able to prevent changesin the identity of his external shareholders)

Thus if the initial owner of the company wants to sell out tomany small shareholders he must go public If he keeps thecompany private instead he cannot sell minority stakes to morethan a few large shareholders As a result he saves the cost oflisting the company on the exchange but has to accept a degree ofmonitoring far greater than that which minimizes agency costsThis is consistent with evidence that in going public companiesattempt to limit the sizes of individual stakes Booth and Chuarsquos[1996] evidence suggests that U S issuers underprice in order toachieve dispersed ownership Similarly Brennan and Franks[1997] nd that in U K initial public offerings owners userationing in the share allocation process in order to reduce the sizeof the largest shareholdings thus making it less likely that theywill be subject to the monitoring of a large shareholder (or to ahostile takeover)

The idea that minority shareholders may monitor too muchseems at rst sight surprising Most of the relevant literature isconcerned with the opposite problem shareholders generallymonitor too little to maximize the value of the company because offree-riding problems But our perspective is different because wetake the viewpoint of the initial owner of the rm He cares notonly about the market value of the company but also about hisfuture private benets as manager of the rm5 So from his pointof view a level of monitoring that maximizes the market value ofthe rm may be excessive This overmonitoring result parallelsthat obtained by Burkart Gromb and Panunzi [1997] whopredict that a concentrated ownership structure leads to highmonitoring and low managerial initiative

Our model has several predictions First the danger ofovermonitoring increases with the amount of outside nance to beraised So companies are more likely to go public if they need alarge amount of new funding relative to their value This empiri-

5 A few other papers take this perspective notablyAghion and Bolton [1992]where capital structure is designed to optimally allocate control to the entrepre-neur in those states where outside investors would inefficiently destroy his privatebenets


cal prediction is supported by Pagano Panetta and Zingales[forthcoming] who nd that independent companies tend to gopublic in the wake of major investment programs A secondprediction of the model is that going public is attractive ifextracting private benets is not very wasteful since then thebenets of monitoring by a large shareholder are lower A thirdprediction is that the incentive to go public is greater if publiccompanies are subject to more stringent disclosure rules and moretransparent accounting standards than private companies thusmaking monitoring more effective

The choice between public and private nancing is quitedifferent if the controlling shareholder can pay off large minorityshareholders to induce them to monitor less at the expense ofother minority shareholders Then private nancing from a singleexternal shareholder no longer implies overmonitoring and strictlydominates going public This may help to explain why in manyEuropean countries companies seldom go public and prefer torely on private partners or internally generated earnings tonance investment in those countries minority shareholdersreceive so little legal protection that the controlling shareholdercan use corporate resources to placate other large shareholders

Finally we investigate whether the ownership structuredesigned by the initial owner of the company will persist overtime ie whether minority shareholders have an incentive toalter their relative stakes by subsequent trading of shares Weshow that this incentive is absent if all the investors have perfectinformation about the amount traded and the identity of tradersand are forward-looking Then prices move so as to discourage areshuffle of the ownership structure If the secondary market isinstead less than fully transparent or some shareholders aremyopic the ownership structure will not be stable

The paper is structured as follows Section II lays out themodel and characterizes the optimal shareholder structure cho-sen by the initial owner In Section III we discuss the costs ofwidening the shareholder base and relate the analysis of theshareholder structure to the going public decision In Section IVwe show how the situation changes when the external sharehold-ers can act cooperatively (subsection IV1) and when the monitor-ing shareholder is able to collude with the controlling shareholder(subsection IV2) In Section V we study the stability of the initialownership structure and in Section VI we briey discuss the


implications of allowing for alternatives to equity nancingSection VII concludes

II THE MODEL

II1 Assumptions

We consider an entrepreneur who is the single owner of acompany worth V0 in which he has invested all his wealthSuppose that he needs to raise an amount of external nance I totake advantage of a new investment opportunity (or alternativelyto use for personal consumption) External nance is assumed totake the form of straight equity (the use of debt or hybridsecurities is discussed in Section VI) Outside shareholders haveno market power they provide the funds I at a price that justcovers their costs and the initial owner of the company appropri-ates the entire surplus from the investment

Utility function The entrepreneurrsquos utility is linear in wealthand private benets (B ) These benets can be widely interpretedmanagerial perks excessive salaries a quiet life nepotisticappointments unprotable lsquolsquopetrsquorsquo projects advantageous dealswith other rms in which he has a stake etc For simplicity anyvalue reduction D that he inicts on the company is assumed toyield private benets at a constant rate b 1 The parameter bcan be thought of as the value that the entrepreneur places oneach dollar he diverts from the company The fact that b 1means that diversion is inefficient there is a loss of 1 2 b for eachunit diverted He cannot extract more than a maximum amountD so that his private benets are

(1) B 5 bD for D D

When the owner diverts an amount D the security value of therm is V 5 V 2 D where V is the maximum (zero-diversion) valueof the rm

Supply of external nance The supply of external capital isperfectly elastic potential shareholders are risk neutral so thattheir required return is independent of the size of their stake Werule out rationing of external funds by assuming that the securityvalue of the company is always larger than the amount of nanceto be raised irrespective of the severity of the diversion problem

(2) V 2 D $ I


Control The owner of the company is assumed to retainenough voting stock that his position as manager of the companyis unassailable for instance by issuing nonvoting stock to outsideinvestors Alternatively his tenure as manager may be assuredbecause he is the only person capable of running the companysuccessfully So we abstract from the issue of changes in corporatecontrol which are very rare events for the type of companies thatwe are trying to analyze

Monitoring Any shareholder can monitor the owner of therm and hold down the amount of diversion to D 5 D(M ) whereM is the amount he spends on monitoring This monitoring may beexerted via a direct involvement in the company when venturecapital is raised lsquolsquoalmost invariably the investor will insist on theright to appoint a nonexecutive directorrsquorsquo especially if the invest-ment is large or particularly risky [Sharp 1991 p 160] Alterna-tively external shareholders may spend time and effort gatheringinformation about the company insisting on audits orchestratingvotes at shareholder meetings generating publicity or takinglegal action against managementrsquos policies The extent to whichthese monitoring activities can be pursued depends on the legaland regulatory protection of shareholder rights which variesgreatly across countries [La Porta Loeacutepez-de-Silanes Shleifer andVishny 1996] We assume that the more monitoring there is theless the owner can divert (D 8(M ) 0) and that there is a xed costM to monitoring so that shareholders with small enough stakeshave no incentive to monitor In the absence of monitoring theentrepreneur can divert the maximal amount ie D(M ) 5 D forM M The monitoring technology has increasing marginal costs(D 9(M ) 0 for M $ M ) Duplicate monitoring is assumed to beuseless if two shareholders spend M0 and M1 M0 respectivelyon monitoring then diversion will be held down to D(M1) and M0

will be wasted The baseline version of the model is developedunder the assumption that external shareholders take theirmonitoring decision noncooperatively so that in Nash equilibriumone of them monitors and the others free ride

II2 The Entrepreneurrsquos Financing Decision

The entrepreneur designs his shareholder base in such a wayas to maximize the sum of his monetary wealth and his privatebenets In particular he decides whether to go ahead with theinvestment and if so how to allocate stakes to external sharehold-


ers In so doing the entrepreneur must try to mitigate the agencyproblem between himself and the outside shareholders possiblyvia the presence of a monitoring shareholder

Formally the entrepreneurrsquos problem is to choose his ownstake in the rm a and a prospective monitorrsquos stake b so as tomaximize the total surplus that arises in the rm dened as themarket value of the whole rm minus the cost of the investmentplus his private benets minus monitoring costs Since he extractsall the surplus and leaves the outside shareholders only with theirreservation value maximizing the total surplus is equivalent tomaximizing his utility U

(3) maxa b

U V 2 I 1 B 2 M 5 V 2 I 2 (1 2 b )D 2 M

subject to incentive compatibility constraints for the entrepreneurand the monitor and to participation constraints for the entrepre-neur and the outside shareholders

The solution to this problem yields the prices at which theentrepreneur sells stakes to the outside shareholders Sharehold-ers who are expected to incur monitoring costs in equilibriumobtain shares at a compensating discount relative to other exter-nal shareholders Thus we assume that the entrepreneur is ableto sell shares to different outside investors at different pricesreecting their willingness to pay (in contrast to Stoughton andZechner [forthcoming])

To solve problem (3) there are three decision stages to beanalyzed

(i) the entrepreneur chooses the companyrsquos ownership struc-ture ( a and b )

(ii) external shareholders choose the level of monitoring(M )

(iii) the entrepreneur chooses the level of diversion (D )This three-stage problem is to be solved by backward

inductionIn stage (iii) the entrepreneur chooses the level of diversion D

so as to maximize the value of his stake plus his private benet

(4) D 5 arg max0 D D (M )

a (V 2 D ) 1 bD 5D (M ) if a b

0 if a $ b

Due to the linearity of the diversion technology the entrepre-neurrsquos policy is very simple Either he extracts as many privatebenets as he can or he extracts none the agency problem


disappears (D 5 0) if the entrepreneurrsquos stake a is large enoughThis depends on the size of the required investment and on thevalue of the rm because a is determined by the participationconstraint of the outside shareholders The latter must receiveenough shares to compensate them for the money they invest Iand spend on monitoring M

(5) (1 2 a )V $ I 1 M

We now turn to stage (ii) the monitoring decision A largeshareholder with stake b will choose the monitoring level M so asto minimize the sum of his monitoring costs and his share of thermrsquos loss from diversion

(6) minM

M 1 b D

where D is given by equation (4)Obviously there is no monitoring if the entrepreneur has no

incentive to divert resources M 5 0 if a $ b ie if

(7) I (1 2 b)V

This is the rst-best case In what follows we focus instead on themore interesting opposite situation where the investment is solarge that the entrepreneurrsquos stake a b and he wants to divertresources from the company (the no-rst-best assumption)

(8) I (1 2 b)V

Recall that we have also assumed V 2 D $ I in equation (2) Notethat our analysis therefore applies to a range of values of I suchthat (1 2 b)V I V 2 D Thus b must be sufficiently large if ouranalysis is to be relevant we must have b DV6

Let M( b ) denote the monitoring level chosen by the externalshareholder ie the solution to problem (6) It is readily shownthat the larger the external shareholderrsquos stake the more he willwant to monitor ie the optimal M( b ) is increasing in b 7

6 However note that the assumption that V 2 D $ I is sufficient but notnecessary to ensure that nancing is available because thanks to monitoring D isgenerally below its maximum level D An alternative assumption that can rule outrationing is U0 bD where U0 is the entrepreneur rsquos reservation utility ie theprojectrsquos private benets alone are not enough to make it attractive to theentrepreneur Thus the entrepreneur will not even demand funding for projectsthat yield insufficient security benets to compensate the investors

7 The problem at hand is nonconvex because we do not want to rule out thecase of a xed cost of monitoring Therefore we resort to revealed preferencearguments rather than to rst-order conditions for comparative statics If M 0


Monitoring at level M reduces diversion by D 2 D(M ) andaccordingly reduces the entrepreneurrsquos private benet byb (D 2 D (M ) The total net gain from monitoring G( b ) is there-fore (1 2 b)(D 2 D (M )) minus the cost of monitoring M

(9) G ( b ) 5 (1 2 b)[D 2 D (M ( b ))] 2 M ( b )

It is worth noting that8

LEMMA 1 For all b 1 2 b the net gain from monitoring isdecreasing in the monitoring shareholderrsquos stake ie G( b ) isdecreasing in b

Intuitively via the IPO price the entrepreneur eventuallypays for the private benets he will extract later and for anymonitoring expenses incurred by shareholders From the entrepre-neurrsquos viewpoint $1 diverted yields $b private benets so that exante he is willing to fund no more than $1 2 b in monitoringexpenses for every $1 diverted But the monitoring shareholderloses $ b per $1 diverted so he is willing to spend up to $ b toprevent it Once the monitorrsquos stake b exceeds 1 2 b from theentrepreneurrsquos ex ante viewpoint there is too much monitoringand the higher the stake b the worse the problem becomes

Stage (i) is the choice of ownership structure This amounts tochoosing the large investorrsquos stake b so as to induce the amount ofmonitoring that maximizes the utility dened by (3)

(10) maxb [ [01]

U 5 V 2 I 2 (1 2 b)D (M ( b )) 2 M ( b )

5 V 2 I 2 (1 2 b)D 1 G ( b )

subject to participation constraints for both the external investors

minimizes M 1 b 0D and M 1 minimizes M 1 b 1D then

M 0 1 b 0D (M 0 ) M 1 1 b 0D (M 1 )

M 1 1 b 1D (M 1 ) M 0 1 b 1D (M 0 )

Hence

( b 1 2 b 0 )[D (M 1 ) 2 D (M 0 )] 0

and since D(M ) is a decreasing function of M

b 1 b 0 THORN M 1 $ M 0

8 The Appendix contains the proofs of this lemma and of all propositions(except Proposition 1)


and the entrepreneur First the external investors will providefunding only if the value of the company net of monitoringexpenses is large enough to reward them for their investment

(11) V 2 D (M ( b )) 2 M ( b ) $ I

Second the entrepreneurrsquos participation constraint is

(12) U $ U0

where U0 is the reservation utility which includes the value of thecompany if the investment opportunity is forgone V0

Suppose that the nancing constraint (11) does not bindThen the entrepreneur chooses b to maximize the net gain frommonitoring G( b ) This is achieved by setting the monitorrsquos stake bso as to align his private incentive with the maximization of theentrepreneurrsquos utility From (6) the monitorrsquos private objective is

minM

M 1 b D (M )

The entrepreneurrsquos objective is

(13) maxb

G ( b ) 5 (1 2 b)D 2 (1 2 b)D (M ( b )) 2 M ( b )

These two problems coincide if b 5 1 2 b This establishes

PROPOSITION 1 The monitorrsquos optimal stake is

(14) b 5 1 2 b

Intuitively the monitor must be given a percentage stake inthe company that induces him to maximize the entrepreneurrsquosutility U For every dollar diverted only a fraction 1 2 b is wasted(the remaining fraction b goes into private benets) If themonitorrsquos stake b is set equal to 1 2 b every dollar that is divertedimplies a loss of 1 2 b dollars for him so that in his monitoringdecision his incentives are perfectly aligned with maximization ofthe entrepreneurrsquos utility

Proposition 1 has an important implication If the amount ofmoney to be raised is so large that external nancing generates anagency problem (equation (8)) then the money should not beraised from a single external shareholder For then his stakewould exceed the optimal value 1 2 b and he would have theincentive to overmonitor the entrepreneur By Lemma 1 anyreduction in his stake would then improve the situation Thus theexternal funding must be provided by at least two shareholders


More specically the proposition tells us that at least (1 2 a )(1 2 b) shareholders are required to achieve an optimal owner-ship structure This ensures that the largest shareholderrsquos stakecan be set to 1 2 b (since the total external shareholdersrsquo stake is1 2 a ) This is optimal since it is reasonable to assume that thelargest shareholder will monitor in equilibrium9

To summarize our main ndings so far we write down theexpression for the entrepreneurrsquos utility when the external stakesare allocated optimally

(15) U 5max (V 2 I U0 ) if I (1 2 b)V

max (V 2 I 2 (1 2 b)D 1 G U0 ) if I (1 2 b)V

where G denotes the maximum gain from monitoring ie G 5G(1 2 b) by Proposition 1 So if the investment is low relative tothe value of the company the utility reaches its rst-best level V2 I Otherwise it is reduced by the agency cost (1 2 b)D but thisreduction is mitigated by the gain from monitoring The maxi-mum gain from monitoring G is an increasing function of theeffectiveness of the monitoring technology and of the wastefulnessof diversion 1 2 b10

Figure I provides a graphical representation of the modelThe entrepreneurrsquos private benets (B ) are measured along the

9 Suppose that there are two or more shareholders whose stakes are largeenough that each of them would want to monitor if nobody else does (that is forsome level of his monitoring expense M the resulting increase in the value of hisstake b (D 2 D (M )) exceeds M ) If their stakes are not too different in size anyone of them could act as the monitor in Nash equilibrium The equilibrium inwhich the largest shareholder monitors seems a natural lsquolsquofocal pointrsquorsquo because allthe other shareholders would most prefer this outcome In contrast if their stakesdiffer substantially there is a unique Nash equilibrium in which the largestshareholder monitors The relevant condition is that the two largest shareholdersrsquostakes ( b 1 and b 2 where b 1 b 2) differ sufficiently so that b 1D(M( b 1)) 1 M( b 1) b 1D(M( b 2)) Then the largest shareholder always chooses to monitor even if anyother shareholder does Hence no other shareholder monitors in Nash equilib-rium

10 This is shown as follows When monitoring is more effective the samelevel of monitoring M implies less diversion D 1(M ) D 0(M ) M Let M 0 and M 1

be the respective choice of M when b 5 1 2 b Then from the expression for G

G1 5 (1 2 b )D 2 (1 2 b )D 1 (M 1 ) 2 M 1

$ (1 2 b )D 2 (1 2 b )D 1 (M 0 ) 2 M 0

$ (1 2 b )D 2 (1 2 b )D 0 (M 0 ) 2 M 0 5 G0

where the rst inequality follows from the optimality of M 1 and the second fromD 1(M ) D0(M ) M

To show that G is increasing in 1 2 b one can again use a revealed preferenceargument For brevity denote 1 2 b c and the values of M(c i) D(M(ci)) and G(ci)


vertical axis and the companyrsquos value along the horizontal axis Ifthe entrepreneur were the sole owner of the company his utilitywould be the sum of his private benets and the security value of

by M i D i and G i where i 5 0 1 Suppose that c1 c0 Then

G 1 2 G 0 5 (c 1 2 c 0 )D 2 (c 1D 1 2 c0D 0 ) 1 (M 0 2 M 1 )

but

M 1 1 c 1D 1 M 0 1 c 1D 0

so that

G 1 2 G 0 $ (c 1 2 c 0 )(D 2 D 0 ) $ 0

So 1 2 b1 1 2 b0 implies that G1 $ G 0

FIGURE IPrivate Benets Monitoring and Firm Value


the company (V 1 B ) and his indifference map would be a eld ofstraight lines with slope 2 1 If he instead has a stake a 1 in thecompany his indifference curves have slope 2 a

With no monitoring the segment AOmdashwith slope 2 bmdashis thepossibility frontier indicating how the entrepreneur can trans-form corporate resources into private benets If his stake a in thecompany is greater than b there is no agency problem theentrepreneur does not divert resources the security value of thecompany is V and so the unconstrained optimum (point O ) isreached Conversely if his stake a falls short of b (the no-rst-bestassumption) he will go to the other corner solution (point A )where he extracts maximum private benets (B 5 bD ) and thesecurity value of the company falls to its minimum (V 5 V 2 D )

With monitoring the possibility frontier net of the monitoringcost M shifts to the left and becomes the curved locus in the gureEach point along this locus corresponds to a different level ofmonitoring with the security value of the rm minus the monitor-ing cost shown horizontally Point E represents the constrainedsocial optimum ie the tangency point between the new produc-tion possibility frontier and a line with slope 2 1 If this line liesabove point A then at point E the entrepreneur achieves a highertotal utility U than with no monitoring in this case monitoringhelps to reduce the agency problem Otherwise monitoring isinefficient and point A is preferable

If the monitorrsquos stake is very small he has little incentive tomonitor diversion is maximal and point A is reached If he owns100 percent of the rmrsquos equity he chooses his monitoring level soas to attain the rightmost point F at which the security value ofthe rm is maximized net of monitoring costs With a stakeintermediate between 0 and 100 percent he selects monitoringand corresponding diversion levels somewhere between A and FAs shown in Proposition 1 the constrained optimum E is reachedif the monitorrsquos stake b is equal to 1 2 b

So far we have taken for granted that the entrepreneur cancostlessly implement the optimal ownership structure just de-scribed In practice there are costs involved The next sectiondiscusses this point

III OPTIMAL MONITORING AND THE DECISION TO GO PUBLIC

III1 Costs of Widening the Shareholder Base

Private companies tend to have very few external sharehold-ers compared with publicly listed companies For example in


Italian private companies that were subsequently listed in 1982mdash1992 the median number of shareholders was three rising to over3000 after the initial public offer [Pagano Panetta and Zingalesforthcoming] This suggests that there may be substantial costs towidening the shareholder base of a private company

These costs can arise in several ways First each investorneeds to verify some basic facts about the value of the companyexistence of a sound business plan adequacy of the asset basecompetence of the management etc Evaluating such informationdemands considerable time and effort from each shareholderSecond each investor anticipates that at some stage he may haveto liquidate his stake and thus incur transaction and search costsin nding a counterpart These costs have a large xed componentper shareholder which is independent of the size of his stake Wecapture this element by assuming that there is a xed cost c foreach additional shareholder

For every additional shareholder the entrepreneur bearsthese costs in the form of a lower total issue price for the newequity Thus there is a cost to increasing the number of sharehold-ers in a private company

Once the number of external shareholders passes a criticallevel it becomes more cost-effective to go public11 Listing providesa certication function a stock price to inform all shareholders ofthe companyrsquos value and a centralized forum for trading Thenthe number of shareholders can be drastically increased at noextra cost all duplication in gathering and evaluating informa-tion as well as in search activity is virtually eliminated

But not every company would want to go public There is asubstantial xed cost to listing which includes the present valueof listing fees the cost of complying with stock exchange disclo-sure requirements and the loss of exibility due to the stricterregulations applying to public companies Ritter [1987] estimatesthat for U S companies listing fees and commissions at the timeof going public alone amount to $250000 plus 7 percent of the

11 Indeed once the shares of a company become sufficiently dispersedregulators may effectively force it to go public In Italy the stock exchangesurveillance commission (CONSOB) can list a company ex officio if it meets thelegal criteria for going public (which include a minimum of 500 shareholders) andif its shares are widely traded informally This procedure can be triggered even ifthese criteria are not met whenever CONSOB considers the listing to be lsquolsquoin theinterest of the general investing publicrsquorsquo (articles 25mdash27 of Regulation 4088 4 May1989 as modied in 1990 1991 and 1992) In the Netherlands there is a similarprocedure of ex officio listing In the United States companies with more than 750shareholders and $1000000 of total assets are subject to the same disclosurerequirements as publicly listed companies and thus already bear one of the maincosts of a public listing


money raised In addition there is the important implicit costassociated with the loss of control over the identity of the externalshareholdersmdasha point taken up in subsection III3

Formally these considerations can be captured by assumingthat the total cost of having n external shareholders is

C 5cn for a private company

L for a public company

where L denotes the xed cost of listing It seems plausible toassume that c L ie that with a single external shareholder itis cheaper not to list12

III2 The Decision to Go Public

We now integrate these costs of increasing the number ofshareholders into the analysis of optimal monitoring conducted inSection II There we argued that a single external shareholdergenerally has an incentive to overmonitor There is clearly atrade-off here increasing the number of shareholders beyond onealleviates the overmonitoring problem but it also carries a costWe now analyze this trade-off and its implications for the decisionto go public

Private nance If the company stays private the entrepre-neurrsquos utility is given by

(16) U V 2 I 2 (1 2 b)D 2 M 2 cn

Consider the entrepreneurrsquos problem in choosing sharehold-ersrsquo stakes Let us assume that the entrepreneur cannot designatea particular shareholder to do the monitoring13 As argued in theprevious subsection it is then reasonable to assume that in Nashequilibrium (any one of) the largest shareholder(s) will monitorLet us denote the optimal number of external shareholders underprivate nance by n and their stakes by b ˜ i for i 5 1 n where

12 The linear specication C 5 cn is analytically convenient but the generalndings of the paper hold for any specication where the cost C arising underprivate nancing is increasing in the number of shareholders

13 If he were able to do so he would designate the monitor assign him astake b 0 and give all the other shareholders larger stakesmdashjust large enough toensure that they will not want to monitor too In this way the stakes are as large aspossible to reduce the number of shareholders and hence cn without affecting theequilibrium level of monitoring This solution has the unattractive property thatthe smallest shareholder does the monitoring which is the worst outcome from thepoint of view of the outside shareholders collectively If these were able tocoordinate in selecting who monitors they would assign the task to the largestshareholder


b ˜ max and b ˜ min are the highest and lowest of these stakes respec-tively

The external equity raised by the entrepreneur must coverthe cost of the investment the implied transaction costs and thecost of monitoring I 1 cn 1 M So the fraction 1 2 a of thecompany to be sold to the external shareholders to satisfy theirparticipation constraint is

(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))

The optimal largest stake b ˜ max and the optimal number ofshareholders n are determined by maximizing

(18) G ( b max ) 2 cn

The number of shareholders should be kept as low as possible toeconomize on the shareholder cost cn This is achieved by choosingn to be the smallest integer greater than or equal to (1 2 a ) b ˜ maxAssuming that the entrepreneurrsquos participation constraint (12)holds his utility under private nance is

(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c

V 2 I 2 (1 2 b)D 1 G(b ˜ max) 2 cn if I (1 2 b)V 2 c

where n and b ˜ max are obtained by maximizing (18) subject to (17)and to the requirement that n is an integer greater than or equalto (1 2 a )b ˜ max As in equation (15) the rst line of the expressionis the entrepreneurrsquos utility if the investment is small enough sothat the rst best is obtained The second line indicates his utilityin the no-rst-best scenario where the monitorrsquos stake b ˜ max ischosen to minimize the sum of the agency costs and the costs of theshareholder base cn

It can be shown that

PROPOSITION 2 (OVERMONITORING PROBLEM) Under private -nance two cases can arise

(i) the individual stakes of external shareholders are allequal and greater than 1 2 b so that monitoring exceedsthe optimal level

(ii) the largest external shareholderrsquos stake is equal to 1 2 band there are at least two external shareholders so thatmonitoring is at the optimal level


If the optimal number of shareholders is sufficiently largecase (i) prevails

If the company stays private it is costly to increase thenumber of shareholders To keep this cost down the entrepreneurmaymdashas described in case (i)mdashchoose a concentrated ownershipstructure and therefore accept some overmonitoring Then interms of Figure I as the stake of each external shareholderincreases above 1 2 b we move downward along the curved locusfrom E toward F shifting the entrepreneur to a lower indifferencecurve Alternativelymdashas described in case (ii)mdashthe entrepreneurmay prefer to accept a costly increase in the number of sharehold-ers and keep monitoring at the optimal level Either way thedanger of overmonitoring imposes a cost

Public nance Once the entrepreneur decides to go publicthere is no incremental cost attached to having more sharehold-ers Therefore if feasible we will assign a stake of size 1 2 b to themonitoring shareholder and achieve optimal monitoring theother shareholders are simply assigned smaller stakes of arbi-trary size Adjusting equation (15) for the listing cost L andassuming that the entrepreneurrsquos participation constraint (12)holds we obtain the entrepreneurrsquos utility under public nance

(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )

Comparison between the two nancing regimes Now we canevaluate the pros and cons of going public by comparing theutility that the entrepreneur obtains from the two nancingregimes We shall focus on the case where an agency problemexists ie on the second line of equations (19) and (20) respec-tively The other case where the rst best is attainable is lessinteresting since the choice between the two regimes reduces to apure comparison of the listing cost L with that of the privatenanciers cn14

When an agency problem exists the entrepreneur faces atrade-off between avoiding excessive monitoring and containingthe cost of the companyrsquos shareholder base If overmonitoring is a

14 If there is no agency problem (rst line in equations (19) and (20)) thechoice of nancial regime involves simply comparing the listing cost L with that ofa single private nancier c (for in that case the whole external stake might as wellbe held by a single shareholder) Since by assumption c L the company willnever go public


serious problem he will want to have many small shareholdersand thus it will be more cost-effective for the company to go publicIf overmonitoring is not a major consideration then the entrepre-neur will prefer to stay private and put up with few largeshareholders thus avoiding the cost of a wide shareholder baseThe gain from going public to be denoted by D is composed ofthree terms

(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L

The rst term in square brackets is the gain from avoiding theovermonitoring that can arise under private nancing It isnonnegative because when the company is public the gain frommonitoring if any is maximal and therefore is at least as large asunder private nancing The second term is the cost saving fromnot having a private shareholder base These two benets are tobe weighted against the third term which is the cost of listingThis trade-off determines the decision to go public

We can then characterize how the propensity to go public isaffected by changes in the parameters of the model

PROPOSITION 3 (PROPENSITY TO GO PUBLIC) The propensity to seekpublic rather than private funding is increasing in the size ofthe investment I (holding constant either V or the net presentvalue of the investment by setting D V 5 D I ) and in themarginal cost of private shareholders c It is decreasing inthe rst-best value of the company V the cost of listing L andthe wastefulness of diversion 1 2 b

The model predicts that a company is more likely to go publicif the external funding required is large relative to the value of thecompany This is because external shareholders will have to take avery large stake in the company exacerbating the overmonitoringproblem associated with private nance A second prediction ofthe model is that if diversion is very efficient in the sense thatevery dollar diverted yields a large private benet then ex antethe entrepreneur will not want to see many resources going intomonitoring and will prefer a dispersed shareholder base so thatthe company will tend to go public

In Proposition 3 we have considered changes in the scale ofinvestment that are not accompanied by commensurate increasesin opportunities for diversion and monitoring We could conductan alternative thought experiment in which all parameters of the


model are scaled up proportionately with investment I Thiswould obviously leave the decision to go public unaffected If therst-best value of the rm after the investment V were instead toincrease less than proportionately (after all a part V0 of the rmrsquostotal value derives from its preexisting assets and not from newinvestment I ) then analytically the effect would be the same asthat of a reduction in V holding all else constant Then as astraight corollary of Proposition 3 the increase in investmentwould again lead to a greater propensity to go public So our resultthat a larger investment makes it more likely that a company willgo public appears robust

The same variables affecting the choice between nancingregimes also affect the decision to carry out the project at all viathe entrepreneurrsquos participation constraint U $ U0

PROPOSITION 4 (PROPENSITY TO INVEST) The propensity to carryout the investment is decreasing in the size of the investmentI (holding its net present value constant) in the costs L and cand in the wastefulness of diversion 1 2 b It is increasing inthe net present value of the investment opportunity V 2 I andin the rst-best value of the company V

If the scale of the investment relative to the company size isvery large a large external stake is required As a result thedanger of overmonitoring is more acute and mitigating this via awider shareholder base is costly This makes the project lessappealing to the entrepreneur As for the other comparativestatics experiments considered in the proposition they all directlyconcern the NPV of the security or the private benets from theinvestment project or both

III3 Loss of Control over Shareholder Base as a Costof Going Public

An important implicit cost of going public that we haveneglected in our analysis so far is the loss of control over theidentity of the external shareholders Private companies routinelyinclude in their statutes clauses restricting the transferability oftheir shares For instance often transactions must be sanctionedby the controlling shareholder or shares must be offered toexisting shareholders before being sold to outsiders Unsanc-tioned transactions can be declared null and void When acompany goes public such clauses must be erased from itsstatutes Groups of shareholders can still agree not to buy or sell


shares without each otherrsquos prior consent But trades violatingthese agreements cannot be reversed (at best compensatorypayments for identiable damages can be extracted) As a resultthe controlling shareholders of public companies are vulnerable tounwelcome changes in the composition of their shareholderregister

Our model can illustrate why the controlling shareholdermay want to limit the transferability of shares Such restrictionscan be used to ensure that the controlling shareholder captures abigger share of any rents that may result from future changes inshareholdersrsquo stakes

Consider the following illustrative example At time 0 exter-nal nancing is raised by selling a stake of size b to a largemonitoring shareholder (A ) and the rest (1 2 a 2 b ) to atomisticshareholders At time 1 with some probability a third party (B )arrives on the scene who has exclusive access to a much superiormonitoring technology For simplicity assume that he is able toreduce diversion to zero at virtually no cost to himself He willbargain with the existing large shareholder to buy his stake(Proposition 6 argues that he will not be able to build up a stake bytrading with atomistic shareholders)

Let us compare what happens if the company is public or if itis private

If the company is public the controlling shareholder cannotprevent A from selling his stake to B The joint gain from tradeaccruing to these two agents is

(22) b D (M ( b )) 1 M ( b )

namely the increase in the values of the stake of size b due to theelimination of all diversion plus the saving in monitoring costThey bargain bilaterally over this gain and depending on theirrelative bargaining power the outsider B captures some fractionl 0 of it

If the company is private instead the entrepreneur will vetothis deal unless he is compensated for his net loss (b 2 a )D(M( b ))namely his loss of private benet bD(M( b )) net of the gain in thevalue of his stake a (note that a b or we would be in therst-best case) He will demand a side payment of at least thisamount if he is to let the deal go through As a result the gainfrom trade shrinks to

(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))


Let us assume that b is large enough so that this expression ispositive Even so it is smaller than the net gain in the publiclisting case (22) Thus the size of the pie bargained over shrinksmoreover three parties are involved in bargaining (A B and thecontrolling shareholder) rather than two Thus one would expectparty B to extract a lower rent from the transactionmdasha smallerfraction (less than l ) of a smaller pie (expression (23))

The fact that third parties can extract more rents from apublic company than from a private one is a deterrent to goingpublic because it is ultimately borne by the original owner of thecompany Let us see why In our example the total social gainassociated with the intervention of party B is the same in bothnancing regimes (if B appears on the scene the same improve-ment in monitoring occurs in both regimes) But its ultimatedistribution is different Under our assumptions the initial largeshareholder A and the atomistic shareholders only get theirreservation values in both regimes private and public15 So thetotal social gain is shared between the controlling shareholderand the outsider B We have shown that B gets more if thecompany is public Hence the controlling shareholder is worse offThis can be seen as a cost of going public16

III4 The Effect of Tighter Disclosure Rules for Public Companies

So far we have assumed that except for the trading costs theparameters of the entrepreneurrsquos problem are the same irrespec-tive of whether his company goes public or stays private In fact inmost countries publicly traded companies are subject to consider-ably tighter disclosure requirements and more transparent ac-counting standards than private companies Therefore in publiccompanies it should be easier for a minority shareholder to verifythat the controlling shareholder is not cheating In terms of ourmodel this can be expressed by saying that going public reducesthe cost of monitoring

It turns out that tougher regulatory standards encourage

15 Any rent that they expect to receive at time 1 is fully discounted into theprice they pay at time 0 (for example in the public listing regime A expects to earna larger rent at time 1 but he pays correspondingly more for his stake at time 0)

16 Naturally the fact that third parties can rip off larger rents from a publiccompany provides them with a greater incentive to seek improvements (say in themonitoring technology) which can in turn benet other shareholders and ulti-mately the entrepreneur ex ante Thus more generally giving up control over theshareholder register can have benets as well as costs for the initial owner of thecompany


companies to go public rather than discouraging them from do-ing so

PROPOSITION 5 If under public nancing there is less scope fordiversion in the sense that for any level of monitoringexpense there is less diversion then the incremental gainfrom going public increases

At rst sight this result may seem surprising consideringthat this parameter change reduces the extraction of privatebenets and encourages monitoring even under public nancingBut the rationale of the result is clear if one considers that thisreduction of private benets stems from more efficient monitor-ing rather than from overmonitoring as under private nancingFormally what happens is that the entrepreneurrsquos ex ante utilityfrom listing UL increases because the net gain from monitoringG increases So stringent disclosure rules that make it easy toverify the conduct of controlling shareholders may actually encour-age new listings This result provides a counterweight to the usualargument that the burden and cost of stricter disclosure regula-tion tend to discourage new listings

IV COOPERATION AMONG SHAREHOLDERS

Up to this point our discussion has assumed that all sharehold-ers behave noncooperatively But one can envisage circumstancesin which they might strike informal bargains over the level ofmonitoring Monitoring is to some extent a veriable activityexamples of certiable monitoring expenses are payments toaccountants and lawyers time spent in board meetings andinformation dissemination costs17 And even when monitoring isnoncontractible a subset of shareholders may be able to sustain acooperative agreement in their monitoring activity if their interac-tions are sufficiently stable over time (see Tirole [1992]) Thensome large external shareholders may team up together to x thelevel of monitoring activity so as to maximize their joint wel-faremdasha scenario explored in subsection IV1 Similarly the control-ling shareholder might be able to strike and enforce a collusiveagreement with a large shareholder in order to reduce his

17 In contrast diversion by the entrepreneur is much less easy to measureand contract upon because it involves a very wide array of nonvalue-maximizingactivities such as pursuing lsquolsquopet projectsrsquorsquo or appointing congenial but incompetentemployees


monitoring activity Subsection IV2 analyzes this possibility Inboth cases the feasibility of cooperative behavior also depends onthe regulatory environment A stronger legal protection of minor-ity shareholders promotes cooperation among external sharehold-ers and hinders collusion between the entrepreneur and potentialmonitors As we shall see our model predicts that via both of theseeffects improved legal protection of minority shareholders in-creases companiesrsquo propensity to go public This prediction isconsistent with the cross-country evidence reported by La PortaLoeacutepez-de-Silanes Shleifer and Vishny [1997] who show thatbetter legal protection of shareholdersrsquo rights is associated with alarger value and number of publicly listed companies and of initialpublic offerings

IV1 Cooperation among External Shareholders

In the previous sections the level of monitoring was assumedto be the noncooperative equilibrium outcome of a game in whicheach shareholder simultaneously chooses how much to monitorBut external shareholders may be able to sustain cooperativebehavior in the choice of their monitoring level If so outsideshareholders can set their joint monitoring effort at the level thatwould be chosen by a single large shareholder owning their jointtotal stake 1 2 a As a result the danger of overmonitoring for theentrepreneur becomes much more serious than in the noncoopera-tive case

External shareholders are likely to cooperate if their interac-tions are sufficiently stable over time Their ability to cooperatealso depends on the legal protection afforded to minority sharehold-ers it will be greater if the law facilitates collective action inexerting voting rights (for instance voting by mail or cumulativevoting for directors) and in reacting to perceived mismanagementby directors (for instance allowing class action suits) But suchcooperation is unlikely to emerge if shareholders are sufficientlysmall both because shareholders with small stakes have a greaterincentive to free ride and because they are less likely to maintainstable mutual relationships This points to an additional benet ofgoing public the ownership structure can be arranged so thatthere is one large shareholder who monitors while all the othersare atomistic and therefore too small to sustain cooperation inmonitoring

If the rm stays private as argued before it is too costly toexpand the number of shareholders so much that they will be


small enough to behave atomistically In this situation the initialowner might just as well seek nance from just one large outsideshareholder to economize on the costs of adding more sharehold-ers As a result the monitor will hold the whole external stakeb 5 1 2 a Relative to our previous analysis therefore theovermonitoring problem under private nance is much moreacute and correspondingly the incentive to go public is strength-ened The net benet from going public in fact becomes

D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))

Intuitively b ˆ is determined by the investorrsquos participationconstraint The large outside shareholderrsquos stake is determined bythe size of the investment to be nanced Reducing it has no effecton the amount of monitoring because external shareholders willcooperate anyway Compared with the noncooperative case consid-ered in the previous section the entrepreneur can no longerregulate the incentives to monitor by ne-tuning individualshareholdersrsquo stakes Thus if large external shareholders are ableto cooperate in monitoring going public becomes more attractivebecause the overmonitoring problem under private nancing ismore acute

IV2 Collusion between the Entrepreneur and the MonitoringShareholders

So far we have portrayed the large monitoring shareholder asa boon to minority shareholders as in much of the literature onagency problems in corporations (eg Shleifer and Vishny [1986])This presupposes that the entrepreneur cannot pay the share-holder to curtail his monitoring activity This assumption isappropriate if the entrepreneur has no private wealth and thelegal system prevents him from sharing private benets withpotential monitors In many countries such as the United Statesand the United Kingdom to protect minority shareholders the lawimposes a duciary duty on managers making it difficult for themto pay off potential monitoring shareholders out of company funds


in return for reduced monitoring activity18 (In contrast the lawdoes not prevent noncontrolling shareholders from cooperatingwith each other in their monitoring activities Indeed suchcooperation generates more monitoring and thus tends to benetother shareholders)

However the view that large monitoring shareholders help toprotect minority shareholders is far less appropriate in manyEuropean countries such as Italy where large noncontrollingshareholders are often bought off by the companyrsquos managingshareholder via disguised side payments in the form of favorablesupply contracts reciprocal share deals etc In these circum-stances the presence of a large outside shareholder may harmrather than help the other minority shareholders

In our model this possibility can be captured by assumingthat once the entrepreneur has funded the investment by sellingequity to external shareholders he can collude with any potentialmonitor to set the level of monitoring activity Suppose that theentrepreneur (with stake a ) and a large shareholder or set ofcooperating shareholders (with stake b 1 2 a ) choose the levelof monitoring M so as to maximize their joint utility

maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M

It is readily shown that the level of monitoring that solves thisproblem is an increasing function of a 1 b 19 Moreover the jointutility of the entrepreneur and the monitoring shareholder(s) ismaximized if and only if a 1 b 5 1 Intuitively they must together

18 For example in the United States lsquolsquogreenmailrsquorsquo payments to see off a largeshareholder are highly controversial and have declined sharply in recent years[Herzel and Shepro 1990]

19 Let a 1 b g and consider the two values g 8 and g 9 with g 9 g 8 Denotethe corresponding optimal choices of M by M 8 and M 9 respectively Then byrevealed preference

g 9 (V 2 D (M 9 )) 1 bD (M 9 ) 2 M 9 $ g 9 (V 2 D (M 8 )) 1 bD (M 8 ) 2 M 8

and

g 8 (V 2 D (M 9 )) 1 bD (M 9 ) 2 M 9 g 8 (V 2 D (M 8 )) 1 bD (M 8 ) 2 M 8

Subtracting the second inequality from the rst

( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0

So D(M 8) $ D(M 9) and therefore M 8 M 9 since D(M ) is a decreasingfunction of M


own the whole company if they are to set the best level ofmonitoring If a 1 b 1 there will be undermonitoring In thiscase the 1 2 a 2 b stake sold to other minority shareholderswould command a low price in anticipation of collusion betweenthe entrepreneur and other large shareholder(s)

Since diffuse shareholders are redundant in preventing over-monitoring and if anything lead to undermonitoring companieshave no incentive to go public This may help to explain why incountries where the law offers little protection to minority share-holders such as Italy entrepreneurs are reluctant to nanceexpansion of their companies by listing them and selling shares todispersed shareholders

V STABILITY OF THE OWNERSHIP STRUCTURE

In the previous sections we have assumed that the stakesallotted by the initial owner to outside parties cannot be changedObviously he can ensure this by inserting in the corporate charterlimits on assembling or splitting up large share stakes But doeshe need to do so In this section we investigate the incentives ofshareholders to alter their stakes via subsequent trading andshow that subsequent trading is not a problem if market partici-pants are forward-looking and observe who is trading and howmuch This is because they will value the shares at a price thatreects the value of the rm under the new ownership structurewith its implied incentives to divert (for the entrepreneur) and tomonitor (for large shareholders)

PROPOSITION 6 (INCENTIVES TO RETRADE I) Neither the monitoringshareholder (if there is one) nor the entrepreneur have anincentive to alter their stakes by subsequent trading if allinvestors are forward-looking and have perfect knowledge ofthe size of the trades and of the identity of the traders

This extreme stability of the corporate ownership structure isa special case of Proposition 1 of Admati Peiderer and Zechner[1994] for risk-neutral investors It is based on the implicitassumption that atomistic shareholders think that even if theypersonally refuse to trade the planned reallocation of ownershipstakes will still go through As in Grossman and Hart [1980] themonitoring shareholder cannot increase his stake at a protableprice each of the atomistic shareholders will refuse to sell to himfor a price lower than that reecting the higher posttrade rmrsquos


value without realizing that their collective refusal will preventthe trade from going through

A contrasting approach exemplied by the work of Huddart[1993] takes agents to believe that if they do not trade the statusquo ownership structure will be maintained They are willing totrade if they can effect a mutually advantageous change relativeto the current situation We will refer to this as the case of lsquolsquostaticrsquorsquoexpectations For this setting we have the following result

PROPOSITION 7 (INCENTIVES TO RETRADE II) Under static expecta-tions the unique equilibrium ownership structure is onewhere all shares are held by the entrepreneur and oneexternal shareholder (or set of cooperating external sharehold-ers) and the entrepreneurrsquos stake exhausts his wealth

The analysis of this section has maintained throughout thatthe market is fully transparent in the sense that tradersrsquo identi-ties are known An analysis of retrading in a less than fullytransparent market such as one where trading is anonymous andsome of it is liquidity-driven (as in Bolton and von Thadden[forthcoming]) remains outside the scope of this paper but mayyield very different results For instance in the extreme case inwhich the market price does not react at all to orders placed by themonitoring shareholders (so that it reects the value of thecompany under the current ownership structure) it is easy toshow that the monitoring shareholder will always gain fromtrading whether he reduces or increases his stake

VI ALTERNATIVES TO EQUITY FINANCING

The reader may be wondering why we concentrate on equitynance considering that risk-free debt might be issued withoutincurring any agency cost In that case it is clearly best to issuesafe debt rather than equity for then there is no agency problemBut issuing risk-free debt is not always possible if the value of therm V is stochastic And when risky debt or other hybridsecurities are issued the agency problem that we analyze inconnection with the issuance of equity is still present Of coursethis agency problem would be tempered to the extent that the rmcan still issue securities that are as insensitive as possible to thermrsquos value this would sharpen the entrepreneur-managerrsquosincentives as much as possible

The general qualitative results of our analysis will still apply


to the choice between the private placement and the public issueof such risky securities to dispersed bondholders (in such anextension our variable V is to be reinterpreted as the expectationof a stochastic rm value) Just like small shareholders dispersedbondholders will not monitor as much as a bank with a large andrisky exposure as a result of a substantial loan or a large bondholding

Capturing these intuitions formally by extending our model ispossible but beyond the scope of this paper

VII CONCLUSIONS

This paper has analyzed the optimal design of the ownershipstructure of a company We have taken the viewpoint of an initialowner who cannot bind himself not to behave opportunisticallyonce he has given up a stake in his rm to outside investors inreturn for investment funds Since ex ante he internalizes allrelevant agency costs he must design shareholdersrsquo stake sizes soas to induce them to do the right amount of costly monitoring Hewill not choose an ownership structure so concentrated that itmaximizes rm value net of monitoring costs because he takesinto account his own future private benets Instead the sharesmust be sufficiently dispersed to ensure the optimal degree ofmonitoring

This has interesting implications for the decision to go publicIf a controlling shareholder wants to sell a minority stake toexternal shareholders without listing the company he cannot aimfor a large and dispersed shareholder base and as a result mayhave to accept more stringent monitoring If the company goespublic instead he can lsquolsquone-tunersquorsquo the stakes of external sharehold-ers to ensure that they monitor optimally But in this case he mustalso bear some costs First listing a company on the stockexchange involves some direct costs with a large xed componentSecond it entails losing control over the identity of the externalshareholders with potentially adverse consequences for the con-trolling shareholder So the initial owner of the company faces atrade-off between the costs of overmonitoring and the costs ofgoing public

The model has a number of empirical predictions First theincentive to go public is an increasing function of the amount ofexternal nance to be raised of the inefficiency of monitoring andof the value of the private benets of control All these factors limit


the degree to which monitoring is worthwhile Second companiesare more willing to list if public companies are subject to tighteraccounting and disclosure standards than private companies thisset of rules enables them to precommit to lsquolsquofair playrsquorsquo by making itharder to extract private benets because it is easier to monitorthe company Third if external shareholders are able to cooperatein their monitoring activity going public tends to become a moreattractive option relative to staying private Finally if minorityshareholders receive so little legal protection that the managingshareholder can bribe large shareholders to monitor less there isno incentive to go public

APPENDIX

Proof of Lemma 1

Again we can use a revealed preference argument DenoteM( b i) D(M( b i)) and G( b i) by M i D i and G i where i 5 0 1 Then

G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )

but by revealed preference

M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)

Note that we are in the no-rst-best region where b i 1 2 b andrecall that b 0 b 1 implies D 0 $ D 1 So b 0 b 1 implies thatG 0 $ G1 j

Proof of Proposition 2

The proof is structured as follows First we show that b ˜ max

cannot be strictly smaller than 1 2 b Second we prove that ifb ˜ max 1 2 b then all stakes are equal b ˜ i 5 b ˜ max for all i (case (i))Third we show that if b ˜ max 5 1 2 b then the number of externalshareholders n 1 (case (ii)) Last we argue that for n largeenough case (i) prevails

Step 1 Proof of b ˜ max $ 1 2 b Suppose not that is b ˜ max 1 2b Then if n 5 1 the rst-best solution obtains a situation ruledout by our assumption (8) If n 1 instead the largest external


stake b ˜ max can be increased by reducing the stakes of othershareholders or by even eliminating some of them In this wayeither we reach b ˜ max 5 1 2 b or we end up again with the rst-bestoutcome (a single external shareholder with a stake below 1 2 b)If we reach b ˜ max 5 1 2 b then we increase the net gain frommonitoring to its maximum G while holding constant or evenreducing the number of shareholders and the implied cost so thatthe initial situation cannot be optimal Hence the entrepreneurchooses b ˜ max $ 1 2 b

Step 2 Proof that if b ˜ max 1 2 b then b ˜ i 5 b ˜ max for all iSuppose not and b ˜ min b ˜ max Then it is possible to reduce b ˜ max

while increasing b ˜ min offsettingly By Lemma 1 this increases thenet gain from monitoring G( b ˜ max) without altering the number ofshareholders and thus their implied cost This shows that theassumed situation cannot be optimal

Step 3 If b ˜ max 5 1 2 b and there is only one shareholder thenthe situation is rst best which is ruled out by our assumption (8)

Step 4 Proof that for n large enough case (i) prevailsSuppose not that is b ˜ max 5 1 2 b and n 1 Hence the total stakeheld by outside shareholders S i 5 1

N b ˜ i is at most n(1 2 b) We wantto show that the entrepreneurrsquos utility can be raised by reducingthe number of shareholders and raising b ˜ max above 1 2 b With n 21 shareholders the entrepreneur can get at least the utility thathe would obtain by assigning to each external shareholder anequal stake n(1 2 b)(n 2 1) 5 (1 2 b)[1 1 1(n 2 1)] His utilitywill be increased if

G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc

which can be rewritten as

G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c

As n becomes large if the G(acute) function is continuous theleft-hand expression tends to zero so that the inequality holdsHence setting b ˜ max 5 1 2 b cannot be optimal To complete theproof we still have to show that assigning n 2 1 equal stakes


summing to a total outside stake n(1 2 b) is feasible in the sensethat the investorsrsquo participation constraint is satised Supposenot that is

I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)

We know that the participation constraint would be met with ninvestors

I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]

Subtracting the rst inequality from the second yields

M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))

Upon rearranging this inequality becomes

2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)

acute D (M (1 2 b)) 2 D Mn

n 2 1(1 2 b) $ c

where the second inequality follows from the fact that D(M(acute)) is anonincreasing function Recalling the denition of the gain frommonitoring G(acute) we have

G (1 2 b) 2 Gn

n 2 1(1 2 b) c

This inequality no longer holds if n is large enough provided thatG(acute) is continuous Thus the participation will hold for largeenough values of n j



The net gain from going public is obtained from equation (21)

(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L

using equation (9) and Proposition 1 Our proof assumes that inthe private case the number of external shareholders n is acontinuous variable neglecting that it can only take integervalues Therefore we focus on case (i) of Proposition 2 where eachexternal shareholder has an equal stake b ˜

We investigate changes in each of the parameters in turnobserving that b ˜ solves

(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c

and where M( b ˜ ) solves

(28) minM

b ˜ D (M ) 1 M

Increases in I (holding V constant)Using the envelope theorem

d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0

from equation (27) So D increases since G 2 L is unaffected bychanges in I

Increases in I (holding the NPV of the investment constant)To keep the NPV of the investment constant consider an

equal increases in I and V The effect on G 2 L is zero By the


envelope theorem the effect on G( b ˜ ) 2 cn( b ˜ ) is

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2

The numerator of this expression is positive because if wemultiply it by n we have

n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0

where the rst inequality holds because n b ˜ 5 1 2 a 1 and thesecond inequality holds by the participation constraint Thus Dincreases

Increases in VBy a similar argument since clearly shy n shy V 0 in equation

(27) D is decreasing in VIncreases in cBy the envelope theorem again D is clearly increasing in c

shy D

shy c5 n 1 c

shy n

shy c 0

because shy n shy c 0 differentiating (27)Increases in LObviously D is decreasing in LIncreased wastefulness of diversion 1 2 bUsing the envelope theorem

shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0

because b ˜ $ 1 2 bj



The entrepreneur will seek nance for the investment projectto be carried out if max (ULUP) $ U0 Consider the public listingand the private nance cases in turn

Public listing (UL)From expression (20)

UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L

Clearly this is increasing in V 2 I (a measure of the NPV of theinvestment) and decreasing in L Using the envelope theorem onesees that

shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0

In the case of a public listing if the NPV of the project(V 2 I 2 V0 ) is held constant any increases in scale ie in V or Ior both have no effect on UL 2 V0

Private nance (UP)From expression (19)

UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]

where b ˜ and n are chosen to minimize the expression in squarebrackets subject to equation (27)

Clearly UP is decreasing in the wastefulness of diversion(1 2 b) as argued for the case of a public listing

Regarding the NPV of the project the rst part of theexpression for UP is clearly increasing in V 2 I As for the part insquare brackets observe that by Proposition 3 UP 2 UL isincreasing in the size of the investment (I ) and the marginalshareholder cost (c) and it is decreasing in V

Combining these observations with the above results for UL

(in the case of a public listing) immediately yields the desiredresult j


Suppose that monitoring technology 1 is more effective thanmonitoring technology 0 or that alternatively there is less scopefor diversion under situation 1 than 0 in the sense that there isless or equal diversion for a given level of monitoring D 1(M ) D 0(M ) M Redene M i and UL

i as the values of M and of UL


corresponding to technology i for i 5 0 1 Then

UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0

where the rst inequality follows by revealed preference and thesecond from the fact that D 1(M ) D 0(M ) for all M j


First the initial owner and the monitoring shareholder willnot trade with each other The only way for them to increase theirtotal utility would be by harming the other shareholders viareduced monitoring since the shareholder base has been designedto maximize the entrepreneurrsquos utility after compensating themonitor for his cost But this is impossible because the entrepre-neur has no personal wealth to buy shares from the monitor andthereby reduce his stake

Second consider the monitorrsquos incentive to trade with otheroutside shareholders Suppose that he wants to change his initialstake b by a fraction D of the companyrsquos shares The farsightedinvestors will then set the stock price at

V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))

The monitorrsquos trading problem is

maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )

But then it is optimal to set D 5 0 To see this observe that byrevealed preference

b (V ( b ) 2 L) 2 M ( b ) $ b (V ( b 1 D ) 2 L ) 2 M ( b 1 D ) D

Third let us see whether the entrepreneur can prot byselling shares to third parties If we are not in the rst-bestsituation the entrepreneurrsquos stake is irrelevant without monitor-ing the amount of diversion is D anyway with monitoring it isthe stake of the monitor that determines the amount of diversionIf we are in the rst-best situation the entrepreneurrsquos stakebecomes relevant if he lowers it below b where the agencyproblem arises and starts diverting resources away from thecompany Suppose that starting from an initial stake a b the


entrepreneur sells a fraction D of the company so large as to endup with a fraction a 2 D b If the third parties to whom he sellsthe shares are rational they will buy them for a price D (V 2 D ) sothat the entrepreneurrsquos utility after the sale is ( a 2 D )(V 2 D ) 1D (V 2 D ) 1 bD 5 a V 2 ( a 2 b)D a V But without the sale hisutility level would have been a V he too has no incentive to sellThe argument is almost identical if his sale were to prompt a largeshareholder to start monitoring ( just replace D with D(M ) in theprevious expressions) j


Suppose that there are at least three independent sharehold-ers the entrepreneur with stake a the (potentially) monitoringshareholder(s) with stake b and a third shareholder with stakeg 0 With static expectations the two external shareholdershave an incentive to trade Let d b 1 g be the sum of theirstakes Then their joint utility sums to

d [V 2 D (M ( b ))] 2 M ( b )

This expression would be higher if b were replaced by d since byrevealed preference

d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )

Thus their joint utility is maximized if the third shareholder sellshis entire stake to the monitor so that the latter can pay himenough to make him better off than in the status quo Thus anownership structure with more than one external shareholder isunstable unless they choose monitoring cooperatively

Now we check that an ownership structure with two share-holders does not permit further mutually advantageous tradingNeither the entrepreneur nor the external shareholder has anincentive to sell shares to third parties as argued in the rst partof the proof of Proposition 6 Regarding the potential for mutuallyadvantageous trades between the entrepreneur and the externalshareholder their joint utility is maximized if the stake of theentrepreneur is as large as he can afford If he is wealthy enoughit will exceed b and the rst-best is reached If he is not it isoptimal to keep the monitoring shareholderrsquos stake as low aspossible (as close to 1 2 b as possible) j

UNIVERSITA DI SALERNO CSEF AND CEPRECARE-ULB TILBURG UNIVERSITY AND CEPR


REFERENCES

Admati Anat Paul Peiderer and Josef Zechner lsquolsquoLarge Shareholder ActivismRisk Sharing and Financial Market Equilibriumrsquorsquo Journal of PoliticalEconomy CII (1994) 1097mdash1130

Aghion Philippe and Patrick Bolton lsquolsquoAn Incomplete Contracts Approach toFinancial Contractingrsquorsquo Review of Economic Studies LIX (1992) 473mdash494

Barca Fabrizio Magda Bianco Luigi Cannari Riccardo Cesari Carlo GolaGiuseppe Manitta Giorgio Salvo and Luigi Signorini Proprieta modelli dicontrollo e riallocazione nelle imprese industriali italiane (Bologna Italy IlMulino 1994)

Bolton Patrick and Ernst-Ludwig von Thadden lsquolsquoBlocks Liquidity and Corpo-rate Controlrsquorsquo Journal of Finance LIII (1998) forthcoming

Booth James R and Lena Chua lsquolsquoOwnership Dispersion Costly Information andIPO Underpricingrsquorsquo Journal of Financial Economics XLI (1996) 291mdash310

Brennan Michael and Julian Franks lsquolsquoUnderpricing Ownership and Control inInitial Public Offerings of Equity Securities in the U Krsquorsquo Journal of FinancialEconomics XLV (1997) 391mdash413

Burkart Mike Denis Gromb and Fausto Panunzi lsquolsquoLarge Shareholders Monitor-ing and the Value of the Firmrsquorsquo Quarterly Journal of Economics CXII (1997)693mdash728

Franks Julian and Colin Mayer lsquolsquoOwnership and Controlrsquorsquo H Siebert ed Trendsin Business Organization Do Participation and Cooperation Increase Competi-tiveness (Tubingen Germany JCB Mohr 1995)

Franks Julian and Colin Mayer lsquolsquoOwnership Control and the Performance ofGerman Corporationsrsquorsquo mimeo London Business School 1997

Franks Julian Colin Mayer and Luc Renneboog lsquolsquoCapital Structure Ownershipand Board Restructuring in Poorly Performing Companiesrsquorsquo Working PaperCatholic University of Leuven 1997

Grossman Sanford and Oliver Hart lsquolsquoTakeover Bids the Free Rider Problem andthe Theory of the Corporationrsquorsquo Bell Journal of Economics XI (1980) 42mdash64

Herzel Leo and Richard W Shepro Bidders amp Targets Mergers and Acquisitionsin the U S (Oxford UK Basil Blackwell 1990)

Huddart Steven lsquolsquoThe Effect of a Large Shareholder on Corporate ValuersquorsquoManagement Science XXXIX (1993) 1407mdash1421

Jensen Michael C and William H Meckling lsquolsquoTheory of the Firm ManagerialBehavior Agency Costs and Ownership Structurersquorsquo Journal of FinancialEconomics XI (1976) 5mdash50

La Porta Rafael Florencio Loeacutepez-de-Silanes Andrei Shleifer and Robert WVishny lsquolsquoLaw and Financersquorsquo NBER Working Paper No 5661 1996

La Porta Rafael Florencio Loeacutepez-de-Silanes Andrei Shleifer and Robert WVishny lsquolsquoLegal Determinants of External Financersquorsquo Journal of Finance LII(1997) 1131mdash1150

Mello Antonio S and John E Parsons lsquolsquoAuctions of Shares with a SecondaryMarket and Tender Offersrsquorsquo CEPR Discussion Paper No 1077 1994

Morck Randall Andrei Shleifer and Robert W Vishny lsquolsquoManagement Ownershipand Market Valuation An Empirical Analysisrsquorsquo Journal of Financial Econom-ics XX (1988) 293mdash315

Pagano Marco Fabio Panetta and Luigi Zingales lsquolsquoWhy Do Companies GoPublic An Empirical Analysisrsquorsquo Journal of Finance LIII (1998) forthcoming

Renneboog Luc lsquolsquoOwnership Corporate Control and Large Shareholder Monitor-ing in Companies Listed on the Brussels Stock Exchangersquorsquo mimeo LondonBusiness School 1996

Ritter Jay R lsquolsquoThe Costs of Going Publicrsquorsquo Journal of Financial Economics XIX(1987) 269mdash281

Sharp Garry The Insiderrsquos Guide to Raising Venture Capital (London UK KoganPage 1991)

Shleifer Andrei and Robert W Vishny lsquolsquoLarge Shareholders and CorporateControlrsquorsquo Journal of Political Economy XCIV (1986) 461mdash488

Stoughton Neal M and Josef Zechner lsquolsquoIPO Mechanisms Monitoring andOwnership Structurersquorsquo Journal of Financial Economics forthcoming

Tirole Jean lsquolsquoCollusion and the Theory of Organizationsrsquorsquo in Jean-JacquesLaffont ed Advances in Economic Theory Sixth World Congress Volume 2


Econometric Society Monographs (Cambridge UK Cambridge UniversityPress 1992)

Yafeh Yishay and Oved Yosha lsquolsquoLarge Shareholders and Banks Who Monitorsand Howrsquorsquo CEPR Discussion Paper No 1178 1995

Wruck Karen H lsquolsquoEquity Ownership Concentration and Firm Value Evidencefrom Private Equity Financingsrsquorsquo Journal of Financial Economics XXIII(1989) 3mdash28

Zingales Luigi lsquolsquoThe Value of the Voting Right A Study of the Milan StockExchangersquorsquo Review of Financial Studies VII (1994) 125mdash148 lsquolsquoInsider Ownership and the Decision to Go Publicrsquorsquo Review of EconomicStudies LXII (1995) 425mdash448


These shareholders also have a greater incentive to place atakeover bid for a badly managed company because they appropri-ate part of the appreciation of the shares due to the improvedmanagement So via better monitoring and takeover bids largeshareholders alleviate the agency problem arising from the sepa-ration between ownership and control as argued by Shleifer andVishny [1986] and Grossman and Hart [1980] Empirical evidenceon the monitoring role of large shareholders is reported by MorckShleifer and Vishny [1988] Wruck [1989] Franks Mayer andRenneboog [1997] Yafeh and Yosha [1995] Renneboog [1996] andothers

In most European countries share ownership is much moreconcentrated than in the United States1 Most companies are notlisted on stock exchanges and even when they are a single largeshareholder or a tightly knit group of shareholders retains acontrolling stake in the company Since this ownership structuremakes companies impervious to takeovers the controlling stake iscommonly retained by the founder of the company and by hisdescendants even when the company is large and publicly listedThe controlling shareholder generally takes active interest inrunning the company by choosing the management and directlytaking executive positions In this situation the main conict ofinterest is that between the controlling shareholder and theminority shareholders2 rather than between hired managers andthe generality of shareholders as in the United States Thisconict can take several formsmdashfrom the diversion of corporateearnings to the advantage of the controlling shareholder to theuse of the companyrsquos assets to favor other companies owned byhim

This conict of interest cannot be analyzed by mechanicallyapplying the models that focus on agency problems betweenmanagement and shareholders In contrast with a manager theinitial owner of a company himself determines the distribution ofshareholding sizes ie designs the companyrsquos ownership struc-ture Assuming that the initial owner retains a controlling stake

1 In Italy 88 percent of the manufacturing companies (accounting for 91percent of the employed labor force) are controlled by one person or family [Barcaet al 1994] and in the majority of publicly listed companies one shareholder ownsmore than 50 percent of the voting rights [Zingales 1994] In Germany 85 percentof the public companies have a major shareholder and in France the correspondinggure is 79 percent to be compared with only 16 percent in the United Kingdom[Franks and Mayer 1995 1997]

2 As in the classic paper by Jensen and Meckling [1976]





















II THE MODEL

II1 Assumptions



(1) B 5 bD for D D



(2) V 2 D $ I










(3) maxa b

U V 2 I 1 B 2 M 5 V 2 I 2 (1 2 b )D 2 M










a (V 2 D ) 1 bD 5D (M ) if a b

0 if a $ b




(5) (1 2 a )V $ I 1 M


(6) minM

M 1 b D



(7) I (1 2 b)V


(8) I (1 2 b)V







(9) G ( b ) 5 (1 2 b)[D 2 D (M ( b ))] 2 M ( b )





(10) maxb [ [01]

U 5 V 2 I 2 (1 2 b)D (M ( b )) 2 M ( b )

5 V 2 I 2 (1 2 b)D 1 G ( b )



M 0 1 b 0D (M 0 ) M 1 1 b 0D (M 1 )

M 1 1 b 1D (M 1 ) M 0 1 b 1D (M 0 )

Hence

( b 1 2 b 0 )[D (M 1 ) 2 D (M 0 )] 0


b 1 b 0 THORN M 1 $ M 0




(11) V 2 D (M ( b )) 2 M ( b ) $ I


(12) U $ U0



minM

M 1 b D (M )


(13) maxb

G ( b ) 5 (1 2 b)D 2 (1 2 b)D (M ( b )) 2 M ( b )



(14) b 5 1 2 b






(15) U 5max (V 2 I U0 ) if I (1 2 b)V

max (V 2 I 2 (1 2 b)D 1 G U0 ) if I (1 2 b)V






G1 5 (1 2 b )D 2 (1 2 b )D 1 (M 1 ) 2 M 1

$ (1 2 b )D 2 (1 2 b )D 1 (M 0 ) 2 M 0

$ (1 2 b )D 2 (1 2 b )D 0 (M 0 ) 2 M 0 5 G0






G 1 2 G 0 5 (c 1 2 c 0 )D 2 (c 1D 1 2 c0D 0 ) 1 (M 0 2 M 1 )

but

M 1 1 c 1D 1 M 0 1 c 1D 0

so that

G 1 2 G 0 $ (c 1 2 c 0 )(D 2 D 0 ) $ 0




























(16) U V 2 I 2 (1 2 b)D 2 M 2 cn







(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))


(18) G ( b max ) 2 cn


(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c











(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES



















































II THE MODEL

II1 Assumptions



(1) B 5 bD for D D



(2) V 2 D $ I










(3) maxa b

U V 2 I 1 B 2 M 5 V 2 I 2 (1 2 b )D 2 M










a (V 2 D ) 1 bD 5D (M ) if a b

0 if a $ b




(5) (1 2 a )V $ I 1 M


(6) minM

M 1 b D



(7) I (1 2 b)V


(8) I (1 2 b)V







(9) G ( b ) 5 (1 2 b)[D 2 D (M ( b ))] 2 M ( b )





(10) maxb [ [01]

U 5 V 2 I 2 (1 2 b)D (M ( b )) 2 M ( b )

5 V 2 I 2 (1 2 b)D 1 G ( b )



M 0 1 b 0D (M 0 ) M 1 1 b 0D (M 1 )

M 1 1 b 1D (M 1 ) M 0 1 b 1D (M 0 )

Hence

( b 1 2 b 0 )[D (M 1 ) 2 D (M 0 )] 0


b 1 b 0 THORN M 1 $ M 0




(11) V 2 D (M ( b )) 2 M ( b ) $ I


(12) U $ U0



minM

M 1 b D (M )


(13) maxb

G ( b ) 5 (1 2 b)D 2 (1 2 b)D (M ( b )) 2 M ( b )



(14) b 5 1 2 b






(15) U 5max (V 2 I U0 ) if I (1 2 b)V

max (V 2 I 2 (1 2 b)D 1 G U0 ) if I (1 2 b)V






G1 5 (1 2 b )D 2 (1 2 b )D 1 (M 1 ) 2 M 1

$ (1 2 b )D 2 (1 2 b )D 1 (M 0 ) 2 M 0

$ (1 2 b )D 2 (1 2 b )D 0 (M 0 ) 2 M 0 5 G0






G 1 2 G 0 5 (c 1 2 c 0 )D 2 (c 1D 1 2 c0D 0 ) 1 (M 0 2 M 1 )

but

M 1 1 c 1D 1 M 0 1 c 1D 0

so that

G 1 2 G 0 $ (c 1 2 c 0 )(D 2 D 0 ) $ 0




























(16) U V 2 I 2 (1 2 b)D 2 M 2 cn







(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))


(18) G ( b max ) 2 cn


(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c











(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES












































II THE MODEL

II1 Assumptions



(1) B 5 bD for D D



(2) V 2 D $ I










(3) maxa b

U V 2 I 1 B 2 M 5 V 2 I 2 (1 2 b )D 2 M










a (V 2 D ) 1 bD 5D (M ) if a b

0 if a $ b




(5) (1 2 a )V $ I 1 M


(6) minM

M 1 b D



(7) I (1 2 b)V


(8) I (1 2 b)V







(9) G ( b ) 5 (1 2 b)[D 2 D (M ( b ))] 2 M ( b )





(10) maxb [ [01]

U 5 V 2 I 2 (1 2 b)D (M ( b )) 2 M ( b )

5 V 2 I 2 (1 2 b)D 1 G ( b )



M 0 1 b 0D (M 0 ) M 1 1 b 0D (M 1 )

M 1 1 b 1D (M 1 ) M 0 1 b 1D (M 0 )

Hence

( b 1 2 b 0 )[D (M 1 ) 2 D (M 0 )] 0


b 1 b 0 THORN M 1 $ M 0




(11) V 2 D (M ( b )) 2 M ( b ) $ I


(12) U $ U0



minM

M 1 b D (M )


(13) maxb

G ( b ) 5 (1 2 b)D 2 (1 2 b)D (M ( b )) 2 M ( b )



(14) b 5 1 2 b






(15) U 5max (V 2 I U0 ) if I (1 2 b)V

max (V 2 I 2 (1 2 b)D 1 G U0 ) if I (1 2 b)V






G1 5 (1 2 b )D 2 (1 2 b )D 1 (M 1 ) 2 M 1

$ (1 2 b )D 2 (1 2 b )D 1 (M 0 ) 2 M 0

$ (1 2 b )D 2 (1 2 b )D 0 (M 0 ) 2 M 0 5 G0






G 1 2 G 0 5 (c 1 2 c 0 )D 2 (c 1D 1 2 c0D 0 ) 1 (M 0 2 M 1 )

but

M 1 1 c 1D 1 M 0 1 c 1D 0

so that

G 1 2 G 0 $ (c 1 2 c 0 )(D 2 D 0 ) $ 0




























(16) U V 2 I 2 (1 2 b)D 2 M 2 cn







(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))


(18) G ( b max ) 2 cn


(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c











(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES






































II THE MODEL

II1 Assumptions



(1) B 5 bD for D D



(2) V 2 D $ I










(3) maxa b

U V 2 I 1 B 2 M 5 V 2 I 2 (1 2 b )D 2 M










a (V 2 D ) 1 bD 5D (M ) if a b

0 if a $ b




(5) (1 2 a )V $ I 1 M


(6) minM

M 1 b D



(7) I (1 2 b)V


(8) I (1 2 b)V







(9) G ( b ) 5 (1 2 b)[D 2 D (M ( b ))] 2 M ( b )





(10) maxb [ [01]

U 5 V 2 I 2 (1 2 b)D (M ( b )) 2 M ( b )

5 V 2 I 2 (1 2 b)D 1 G ( b )



M 0 1 b 0D (M 0 ) M 1 1 b 0D (M 1 )

M 1 1 b 1D (M 1 ) M 0 1 b 1D (M 0 )

Hence

( b 1 2 b 0 )[D (M 1 ) 2 D (M 0 )] 0


b 1 b 0 THORN M 1 $ M 0




(11) V 2 D (M ( b )) 2 M ( b ) $ I


(12) U $ U0



minM

M 1 b D (M )


(13) maxb

G ( b ) 5 (1 2 b)D 2 (1 2 b)D (M ( b )) 2 M ( b )



(14) b 5 1 2 b






(15) U 5max (V 2 I U0 ) if I (1 2 b)V

max (V 2 I 2 (1 2 b)D 1 G U0 ) if I (1 2 b)V






G1 5 (1 2 b )D 2 (1 2 b )D 1 (M 1 ) 2 M 1

$ (1 2 b )D 2 (1 2 b )D 1 (M 0 ) 2 M 0

$ (1 2 b )D 2 (1 2 b )D 0 (M 0 ) 2 M 0 5 G0






G 1 2 G 0 5 (c 1 2 c 0 )D 2 (c 1D 1 2 c0D 0 ) 1 (M 0 2 M 1 )

but

M 1 1 c 1D 1 M 0 1 c 1D 0

so that

G 1 2 G 0 $ (c 1 2 c 0 )(D 2 D 0 ) $ 0




























(16) U V 2 I 2 (1 2 b)D 2 M 2 cn







(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))


(18) G ( b max ) 2 cn


(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c











(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES

































II THE MODEL

II1 Assumptions



(1) B 5 bD for D D



(2) V 2 D $ I










(3) maxa b

U V 2 I 1 B 2 M 5 V 2 I 2 (1 2 b )D 2 M










a (V 2 D ) 1 bD 5D (M ) if a b

0 if a $ b




(5) (1 2 a )V $ I 1 M


(6) minM

M 1 b D



(7) I (1 2 b)V


(8) I (1 2 b)V







(9) G ( b ) 5 (1 2 b)[D 2 D (M ( b ))] 2 M ( b )





(10) maxb [ [01]

U 5 V 2 I 2 (1 2 b)D (M ( b )) 2 M ( b )

5 V 2 I 2 (1 2 b)D 1 G ( b )



M 0 1 b 0D (M 0 ) M 1 1 b 0D (M 1 )

M 1 1 b 1D (M 1 ) M 0 1 b 1D (M 0 )

Hence

( b 1 2 b 0 )[D (M 1 ) 2 D (M 0 )] 0


b 1 b 0 THORN M 1 $ M 0




(11) V 2 D (M ( b )) 2 M ( b ) $ I


(12) U $ U0



minM

M 1 b D (M )


(13) maxb

G ( b ) 5 (1 2 b)D 2 (1 2 b)D (M ( b )) 2 M ( b )



(14) b 5 1 2 b






(15) U 5max (V 2 I U0 ) if I (1 2 b)V

max (V 2 I 2 (1 2 b)D 1 G U0 ) if I (1 2 b)V






G1 5 (1 2 b )D 2 (1 2 b )D 1 (M 1 ) 2 M 1

$ (1 2 b )D 2 (1 2 b )D 1 (M 0 ) 2 M 0

$ (1 2 b )D 2 (1 2 b )D 0 (M 0 ) 2 M 0 5 G0






G 1 2 G 0 5 (c 1 2 c 0 )D 2 (c 1D 1 2 c0D 0 ) 1 (M 0 2 M 1 )

but

M 1 1 c 1D 1 M 0 1 c 1D 0

so that

G 1 2 G 0 $ (c 1 2 c 0 )(D 2 D 0 ) $ 0




























(16) U V 2 I 2 (1 2 b)D 2 M 2 cn







(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))


(18) G ( b max ) 2 cn


(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c











(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES








































(3) maxa b

U V 2 I 1 B 2 M 5 V 2 I 2 (1 2 b )D 2 M










a (V 2 D ) 1 bD 5D (M ) if a b

0 if a $ b




(5) (1 2 a )V $ I 1 M


(6) minM

M 1 b D



(7) I (1 2 b)V


(8) I (1 2 b)V







(9) G ( b ) 5 (1 2 b)[D 2 D (M ( b ))] 2 M ( b )





(10) maxb [ [01]

U 5 V 2 I 2 (1 2 b)D (M ( b )) 2 M ( b )

5 V 2 I 2 (1 2 b)D 1 G ( b )



M 0 1 b 0D (M 0 ) M 1 1 b 0D (M 1 )

M 1 1 b 1D (M 1 ) M 0 1 b 1D (M 0 )

Hence

( b 1 2 b 0 )[D (M 1 ) 2 D (M 0 )] 0


b 1 b 0 THORN M 1 $ M 0




(11) V 2 D (M ( b )) 2 M ( b ) $ I


(12) U $ U0



minM

M 1 b D (M )


(13) maxb

G ( b ) 5 (1 2 b)D 2 (1 2 b)D (M ( b )) 2 M ( b )



(14) b 5 1 2 b






(15) U 5max (V 2 I U0 ) if I (1 2 b)V

max (V 2 I 2 (1 2 b)D 1 G U0 ) if I (1 2 b)V






G1 5 (1 2 b )D 2 (1 2 b )D 1 (M 1 ) 2 M 1

$ (1 2 b )D 2 (1 2 b )D 1 (M 0 ) 2 M 0

$ (1 2 b )D 2 (1 2 b )D 0 (M 0 ) 2 M 0 5 G0






G 1 2 G 0 5 (c 1 2 c 0 )D 2 (c 1D 1 2 c0D 0 ) 1 (M 0 2 M 1 )

but

M 1 1 c 1D 1 M 0 1 c 1D 0

so that

G 1 2 G 0 $ (c 1 2 c 0 )(D 2 D 0 ) $ 0




























(16) U V 2 I 2 (1 2 b)D 2 M 2 cn







(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))


(18) G ( b max ) 2 cn


(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c











(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES


































(3) maxa b

U V 2 I 1 B 2 M 5 V 2 I 2 (1 2 b )D 2 M










a (V 2 D ) 1 bD 5D (M ) if a b

0 if a $ b




(5) (1 2 a )V $ I 1 M


(6) minM

M 1 b D



(7) I (1 2 b)V


(8) I (1 2 b)V







(9) G ( b ) 5 (1 2 b)[D 2 D (M ( b ))] 2 M ( b )





(10) maxb [ [01]

U 5 V 2 I 2 (1 2 b)D (M ( b )) 2 M ( b )

5 V 2 I 2 (1 2 b)D 1 G ( b )



M 0 1 b 0D (M 0 ) M 1 1 b 0D (M 1 )

M 1 1 b 1D (M 1 ) M 0 1 b 1D (M 0 )

Hence

( b 1 2 b 0 )[D (M 1 ) 2 D (M 0 )] 0


b 1 b 0 THORN M 1 $ M 0




(11) V 2 D (M ( b )) 2 M ( b ) $ I


(12) U $ U0



minM

M 1 b D (M )


(13) maxb

G ( b ) 5 (1 2 b)D 2 (1 2 b)D (M ( b )) 2 M ( b )



(14) b 5 1 2 b






(15) U 5max (V 2 I U0 ) if I (1 2 b)V

max (V 2 I 2 (1 2 b)D 1 G U0 ) if I (1 2 b)V






G1 5 (1 2 b )D 2 (1 2 b )D 1 (M 1 ) 2 M 1

$ (1 2 b )D 2 (1 2 b )D 1 (M 0 ) 2 M 0

$ (1 2 b )D 2 (1 2 b )D 0 (M 0 ) 2 M 0 5 G0






G 1 2 G 0 5 (c 1 2 c 0 )D 2 (c 1D 1 2 c0D 0 ) 1 (M 0 2 M 1 )

but

M 1 1 c 1D 1 M 0 1 c 1D 0

so that

G 1 2 G 0 $ (c 1 2 c 0 )(D 2 D 0 ) $ 0




























(16) U V 2 I 2 (1 2 b)D 2 M 2 cn







(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))


(18) G ( b max ) 2 cn


(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c











(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES

































(5) (1 2 a )V $ I 1 M


(6) minM

M 1 b D



(7) I (1 2 b)V


(8) I (1 2 b)V







(9) G ( b ) 5 (1 2 b)[D 2 D (M ( b ))] 2 M ( b )





(10) maxb [ [01]

U 5 V 2 I 2 (1 2 b)D (M ( b )) 2 M ( b )

5 V 2 I 2 (1 2 b)D 1 G ( b )



M 0 1 b 0D (M 0 ) M 1 1 b 0D (M 1 )

M 1 1 b 1D (M 1 ) M 0 1 b 1D (M 0 )

Hence

( b 1 2 b 0 )[D (M 1 ) 2 D (M 0 )] 0


b 1 b 0 THORN M 1 $ M 0




(11) V 2 D (M ( b )) 2 M ( b ) $ I


(12) U $ U0



minM

M 1 b D (M )


(13) maxb

G ( b ) 5 (1 2 b)D 2 (1 2 b)D (M ( b )) 2 M ( b )



(14) b 5 1 2 b






(15) U 5max (V 2 I U0 ) if I (1 2 b)V

max (V 2 I 2 (1 2 b)D 1 G U0 ) if I (1 2 b)V






G1 5 (1 2 b )D 2 (1 2 b )D 1 (M 1 ) 2 M 1

$ (1 2 b )D 2 (1 2 b )D 1 (M 0 ) 2 M 0

$ (1 2 b )D 2 (1 2 b )D 0 (M 0 ) 2 M 0 5 G0






G 1 2 G 0 5 (c 1 2 c 0 )D 2 (c 1D 1 2 c0D 0 ) 1 (M 0 2 M 1 )

but

M 1 1 c 1D 1 M 0 1 c 1D 0

so that

G 1 2 G 0 $ (c 1 2 c 0 )(D 2 D 0 ) $ 0




























(16) U V 2 I 2 (1 2 b)D 2 M 2 cn







(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))


(18) G ( b max ) 2 cn


(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c











(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES

































(9) G ( b ) 5 (1 2 b)[D 2 D (M ( b ))] 2 M ( b )





(10) maxb [ [01]

U 5 V 2 I 2 (1 2 b)D (M ( b )) 2 M ( b )

5 V 2 I 2 (1 2 b)D 1 G ( b )



M 0 1 b 0D (M 0 ) M 1 1 b 0D (M 1 )

M 1 1 b 1D (M 1 ) M 0 1 b 1D (M 0 )

Hence

( b 1 2 b 0 )[D (M 1 ) 2 D (M 0 )] 0


b 1 b 0 THORN M 1 $ M 0




(11) V 2 D (M ( b )) 2 M ( b ) $ I


(12) U $ U0



minM

M 1 b D (M )


(13) maxb

G ( b ) 5 (1 2 b)D 2 (1 2 b)D (M ( b )) 2 M ( b )



(14) b 5 1 2 b






(15) U 5max (V 2 I U0 ) if I (1 2 b)V

max (V 2 I 2 (1 2 b)D 1 G U0 ) if I (1 2 b)V






G1 5 (1 2 b )D 2 (1 2 b )D 1 (M 1 ) 2 M 1

$ (1 2 b )D 2 (1 2 b )D 1 (M 0 ) 2 M 0

$ (1 2 b )D 2 (1 2 b )D 0 (M 0 ) 2 M 0 5 G0






G 1 2 G 0 5 (c 1 2 c 0 )D 2 (c 1D 1 2 c0D 0 ) 1 (M 0 2 M 1 )

but

M 1 1 c 1D 1 M 0 1 c 1D 0

so that

G 1 2 G 0 $ (c 1 2 c 0 )(D 2 D 0 ) $ 0




























(16) U V 2 I 2 (1 2 b)D 2 M 2 cn







(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))


(18) G ( b max ) 2 cn


(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c











(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES

































(11) V 2 D (M ( b )) 2 M ( b ) $ I


(12) U $ U0



minM

M 1 b D (M )


(13) maxb

G ( b ) 5 (1 2 b)D 2 (1 2 b)D (M ( b )) 2 M ( b )



(14) b 5 1 2 b






(15) U 5max (V 2 I U0 ) if I (1 2 b)V

max (V 2 I 2 (1 2 b)D 1 G U0 ) if I (1 2 b)V






G1 5 (1 2 b )D 2 (1 2 b )D 1 (M 1 ) 2 M 1

$ (1 2 b )D 2 (1 2 b )D 1 (M 0 ) 2 M 0

$ (1 2 b )D 2 (1 2 b )D 0 (M 0 ) 2 M 0 5 G0






G 1 2 G 0 5 (c 1 2 c 0 )D 2 (c 1D 1 2 c0D 0 ) 1 (M 0 2 M 1 )

but

M 1 1 c 1D 1 M 0 1 c 1D 0

so that

G 1 2 G 0 $ (c 1 2 c 0 )(D 2 D 0 ) $ 0




























(16) U V 2 I 2 (1 2 b)D 2 M 2 cn







(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))


(18) G ( b max ) 2 cn


(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c











(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES


































(15) U 5max (V 2 I U0 ) if I (1 2 b)V

max (V 2 I 2 (1 2 b)D 1 G U0 ) if I (1 2 b)V






G1 5 (1 2 b )D 2 (1 2 b )D 1 (M 1 ) 2 M 1

$ (1 2 b )D 2 (1 2 b )D 1 (M 0 ) 2 M 0

$ (1 2 b )D 2 (1 2 b )D 0 (M 0 ) 2 M 0 5 G0






G 1 2 G 0 5 (c 1 2 c 0 )D 2 (c 1D 1 2 c0D 0 ) 1 (M 0 2 M 1 )

but

M 1 1 c 1D 1 M 0 1 c 1D 0

so that

G 1 2 G 0 $ (c 1 2 c 0 )(D 2 D 0 ) $ 0




























(16) U V 2 I 2 (1 2 b)D 2 M 2 cn







(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))


(18) G ( b max ) 2 cn


(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c











(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES


































G 1 2 G 0 5 (c 1 2 c 0 )D 2 (c 1D 1 2 c0D 0 ) 1 (M 0 2 M 1 )

but

M 1 1 c 1D 1 M 0 1 c 1D 0

so that

G 1 2 G 0 $ (c 1 2 c 0 )(D 2 D 0 ) $ 0




























(16) U V 2 I 2 (1 2 b)D 2 M 2 cn







(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))


(18) G ( b max ) 2 cn


(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c











(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES
























































(16) U V 2 I 2 (1 2 b)D 2 M 2 cn







(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))


(18) G ( b max ) 2 cn


(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c











(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES















































(16) U V 2 I 2 (1 2 b)D 2 M 2 cn







(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))


(18) G ( b max ) 2 cn


(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c











(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES








































(16) U V 2 I 2 (1 2 b)D 2 M 2 cn







(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))


(18) G ( b max ) 2 cn


(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c











(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES


































(17) 1 2 a 5 oi 5 1

n

b ˜ i 5I 1 cn 1 M ( b ˜ max)

V 2 D (M ( b ˜ max ))


(18) G ( b max ) 2 cn


(19) UP 5V 2 I 2 c if I (1 2 b)V 2 c











(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES



































(20) UL 5V 2 I 2 L if I (1 2 b)(V 2 L )

V 2 I 2 (1 2 b)D 1 G 2 L if I (1 2 b)(V 2 L )






(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES

































(21) D UL 2 UP 5 [G 2 G ( b ˜ max)] 1 cn 2 L



















(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES












































(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES





































(22) b D (M ( b )) 1 M ( b )



(23) b D (M ( b )) 1 M ( b ) 2 (b 2 a )D (M ( b ))
























D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES






















































D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES














































D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES







































D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES

































D 5 [G 2 G ( b ˆ)] 1 c 2 L

where

(24) b ˆ 1 2 a 5I 1 t 1 M ( b ˆ)

V 2 D (M ( b ˆ))








maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES



































maxM

( a 1 b )(V 2 D (M )) 1 bD (M ) 2 M





and



( g 9 2 g 8 )(D (M 8 ) 2 D (M 9 )) $ 0




















VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES

















































VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES










































VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES


































VII CONCLUSIONS






APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES

































APPENDIX

Proof of Lemma 1


G1 2 G 0 5 (1 2 b)(D 0 2 D1 ) 1 (M0 2 M1 )


M0 1 b 0D 0 M1 1 b 0D 1

so that

G1 2 G 0 (1 2 b 2 b 0)(D0 2 D 1)













G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES






































G (1 2 b) 1 11

n 2 12 (n 2 1)c G (1 2 b) 2 nc


G (1 2 b) 1 11

n 2 12 G (1 2 b) 2 c




I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES

































I 1 Mn

n 2 1(1 2 b) 1 c(n 2 1)

n (1 2 b) V 2 D Mn

n 2 1(1 2 b)


I 1 M (1 2 b) 1 cn n (1 2 b)[V 2 D (M (1 2 b))]


M (1 2 b) 2 Mn

n 2 1(1 2 b)

1 c n (1 2 b) D Mn

n 2 1(1 2 b) 2 D (M (1 2 b))


2 M (1 2 b) 2 (1 2 b)D (M (1 2 b)) 1 Mn

n 2 1(1 2 b)

1 (1 2 b)D Mn

n 2 1(1 2 b) c 1 (n 2 1)(1 2 b)


n 2 1(1 2 b) $ c


G (1 2 b) 2 Gn

n 2 1(1 2 b) c





(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES


































(25) D 5 G 2 G ( b ˜ ) 1 c (n ) 2 L

5 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)]

1 (1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn 2 L



(26) minb ˜

(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn

where

(27) n 5 n ( b ˜ ) 5I 1 M ( b ˜ )

b ˜ [V 2 D (M ( b ˜ ))] 2 c


(28) minM

b ˜ D (M ) 1 M


d [G ( b ˜ ) 2 cn ( b ˜ )]

dI5 c

shy n

shy I 0






shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES

































shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy I1

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy V5 c

shy n

shy I1

shy n

shy V

5 cb ˜ [V 2 D (M ( b ˜ ))] 2 c 2 b ˜ [I 1 M ( b ˜ )]

b ˜ [V 2 D (M ( b ˜ ))] 2 c 2


n b ˜ [V 2 D (M ( b ˜ )) 2 I 2 M ( b ˜ )] 2 nc

$ n b ˜ [V 2 D (M ( b ˜ ))] 2 I 2 M ( b ˜ ) 2 nc $ 0




shy D

shy c5 n 1 c

shy n

shy c 0


shy (G 2 L)

shy (1 2 b)5 2 D (M (1 2 b))

and

shy [G ( b ˜ ) 2 cn ( b ˜ )]

shy (1 2 b)5 2 D (M ( b ˜ ))

Thusshy D

shy (1 2 b)5 D (M ( b ˜ )) 2 D (M (1 2 b)) 0






UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES



































UL 5 V 2 I 2 [(1 2 b)D (M (1 2 b)) 1 M (1 2 b)] 2 L


shy UL

shy (1 2 b)5 2 D (M (1 2 b)) 0



UP 5 V 2 I 2 [(1 2 b)D (M ( b ˜ )) 1 M ( b ˜ ) 1 cn]











UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES

































UL1 5 V 2 I 2 L 2 (1 2 b)D 1(M1(1 2 b)) 2 M1 (1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 1(M 0(1 2 b)) 2 M0(1 2 b)

$ V 2 I 2 L 2 (1 2 b)D 0(M 0(1 2 b)) 2 M0(1 2 b) 5 UL0





V ( b 1 D ) 2 L V 2 (1 2 b)D (M ( b 1 D ))


maxD

( b 1 D )(V ( b 1 D ) 2 L) 2 M ( b 1 D ) 2 D (V ( b 1 D ) 2 L)

5 b (V ( b 1 D ) 2 L) 2 M ( b 1 D )








d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES



































d [V 2 D (M ( b ))] 2 M ( b )


d D (M ( d )) 1 M ( d ) minb

d D (M ( b )) 1 M ( b )





REFERENCES
































REFERENCES





































Date post:	21-Sep-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

THECHOICEOFSTOCKOWNERSHIPSTRUCTURE: M P A R ELL...THECHOICEOFSTOCKOWNERSHIPSTRUCTURE:...

Documents