Implication: Two Shares { One Price
Ken L.Bechmann a
Department of Finance
Copenhagen Business School
Johannes Raaballe b
Department of Management
University of Aarhus
May 1,2000
Key words: DualClassShares; Regulationof Tender Oers;theVoting Premium
JEL Classication: G18; G32; G34; G38
a
SolbjergPlads3,DK-2000 Frederiksberg, Denmark
Phone: +45 38152953 Fax: +45 38153600 e-mail: kb.@cbs.dk
b
UniversityPark 350,DK-8000 AarhusC, Denmark
Phone: +45 89421564 Fax: +45 86135132 e-mail: jraaballe@econ.au.dk
TheauthorsowethankstoGunterFranke,BruceD.Grundy,PeterLchteJrgensen,HenrikLando,Michael
Mller,PradipkumarRamanlal,KristianRydqvist,CarstenSrensen,andtheparticipantsintheNordicWork-
shoponCorporateFinance,theCorporateFinance DiscussionGroupatWharton,UniversityofPennsylvania,
theFinance seminars atUniversity ofAarhus andCopenhagen Business School, the Centrefor Analytical Fi-
nanceseminaratSandbjergManorHouse,the1999FMAEuropeanConference,the1998EIASM/EFADoctoral
Implication: Two Shares { One Price
Abstract
Thispaperexaminestheconsequencesofacertainregulatoryrestrictiononbids
fordual class shares. Shares of dierent classesare often argued to have dierent
pricesbecause apremiumwillbepaidto thesuperiorvotingsharesinthecaseofa
tenderoer. Thispapertakesasgivenasetupwherethesharesinarmarewidely
heldand regulations requirethat atenderoer pays thesame relative premiumto
all share classes. In this setup, it is shown that theshares of dierent classes will
sell at the same price as long as there is a strictlypositive probabilitythat either
thecurrent management issuÆcientlystrongorthata suÆcientlystrongrivalwill
show up. Furthermore, under this condition the regulation is socially optimal in
the sensethat the management that gives the highest total rm value will be the
management of therm. Finally,theregulation is shown to favor (orprotect) the
holders ofrestricted voting sharesand this isnot necessarily at theexpense of the
holdersof superiorvoting shares.
Iftheweakconditionaboveisnotsatised,thepaperdemonstratestheexistence
of a whole range of possible price equilibria. These equilibria can be decisive for
whetherthe current managementwillcontinue ortherivalwilltake over.
The practical interest of this paper derives from the fact that some European
countries have adopted regulatory restrictions on bids for dual class shares. This
hasmore orlessoccurreddueto proposedEUDirectives. Theregulationexamined
inthispaperappliesforexampleto tenderoersinDenmarkand empiricalresults
onthevoting premiuminDenmarkareshownto beconsistent withthetheoretical
resultsinthispaper.
Bydualclasssharesismeant thatarmhasissuedtwo dierenttypesofshares. Theseshares
usuallydierwith respect to thenumberofvotesthat areattachedto each share. Theshares
might also dier with respect to the dividendrights.
1
The dierence in votes meansthat the
shares are divided into superior voting shares (or A-shares) and restricted voting shares (or
B-shares).
Given that the two types of shares receive the same dividends,superior voting shares are
expected to be worth at least asmuch asrestricted voting shares. In addition,the possibility
ofatakeovermightimplythatsuperiorvotingshareswillbeworthmorethanrestricted voting
shares. Intakeover situations investors willinparticular be interestedin voting rights. Inthe
debateitisarguedthatadualclassstructurecanmakeittooeasyforacompetingmanagement
to obtainthemajorityof thevotesbyonlybuyingsuperiorvoting shares. The takeover might
thenhappenat theexpense ofthe non-sellingshareholdersnaturally includingall theholders
ofrestrictedvoting shares. This potentialproblemassociatedwithdualclassshareshasled to
adiscussionabouttheextent to which tenderoersand dualclass sharesshouldbe regulated
by law. In order to considerthe relevance of and needfor such regulations it is important to
knowthe consequences ofthedierent possibleregulations.
Theseissues areespeciallybecomingofimportanceasitnowappearsmoreand morelikely
that theEuropean Councilwill adoptthe 13th Directive on company law concerning takeover
bids.
2
According to thisDirective all member states of the European Union must have rules
thatprovideprotectionof minorityshareholdersinthecase of takeovers. However, the Direc-
tivewillprobablyleave thememberstateswithconsiderablelatitudeonexactlyhowtoachieve
thisprotection. As stated inthe Directive: \Member States should ensure that rules or other
mechanisms or arrangements are in force which either oblige this person [who has obtained
controlof a company] to make a bid in accordance with Article 10 [i.e. to all shareholders for
allorforasubstantialpart oftheirholdingsat apricewhichmeetstheobjective ofprotecting
theirinterests]or oer other appropriate and at least equivalent means in order to protect the
minority shareholders of that company."
3
With this latitude in the Directive it is natural to
expectdiscussionsinthedierentcountriesabouttheprosandconsofthedierentregulations
1
Insomecasesrmshaveissuedmorethantwodierenttypesofshares.
2
TheDirective was discussed ona meeting December 7,1999 where the memberstatesseem to have ap-
proachedacommonagreementabouttheDirective.
3
Article3(1).
voting shares are normally among the minority shareholders. Therefore, the Directive also
requiresregulation on how restrictedvoting sharesshouldbe treatedrelative to holdersof su-
periorvoting sharesinthecase oftakeovers.
Grossmanand Hart(1988)examinedtheoutcomeoftenderoersundertwodierentregu-
lationsoftenderoersforarmnancedwithwidelyhelddualclassshares. Therstregulation
examinedisthecasewherethetenderoercan berestrictedto onlyafractionofashare class.
In thiscase itwill be possible forthe bidder to discriminate betweenshareholders withinthe
same class and between share classes. The second regulation examined is the case where a
restrictedtenderoer withina classisnotpossiblebutwherethebidderisallowed to discrim-
inate freelybetweenshare classes.
However, as the proposed EU directives illustrate, it is possible to make the regulation
of tender oers even more restrictive. In some countries a tender oer cannot discriminate
freelybetweenshareclasses. ExamplesofsuchcountriesareAustria,Denmark,Finland,Great
Britain, Sweden, and Switzerland. Inthese countries a person obtainingcontrolof a rm has
to make an oer to all classes of shares and the prices oered either have to give the \same
premium"to all classesof sharesorthe pricesoered to thedierentclasses ofshares have to
be \reasonable" (see Clausenand Srensen (1998)). In Denmark, for example, a tender oer
is required to give a class of restricted voting shares the same relative premium as oered to
a class of superior voting shares.
4
This means that if the two share classes trade at 100 and
50 respectively, a tenderoer giving 150 to the rst classis required to oer 75 to the second
class. Furthermore,partialbidsare notallowed.
Thispaperexaminestheconsequences oftheregulationadoptedinDenmark,i.e.thepaper
considers the case where a tender oer is required to give the same relative premium to all
classes of shares. The main theoretical result of the paper is that the regulation under weak
conditions implies that there will be no price dierence between share classes. We will later
see that theconditionfor thisto be the case is robustwith respectto changes inthe model's
assumptions.
A consequenceofidenticalpricesforA-and B-shares isthatthemanagement underwhich
thermwillhave thehighesttotal valuewillbeable tooer thehighest pricefortheA-shares
4
Thisfollowsfrom theLawonSecurity TradeNumber1072,December20, 1995,xx31{32, and Fondsradets
legalnoticeno.333, April23,1996,xx1{10. Theintroductionoftheregulationisfurtherdescribedinsection6.
and thereby controlthe company. Therefore, the regulation leadsto socialoptimality inthe
sense that the company will be controlled by the management that can contribute with the
highest total value. Finally, the results show that when there is no price dierence between
shareclassestheregulationwillalwaysfavor(orprotect)theholdersofrestrictedvotingshares.
Whether theregulation favors theholders of A-sharesand/or theshareholdersas a wholede-
pendson the behavior of the losing management team. It is especiallypossible to have cases
wheretheholdersofA-sharesandtheshareholdersasawholealsobenetfromtheregulation.
Dual class shareshave beenconsideredin anumberof papers.
6
From theempiricallitera-
ture follows that superiorvoting shares sell at higher prices than restricted voting shares but
also that the premiums dier across countries.
7
Several papers including Lease, McConnell,
andMikkelson(1983),DeAngeloandDeAngelo(1985), Megginson(1990), SmithandAmoako-
Adu (1995), and Rydqvist (1996) explained this price dierence between share classes by a
premiumpaid to superiorvoting shares inthe case of tenderoers and providedevidence for
thisexplanation. Adierentexplanationforthepricedierencebetweenshareclassesisoered
by Bergstrom and Rydqvist (1992). In a model with a pivotal blockholder and a bidderthat
wants(has)tobuyalltheshares,BergstromandRydqvistpointtothefactthatitistheability
to pricediscriminate (not the voting power) between share classes that gives rise to dierent
prices. IfandonlyiftheblockholderisendowedprimarilywithA-shares,theA-shareswillsell
at a premium relative to the B-shares. In addition, the wealth consequences of a regulation
requiringthat thesame oer has to begiven to bothclassesof sharesare discussedinthepa-
per. Theeect ofachangeinregulationsonthepricedierencebetweenshareclasseshasbeen
examined empirically in Maynes (1996). Maynes (1996) considered the introduction of a so-
calledcoattail requirementfor theToronto Stock Exchange in1984. According to thecoattail
requirement, holders of restricted voting shares must be given an oer equivalent to the oer
made forthesuperiorvoting shares. Consistent withthe resultsinthepresent paper, Maynes
foundasignicantdeclineinthepremiumpaidforsuperiorvoting sharesattheannouncement
ofthe coattail requirement.
5
Heretotal value refersto thesumofthe value of theshares(thesecurityvalue)and thevalue derivedby
themanagementfromhavingcontrolovertherm(theprivatebenetofcontrol).
6
Rydqvist(1992)providedareviewofthetheoryandempiricalevidenceondualclassshares.
7
Someof thepremiumsfoundintheliterature are: Canada 10% (Smithand Amoako-Adu(1995)), France
54% (Muus (1998)), Israel46% (Levy (1983)), Italy 80% (Zingales (1994)), Norway 10% (degaard (1998)),
Sweden12%(Rydqvist(1996)), Switzerland10% (Horner(1988)), UK13% (Megginson (1990)), andUSA5%
mainresults. Section 3 illustratesa case with multipleprice equilibria. Section 4 derivesthe
distributionalconsequences oftheregulation. Section5 showsthatthemainresultsarerobust
tochangesinthebasicmodel. Section6demonstratesthatthechangesovertimeinthevoting
premiuminDenmarkareconsistent withthetheoreticalresultsinthispaper. Theconclusions
aregiven insection 7.
2 The basic model
ThebasicmodelinthispaperissimilartothemodelinGrossmanandHart(1988). Weassume
theexistence ofa rm whichunder thecurrent (incumbent)management, I,hasa total value
given as y
I +z
I . y
I
> 0 is the security value and z
I
0 is the private benet of control
derived bythecurrent management. The rm isnanced witha dualclassshare structure. It
is assumed that one of the classes denoted class A hasthe majority of the votes. In order to
change the management of the rm, the majority of the votes is required. The other class is
denoted class B. Without loss of generality, we assume that the total sum of the number of
A-shares (s
A
) and the number of B-shares (s
B
) is equal to one, i.e. that s
A +s
B
= 1. The
securityvalue of the rm,y
I
,is distributedbetween thetwo classeswith the amount s
A y
I to
classA and s
B y
I
to classB.
At some time denoted time one, it is possible that a competing management (a rival), R ,
shows up. If the competing management takes over the rm, the total value of the rm will
change to y
R +z
R
. The security valuey
R
>0 is again distributed betweenthe two classes of
shares,andz
R
0istheprivatebenetofcontrolderivedbythecompetingmanagementwhen
thecompetingmanagement is controllingtherm.
It is assumed that the shares are widely held, i.e. that all shareholders are small. This
meansthat neither the incumbent northe competing management own largeblocks of shares
beforethepotentialbiddingcontest. Finally,itis assumedthatI,R ,and theshareholdersare
allriskneutral,thattheinterestrateiszero,and thatthereisnocostassociatedwithatender
oer.
8
At time one when the competing management shows up, thevaluesof y
I
; z
I
; y
R
;and
z
R
willberealizedbutseen from timezero, they can all be stochastic variables.
8
Section5considers thecase where the two competingmanagement teams ownshares beforethe takeover
A B
B-sharesrespectively.
9
Becausebothtypesofshareshaveequalclaimto thesecurityvalueand
theclass A-sharesdeterminecontrol, we shouldexpectp
A p
B
. Therefore, in therest of this
paperwe willrestricttheanalysis to thenaturalcase with p
A
p
B 1.
Similarly,we willletp 1
A and p
1
B
denotethepricepershare oeredina tenderoerat time
one for the class A- and class B-shares respectively. In this setup, the requirement that the
same relative premiumshouldbeoered to bothclassesof sharescan bewritten asfollows
p 1
A
p
A
= p
1
B
p
B
, p
1
B
= p
B
p
A p
1
A
: (1)
Themaintheoreticalresultofthepaperisthattheregulationunderweakconditionsimplies
that there will be no price dierence between share classes. Before going into the technical
details we will give the basic intuition for this result. Just before time one, there will be a
competition for control of the company. The outcome of this competition is either that the
incumbent management stays in control or that a rival (competing management) takes over.
Thepricesforan A-anda B-shareattimezero(p
A andp
B
)aregivenastheexpectedvalueof
theA-andtheB-sharerespectivelyat timeone. Assumenowthatthere existsan equilibrium
characterizedby p
A
p
B
>1. Inordertocontrolthecompanyattimeone,thewinnerhastoacquire
thecontrolling A-shares{ i.e. at time one the winnerhas to give an oer that is accepted by
the atomistic holders of the A-shares. In addition, because of the regulation the winner will
have to give a pro rata lower oer to the holders of the B-shares ( p
B
p
A
the oer given to the
holdersoftheA-shares). IftheholdersoftheB-sharesacceptthisoer,thepriceofanA-share
at time one willbe precisely p
A
p
B
the priceof a B-share at time one. If all the winningoers
arecharacterized bythat theholders of B-shares accept the prorata lower oer,we willhave
thatthepriceofan A-shareat timeone always willbe p
A
p
B
thepriceof aB-shareat timeone.
Therefore, p
A
p
B
>1willinthiscase be arationalequilibrium. Ontheotherhand,iftheholders
of the B-shares do not always accept the pro rata lower oer (because it is optimal for them
instead to get their share of therm's security value) we have that theprice of an A-share is
lessthan p
A
p
B
the priceof a B-shareat time one. Therefore, in such a case p
A
p
B
>1 cannot be
arationalequilibrium. Thisimpliesthat ifthere forall p
A
p
B
>1is justa smallprobabilitythat
the winning oer is rejected by the holders of the B-shares, then we will have that the only
A B
securityvalueof theirshares. Itturnsout thatthisconditionwillbe fullledifthere isjust a
slight chancethat asuÆcientlystrongwinnerexists.
2.1 The analysis
Theanalysiswillnowproceedinthefollowingway. Firstly,for p
A
p
B
given,wecalculatethemax-
imumpricesthatthe twocompeting management teamswillbe ableto oer forthetwo share
classesat time one. The management with thehighest maximumprice forthe A-shares (and
therebyalso fortheB-shares)willwinthetakeovercontest. Secondly,weusethisto derivethe
possiblepricesattimeonefortheclassA-andtheclassB-sharesrespectively. Finally,thiswill
allow us to go back to time zero and derive the share prices, ifany, that are consistent both
withthevalueof theshares at timeone and the p
A
p
B
thatwastaken asgiven.
Thefollowinglemmastatesthemaximumpricesthatthetwocompetingmanagementteams
areable to oerat timeone.
Lemma 1 (The incumbent's and the rival's maximum prices).
a) Incumbent:
There exist maximum prices (p I
A
;p I
B
) given by
p I
A
= y
I +z
I
s
A +s
B p
B
p
A
for p
A
p
B 1+
z
I
s
A y
I
(2a)
p I
A
=y
I +
z
I
s
A
for p
A
p
B
>1+ z
I
s
A y
I
(2b)
p I
B
= p
B
p
A p
I
A
such that the incumbent only will launch winning oers p for the A-shares characterized by
y
I
pp I
A .
b) Rival:
There exist maximum prices (p R
A
;p R
B
) given by
p R
A
= y
R +z
R
s
A +s
B p
B
p
A
for p
A
p
B 1+
z
R
s
A y
R
(3a)
p R
A
=y
R +
z
R
s
A
for p
A
p
B
>1+ z
R
s
A y
R
(3b)
p R
B
= p
B
p
A p
R
A
such thatthe rivalonlywill launchwinning oers pfor theA-sharescharacterized byy
R
p
Since theresultsin thelemma aresymmetric withrespectto the incumbent and therival,
we will only give comments to the results for the incumbent. We note that the incumbent's
maximum price for the controlling A-shares is continuous in p
A
p
B
and strictlyincreasing in p
A
p
B
until p
A
p
B
reachesthecriticalvalue1+ z
I
s
A y
I . For
p
A
p
B
largerthanthiscriticalvalue,themaximum
price is independent of p
A
p
B
. The explanation is as follows. For low values of p
A
p
B
the holders
of A- and B-shares will all accept the maximum oer given by the incumbent. Thereby, the
incumbent willlose moneyon buying boththe A-and theB-shares because the shareholders
areoered aprice higherthanthesecurity value. Theloss ofmoney is covered bythe private
benet. Forlarger valuesof p
A
p
B
,theoer givento theholders ofB-shares islowered. This will
decrease the incumbent's loss on the B-shares and make it possible for him to pay a higher
pricefortheA-shares. Fora suÆcientlyhigh p
A
p
B
(>1+ z
I
s
A y
I
) theholders of B-shares willnot
accepttheincumbent's oerbecauseit insteadwillbeoptimalforthemto receive thesecurity
valueoftheshares. Inthiscase, theincumbent canuse allhis privatebenetto coverthe loss
from buyingtheA-shares z
I
=s
A (p
I
A y
I
) , p
I
A
=y
I +
z
I
s
A
.
We will now choose a xed p
A
p
B
1. At time one we can then use lemma 1 to determine
which management that will win the takeover contest. The incumbent will stay in control if
and only if p I
A p
R
A
. In order to determine the precise size of the winning oer, we must
knowsomethingaboutthebehaviorofthelosingmanagement. Let usconsiderthecasewhere
p I
A
>p R
A
. Iftherivaldoesnotgiveanyoer therivalwillhave zeroprot. Iftherivalgivesan
oerp^ R
A
>p I
A
,theincumbentwillprefertolet therivalwin. This willlead toa negative prot
fortherival. Ifinsteadtherivalgivesanoerp^ R
A 2]p
R
A
;p I
A
],therivalwillgetazeroprotifthe
incumbentmatcheshisoerandtherivalwillgetanegativeprotiftheincumbentmistakenly
(has a tremblinghand and) lets R win. If R gives an oer p^ R
A p
R
A
, R will get zero prot if
his oer is matched by the incumbent and a positive prot if the incumbent mistakenly (has
a tremblinghandand) letsR win. Thereby, we have argued that R 'sloser reply, p^ R
A
,mustbe
givenbyp^ R
A p
R
A
and correspondingly,that I'sloser reply,p^ I
A
,must be given byp^ I
A p
I
A .
10
We will later in this paper show that the main theoretical results are independent of the
precisespecicationof theloser'sbehavior. However, whenwe inthefollowingdiscussthedis-
10
Thishingesontheassumptionthat theloserdoesnotalreadyownanyA-shares. Iftheloseralreadyowns
someshares,theloserwillhaveastrategicincentivetooerabovehisownmaximumpricebutbelowthewinner's
maximumprice. Theloserwillhavethisincentivebecausethiswillforcethewinnertobuyhissharesatahigher
price. Thisfactwillbefurtherdiscussedinsection 5.
to have a more specic descriptionof the behavior of the losing management. Therefore, we
willhere briey describe the behavior where R doesnotgive anyreply (R is passive,p^ R
A
=0)
butI givesa replycorrespondingto his maximumprice(I ismaximallyaggressive,p^ I
A
=p I
A ).
The reason for such a behavior could be as follows. When R knows that he will never win,
R willnever give an oer iffor examplehe incursjust a smallcost of bidding. In contrast, it
can be argued that I willnot incur the same cost of bidding and that I at the same time is
more loyal to the existing shareholdersthanto R . Ifthese relations areknown to both ofthe
competing management teams, R 'slosingbehavioris never to give an oer whileR aswinner
always willhave to oer at leastI's maximumprice.
For axed p
A
p
B
1,an arbitraryrealizationof (y
I
;z
I
;y
R
;z
R
) at timeone,and an arbitrary
loser reply p^ I
A
;p^ R
A
, table 1 lists the winner of the control contest, the winner's oer for the
A-shares (and thereby also for the B-shares), and the condition for the holders of B-shares
to accept the oer from the winning management. The table will later be used to obtainthe
prices of theA- and B-shares at time one and based on rationalexpectationsthiswill enable
usto calculatetheprices at timezero.
We will now briey explain table 1. Assume that we at time one (for a xed p
A
p
B ) have
a realizationof (y
I
;z
I
;y
R
;z
R
) fulllingconditiona) in table 1. In thiscase I's and R 's maxi-
mum prices are given by (2a) and (3a) respectively. This gives that R willwin if and only if
y
R +z
R
>y
I +z
I
. IfR oerslessthany
I
fortheA-shares, theshareholderswillnottenderto
R . Hence,we can saythat I always givesan implicitreply of y
I
andaccordingly we willhave
^ p I
A 2 [y
I
;p I
A
]. Because R must buy the A-shares, R has to: i) match I's reply and ii) ensure
that his oer is at least y
R
such that theholders of the A-shares willpreferto sell theshares
rather than free riding. Thereby, R 'soer for the A-shares has to be given by maxfy
R
;p^ I
A g.
TheholdersoftheB-sharesthenreceiveanoerof p
B
p
A
max fy
R
;p^ I
A
g. Theywillacceptthisoer
ifand onlyiftheoeris abovey
R
,because theybyrejectingtheoer insteadwillreceivetheir
share of the security value, y
R
, under R 's management. For p
A
p
B
> 1 or p
A
p
B
= 1 and p^ I
A
> y
R
theconditionfortheholdersofB-sharesto acceptisequivalentto p
B
p
A
^ p I
A y
R . For
p
A
p
B
=1and
^ p I
A y
R
we willjust assume that the holders of B-shares reject the oer and instead receive
theirshare ofthe securityvalue, whichgivesthe same result.
The winner'soer holders of B-shares
fortheA-shares to accept the oer
a) p
A
p
B
minf1+ z
I
s
A y
I
;1+ z
R
s
A y
R g
a
R
) R wins: y
R +z
R
>y
I +z
I
maxfy
R
;p^ I
A g
p
B
p
A
^ p I
A y
R
^ p I
A 2
y
I
; y
I +z
I
s
A +s
B p
B
p
A
a
I
) I wins: y
I +z
I y
R +z
R
maxfy
I
;p^ R
A g
p
B
p
A
^ p R
A y
I
^ p R
A 2
0;
y
R +z
R
s
A +s
B p
B
p
A
b) 1+ z
I
s
A y
I
<
p
A
p
B
<1+ z
R
s
A y
R
z
I
y
I
<
z
R
y
R
b
R
) R wins:
y
R +z
R
s
A +s
B p
B
p
A
>y
I +
z
I
s
A
maxfy
R
;p^ I
A g
p
B
p
A
^ p I
A y
R
^ p I
A 2
h
y
I
;y
I +
z
I
s
A i
b
I
) I wins: y
I +
z
I
s
A
y
R +z
R
s
A +s
B p
B
p
A
maxfy
I
;p^ R
A g
p
B
p
A
^ p R
A y
I
^ p R
A 2
0;
y
R +z
R
s
A +s
B p
B
p
A
c) 1+ z
R
s
A y
R
<
p
A
p
B
<1+ z
I
s
A y
I
z
R
y
R
<
z
I
y
I
c
R
) R wins: y
R +
z
R
s
A
>
y
I +z
I
s
A +s
B p
B
p
A
maxfy
R
;p^ I
A g
p
B
p
A
^ p I
A y
R
^ p I
A 2
y
I
; y
I +z
I
s
A +s
B p
B
pA
c
I
) I wins:
y
I +z
I
s
A +s
B p
B
p
A y
R +
z
R
s
A
maxfy
I
;p^ R
A g
p
B
p
A
^ p R
A y
I
^ p R
A 2
h
0;y
R +
z
R
s
A i
d) p
A
p
B
maxf1+ z
I
s
A y
I
;1+ z
R
s
A y
R g
d
R
) R wins: y
R +
z
R
s
A
>y
I +
z
I
s
A
maxfy
R
;p^ I
A g
p
B
p
A
^ p I
A y
R
^ p I
A 2
h
y
I
;y
I +
zI
s
A i
d
I
) I wins: y
I +
z
I
s
A y
R +
z
R
s
A
maxfy
I
;p^ R
A g
p
B
p
A
^ p R
A y
I
^ p R
A 2
h
0;y
R +
zR
s
A i
Table 1: Fora xed p
A
p
B
1, an arbitrary realizationof (y
I
;z
I
;y
R
;z
R
), and an arbitrary reply
of the losing management given by p^ I
A and p^
R
A
, the table shows which management that will
winthetakeovercontest, thewinningoer forthe A-shares,and the conditionfortheholders
ofthe B-shares to accept theoer given bythewinningmanagement.
remember is thedierence between the cases where I winsand thecases where R wins. If R
wins,he hasto oer at leasty
I
because I otherwisewillstayincontrol, i.e. I always givesan
implicitoer ofy
I
. Inthecases whereI wins,there isno suchimplicitoerfrom Rbecause R
willhaveto give an explicitoer inorder to obtaincontrol.
2.2 Share prices at time zero
Attimeone, thevalue of(y
I
;z
I
;y
R
;z
R
)willberealized. Wewilldenotethese possiblerealiza-
tionsby(y i
I
;z i
I
;y i
R
;z i
R
),i=1;:::;n. Thecorrespondingprobabilities,knownattimezero, are
denoted i
. We willnowfora given p
A
p
B
1 denethefollowingdisjointsets:
A
R
=fijR winscontroland theB-shareholdersreject R 'soerg
AB
R
=fijR winscontroland theB-shareholdersaccept R 'soerg
A
I
=fijI winscontroland theB-shareholders reject I's oerg
AB
I
=fijI winscontroland theB-shareholders accept I's oer g:
Fromriskneutrality,aninterestrateofzero, and theresultsintable 1,theshareprices at
timezero aregiven as
p
A
= X
i2A
R
i
maxfy i
R
;p^ I
A (i)g+
X
i2AB
R
i
maxfy i
R
;p^ I
A (i)g
+ X
i2A
I
i
max fy i
I
;p^ R
A (i)g+
X
i2AB
I
i
maxfy i
I
;p^ R
A (i)g
= X
i2A
R
i
maxfy i
R
;p^ I
A (i)g+
X
i2AB
R
i
^ p I
A (i)+
X
i2A
I
i
maxfy i
I
;p^ R
A (i)g+
X
i2AB
I
i
^ p R
A
(i) (4)
p
B
= X
i2A
R
i
maxfy i
R
; p
B
p
A
^ p I
A (i)g+
X
i2AB
R
i
maxfy i
R
; p
B
p
A
^ p I
A (i)g
+ X
i2A
I
i
max fy i
I
; p
B
p
A
^ p R
A (i)g+
X
i2AB
I
i
maxfy i
I
; p
B
p
A
^ p R
A (i)g
= X
i2A
R
i
y i
R +
X
i2AB
R
ip
B
p
A
^ p I
A (i)+
X
i2A
I
i
y i
I +
X
i2AB
I
ip
B
p
A
^ p R
A
(i): (5)
We will now examine if a given p
A
p
B
1 is consistent with (4) and (5) . First we observe
that p
A
p
B
= 1 is consistent with (4) and (5) (can be seen directly byinserting in(4) and (5) ).
The intuition is that there inthis case willbe given thesame oer forbothA- and B-shares.
B-shares. Therefore,thevalueofanA-shareat timeonewillbeequalto thevalueofaB-share
at timeone realizationforrealization. Thisgivesp
A
=p
B .
We willnowassumethat p
A
p
B
>1andexaminewhenthiscanbeconsistentwith(4)and(5) .
For p
A
p
B
>1 we have that
i2A
R
)
p
B
p
A
maxfy i
R
;p^ I
A
(i)g <y i
R
, maxfy
i
R
;p^ I
A (i)g<
p
A
p
B y
i
R
(6)
i2A
I
)
p
B
p
A
maxfy i
I
;p^ R
A
(i)g<y i
I
, maxfy
i
I
;p^ R
A (i)g<
p
A
p
B y
i
I
: (7)
By usingthistogether with(4)and (5) we obtain
p
A
p
A
p
B 8
<
: X
i2A
R
i
y i
R +
X
i2AB
R
i p
B
p
A
^ p I
A (i)+
X
i2A
I
i
y i
I +
X
i2AB
I
i p
B
p
A
^ p R
A (i)
9
=
;
=p
A :
For p
A
p
B
>1wehave thatthe weakinequalityabovewillhold asastrict inequality(leading
to a contradiction)ifand onlyifthere is a realizationiinone of the two cases in(6) and (7) .
Therefore, we have that p
A
p
B
>1 will be an equilibrium ifand onlyif P
i2AB
R [AB
I
i
=1, i.e.
when the holders of B-shares always accept the winning oer. The intuition for this is fairly
simple. AssumeforexamplethatthepriceofanA-shareis30%abovethepriceofaB-shareat
timezero. Inthecases wheretheholdersof theB-shares alsoaccept thewinningoerat time
one,thepriceofan A-sharewillbe30% above thepriceof aB-shareat timeone. However, in
thecases where theholders ofB-shares do notaccept thewinningoer at time one,the price
of an A-share will be less than30% above the price of a B-shareat time one. The price of a
share at timezero is theexpectedvalue ofthe share at timeone. Therefore, ifthere arecases
wherethe holdersof B-sharesdo not accept thewinningoer, theprice of anA-share willbe
lessthan 30%above thepriceof a B-share. Hence, p
A
p
B
=1:3cannot bean equilibriumin this
case.
In thecase where p
A
p
B
=1is theonlypossibleequilibrium,all realizations of(y
I
;z
I
;y
R
;z
R )
willbeinregiona)intable1. ThisgivesthatRwillobtaincontrolifandonlyify
R + z
R
>y
I + z
I .
Therefore, the management that can provide the highest total value of the rm will end up
managingthe rm,i.e.theregulation impliessocialoptimality.
We summarizetheabove inthefollowingtheorem.
Independent of the reply of the losingmanagement we have the following results:
a) Multiple equilibria:
Therewillbemultipleequilibria ifandonlyifthereexistsa p
A
p
B
>1such that P
i2AB
R [AB
I
i
=
1, i.e. if and onlyif the holders of B-shares always accept the winning oer. Furthermore, we
can say that
p
A
p
B
=1 will always beone equilibrium.
b) Unique equilibrium:
If there does not exist a p
A
p
B
>1 such that P
i2AB
R [AB
I
i
=1 then
p
A
p
B
=1 is the onlyequilibrium.
There is social optimality in the sense that the management giving the highest total rm
value (y
j +z
j
) will bethe management after the takeover contest.
Theprices at time zero for the two classes of shares will be givenby
p
A
=p
B
= X
i2A
R
i
y i
R +
X
i2AB
R
i
^ p I
A (i)+
X
i2A
I
i
y i
I +
X
i2AB
I
i
^ p R
A (i);
where the reply of the losingmanagement is characterized by
^ p I
A (i)y
i
R
^ p R
A (i)y
i
I :
Theorem1doesnotspecifytheexact replyofthelosingmanagement. Wewillnowexamine
howthe reply of the losing management inuences the possibilityof multipleprice equilibria.
Thereafter,we willsimplifytheconditionabove forexcludingmultiplepriceequilibria.
Denition: Areply
^
^ p I
A
;
^
^ p R
A
issaidtobemoreaggressivethanp^ I
A
;p^ R
A
ifandonlyif
^
^ p I
A (i)p^
I
A (i)
and
^
^ p R
A (i)p^
R
A
(i) forall i.
Lemma 2 (Equilibrium under dierent replies).
If p
A
p
B
>1 isan equilibrium underone reply p^ I
A
;p^ R
A
, the same p
A
p
B
will also bean equilibrium for
all replies that are more aggressive than p^ I
A
;p^ R
A .
Proof. If A
p
B
>1 isan equilibriumunder thereply p^ I
A
;p^ R
A
,it followsfrom table 1 and theorem
1that we forevery realization(y i
I
;z i
I
;y i
R
;z i
R
) eitherhave that
i) R wins and p
B
p
A
^ p I
A (i)y
i
R
orthat
ii) I wins and p
B
p
A
^ p R
A (i)y
i
I :
But if R (I) wins for a given realization under the reply p^ I
A (^p
R
A
), then for xed p
A
p
B
> 1 R
(I) willalso win underthe more aggressive reply
^
^ p I
A (
^
^ p R
A
) (see table 1). Because p
B
p
A
^
^ p
I(R)
A
(i)
p
B
p
A
^ p
I(R)
A
(i) y i
R (y
i
I
), we have that the holders of B-shares also willaccept thewinner's oer
underthemore aggressive reply.
Fromlemma 2itfollowsthatamoreaggressivereplyfromtheloserwillmakeitmorelikely
thatthere existmultiplepriceequilibria. Therefore,ifwecan excludemultiplepriceequilibria
forthemaximallyaggressivereply,wewillhaveexcludedmultiplepriceequilibriaforallreplies.
Theconditionformultiplepriceequilibriagivenintheorem1 isdiÆculttousebecausethe
determination of whether a realization (y i
I
;z i
I
;y i
R
;z i
R
) belongs to a certain region or not will
dependon p
A
p
B
. The following theorem will give the necessary and suÆcient conditionfor one
realization (y i
I
;z i
I
;y i
R
;z i
R
) to exclude multipleprice equilibriaunder the maximally aggressive
reply. From lemma 2 follows that thiscondition willalso be a suÆcient conditionto exclude
multiplepriceequilibriaforlessaggressive replies.
Theorem 2 (Excluding multiple price equilibria).
a) Forallreplies,we havethat p
A
p
B
=1 will betheuniqueequilibriumifthereisa strictlypos-
itiveprobability for a realization (y i
I
;z i
I
;y i
R
;z i
R
) fulllingone of the following conditions:
i) z
i
R s
B z
i
I
y i
R
>y i
I +z
i
I
(R wins)
ii) z
i
R
<s
B z
i
I
y i
R +
z i
R
s
A
>y i
I +
z i
I
s
A
(R wins)
iii) z
i
I s
B z
i
R
y i
I y
i
R +z
i
R
(I wins)
iv) z
i
I
<s
B z
i
R
y i
I +
z i
I
s
A y
i
R +
z i
R
s
A
(I wins)
conditions for one single realization to exclude multiple priceequilibria.
Proof. The proofis given intheappendix.
From theorem 2 it follows that multiple price equilibria will be excluded if there is just
a small probability for a realization where there exists a suÆciently strong winner. For the
maximallyaggressivereply,theconditioninthetheoremisalsothenecessaryconditionforone
realizationto exclude multiple priceequilibria. However, it is possiblefor several realizations
together to exclude multipleprice equilibriaeven though each realization alone is not able to
excludemultiplepriceequilibria(seetheexampleinsection3). Therefore,thetheorem canbe
strengthenedinthe generalcasewith respectto thenecessaryconditions.
3 A numerical example with multiple price equilibria
Thefollowingnumericalexamplewillillustratethepossibilityofmultiplepriceequilibriaandgo
throughtheeconomic argumentsleading to these equilibria. Inaddition we willalsoillustrate
howseveralrealizationof(y
I
;z
I
;y
R
;z
R
)togethercanreducethesetofmultiplepriceequilibria.
Especially, we will see how several realizations together can exclude multiple price equilibria
even though each realizationalone is notableto exclude multiplepriceequilibria.
In theexamplewe willassume thatthe behavior ofthe losingmanagement is asdiscussed
insubsection2.1, i.e. thatR istotally passiveand thatI ismaximallyaggressive.
11
We assume that s
A
= s
B
= 1
2
. Furthermore, we will start by assuming that there is only
one possiblerealizationgiven by (y 1
I
;z 1
I
;y 1
R
;z 1
R
)=(120;30;100;45). We observe thatthisreal-
izationdoesnotsatisfytheconditionintheorem2{i.e.wemusthavethatthereexist multiple
priceequilibriaunder some replies.
From lemma 1 we get that I's maximum price is higher than R 's maximum price if and
onlyif p
A
p
B
1:6364. The multiplepriceequilibriaare givenin thefollowingtable.
11
Herewenotethat ifwehavemultiplepriceequilibriaunderthisbehavior,wewillalsohavemultipleprice
equilibriaifRhasamoreaggressivebehavior(lemma2).
A
p
B
p
A p
B
Winner
1 120 120 I
2]1:6364;1:8] 180 p
B
p
A
180 R
If the market sets p
A
p
B
= 1, I will win the takeover contest. If the market sets 1:6364 <
p
A
pB
1:8, R will win. In addition these prices are the only possible rational prices. On the
other hand, if themarketexpects I to win, we will have p
A
p
B
=1,and I will win. Similarly,if
the market expects R to win, we will have 1:6364 <
p
A
p
B
1:8 and R will actually win. The
explanationforthe priceequilibriagoesasfollows:
1. If p
A
p
B
1:6364I willbeabletogivethehighestoerandtherebyI willwin. Risassumed
tobepassivesoRwillnotgiveanycounteroerandIwilljuststayincontrol. Therefore,
we willhave thatp
A
=p
B
=120 and 1<
p
A
p
B
1:6364willnotbearationalequilibrium.
2. If 1:6364 <
p
A
p
B
1:8 R will be able to give the highest oer and thereby R will win.
BecauseI isassumed to be maximallyaggressive, I's reply is (^p I
A
;p^ I
B
)=(180;
p
B
p
A 180).
Since p
B
p
A
180 <120 forall p
A
p
B
in theinterval, we have thatthe holders ofB-shares will
rejectthe oer. Hereby,I willbuyonlytheA-sharesandhis protwillbe30+ 1
2 120
1
2
180 = 0. This shows that it willbe possible for I to give such an oer. In order to
win,R hasto match I's oer. Since p
B
p
A
180100 forall p
A
p
B
intheinterval, theholders
of B-shares will accept R 's oer. Therefore, we have that p
A
=180 and p
B
= p
B
p
A 180
and therebythat p
A
p
B
intheintervalwillbe arationalequilibrium.
3. If p
A
p
B
>1:8we willstillhave that R wins and thatR must match I's oer. However, in
thiscaseR 'soertotheB-shareholderswillbe p
B
p
A
180<100forall p
A
p
B
>1:8. Therefore,
the holders of B-shares will reject the oer given by R and we get that p
A
= 180 and
p
B
=100. Butthis leadsto pA
p
B
=1:8 why pA
p
B
>1:8 cannotbe arationalequilibrium.
We nowchangethe examplebyassuming thatwe have two possiblerealizations:
(y 1
I
;z 1
I
;y 1
R
;z 1
R
)=(120;30;100;45 ) withprobability 1
= 1
2
(y 2
I
;z 2
I
;y 2
R
;z 2
R
)=(120;30;42:941;10 0) withprobability 2
= 1
2 :
by
n
p
A
p
B j
p
A
p
B
=1 _ p
A
p
B
2]1:7;1:8]
o
:
We observe that the extension of the example with another realization has decreased the
set of price equilibria. Furthermore, we note that realization 2 is not alone able to exclude
multiplepriceequilibria.
Finally,we nowchangerealization2 to
(y 2
I
;z 2
I
;y 2
R
;z 2
R
)=(120;30;37:368;10 0) withprobability 2
= 1
2 :
By going through the calculations again we now get that p
A
p
B
= 1 is the onlyequilibrium.
Thisshows thatrealization1 and2 togethercan exclude multiplepriceequilibriaeven though
none of the two realizations fulll the conditions in theorem 2. For realization 1 we have al-
readyseenthat wefor thisrealizationalone ( 1
=1) willhave multiplepriceequilibrium. For
realization2 alone( 2
=1) we can show thatwe aspriceequilibriainaddition to p
A
p
B
=1 will
have p
A
p
B
2]1:9;4:817].
4 Distributional consequences of the regulation
Section2 showed that theregulation underweak conditionsleads to p
A
p
B
=1 being theunique
priceequilibrium. From thisfollowed that theregulationalso impliessocialoptimality. Based
on p
A
p
B
=1we willinthissectioninfurtherdetaildiscusstheconsequences oftheregulation(in
thissection called regulation1) with respectto share prices and thetotal value of the shares.
The consequenceswill be compared to thelessrestrictive regulation (calledregulation 2) that
allows tenderoersto discriminatebetweenshare classesbutnotwithinthesame class.
12
Under regulation1,R willtake overifand onlyify
R +z
R
>y
I +z
I
. Under regulation2,I
and R willonlycompeteabouttheA-shares to which they areable to oer y
j +
z
j
s
A
,j =I;R .
Therefore,R willwinifandonlyify
R +
z
R
s
A
>y
I +
z
I
s
A
. Hence,regulation2doesnotingeneral
implysocialoptimalitysincetoo muchweight isputon theprivatebenet. Inorder to further
discuss the distributional consequences, it is advantageous to make an assumption regarding
12
Forregulation2wealsoassumethatpartialbidswithinaclassisnotallowed(asalsoassumedforregulation
losing behavior is passive whileI's losing behavior is maximally aggressive. The results that
willfollow from assuminganother behavior of the losing management willalso be listed, and
we willseethat there isno qualitativedierencein theresults.
From the losing behavior assumed it follows that R will not give any oer under regula-
tion 1 when y
I +z
I y
R +z
R
. Therefore, in this case I will just stay in control. When
y
R +z
R
>y
I +z
I
,R willhave to oer maxfy
I +z
I
;y
R
g forboththeA-and theB-shares due
tothelosingbehaviorof I. Similarly,underregulation2wehavethatR willnotgiveanyoer
when y
I +
z
I
s
A y
R +
z
R
s
A
why I willjust stay incontrol. When instead y
R +
z
R
s
A
>y
I +
z
I
s
A , R
willhave to oer maxfy
I +
z
I
s
A
;y
R
gfortheA-sharesduetothelosingbehaviorof I. From this
we obtainthefollowinglemma.
Lemma 3 (Results from a tender oer).
Assume that we have given an incumbent management with (y
I
;z
I
) and a rival with (y
R
;z
R )
both known at time zero. Furthermore,assume that the losingbehavior of I is tobemaximally
aggressive while the losing behavior of R is to bepassive. Under the two dierent regulations,
the share prices and the market value of the rm attime zero will then be as follows.
a) If the same relative premium has to be oered to both share classes (regulation 1), we will
have:
The managementwith maximum y
j +z
j
will win the takeover contest.
i) If I is the winner (y
I +z
I y
R +z
R ):
p
A
=y
I
p
B
=y
I
V
firm
=y
I :
ii) If R is the winner (y
I +z
I
<y
R +z
R ):
p
A
=maxfy
I +z
I
;y
R g
p
B
=maxfy
I +z
I
;y
R g
V
firm
=maxfy
I +z
I
;y
R g:
The managementwith maximum y
j +
z
j
s
A
will win the takeover contest.
i) If I is the winner (y
I +
z
I
s
A y
R +
z
R
s
A ):
p
A
=y
I
p
B
=y
I
V
firm
=y
I :
ii) If R is the winner (y
I +
z
I
s
A
<y
R +
z
R
s
A ):
p
A
=maxfy
I +
z
I
s
A
;y
R g
p
B
=y
R
V
firm
=s
A maxfy
I +
z
I
s
A
;y
R g+s
B y
R :
By usinglemma 3 we can now comparethe distributionalconsequences of the two regula-
tions.
Theorem 3 (Distributional consequences).
Ifweundertheassumptionsstatedinlemma3comparethecasewheretenderoersarerequired
togivethesamerelativepremiumtobothclassesofshares(regulation1)tothecasewheretender
oers can discriminate freely betweenshare classesbut not withinthe shareclasses (regulation
2), we have the following:
a) Regulation 1 will always favor the holders of the class B-shares.
b) Theeect of regulation 1 on the value of the class A-shares is mixed.
c) Theeect of regulation 1 on the market value of the rm ismixed.
Proof. The proofis given intheappendix.
Further analysis shows that the holders of the B-shares independently of the losing be-
havior will prefer regulation 1. Only when both management teams are passive (maximally
aggressive)aslosingmanagement willtheholdersof A-sharesunambiguouslypreferregulation
regulation2. Furthermore,onlyin thiscasewillthe conclusionunder c) intheorem 3 be that
themarket value ofthe rmis unambiguouslyhighest underregulation 1.
Allinallwecanconcludethatregulation1issociallyoptimalinthesensethatregulation1
ensuresthat themanagement of thermwillbe theone underwhichtherm hasthehighest
total value. In addition regulation 1 favors the holders of B-shares. However, the conclusion
regarding the holders of the A-shares and the total market value of the rm depends on the
specic circumstances.
In the theorem above, we have notsaid anything about how s
A
willbe chosen under reg-
ulation 2. Let us assume that an entrepreneur (E) is considering to start up a new rm and
hence,E wantsto chooses
A
suchthathisexpectedprotfromthermismaximized. Further-
more, we willassume that I's losing behaviour is to be maximally aggressive, that R 'slosing
behaviouris passive,andthat E canappropriateI'sprivatebeneteitherwhenhiringI orby
employinghimselfasI. Independentlyofwhichs
A
Echooses,E willnotbeabletoappropriate
all R 's private benet in the cases where R wins control.
13
If the probabilityfor z i
R z
i
I
> 0 is
small i.e. that it is unlikely that both management teams have large private benet, then E
willoptimally choose s
A
= 1. Hence, under this conditionand with regulation 2 applying, it
willbe optimalfor E voluntarily to enforce regulation 1. However, if instead the probability
for z i
R z
i
I
> 0 is large, 14
E will choose s
A
< 1 under regulation 2. The reason for this is that
E then willbe able appropriate a larger fraction of R 's private benet in the cases where R
obtains control. Because it under regulation 2 can be optimal for E to choose s
A
< 1, it is
no longer certain that the rm will be controlled by the management providing the highest
total value. Hence, forexisting rmsregulation1 is sociallypreferred relative to regulation2.
However, inthecase of upstartof a rm itmay besocially optimalonlyto enforce regulation
2. This is because regulation 2 can make it possiblefor E to appropriate more of R 's private
benet compared to regulation 1. Hereby, we can come in situations where E will start up a
rm under regulation 2 but not under regulation 1. This disadvantage of regulation 1 has to
beweighted against the advantage regulation1 will have for existing rms. Finally, it should
be noted that it is still notpossible to appropriate all R 's private benet under regulation2,
whyregulation 2also givesa toosmallincentive to startup rms.
13
Theproofs forthisresultandtheremainingresults inthissectionareomitted. However,theproofs canbe
obtainedfromtheauthors.
14
Comparewithsituation3inGrossmanandHart(1988).
In thissection we willexamine and discussifthe resultsabove also hold when theincumbent
andtherivalalreadyownsomeoftheA-sharesbeforethetakeovercontest.
15
Furthermore,this
sectionwilldiscusstheconsequences ofthecasewhereinadditiontoownershipofA-sharesthe
rival incurs a xed cost when bidding for the shares. Finally, this section willbriey discuss
thedistributionalaspects of thesechanges inthemodel.
16
5.1 Incumbent/Rival owns A-shares before the takeover contest
Wewillnowassumethatthetwopossiblemanagement teamsownanumberofA-sharesbefore
thetakeovercontest. However,wewillassume thatnoneofthetwo controlthemajorityofthe
votes.
The maximum pricesthat thetwo management teams are ableto oer areindependent of
theirshareholdingsandaretherefore stillgivenbylemma1. Forapricebelowamanagement's
maximumpricethemanagementwillprefertobuyandinthiswayobtaincontrol. Foranoer
above the maximum price, the management will prefer to sell his A-shares. Given that it is
costless to make an oer, the losing management willnow always have an incentive to make
an oer at least corresponding to his own maximum price. This way the losing management
willincrease the price that the winningmanagement will have to pay for the loser's position
in A-shares. The loser will preferan oer that is as close to the winner'smaximum price as
possiblebecausehe inthiswaywillobtainthehighestpriceforhisA-shares. Thelosercannot
hopeto pressthepriceabove thewinner'smaximumpricebecausesuchan oerwillmakethe
winnersell his shares. Therefore, we must ingeneral expectthat the losing behavior leads to
a winning oer between the maximum price of the loser and the winner. We will not model
exactlywhat thepricewillbe.
With respectto the earlierresults, we willalso have that lemma 2 holdsunchanged. Fur-
thermore, we also have that p
A
p
B
= 1 always is an equilibrium and that there willbe multiple
price equilibria if and only if there exists a p
A
p
B
> 1 such that the winner always wins with
an oer that is accepted bythe holders of theB-shares. Therefore, it is still mostdiÆcult to
excludemultiplepriceequilibriawhentheloser'sreplyisequaltothewinner'smaximumprice.
15
This will among other things be of importance in the following empirical section. This is because the
ownershipofsharesinDenmarkisnotingeneralpublicinformation. Therefore,itwillnotbepossibletodivide
thedata-setintogroupsdependentonhowmanyA-sharestheincumbentorapotentialrivalowns.
16
libria.
Theorem 4 (Excluding multiple price equilibria { no cost of bidding).
ConsiderthecasewhereI andR bothhaveapositionintheA-sharesbeforethetakeovercontest
andwherethereisno cost associatedwithbidding. In this case thefollowinggivesthenecessary
andsuÆcient conditionfor one realization to exclude multiple priceequilibria when the loser's
reply is equal to the winner's maximum price (maximally aggressive). The condition is at the
same time a suÆcientcondition for excluding multipleprice equilibria for all weaker replies.
The condition is that there exist a realization (y i
I
;z i
I
;y i
R
;z i
R
) such that either
a) z
i
I
=0 and y i
I y
i
R +
z i
R
s
A
(I wins)
or
b) z
i
R
=0 and y i
R
>y i
I +
z i
I
s
A
(R wins) :
Beforegiving the comments to thetheorem wewill rst brieysketch theproofof part b)
in the theorem: When y i
R
> y i
I +
z i
I
s
A
, R will for all p
A
p
B
be able to match any oer given by
I and therefore R will win. It is most diÆcult to exclude multiple price equilibriawhen I is
maximally aggressive. When z i
R
= 0, it is only possible for I to press R to oer y i
R
for the
A-shares. For p
A
p
B
>1 we willhave p
B
p
A y
i
R
<y i
R
whytheholdersof B-shares willreject theoer
and both classes of shares will be worth y i
R . If z
i
R
> 0 it will be possiblefor I to press R to
give an oer higherthany i
R
for theA-shares. But inthat case, there willexist a p
A
p
B
>1 such
thattheholders oftheB-shares alsowillaccept theoer. Therefore, we have thatz i
R
=0 isa
necessarycondition.
Considernowacompany,wheretheincumbentalwaysderivespositiveprivatebenet,hasa
positionintheA-shares,andismaximallyaggressivewhenlosing. Inordertoexclude multiple
priceequilibriafor thiscompany there must be a positive probabilityfor a suÆcientlystrong
rivalwithoutanyprivatebenet. Wealso notethatiftheincumbentdoesnotownanyshares,
condition b) above is replaced with theorem 2's conditions i) and ii). The reason for this is
the shares, his maximum reply will increase discontinuously to R 's maximum price. Similar
arguments holdsforR .
Comparedtotheorem 2whereitnearlyseemstooeasy toexcludemultiplepriceequilibria,
wearewhenbothmanagementteamsalwaysownA-sharesinasituationwhereitismuchmore
diÆcultto excludeequilibriawith p
A
p
B
>1.
Inthefollowingsubsectionwe willsee thattheexistenceofcost ofbiddingagainwillmake
iteasier to excludepriceequilibriawith p
A
p
B
>1.
5.2 Cost of bidding
Now assume that it costs a xed amount c for R to bid for the shares. The maximum price
for therival is stillgiven by lemma 1 if we change the denitionof the rival's privatebenet
to being privatebenet netof costs, i.e.z 0
R z
R
c. We note that it isnow possibleforthe
private benet net of costs to be negative.
17
If we let e
I and e
R
denote the fractions of the
A-sharesownedbytheincumbentandtherivalrespectively,wecanshowthefollowingtheorem.
Theorem 5 (Excluding multiple price equilibria { R incurs cost of bidding).
ConsiderthecasewhereI andR bothhaveapositionintheA-sharesbeforethetakeovercontest
andwhereR incurs acost of cwhen bidding. In this case the following givesthe necessary and
suÆcientconditionfor onerealizationto excludemultiplepriceequilibria whentheloser's reply
isequaltothewinner'smaximumprice. TheconditionisatthesametimeasuÆcientcondition
for excluding multipleprice equilibria for all weaker replies.
The condition is that there exist a realization (y i
I
;z i
I
;y i
R
;z i
R
) satisfying one of the following
ve conditions:
a) z
i
R 0
0; z i
I s
B z
i
R 0
; y i
I y
i
R +z
i
R 0
; e
R z
i
I c
(I wins and p 1
A
=p 1
B
=y i
I ):
b) z
i
R 0
0; z i
I
<s
B z
i
R 0
; y i
I +
z i
I
s
A y
i
R +
z i
R 0
s
A
; e
R z
i
I c
(I wins and p 1
A
=p 1
B
=y i
I ):
17
Wewill, whenR obtains control, assume thatI'sunmodeledreply is knownby R . Alternatively,wecan
assume that R doesnot incurthe cost c until after I has given his last reply. In both cases, it will not be
protableforI tomakeanoeraboveR 'smaximumprice.
c) z i
R 0
=0; y i
R
>y
I +
z
I
s
A
(R wins andp 1
A
=p 1
B
=y i
R ):
d) z
i
R 0
<0; z i
R 0
e
R s
A (y
i
I y
i
R ); y
i
I +
z i
I
e
I s
A +e
R s
A y
i
R +
z i
R 0
e
I s
A +e
R s
A
(I allows R towin and p 1
A
=p 1
B
=y i
R ):
e) z
i
R 0
<0; y i
I +
z i
I
e
I s
A +e
R s
A
>y i
R +
z i
R 0
e
I s
A +e
R s
A
; e
R z
i
I c
(I wins andp 1
A
=p 1
B
=y i
I ):
We observe that if we in theorem 5 set c = 0 (and have e
R
> 0), the theorem is reduced to
theorem 4. Therefore,asexpected wehave that theorem4 is justaspecialcaseof theorem 5.
Ifwecomparetheorem 4case a) withtheorem5cases a), b),and e)weseethat itiseasier
for the incumbent to prevent multiple price equilibriawhen the rival incurs cost of bidding.
Thisiscaused bythefactthat inthiscase R willnotbeable to pressI soaggressively. When
I wins independentlyof p
A
p
B
,R 's reasonfor pressingI is that he willobtaina higherprice for
his A-shares. If R does not press I, R 'sA-shares willbe worth e
R s
A y
i
I
. The maximum price
that R will be able to press I to pay is I's maximum price, in which case the shares will be
worthatmoste
R s
A y
i
I +
z i
I
s
A
. Thereby,R 'sprotonhissharesfrompressingI isatmoste
R z
i
I .
Therefore, ifwe have thatR 'spositionintheA-sharesis smallrelative to thecost of bidding,
R willchoose to be a passiveloser andI willstay asmanagement withoutany competition.
Asintheorem4wehavethatarivalwitha(net)privatebenetofzerowillexcludemultiple
priceequilibria. Becauseof thecost of biddingwe can have thatthe(net) privatebenetnow
becomesnegative. In thiscase it is possibleto come in a situationwhere the rival is notable
to make an oer above the security value of the A-shares. Given such an oer neither of the
two groups of atomistic shareholderswill accept such an oer. However, it is possiblethat it
willbeprotable for theincumbent to accept such an oer. By accepting theoer I will lose
his privatebenet buton theother hand I may receive a higherpricefor his shares.
18
When
R hasboughtall I'sA-shareswe have two possiblecases. First,R maynowown themajority
of the votes and has thereby obtained control. Second, R may still not own the majority of
thevotes,but he willstill obtaincontrol because noneof the atomisticshareholders willvote
againstR . Thisisbecause case d) impliesthaty i
R
>y i
I .
equilibriathan p
A
p
B
=1. Beforewe in thenext section examine thisresult empirically,we will
endthissectionbylistingthedistributionalconsequencesof p
A
p
B
=1inthismoregeneralsetup.
Based on z 0
R
0,and p
A
p
B
=1 underregulation1, we have 19
a) The management that can contribute with the highest total value net of bidding cost
willendup managing the company. Therefore, we have thatregulation 1 in contrast to
regulation2issociallyoptimalinthesensethatregulation1ensuresthatthemanagement
oftherm willbetheone under which therm hasthehighesttotal value.
b) ThepriceoftheB-shareswillbe(weakly)higherunderregulation1thanunderregulation
2.
c) It is not possible to say anything general about the price of the A-shares and the total
marketvalueof therm underregulation 1versus regulation2.
6 The time pattern in the voting premium in Denmark
Therequirement thatthesame relative premiumshouldbeoered to all classesof shareswas
rstmentionedinDenmarkintheEthicRulesfortheCopenhagenStockExchange(Brsetiske
regler) on November 3, 1987. However, there were no sanctions or punishments for breaking
these rules. Therefore, it was not until the Law on Security Trade No. 1072, December 20,
1995, xx31{32, and Fondsradets legal notice No. 333, April 23, 1996, xx1{10, that the regula-
tionbecame requiredbylaw.
The results in this paper imply that the regulation under weak conditions leads to equal
prices forclass A-and class B-shares. The followingwillexamine ifthisresult can be seen in
thetimepatternof thepriceratiobetween thedierentshare classesinDenmark.
OnlyafewotherstudieshavelookedatthepriceratiobetweenA-andB-sharesinDenmark
and noneofthese examinedwhether there isa changeinthisratioovertime.
20
19
Whenz 0
R
<0and y
R +z
0
R
>y
I +z
I
it is notcertain thatR will obtain control overthe company. Ife
I
is suÆcientlysmall, it will notbepossible for R to buythe A-shares from I. This is because I will lose zI
and will onlyreceivea smallprot onhis own positioninA-shares. Therefore, R mustobtain control based
ontheatomistic shareholders. Becauseof freeriding this requires anoer ofat leastyR for theshares. If R
givessuchanoer,hewillobtaincontrolandhisprotwillbez 0
R +e
R s
A y
R
. IfRdoesnotgivesuchanoerI
willstayincontrolandR 'sprotwillbeeRsAyI. Therefore,theincrementalprotfromgivingsuchanoeris
eRsA(yR yI)+z 0
R
=eRsA(yR yI)+zR c. Butbecause cisassumedtobelargerthanR 'sprivatebenet,
we havethatifR onlyhasasmall positionintheA-sharesitwill notbepossiblefor Rto earnenoughonhis
