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\begin{center}
ECONOMIC ANALYSIS GROUP

DISCUSSION PAPER

\vspace{3cm}

Bargaining Power and the Effects of Joint Negotiation: The "Recapture Effect"

\vspace{0.25cm} by \vspace{0.25cm}

Craig T. Peters* \\[0pt]

EAG 14-3 $\quad$ September 2014
\end{center}

\vspace{2cm}

\noindent EAG Discussion Papers are the primary vehicle used to disseminate
research from economists in the Economic Analysis Group (EAG) of the
Antitrust Division. These papers are intended to inform interested
individuals and institutions of EAG's research program and to stimulate
comment and criticism on economic issues related to antitrust policy and
regulation. The Antitrust Division encourages independent research by its
economists. The views expressed herein are entirely those of the author and
are not purported to reflect those of the United States Department of
Justice.

\vspace{0.25cm}

\noindent Information on the EAG research program and discussion paper
series may be obtained from Russell Pittman, Director of Economic Research,
Economic Analysis Group, Antitrust Division, U.S. Department of Justice, LSB
9004, Washington, DC 20530, or by e-mail at russell.pittman@usdoj.gov.
Comments on specific papers may be addressed directly to the authors at the
same mailing address or at their e-mail addresses.

\vspace{0.25cm}

\noindent To obtain a complete list of titles or to request single copies of
individual papers, please write to Kathy Burt at kathy.burt@usdoj.gov or
call (202) 307-5794. In addition, recent papers are now available on the
Department of Justice website at
http://www.justice.gov/atr/public/eag/discussion-papers.html.

\vspace{0.25cm}

\noindent\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_%
\_\_\_\_\_\_\_\_\_

{\small \noindent *Economic Analysis Group, Antitrust Division, U.S.
Department of Justice. Email: craig.peters@usdoj.gov. \ The views expressed
in this paper are those of the author and are not purported to reflect the
views of the U.S. Department of Justice.\ Thanks to Beth Armington, Dave
Balan, Leemore Dafny, Gautam Gowrisankaran, Patrick Greenlee, Helen Knudsen,
Aviv Nevo, Alex Raskovich, Fiona Scott Morton, Bob Town, Greg Vistnes, and
seminar participants at the U.S. Department of Justice for helpful
discussions about the ideas in this paper. }

{\normalsize \newpage }

\begin{abstract}
{\normalsize This paper considers the effects of joint negotiation when
suppliers and intermediaries engage in bilateral negotiation over inclusion
of a supplier's product in an intermediary's network. I identify conditions
under which joint negotiation by two suppliers increases the suppliers'
bargaining power even when the suppliers' products are not substitutes for
each other. }In particular, joint negotiation increases the suppliers'
bargaining power if suppliers face smaller losses from disagreement when
they negotiate jointly. If joint negotiation causes an intermediary to lose
more of its consumers to competing intermediaries in the event of
disagreement, and if the suppliers sell their products through these
competing intermediaries, the suppliers will be able to recapture more of
the sales that they would otherwise have lost in the event of disagreement.
As a result, joint negotiation reduces the suppliers' losses from
disagreement, and thus enhances their bargaining power. I show that these
conditions arise under a wide range of assumptions about consumer
preferences. 
\end{abstract}

{\normalsize \newpage \setcounter{page}{1} }

\section{\protect\normalsize Introduction}

{\normalsize In many industries, suppliers and intermediaries bargain over
whether the supplier's product will be included among a range of products
that the intermediary makes available to consumers. For example,
manufacturers negotiate with retailers for access to shelf space; record
companies and publishers negotiate with owners of digital media platforms to
have their content available on the platform; producers of video programming
negotiate with cable companies; and health care providers negotiate with
commercial health insurers for inclusion in the insurer's managed care
network. In all of these industries, an intermediary's bargaining leverage
derives from the threat that the supplier will make fewer sales to final
consumers if it does not reach agreement with the intermediary, and a
supplier's bargaining leverage derives from the threat that the
intermediary's offering will become less attractive to consumers if it does
not include the supplier's products. }

{\normalsize A potentially important issue is how the relative bargaining
power of the negotiating parties may be affected if two or more suppliers
negotiate as a single entity. For example, this issue may arise in the
context of a merger between two suppliers. \ More generally, any arrangement
between two suppliers that commits them to negotiating jointly will raise
the issue. \ A substantial literature has investigated the effects of joint
negotiation on bargaining power, with industry applications ranging from
health care to cable television.\footnote{{\normalsize For example, a large
literature, including Town and Vistnes (2001), Capps, Dranove, and
Satterthwaite (2003), and Gowrisankaran, Nevo, and Town (2014), considers
the effects of hospital mergers on the merged entity's bargaining position.
Similarly, Chipty and Snyder (1999) consider the effects of \textit{buyer }%
mergers on bargaining power in the cable television industry. O'Brien and
Shaffer (2005) examine the bargaining power effects of mergers between
upstream suppliers who sell differentiated products to a downstream retail
monopolist.}} }

{\normalsize A central conclusion to emerge from this literature is that
joint negotiation between two suppliers is likely to enhance the suppliers'
bargaining power if the loss to the intermediary from failing to reach
agreement with both suppliers is greater than the sum of the losses from
failing to reach agreement with each supplier individually; that is, if the
intermediary's value function is concave. The most natural reason for the
intermediary's value function to be concave is that final consumers view the
suppliers as substitutes for one another. }

{\normalsize To see the intuition for this, consider a hypothetical example
from the health care industry, in which two similar hospitals are located
close to one another. If patients view these hospitals as close substitutes,
an insurer will be able to offer an attractive network to potential members
even if it includes only one of the two hospitals. There will be little cost
to the insurer from failing to reach an agreement with one of the hospitals,
since potential enrollees will be reasonably satisfied as long as they have
in-network access to the other hospital. If, however, the insurer fails to
reach an agreement with either hospital, any patients that viewed the two
hospitals as their first and second choices will need to switch to a less
preferred hospital (or change to a different health plan). Thus, as a direct
result of patients' viewing the hospitals as substitutes, the loss to the
insurer from dropping both hospitals is greater than the sum of the losses
from dropping only one of them, and hence joint negotiation by the hospitals
would weaken the insurer's bargaining position. }

{\normalsize Based on this result, the existing literature on identifying
the likely competitive effects of hospital mergers has placed considerable
importance on measuring the degree of patient substitution between
hospitals. Indeed, in many widely used models, a merger's competitive
effects are determined entirely by the extent to which patients view the
merging providers as substitutes. \ For example, the discussion of
bargaining theory in Farrell et al. (2011) states "If the hospitals are not
substitutes, ... the cost of failing to reach an agreement with both
hospitals is equal to the sum of the costs of failing to reach agreements
with them separately, and so there will be no effect on price."\footnote{%
{\normalsize Farrell et al. (2011) at 275. \ See also Gaynor and Town
(2012): "The merger will increase price as long as the additional loss in
per-patient welfare [from a network without hospitals j and k] is greater
than the loss in welfare from hospital network [without k only]. This will
be the case if and only if patients view hospitals j and k as substitutes."}}
}

{\normalsize There is thus a well-established consensus in the literature
that the degree of substitution plays an important role in determining the
effects of joint negotiation on bargaining power. \ In some instances,
however, two suppliers with little or no consumer substitution between their
products may enter into a joint contracting arrangement, and well-informed
industry observers and participants nonetheless expect the joint negotiation
to result in enhanced bargaining power. \ For example, in the health care
industry, many would expect commercial insurers to pay increased
reimbursement rates when a large hospital acquires an independent physician
practice, even if patients are not able to substitute between the services
of the hospital and those of the physician group.\footnote{{\normalsize %
Based on interviews with representatives of hospitals, physician groups,
health plans, and other health industry participants, Berenson \textit{et
al. }(2010) conclude that "one clear goal of an alliance between hospitals
and physicians is to improve negotiating clout for both." Similarly,
Berenson \textit{et al.} (2012) report that "Respondents from health plans
and provider organizations agreed that hospitals negotiating on behalf of
their employed physicians are able to obtain higher prices for physician
services than can be achieved by independent physician practices. Some plan
respondents reported that having a large employed physician contingent also
increased hospital leverage over rates for hospital services."}} \ This
creates something of a puzzle for economic theory. Either these observers
are incorrect about the likely consequences of joint negotiation in these
contexts, or there is something important missing from a theoretical
framework that relies solely on consumer substitution\ to produce effects. }

{\normalsize A few recent papers, focused on the health care industry, have
proposed theories that could address this puzzle. Vistnes and Sarafidis
(2013) consider the effects of mergers between hospitals in geographically
distinct areas, and identify certain conditions under which an insurer's
value function may be concave even if patients do not view the hospitals as
substitutes. Under these conditions, "cross-market hospital mergers may
reduce competition even in the absence of any significant patient
substitution between the merging hospitals."\footnote{{\normalsize Vistnes
and Sarafidis (2013) at 259.}} \ As an alternative explanation for an
observed empirical relationship between a hospital's negotiated prices and
its membership in a large hospital system, Lewis and Pflum (2014) argue that
large systems may have informational or other advantages that enhance their
bargaining effectiveness. \ }

{\normalsize In this paper, I describe an alternative mechanism that can
explain an increase in bargaining power resulting from joint negotiation by
two suppliers that do not offer substitute products. \ The mechanism does
not depend on concavity of the intermediary's value function; that is, there
is no requirement that the intermediary's losses from failing to reach
agreement with both suppliers be greater than the sum of its losses from
failing to reach agreement with each one individually. \ Instead, the
mechanism relies on differences in the \textit{suppliers' }pay-offs. \ In
particular, joint negotiation increases the suppliers' bargaining power if
suppliers face smaller losses from disagreement when they negotiate
jointly. An intermediary's leverage over a supplier is derived from the
threat that the supplier will lose access to the intermediary's consumers in
the event of disagreement. But if disagreement with one supplier also
triggers disagreement with the other (because of joint negotiation), the
intermediary will lose more of its consumers to competing intermediaries in
the event of disagreement. \ If the supplier sells its products through
these competing intermediaries, the supplier will be able to recapture more
of the sales that it would otherwise have lost. \ As a result, joint
negotiation reduces the suppliers' losses from disagreement, and thus
enhances their bargaining power. }

{\normalsize It is important to note that this mechanism relies on the
presence of some degree of competition among intermediaries. \ That is,
consumers that substitute away from one intermediary have the option of
choosing an alternative intermediary to access their preferred suppliers. \
If, instead, suppliers negotiate with a single intermediary and consumers
have no alternative means of accessing the suppliers, the effect identified
in this paper will not arise. \ In that setting, concavity of the
intermediary's value function is a necessary condition for joint negotiation
to increase suppliers' bargaining power. }

{\normalsize A natural question is why this effect does not violate the
principle of "one monopoly rent." Unsurprisingly, the explanation is that
joint negotiation allows the suppliers to internalize contracting
externalities. An analogy with suppliers of substitute products illustrates
this point. When an intermediary reaches an agreement with one supplier,
that agreement has the (external) effect of reducing the value to that
intermediary of reaching an agreement with a different supplier that sells
substitute products. In the familiar case in which two suppliers of
substitute products negotiate jointly, joint negotiation allows the
suppliers to internalize that externality and hence increase their
bargaining leverage. The effect identified in this paper can be viewed
similarly. When an intermediary reaches an agreement with one supplier, that
agreement has the effect of increasing the value to another supplier of
reaching an agreement with that intermediary, because the volume of
consumers that can be accessed only through that intermediary is higher as a
result of the initial agreement. \ Again, joint negotiation allows the
suppliers to internalize this externality and thus increase their bargaining
leverage. }

{\normalsize The intuition behind this idea can be illustrated by an example
of a hospital and a physician practice entering into an agreement to
negotiate jointly with insurers. \ Negotiating on its own, the price that
the physician practice can negotiate will be constrained by the threat that
substantial patient volume will be diverted to competing physicians in the
event of disagreement. \ By contrast, joint contracting between the
physician practice and the hospital diminishes this threat, because more of
the insurer's members will switch to another health plan rather than forego
access to the hospital. \ In this way, joint negotiation increases the
providers' bargaining power by preventing the insurer from threatening to
steer as much volume away from each provider in the event of disagreement. }

{\normalsize This concept is applicable in a number of settings in which
consumers demand access to multiple products. \ Because the mechanism relies
on consumers who value both of the suppliers' products, it will not
generally account for any bargaining power effects that may arise when
suppliers in geographically distinct areas negotiate collectively. In this
sense, the ideas in this paper may be viewed as complementary to the ideas
proposed by Vistnes and Sarafidis (2013) and Lewis and Pflum (2014). Earlier
work by Gal-Or (1999) considers the effects of hospital acquisitions of
physician practices, identifying certain conditions under which such
acquisitions increase the providers' bargaining power. Gal-Or's model is in
fact a special case of the framework I consider here, but the mechanism
underlying its effects does not appear to have been well-understood. I
consider the model in detail below. }

{\normalsize The remainder of the paper is organized as follows: In Section
2, I describe a general model of bargaining between suppliers and
intermediaries, and derive expressions for the effects of joint negotiation
on the equilibrium transfer from intermediaries to suppliers in this model.
\ In Section 3, I consider a special case of this framework, the model
introduced by Gal-Or (1999); I show that this model's support for the
conclusion that joint negotiation enhances bargaining power is in fact more
robust than Gal-Or's paper suggests. \ In Section 4, I consider an
alternative model, based on logit demand, illustrating that the effect
identified in this paper arises under a wide range of assumptions about the
distribution of consumer preferences. \ Section 5 is a brief conclusion. }

\section{\protect\normalsize A general model of bargaining between suppliers
and intermediaries}

{\normalsize In the general framework for the models in this paper,
suppliers produce goods or services that consumers obtain through
intermediaries. There are two stages to the analysis: In the first stage,
each supplier-intermediary pair engages in simultaneous bilateral
negotiations to determine whether the supplier's products will be available
to consumers at that intermediary, as well as the terms of any financial
exchange between the supplier and the intermediary. The outcome of each
negotiation is given by the (asymmetric) Nash bargaining solution. Following
the bargaining framework introduced by Horn and Wolinsky (1988), the parties
to each negotiation take the equilibrium outcomes of all other negotations
as fixed when considering whether to enter into an agreement.\ \
Intermediaries simultaneously set the prices that they charge to consumers
for access to their portfolio of products (e.g. membership or subscription
fees, or insurance premia). If suppliers charge prices directly to
consumers, these prices are also determined at the first stage.\ (I also
consider variants of the model in which suppliers do not charge prices
directly to consumers, but rather receive reimbursement solely through the
terms of their agreement with the intermediary). \ Under this assumption on
the timing of price determination, these prices are also treated as fixed
during supplier-intermediary negotiations. (I also consider an extension to
the model in which each intermediary expects to be able to adjust its price
in the event of a disagreement with a supplier.) In the second stage,
consumers observe the set of products available at each intermediary (i.e.,
the intermediary's "network") and all prices, choose one intermediary, and
consume products available at that intermediary. Based on these consumer
choices, the intermediaries and suppliers realize profits. }

\subsection{\protect\normalsize Lump sum transfers}

{\normalsize Consider a bilateral negotiation between an intermediary $I$
and a supplier $A$. If (and only if) the negotiation ends in agreement,
supplier $A$ will be included in intermediary $I$'s network, and the parties
agree to the terms of payment between them. \ In this section, I assume
payment consists of a lump sum transfer $T_{A}$. $\ T_{A}$ is defined to be
positive if the intermediary pays the supplier, and negative if the reverse
occurs. \ }

{\normalsize Let $V_{I}\left( \Omega \right) $ and $V_{A}\left( \Omega
\right) $ denote the gross payoffs for intermediary $I$ and supplier $A$,
excluding the value of $T_{A}$, given $I$'s network $\Omega $. Let $\Omega
^{\ast }$ denote the network that both parties expect in the event of
agreement (i.e., a network that includes supplier $A$), and let $\Omega ^{-A}
$ denote the same network without supplier $A$ (the disagreement outcome).
The (asymmetric) Nash bargaining solution will solve the following
maximization problem:%
\begin{equation*}
\max_{T_{A}}\left[ V_{I}\left( \Omega ^{\ast }\right) -V_{I}\left( \Omega
^{-A}\right) -T_{A}\right] ^{\left( 1-\beta \right) }\cdot \left[
V_{A}\left( \Omega ^{\ast }\right) -V_{A}\left( \Omega ^{-A}\right) +T_{A}%
\right] ^{\beta }
\end{equation*}
}

{\normalsize where $\beta \in \lbrack 0,1]$ is the Nash bargaining parameter
indexing the unmodeled bargaining effectiveness of the two parties. \ If $%
\beta =1$, the supplier has all the bargaining power, and so can effectively
make a take-it-or-leave-it offer to the intermediary. \ If $\beta =0$, the
intermediary has all the bargaining power. }

{\normalsize The solution to this problem is:%
\begin{equation*}
T_{A}=\beta \left[ V_{I}\left( \Omega ^{\ast }\right) -V_{I}\left( \Omega
^{-A}\right) \right] +\left( 1-\beta \right) \left[ V_{A}\left( \Omega
^{-A}\right) -V_{A}\left( \Omega ^{\ast }\right) \right]
\end{equation*}
}

{\normalsize Here, the equilibrium transfer is expressed as a weighted
average of two extremes: the highest possible transfer that the insurer
would be willing to pay ($V_{I}\left( \Omega ^{\ast }\right) -V_{I}\left(
\Omega ^{-A}\right) $) and the lowest possible transfer that the provider
would be willing to accept ($V_{A}\left( \Omega ^{-A}\right) -V_{A}\left(
\Omega ^{\ast }\right) $). }

{\normalsize The outcome of bilateral negotiations between intermediary $I$
and supplier $B$ produces a similar expression for transfer $T_{B}$: \ 
\begin{equation*}
T_{B}=\beta \left[ V_{I}\left( \Omega ^{\ast }\right) -V_{I}\left( \Omega
^{-B}\right) \right] +\left( 1-\beta \right) \left[ V_{B}\left( \Omega
^{-B}\right) -V_{B}\left( \Omega ^{\ast }\right) \right]
\end{equation*}%
I assume that $\beta $\ is the same for every supplier-intermediary pair.
(If some suppliers have greater bargaining effectiveness than others, the
effects of joint negotiation will partly reflect these differences. For
example, if a supplier with high negotiating skill agrees to negotiate on
behalf of a less-skilled supplier, the jointly negotiating suppliers may be
able to extract greater rents as a result of their contracting arrangement.
In this paper, in order to focus on effects of joint negotiation that arise
even in the absence of such differences in bargaining effectiveness, I
abstract away from this possibility). }

{\normalsize Now consider the change in bargaining if suppliers $A$ and $B$
negotiate jointly. In particular, assume that suppliers $A$ and $B$ make a
binding commitment to negotiate as a single entity, so that an intermediary
must agree to include both of them in its network if it wishes to include
either one.\ In this case, a failure to reach agreement with intermediary $%
I\,$\ will result in the network $\Omega ^{-A-B}$, the network without
suppliers $A $ and $B$. \ Under Nash bargaining, the combined transfer to
the two providers, $T_{A+B}$, will be given by the following expression:%
\begin{eqnarray*}
T_{A+B} &=&\beta \left[ V_{I}\left( \Omega ^{\ast }\right) -V_{I}\left(
\Omega ^{-A-B}\right) \right] \\
&&+\left( 1-\beta \right) \left[ V_{A}\left( \Omega ^{-A-B}\right)
+V_{B}\left( \Omega ^{-A-B}\right) -V_{A}\left( \Omega ^{\ast }\right)
-V_{B}\left( \Omega ^{\ast }\right) \right]
\end{eqnarray*}
}

{\normalsize The expressions for $T_{A}$, $T_{B}$, and $T_{A+B}$ include
terms for the parties' payoffs under the equilibrium network, $V_{I}\left(
\Omega ^{\ast }\right) $, $V_{A}\left( \Omega ^{\ast }\right) $, and $%
V_{B}\left( \Omega ^{\ast }\right) $. In principle, these payoffs may be
different depending on whether $A$ and $B$ negotiate jointly. For example,
if $A$ and $B$ are able to obtain higher payments from other intermediaries
as a result of joint negotiation, $V_{A}\left( \Omega ^{\ast }\right) $ and $%
V_{B}\left( \Omega ^{\ast }\right) $ will be higher in the expression for $%
T_{A+B}$ than in the expressions for $T_{A}$ and $T_{B}$. \ However, as long
as the only effect of joint negotiation is to change the value of lump sum
transfers, then under Nash bargaining (in which the terms of other
agreements are treated as fixed in the event of disagreement), these
differences will cancel out of the expressions above. \ Thus, for the
purposes of the analysis with lump sum transfers, I treat $V_{I}\left(
\Omega ^{\ast }\right) $, $V_{A}\left( \Omega ^{\ast }\right) $, and $%
V_{B}\left( \Omega ^{\ast }\right) $ as invariant to whether $A$ and $B$
negotiate jointly. }

{\normalsize Combining these results, the change in the combined transfer
that results from joint contracting is:%
\begin{eqnarray}
\Delta T &\equiv &T_{A+B}-\left( T_{A}+T_{B}\right)
\label{Delta T - General} \\
&=&\beta \left\{ \left[ V_{I}\left( \Omega ^{-B}\right) -V_{I}\left( \Omega
^{-A-B}\right) \right] -\left[ V_{I}\left( \Omega ^{\ast }\right)
-V_{I}\left( \Omega ^{-A}\right) \right] \right\}  \notag \\
&&+\left( 1-\beta \right) \left[ V_{A}\left( \Omega ^{-A-B}\right)
-V_{A}\left( \Omega ^{-A}\right) +V_{B}\left( \Omega ^{-A-B}\right)
-V_{B}\left( \Omega ^{-B}\right) \right]  \notag
\end{eqnarray}
}

{\normalsize The first part of this expression (in braces) reflects the
effect of the intermediary's payoff. \ It says that joint negotiation will
tend to increase the transfer if the incremental value to the intermediary
of adding $A$ to its network is greater when $B$ is not also in the network: 
$\left[ V_{I}\left( \Omega ^{-B}\right) -V_{I}\left( \Omega ^{-A-B}\right) %
\right] >\left[ V_{I}\left( \Omega ^{\ast }\right) -V_{I}\left( \Omega
^{-A}\right) \right] $. This is a familiar concavity condition: Joint
negotiation by two suppliers will tend to increase bargaining power when the
loss of both suppliers is more costly to the intermediary than the sum of
the losses of each supplier individually. \ }

{\normalsize The second part of the expression shows that joint negotiation
will also tend to increase the transfer if the payoff to $A$ (or $B$) when
it is excluded from the network is greater if $B$ (or $A$) is also excluded: 
$V_{A}\left( \Omega ^{-A-B}\right) >V_{A}\left( \Omega ^{-A}\right) $ (or $%
V_{B}\left( \Omega ^{-A-B}\right) >V_{B}\left( \Omega ^{-B}\right) $). In
other words, if an agreement between an intermediary and a supplier has a
negative externality on another supplier's payoff under disagreement,
coordination by the two suppliers can internalize this effect and allow the
suppliers jointly to negotiate higher payment. This externality is the
feature of the model that has generally been absent from applied work in the
literature.\footnote{{\normalsize For example, Gaynor and Town (2012)
specify a model of hospital-insurer bargaining in which the combined
disagreement payoff of two hospitals negotiating jointly is assumed to be
equal to the sum of the disagreement payoffs of each hospital when they
negotiate separately. \ Under this assumption, the terms in the second part
of equation \ref{Delta T} cancel out, and the effect of joint negotiation
depends only on the concavity of the insurer's value function.}} \ This
paper's primary contributions are to introduce this externality as a
potential feature of bilateral bargaining environments and to identify
conditions under which the externality is likely to be present. }

{\normalsize The expression for $\Delta T$ above is very general, relying
only on the assumption of Nash bargaining with lump sum transfers and
constant $\beta $.\ To investigate the conditions that determine the sign
and magnitude of $\Delta T$, I introduce the following specification for the
players' value functions: \ 
\begin{equation}
V_{j}\left( \Omega \right) =\pi _{j}N_{j}\left( \Omega \right) ,\text{ }%
j=A,B,I  \label{Value function}
\end{equation}
}

{\normalsize where $\pi _{j}$ is a measure of the profit per consumer for
player $j$, and $N_{j}\left( \Omega \right) $ represents a number of
consumers as a function of network $\Omega $. \ For intermediaries, $\pi _{I}
$ is interpreted as the price intermediary $I$ charges consumers for access
to its products (less any unit costs), and $N_{I}\left( \Omega \right) $ is
the number of consumers choosing intermediary $I$ when $I$'s network is $%
\Omega $. Intermediary $I$ is assumed here to set $\pi _{I}$ simultaneously
with the negotiations over network participation. As a result, $\pi _{I}$
does not vary with $\Omega $. (I consider relaxing this assumption below).
Also, for the purposes of the analysis with lump sum transfers, $\pi _{I}$
is the same whether the suppliers negotiate separately or jointly. }

{\normalsize For suppliers, the expression for $\Delta T$ \ depends only on
suppliers' payoffs in the event of disagreement. \ For the purposes of this
analysis, then, $\pi _{j}$ measures the profit per consumer that supplier $j$
earns from consumers that choose intermediaries\ other than $I$.\ This could
reflect profits from prices the supplier charges directly to consumers, or
profits from other sources of revenue that vary with the number of
consumers. For example, if the supplier is a video programming distributor, $%
\pi _{j}$ may reflect advertising revenues. (In the next section, I consider
a variant of this model in which a supplier's profit per consumer is the
outcome of the negotiation with the intermediary.) If not all of the
consumers with access to supplier $j$ choose to consume its products, $\pi
_{j}$ should be interpreted as the expected profit per consumer; that is,
the profit earned from each consumer that chooses supplier $j$'s products
multiplied by the probability that a consumer with access to supplier $j$
does so. For suppliers, $N_{j}\left( \Omega \right) $ measures the total
number of consumers that have access to supplier $j$ through intermediaries
other than $I$, given that $I$'s network is $\Omega $. $N_{j}\left( \Omega
\right) $ varies with $I$'s network because a change in the network may
induce consumers to choose a different intermediary. I will restrict
attention to scenarios in which any consumers who leave intermediary $I$ in
response to a change in network will switch to another intermediary that
offers both $A$ and $B$ in equilibrium. This condition will hold, for
example, if in equilibrium every intermediary offers a comprehensive network
that includes all suppliers. }

{\normalsize Under general conditions on consumer preferences \.{(}in
particular, broader networks are always preferred to narrower), the set of
consumers that choose intermediary $I$ in equilibrium, $N_{I}\left( \Omega
^{\ast }\right) $, can be divided into five mutually exclusive and
exhaustive groups as follows: }

{\normalsize $N_{0}=$ consumers who would choose $I$ even if both $A$ and $B$
are excluded from $I$'s network. }

{\normalsize $N_{1}=$ consumers who would choose $I$ if and only if $A$ is
included in $I$'s network. }

{\normalsize $N_{2}=$ consumers who would choose $I$ if and only if $B$ is
included in $I$'s network. }

{\normalsize $N_{3}=$ consumers who would choose $I$ if and only if at least
one of $A$ or $B$ is included in $I$'s network (I refer to these consumers
as "substitute-type"). }

{\normalsize $N_{4}=$ consumers who would choose $I$ if and only if both $A$ 
\textit{and} $B$ are included in $I$'s network ("complement-type"). }

{\normalsize The consumers in group 3 are called "substitute-type" because
their choice of intermediary is consistent with preferences that treat
access to $A$ as a substitute for access to $B$. \ It is not necessary that
such consumers view $A$'s and $B$'s products as substitutes at the time of
consumption. Even if $A$ and $B$ produce distinct types of product (e.g.
physician services and hospital services), there may be some consumers where
the exclusion of one or the other from a network is not sufficient to induce
the consumer to switch intermediaries, but the exclusion of both is.
Similarly, the consumers in group 4 are called "complement-type" based on
how their choice of intermediary responds to the exclusion of $A$ or $B$,
which need not correspond to complementarity between $A$'s and $B$'s
products in the usual sense. }

{\normalsize Letting $M_{j}$ denote the equilibrium number of consumers at
other intermediaries that offer supplier $j$, we can express the number of
consumers for each network configuration as follows: }

{\normalsize \bigskip }

{\normalsize 
\begin{tabular}{lll}
$I$'s Network ($\Omega $) & $N_{I}\left( \Omega \right) $ & $N_{j}\left(
\Omega \right) ,$ $j=A,B$ \\ \hline
$\Omega ^{\ast }$ & $N_{0}+N_{1}+N_{2}+N_{3}+N_{4}$ & $M_{j}$ \\ 
$\Omega ^{-A}$ & $N_{0}+N_{2}+N_{3}$ & $M_{j}+N_{1}+N_{4}$ \\ 
$\Omega ^{-B}$ & $N_{0}+N_{1}+N_{3}$ & $M_{j}+N_{2}+N_{4}$ \\ 
$\Omega ^{-A-B}$ & $N_{0}$ & $M_{j}+N_{1}+N_{2}+N_{3}+N_{4}$%
\end{tabular}
}

{\normalsize \bigskip }

{\normalsize Substituting the appropriate values for the players' value
functions into equation \ref{Delta T - General} yields the following
expression for $\Delta T$ :%
\begin{equation}
\Delta T=\beta \pi _{I}\left( N_{3}-N_{4}\right) +(1-\beta )\left[ \pi
_{A}\left( N_{2}+N_{3}\right) +\pi _{B}\left( N_{1}+N_{3}\right) \right]
\label{Delta T}
\end{equation}
}

{\normalsize This expression identifies the key factors that determine the
effect on joint negotiation on bargaining power. \ The first term
(multiplied by $\beta $) reflects the component of $\Delta T$ based on how
joint contracting changes the insurer's disagreement payoff. Joint
contracting will tend to benefit the suppliers if the number of
substitute-type consumers is larger than the number of complement-type
consumers (i.e., $N_{3}>N_{4}$). A large number of substitute-type consumers
means that the two suppliers can threaten to drive more consumers away from
the intermediary when they negotiate jointly than they can when they
negotiate separately. \ The presence of complement-type consumers offsets
this effect, because each supplier can individually threaten to divert these
consumers away from the intermediary. Under separate negotiation, the value
of these consumers to the intermediary is reflected in each of transfers to
the two suppliers, while under joint negotiation it is reflected only in a
single joint transfer. As a result, if the number of complement-type
consumers is large, joint negotiation will tend to reduce the transfer. This
is analogous to Cournot's (1838) result that joint ownership of
complementary goods reduces prices. In general, there may be some consumers
of both types. The net effect of this component of $\Delta T$ will depend on
which group of consumers is larger. }

{\normalsize The second term in equation \ref{Delta T} (multiplied by $%
(1-\beta )$) reflects the component of $\Delta T$ based on how joint
contracting changes the suppliers' disagreement payoffs. Consider the first
part of this term, $\pi _{A}\left( N_{2}+N_{3}\right) $. The sum of $N_{2}$
and $N_{3}$ is the number of consumers that stay with intermediary $I$ when $%
A$ alone is excluded from the network, but that switch to a different
intermediary when both $A$ and $B$ are excluded. Under separate negotiation, 
$A$ will lose access to these consumers in the event of disagreement with $I$%
. \ By contrast, with joint negotiation, $A$ will still have access to these
consumers in the event of disagreement with $I$, through a different
intermediary. Joint negotiation thus prevents $A$ from losing access to
those consumers in the event of disagreement. As a result, the effect of
joint negotiation on $A$'s disagreement payoff is equal to the value that $A$
receives from having access to those consumers, $\pi _{A}\left(
N_{2}+N_{3}\right) $. In other words, this is the value of the externality
that $B$'s participation in $I$'s network has on $A$'s disagreement payoff.
A similar logic applies to $B$, reflected in the term $\pi _{B}\left(
N_{1}+N_{3}\right) $. Because the effect of joint negotiation is that each
supplier can expect to "recapture" some consumers that it otherwise would
have lost access to in the event of disagreement, I will refer to this
component of the effect as the "recapture effect."\footnote{{\normalsize I
am indebted to Patrick Greenlee for suggesting this terminology.}} \ }

{\normalsize A key contribution of this paper is to show how joint
negotiation can increase suppliers' bargaining power even if the
intermediary's value function is not concave, i.e. even if the loss to the
intermediary of both suppliers is no worse than the sum of the losses of
each individually. \ Equation \ref{Delta T} illustrates this point clearly: $%
\Delta T$ may be positive even if $N_{3}\leq N_{4}$, as long as there are
consumers whose choice of intermediary depends on access to $B$ (or $A$) but
who are also valuable to $A$ (or $B$), so that $\pi _{A}\left(
N_{2}+N_{3}\right) >0$ (or $\pi _{B}\left( N_{1}+N_{3}\right) >0$). }

\subsection{\protect\normalsize Negotiated unit prices}

{\normalsize In the analysis above, I assumed that suppliers and
intermediaries negotiated over a lump sum transfer. \ At the same time, an
important driver of the result was the assumption that suppliers benefit
from incremental unit sales, i.e. $\pi _{j}>0$ for $j=A,B$. \ In some
contexts, this may be a reasonable assumption; for example, suppliers may
charge a price directly to consumers. In other contexts, it may be more
appropriate to model suppliers and intermediaries as negotiating over a unit
price. As a result, the suppliers' profit per consumer, $\pi _{j}$, will
depend on the outcome of the negotiation. \ In this section, I derive an
expression for the effect of joint negotiation on the total payment from an
intermediary to the jointly negotiating suppliers, under the assumption that
intermediaries and suppliers negotiate over unit prices rather than over a
lump sum transfer. \ (A similar expression can be derived for the case of
negotiated \textit{per-consumer }prices, where the total payment to
suppliers is proportional to the total number of consumers choosing the
intermediary.) }

{\normalsize For this analysis, I focus on the \textit{first-order }effect
of joint negotiation. That is, I consider how the total payment from an
intermediary to the jointly negotiating suppliers would change as a result
of joint negotiation, under the assumption that all prices for transactions
involving other intermediaries or other suppliers remain fixed at the level
of the equilibrium with separate negotiations, as does the per-unit price
the intermediary charges directly to consumers. As a result, for the moment
I do not solve for a new equilibrium with joint negotiation. \ Doing so
would require accounting for the reactions of other prices, which I will
defer for a later section of the paper when I consider a fully specified
model of consumer demand. The advantage of focusing on the first-order
effect is that it substantially simplifies the analysis and allows for a
clear illustration of the primary mechanism driving the effects of joint
negotiation. Also, because prices are usually strategic complements, higher
order effects are likely to reinforce the effects identified in this
section. }

{\normalsize As described in the previous section, the consumers that choose
intermediary $I$ in equilibrium are divided into five groups, indexed by $%
k\in \left\{ 0,\ldots ,4\right\} .$ Let $\lambda _{jk}\left( \Omega \right) $
denote the average number of units of supplier $j$'s products consumed by a
consumer in group $k$, given that intermediary $I$'s network is $\Omega $.
The usage level depends on the network because if supplier $j$ is excluded
from $I$'s network, any consumers that remain with intermediary $I$ will
reduce their usage of supplier $j$'s products to zero, and will generally
increase their usage of other (substitute) suppliers' products. Let $r_{Ij}$
denote the negotiated unit price paid by intermediary $I$ to supplier $j$. \
Let $\overline{r}_{jk}$ denote the average unit price paid to supplier $j$
by intermediaries other than $j$ for consumers in group $k$ in the event
that those consumers switch to another intermediary. \ In general, this
average price may vary across consumer groups because different consumers
may switch to different intermediaries and different intermediaries may pay
different prices. For simplicity, however, I will treat $\overline{r}_{jk}$
as constant across consumer groups, and so suppress the subscript $k$. Also
for simplicity, I will assume that suppliers have zero costs. Finally, $\pi
_{I}$ denotes the profit per consumer earned by intermediary $I$, excluding
the costs of any payments to suppliers. For example, $\pi _{I}$ could
represent the price charged by the intermediary. I assume here that $\pi _{I}
$ is determined simultaneously with bargaining over network participation,
so that the value of $\pi _{I}$ is taken as fixed during the negotiations.
(In the next section, I consider how the results would vary if the
intermediary expects to be able to adjust $\pi _{I}$ in the event of
disagreement). }

{\normalsize The payoffs for intermediary $I$ as a function of its network
are given by the following expressions:%
\begin{equation*}
V_{I}(\Omega ^{\ast })=\underset{k=0}{\overset{4}{\sum }}N_{k}\left( \pi
_{I}-\underset{j}{\sum }r_{Ij}\lambda _{jk}\left( \Omega ^{\ast }\right)
\right)
\end{equation*}%
\begin{equation*}
V_{I}(\Omega ^{-A})=\underset{k=0,2,3}{\sum }N_{k}\left( \pi _{I}-\underset{%
j\neq A}{\sum }r_{Ij}\lambda _{jk}\left( \Omega ^{-A}\right) \right)
\end{equation*}%
\begin{equation*}
V_{I}(\Omega ^{-B})=\underset{k=0,1,3}{\sum }N_{k}\left( \pi _{I}-\underset{%
j\neq B}{\sum }r_{Ij}\lambda _{jk}\left( \Omega ^{-B}\right) \right)
\end{equation*}%
\begin{equation*}
V_{I}(\Omega ^{-A-B})=N_{0}\left( \pi _{I}-\underset{j\neq A,B}{\sum }%
r_{Ij}\lambda _{j0}\left( \Omega ^{-A-B}\right) \right)
\end{equation*}
}

{\normalsize The payoffs for suppliers $A$ and $B$ are:%
\begin{equation*}
V_{j}(\Omega ^{\ast })=r_{Ij}\underset{k=0}{\overset{4}{\sum }}N_{k}\lambda
_{jk}\left( \Omega ^{\ast }\right) ,\text{ }j=A,B
\end{equation*}%
\begin{equation*}
V_{A}(\Omega ^{-A})=\overline{r}_{A}\underset{k=1,4}{\sum }N_{k}\lambda
_{Ak}\left( \Omega ^{-A}\right)
\end{equation*}%
\begin{equation*}
V_{B}(\Omega ^{-B})=\overline{r}_{B}\underset{k=2,4}{\sum }N_{k}\lambda
_{Bk}\left( \Omega ^{-B}\right)
\end{equation*}%
\begin{equation*}
V_{j}(\Omega ^{-A-B})=\overline{r}_{j}\underset{k=1}{\overset{4}{\sum }}%
N_{k}\lambda _{jk}\left( \Omega ^{-A-B}\right) ,\text{ }j=A,B
\end{equation*}
}

{\normalsize Under separate negotiation, the price $r_{IA}$ that solves the
Nash bargaining problem will satisfy the following expression for the total
payment from $I$ to $A$:%
\begin{eqnarray}
T_{A} &\equiv &r_{IA}\underset{k=0}{\overset{4}{\sum }}N_{k}\lambda
_{Ak}\left( \Omega ^{\ast }\right)  \label{T A} \\
&=&\beta \left[ \underset{k=1,4}{\sum }N_{k}\left( \pi _{I}-\underset{j\neq A%
}{\sum }r_{Ij}\lambda _{jk}\left( \Omega ^{\ast }\right) \right) +\underset{%
k=0,2,3}{\sum }N_{k}\left( \underset{j\neq A}{\sum }r_{Ij}\left( \lambda
_{jk}\left( \Omega ^{-A}\right) -\lambda _{jk}\left( \Omega ^{\ast }\right)
\right) \right) \right]  \notag \\
&&+\left( 1-\beta \right) \text{ }\overline{r}_{A}\underset{k=1,4}{\sum }%
N_{k}\lambda _{Ak}\left( \Omega ^{-A}\right)  \notag
\end{eqnarray}
}

{\normalsize The first term in the square brackets reflects the profits
intermediary $I$ earns on the consumers it would lose if $A$ were excluded
from its network. \ The second term in the square brackets represents the
additional payments to other suppliers that intermediary $I$ would have to
pay if $A$ were excluded, as a result of its retained consumers increasing
usage of other suppliers when they lose access to $A$. The final term in the
expression is the disagreement payoff for supplier $A$: the earnings from
consumers that would switch to alternative intermediaries in order to retain
access to $A$. }

{\normalsize Similarly, $r_{IB}$ under separate negotiation will satisfy: }

{\normalsize 
\begin{eqnarray}
T_{B} &\equiv &r_{IB}\underset{k=0}{\overset{4}{\sum }}N_{k}\lambda
_{Bk}\left( \Omega ^{\ast }\right)  \label{T B} \\
&=&\beta \left[ \underset{k=2,4}{\sum }N_{k}\left( \pi _{I}-\underset{j\neq B%
}{\sum }r_{Ij}\lambda _{jk}\left( \Omega ^{\ast }\right) \right) +\underset{%
k=0,1,3}{\sum }N_{k}\left( \underset{j\neq B}{\sum }r_{Ij}\left( \lambda
_{jk}\left( \Omega ^{-B}\right) -\lambda _{jk}\left( \Omega ^{\ast }\right)
\right) \right) \right]  \notag \\
&&+\left( 1-\beta \right) \text{ }\overline{r}_{B}\underset{k=2,4}{\sum }%
N_{k}\lambda _{Bk}\left( \Omega ^{-A}\right)  \notag
\end{eqnarray}
}

{\normalsize Under joint negotiation, the total payment from intermediary $I$
to $A$ and $B$ together will be: }

{\normalsize 
\begin{eqnarray}
T_{A+B} &\equiv &\underset{k=0}{\overset{4}{\sum }}N_{k}\left( r_{IA}\lambda
_{Ak}\left( \Omega ^{\ast }\right) +r_{IB}\lambda _{Bk}\left( \Omega ^{\ast
}\right) \right)  \label{T A+B} \\
&=&\beta \left[ \underset{k=1}{\overset{4}{\sum }}N_{k}\left( \pi _{I}-%
\underset{j\neq A,B}{\sum }r_{Ij}\lambda _{jk}\left( \Omega ^{\ast }\right)
\right) +N_{0}\left( \underset{j\neq A,B}{\sum }r_{Ij}\left( \lambda
_{j0}\left( \Omega ^{-A-B}\right) -\lambda _{j0}\left( \Omega ^{\ast
}\right) \right) \right) \right]  \notag \\
&&+\left( 1-\beta \right) \text{ }\underset{k=1}{\overset{4}{\sum }}%
N_{k}\left( \overline{r}_{A}\lambda _{Ak}\left( \Omega ^{-A-B}\right) +%
\overline{r}_{B}\lambda _{Bk}\left( \Omega ^{-A-B}\right) \right)  \notag
\end{eqnarray}
}

{\normalsize While this expression must be satisfied in equilibrium, a
limitation of the Nash bargaining solution for jointly negotiated per-unit
prices is that the framework does not yield unique solutions for $r_{IA}$
and $r_{IB}$. For any set of values for the other prices in the model, there
is a continuum of pairs of $r_{IA}$ and $r_{IB}$ that satisfy equation \ref%
{T A+B}. \ In other words, the model results in one equation with two
unknowns. While one might apply some equilibrium refinement to derive an
explicit solution for the individual prices, I will focus instead on what we
can learn from the analysis without going beyond the implications of the
Nash bargaining solution. (As I discuss further below, imposing unrealistic
restrictions to obtain a unique solution is a pitfall that can result in
misleading conclusions). }

{\normalsize The equilibrium values for $r_{Ij}$, $\overline{r}_{j}$, and $%
\pi _{I}$ will vary depending on whether negotiation is separate or joint.
Accounting for these higher order effects would require more structure on
the model. However, as noted above, I focus here only on the first order
effects of joint negotiation. Accordingly, I define $r_{IA}^{Joint}$ and $%
r_{IB}^{Joint} $ to be a pair of values for $r_{IA}$ and $r_{IB}$ that
satisfy equation \ref{T A+B} under the assumption that all other prices are
equal to their equilibrium level under separate negotiation, and $%
T_{A+B}^{Joint}$ to denote $T_{A+B}$\ evaluated at these values.\ I use $%
r_{Ij}^{\ast }$, $\overline{r}_{j}^{\ast }$, and $\pi _{I}^{\ast }$ to
denote the initial equilibrium values, and $T_{j}^{\ast }$ the corresponding
values of $T_{j}$, for $j=A,B$. \ The first order effect of joint
negotiation on the total payment from intermediary $I$ to $A$ and $B$ is
then defined as: 
\begin{equation*}
\widehat{\Delta T}\equiv T_{A+B}^{Joint}-\left( T_{A}^{\ast }+T_{B}^{\ast
}\right) =\underset{k=0}{\overset{4}{\sum }}N_{k}\left[ \left(
r_{IA}^{Joint}-r_{IA}^{\ast }\right) \lambda _{Ak}\left( \Omega ^{\ast
}\right) +\left( r_{IB}^{Joint}-r_{IB}^{\ast }\right) \lambda _{Bk}\left(
\Omega ^{\ast }\right) \right]
\end{equation*}
}

{\normalsize In order to obtain a parsimonious expression for $\widehat{%
\Delta T}$, I impose two simplifying assumptions:%
\begin{equation*}
\text{Assumption (1): }\underset{j}{\sum }\lambda _{jk}\left( \Omega \right)
=\underset{j}{\sum }\lambda _{jk}\left( \Omega ^{\prime }\right) \text{ for
all }\Omega ,\Omega ^{\prime },k
\end{equation*}%
\begin{equation*}
\text{Assumption (2): }r_{Ij}^{\ast }=r_{Ij^{\prime }}^{\ast }\text{ for all 
}j,j^{\prime }s.t.\lambda _{jk}\left( \Omega \right) >\lambda _{jk}\left(
\Omega ^{-j^{\prime }}\right) \text{ for any }k
\end{equation*}
}

{\normalsize Assumption (1) states that the total number of units consumed
by any group of consumers is invariant to network changes. If a supplier
becomes unavailable to a group of consumers as a result of its exclusion
from a network, those consumers will divert their consumption to other
suppliers, with no change in the total number of units consumed. Note that I
do not assume that there is any diversion from $A$ to $B$ if $A$ is excluded
from the network, or vice versa. For the purposes of this analysis, $A$ and $%
B$ may or may not be substitutes for one another. Whether the assumption of
invariant total consumption is appropriate in an applied setting will
generally depend on the application. For example, it is often assumed that
patients' overall demand for health care services does not vary with the
breadth of the insurance network. Here, the assumption allows me to abstract
away from any changes in overall consumption, and focus instead on the
effects of consumers switching to alternative suppliers or intermediaries. }

{\normalsize Assumption (2) assumes that the equilibrium price for two
different suppliers will be the same if those suppliers are substitutes for
one another. This is certainly a restrictive assumption; in general,
different suppliers may have different costs or demands, so that their
equilibrium prices will be different. The effect on an intermediary of
excluding a supplier from its network will then depend on whether the
retained consumers are likely to substitute to a more or less expensive
alternative. By imposing this assumption, I avoid cluttering the analysis
with terms that reflect these differences. Note that I do not assume
uniformity for the prices of products that are not substitutes for one
another. \ In particular, if consumers do not view $A$ and $B$ as
substitutes, they may receive different per-unit payments from
intermediaries. }

{\normalsize Under these two assumptions, the first order effect of joint
negotiation is given by the following expression:%
\begin{eqnarray}
\widehat{\Delta T} &=&\beta \left[ N_{3}\left( \pi _{I}^{\ast }-\underset{j}{%
\sum }r_{Ij}^{\ast }\lambda _{j3}\left( \Omega ^{\ast }\right) \right)
-N_{4}\left( \pi _{I}^{\ast }-\underset{j}{\sum }r_{Ij}^{\ast }\lambda
_{j4}\left( \Omega ^{\ast }\right) \right) \right]  \label{Delta T hat} \\
&&+\left( 1-\beta \right) \left[ \overline{r}_{A}^{\ast }\left( \underset{%
k=2,3}{\sum }N_{k}\lambda _{Ak}\left( \Omega ^{-A-B}\right) \right) +%
\overline{r}_{B}^{\ast }\left( \underset{k=1,3}{\sum }N_{k}\lambda
_{Bk}\left( \Omega ^{-A-B}\right) \right) \right] \text{ }  \notag
\end{eqnarray}
}

{\normalsize Not surprisingly, equation \ref{Delta T hat} is quite similar
to equation \ref{Delta T}, the expression for $\Delta T$ with lump sum
transfers. \ The similarity between these two expressions shows that the
basic mechanism driving the effects of joint negotiation on bargaining power
is the same whether we model negotiation over lump sum transfers or unit
prices. The two effects may of course be different to some extent, but the
underlying intuition is the same: The effect of joint negotiation acting
through the intermediary's value function depends on the value to the
intermediary of substitute-type consumers relative to complement-type
consumers (i.e. the concavity or convexity of the intermediary's value
function); while the effect acting through the suppliers' value functions
depends on the value to each supplier of the consumers it would lose if
excluded individually but would recapture if excluded together with the
other supplier (i.e. the recapture effect). }

\subsection{\protect\normalsize Intermediary price adjustment after
disagreement}

{\normalsize A key assumption in the analysis above is that the
intermediary's profit per enrollee, $\pi _{I}$, is determined simultaneously
with the negotiation over network participation. As a result, $\pi _{I}$ is
treated as fixed across agreement and disagreement outcomes. In this
section, I consider the implications of an alternative assumption, that each
intermediary expects to be able to adjust its price in the event of
disagreement. \ Returning to the model with lump sum transfers described
above, I generalize the expression for the value function for intermediary $I
$ as follows:%
\begin{equation*}
V_{I}\left( \Omega \right) =\pi _{I}\left( \Omega \right) N_{I}\left( \Omega
\right)
\end{equation*}%
\bigskip \qquad \qquad\ }

{\normalsize For any set of consumer preferences, the values of $N_{I}\left(
\Omega \right) $ will generally be different if $\pi _{I}$ is permitted to
vary, because the number of consumers that switch intermediaries in response
to a change in network will be influenced by the corresponding change in the
intermediary's price. In this context, then, the function $N_{I}\left(
\Omega \right) $ represents the number of consumers intermediary $I$ expects
to have if its network is $\Omega $, after taking into account consumers'
response to any change in $\pi _{I}$. Formally, if $\widetilde{N}_{I}\left(
\Omega ,\pi _{I}\right) $ represents intermediary $I$'s consumers as a
function of both its network and its price, then $N_{I}\left( \Omega \right)
\equiv $ $\widetilde{N}_{I}\left( \Omega ,\pi _{I}\left( \Omega \right)
\right) $. Similarly, in the value functions for suppliers $A$ and $B$,
given by equation \ref{Value function} above, $N_{j}\left( \Omega \right) $
represents the number of consumers after accounting for changes in $\pi _{I}$%
. The values $N_{0},\ldots ,N_{4}$ will be defined as in the previous
sections, taking into account consumers' responses to changes in the
network. }

{\normalsize As before, the values of $\pi _{A}$ and $\pi _{B}$ are taken as
fixed. Recall that these values represent the per-consumer profits that the
suppliers earn from intermediaries other than $I$. \ Because the Nash
bargaining solution takes the outcomes of all other negotiations as fixed,
and because $\pi _{A}$ and $\pi _{B}$ may be interpreted as proxies for the
outcomes of negotiations with other intermediaries, allowing the suppliers
to adjust $\pi _{A}$ and $\pi _{B}$ in the event of disagreement would seem
to be in tension with the underlying bargaining framework. }

{\normalsize Also as before, the values of $\pi _{I}\left( \Omega \right) $
and $N_{I}\left( \Omega \right) $ do not depend on whether suppliers
negotiate jointly or separately. }

{\normalsize Under these conditions, the effect of joint negotiation on the
transfer is given by the following expression:%
\begin{eqnarray}
\Delta T &=&\beta \left\{ 
\begin{array}{c}
\pi _{I}\left( \Omega ^{\ast }\right) \left( N_{3}-N_{4}\right) \\ 
+\left[ \pi _{I}\left( \Omega ^{-B}\right) -\pi _{I}\left( \Omega
^{-A-B}\right) -\left( \pi _{I}\left( \Omega ^{\ast }\right) -\pi _{I}\left(
\Omega ^{-A}\right) \right) \right] N_{0} \\ 
-\left[ \pi _{I}\left( \Omega ^{\ast }\right) -\pi _{I}\left( \Omega
^{-A}\right) \right] \left( N_{2}+N_{3}\right) -\left[ \pi _{I}\left( \Omega
^{\ast }\right) -\pi _{I}\left( \Omega ^{-B}\right) \right] \left(
N_{1}+N_{3}\right)%
\end{array}%
\right\}  \label{Delta T - adj} \\
&&+(1-\beta )\left[ \pi _{A}\left( N_{2}+N_{3}\right) +\pi _{B}\left(
N_{1}+N_{3}\right) \right]  \notag
\end{eqnarray}
}

{\normalsize The first line of this expression is a term that appears in
equation \ref{Delta T}: the effect of joint negotiation depends on the
relative numbers of substitute-type and complement-type consumers. The
second line is also quite intuitive: For the $N_{0}$ consumers that $I$ will
retain even in the event of disagreement with both $A$ and $B$, the effect
of joint negotiation depends on whether the exclusion of supplier $A$ from $I
$'s network causes $\pi _{I}$ to fall by more (or less) if supplier $B$ is
also excluded.\ These effects are straightforward examples of the general
principle that the effect of joint negotiation depends on the concavity (or
convexity) of the insurer's value function. }

{\normalsize The third line of equation \ref{Delta T - adj} is of particular
interest. It shows that joint negotiation tends to reduce the transfer to
the suppliers, to the extent that the exclusion of $A$ from $I$'s network
reduces the profits $I$ earns from the consumers it retains when $A$ (or $B$%
) alone is excluded from its network but loses when both $A$ and $B$ are
excluded, $N_{2}+N_{3}$ (or $N_{1}+N_{3}$). The intuition is that adding
supplier $A$ to the network has two effects on the intermediary's profits:
first, the intermediary gains consumers; second, the intermediary earns more
profit from consumers that it would have retained even without supplier $A$.
This second category includes consumers that the intermediary would lose if
it did not have $B$ in network. Thus, a portion of the profits the
intermediary earns from these consumers requires the intermediary to include 
\textit{both }$A$ and $B$ in its network. In this way, these profits are
similar to the profits earned from complement-type consumers. Joint
negotiation allows the intermediary to avoid negotiating over this portion
of the surplus \textit{twice}, and thus reduces the total amount the
intermediary pays in equilibrium. This effect will tend to offset the effect
of allowing $A$ and $B$ to internalize the externality that each one's
participation in the network has on the other (i.e., the recapture effect,
reflected in the fourth line of equation \ref{Delta T - adj}). }

{\normalsize Because these two offsetting effects work through precisely the
same groups of consumers, it is natural to ask whether there is a
relationship between their magnitudes. In particular, if $\beta \left( \pi
_{I}\left( \Omega ^{\ast }\right) -\pi _{I}\left( \Omega ^{-j}\right)
\right) =(1-\beta )\pi _{j}$ for $j=A,B$, then the third and fourth lines of
equation \ref{Delta T - adj} will exactly cancel each other out. In other
words, the relevant comparison is between the incremental effect on $I$'s
per-consumer profits of adding $A$ (or $B$) to its network and the
per-consumer profits that $A$ (or $B$) earns from other intermediaries. It
may seem intuitive that these amounts should be closely related. In
particular, suppose that adding supplier $j$ to any intermediary's network
creates $S_{j}$ of gross surplus per consumer available to be captured by
the intermediary and the supplier. If that surplus is divided between
supplier $j$ and the intermediary according to their relative bargaining
effectiveness, then we might intuitively expect: $\pi _{j}=\beta S_{j}$ and $%
\pi _{I}\left( \Omega ^{\ast }\right) -\pi _{I}\left( \Omega ^{-j}\right)
=(1-\beta )S_{j}$.\ Under these conditions, the equality above will hold,
and the two effects will exactly cancel out. }

{\normalsize It turns out that this intuition does not hold generally. In
the remainder of the paper, I consider two fully specified models that are
special cases of the general framework described above. In both models, I
find that allowing the intermediary to adjust its price in the event of
disagreement will partially, but not fully, offset the recapture effect. I
turn now to these specific models. }

\section{\protect\normalsize Gal-Or's (1999) model}

{\normalsize Gal-Or (1999) introduced a model to study the effects of
mergers between hospitals and physician practices on the joint profits of
the merged entity. In this model, bilateral bargaining between health care
providers and insurers determines the prices of health care services, with
the equilibrium prices satisfying the Nash bargaining solution. The model is
thus a special case of the general framework outlined above, specifically
the variant with negotiated unit prices.\ In this section, I examine the
implications of this model in detail, for four reasons. First, the analysis
demonstrates that the recapture effect identified above arises in a fully
specified model of consumer demand. Second, while the general framework
above allowed me to consider only the first-order effects of joint
negotiation in the model with negotiated unit prices, using a full model
allows me to derive an expression (which does not appear in Gal-Or's paper)
for the full equilibrium effects of joint negotiation on the payment to the
combined entity, distinguishing the first-order and higher order components
of the effect. Third, I address a shortcoming in Gal-Or's paper: One of the
central conclusions of that paper is that joint negotiation by a hospital
and a physician group may fail to increase the combined entity's bargaining
power if the degree of competitiveness of the two markets is sufficiently
different. I show that this conclusion relies on an unstated and arbitrary
equilibrium refinement assumption. Under a wide range of alternative
assumptions, Gal-Or's model implies that joint negotiation increases the
combined entity's bargaining power for any parameter values. Finally,
modifying Gal-Or's model to permit the insurer to adjust its premium in the
event of disagreement demonstrates that joint negotiation continues to
increase bargaining power under this variant of the model. }

\subsection{\protect\normalsize Separate negotiation}

{\normalsize There are two insurers, $m$ hospitals, and $n$ physicians. Each
insurer-provider pair engages in simultaneous negotiation over whether the
insurer will include the provider in its network. If the parties to the
negotiation reach agreement, they agree to a unit price that the insurer
will pay to the provider for each patient that receives treatment from that
provider. The agreed-upon prices satisfy the symmetric Nash bargaining
solution ($\beta =%
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
$). Simultaneously with these negotiations, each insurer sets its premium to
maximize its profits. After the networks and premia have been determined,
each consumer chooses one of the two insurers. The population of consumers
has unit mass and preferences across insurers represented by a uniform
distribution around a unit circle with a transportation cost $M$ and with
the insurers located on opposite sides of the circle. After choosing an
insurer, a consumer becomes sick with probability $\theta $, at which point
she chooses the services of one hospital and one physician from among the
providers participating in her insurer's network. Preferences across
hospitals and physicians are represented by uniform distributions around two
respective unit circles, with transportation costs $t$ (for hospitals) and $%
s $ (for physicians) and with the providers in the insurer's network
distributed evenly around the two circles. In choosing among insurers,
consumers take into account not only the insurer's premium and its location
on the unit circle relative to the consumer, but also the expected
transportation cost that the consumer will bear if she becomes sick. \ In a
symmetric equilibrium with separate negotiations, the hospital price is
denoted by $y^{\ast }$, the physician price by $z^{\ast }$, and the
insurance premium by $F^{\ast }$. }

{\normalsize This model is a special case of the model with negotiated unit
prices described above. If we treat supplier $A$ as a hospital and supplier $%
B$ as a physician, the expressions derived above can thus be used to
evaluate Gal-Or's model. \ In equilibrium, each insurer has one-half of the
population of consumers, i.e. $N_{I}\left( \Omega ^{\ast }\right) =%
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
$. In the event of disagreement with one hospital, the insurer loses $\frac{%
\theta t}{2m^{2}M}$ members; disagreement with one physician results in a
loss of $\frac{\theta s}{2n^{2}M}$ members; and disagreement with one of
each results in a loss of $\frac{\theta t}{2m^{2}M}+\frac{\theta s}{2n^{2}M}$
members. These figures imply the following expressions for $N_{0},\ldots
N_{4}$: }

{\normalsize $\bigskip $ }

{\normalsize $N_{0}=%
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
-\left( \frac{\theta t}{2m^{2}M}+\frac{\theta s}{2n^{2}M}\right) $ }

{\normalsize $N_{3}=N_{4}=\min \left( \frac{\theta t}{2m^{2}M},\frac{\theta s%
}{2n^{2}M}\right) $ }

{\normalsize $N_{1}+N_{3}=N_{1}+N_{4}=\frac{\theta t}{2m^{2}M}$ }

{\normalsize $N_{2}+N_{3}=N_{2}+N_{4}=\frac{\theta s}{2n^{2}M}$ }

{\normalsize \bigskip }

{\normalsize To ensure that an equilibrium exists, the parameters are
assumed to be such that $N_{0}>0$. That is, the insurer must retain at least
some portion of its members if it excludes both a hospital and a physician
from its network.\footnote{{\normalsize This assumption is equivalent to the
restriction in equation (16) in Gal-Or (1999).}}%
\begin{equation*}
\text{Assumption (3): \ \ }\frac{\theta t}{2m^{2}M}+\frac{\theta s}{2n^{2}M}<%
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
\end{equation*}
}

{\normalsize The average number of treatments that a consumer receives from
a given hospital is $\frac{\theta }{m}$ if the consumer's insurance network
includes all $m$ hospitals, and $\frac{\theta }{m-1}$ if one hospital is
excluded from the network (the probability the consumer becomes sick
multiplied by the probability the consumer chooses the hospital). Similar
expressions ($\frac{\theta }{n}$ and $\frac{\theta }{n-1}$) represent the
average number of treatments a consumer receives from a given physician.
These expressions correspond to the terms $\lambda _{jk}\left( \Omega
\right) $ in the general framework. Note that assumption (1) is therefore
satisfied in this model. \ Also, in the symmetric equilibrium, providers'
unit prices are equal to $r_{Ij}^{\ast }=\overline{r}_{j}^{\ast }=y^{\ast }$
if $j$ is a hospital, $z^{\ast }$ if $j$ is a physician, so that assumption
(2) is satisfied as well.\ As in the general framework, providers are
assumed for simplicity to have zero marginal cost, so $y^{\ast }$ and $%
z^{\ast }$ represent providers' profits per treatment. Finally, the
equilibrium insurance premium is given by the following expression:%
\begin{equation}
\pi _{I}^{\ast }=F^{\ast }=\frac{M}{2}+\theta \left( y^{\ast }+z^{\ast
}\right)  \label{F*}
\end{equation}%
Thus, each insurer's equilibrium profit per consumer after payments to
providers is $\frac{M}{2}$. }

{\normalsize Substituting these expressions into equations \ref{T A} and \ref%
{T B} above yields the expressions given in Gal-Or's paper for the
equilibrium values of $y^{\ast }$ and $z^{\ast }$ under separate
negotiation. \ Alternatively, these expressions can be manipulated into the
following forms:%
\begin{equation}
\frac{\theta }{m}y^{\ast }\left( 
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
-\frac{\theta t}{2m^{2}M}\right) =\frac{M}{2}\frac{\theta t}{2m^{2}M}
\label{y*}
\end{equation}%
\begin{equation}
\frac{\theta }{n}z^{\ast }\left( 
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
-\frac{\theta s}{2n^{2}M}\right) =\frac{M}{2}\frac{\theta s}{2n^{2}M}
\label{z*}
\end{equation}
}

{\normalsize In other words, in equilibrium, each provider's average profit
per consumer multiplied by the number of consumers the provider will lose if
excluded from an insurer's network will be equal to the insurer's average
profit per consumer multiplied by the number of consumers the insurer will
lose if it excludes the provider from its network. \ (Under asymmetric
bargaining, each side of these equations would be multiplied by the
corresponding bargaining weight: $\left( 1-\beta \right) $ on the left, $%
\beta $ on the right). }

\subsection{\protect\normalsize Joint negotiation}

{\normalsize The terms above can also be substituted into equation \ref%
{Delta T hat} to obtain the first-order effect of joint negotiation on the
total transfer from each insurer to the jointly negotiating providers. With
joint negotiation by hospital $k$ and physician $v$, these providers'
equilibrium prices are denoted by $y_{k}$ and $z_{v}$, respectively, while
the equilibrium prices of other providers are denoted by $y_{-k}$ and $z_{-v}
$. \ In the expression for the first order effect, $\widehat{y}_{k}$ and $%
\widehat{z}_{v}$ denote the prices after taking into account first order
effects, which will not be equilibrium prices. \ Generalizing to the
asymmetric bargaining case, the first-order effect is given by the following
expression:%
\begin{equation}
\widehat{\Delta T}=\frac{\theta }{2}\left( \frac{\widehat{y}_{k}-y^{\ast }}{m%
}+\frac{\widehat{z}_{v}-z^{\ast }}{n}\right) =\left( 1-\beta \right) \left( 
\frac{\theta }{m}y^{\ast }\frac{\theta s}{2n^{2}M}+\frac{\theta }{n}z^{\ast }%
\frac{\theta t}{2m^{2}M}\right)  \label{Gal-Or Delta T hat}
\end{equation}
}

{\normalsize This expression can also be derived directly from Gal-Or's
model, by solving for values of $y_{k}$ and $z_{v}$ that satisfy the Nash
bargaining solution under the assumption that all other prices are equal to
their equilibrium values under separate contracting. Note that because $%
N_{3}=N_{4}$ in this model, the insurer's value function is linear and the
first line of equation \ref{Delta T hat} is zero. The remaining component of
the first-order effect is the recapture effect identified in this paper:
hospital $k$'s average profit per consumer multiplied by the number of
consumers that the hospital loses access to if excluded alone but not if
excluded jointly, plus the corresponding term for physician $v$. Because
this expression is positive for all parameter values, we can conclude that
the first-order effect of joint negotiation in Gal-Or's model is
unambiguously to increase the bargaining power of the jointly negotiating
providers. }

{\normalsize Unlike the general framework above with negotiated unit prices,
the use of a specific model permits analysis of the full equilibrium effect
of joint negotiation. In a Nash bargaining equilibrium with joint
negotiation by hospital $k$ and physician $v$, the change in the transfer
must satisfy the following expression:%
\begin{eqnarray}
\Delta T &=&\frac{\theta }{2}\left( \frac{y_{k}-y^{\ast }}{m}+\frac{%
z_{v}-z^{\ast }}{n}\right)  \label{Gal-Or Delta T} \\
&=&\frac{1}{\gamma }\left( 1-\beta \right) \left( \frac{\theta }{m}y^{\ast }%
\frac{\theta s}{2n^{2}M}+\frac{\theta }{n}z^{\ast }\frac{\theta t}{2m^{2}M}%
\right) +\beta \frac{\theta }{2}\left( \frac{y_{-k}-y^{\ast }}{m}+\frac{%
z_{-v}-z^{\ast }}{n}\right)  \notag
\end{eqnarray}%
\begin{equation*}
\text{where }\gamma =\frac{%
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
-\left( \frac{\theta t}{2m^{2}M}+\frac{\theta s}{2n^{2}M}\right) }{%
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
}\text{ }
\end{equation*}
}

{\normalsize This expression shows that the equilibrium effect consists of
the first-order effect combined with two distinct higher order effects. \
First, the first-order effect is multiplied by $\frac{1}{\gamma }$, where $%
\gamma $ is equal to the number of consumers the insurer will retain if it
excludes both the hospital and the physician from its network ($N_{0}$),
expressed as a percentage of the insurer's total equilibrium membership (%
%TCIMACRO{\U{bd}}%
%BeginExpansion
$\frac12$%
%EndExpansion
).\ This term appears because any first-order effect on the payment from
insurer to providers has a reinforcing effect on both the insurer's and the
providers' disagreement payoff. In particular, an increase (or decrease) in
the prices paid by one insurer will be accompanied in equilibrium by a
corresponding increase (or decrease) in the prices paid by the other
insurer, affecting the providers' disagreement payoff, and in the insurance
premium, affecting the insurer's disagreement payoff. The more patients the
providers can retain after exclusion from the insurer's network, the larger
these reinforcing effects will be. }

{\normalsize The second higher order effect is reflected in the term $\frac{%
\theta }{2}\left( \frac{y_{-k}-y^{\ast }}{m}+\frac{z_{-v}-z^{\ast }}{n}%
\right) $, the equilibrium change in payments to any separately contracting
physician-hospital pair. These providers' prices adjust in equilibrium in
response to any change in the jointly negotiating providers' prices. An
increase (or decrease) in these prices alters the insurer's outside option,
because it must pay more (or less) to other providers to replace the
services of hospital $k$ and physician $v$ in the event of disagreement. The
equilibrium changes in other providers' prices are given by the following
expressions:%
\begin{equation}
y_{-k}-y^{\ast }=\frac{y_{k}-y^{\ast }}{m}  \label{y-k - y*}
\end{equation}%
\begin{equation}
z_{-v}-z^{\ast }=\frac{z_{v}-z^{\ast }}{n}  \label{z-v - z*}
\end{equation}
}

{\normalsize In other words, in equilibrium, a price change by one of the
jointly negotiating providers will be accompanied by a change in the prices
of all other providers of that type, where a rival hospital's (or
physician's) price change relative to the jointly negotiating hospital's (or
physician's) price change will be equal to the inverse of the number of
hospitals (physicians) in the market.\ Intuitively, the more competing
providers there are in the market, the smaller the response to an increase
in one provider's price. }

{\normalsize As noted previously, the equilibrium conditions defined by
equations \ref{Gal-Or Delta T}, \ref{y-k - y*}, and \ref{z-v - z*} do not
uniquely identify $y_{k}$ and $z_{v}$. In particular, there is a continuum
of $\left( y_{k},z_{v}\right) $ pairs that satisfy these conditions. \ If we
do not impose any constraints on the possible prices (including
non-negativity constraints), then for any set of parameters with $m\neq n$,
there are solutions that increase the transfer and solutions that reduce it.
\ (For the case $m=n$, all solutions imply a unique positive value for $%
\Delta T$). As a result, it is necessary to impose some criterion in
addition to Nash bargaining to refine the predictions of the model. One
possible criterion is to consider the payoffs of the providers and insurers.
If the continuum of solutions to the Nash bargaining problem are, in effect,
multiple equilibria, then one might expect the players to coordinate on an
equilibrium that maximizes collective payoffs. Without any constraints, this
approach would imply that the price in the market with fewer participants
would increase without bound, while the price in the other market falls to
compensate. Providers collectively prefer equilibria with higher prices in
the market with fewer participants because the jointly negotiating
provider's price increase has a larger impact on rivals' prices in this
market. Insurers are indifferent among all possible equilibria, because
insurers always pass provider price increases through to consumers in the
form of a higher premium, with no change in insurer profits. This is
sustainable only as long as consumers' aggregate demand for health insurance
remains perfectly inelastic. Under the assumptions of the model, this will
hold until premia rise to the point that marginal consumers are on the
margin between insurance and no insurance, rather than on the margin between
the two insurers. Thus, one possible constraint on the set of feasible
equilibria could be that prices in the market with fewer participants should
not rise so high that consumers begin to drop insurance coverage.
Alternatively, we could consider a non-negativity constraint on the prices
in the market with more participants, or that prices in this market should
not go so low that providers are unable to cover their fixed costs.
Regardless of the constraint we impose to prevent prices from rising without
bound, if the equilibrium selection criterion is based on the idea that
players are more likely to coordinate on an equilibrium with higher payoffs
than lower, the transfer to the jointly negotiating providers will increase.
(A sufficient condition for the transfer to increase is that price does not
fall in the market with fewer participants.) }

{\normalsize Alternatively, it could be reasonable to impose an equilibrium
selection criterion based on symmetry. For example, one could assume that
the jointly negotiating providers' change in hospital price is equal to
their change in physician price, on an absolute basis ($y_{k}-y^{\ast
}=z_{v}-z^{\ast }$) or on a percentage basis ($\frac{y_{k}-y^{\ast }}{%
y^{\ast }}=\frac{z_{v}-z^{\ast }}{z^{\ast }}$). \ Either of these symmetry
conditions will yield a unique equilibrium in which both prices strictly
increase for any parameter values. A weaker symmetry condition could be $%
sign\left( y_{k}-y^{\ast }\right) =sign\left( z_{v}-z^{\ast }\right) $, also
sufficient to guarantee that both prices increase. Some reasonable
asymmetric selection criteria will also yield unambiguous increases in the
transfer to providers. For example, either of the two prices could be fixed
at the initial equilibrium price while the other adjusts to satisfy the
equilibrium conditions. }

{\normalsize While any of these criteria support an unambiguous increase in
provider bargaining power, Gal-Or imposes a different symmetry condition: $%
y_{k}=z_{v} $. This condition means that, if the equilibrium hospital and
physician prices are very different from one another under separate
negotiation, the jointly negotiating providers must select an equilibrium in
which the higher price is reduced, possibly substantially, to equalize the
two prices. If the higher price occurs in the market with fewer
participants, as will often be the case (for example, if $t=s$), this
equality requirement has the potential to result in an equilibrium in which
the jointly negotiating providers are strictly worse off relative to
separate negotiation, because the negative second order effects of the lower
price in the more concentrated market can outweigh the positive first order
effects. Gal-Or's symmetry condition is the basis for her conclusion that
"when one provider's market is much more competitive than the other a
vertical merger may reduce the joint profits of the merged entity."\footnote{%
{\normalsize Gal-Or(1999), p. 623.}} \ This conclusion has been cited in
subsequent literature on hospital-physician integration,\footnote{%
{\normalsize See, e.g., Gaynor (2006).}} but it appears that the underlying
mechanisms behind the results have not been well-understood. Absent the
somewhat arbitrary symmetry condition, Gal-Or's model implies the following
conclusions: The first-order effect of joint negotiation is always to
increase the combined entity's bargaining power; prices always exist that
satisfy the Nash bargaining solution and result in higher profits for the
jointly negotiating providers; and under a wide range of reasonable
equilibrium refinement criteria, including alternative symmetry assumptions,
the equilibrium outcome for any set of parameters will be to increase the
payment from insurers to providers. }

{\normalsize Before turning to the extension of the model with
post-disagreement price adjustment, I consider the potential range of
magnitudes of the effect on provider profits. \ Substituting equations \ref%
{y*} and \ref{z*} into equation \ref{Gal-Or Delta T hat} shows that, for any
value of the sum $\frac{\theta t}{2m^{2}M}+\frac{\theta s}{2n^{2}M}$, the
first-order effect on the transfer is increasing in the product $\frac{%
\theta t}{2m^{2}M}\cdot \frac{\theta s}{2n^{2}M}$. Since the sum is
constrained by assumption (3) to be strictly less than 
%TCIMACRO{\U{bd}}%
%BeginExpansion
$\frac12$%
%EndExpansion
, the first-order effect is bounded above by the value for $\frac{\theta t}{%
2m^{2}M}=\frac{\theta s}{2n^{2}M}=%
%TCIMACRO{\U{bc}}%
%BeginExpansion
{\frac14}%
%EndExpansion
$. \ I define the first-order \textit{percentage} increase in the transfer
as follows:%
\begin{equation*}
\widehat{\%\Delta T}=\frac{\widehat{\Delta T}}{\frac{\theta }{2}\left( \frac{%
y^{\ast }}{m}+\frac{z^{\ast }}{n}\right) }
\end{equation*}
}

{\normalsize Evaluating this expression at $\frac{\theta t}{2m^{2}M}=\frac{%
\theta s}{2n^{2}M}=%
%TCIMACRO{\U{bc}}%
%BeginExpansion
{\frac14}%
%EndExpansion
$ shows that the upper bound for $\widehat{\%\Delta T}$ is $\frac{\left(
1-\beta \right) }{2}$. \ For symmetric bargaining ($\beta =%
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
$), then, the maximum possible first-order effect is to increase the payment
to providers by 25\%. \ }

{\normalsize Without imposing additional constraints, there is no upper
bound for the \textit{equilibrium} effect for the range of parameters
defined by assumption (3), because as $\frac{\theta t}{2m^{2}M}+\frac{\theta
s}{2n^{2}M} $ approaches 
%TCIMACRO{\U{bd}}%
%BeginExpansion
$\frac12$%
%EndExpansion
, $\gamma $ goes to zero, so that the second-order effects cause prices (and
insurance premia) to increase without bound. For some level of premia,
marginal consumers would no longer switch to the rival insurer in the event
of disagreement, which would eliminate providers' ability to recapture lost
patients, thus constraining the effect of joint contracting. Assuming that
this constraint does not bind when the jointly negotiating providers' prices
increase by up to 100\%, the figure below shows the relationship between the
percentage increase in the transfer to the jointly negotiating providers and
the share of members an insurer would lose with the exclusion of one
hospital or one physician. \ For example, the chart shows that if the loss
of one hospital or the loss of physician would cause an insurer to lose 40\%
of its members (i.e. $\frac{\theta t}{2m^{2}M}=\frac{\theta s}{2n^{2}M}=%
\frac{1}{5}$), then joint negotiation would increase the transfer to the two
providers by 100\%. }

{\normalsize For simplicity, the effects shown in the chart incorporate only
the first-order effect and the first of the two higher order effects
identified above (division by $\gamma $); the second higher order effect
depends on the number of providers in each market ($m$ and $n$), and will
vary depending on the equilibrium selection rule used to choose a unique
outcome (if $m\neq n$). For a reasonable number of providers in each market,
this effect is relatively small; for example, if there are 13 hospitals and
13 physicians, this component of the effect will increase the absolute
change in the transfer by 4\%). \ (Also, the chart assumes symmetric
bargaining.) }

{\normalsize \FRAME{dtbpFUX}{408.375pt}{299.5pt}{0pt}{\Qcb{This figure
shows, for the Gal-Or (1999) model, isoquant lines for the percentage
increase in the total payment from an insurer to the jointly negotiating
providers as a function of the share of its members the insurer would lose
in the event of disagreement with one physician or one hospital. The
increases included in the chart do not include the component of the effect
due to competing providers' adjusting their equilibrium prices. \ This
component depends on the number of competing physicians and hospitals, and
will vary depending on what criterion is used to allocate the bargaining
power effect between hospital and physician prices.}}{}{Figure}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "T";width 408.375pt;height 299.5pt;depth
0pt;original-width 404.1875pt;original-height 295.9375pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";tempfilename
'NCEZXE00.wmf';tempfile-properties "XPR";}} }

\subsection{\protect\normalsize \ Insurer price adjustment after disagreement%
}

{\normalsize In this section, I assume that during each bilateral
negotiation, the parties expect that in the event of disagreement the
insurer will be able to adjust its premium before consumer demand is
realized. \ The new premium will be the profit-maximizing level given that
the negotiation ended in disagreement, holding all other prices (including
the rival insurer's premium) and all other network participation decisions
at their equilibrium levels. }

{\normalsize Under separate negotiation, the difference between the
equilibrium premium ($F^{\ast }$) and the premium an insurer will set in the
event of disagreement with a hospital, denoted $F^{-H}$, is given by the
following expression:%
\begin{equation}
F^{\ast }-F^{-H}=\frac{M}{2}\frac{\theta t}{2m^{2}M}  \label{dF*}
\end{equation}
}

{\normalsize Similarly, the premium reduction in the event of disagreement
with a physician is:%
\begin{equation*}
F^{\ast }-F^{-D}=\frac{M}{2}\frac{\theta s}{2n^{2}M}
\end{equation*}
}

{\normalsize The premium reduction in the event of disagreement with a
jointly negotiating hospital-physician pair is:%
\begin{equation*}
F^{\ast }-F^{-H-D}=\frac{M}{2}\left( \frac{\theta t}{2m^{2}M}+\frac{\theta s%
}{2n^{2}M}\right)
\end{equation*}
}

{\normalsize In other words, the premium reduction will equal the insurer's
average profit per consumer ($\frac{M}{2}$) multiplied by the number of
consumers the insurer would lose in the event of disagreement absent the
premium reduction. }

{\normalsize The effect of this premium reduction is to reduce by one-half
the number of members that switch insurers in the event of disagreement. \
That is, in the event of disagreement with one hospital, the insurer loses $%
\frac{1}{2}\frac{\theta t}{2m^{2}M}$ members; disagreement with one
physician results in a loss of $\frac{1}{2}\frac{\theta s}{2n^{2}M}$
members; and disagreement with one of each results in a loss of $\frac{1}{2}%
\left( \frac{\theta t}{2m^{2}M}+\frac{\theta s}{2n^{2}M}\right) $members. }

{\normalsize The equilibrium hospital price under separate bargaining, $%
y^{\ast }$, will satisfy the following expression:%
\begin{equation}
\left( 1-\beta \right) \frac{\theta }{m}y^{\ast }\left( 
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
-%
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
\frac{\theta t}{2m^{2}M}\right) =\beta \left[ \frac{M}{2}%
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
\frac{\theta t}{2m^{2}M}+\frac{M}{2}\frac{\theta t}{2m^{2}M}\left( 
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
-%
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
\frac{\theta t}{2m^{2}M}\right) \right]  \label{y*_adj}
\end{equation}
}

{\normalsize This expression is analogous to equation \ref{y*}: Just as in
that expression, the left-hand side is the hospital's average profit per
consumer multiplied by the number of consumers it would lose access to in
the event of disagreement (multiplied by the bargaining weight), and the
first term in the brackets on the right-hand side is the insurer's
equilibrium profit per consumer multiplied by the number of consumers it
would lose in the event of disagreement. The second term in the brackets on
the right-hand side shows an additional effect due to post-disagreement
price adjustment: the change in the insurer's profit per consumer ($F^{\ast
}-F^{-H}$) multiplied by the number of consumers the insurer will retain in
the event of disagreement. This expression shows that equilibrium price is
strictly lower when the insurer is able to adjust its premium in the event
of disagreement. \ (A similar expression can be derived for the physician
price, $z^{\ast }$). }

{\normalsize Turning now to the effect of joint negotiation, recall that in
the discussion of the general framework above, equation \ref{Delta T - adj}
showed that post-disagreement price adjustment would fully offset the
recapture effect if $(1-\beta )\pi _{j}\leq \beta \left( \pi _{I}\left(
\Omega ^{\ast }\right) -\pi _{I}\left( \Omega ^{-j}\right) \right) $, where
the left-hand side is the supplier's per-consumer profit and the right-hand
side is the change in the intermediary's per-consumer profit in the event of
disagreement. In the notation of the Gal-Or model, the left-hand side is $%
\left( 1-\beta \right) \frac{\theta }{m}y^{\ast }$ and the right-hand side is%
$\ \beta \left( F^{\ast }-F^{-H}\right) $. Equations \ref{dF*} and \ref%
{y*_adj} show that the condition does not hold: 
\begin{equation}
\left( 1-\beta \right) \frac{\theta }{m}y^{\ast }=\beta \left( F^{\ast
}-F^{-H}+\frac{M}{2}\frac{%
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
\frac{\theta t}{2m^{2}M}}{%
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
-%
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
\frac{\theta t}{2m^{2}M}}\right) >\beta \left( F^{\ast }-F^{-H}\right)
\label{y* inequality}
\end{equation}
}

{\normalsize This indicates that the recapture effect will outweigh the
effect of post-disagreement price adjustment, so that the net effect of
joint negotiation will be to increase the profits of the jointly negotiating
providers. This conclusion can be verified in the expression for the
first-order effect of joint negotiation on the transfer:%
\begin{eqnarray}
\widehat{\Delta T} &=&\frac{\theta }{2}\left( \frac{y_{k}-y^{\ast }}{m}+%
\frac{z_{v}-z^{\ast }}{n}\right)  \label{Gal-Or Delta T hat - adj} \\
&=&\left( 1-\beta \right) 
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
\left( \frac{\theta }{m}y^{\ast }\frac{\theta s}{2n^{2}M}+\frac{\theta }{n}%
z^{\ast }\frac{\theta t}{2m^{2}M}\right) -\beta \frac{M}{2}\frac{\theta t}{%
2m^{2}M}\frac{\theta s}{2n^{2}M}  \notag
\end{eqnarray}
}

{\normalsize The first term in the second line is the recapture effect,
equal to one-half the effect in the case with no post-disagreement price
adjustment (as given in equation \ref{Gal-Or Delta T hat}), since price
adjustment allows the insurer to retain half he consumers it would otherwise
lose in the event of disagreement. \ The second term in the second line is
equal to $\beta \left[ \left( F^{\ast }-F^{-H}\right) \left(
N_{2}+N_{3}\right) +\left( F^{\ast }-F^{-D}\right) \left( N_{1}+N_{3}\right) %
\right] $, which corresponds to the term in the third line of equation \ref%
{Delta T - adj}. \ By the same logic as in inequality \ref{y* inequality},
this expression is strictly positive. A similar expression for the
equilibrium change in the transfer can be easily derived by extending
equation \ref{Gal-Or Delta T hat - adj} to include higher order terms as in
equation \ref{Gal-Or Delta T}. \ }

{\normalsize We can therefore conclude, based on both the first-order effect
and the equilibrium effect (subject to the caveat above about equilibrium
selection), joint negotiation in this model unambiguously increases the
transfer to the jointly negotiating providers, even when the insurer is able
to adjust its premium in the event of disagreement. }

\section{\protect\normalsize A logit-based model}

{\normalsize The Gal-Or model described above is useful for demonstrating
that the recapture effect can arise in a fully specified model of consumer
demand, and for investigating the properties of the Nash equilibrium in a
model with negotiated unit prices. While the model is convenient in that it
allows for closed form solutions, some of the restrictive features of the
model may raise questions about how general the conclusions are. \ In
particular, the assumption that consumer preferences are distributed
uniformly gives rise to the linearity of the insurer's value function, so
that the only effect of joint negotiation occurs as a result of changes in
the providers' value functions. Since the prior literature on the effects of
joint negotiation has focused on the curvature of the insurer's value
function, it is natural to ask whether the recapture effect that arises in
Gal-Or's model also plays an important role in models with alternative
assumptions about consumer demand. }

{\normalsize In this section, I introduce a simple model in which consumer
preferences across different suppliers and intermediaries are distributed
according to the logit model. \ While I do not obtain closed form solutions
to the model, I numerically solve for the effects of joint negotiation on
the profits of the jointly negotiating suppliers. In particular, I consider
the effects of joint negotiation by two suppliers of products that are not
substitutes for one another, but that are consumed by the same consumers. \
Under the logit demand assumption, the intermediary's value function is
generally \textit{convex}: the number of consumers the intermediary stands
to lose from the loss of both suppliers is strictly \textit{less} than the
sum of its losses from losing each supplier individually. As a result, under
asymmetric bargaining with sufficient weight on the intermediary's value
function (i.e., when the suppliers have greater bargaining effectiveness),
the effect of joint negotiation is to \textit{reduce }the profits of the
suppliers. However, the recapture effect arises in this model as well: each
supplier has less to lose from disagreement when negotiating jointly with
the other. \ Under symmetric bargaining, this effect predominates, so that
the net effect of joint negotiation is to increase the suppliers' profits,
in spite of the convexity of the intermediary's value function. While the
percentage increase in the suppliers' profits is generally smaller in
magnitude than the changes that arise in Gal-Or's model, the positive net
effect demonstrates that the recapture effect can play an important role
under alternative demand specifications. }

\subsection{\protect\normalsize The model}

{\normalsize There are two types of product, A and B. \ Suppliers A1 and A2
(B1 and B2) each produce a differentiated product of type A (B). \ Each
consumer demands exactly one unit of each type of product. In order to
obtain these products, each consumer chooses one of two differentiated
intermediaries, I1 and I2. Consumers always use the same intermediary to
obtain both products. Consumers have heterogeneous preferences across the
two types of product and across intermediaries. \ In particular, consumer $i$%
's utility from consuming product $j$ of type A, product $k$ of type B,
obtained through intermediary $m$, is given by the following expression:%
\newline
\begin{equation*}
u_{ijkm}=\delta _{j}^{A}+\delta _{k}^{B}+\delta _{m}^{I}-\mu \left(
p_{j}^{A}+p_{k}^{B}+p_{m}^{I}\right) +\sigma ^{A}\varepsilon
_{ij}^{A}+\sigma ^{B}\varepsilon _{ik}^{B}+\sigma ^{I}\varepsilon _{im}^{I}
\end{equation*}
}

{\normalsize where $\delta _{j}^{A}$, $\delta _{k}^{B}$, and $\delta _{m}^{I}
$are the mean utilities from consuming product $j$ of type A, consuming
product $k$ of type B, and using intermediary $m$, respectively; $p_{j}^{A}$%
, $p_{k}^{B}$, and $p_{m}^{I}$ are the prices the consumer must pay for
product $j$ of type A, product $k$ of type B, and intermediary $m$,
respectively; $\mu $ is the marginal utility of income; $\varepsilon
_{ij}^{A}$, $\varepsilon _{ik}^{B}$, and $\varepsilon _{im}^{I}$ are
idiosyncratic components of consumer $i$'s utility function, distributed
i.i.d. extreme value; and $\sigma ^{A}$, $\sigma ^{B}$, and $\sigma ^{I}$
are parameters indexing the variance in the distribution of these
idiosyncratic components. \ In all of the following analysis, $\mu $ is
normalized to one; other values would simply scale the units of prices
without affecting any other results. \ Similarly, I normalize the sum of the
mean utilities for each product type and for intermediaries to be equal to
zero: $\delta _{1}^{A}+\delta _{2}^{A}=\delta _{1}^{B}+\delta
_{2}^{B}=\delta _{1}^{I}+\delta _{2}^{I}=0$. \ Since there is no outside
good in this model (consumers always consume one product of each type), this
normalization is equivalent to the usual practice of normalizing the utility
of one product to be equal to zero. }

{\normalsize As in the general model described above, each
supplier-intermediary pair initially engages in simultaneous bilateral
negotiations over whether the supplier's product will be included in the
intermediary's network, and a lump sum transfer is exchanged. Suppliers and
intermediaries simultaneously set the prices that they charge to consumers.
(Each supplier charges a single price that is charged to consumers at any
intermediary that carries that product.) Next, consumers observe the set of
products available at each intermediary and all prices, choose an
intermediary, and consume products available at that intermediary. \ }

{\normalsize I consider equilibria in which both intermediaries carry both
products of each type. \ (In principle, there may also be equilibria in
which intermediaries and suppliers differentiate themselves by specializing,
but I do not consider those here). \ While the model does not allow for
closed form solutions under the disagreement scenarios, I numerically solve
for equilibrium prices, then calculate profits for each agent under each
scenario by simulating a population of consumers and identifying their
optimal choices. (For the moment, I assume that agents treat prices as fixed
in the event of disagreement). }

{\normalsize To illustrate the effects of joint negotiation, I construct a
simple numerical example by making the following baseline assumptions: $%
\delta _{1}^{A}=\delta _{1}^{B}=\delta _{1}^{I}=0$, so that each market is
divided equally among the two firms; and $\sigma ^{A}=\sigma ^{B}=\sigma
^{I}=1,$ so that the $A$, $B$, and $I$ markets all have an equal degree of
differentiation. \ Under these assumptions, the profit-maximizing price for
all suppliers and intermediaries is 2. \ In equilibrium one-half of the
consumers choose each intermediary, and one-half of the consumers at each
intermediary choose each product of each type. \ In the event of
disagreement between an intermediary and a supplier of either type, each of
the two parties will lose 25\% of its customers, and thus 25\% of its gross
profits (i.e. profits before accounting for the lump sum transfer). In
particular, if intermediary 1 fails to reach agreement with supplier A1,
one-half of the consumers that initially purchased product A1 and
intermediary 1 will prefer to stay at intermediary 1 and switch to product
A2, while the other half will switch to intermediary 2 and continue to
purchase product A1. Since both B products are available at both
intermediaries, the disagreement has no effect of the B suppliers' profits.
\ Under symmetric bargaining, there will be no transfer between
intermediaries and suppliers, since each stands to lose the same amount from
disagreement. In the extremes of asymmetric bargaining ($\beta =0$ or $\beta
=1$), the party with all of the bargaining power will receive a lump sum
transfer equal to 25\% of the other party's gross profits. }

{\normalsize If the intermediary fails to reach agreement with suppliers A1
and B1 (when they are negotiating jointly), the intermediary will lose
roughly 44\% of its customers (and thus profits). Note that this is less
than the sum of its losses from losing each supplier individually (50\%), so
that the intermediary's value function is convex. \ These customers switch
to the other intermediary, and continue to purchase the same products as
before. \ Because a greater number of consumers switch intermediaries, the
two suppliers lose fewer customers than when only one is excluded from the
intermediary's network. In particular, each supplier loses roughly 18\% of
its customers, rather than 25\%. \ The effect of joint negotiation on the
lump sum transfer depends on the assumed bargaining parameter. \ For the
case of $\beta =0$ (the intermediary has all of the bargaining power), joint
negotiation means that the intermediary's ability to steer consumers away
from the suppliers is diminished. \ Instead of extracting 25\% of each
supplier's profits as it would under separate negotiation, the intermediary
can extract only 18\% of the two suppliers' joint profits. \ Joint
negotiation thus allows the suppliers to increase their combined profits,
net of the transfer to the intermediary, by 9\%. \ }

{\normalsize By contrast, if $\beta =1$ (suppliers have all of the
bargaining power), the jointly negotiating suppliers can extract only 44\%
of the intermediary's profits, rather than a combined 50\% under separate
negotiation. \ Thus, the suppliers' combined profits will fall (by just over
2\%) as a result of the joint negotiation. \ As a practical matter, the Nash
bargaining solution makes little sense when the intermediary's value
function is convex and the suppliers have all of the bargaining power. \
Under these conditions, the intermediary would be better off refusing to
contract with both of the suppliers than paying each one its full marginal
value. This is a general limitation of the Nash bargaining solution in the
case of complements, and could be viewed as putting an upper bound on the
value of $\beta $. Nonetheless, considering the polar case of $\beta =1$ is
useful in isolating the components of the net effect that will arise for
intermediate bargaining parameters. }

{\normalsize Under symmetric bargaining, the recapture effect outweighs the
effect of convexity in the intermediary's value function: the net effect of
joint negotiation in this example is to increase the suppliers' combined
profits by roughly 2\%. This is substantially smaller in magnitude than in
the comparable example for the Gal-Or model. \ Figure 1 in the previous
section showed that when the loss of a single provider would cause the
insurer to lose 25\% of its members, joint negotiation by a physician and a
hospital would result in an increase in the providers' profits of roughly
25\%. In part, this reflects the higher order effects that arise only when
negotiating unit prices rather than a lump sum transfer, and in part, the
linearity of the insurer's value function that results from the assumed
uniform distribution of consumers. The difference may also reflect the fact
that in the current model, suppliers independently choose their prices,
giving them a source of profits that is not subject to negotiations.
Overall, this suggests that the magnitude of the effect of joint negotiation
may be quite sensitive to specific assumptions about the nature of
competition.\ Nonetheless, the results for both models support the
conclusion that the recapture effect can be an important determinant of the
effect of joint negotiation. }

\subsection{\protect\normalsize Comparative statics}

{\normalsize To examine the implications of this model in more detail, I
consider how the effect of joint negotiation varies with the parameters of
the model. In particular, I consider three comparative statics exercises. \
First, I vary the mean utility parameters of the suppliers, and hence their
market shares; second, the mean utility (and thus market share) of the
intermediary; and third, the degree of differentiation in the suppliers'
markets relative to the intermediary market. \ For each exercise, I report
results for the symmetric bargaining case and for the two polar asymmetric
cases. }

{\normalsize Figures 2-4 depict the percentage increase in the suppliers'
joint profits due to joint negotiation as a function of the suppliers'
market shares.\footnote{{\normalsize For these charts, $\delta _{1}^{I}$ and
all of the variance parameters are fixed at the baseline values. $\delta
_{1}^{B}$ takes one of three possible values, -1, 0, and 1, corresponding to
market shares for supplier B1 of 35\%, 50\%, and 65\%, respectively. $\delta
_{1}^{A}$ takes on a range of values from -4 to 4, in increments of 0.1. \
For each pair ($\delta _{1}^{A},\delta _{1}^{B}$), I simulate a population
of 1,000,000 consumers and calculate the equilibrium profits and
disagreement payoffs; then for each bargaining parameter in $\left\{ 0,%
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
,1\right\} $, I calculate the transfers under separate and joint\
negotiation. The curves in the chart reflect the percentage change in net
profits for each set of parameters, smoothed using a lowess smoother with
bandwidth 0.1.}} Figure 2, illustrating the case where the intermediary
makes take-it-or-leave-it offers ($\beta =0$), shows that the recapture
effect, expressed as a percentage of joint profits, does not increase
monotonically with the suppliers' shares. \ The inverted U-shape reflects
the fact that, as a supplier becomes the first choice for a large majority
of consumers, its bargaining power is relatively large even under separate
negotiation. \ As supplier A1 grows large, the incremental value of joint
negotiation with a smaller supplier B1 diminishes because many of the
consumers that switch when both suppliers leave the intermediary would also
switch when only A1 leaves. \ The largest recapture effect occurs when 
\textit{both} suppliers are relatively large: for the ranges depicted in the
chart, the percentage increase in profits peaks at more than 10\% when both
suppliers have roughly 65-70\% share of their respective markets. }

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{\normalsize \bigskip }

{\normalsize Figure 3 shows the effect of convexity of the intermediary's
value function. As before, the absolute magnitude of the effect of joint
negotiation on the suppliers' profits is greatest when both suppliers are
large, but tapers as any one supplier becomes significantly larger than the
other. \ Figure 4, depicting the net effect under symmetric bargaining,
shows that the recapture effect predominates, with the net effect always
positive with an inverted U shape as in Figure 2, but with effects smaller
in magnitude as a result of the offsetting effect of convexity. }

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{\normalsize Figure 5 depicts the effect of joint negotiation as a function
of the \textit{intermediary's }market share.\footnote{{\normalsize For this
chart, $\delta _{1}^{A}$, $\delta _{1}^{B}$, and all variance parameters are
fixed at the baseline values, while $\delta _{1}^{I}$ takes on a range of
values from -4 to 4, in increments of 0.1. Construction of the chart then
follows the description in the previous footnote.\ }}\ \ For the case with $%
\beta =0$, the recapture effect in percentage terms is always increasing in
the intermediary's share, rising to above 30\% for the range of parameters
considered here. \ The intuition is that, when the intermediary has a large
share and is able to make take-it-or-leave-it offers, it is able to extract
a large share of the suppliers' profits. \ In this setting, the value of
joint negotiation is particularly valuable to the suppliers as a means of
retaining some portion of the surplus. \ The net effect for symmetric
bargaining remains relatively small, however, generally less than 5\%. }

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{\normalsize Finally, Figures 6-8 show how the effects vary when the
suppliers' markets have different degrees of differentiation relative to the
intermediary's market.\footnote{{\normalsize For these charts, $\sigma ^{I}$
and all mean utilities are fixed at the baseline level. $\sigma ^{B}$ takes
one of three possible values (0.5, 1, and 2), while $\sigma ^{A}$ takes on a
range of 80 different values between 0.25 and 4. \ For each set of
parameters, construction of the chart proceeds as before.}} \ In these
charts, the degree of differentiation is indexed by the equilibrium prices
in each supplier market (A and B), relative to the prices in the
intermediary market. \ Each supplier and intermediary has a share fixed at
50\%, but competing suppliers may be closer or more distant substitutes
compared to the intermediaries, so that their relative prices may be lower
or higher, respectively. \ Figure 6 shows that, again, the recapture effect
does not vary monotonically. \ The largest effects (in percentage terms)
occur when the supplier markets have roughly the same intensity of
competition as the intermediary market, or modestly more. \ As the supplier
markets become either substantially more or substantially less competitive
than the intermediary market, the percentage effect of joint negotiation
drops significantly. \ Figure 7 shows that a reverse relationship holds for
the effect caused by the curvature of the intermediary's value function,
while Figure 8 shows that the recapture effect predominates across the full
range of parameters. }

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{\normalsize These comparative statics results show that the effects of
joint negotiation by two suppliers that do not sell substitute products can
have a modest impact on the bargaining power of the suppliers, with a
magnitude that varies with the specific features of the market in ways that
are not always predictable. \ But a robust feature of these results is that,
under symmetric bargaining, the impact of joint negotiation on suppliers'
profits is positive for any parameter values, despite the presence of a
convex intermediary value function. }

\subsection{\protect\normalsize Post-disagreement price adjustment}

{\normalsize In the analysis above, prices were set simultaneously with
negotiation over networks, so that prices were treated as fixed in the event
of disagreement. \ In this subsection, I relax this assumption. \ In
particular, I assume that, in the event of disagreement, the intermediary is
able to re-optimize its price to offset the deterioration in its network. \
As discussed above, this is the particular re-optimization that will tend to
offset the recapture effect, so I restrict attention to this issue only. \
Suppliers' prices and the other intermediary's price are held fixed at the
equilibrium level. \ Under the baseline parameters in the numerical example
described above, I find that the effect of relaxing this assumption is quite
modest. \ Under separate negotiation, if intermediary I1 fails to reach an
agreement with supplier A1, instead of losing 25\% of its consumers, the
intermediary reduces its price so that on net it loses only about 20\%. \
However, some of the consumers that the intermediary gains as a result of
the price reduction would otherwise have used intermediary I2 to purchase
product A2. \ Thus, A1's losses due to disagreement are only slightly
greater than the 25\% it would lose if the intermediary's price were fixed
at the equilibrium level. \ Under joint negotiation, in the event of
disagreement, the intermediary cuts its price so that instead of losing 44\%
of its consumers, it loses only 38\% (on net). \ Again, many of the
consumers gained as a result of the price decrease are switching
intermediaries, not switching products, so the suppliers lose roughly the
same number of consumers as before. \ The net effect of joint negotiation in
this example is to increase the suppliers' profits by roughly 1.6\%,
somewhat less than the 2\% increase that occurs when prices are held fixed.
\ This result is consistent with the conclusion from the Gal-Or model, that
allowing alternative beliefs about prices in the event of disagreement will
partially, but not fully, offset the positive effect of joint negotiation on
suppliers' profits. }

\section{\protect\normalsize Conclusion}

{\normalsize The recapture effect introduced in this paper appears to be a
fairly robust outcome of joint negotiation, under the assumption that the
equilibrium is characterized by the Nash bargaining solution. One limitation
of this bargaining framework is its essentially static nature. \ Recent
work, such as Lee and Fong (2013), has begun to explore alternative
bargaining frameworks that feature explicit modeling of dynamic bargaining
environments. It would be an interesting topic for future work to consider
whether similar effects arise in these alternative models. In the mean time,
the analysis in this paper suggests that it may be desirable for applied
researchers relying on the workhorse Nash bargaining framework to allow for
bargaining power effects to arise when suppliers negotiate jointly, even
when the suppliers do not sell products that consumers view as substitutes.
It would be interesting to learn the extent of empirical support for the
effects identified in this paper. }

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