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%From: Neil Gandal <GANDAL@VM.TAU.AC.IL>
%Date: Wed, 03 Nov 93 17:10:22 IST

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\pagestyle{myheading}\markright{NEVER WORKS}
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\begin{titlepage} \vspace*{0.05in}
\begin{center}
{\Large\bf  Equilibrium Foreclosure and Complementary Products  }
\\
\vspace{.15in} Jeffrey Church \\
University of Calgary \\ and \\ Neil Gandal\\
Tel-Aviv University \\
\vspace{0.15in}\today \\
\vspace{.1in}{\bf Abstract}\\\end{center}
{\em  In this paper we address the possibility of horizontal
foreclosure in
markets for complementary services (software) where the
consumption
value of
durables (hardware) depends on the availability of software.
Horizontal
foreclosure occurs when a hardware firm merges with a software
firm
and the
integrated firm ceases to supply compatible software for a rival
technology.
We find that horizontal foreclosure can be an equilibrium outcome
where both the merger and compatibility decisions are part of a
multistage
game which permits the foreclosed firm to play a number of
counter-
strategies.  Moreover, foreclosure may result in monopolization of
the
hardware market.
We find that the foreclosure equilibrium is inefficient:  total
surplus would
be higher without foreclosure.}


\vspace{3 mm}

\noindent We would like to thank C. Fershtman, A. Fishman, P.
Kennedy, O.
Shy, D.
Sibley and seminar participants at the Summer in Tel-Aviv
Mini-Conference on Networks, Simon Fraser University, and the
University of Victoria.  Church gratefully acknowledges the
financial support of the
Research Grants Committee at the University of Calgary.

\vspace{0.05in}\noindent
\begin{description}
\item {\bf Keywords:} Integration, Complementary Products,
Foreclosure
\end{description}
\end{titlepage}
\baselineskip=0.25in

\section{Introduction}
\hspace*{6 mm} In recent years, economists have explored  a wide
range of
 exclusionary strategies that can be used by firms to increase
their market
 share and profits.\footnote{In general
these strategies work by disadvantaging rivals either by raising
the costs of
rivals or reducing demand for a rival's product.  This work is
surveyed in
Ordover and
 Saloner (1989).  Seminal contributions are Salop and Scheffman
(1983) and two
papers by Krattenmaker and Salop (1986a, 1986b).}  The hallmark
of
recent
 work has been the emphasis on examining the effectiveness of
these
strategies
in a fully specified game-theoretic model.  In particular the
thorny questions
of the foreclosure effects of vertical integration and tying have
been
 recently addressed in equilibrium models by Ordover, Saloner,
and
Salop (OSS,
1990) and Whinston (1990), respectively.\footnote{See also Hart
and Tirole (1990) for the
foreclosure
effects of vertical integration.}
In this paper we ask whether horizontal foreclosure can emerge in
 equilibrium and if it does what are the efficiency effects in
oligopolistic
markets where complementary products are important.   In
particular
we are
 interested in industries where the consumption benefit of a
durable or
hardware good is a function of the variety of complementary
products or
 software available.  We define horizontal foreclosure to occur
when a
software firm merges with a hardware firm and the integrated firm
ceases to
 supply compatible software for a rival hardware technology.
This
action is
demand impairing or revenue reducing since the reduction in the
number of
software varieties
 compatible with the competing hardware technology decreases its
value to
consumers.

The recent trend toward integration in the entertainment industry
highlights
 the possibilities for this type of behaviour.  In the music
industry,
three of the largest electronics firms control or own three of
the
six big
record
 companies.\footnote{In the late 1980s, Sony  purchased Columbia
Records
and Columbia Pictures  (Business Week,
              $10/09/89$).
Matsushita acquired MCA, an entertainment conglomerate which
included a
record company and a major film studio  (Economist, $12/01/90$).
Philips owns Polygram  (Economist, $8/11/90$). Moreover, Toshiba
has a
minority stake in Time-Warner (Economist, $11/2/91$).  The other
two
major record companies are Bertelsmann and Thorn-EMI.} In 1993, the
next
great
format war
  will
start over the technology that will replace the audio
cassette.\footnote{See Economist, $11/07/92$; Business Week,
$06/15/92$; and Music-Week, $06/20/92$; for details.}
The two
combatants are the Digital Compact Cassette (DCC) developed by
Philips and
 Matsushita and the Mini Disc (MD), a Sony product.  The outcome of
the ``war" will largely
depend on the availability of software.
At this juncture, it appears that all of the major record
companies,
including Sony, will release titles
(software) in the DCC format, but Phillips, Matsushita,
Time-Warner, and
Bertelsmann  are hedging on whether or not they will
release titles in the MD format.\footnote{Music
Week, $06/20/92$.}

Video delivery in the home provides a second example.  At present
this service
is almost exclusively provided by the cable television industry
in
Canada and
 the United States. Integration between film
studios, cable
channels, and local cable franchise operators is quite prevalent.
Moreover,
there are numerous instances where a film studio and a network
have
entered an
exclusive licensing agreement  and where
local
cable operators have exclusive broadcast rights within their
franchise areas
 for cable channels.\footnote{For more on the
prevalence of
exclusive contracts in the cable television industry and on the
incentives to acquire programming on an exclusive basis, see
Gandal and Salant
 (1991).  An exclusive licensing
agreement
provides a network with
 exclusive broadcast rights and is very similar in effect to
integration
and foreclosure.  See Mathewson and Winter (1984).}  The prevalence
of
integration and its substitute, exclusive contracting, may make
entry by
new hardware technologies very difficult.  The ability of both
Direct
Broadcast Satellite and the telephone companies to provide
a
competitive home video delivery alternative will be impaired if
they offer
insufficient programming.

We
address the
question of horizontal foreclosure by modelling both the merger
 and foreclosure decisions as part of a multistage game.   As
Rasmussen et al
(1991) and OSS (1990) emphasize, only in a fully specified
equilibrium model
can the rationality and the welfare implications of exclusionary
strategies be
assessed.  Developing an equilibrium model permits explicit
consideration of
the objections of the `Chicago School' to the theoretic
possibility, welfare
implications, and jurisprudence of exclusionary
practices.\footnote{For a summary of the objections of the
`Chicago
School' to foreclosure theory, see the
discussions in OSS (1990) and Whinston (1990) and the references
therein.}

In the context of the possibility of horizontal foreclosure of
compatible
software there are three prima facie reasons for why it is
unlikely
to be an
equilibrium phenomena.  The first is the availability of a {\em
counter-strategy}.  If a foreclosed hardware firm has an
opportunity to
respond in a
similar manner, the initial foreclosure will be ineffective.  The
second is
that there is likely to be a {\em commitment problem}.  The
hardware firm that
forecloses is forgoing a potentially very profitable software
market.  It may
well be more profitable for the integrated firm to supply
software
in both
formats, and if so, it will be difficult to credibly commit to
foreclose.  The
last objection is that there is likely to be a {\em hold-up}
problem.
Suppose initially there are two software firms and one merges
with
a hardware
firm and the integrated firm forecloses on the second hardware
technology.  The remaining software firm will be a monopoly
provider of
software
for the
second hardware technology as well as a duopolist in the software
market for
the integrated firm's hardware.  This may be very profitable,
suggesting the
possibility that it is better not to be the software firm that
merges and
forecloses, but to remain independent.

In the model we develop there are two hardware firms who produce
differentiated products.  The
 willingness of a consumer to pay for the hardware goods depends
on
her
preferences about hardware, software and the number of compatible
software
varieties
 available.  Software is provided by two multiproduct software
firms.  Each
software firm, in the absence of foreclosure, can supply software
in
formats
 compatible with both hardware technologies.

In the initial stage of the game, the hardware firms have the
opportunity to
acquire one of the two independent software firms.  If a bid for a
software firm is accepted, then the
integrated  or
 merged firm  must determine
the formats in which it will offer software.  The integrated
firm can
 elect either to make the
software available only in a format compatible with its own
hardware or it can
make its software available in formats compatible with both
hardware
 technologies.

Having observed both the integration and foreclosure decisions of
the
first
hardware firm, the remaining hardware firm then has the
opportunity
in the
 second stage to acquire the other independent software firm and if
it
does so, it
also makes a compatibility decision.   Thus, there are two
strategic
decisions hardware
firms make.  The
first is whether or not to integrate or acquire a software firm.
The
second is, having merged with a software firm, whether or not to
make
software compatible with the competing hardware technology.
Foreclosure occurs when an integrated firm makes its software
incompatible.

We find that there are two possible equilibria.   In the first,
 the industry structure which emerges endogenously is
unintegrated.
In this
equilibrium neither hardware firm merges or acquires a software
firm.  In the
 second, we get a foreclosure outcome.  One of the hardware firms
merges with
one of the software firms and forecloses on the remaining
hardware
firm.  The
 remaining hardware firm does not respond by integrating with
the remaining independent software firm and
foreclosing.  Horizontal foreclosure
emerges
as an
equilibrium even though we have explicitly allowed for the
remaining hardware
firm to counter by integrating
 and foreclosing, addressed the potential for a hold-up problem,
and
explicitly
considered the format choice by the foreclosing firm.

There are three effects associated with unilateral foreclosure
which
underlie our results:  a demand effect, a
software
profit effect and a hardware price effect.  Unilateral foreclosure
provides a hardware firm
with a
relative advantage since there will be twice as many software
varieties
available for it.  This demand effect increases the hardware market
share
of the foreclosing
firm.

Unilateral foreclosure also has an effect on the
software
markets for the two technologies.  The foreclosing technology
remains a
duopoly, but the software market for the foreclosed technology is
now a
monopoly.  The integrating firm which forecloses forgoes duopoly
software profits in the other technology.

There is also a price effect associated with integration and
foreclosure. Integration and foreclosure  results in an increase in
price
competition in the hardware market.  Equilibrium hardware prices
are
lower when hardware firms integrate and foreclose than in the
unintegrated market structure.

If the degree of hardware differentiation is not too large relative
to the
preferences of consumers for variety, then a hardware firm will
find it
profitable to integrate and foreclose if it anticipates no response
from
the other hardware firm.  In these circumstances, the increase in
hardware market share and profits from the demand effect more than
compensates for the lost software profits.

We show that it is never optimal for a foreclosed hardware firm to
respond by integrating and foreclosing.  While this nullifies the
demand
effect, the resulting low hardware pricing (the price effect)
and the
loss in software duopoly profits result in lower profits than the
hardware
firm and the software firm would earn if they remained independent.

An alternative counter-strategy is for the foreclosed hardware firm
to
integrate  but not foreclose.  This allows it to credibly commit to
low
hardware prices, since by charging a low hardware price it extends
its
hardware market share and thus the software market in which it is
a
monopolist.  This will be a profitable counter-strategy if demand
for
hardware is relatively price elastic, ie the degree of hardware
differentiation is not too large.

The foreclosure equilibrium arises if the degree of hardware
differentiation is small enough relative to the preferences of
consumers
for software variety so that the increase in hardware market share
from
the demand effect is relatively significant.  However, the degree
of
hardware differentiation must also be large enough so that the
increase in
the hardware market share from the price effect of the
counter-strategy
is small.  Essentially what is required is that the demand for
hardware be
elastic with respect to changes in the number of software varieties
(thus
making foreclosure profitable), but relatively inelastic with
respect to
changes in the price of hardware (thus making the counter-strategy
ineffective).

Moreover if the demand effect is sufficiently large, the
foreclosure
equilibrium will involve standardization on the hardware technology
of
the foreclosing hardware firm.  Foreclosure can be an effective
means to
monopolize the hardware market.

We show that the foreclosure equilibrium is inefficient.
Total surplus
is greater in the unintegrated industry structure than in the
foreclosure
 equilibrium.

The structure of our model is closely related to that of OSS
(1990).  They
consider whether
vertical or
supply side foreclosure can emerge in an industry initially
characterized by
 two upstream suppliers, who provide an input used in production
by
 two
downstream firms.  In their paper, firms have an opportunity to
acquire one of
 the upstream suppliers in stage one.  Then in stage two, input
prices are
determined.  If a merger took place in stage one, the other
downstream firm
 has the opportunity to acquire the remaining upstream firm in
stage three.
Finally, downstream prices are set in stage four.  They find that
foreclosure
 can emerge in equilibrium.

Hart and Tirole (1990) and Reiffen (1992) argue that the OSS
result (that foreclosure can emerge in equilibrium) depends  on the
ability
of a merged firm to commit to a price in stage two.  That is, the
merged
firm must be able to commit not to compete aggressively with the
remaining supplier to supply the other downstream firm for
foreclosure to
be an equilibrium.  If the input prices were determined by Bertrand
competition (Nash equilibrium in prices), foreclosure would not
arise in
equilibrium.

In our model, the long-run strategic decisions (integration and
compatibility) are undertaken before any prices are determined.
The desirability of either integration or foreclosure is based on
the effect that it has on price competition in the hardware market.
An integrated
hardware firm that has foreclosed has no incentive to supply
software to
the other hardware firm, even after the second hardware firm has
decided
not to integrate: to do so would lower its profits and thus  the
decision not to supply compatible software is credible.

The plan of the paper is as follows.    In  section  2,  we
specify
technology
and the  preferences  of consumers.  In section 3, we derive
the equilibria to
the hardware pricing game for all possible industry structures.
In
section 4
we determine the equilibrium industry structure when foreclosure
does
not result in standardization.  In section 5 we consider whether
foreclosure can result in monopolization and standardization in the
hardware market. We address
the welfare implications of the foreclosure equilibrium in section
6 and section 7 provides
brief concluding remarks.

\section{The Model}

In this section we develop the model.  We begin by describing
technology.
\subsection{Technology}
We consider a situation where there are four firms:  two
manufacturers of
hardware and two software firms.  We denote the two hardware
firms
by A and B.
The hardware products are differentiated along the unit interval.
The
locations of the hardware firms are fixed: technology A is at the
left-end
point, technology B at the right-end point.

The software firms are multiproduct and each has a stock or
inventory
of $N$ software products, where $N>1$.  The development costs for
this
inventory of software products are sunk and we assume that the
number cannot
be augmented.  The software of both software firms is initially
available in
formats
compatible with both of the hardware technologies.  Finally, with
no loss of
generality, we assume
that the marginal cost of producing a unit of hardware or a unit
of
software
is equal to zero.

\subsection{Preferences of Consumers}

We now specify the preferences of consumers over
hardware,   software,   and   an  outside  good.
In  modelling preferences over hardware and software  we
explicitly  recognize the following:

\vspace{3 mm}
$(i)$   The  value  of  hardware depends primarily on the
availability of
software.  Without the provision of compatible software,
hardware
provides minimal consumption benefit.

$(ii)$  The  greater  the  variety of software,
the greater the  benefit or value of a hardware  technology.
However, the   marginal   value   of   additional
software is decreasing.

$(iii)$ The demand for both hardware and a variety of  software
is perfectly inelastic.
\vspace{3 mm}

\noindent From  $(iii)$ consumers purchase only
 one unit of hardware and one
unit of each variety of software.  However, from $(ii)$ they
will,
in
general, purchase more than  one  variety  of software.  We assume
that
the benefit consumers receive from consuming $N$ varieties of
software
is given by  $N^{\b}$, where,  $\b$, the value of variety, is less
than one
but greater than zero by $(ii).$

The preferences of consumers for hardware are represented using
an
address model.  The tastes of consumers are distributed uniformly
along a line of unit length, the population is normalized to one,
and  all consumers have income  $y$.  The consumption of
a hardware
technology different from the most preferred type
imposes a utility
cost  on the consumer that is proportional
 to the distance separating the
two types.

The utility function of a consumer located
distance  $t_i$  from hardware $i$ is therefore\footnote{The
network
benefit, $N_i^{\b}$,
is only obtained if hardware $i$ is purchased.}

\begin{equation}
U_i =  N_i^{\b}+x+ \alpha -kt_i,
\label{uti}
\end{equation}

\noindent where $\alpha$ is the stand-alone benefit of hardware,
$N_i$ is the number
 of software varieties consumed,
$x$ is consumption of a competitively supplied outside good, and
$k$ measures
the extent of hardware differentiation.  The
budget constraint for
a consumer is:

\begin{equation}
\sum_{j=1}^N \rho_{ji} + x=y-p_i,
\label{bud}
\end{equation}

\noindent where $\rho_{ji}$ is the price of a unit of software
variety $j$ available for hardware technology  $i$,  $y$  is
the income of the   consumer,  $p_i$   is the price of hardware
technology  $i$, and $N$ is the number of software varieties
purchased.
Maximization  of (\ref{uti}) is a two-stage procedure.  In the
second stage,  (\ref{uti})  is  maximized subject  to
(\ref{bud})
for  each  hardware technology $i$.  In  this  stage,  the
consumer selects  which varieties of software and the total
number
of varieties to consume.  Substituting this into (\ref{uti})
gives
indirect utility for  technology $i$.  In the first stage the
consumer selects the hardware variety for which indirect
utility is greatest.\footnote{We
 assume that $\alpha$ is large enough so that the purchase of one
of the two hardware technologies is optimal.}

To solve the second-stage for technology $i$,
rank the software varieties in ascending order by price.  The
marginal benefit  of  another  software  variety is $\beta
N^{\b-1}$.\footnote{To reduce notational clutter, we temporarily
drop the $i$ subscript.}  Ignoring the integer problem, the
number
of varieties consumed is then implicitly defined by $\rho_N=\b
N^{\b-1}$, where $\rho_N$ is the price of the $N^{th}$ most
expensive software variety.

In other words, the consumer purchases one unit of the $N$
lowest-priced varieties, where $N$ is  such that the
marginal benefit of the $N^{th}$
software product equals $\rho_N$.  If  the
marginal benefit of the
$N^{th}$ software product exceeds $\rho_N$, one unit of all
software varieties is consumed.

We assume that consumers purchase hardware first,
 then software.\footnote{The import of this assumption
is that software vendors cannot affect
the market share of a hardware technology by their
pricing decision.}
Before considering the indirect utility or
consumption benefit that
a consumer receives from consuming technology $i$, we state the
following lemma regarding the price of software.\footnote{The
proofs of all lemmas and propositions are in the appendix.}

\begin{lem}
If $N$ software varieties are supplied for technology $i$, then
the symmetric Nash equilibrium software price is
\begin{equation}
\rho_i =\b N_i^{\b-1}.
\label{s}
\end{equation}
\label{sp}
\end{lem}

In equilibrium, software is priced such that
each consumer who purchased technology $i$ is just willing to
purchase one unit of each software variety provided.
{}From Lemma \ref{sp}, the price of
software is determined solely by the number of software
varieties supplied and it is a decreasing function of the number
of software varieties.

A symmetric software price equal to $\b N_i^{\b-1}$  implies
that
the  budget constraint can be written as $\b N_i^{\b} + x = y -
p_i.$  Solving for $x$ and substituting into (\ref{uti}) gives
the
indirect utility function of a consumer located
distance $t_i$ from technology $i$:
\begin{equation}
V_i = (1-\b)N_i^{\b}+y+ \alpha -p_i -kt_i.
\label{iu}
\end{equation}

A consumer purchases hardware A if the benefit
from adopting system    A $(V_A )$   exceeds the benefit
from adopting system  B $(V_B)$.  If we measure $t$ from the
left-end
point,\footnote{In other words, $t_A=t$ and $t_B=1-t$.} then
the marginal consumer is defined implicitly by $V_A=V_B$.  Using
equation (\ref{iu}) and rearranging terms, the value for the
equilibrium market share for technology A is
\begin{equation}
t= {((1-\b)(N_A^{\b}-N_B^{\b}) -(p_A-p_B) +k)
\over (2k)}.
\label{mar}
\end{equation}

The equilibrium market share for technology B is simply ($1-t$).
The market share for either hardware firm cannot exceed 1.  If
$t=1$ then technology A is exclusively adopted.

\section{Timing of the game}

We model a three-stage game.
In the initial stage, the hardware
firms have the opportunity to bid for one of two independent
software firms.  If the software firm is acquired, we assume
without loss of generality that it is acquired by hardware firm
A.
The integrated  or merged firm then must determine the formats in
which it will offer software.  The integrated firm can
elect
either to make the
software available only in a format compatible with its own
hardware or
it can market software available in formats compatible with both
hardware technologies.
If one of the hardware firms acquires one of the software firms
in
the first stage, the remaining hardware firm then has the
opportunity in the second stage to acquire the other independent
software firm, and if it does so, it also makes a compatibility
decision.  It does this after observing the compatibility
decision
of the first integrated firm.  In the final stage hardware prices
are determined and consumers
make their adoption decisions.  The subgame perfect equilibrium
is
found by backwards induction.

\subsection{Hardware Price Competition}

There are four possible industry structures.\footnote{There might
appear to be two additional industry structures: (1) both
hardware firms integrate, but neither foreclose and (2) one
hardware firm integrates, but does not foreclose, and the other
hardware firm does not integrate.   These cases are
formally equivalent to the unintegrated case since they do not
change the
number of software varieties available for either technology or the
pricing behaviour of the hardware firms. There is neither a demand
nor a
pricing effect.} These four subcases are:\newline

\noindent (1)  Unintegrated:  Neither hardware firm
acquires a software firm.\newline
(2)  Bilateral Foreclosure:  Both hardware firms
merge with an independent software firm and foreclose.\newline
(3)  Integrated:  Both hardware firms have
acquired independent software firms, but only one has
foreclosed.\newline
(4)  Foreclosure:  Only one hardware firm
has
acquired a software firm and it has foreclosed.

Without loss of generality we assume that whenever only one of
the
hardware firms integrates and forecloses, it is firm A.  In
cases (3) and
(4), firm A forecloses.  The difference between the two is
that in (3) firm B integrates, but continues to make its software
compatible, whereas in (4), firm B does not integrate.
If the industry structure is (1), then
$N_A=N_B=2N$,\footnote{Recall that in
the absence of integration, all software products are compatible
with both hardware technologies.} in (2),
$N_A=N_B=N$, while for (3) and (4) $N_A=2N$, but $N_B=N$.  The
marginal
 consumer and the market shares of the two hardware firms as a
function of the prices of hardware are found by substituting in
the
relevant number of software varieties available for each hardware
 technology into (\ref{mar}).  In the first two industry
structures
the number of software products supplied for each hardware
platform
is the same and thus,
\begin{equation}
%]|Expr|[($:4;,=b""t,]<2(%$^p^;)B_;,,M$^p^;)A_;,,Kk("2k>,N]|[
t\rm ={{\mit p}_{\mit B}\rm -{\mit p}_{\mit A}\rm +\mit k \over \rm
2\mit
k}\rm.
\label{t1}
\end{equation}
In cases (3) and (4), the foreclosure cases,
\begin{equation}
%]|Expr|[($:4;,=b""t,]<2('<c!$1(#1/0:7b><c!$1(#$^:42_^:7;)b:4;,,M|
%|1>$^N_^:7;)b:4;,,M<c!$1(#$^p^;)B_;,,M$^p^;)A_>;,,Kk("2k>,N]|[
t\rm ={\left({1-\mit \beta }\right)\left({{\rm 2}^{\mit \beta }\rm
-
1}\right){\mit N}^{\beta }\rm -\left({{\mit p}_{\mit A}\rm -{\mit
p}_{\mit B}}\right)\rm +\mit k \over \rm 2\mit k}\rm .
\label{t2}
\end{equation}
Foreclosure provides firm A with a competitive advantage.
Comparing (\ref{t1})
 and (\ref{t2}) illustrates that a greater supply of software
leads to a market share advantage since \linebreak
$\left({\rm 1-\beta
}\right)\left({{2}^{\beta }-1}\right)>0$ for all $0< \b < 1$.
This is the demand effect.

\subsection{Industry Structure Subgames}
We now consider each of the four possible industry structures
(subgames) in turn.

\subsubsection{Unintegrated Industry}

In this case each hardware firm only derives profits from sales
of
hardware.  The respective profits of hardware
firms A and  B are
\begin{equation}
\p_A=tp_A= {[(p_B -p_A)+k] \over 2k}p_A,
\label{profa}
\end{equation}
and
\begin{equation}
\p_B=(1-t)p_A= {[(p_A -p_B)+k] \over 2k}p_B.
\label{profb}
\end{equation}

The following Lemma summarizes the equilibrium hardware prices and
profits in the unintegrated case.

\begin{lem}
The unique equilibrium hardware prices in the unintegrated
industry
structure are $p_U=p_A=p_B=k$,
 equilibrium
market share of each firm is $t=1/2$, and each hardware firm
earns
profits of $\p_U=k/2$.
\label{nofore}
\end{lem}

\noindent The profits earned by each of the
independent software firms in this case are
\begin{eqnarray}
%]|Expr|[()$^:7;,=b""p^:4;)s^i;,,]$^:7r^:4;)A_;,tN,K$^:7r^:4;)B|
%|_<c!$1(#;,1/0t>N]|[
{\pi }_{s}^{U}\rm& =&{\mit \rho }_{A}tN\rm +{\mit \rho
}_{\mit B}\left({\rm 1-\mit t}\right)N\nonumber \\
%]|Expr|[(*:4;,=b"",]:7b$^<c!$1(":42N>_(#:7;)b:4/01;,tN,K:7b$^|
%|<c!$1(":42N>_(#:7;)b:4/01<c!$1(#;,1/0t>N]|[
&\rm =&\mit \beta {\left({\rm 2\mit N}\right)}^{\beta \rm -1}\mit
tN\rm +\mit \beta {\left({\rm 2\mit N}\right)}^{\beta \rm
-1}\left({\rm 1-\mit t}\right)N\nonumber\\
& = &\b (2N)^{(\b-1)}N,
\label{unintsof}
\end{eqnarray}
where we have made use of $N_A=N_B=2N$ and Lemma \ref{sp}.

\subsubsection{Bilateral Foreclosure}

In this setting, each hardware firm earns profits from the sales
of
both hardware and software.  The software provided by each
hardware
firm is only compatible with its hardware and consequently, each
hardware firm is the monopoly supplier of software for its
technology.
{}From Lemma \ref{sp}, hardware firm
A will charge
%]|Expr|[($$^:7;,=b""r^:4;)A_;,,]:7b$^:4N_(%;),H:7b:4/01,I]|[
${\rho }_{\mit A}\rm =\mit \beta {N}^{\rm (\mit \beta \rm -1)}$
 for each of its software products and
software revenues will be $t\b N^\b$.  The profits
of hardware firm A,  after substituting (\ref{t1})
for $t$, are
\begin{equation}
%]|Expr|[(3$^:7;,=b""p^:4;)A_;,,]t,H$^p^;)A_;,,K:7b$^:4N_^:7;)b|
%|:4;,,I ,] <c!$1^<2(',H$^p^;)B_;,/0$^p^;)A_;,,I,Kk("2k>>,H$^p^|
%|;)A_;,,K:7b$^:4N_^:7;)b:4;,,I]|[
{\pi }_{\mit A}\rm =\mit t\rm ({\mit p}_{A}\rm +\mit \beta
{N}^{\beta
}\rm )=\left({{({\mit p}_{\mit B}\rm -{\mit p}_{\mit A}\rm )+\mit
k \over
\rm
2\mit k}}\right)\rm ({\mit p}_{\mit A}\rm +\mit \beta {N}^{\beta
}\rm
).
\label{profac}
\end{equation}


The following Lemma provides the equilibrium prices and profits in
the bilateral foreclosure case.
\begin{lem}
The equilibrium hardware price in the bilateral
foreclosure (BF) case is
%]|Expr|[(*$^:4;,=b""p^;)i_;,,]$^p^;)A_;,,]$^p^;)B_;,,]k/0:7b$|
%|^:4N_^:7;)b]|[
${p}_{BF}\rm ={\mit p}_{\mit A}\rm ={\mit p}_{\mit B}\rm =\mit k\rm
-
\mit
 \beta {N}^{\beta }$
,
equilibrium market share is $t=1/2$, and $ \p_{BF}=\p_A= \p_B=k/2$.
\label{pri}
\end{lem}

\subsubsection{Integrated}

In this case both hardware firms have acquired control of a
software firm.  However, hardware firm A makes its software
available in a format compatible only with its hardware.
Hardware
firm B on the other hand elects to make its software available in
formats compatible with both hardware technologies.  The two
hardware firms compete as duopolists in the software market for
technology A and from Lemma \ref{sp}, the price of software will
be ${\rho }_{A}\rm =\mit \beta {\rm \left({2\mit N}\right)}^{\beta
\rm
 -1},$ while in the software market for technology B, firm B is a
monopolist and the price of software will be ${\rho }_{B}\rm =\mit
\beta {\rm \left({\mit N}\right)}^{\beta \rm -1}.$  We can state
the following lemma, which summarizes the equilibrium prices,
market shares and profits in the integrated case.  In
advance of the lemma, define
\begin{equation}
%]|Expr|[(#$^:4;,=b""k^;)i_;,,]<2("<c!$1(#$^2_^:7;)b:4;,,M1>$^|
%|N_^:7;)b^:4;,3>]|[
{k}_{i}\rm ={\left({{2}^{\mit \beta }\rm -1}\right){\mit N}^{\beta
} \over
\rm 3}.
\label{it}
\end{equation}


\begin{lem}
When $k > ( \leq ) k_i,$ an interior (standardization) equilibrium
obtains.  The equilibrium hardware prices, market shares, and
profits when both hardware firms integrate, but only A forecloses
in an interior equilibrium are:
%\begin{equation}
%]|Expr|[(%$^:4;,= p^;)A^I;, ,] <2()<c!$1('!$^2_(%:7;*=b""b:4=  |
%|/0 1<c!$1(%;,!2/03:7b>:4 /0 1>!$^=b""N_^:7;*b:4;,=  ,K 3!=b""k|
%|^= 3>]|[
$${\rm p}_{A}^{I}={\left({{2}^{\mit \beta \rm -1}\left({2-3\beta
}\right)-
1}\right){\mit N}^{\beta }\rm +3\mit k \over \rm 3},$$
%\label{pafipf}
%\end{equation}
%\begin{equation}
%]|Expr|[(&$^:4;,= p^;)B_;, ,]
%!<2((<c!$1(+<c!$1()$^2_(%:7;*=b""b|
%|:4=  /0 1;, ,K $^2_^:7;*=b""b:4;,=  /0 3>!:7=b""b:4=  /0 $^2_|
%|^:7;*=b""b:4;,=  ,K 1>!$^=b""N_^:7;*b:4;,=  ,K 3=b""k^= 3>]|[
$${\rm p}_{B}^{I}={\left({\left({{2}^{\mit \beta \rm -1}+{2}^{\mit
\beta
}\rm -3}\right)\mit \beta \rm -{2}^{\mit \beta }\rm
+1}\right){\mit
N}^{\beta }\rm +3\mit k \over \rm 3},$$
%\label{pbfipf}
%\end{equation}
%\begin{equation}
%]|Expr|[(&$^:4;,=b""t_^;)I;,=  ,] !<2()<c!$1(& $^2_^:7;*=b""b|
%|:4;,=  /0 1>!$^=b""N_^:7;*b:4;,=  ,K 3!=b""k("6k>]|[
$${t}^{I}\rm ={\left({{2}^{\mit \beta }\rm -1}\right){\mit
N}^{\beta
}\rm
+3\mit k \over \rm 6\mit k},$$
%\label{tfipf}
%\end{equation}

%\begin{equation}
%]|Expr|[(&$^:7;,=b""p^:4;)A_;, ,]
%!<2^$^<c!$1(%<c!$1(#$^2_^:7;)b|
%|:4;,/01>$^N_^:7;)b:4;,,K3k>_^;)2(#;,18k>]|[
$${\pi }_{A}^{I}\rm ={{\left({\left({{2}^{\mit \beta }\rm
-1}\right){\mit
N}^{\beta }\rm +3\mit k}\right)}^{\rm 2} \over 18\mit k},$$
%\label{prafipfe}
%\end{equation}
%\begin{eqnarray}
%]|Expr|[(%$^:7;,=b""p^:4;)B_;,,]<2("<c!$1()3k/0,H$^2_^:7;)b:4;,/
%0|
%|1,I$^N_^:7;)b><c!$1(%:4;,3k,K<c!$1($$^2_(#:7;)b:4/01<c!$1($;,3|
%|:7b:4/02>,K1>$^N_^:7;)b>(#:4;,18k>,K]|[
$${\pi }_{B}^{I}\rm ={\left({3\mit k\rm -({2}^{\mit \beta }\rm
-1){\mit
N}^{\beta }}\right)\left({\rm 3\mit k\rm +\left({{2}^{\mit \beta
\rm -1}\left({3\mit \beta \rm -2}\right)+1}\right){\mit N}^{\beta
}}\right) \over \rm 18\mit k}\rm +$$
%& &
%]|Expr|[^<2(%:4;,=b""3:7b$^:42_(#:7;)b:4/01$^;,N_^:7;)b<c!$1()|
%|:4;,3k,K,H$^2_^:7;)b:4;,/01,I$^N_^:7;)b>(#:4;,18k>]|[
$${\rm 3\mit \beta {\rm 2}^{\mit \beta \rm -1}{\mit N}^{\beta
}\left({\rm 3\mit k\rm +({2}^{\mit \beta }\rm -1){\mit N}^{\beta
}}\right) \over \rm 18\mit k}.$$
%\label{prbfipfe}
%\end{eqnarray}

\noindent In a standardization equilibrium, the equilibrium prices,
market shares and profits are:\footnote{In a standardization
equilibrium, an integrated firm B continues to earn profits from
its ``technology A" compatible software.}

$$p_A^{IS}=((1-\b)(2^{\b}-1)+(2^{\b-1}-1) \b)N^{\b}-k,$$
$$p_B^{IS}=(2^{\b-1}-1) \b N^{\b},$$
%\label{pbfipfs}
%\end{equation}
%\begin{equation}
%]|Expr|[(&$^:4;,=b""t_(";)= IS;, ,] !1]|[
$${t}^{\rm IS}=1,$$
%\label{tfipfs}
%\end{equation}
%\begin{equation}
%]|Expr|[(&$^:7;,=b""p^:4;)A("SI;,,]<c!$1(#$^2_^:7;)b:4;,,M1>$|
%|^N_^:7;)b:4;,,Mk]|[
$${\pi }_{A}^{SI}\rm =\left({{2}^{\mit \beta }\rm -1}\right){\mit
N}^{\beta
}\rm -\mit k,$$
%\label{prafipfes}
%\end{equation}
%and
%\begin{equation}
%]|Expr|[(%$^:7;,=b""p^:4;)B("SI;,,]:7b$^:42_(#:7;)b:4,M1$^;,N|
%|_^:7;)b]|[
$${\pi }_{B}^{IS}\rm =\mit \beta {\rm 2}^{\mit \beta \rm -1}{\mit
N}^{\beta
}.$$
%\label{prbfipfes}
%\end{equation}
\label{4}
\end{lem}

Foreclosure provides firm A with a market share advantage in
equilibrium since \\
%]|Expr|[(%<c!$1($$^:4;,=b""2_^:7;)b:4;,/0 1> ,^ 0]|[
$\left({{\rm 2}^{\mit \beta }\rm -1}\right)>0$.  If the parameters
are such
that $k_i \geq k$, then the variety advantage accorded to
technology A
is large enough vis-a-vis the degree of hardware differentiation
that all
consumers purchase technology A, ie, the standardization
equilibrium obtains.


\subsubsection{Foreclosure}

This scenario is very similar to the
integrated case. In both cases hardware firm A acquires one of
 the software firms and forecloses on firm B.  However, in this
 scenario firm B does not respond by acquiring the second
software
 firm.  The remaining independent software vendor provides
 software for both hardware technologies.  Thus, in both of these
 industry structures there are $2N$ software products provided
for
 technology A and only $N$ software varieties provided for
 technology B.   In advance
of the following lemma, define
\begin{equation}
%]|Expr|[(#$^:4;,=b""k^;)f_;,,]<2("f$^N_^:7;)b^:4;,3>]|[
{k}_{f}\rm ={\mit f{N}^{\beta } \over \rm 3},
\label{ft}
\end{equation}
where
%]|Expr|[(*:4;,=b""f ,] $^2_(#:7;)b:4/01<c!$1(#;,2/0:7b>:4,K:7b|
%|:4/01]|[
$f\rm ={2}^{\mit \beta \rm -1}\left({2-\mit \beta }\right)\rm
+\mit
\beta \rm -1$.

\begin{lem}
When $k > ( \leq ) k_f,$ an interior (standardization) equilibrium
obtains.  The equilibrium hardware prices, market shares, and
profits in an interior equilibrium when only firm A integrates and
forecloses are:
%\begin{equation}
%]|Expr|[(&$^:4;,=b""p^;)A^f;,=  ,] !<2('<c!$1((<c!$1(&=b""1=  |
%|/0 2:7=b""b>$^:4= 2_(":7;*=b""b:4=  ;, /0 1,K:7b>:4!$^=b""N_^|
%|:7;*b:4;,=  ,K3=b""k^= 3>]|[
$${p}_{A}^{f}\rm ={\left({\left({1-2\mit \beta }\right){\rm
2}^{\mit
\beta
\rm }-1+\beta }\right){\mit N}^{\beta }\rm +3\mit k \over \rm 3},$$
%\label{pricea}
%\end{equation}
%\begin{equation}
%]|Expr|[(&$^:4;,=b""p^;)B^f;,=  ,] !<2((3 =b""k/0f= !$^=b""N_|
%|^:7;*b:4;,=  ^3>]|[
$${p}_{B}^{f}\rm ={3\mit k\rm -\mit f\rm {\mit N}^{\beta }\rm
\over
3},$$
%\label{priceb}
%\end{equation}
%\begin{equation}
%]|Expr|[($$^:4;,=b""t_^;)f;, ,]<2(%f$^N_^:7;)b:4;,,K3k("6k>]|[
$${t}^{f}\rm ={\mit f{N}^{\beta }\rm +3\mit k \over \rm 6\mit k},$$
%\label{tforeq}
%\end{equation}

%\begin{equation}
%]|Expr|[(#$^:7;,=b""p^:4;)A^f;,,]<2^$^<c!$1(%;*3k,Kf$^N_^:7;'b>|
%|_^:42(#;,18k>]|[
$${\pi }_{A}^{f}\rm ={{\left({3\mit k\rm +\mit f{N}^{\beta
}}\right)}^{\rm 2}
\over 18\mit k},$$
%\label{prforae}
%\end{equation}
%\begin{equation}
%]|Expr|[(#$^:7;,=b""p^:4;)B^f;,,]<2^$^<c!$1(%;*3k/0f$^N_^:7;'b>|
%|_^:42(#;,18k>]|[
$${\pi }_{B}^{f}\rm ={{\left({3\mit k\rm -\mit f{N}^{\beta
}}\right)}^{\rm 2}
\over 18\mit k}.$$
%\label{prforbe}
%\end{equation}

\noindent In a standardization equilibrium, the equilibrium prices,
market shares and profits are:
%\begin{equation}
%]|Expr|[(,$^:4;,=b""p^;)A("fs;,=  ,] !<c!$1($=b""1=  /0:7=b""b>|
%|<c!$1(%$^:4= 2_(":7;*=b""b:4=  ;, /0 1>!$^=b""N_^:7;*b:4;,=  |
%|,M=b""k]|[
$${p}_{A}^{fs}\rm =\left({1-\mit \beta }\right)\left({{\rm 2}^{\mit
\beta
\rm }-1}\right){\mit N}^{\beta }\rm -\mit k,$$
%\label{priceas}
%\end{equation}
%\begin{equation}
%]|Expr|[(&$^:4;,=b""p^;)B("fs;,=  ,] !0]|[
$${p}_{B}^{fs}\rm =0,$$
%\label{pricebs}
%\end{equation}
%\begin{equation}
%]|Expr|[($$^:4;,=b""t_(";)fs;, ,]1]|[
$${t}^{fs}\rm =1,$$
%\label{tforeqs}
%\end{equation}
%\begin{equation}
%]|Expr|[(&$^:7;,=b""p^:4;)A("fs;,,]f$^N_^:7;)b:4;,,Mk]|[
$${\pi }_{A}^{fs}\rm =\mit f{N}^{\beta }\rm -\mit k,$$
%\label{prforaes}
%\end{equation}
%\begin{equation}
%]|Expr|[(#$^:7;,=b""p^:4;)B("fs;,,];*0]|[
$${\pi }_{B}^{fs}\rm =0.$$
%\label{prforbes}
%\end{equation}
\label{5}
\end{lem}

Foreclosure increases the market share of
firm A since $f>0$.  If the variety advantage is large relative to
the degree of hardware differentiation,  then
a standardization equilibria results and all consumers purchase
technology  A.

In the interior equilibrium, the profit of the independent software
firm
equals the sum of its monopoly profits in the software market for
technology B and
its duopolist profits in the software market for technology A:
\begin{eqnarray}
%]|Expr|[()$^:7;,=b""p^:4;)s^i;,,]$^:7r^:4;)A_;,Nt,K$^:7r^:4;)B|
%|_;,N<c!$1(#1/0t>]|[
{\pi }_{s}^{f}\rm &=&{\mit \rho }_{A}Nt\rm +{\mit \rho
}_{B}N\left({\rm 1-\mit t}\right)\nonumber\\
%]|Expr|[(0$^:7;,=b""p(":4;)is_;, ,] :7b$^<c!$1(":42N>_(#:7;)b|
%|:4/01;,Nt,K$(":7b<c!$1^:4N>_(#:7;)b:4/01;,N,H1/0t,I]|[
&=&\mit \beta {\left({\rm 2\mit N}\right)}^{\beta \rm -1}\mit
Nt\rm
+{\mit \beta \left({N}\right)}^{\beta \rm -1}\mit N\rm (1-\mit
t\rm)\nonumber\\
&=&{\left({\left({{2}^{\mit \beta \rm
-1}-1}\right){\mit N}^{\rm \mit \beta }\rm \mit f\rm
+3\left({{2}^{\mit \beta \rm -1}+1}\right)\mit k}\right)\rm \mit
\beta {N}^{\beta } \over \rm 6\mit k,}
\label{indsof}
\end{eqnarray}
where the last equality is obtained by substituting the reduced
form expression for the equilibrium
market share from Lemma \ref{5}.

In the standardization case, the profits of the independent
software firm are simply duopoly profits from the entire market:
\begin{equation}
%]|Expr|[(%$^:7;,=b""p^:4;)s("fs;,,]:7b$^:42_(#:7;)b:4,M1$^;,N|
%|_^:7;)b]|[
{\pi }_{s}^{fs}\rm =\mit \beta {\rm 2}^{\mit \beta \rm -1}{\mit
N}^{\beta
}.
\label{indsofes}
\end{equation}

\subsection{Implications of the Alternative Structures}
In both the unintegrated and bilateral foreclosure cases the two
firms split the market in half.  The demand effect ensures that in
the
integrated and foreclosure
cases,
the market share of firm A is greater than one
half and in both of these cases, it may be large enough that
standardization may arise.

We can also rank equilibrium hardware prices.  The highest hardware
prices occur in the unintegrated market structure, the lowest in
the
bilateral foreclosure case.  In the bilateral foreclosure case,
integration
and foreclosure
increase the
incentives for a
hardware firm to price its hardware more competitively relative
to
a non-integrated hardware firm.  In this case a hardware firm which
has
integrated
and foreclosed has an additional incentive to lower the price of
hardware since this increases the market for its software sales.
An
increase in the number of software products makes a hardware
firm `tougher' in the terminology of Fudenberg and Tirole
(1984) .

In the foreclosure and integrated case, firm A (the foreclosing
firm) has
an incentive to price its hardware more competitively only when the
demand effect is large.  If the demand effect is small, increases
in the
number of software products make firm A 'softer:'  it is more
profitable to
charge higher prices to captive customers than fight for a larger
market
share.\footnote{The hardware price reaction function of firm A in
both of
these cases is given by (\ref{raf}) in the appendix.  When
$\b<.22179$
increases in the number of software varieties makes firm A softer,
when
$\b>.22179$, increases in the number of software varieties makes
firm A
tougher.}

In both of these cases, however, firm B (the foreclosed firm) has
an
incentive to price its hardware more competitively relative to the
unintegrated market structure.  In the foreclosure case, it has an
incentive to compete
more
aggressively because its market share is smaller and hence the loss
on
inframarginal units less.  This effect is such that even if the
demand
effect is small, prices in the foreclosure equilibrium are lower
than in the
unintegrated case.

In the integrated case, firm B is a monopoly provider of software
for its
own
technology and
a duopolist in the software market for technology A.
This differential in software market structure provides it with an
incentive to increase its hardware market share by lowering its
hardware
price in order to extend its monopoly in software. The market share
of
technology A in the foreclosure case is greater than
in the
integrated case, because the price effect when B is integrated is
greater
than when it is not.

\section{Non-Standardization Equilibrium}
In the preceding section, we determined the equilibrium
prices and profits for each of the four possible industry
structures.  In this section we use that analysis to
determine the equilibrium to the full three-stage game under the
assumption that the resulting equilibrium in any of the subgames
does not
does not entail standardization.  Since $k_f > k_i$, this means
we restrict the
parameter space to $k > k_f.$
We begin
by considering the effect of the two counter-strategies available
to firm
B if firm A integrates
and forecloses.
\subsection{Optimal Response to Foreclosure}

Firm B has three options when faced with integration and
 foreclosure by Firm A.  It can integrate and foreclose,
integrate
 but not foreclose, or remain unintegrated.  We determine the
optimal response of firm B in the following two propositions.  In
Proposition 1, we compare the stand-alone profits of firm B and
the remaining independent software firm to the profits that firm
B would earn if it integrated and foreclosed.  In Proposition 2,
we compare these stand-alone profits to the profits that firm B
would earn if integrated but did not foreclose.

\begin{prop}
If firm A forecloses, firm B will never find it optimal to
acquire
the remaining independent software firm and foreclose on firm A.
\label{retaliate}
\end{prop}

The intuition for the non-optimality of an integration and
foreclosure response by firm B
is straight forward.  Integration and foreclosure
significantly increase the degree of price
competition in the hardware market, since increasing
hardware market share increases the size of the software
market in which each integrated firm has a monopoly.
However, the increased hardware price competition dissipates any
software profits.
Moreover, the profitability of the independent software firm has
increased since if it remains independent it will earn monopoly
profits in the software market for technology B as well as
duopoly
profits in the market for technology A.

Firm B has at its disposable a second counter-strategy, which is
to
integrate, but not foreclose.  In advance of the following
proposition,
define
\begin{equation}
%]|Expr|[(*$^:4;,= k(#;)min_;,,] !<2^;)1^3>;,!<2^<c!$1(1<c!$1(*|
%|!$^2_(#:7;*=b""b:4,K1;,=  /0 $^2_(';*2!:7=b""b:4=  /0 1;, /0 |
%|2>!:7=b""b:4=  ,K $^2_(';*2!:7=b""b:4=  /0 1;, /0
%3!$^2_(%:7;*=b""b|
%|:4=  /0 1;, ,K 1>($$^2_(%:7;*=b""b:4=  /0 1;,/01
%>!$^=b""N_^:7;*b|
%|:4;, ]|[
{k}_{min}={1 \over 3}{\left({\left({{2}^{\mit \beta \rm
+1}-{2}^{2\mit
\beta \rm -1}-2}\right)\mit \beta \rm +{2}^{2\mit \beta \rm -1}-
3{2}^{\mit \beta \rm -1}+1}\right) \over {2}^{\mit \beta \rm
-1}-1}{\mit
N}^{\beta }\rm.
\label{kmin}
\end{equation}

\begin{prop}
If a merged firm forecloses, the unintegrated hardware firm will
find it
profitable not to integrate if %]|Expr|[(#:4;,= k,^$^k(#;)min_]|[
$ k>{k}_{min}$.  If $ k<{k}_{min}$, the unintegrated
hardware firm
will integrate with the remaining independent software firm and
continue
to supply compatible software for both hardware technologies.
\label{retaliate2}
\end{prop}

Proposition 2 indicates that if the degree of hardware
differentiation is
relatively small then firm B will find it optimal to respond to
foreclosure
by firm A by integrating but not foreclosing.  The intuition for
this is that
if the degree of differentiation is relatively insignificant, the
market
share increase for firm A from the demand effect associated with
foreclosure is relatively large.  Moreover, B can effectively
restore its
market share by lowering its price of hardware since hardware will
be
relatively price elastic.  Integrating into software provides it
with a
means to credibly lower its hardware price, increasing both
hardware
market share and extending the market of its software monopoly.
%The following corollary, which we prove in the appendix, shows how
%the critical degree of hardware
%differentiation depends on the number of software varieties and
%the preferences for variety of consumers.

%\begin{cor}
%For a given degree of hardware differentiation, if the value
%consumers
%place on software variety increases or the number of software
%products increases it is more likely that firm B will respond to
%integration and foreclosure by firm A by integrating but not
%foreclosing.
%\label{c1}
%\end{cor}


\subsection{Optimality of Foreclosure}

In this section we consider the incentives of a merged firm A to
foreclose.  The advantages of foreclosure arise from the demand
effect:
the favorable differential in software variety increases hardware
market
share.  The cost of foreclosure is the forgone software profits for
the
other technology.  The circumstances when a merged firm will find
it
profitable to foreclose are detailed
in the next proposition.  Define
\begin{equation}
%]|Expr|[($$^:4;,=b""k(#;)max_;,
%,]<2("$^f_^;)2$^;,N_^:7;)b(':4;,1|
%|8:7b$^:42_(#:7;)b:4/01;,/06f>]|[
{k}_{max}\rm ={{\mit f}^{\rm 2}{\mit N}^{\beta } \over \rm 18\mit
\beta
{\rm 2}^{\mit \beta \rm -1}-6\mit f}.
\label{kmax}
\end{equation}

\begin{prop}
Suppose that hardware firm A and one of the independent software
firms have merged and that firm B will not integrated with the
remaining independent software firm.  The merged firm will
foreclose (not foreclose) if
%]|Expr|[(#$^:4;,=b""k(#;)max_;,,^k]|[
$k < k_{max} $ ($\mit k>{k}_{max}$).
\label{foreclosure}
\end{prop}
Proposition \ref{foreclosure} indicates that if the hardware
products
are highly differentiated firm A will prefer
not to
foreclose.  The reason is that in these circumstances, the market
share gain
from the demand effect of foreclosure is small and is more than
offset by
the
decrease in
hardware prices and lost software profits.  If the hardware
products are
not highly differentiated then the demand effect will be large and
a hardware firm will find it profitable to integrate and
foreclose, since
the increase in the hardware market from the demand effect more
than
makes up for the
decrease
in hardware prices and lost software profits.
%\begin{cor}
%{\em  For a given degree of hardware differentiation, if the value
%consumers place on software variety increases (provided $N \ge
%2.25$) or
%the number of software products increases it is more likely that
%firm A
%will integrate and foreclose.}
%\label{c2}
%\end{cor}

%\noindent{\em Proof.}  Provided $N \ge 2.25$ and
%%]|Expr|[():4;,=b""0 ,\ :7b:4 ,\ 1]|[
%$\rm 0<\mit \beta \rm <1$, %]|Expr|[^$^:4;,=b""k(#;)max_]|[
%${k}_{max}$ is increasing in both $\b$ and $N$.
%\hfill {\em Q.E.D.}
%\vspace{3 mm}


The premise of Proposition \ref{foreclosure} is that firm B will
not respond by integrating.  Proposition \ref{retaliate2}
indicated
that for certain parameter values, firm B will in fact respond to
integration and foreclosure by integrating but not foreclosing.
The next proposition shows that
if Firm A anticipates this response, it will not integrate and
foreclose.

\begin{prop}
If firm B finds it profitable to respond to integration and
foreclosure by firm A by integrating but not foreclosing, then
firm
A will not integrate and foreclose.
\label{noforeclosure}
\end{prop}
Proposition \ref{noforeclosure} indicates that if the foreclosed
hardware
firm responds with the counter-strategy of integrating but not
foreclosing,
then the foreclosure strategy of the first hardware firm to
foreclose is in
fact ineffective.  The incentive that the second firm has to
lower its
hardware price in order to expand the software market in which it
has a
monopoly sufficiently mitigates the demand effect associated with
foreclosure.

Combining Propositions 2, 3, and 4, a necessary condition for
foreclosure to be an equilibrium is
%]|Expr|[('$^:4;,=b""k(#;)max_;,,^k ,^ $^k(#;)min_]|[
${k}_{max}\rm >\mit k\rm >{\mit k}_{\mit min}$.  In other words,
the
degree of hardware differentiation must be low enough
($k< k_{max}$) that the demand
effect of foreclosure makes foreclosure profitable, but high enough
($k>k_{\mit min}$) that
the price effect of the counter-strategy is relatively
small.\footnote{It is easy to show that $k_{max} \geq k_{min}$ if
$\b \leq .098733.$}

The condition, ${k}_{max}\rm >\mit k\rm >{\mit k}_{min}$, is not
sufficient for equilibrium, however.  For a foreclosure equilibrium
to exist a software firm
must
voluntarily agree
to merge with one of the hardware firms.  A software firm
will not do so if it anticipates that its profits would be
greater
if it
was the independent software firm in a foreclosure equilibrium.
Since it
will be a monopoly provider of software for technology B and
remain
a duopolist in the software market for technology A, its profits
will increase
relative to the unintegrated market structure.  The maximum a
hardware firm
 will bid to acquire a software firm in the first stage is the
difference
 between what it earns if it forecloses and its profits if it is
foreclosed
upon.  The next proposition
provides a sufficient
 condition for when this difference is greater than the profits of
the
independent software firm in the foreclosure equilibrium.
\begin{prop}
For $0<\b<.17287$,
there is not a hold-up problem.
\label{noholdup}
\end{prop}

Proposition \ref{noholdup} indicates that when $\b$ is
small, there is no hold-up problem and software firms prefer
to merge and foreclose, rather than remain independent.  For
greater values of $\b$, the software firm prefers to remain
independent and become a monopolist in the software market
for technology B.  Large values of $\b$ imply a high willingness
to pay for
software and hence software prices and profits are high.
Another implication of Proposition  \ref{noholdup} is that if there
is in
fact a bidding process for the initial software firm, we would
expect that
the winning bid would be the difference
between the
profits a hardware firm obtains if it forecloses and the profits it
earns if
it is foreclosed upon.  Thus, the payoffs of the two hardware firms
will be
identical regardless of whether they foreclose or are foreclosed
upon if
foreclosure is in fact an equilibrium outcome.  In the next
proposition we
show that there are parameter values
such that foreclosure is an equilibrium outcome.
\begin{prop}
Integration and foreclosure by firm A and no response by firm B
will be an equilibrium outcome when foreclosure leads to an
interior
equilibrium when
%]|Expr|[(%$^:4;,=b""k(#;)max_;,,^k,^$^k(#;)min_]|[
$\rm 0<\mit \beta \rm \le .098733$,
and
${k}_{max}\rm >\mit k\rm >{\mit k}_{\mit min}.$
If either of these parameter restrictions do not hold, the
equilibrium industry structure will be unintegrated for all
%]|Expr|[(%:4;,=b""k ,^ $^k^;)f_]|[
$k >{\mit k}_{f}.$
\label{equilibrium}
\end{prop}


Proposition \ref{equilibrium} indicates that there are two
possible
equilibria to the full game when the parameters are restricted such
that
foreclosure leads to an interior equilibrium.  If the
parameter restrictions
hold, then in equilibria foreclosure occurs:  firm A integrates and
forecloses on firm B, firm B
remains unintegrated.
If the parameter restrictions are not satisfied, then neither
firm
integrates and the equilibrium industry structure is
unintegrated.


%For a given degree of hardware differentiation, if the value
%consumers
%place on software variety increases or the number of software
%products increases it is more likely that firm B will respond to
%integration and foreclosure by firm A by integrating but not
%foreclosing.  This is true for two reasons.  The first is that the
%demand
%effect associated with foreclosure is increasing in $N$ and $\b$.
%The
%second is that software is profitable, so it is advantageous to
%extend the
%monopoly in the technology for firm B by lowering the hardware
%price of
%firm B.

%Increases in $\b$ and $N$ when both are
%large
%relative to $k$,  will make it more likely that firm A will
%foreclose, since
%the demand effect will increase.
%However,
%if $\b$ is large and $N$ small, increases in $\b$ may make it
%less
%likely that
%firm A will find it profitable to foreclose.  Under these
%circumstances, the
%expansion in market share from foreclosure is small, and since the
%price of software,
% is increasing in $\b$,  foreclosure is not
%profitable because
%it involves forgoing (large) duopoly profits in the (large)
%software market for  technology B.

\section{Standardization Equilibrium}
In this section we briefly consider the case in which
foreclosure results in standardization.  If the degree of hardware
differentiation is
sufficiently small, in particular if,
$k<k_f$, then the relative variety advantage provided
to a
foreclosing hardware firm is significant enough that the market
share of
the rival technology is reduced to zero if it does not retaliate.
We can state the following proposition.
\begin{prop}
For values of
%]|Expr|[(($^:4;,=b""k^;)f_;, ,^k ,^ $^k^;)i_]|[
${k}_{f}\rm >\mit k\rm >{\mit k}_{\mit i},$
the equilibrium industry structure is unintegrated.
\label{rs}
\end{prop}
The intuition for this result is that for $k> k_i$, by responding
to foreclosure by integrating, firm B obtains
a positive market share in the hardware market.  Similar to
Proposition 4, this makes foreclosure unattractive to the merged
firm.

We now consider the equilibrium for values of
${k}_{i}\rm >\mit k.$   In advance of the following proposition,
define
\begin{equation}
%]|Expr|[(#$^:4;,=b""k^;)S_;,,]<2($2<c!$1(#1,M:7b><c!$1(#$^:42|
%|_^:7;)b:4;,,M1>$^N_^:7;)b^:4;,3>]|[
{k}_{s}\rm ={2\left({1-\mit \beta }\right)\left({{\rm 2}^{\mit
\beta }\rm
-1}\right){\mit N}^{\beta } \over \rm 3}.
\label{ks}
\end{equation}

\begin{prop}
For $k_s>k$, firm A will always find it profitable to foreclose if
B does not retaliate.  For $k<k_i$, it is not profitable for firm
B to
retaliate by acquiring the remaining independent
software firm.
\label{1s}
\end{prop}

The intuition for the optimality of a non-response by  firm B is
that even
if it responds by
merging and not foreclosing, it does not make any hardware sales
and hence the combined profits of the merged firm are equivalent to
the profits earned  by an independent software firm in a
standardization equilibrium.  If $k$ is relatively small vis-a-vis
$\b$ and
$N$, then de facto standardization by foreclosing is profitable for
firm A.
Whether firm A forecloses or not its software profits are the same
since
in both cases it is a duopoly supplier of software to the entire
market.
Hence whether or not it forecloses depends on the impact
foreclosure has
on hardware profits.  If the degree of hardware differentiation is
large ($k>k_s$), then the price effect makes foreclosure
unprofitable.  If
the degree of hardware differentiation is small ($k_s>k$), then the
price
effect is small and foreclosure profitable.
There is no hold-up problem, since the profits that the independent
software firm will earn in a foreclosure equilibrium are simply
duopoly profits in the software market for technology A.  This is
also the profit
that firm A earns from software sales, whether it forecloses or
not.  The
increase in hardware profits when firm A forecloses provides the
requisite surplus to make a merger between firm A and a software
firm
voluntary.
%\begin{prop}
%When $\b$ is relatively small ($\b<.56421$), there is no holdup
%problem.
%\label{2s}
%\end{prop}
%The intuition for this proposition is similar to Proposition 5.
%For large values of $\b$, the software firm prefers to remain
%independent and become a monopolist in the software market
%for technology B.  Large values of $\b$ imply a high willingness
%to pay for software and hence software prices and profits are
%high.
Finally, we can state the following proposition, which follows
immediately from Propositions (\ref{rs}) and (\ref{1s}).


\begin{prop}
For values of
%]|Expr|[(%$^:4;,=b""k^;)i_;, ,^ k]|[
$k < min[k_i,k_s]$ there is a foreclosure equilibrium in which
technology A is exclusively adopted.  Otherwise, the equilibrium
industry structure is unintegrated.
\label{3s}
\end{prop}

\section{Social Welfare}
In this section we evaluate the social desirability of the
foreclosure equilibrium.  We do this by comparing the total
surplus
associated with the unintegrated equilibrium
to the total surplus associated with the foreclosure equilibrium.

\subsection{Unintegrated Total Surplus}
Lemma 2 provides details on the unintegrated equilibrium.  Total
surplus is the sum of hardware profits, software profits, and
consumers' surplus.  Using (\ref{iu}), the surplus of consumers
is
\begin{eqnarray}
%]|Expr|[(&:4;,=b""C$^S_^;)U;,,] 2<c"
%#($<c!$1(-$(#<c!$1(#1/0:7b>|
%|:4 <c!$1("2N>_^:7;)b:4;, ,K y /0 k /0kt> dt^;)0(#1,O2>]|[
C{S}^{U}\rm &=&2\int_{0}^{1/2}\left({{\left({1-\mit \beta
}\right)\rm \left({2\mit N}\right)}^{\beta }\rm +\mit y\rm
+\alpha
-\mit k\rm -\mit kt}\right)\rm \mit dt\nonumber\\
%]|Expr|[(+:4;,=b"",] $(#<c!$1(#1/0:7b>:4 <c!$1("2N>_^:7;)b:4;, |
%|,K y /0 <2("5k^4>]|[
& \rm=&{\left({1-\mit \beta }\right)\rm \left({2\mit
N}\right)}^{\beta }\rm +\mit y\rm+\alpha -{5\mit k \over \rm 4}.
\label{csu}
\end{eqnarray}
The combined profits of the four firms are, using Lemma
5 and (\ref{unintsof}):
\begin{equation}
%]|Expr|[(*$^:7;,=b""P_^:4;)U;, ,] k ,K
%:7b$^<c!$1(":42N>_^:7;)b]|[
{\Pi }^{U}\rm =\mit k\rm +\mit \beta {\left({\rm 2\mit
N}\right)}^{\beta }.
\label{pu}
\end{equation}
Consequently, total surplus in the unintegrated industry
structure
is, summing (\ref{csu}) and (\ref{pu}):
\begin{equation}
%]|Expr|[(-:4;,=b""T$^S_^;)U;,,] $(" <c!$1("2N>_^:7;)b:4;, ,K |
%|y /0 <2^k^4>]|[
T{S}^{U}\rm ={\left({2\mit N}\right)}^{\beta }\rm +\mit y\rm
+\alpha -{\mit k \over \rm 4}.
\label{tsu}
\end{equation}
\subsection{Total Surplus, Foreclosure Equilibrium}

The surplus of consumers in the foreclosure equilibrium is, using
(\ref{iu}), $N_A=N$, and $N_B=2N$
\begin{eqnarray}
%]|Expr|[(':4;,=b""C$^S_^;)F;,,] <c" #(#<c!$1(,$($
%<c!$1(#1/0:7b>|
%|:4 <c!$1("2N>_^:7;)b:4;, ,K y /0 $^p^;)A_;,/0kt>dt^;)0^t>;,
%,K]|[
C{S}^{F}\rm &=&\int_{0}^{\mit t}\left({{\rm \left({1-\mit \beta
}\right)\rm \left({2\mit N}\right)}^{\beta }\rm +\mit y\rm
+\alpha
-{\mit p}_{\mit A}\rm -\mit kt}\right)dt\rm +\nonumber\\
%]|Expr|[^<c" #(#<c!$1(,$($:4;,=b"" <c!$1(#1/0:7b>:4 <c!$1^N>_|
%|^:7;)b:4;, ,K y /0 $^p^;)B_;,/0k<c!$1(#1/0t>>dt^;)t^1>]|[
&&\int_{t}^{\rm 1}\left({{\left({1-\mit \beta }\right)\rm
\left({\mit N}\right)}^{\beta }\rm +\mit y\rm +\alpha -{\mit
p}_{\mit B}\rm -\mit k\left({\rm 1-\mit t}\right)}\right)dt
\nonumber\\
%]|Expr|[((:4;,=b"",] <c!$1()$($ <c!$1(#1/0:7b>:4 <c!$1("2N>_^|
%|:7;)b:4;, ,K y /0 $^p^;)A_>;,t,K<c!$1()$($ <c!$1(#1/0:7b>:4
%<c!$1|
%|^N>_^:7;)b:4;, ,K y /0 $^p^;)B_><c!$1(#;,1/0t>/0]|[
&\rm =&\left({{\left({1-\mit \beta }\right)\rm \left({2\mit
N}\right)}^{\beta }\rm +\mit y\rm +\alpha -{\mit
p}_{\mit A}}\right)t\rm
+\left({{\left({1-\mit \beta }\right)\rm \left({\mit
N}\right)}^{\beta }\rm +\mit y\rm +\alpha -{\mit
p}_{\mit B}}\right)\left({\rm
1-\mit t}\right)\rm -\nonumber\\
%]|Expr|[("<2^:4;,=b""k^2><c!$1(#$^t_^;)2;,,K$^<c!$1(#1/0t>_^;)2>
%]|[
&&{k \over \rm 2}\left({{\mit t}^{\rm 2}+{\left({1-\mit
t}\right)}^{\rm 2}}\right)
\label{csf}
\end{eqnarray}
Hardware and software profits equal
\begin{equation}
%]|Expr|[(6$^:7;,=b""P_^:4;)f;,,]$^p^;)A_;,t,K:7b$($:4,H2N,I_(#|
%|:7;)b:4/01;,Nt,K<c!$1(#1/0t>$^p^;)B_;,,K:7b$($:4,H2N,I_(#:7;)b|
%|:4/01;,Nt,K:7b$(#:4,HN,I_(#:7;)b:4/01;,N<c!$1(#1/0t>]|[
{\Pi }^{f}\rm ={\mit p}_{\mit A}t\rm +\mit \beta {\rm (2\mit N\rm
)}^{\mit \beta \rm -1}\mit Nt\rm +\left({1-\mit
t}\right){p}_{\mit B}\rm
+\mit \beta {\rm (2\mit N\rm )}^{\mit \beta \rm -1}\mit Nt\rm
+\mit
\beta {\rm (\mit N\rm )}^{\mit \beta \rm -1}\mit N\left({\rm
1-\mit
t}\right)
\label{pf}
\end{equation}
Total surplus in the foreclosure equilibrium is the sum of
(\ref{csf}) and (\ref{pf}):
\begin{equation}
%]|Expr|[(*:4;,=b""T$^S_^;)f;,,]y,K$^N_^:7;)b<c!$1($<c!$1(#$^:4;,
%2|
%|_^:7;)b:4;,/01>t,K1>/0<2^k^2><c!$1(#$^t_^;)2;,,K$^<c!$1(#1/0t>|
%|_^;)2>]|[
T{S}^{f}\rm =\mit y\rm+ \alpha
 +{\mit N}^{\beta }\left({\left({{\rm
2}^{\mit \beta }\rm -1}\right)\mit t\rm +1}\right)-{\mit k \over
\rm 2}\left({{\mit t}^{\rm 2}+{\left({1-\mit t}\right)}^{\rm
2}}\right).
\label{tsf}
\end{equation}
Setting $t=1$ in  (\ref{tsf}) yields total surplus if foreclosure
involves
standardization.
\begin{prop}
The foreclosure equilibrium is inefficient.
\label{sw}
\end{prop}


It is perhaps not surprising that the foreclosure equilibrium is
inefficient.
While the hardware firm and software firm which merge clearly
gain,
as does
the
remaining independent software firm, these gains are more than
offset by
the reduction in the surplus of consumers and the profits of the
unintegrated
hardware firm.   In the foreclosure equilibrium, customers of the
unintegrated
hardware firm are supplied with only half the number of software
products and
 at significantly higher prices, compared to the unintegrated
industry
equilibrium.  The reduction in the hardware price for  both
technologies
engendered by the
integration of firm A is not enough to compensate.  The decrease
in
the price
of hardware also impacts negatively on the profits of the
unintegrated
hardware
firm.

\section{Conclusion}

In this paper we developed a model to address whether or not
horizontal
foreclose in the markets for complementary services for consumer
durables is
an equilibrium outcome and, if it is, what the welfare
implications
are.  We
find that horizontal equilibrium foreclosure can occur as the
outcome of a
fully specified equilibrium model, and that when it does
occur,
it is
inefficient.  We find that for certain parameter values, in
equilibrium, one
hardware will merge with a software firm and discontinue software
support for
a rival hardware technology.  Moreover, the remaining hardware
firm
will not
respond in kind.  Thus, foreclosure occurs even though a
successful
counter-strategy is available.  Furthermore, the integrated firm
has no
incentive to
supply software compatible with the hardware of its rival and
neither of the
software firms has an incentive to hold-out when approached to
merge.  In addition, we found that when the degree of
differentiation of
the hardware products is small the foreclosure equilibrium can
result in de
facto standardization.  The technology of the foreclosing firm is
exclusively adopted and foreclosure is an effect monopolization
strategy.


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\pagebreak

\begin{center}
{\bf Appendix}
\end{center}
\vspace{4 mm}

\noindent{\em Proof of Lemma 1.}
If the price of a software variety exceeds $\b
N_i^{\b-1}$, consumers will not purchase it.  A price less
than $\b N_i^{\b-1}$ reduces profits: sales are unchanged
as the demand by a consumer
for a variety of software is perfectly inelastic
and the market size is determined
by hardware sales.
\hfill {\em Q.E.D.}
\vspace{3 mm}

\noindent{\em Proof of Lemma 2.}
Maximizing
(\ref{profa}) and (\ref{profb}) with respect to the relevant
hardware price yields the
following best response functions:
%\begin{equation}
%]|Expr|[(6$^:4;,=b""p^;)i_;, ,] <2(#$^p^;)j_;,,Kk^2>,L i,Lj ,]|
%| A,LB,L i.Mj,N]|[
$${p}_{i}\rm ={{\mit p}_{j}\rm +\mit k \over \rm 2},\mit i\rm ,\mit
j\rm =\mit A\rm ,\mit B\rm ,\mit i\rm \ne \mit j\rm .$$
%\label{best1}
%\end{equation}
The expressions in Lemma \ref{nofore} follow immediately from the
best-response functions.
\hfill {\em Q.E.D.}
\vspace{3 mm}

\noindent{\em Proof of Lemma 3.}
Maximizing (\ref{profac}) with respect to $p_A$ and the analogous
profit function for firm B with respect to $p_B$ yields price
best
response functions for firms A and B:
%\begin{equation}
%]|Expr|[(3$^:4;,=b""p^;)i_;,,]<2(&$^p^;)j_;,,Kk/0:7b$^:4N_^:7;)b
%|
%|^:4;,2>,L i,Lj ,] A,LB,L i.Mj]|[
$${p}_{i}\rm ={{\mit p}_{j}\rm +\mit k\rm -\mit \beta {N}^{\beta }
\over \rm 2},\mit i\rm ,\mit j\rm =\mit A\rm ,\mit B\rm ,\mit
i\rm
\ne \mit j.$$
%\label{besti}
%\end{equation}
The equilibrium expressions in Lemma \ref{pri} follow immediately
from the best-response functions.
\hfill {\em Q.E.D.}
\vspace{3 mm}

\noindent {\em Proof of Lemma \ref{4}.}
The profits of firm A from its hardware and sales of $N$ software
products are
\begin{equation}
%]|Expr|[(&$^:7;,=b""p^:4;)A_;, ,] <c!$1(%$^p^;)A_;,,K:7b$($:4,H|
%|2N,I_(#:7;)b:4/01;,N>t]|[
{\pi }_{A}\rm =\left({{\mit p}_{A}\rm +\mit \beta {\rm (2\mit N\rm
)}^{\mit \beta \rm -1}\mit N}\right)\mit t.
\label{prafipf}
\end{equation}
\noindent The best-response function for firm A is found by
 substituting (\ref{t2}) for $t$ into (\ref{prafipf}) and
maximizing
with respect to $p_A$:
\begin{equation}
%]|Expr|[(#$^:4;,=b""p^;)A_;,,]<2(&<c!$1(-,H1/0:7b:4,I,H$^2_^:7;)
%b|
%|:4;,/01,I/0:7b$^:42_(%;),H:7b:4/01,I>$^;,N_^:7;)b:4;,,K$^p^;)B|
%|_;,,Kk^2>]|[
{p}_{A}\rm ={\left({(1-\mit \beta \rm )({2}^{\mit \beta }\rm
-1)-\mit \beta {\rm 2}^{(\mit \beta \rm -1)}}\right){\mit
N}^{\beta
}\rm +{\mit p}_{\mit B}\rm +\mit k \over \rm 2}
\label{raf}
\end{equation}

%Whether an increase in the number of software products provided
%by
%firm A makes it tougher or softer depends on the preferences of
%consumers for variety.  If $\b > .22179$, then increases in the
%number
% of software varieties leads to more aggressive hardware pricing
%behaviour on the part of an integrated firm A.  When $\beta$ is
%large, software prices are high and increasing the market for
%software by increasing the market share of hardware is
%profitable.
%When $\b<.22179$, firm A is a `fat cat,' and it uses its software
%advantage
%to charge a higher price for hardware.
Firm B derives profits from hardware, software for technology A,
and software compatible with technology B.  The profits of firm B
are
\begin{equation}
%]|Expr|[()$^:7;,=b""p^:4;)B_;,,]<c!$1($$^p^;)B_;,,K:7b$^:4N_^|
%|:7;)b><c!$1(#:4;,1/0t>,K:7b$^<c!$1(":42N>_(#:7;)b:4/01;,Nt]|[
{\pi }_{\mit B}\rm =\left({{\mit p}_{B}\rm +\mit \beta {N}^{\beta
}}\right)\left({\rm 1-\mit t}\right)\rm +\mit \beta {\left({\rm
2\mit N}\right)}^{\beta \rm -1}\mit Nt
\label{prbfipf}
\end{equation}
Substituting in (\ref{t2}) for $t$ into (\ref{prbfipf}) and
maximizing with respect to $p_B$ gives the reaction function for
firm B:
\begin{equation}
%]|Expr|[(%$^:4;,= p^;)B_;,,] !<2^<c!$1(+$^p^;)A_;, ,K =b""k=  |
%|,K <c!$1((<c!$1(%$^2_(%:7;*=b""b:4=  /0 1;, /0 1>!:7=b""b:4=
%/0|
%|<c!$1(%1/0!:7=b""b:4=  ><c!$1(%$^2_^:7;*=b""b:4;,=  /0 1>! >!|
%|$^=b""N_^:7;*b>^:4;,= 2>]|[
{\mit p}_{\mit B}={\left({{p}_{A}+\mit k\rm
+\left({\left({{2}^{\mit
\beta \rm -1}-1}\right)\mit \beta \rm -\left({1-\mit \beta \rm
}\right)\left({{2}^{\mit \beta }\rm -1}\right)}\right){\mit
N}^{\beta }}\right) \over \rm 2}
\label{rbinf}
\end{equation}

%\noindent The coefficient on $N$ in (\ref{rbinf}) is always
% negative for $0<\b<1$.  An increase in the
% number of software products provided by firm B always reduces
%the hardware price of firm B.
%Decreases in the price of
% hardware by firm B increase its hardware market share and thus
% the extent of the software market in which it has a monopoly.

Then, the equilibrium prices and market shares in an interior
equilibrium follow immediately from the best-response
functions,
 (\ref{raf}) and  (\ref{rbinf}).  Substitution of
the equilibrium values for $p_A^{I}$, $p_B^{I}$, and $t^{I}$ into
the profit functions for the two hardware firms,
(\ref{prafipf}) and (\ref{prbfipf}), gives profits in the interior
equilibrium.

For parameter values where
%]|Expr|[(%$^:4;,=b""k^;)i_;, .S k]|[
${k}_{i}\rm \ge \mit k,$
there will be a standardization equilibrium since
%]|Expr|[(%$^:4;,=b""t_^;)I;, ,] 1]|[
${t}^{I}\rm \ge 1.$  The equilibrium price in the standardization
equilibria for firm A follows from substituting B's best response
function, (\ref{rbinf}) into the expression for market share,
(\ref{mar}), and setting
the resultant expression equal to one.  Substituting the
standardization equilibrium price of firm A into
(\ref{rbinf}) yields the standardization
equilibrium price
for firm B.  The standardization profits follow immediately.
\hfill {\em Q.E.D.}
\vspace{3 mm}

\noindent {\em Proof of Lemma \ref{5}.}
The profits of firm B arise only from the sale of hardware and
are simply
\begin{equation}
%]|Expr|[()$^:7;,=b""p^:4;)B_;, ,],H1/0t,I$^p^;)B_]|[
{\pi }_{B}\rm =(1-\mit t\rm ){\mit p}_{B}
\label{prforb}
\end{equation}
Substituting in (\ref{t2}) for $t$ and maximizing (\ref{prforb})
with respect to $p_B$ yields the best-response function for firm
B:
\begin{equation}
%]|Expr|[(#$^:4;,=b""p^;)B_;,,]<2(/$^p^;)A_;,,Kk/0,H1/0:7b:4,I|
%|,H$^2_^:7;)b:4;,/01,I$^N_^:7;)b^:4;,2>]|[
{p}_{B}\rm ={{\mit p}_{A}\rm +\mit k\rm -(1-\mit \beta \rm
)({2}^{\mit
\beta }\rm -1){\mit N}^{\beta } \over \rm 2}.
\label{rbf4}
\end{equation}

%\noindent Even though firm B is not integrated in this subgame,
%increases in the number of software products makes its pricing
%behaviour more
%aggressive.  The lost revenue from decreasing price on
% inframarginal units is less since the market share of B is less
% when A has foreclosed and as a result Firm B prices more
%aggressively in order to extend its own market share.

The best response function for firm A is again given by
(\ref{raf}).  The interior equilibrium expressions follow from the
two best response functions, and equations (\ref{prforb}) and
(\ref{prafipf}).

A standardization equilibrium will result if
%]|Expr|[(#$^:4;,=b""k^;)f_:7;,.S:4k]|[
${k}_{f}\rm \ge \mit k$
since
${t}^{f}\rm \ge 1.$
The standardization price for technology A is
found by substituting B's reaction function, (\ref{rbf4}), for
$p_B$ into the
expression for market share, setting the result equal to one, and
solving
for $p_A$.  Substituting into B's reaction function
yields $p_B=0.$  The standardization profits follow
immediately.
\begin{flushright}
\hfill {\em Q.E.D.}
\end{flushright}
\vspace{3 mm}

\noindent{\em Proof of Proposition \ref{retaliate}.}
Denote the sum of the profits of firm B and the independent
software firm in the foreclosure
equilibrium as
stand-alone (SA) profits.  Using  Lemma \ref{5},
\begin{equation}
%]|Expr|[(/$^:7;,=b""p_(":4;)SA;, ,] $^:7p^:4;)B_;,,K$^:7p^:4;)S|
%|^i;,=  ,] !<2(0<c!$1()!<c!$1(%$^2_(%:7;*=b""b:4=  /0 1;, /0 1>|
%|!:7=b""b:4,Kf,O3=  >!$^=b""N_(#;*= 2!:7=b""b:4;,= !=b""f=   ,K|
%|! <c!$1(*3!<c!$1(%$^2_(%:7;*=b""b:4=  /0 1;, ,K 1>!:7=b""b:4=
%/0|
%|!2!=b""f>=  !$^=b""N_^:7;*b:4;,= !=b""k("6k>=  ,K<2^k^2>]|[
{\pi }_{SA}^{f}\rm ={\mit \pi }_{\mit B}^{\mit f}\rm +{\mit \pi
}_{\mit
S}^{\mit f}\rm
={\left({\left({{2}^{\mit \beta \rm -1}-1}\right)\mit \beta \rm
+\mit f\rm /3}\right){\mit N}^{\rm 2\mit \beta }\rm \mit f\rm
+\left({3\left({{2}^{\mit \beta \rm -1}+1}\right)\mit \beta \rm
-2\mit f}\right)\rm {\mit N}^{\beta }\rm \mit k \over \rm 6\mit
k}\rm +{k \over 2}.
\label{sa}
\end{equation}
{}From Lemma \ref{pri}, the profits of firm B if it responds by
integrating and foreclosing
are $k/2$.  The difference between stand alone profits and
integrate and foreclose profits is
\begin{equation}
%]|Expr|[(*$^:7;,=b""p_(":4;)SA;, /0 $^:7p^:4;)B^f;,=  ,] !<2(0|
%|<c!$1()!<c!$1(%$^2_(%:7;*=b""b:4=  /0 1;, /0 1>!:7=b""b:4,Kf,O|
%|3=  >!$^=b""N_(#;*= 2!:7=b""b:4;,= !=b""f=   ,K!
%<c!$1(*3!<c!$1|
%|(%$^2_(%:7;*=b""b:4=  /0 1;, ,K 1>!:7=b""b:4= /0!2!=b""f>=  !|
%|$^=b""N_^:7;*b:4;,= !=b""k("6k>]|[
{\pi }_{SA}^{f}\rm -{\mit \pi }_{B}^{\mit BF}\rm
={\left({\left({{2}^{\mit
\beta \rm -1}-1}\right)\mit \beta \rm +\mit f\rm /3}\right){\mit
N}^{\rm 2\mit \beta }\rm \mit f\rm +\left({3\left({{2}^{\mit
\beta
\rm -1}+1}\right)\mit \beta \rm -2\mit f}\right)\rm {\mit
N}^{\beta
}\rm \mit k \over \rm 6\mit k}.
\label{difference}
\end{equation}

Substituting the value for the lower bound on $k$,
%]|Expr|[(#$^:4;,=b""k^;)f_;,,]<2("f$^N_^:7;)b^:4;,3>]|[
$${k}_{f}\rm ={\mit f{N}^{\beta } \over \rm 3}$$
into
(\ref{difference}) gives
%]|Expr|[(&$^:7;,=b""p_(":4;)SA;,/0$^:7p^:4;)B^f;,,]= !<2(%<c!$1|
%|(*!$^2_^:7;*=b""b:4;,= !:7=b""b:4=  /0 =b""f,O3>= !$^=b""N_(#|
%|;*= 2!:7=b""b:4;,= !=b""f("6k>]|[
$${\pi }^{SA}\rm -{\mit \pi }_{\mit B}^{f}\rm ={\left({{2}^{\mit
\beta
}\rm \mit \beta \rm -\mit f\rm /3}\right){\mit N}^{\rm 2\mit
\beta
}\rm \mit f \over \rm 6\mit k},$$
\noindent which is greater than zero for
$\rm 0<\mit \beta \rm <1$.
Since the numerator of (\ref{difference}) is increasing in $k$,
it
will still
be positive for values of $k$ greater than the lower bound.
\hfill {\em Q.E.D.}
\vspace{3 mm}

\noindent{\em Proof of Proposition \ref{retaliate2}.}
{}From Lemma \ref{4},
the difference between the
profits of
following this strategy and stand alone profits is:
\begin{eqnarray}
%]|Expr|[(($^:7;,=b""p^:4;)B^i;,/0$^:7p(":4;)SA_;,=  ,]
%!<2($<c!$1|
%|(#<c!$1(/<c!$1(*$^2_(';*2!:7=b""b:4=  /0
%1;,/0!2!$^2_(#:7;*=b""b|
%|:4,K1;,= ,K 2>!:7=b""b:4=  ,K $("32_(%:7;*=b""b:4=  /0 1;, /0|
%| $^2_(';*2!:7=b""b:4=  /0 1;, /0 1>!:7=b""b>:4= !$^=b""N_(#;*=
%2|
%|!:7=b""b:4;,=  (#=b""18k>]|[
{\mit \pi }_{SA}^{f}\rm-\mit{\pi }_{B}^{i}\rm&
=&-{\left({\left({\left({{2}^{2\mit \beta \rm -1}-2{2}^{\mit \beta
\rm +1}+2}\right)\mit \beta \rm +{32}^{\mit \beta \rm
-1}-{2}^{2\mit \beta \rm -1}-1}\right)\mit \beta }\right)\rm
{\mit
N}^{\rm 2\mit \beta }\rm  \over 18\mit k}-\nonumber\\
%]|Expr|[^<2(':4;,= 3!<c!$1($<c!$1($$^2_(%:7;*=b""b:4=  /0 1;,/0|
%|!1>  :7=b""b>:4= !$^=b""N_^:7;*b:4;,= !=b""k(#= 18k>]|[
&&{\rm 3\left({\left({{2}^{\mit \beta \rm -1}-1}\right)\mit \beta
}\right)\rm {\mit N}^{\beta }\rm \mit k \over \rm 18\mit k},
\end{eqnarray}
which is greater than zero if $ k>{k}_{min}$.
\hfill {\em Q.E.D.}
\vspace{3 mm}

%\noindent{\em Proof of Corollary \ref{c1}.}
%Provided (as assumed) that $N\rm >1$, ${k}_{min}$ is increasing
%in both
%$\b$ and
%$N$.
%\hfill {\em Q.E.D.}
%\vspace{3 mm}
\noindent{\em Proof of Proposition 3.}
If the merged firm did not foreclose on firm B it would
derive profits from hardware sales and software sales for both
technologies:
%]|Expr|[(-$^:7;,=b""p^:4;)A_;, ,] $^p^;)A_;,t ,K$^:7r^:4;)A_;,t|
%|,K$^:7r^:4;)B_<c!$1(#;,1,Mt>]|[
$${\pi }_{\mit A}\rm ={\mit p}_{\mit A}\mit t\rm +{\mit \rho
}_{\mit
A}\mit t\rm
+{\mit
\rho
}_{\mit B}\left({\rm 1-\mit t}\right)$$
or recognizing that the number of software varieties available
for
each technology will be the same and substituting in from Lemma
\ref{sp} for the price of software,
\begin{equation}
%]|Expr|[(+$^:7;,=b""p^:4;)A_;, ,] <c!$1('$^p^;)A_;, ,K :7b$^:42|
%|_(#:7;)b:4/01$^;,N_^:7;)b>:4;,t,K:7b$^:42_(#:7;)b:4/01$^;,N_^|
%|:7;)b<c!$1(#:4;,1,Mt>]|[
{\pi }_{A}\rm =\left({{\mit p}_{\mit A}\rm +\mit \beta {\rm
2}^{\mit
\beta \rm -1}{\mit N}^{\beta }}\right)\mit t\rm +\mit \beta {\rm
2}^{\mit \beta \rm -1}{\mit N}^{\beta }\left({\rm 1-\mit
t}\right)
\end{equation}
or
\begin{equation}
%]|Expr|[(*$^:7;,=b""p^:4;)A_;, ,] $^p^;)A_;,t,K:7b$^:42_(#:7;)b|
%|:4/01$^;,N_^:7;)b]|[
{\pi }_{A}\rm ={\mit p}_{\mit A}\mit t\rm +\mit \beta {\rm 2}^{\mit
\beta
\rm
-1}{\mit N}^{\beta }.
\label{saa}
\end{equation}
Merger with non-foreclosure yields an expression for the profit
of firm A
identical to the unintegrated case except for an additive
constant
equal to the profits of an independent software firm in the
unintegrated case.\footnote{It is for this reason that
integration
without foreclosure by both firms and integration without
foreclosure by one firm when the other firm is unintegrated are
formally identical to the
unintegrated
 industry structure, ie there is no price effect.}  Consequently,
substituting into
(\ref{saa})
 from the results of Lemma \ref{nofore} for the hardware profits
of
firm A, the profits of the merged firm when it does not foreclose
and
B does not integrate, are simply the sum of an independent
software
firm and an unintegrated hardware
firm in the unintegrated industry structure:
\begin{equation}
%]|Expr|[()$^:7;,=b""p^:4;)A("nf;, ,] <2^;.k^2>;,,K:7b$^:42_(#|
%|:7;)b:4/01$^;,N_^:7;)b]|[
{\pi }_{A}^{nf}\rm ={\mit k \over \rm 2}+\mit \beta {\rm 2}^{\mit
\beta \rm -1}{\mit N}^{\beta }.
\label{saae}
\end{equation}
The incentive for the merged firm to foreclose is the difference
between
foreclosure profits for firm A from Lemma \ref{5} and
non-foreclosure profits, from (\ref{saae}):
\begin{equation}
%]|Expr|[(1$^:7;,=b""p^:4;)A^f;, /0$^:7p^:4;)A("nf;, ,]
%<2^$^<c!$1|
%|(%f$^N_^:7;)b:4;,,K3k>_^;)2(#;,18k> /0
%<2^k^2>/0:7b$^:42_(#:7;)b|
%|:4/01$^;,N_^:7;)b]|[
{\pi }_{A}^{f}\rm -{\mit \pi }_{\mit A}^{\mit nf}\rm ={{\left({\mit
f{N}^{\beta }\rm +3\mit k}\right)}^{\rm 2} \over 18\mit k}\rm
-{\mit k \over \rm 2}-\mit \beta {\rm 2}^{\mit \beta \rm -1}{\mit
N}^{\beta }.
\end{equation}
This is positive if ${k}_{max}\rm >\mit k$.
\hfill {\em Q.E.D.}
\vspace{3 mm}

\noindent{\em Proof of Proposition \ref{noforeclosure}.}
If firm A does not integrate and foreclose, then the profits
earned by it and the independent software firm equals
(\ref{saae})
when B does not integrate.  If A forecloses and
B retaliates by merging but not foreclosing, the profits of firm
A are given by Lemma \ref{4}.  The difference between these
indicates
whether A will foreclose even if B retaliates.  This difference
is
equal to
\begin{equation}
%]|Expr|[(*$^:7;,=b""p^:4;)A^f;, /0$^:7p^:4;)A("nf;, ,] =  !<2|
%|(-<c!$1(*$^2_(#;*2!:7=b""b:4;,=  /0 !$^2_(#:7;*=b""b:4,K1;,=  |
%|,K 1>!$^=b""N_(#;*= 2!:7=b""b:4;,=  ,K 6!<c!$1(,$^2_^:7;*=b""b|
%|:4;,=  /0 1/0!3!$^2_(%:7;*=b""b:4=  /0 1;,!:7=b""b>:4=
%!$^=b""N|
%|_^:7;*b:4;,= !=b""k(#18k>]|[
{\pi }_{A}^{f}\rm -{\mit \pi }_{\mit A}^{\mit nf}\rm
={\left({{2}^{2\mit
\beta }\rm -{2}^{\mit \beta \rm +1}+1}\right){\mit N}^{\rm 2\mit
\beta }\rm +6\left({{2}^{\mit \beta }\rm -1-3{2}^{\mit \beta \rm
-1}\mit \beta }\right)\rm {\mit N}^{\beta }\rm \mit k \over \rm
18\mit k}.
\label{deltanf}
\end{equation}
For an interior equilibrium, the market share of firm A cannot
exceed 1.
Thus a lower bound on $k$ is given by (\ref{ft}),
%]|Expr|[^$^:4;,=b""k^;)f_]|[
${k}_{f}$
, which is greater than %]|Expr|[^$^:4;,=b""k^;)i_]|[
${k}_{i}$.  From (\ref{it}),
\begin{equation}
%]|Expr|[(%$^:4;,=b""k^;)i_;,=  ,] <2(),H$^2_^:7;*=b""b:4;,=  |
%|/0 1,I!$^=b""N_^:7;*b^:4;,= 3>]|[
{k}_{i}\rm ={({2}^{\mit \beta }\rm -1){\mit N}^{\beta } \over \rm
3}.
\label{k3}
\end{equation}
Substituting this lower bound for $k$ into (\ref{deltanf}) gives
%]|Expr|[('$^:7;,=b""p^:4;)A^f;, /0$^:7p^:4;)A("nf;, ,]<2('3= !|
%|<c!$1(%$^2_^:7;*=b""b:4;,=  /0 1>!<c!$1('!$^2_^:7;*=b""b:4;,=
%|
%|<c!$1(#1/0:7b>:4/0 1>!$^=b""N_(#;*= 2!:7=b""b(#:4;,18k>]|[
$${\pi }_{A}^{f}\rm -{\mit \pi }_{A}^{nf}\rm ={3\left({{2}^{\mit
\beta }\rm -1}\right)\left({{2}^{\mit \beta }\rm \left({1-\beta
}\right)-1}\right){\mit N}^{\rm 2\mit \beta } \over \rm 18\mit
k},$$
which is less than zero for $0<\beta<1$.  Since
the numerator of (\ref{deltanf}) is decreasing in $k$, as $k$
increases, the
numerator remains negative.
\hfill {\em Q.E.D.}
\vspace{3 mm}

\noindent{\em Proof of Proposition \ref{noholdup}.}
The maximum a hardware firm will bid for a software firm is the
difference between its profits if it forecloses and if it is
foreclosed upon.
In order for a bid to be accepted, it must be greater than the
profits a
software firm would earn if it was the independent software firm in
the
foreclosure equilibrium.  Thus in order for a successful bid to be
made,
\begin{equation}
%]|Expr|[('$^:7;,=b""p^:4;)A^f;,/0$^:7p^:4;)B_;, ,^ $^:7p^:4;)S|
%|^i]|[
{\pi }_{A}^{f}\rm -{\mit \pi }_{\mit B}^{\mit f}\rm >{\mit \pi
}_{\mit
S}^{\mit f}
\end{equation}
or
%]|Expr|[(($^:7;,=b""p^:4;)A^f;, ,^ $^:7p^:4;)S^i;, ,K$^:7p^:4;)B|
%|_]|[
$${\pi }_{A}^{f}\rm >{\mit \pi }_{\mit S}^{\mit f}\rm +{\mit \pi
}_{\mit
B}^{\mit f}$$
or
%]|Expr|[(%$^:7;,=b""p^:4;)A^f;, ,^ $^:7p^:4;)S("SA]|[
$${\pi }_{A}^{f}\rm >{\mit \pi }_{\mit f}^{\mit SA}.$$
{}From Lemma \ref{5},
\begin{equation}
%]|Expr|[(#$^:4;,=b"".Y^;)A^f;,,]<2^$^<c!$1(%;*3k,Kf$^N_^:7;'b>|
%|_^:42(#;,18k>]|[
{\rm \pi }_{\mit A}^{f}\rm ={{\left({3\mit k\rm +\mit f{N}^{\beta
}}\right)}^{\rm 2} \over 18\mit k}.
\end{equation}
Stand alone (hardware and software) profits in the foreclosure
industry
structure are given by (\ref{sa}):
%]|Expr|[(*$^:7;,=b""p_(":4;)SA;, =  ,] !<2(0<c!$1()!<c!$1(%$^|
%|2_(%:7;*=b""b:4=  /0 1;, /0 1>!:7=b""b:4,Kf,O3=  >!$^=b""N_(#|
%|;*= 2!:7=b""b:4;,= !=b""f=   ,K!
%<c!$1(*3!<c!$1(%$^2_(%:7;*=b""b|
%|:4=  /0 1;, ,K 1>!:7=b""b:4= /0!2!=b""f>=
%!$^=b""N_^:7;*b:4;,=!|
%|=b""k("6k>=  ,K<2^k^2>]|[
$${\pi }_{SA}^{f}\rm ={\left({\left({{2}^{\mit \beta \rm
-1}-1}\right)\mit \beta
\rm +\mit f\rm /3}\right){\mit N}^{\rm 2\mit \beta }\rm \mit f\rm
+\left({3\left({{2}^{\mit \beta \rm -1}+1}\right)\mit \beta \rm
-2\mit
f}\right)\rm {\mit N}^{\beta }\rm \mit k \over \rm 6\mit k}\rm
+{\mit k
\over 2}.$$
The difference between these two is
\begin{equation}
%]|Expr|[(($^:7;,= p^:4;)A^f;,/0$^:7p_(":4;)SA;, ,] !<2(/!!<c!$1|
%|($1/0!$^2_(%:7;*=b""b:4=  /0 1>;,!$^=b""N_(#;*= 2!:7=b""b:4;,=
%!|
%|=b""f= !:7=b""b:4,K<c!$1(*= 4!=b""f=  /03
%<c!$1(&!$^2_(%:7;*=b""b|
%|:4=  /0 1;, ,K 1>!:7=b""b>:4= !$^=b""N_^:7;*b:4;,=
%!=b""k("6k>]|[
{\rm \pi }_{A}^{f}-{\pi }_{SA}^{f}={\left({1-{2}^{\mit \beta \rm -
1}}\right){\mit N}^{\rm 2\mit \beta }\rm \mit f\rm \mit \beta \rm
+\left({4\mit f\rm -3\left({{2}^{\mit \beta \rm -1}+1}\right)\mit
\beta
}\right)\rm {\mit N}^{\beta }\rm \mit k \over \rm 6\mit k},
\label{new}
\end{equation}
which simplifies to
\begin{equation}
%]|Expr|[(($^:7;,= p^:4;)A^f;,/0$^:7p_(":4;)SA;, ,] !<2(%<c!$1|
%|(+4!=b""f=  /0 6!$^2_(%:7;*=b""b:4=  /0 1;,!:7=b""b>:4=
%!$^=b""N|
%|_(#;*= 2!:7=b""b:4;,= !=b""f(#18k>]|[
{\rm \pi }_{A}^{f}-{\pi }_{SA}^{f}={\left({4\mit f\rm -6{2}^{\mit
\beta
\rm -
1}\mit \beta }\right)\rm {\mit N}^{\rm 2\mit \beta }\rm \mit f
\over \rm
18\mit k}.
\label{ldifference}
\end{equation}
when we substitute in the lower bound for $k$,
%]|Expr|[^$(":4;,=b"" k^;)f_]|[
${\rm \mit k}_{f},$ where,
%]|Expr|[(':4;,=b""k=  ,] =b""f= !<2^$^=b""N_^:7;*b^:4;,= 3>]|[
$k_{f}\rm =\mit f\rm {{\mit N}^{\beta } \over \rm 3}$.  For
$0<\b<.56421$
(\ref{ldifference}) is positive and the numerator of (\ref{new}) is
increasing in
$k$
for $<\b<.17287$.
\hfill {\em Q.E.D.}
\vspace{3 mm}

\noindent{\em Proof of Proposition \ref{equilibrium}.}
{}From Proposition 3, Firm A finds it profitable to foreclose,
given no
retaliation
when %]|Expr|[(#$^:4;,=b""k(#;)max_;,,^k]|[
${k}_{max}\rm >\mit k$.
 From Proposition 2, Firm B will, however, retaliate if
%]|Expr|[(#:4;,=b""k,^$^k(#;)min_]|[
$k\rm <{\mit k}_{min}$.
For a
foreclosure equilibrium
to exist, %]|Expr|[()$^:4;,=b""k(#;)max_;, .S k .S $^k(#;)min_]|[
${k}_{max}\rm \ge \mit k\rm \ge {\mit k}_{min}$.  Such a range
for
$k$ will exist if ${k}_{max}\rm \ge {\mit k}_{min}$.    This is
true for
%]|Expr|[(+:4;,=b""0,\:7b:4.R,N098733]|[
$\rm 0<\mit \beta \rm \le .098733$.  There is not a holdup
problem
since from
Proposition \ref{noholdup}, a software firm will prefer to merge
and foreclose
if $\b<.17287$.  We also check that
%]|Expr|[($$^:4;,=b""k(#;)min_;,,^ $^k^;)f_]|[
${k}_{min}\rm >{\mit k}_{f}$ for
%]|Expr|[():4;,=b""0 ,\ :7b:4 ,\1]|[
$\rm 0<\mit \beta \rm <1$, so that for $k>{k}_{min}$, the
foreclosure
subgame equilibrium is in fact an interior equilibrium.
Furthermore, we need to check
that
%]|Expr|[(%$^:4;,=b""k($;)max _;, ,^ $^k^;)f_]|[
${k}_{max\rm }>{\mit k}_{f}$ so that when A forecloses the
equilibrium is
in fact an interior equilibrium and not a standardization
equilibrium.  For
%]|Expr|[(.:4;,=b""0 ,\ :7b:4 ,\ ,N22179]|[
$\rm 0<\mit \beta \rm <.22179$, it is straightforward to show that
%]|Expr|[(%$^:4;,=b""k(#;)max_;, ,^ $^k^;)f_]|[
${k}_{max}\rm >{\mit k}_{\mit f}$.  If the two parameter
 restrictions do not hold, then either firm A does not find it
profitable to foreclose, even if firm B does not merge or even if
A finds it
profitable to foreclose if B does not respond, it is profitable
for firm B to respond by integrating and not foreclosing which
makes foreclosure by A non-optimal.
\hfill {\em Q.E.D.}
\vspace{3 mm}

\noindent{\em Proof of Proposition \ref{rs}.}
We first show that for
%]|Expr|[()$^:4;,=b""k^;)f_;, ,^ k ,^$^k^;)i_]|[
${k}_{f}\rm >\mit k\rm >{\mit k}_{\mit i}$, firm B always responds
to
foreclosure by firm A by integrating but not foreclosing.  If
%]|Expr|[(%:4;,=b""k ,] $^k^;)i_]|[
$k\rm ={\mit k}_{\mit i}$, then the variety advantage afforded firm
A if
it
forecloses and B retaliates is such that the retaliation is
ineffective and
%]|Expr|[(%:4;,=b""t ,] 1]|[
$t\rm =1$.  In this case, the profits of B if it integrates arise
only from the sale of software to technology A.  However, this is
also the
value of stand alone  profits.  For
%]|Expr|[(%:4;,=b""k ,^$^k^;)i_]|[
$k\rm >{\mit k}_{\mit i}$, %]|Expr|[($:4;,=b""1,^ $^t_^;)I]|[
$\rm 1>{\mit t}^{\mit I},$ %]|Expr|[^$^:4;,=b""p^;)B^I]|[
${p}_{B}^{I}$ is increasing in $k$, ${\mit t}^{I}$ decreasing in
$k$, and the
profits of firm B,
%]|Expr|[()$^:7;,=b""p^:4;)B^I;,,]<c!$1($$^p^;)B^I;,,K:7b$^:4N|
%|_^:7;)b><c!$1(#:4;,1,M$^t_^;)I>;,,K:7b$^:42_(#:7;)b:4,M1$^;,N|
%|_^:7;)b$^:4;,t_^;)I]|[
$${\pi }_{B}^{I}\rm =\left({{\mit p}_{B}^{I}\rm +\mit \beta
{N}^{\beta
}}\right)\left({\rm 1-{\mit t}^{I}}\right)\rm +\mit \beta {\rm
2}^{\mit
\beta \rm -1}{\mit N}^{\beta }{t}^{I},$$
are increasing and therefore greater than stand alone profits.

{}From Proposition (\ref{noforeclosure}), it will not be profitable
for firm A to foreclose if firm B retaliates when
 %]|Expr|[(%:4;,=b""k ,^ $^k^i_]|[
$k\rm >{\mit k}_{i}$.
\hfill Q.E.D.

\vspace {4 mm}
\noindent{\em Proof of Proposition \ref{1s}.}
Foreclosure profits are, from
Lemma \ref{5}
\begin{equation}
%]|Expr|[(&$^:7;,=b""p^:4;)A("fs;,,]<c!$1(&$^2_(#:7;)b:4,M1<c!$1|
%|(#;,2,M:7b>:4,K:7b:4,M1>$^N_^:7;)b:4;,,Mk]|[
{\pi }_{A}^{fs}\rm =\left({{2}^{\mit \beta \rm -1}\left({2-\mit
\beta
}\right)\rm +\mit \beta \rm -1}\right){\mit N}^{\beta }\rm -\mit k.
\end{equation}
or, breaking out hardware and software profits,
%]|Expr|[(-$^:7;,=b""p^:4;)A("fs;,,]<c!$1(#1,M:7b><c!$1(#$^:42|
%|_^:7;)b:4;,,M1>$^N_^:7;)b:4;,,Mk ,K :7b$^:42_(#:7;)b:4,M1$^;,N|
%|_^:7;)b]|[
$${\pi }_{A}^{fs}\rm =\left({1-\mit \beta }\right)\left({{\rm
2}^{\mit
\beta }\rm -1}\right){\mit N}^{\beta }\rm -\mit k\rm +\mit \beta
{\rm
2}^{\mit \beta \rm -1}{\mit N}^{\beta }.$$
{}From (\ref{saae}) the non-foreclosure profits of firm A are
%]|Expr|[('$^:7;,=b""p^:4;)A("nf;,,]<2^k^2>,K:7b$^:42_(#:7;)b:4,M|
%|1$^;,N_^:7;)b]|[
$${\pi }_{A}^{nf}\rm ={\mit k \over \rm 2}+\mit \beta {\rm 2}^{\mit
\beta
\rm -1}{\mit N}^{\beta }.$$
The difference between these two is positive when $k_s>k$.
We now show that no integration by firm B is the best response to
foreclosure by firm A.  We first show that firm B will never find
it optimal to acquire the remaining software firm and foreclose on
A.  If foreclosure results in standardization, then in the
foreclosure
equilibrium the profits of firm B are zero and hence stand-alone
profits
are simply duopoly software profits.  The difference between
stand-alone
profits and bilateral foreclosure profits is:
%]|Expr|[(/$^:7;,=b""p(":4;)SA^f;, ,M $^:7p_(":4;)BF;, ,] :7b$|
%|^:42_(#:7;)b:4,M1$^;,N_^:7;)b:4;,,Mk,O2]|[
$${\pi }_{SA}^{f}\rm -{\mit \pi }^{\mit BF}\rm =\mit \beta {\rm
2}^{\mit
\beta
\rm -1}{\mit N}^{\beta }\rm -\mit k\rm /2.$$
This is greater than zero if
%]|Expr|[(*$^:4;,=b""2_^:7;)b;,b:4,Mf,O3 ,^ 0]|[
${\rm 2}^{\mit \beta }\beta \rm -\mit f\rm /3>0,$
which is identical to the condition for an interior equilibrium and
it holds
for all %]|Expr|[():4;,=b""0 ,\ :7b:4 ,\ 1]|[
$\rm 0<\mit \beta \rm <1.$

We now argue that firm B will never find it optimal to acquire the
independent software firm and not foreclose.
If firm A has foreclosed, the stand alone
profits of
firm B and the independent software firm are simply duopoly profits
from
software sales to technology A.  If B integrates, the equilibrium
is still a
standardization equilibrium and the integrated firm earns profits
equal to
duopoly profits from software sales to technology A.
\hfill {\em Q.E.D.}
\vspace{3 mm}

\noindent{\em Proof of Proposition \ref{sw}.}
The total surplus associated with the foreclosure equilibrium is
given by (\ref{tsf}).  The middle term,
%]|Expr|[("$^:4;,=b""N_^:7;)b<c!$1($<c!$1(#$^:4;,2_^:7;)b:4;,/0|
%|1>t,K1>]|[
${N}^{\beta }\left({\left({{\rm 2}^{\mit \beta }\rm
-1}\right)\mit
t\rm +1}\right)$, is increasing in $t$ since $2^{\b}-1 >0$.
The last term,
$${k \over \rm 2}\left({{\mit t}^{\rm 2}+{\left({1-\mit
t}\right)}^{\rm
2}}\right),$$
is minimized when
$t=1/2$ and is increasing in $t$ for $t>1/2$.  Substituting in
$t=1$ for $t$ in the middle term and $t=1/2$ for the last, a
strict
upper bound on $TS^{f}$ is,
\begin{equation}
%]|Expr|[():4;,=b""T$^S_^;)f;,,\y,K$^<c!$1("$^2__$^N__>_^:7;)b|
%|:4;,/0 <2^k^4>]|[
T{S}^{f}\rm <\mit y\rm + \alpha
+{\left({{2}^{}{\mit N}^{}}\right)}^{\beta
}\rm -{\mit k \over \rm 4}.
\end{equation}
The right-hand side is identical to (\ref{tsu}), the total
surplus in the unintegrated equilibrium.
\begin{flushright}
\hfill {\em Q.E.D.}
\end{flushright}

\vspace{3 mm}

\end{document}


 jrchurch acs  2/25/93
Jeffrey Robert Chur Neil Gandal          2/25/93 Foreclosure Paper


-------------------------------------------------------------------------
NEIL GANDAL
THE EITAN BERGLAS SCHOOL OF ECONOMICS
TEL AVIV UNIVERSITY
RAMAT AVIV , TEL AVIV 69978, ISRAEL
TEL: 972-3-640-9604  FAX: 972-3-640-9908
HOME: TEL: 972-9-581124
E-MAIL: GANDAL@TAUNIVM.BITNET
-------------------------------------------------------------------------
