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\begin{center}
ECONOMIC ANALYSIS GROUP

DISCUSSION PAPER


\vspace{3.8cm}


The Entry Incentives of Complementary Producers:\\ A Simple Model
with Implications for Antitrust Policy

\vspace{0.25cm} By \vspace{0.25cm}

Juan S. Lleras* and Nathan H. Miller**  \\

EAG 09-7  $\quad$ November 2009

\end{center}

\vspace{0.45cm}

\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 authors 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 9446, 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 email addresses.

\vspace{0.25cm}

\noindent Recent EAG Discussion Paper and EAG Competition Advocacy
Paper titles are listed at the end of this paper.  To obtain a
complete list of titles or to request single copies of individual
papers, please write to Janet Ficco at the above mailing address or
at janet.ficco@usdoj.gov or call (202) 307-3779.  In addition,
recent papers are now available on the Department of Justice website
at http://www.usdoj.gov/atr/public/eag/discussion\_papers.htm.
Beginning with papers issued in 1999, copies of individual papers
are also available from the Social Science Research Network at
www.ssrn.com.

\vspace{0.25cm}

\noindent\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

\noindent * \ University of California, Berkeley.  Email:
jslleras@econ.berkeley.edu.

\noindent ** \ Economic Analysis Group, Antitrust Division, U.S.
Department of Justice. Email: nathan.miller@usdoj.gov.  We thank
Ronald Drennan, Patrick Greenlee, Bob Majure, Mikko Packalen, Alex
Raskovich, and Jeremy Verlinda for valuable comments.

\newpage
\thispagestyle{empty}

\begin{abstract}            % beginning of the abstract
We model competition between two firms in a vertical
upstream-downstream relationship. Each firm can pay a sunk cost to
enter the other's market. For equilibria in which both firms enter,
the downstream price can be lower than the joint profit maximizing
level, and coordination (e.g., through merger) is anticompetitive.
\end{abstract}              % end of the abstract


\newpage
\thispagestyle{empty} \setcounter{page}{1} \onehalfspacing


\onehalfspacing

%\doublespacing

\section{Introduction}

It is a well-known principle of economics that a producer of one
product benefits from enhanced competition among producers of
complementary products.  Yet the implications of this principle for
antitrust policy are less well developed in the academic literature.
We explore these implications using a popular theoretical model in
which two firms exist in a vertical upstream-downstream relationship
-- the upstream firms sets the price of an intermediate product and
the downstream firm then sets the price of the final product.

We augment the model by permitting each firm to enter the other's
market.  Entry is costly and constrains prices in the affected
market.  The model takes the form of a two-stage game -- the firms
make entry decisions in the first stage and set prices in the second
stage.  We demonstrate that equilibria exist in which one or both
firms choose to enter. Double-marginalization is mitigated in these
equilibria. Further, if both firms enter then the downstream price
can be lower than the joint profit maximizing level. The results
imply that coordination (e.g., through merger) between two firms in
a vertical relationship can be anticompetitive provided that each
firm exerts competitive pressure in the other's market.

We develop two auxiliary results. First, we show that entry may be
profitable for one or both firms even when entry would be
unprofitable for a hypothetical third party; the distinction is due
to the positive effect on margins in the complimentary market. In
that sense, each firm in the vertical relationship can be uniquely
positioned to compete in the other's market. Second, we show that
entrants need not be efficient relative to the incumbent firms for
coordination to be anticompetitive. Indeed, some level of entrant
inefficiency is needed to support an equilibrium in which both firms
enter the other's market; the presence of fully efficient entrants
would prevent the firms from recouping the sunk cost of entry
through their margins in their original markets.

Our work contributes to a burgeoning theoretical literature that
examines competition among producers of complementary products.
Packalen (2009) examines a model in which complementary producers
can induce third-party entry, and similarly demonstrates that
coordination between the producers can be anticompetitive. Packalen
does not obtain the auxiliary results. Other recent contributions
examines scenarios in which one producer can intensify competition
in a complementary market (e.g., Farrell and Katz 2000, Chen and
Nalebuff 2006, Chan and Nahm 2007, Casadesus-Masanell, Nalebuff, and
Yoffie 2007). We refer the reader to Packalen (2009) for a more
thorough review of this literature.

Our work also has implications for the theory of divided technical
leadership.  The theory, as espoused by Bresnahan and Greenstein
(1999), states that the struggle for technical leadership among
complementary firms can induce entry, technology races,
and``epochal'' competition.  By contrast, our results suggest that
firms may establish follower positions in complementary markets,
even if those positions are weak and mainly serve to constrain
pricing. Thus, for example, our results could rationalize
Microsoft's recent launch of Bing in the market for online consumer
search, insofar as the existence of a Google competitor creates
positive externalities for Microsoft's positions in operating
systems and applications.

\section{Model}

\subsection{The game}

The model is a two-stage game featuring two firms in a vertical
upstream-downstream relationship.  In the first stage, the firms
simultaneously decide whether to enter the other's market. If
neither firm enters then both firms are monopolists in their
respective markets. In the second stage, the firms set prices and
payoffs are realized.  The solution concept we employ is
pure-strategy subgame perfect Nash equilibrium (SPNE). We compare
the outcomes of this game against an alternative scenario in which
the two firm merge prior to the entry stage.  In this alternative
scenario, entry does not occur and the firms price to maximize joint
profits.  We illustrate the timing of the model in Figure \ref{Fig:
timing}.

\begin{figure}[h]
\caption{Timing of Actions}
\begin{center}
\begin{pspicture}(12,2)\label{Fig: timing}  %size of the picture
%\psgrid  %Makes a grid to help draw.
\psline{->}(1,1)(11,1) \psline(1,1.1)(1,0.9)
\psline[linestyle=dotted](3,1.5)(3,0.5) \psline(5.5,1.1)(5.5,0.9)
\psline(9,1.1)(9,0.9) \rput(1,0.5){Stage 0: Merger}
\rput(5.5,0.5){Stage 1: Entry} \rput(9,0.5){Stage 2: Pricing}
\rput(11.75,1){Payoffs} \rput(1,1.5){$t=0$} \rput(5.5,1.5){$t=1$}
\rput(9,1.5){$t=2$}
\end{pspicture}
\end{center}
\end{figure}

% The figure is an arrow pointing to the right.  The left part of the
% arrow is labeled ``Stage 0: Merger.''  The middle part is labeled
% ``Stage 1: Entry'' and the right part of the arrow is labeled
% ``Stage 2: Pricing''.  At the end of the arrow is the term
% ``Payoffs.''



We represent the entry stage as the following non-cooperative normal 2x2 game: \\

\begin{game}{2}{2}[\underline{Downstream  Firm}][\underline{Upstream Firm}]
 & $\text{Enter} $ & $\text{Do Not Enter}$\\
 $\text{Enter} $ &$\pi_D^{UD},\pi_U^{UD}$& $\pi_D^{U},\pi_U^{U}$\\
 $\text{Do Not Enter}$ &$\pi_D^{D}, \pi_U^{D}$ &$\pi_D^{N}, \pi_U^{N}$
 \end{game}\\ \\

% The figure is a 2x2 chart that lists payoffs from the four possible
% outcomes of the entry stage-game.

\bigbreak \bigbreak \noindent We refer to the outcome of the game as
$(\sigma_U,\sigma_D)$ where $\sigma_U$ and $\sigma_D$ are the
actions of the upstream and downstream firms, respectively. Payoffs
correspond to the Nash equilibrium of the subsequent pricing stage.
We denote the payoffs of player $x$ when there is entry in market
$y$ as $\pi_{x}^{y}$.  As an example, $\pi_D^{UD}$ represents the
profits of the downstream firm when there is entry in both the
upstream and downstream markets.  The exception is $\pi_x^N$, which
denotes payoffs when there is no entry.

\subsection{Payoffs}

In the pricing stage, the upstream firm sets a price $p_U$ for the
intermediate product and the downstream firm sets a price $p_D$ for
the final product. The firms both face a constant marginal cost
$c_0$, which we normalize to zero. Demand for the final product is
$q(p_D) = \a-\b p_D$. Variable profits are given by $\pi_U=p_U
q(p_D)$ and $\pi_D=(p_D-p_U)q(p_D)$, where $\pi_U$ and $\pi_D$
denote the variable profits of the upstream and downstream firms,
respectively. Each firm can pay a sunk cost $f \geq 0$ to enter the
other's market. Entrants are technologically inefficient and face a
constant marginal cost $c \in
\left[0,\frac{\beta}{2}\right]$.\footnote{The upper bound on entrant
inefficiency rules out payoffs that are of little theoretical
interest. Leaving the strict confines of the model momentarily, we
note that these assumptions on price competition imply that entry by
a hypothetical third party would not be profitable. The third party
entrant would earn zero variable profits and could not recoup the
sunk costs of entry.  The two incumbents that we model are therefore
``natural monopolists.'' We exclude third party entry from the model
in the interest of brevity.}

A single Nash equilibrium exists for each of the four possible
outcomes of the entry stage.  We sketch these equilibria in turn:

\begin{enumerate}[i)]
  \item Neither firm enters.  Variable profits are
  $\pi_U^N=\frac{\a^2}{8\b}$ and $\pi_D^N=\frac{\a^2}{16\b}$ and
  the downstream price is $p_D=\frac{3\a}{4\b}$.

  \item The upstream firm enters the downstream market.  The
  downstream firm prices to just undercut the entrant's  marginal cost.  Variable
  profits are $\pi_U^D=\frac{\a^2}{4\b}-\frac{\a c}{2} + \frac{\b c^2}{4}$ and $\pi_D^D=\frac{\a c}{2} - \frac{\b c^2}{2}$.
  The downstream price of $p_D=\frac{\a}{2 \b}+ \frac{c}{2}$ reflects
  the upstream price and a downstream markup of $c$.

  \item The downstream firm enters the upstream market. The
  upstream firm prices to just undercut the entrant's marginal cost.  Variable
  profits are $\pi_U^U=\frac{\a c}{2}-\frac{\b c^2}{2}$ and $\pi_D^U=\frac{\a^2}{4\b}-\frac{\a c}{2} + \frac{\b c^2}{4}$.
  The downstream price of $p_D=\frac{\a}{2\b} + \frac{c}{2}$
  reflects an upstream price of $c$ and a downstream markup.

  \item Both firms enter.  The firms price to just undercut the
  entrants' marginal costs.  Variable profits are
  $\pi_U^{UD}=\pi_D^{UD}=2 \a c - 4 \b c^2 $.  Downstream prices are
  $2c$.

\end{enumerate}

\noindent In the alternative merger scenario, the firms set a
downstream price of $\frac{\alpha}{2\beta}$, which maximizes joint
profits.  Outcomes of the two-stage game are characterized by
double-marginalization whenever the downstream price exceeds this
level.  We refer to the vertical merger as anticompetitive if the
pure-strategy SPNE of the two-stage game produces a downstream price
lower than the joint profit maximizing level.

\subsection{Solutions}
The solution concept is pure-strategy SPNE.  The symmetry of the
game guarantees the existence of at least one such equilibrium:

\begin{proposition}
  There exists at least one pure-strategy SPNE equilibrium in the 2x2 game.
\end{proposition}

\begin{proof}
The proof is by contradiction.  We exploit the equalities
$\pi_D^{UD} = \pi_U^{UD}$, $\pi_D^{U}=\pi_U^{D}$, $\pi_U^{U} =
\pi_D^{D}$, and $\pi^N_U = 2 \pi^N_D$.  Suppose there are no pure
strategy Nash equilibria. Since (Enter, Enter) is not an
equilibrium, it must be that $\pi^U_U>\pi^{UD}_U$. Given this, and
the supposition that (Enter, Do Not Enter) is not an equilibrium, it
must be that $\pi^N_D>\pi^D_U$.  This implies that (Do Not Enter, Do
Not Enter) is an equilibrium.
\end{proof}

\noindent We evaluate the equilibria graphically in Figure \ref{Fig:
Equilibrium} for specific parameter combinations. We consider the
demand parameters $\alpha=10$ and $\beta=1$, and focus on the
two-dimensional parameter space defined by $c \in [0,5]$ and $f \in
[0,30]$.  The solid lines delineate five regions, and we examine
each region in turn.

\begin{figure}[h]
        \centering
                \includegraphics[height=3in]{Entryfig.eps}
        \caption{Equilibrium Regions}
        \label{Fig: Equilibrium}
\end{figure}

% The figure plots five different regions, each of which is
% characterized by different equilibria outcomes.  The horizontal axis
% is marginal costs and the vertical axis is sunk costs.  The body of
% the paper describes each region in detail.

Region I is defined by the parameter space over which (Enter, Enter)
is a unique equilibrium. Within this region, the upstream firm uses
entry to constrain the prices of the downstream  firm, and
vice-versa.  The firms recoup the sunk cost of entry in the
subsequent price competition because entrants only partially dampen
the margins of the incumbents. The bounds on the region are
intuitive.  Sunk costs cannot be too great, and entrants must be
efficient enough to constrain prices but not so efficient that
margins are eliminated.  This region is important for antitrust
policy because downstream prices are lower than the joint profit
maximizing level for entrant marginal costs that are sufficiently
low ($c<\frac{\a}{4\b}$).  We plot this threshold with a dotted
line. The sub-regions I$_A$ and I$_B$ map the parameter regions in
which the downstream price is lower and higher than the joint profit
maximizing price, respectively, so that vertical merger is
anticompetitive in sub-region I$_A$.\\

We now discuss the other regions plotted in Figure \ref{Fig:
Equilibrium}:

\begin{itemize}

\item Region II is defined by the parameter space over which (Enter, Enter) and (Do Not Enter, Do Not
Enter) are the two equilibria. The latter equilibrium yields higher
profits for both firms.  However, if both firms enter then prices
can be lower or higher than the joint profit maximizing level.

\item Region III is defined by the parameter space over which
(Enter, Do Not Enter) and (Do Not Enter, Enter) are the two
equilibria.  The efficiency of the entrants creates a situation in
which each firm can recoup the sunk costs of entry only if the other
firm does not enter. Downstream prices exceed the joint profit
maximizing level.

\item Region IV is defined by the parameter space over which (Do Not
Enter, Enter) is the unique equilibrium (i.e., the downstream firm
enters the upstream market).  Absent entry, the upstream firm earns
greater margins than the downstream firm due to its ability to set a
take-it-or-leave-it price.  These margins induce the downstream firm
to enter.  Downstream prices exceed the joint profit maximizing
level.

\item Region V is defined by the parameter space over which (Do Not
Enter, Do Not Enter) is the unique equilibrium.   Sunk costs are
high and entrants are inefficient.  Downstream prices exceed the
joint profit maximizing level.

\end{itemize}


\section{Discussion}

We formalize the intuition that firms in a vertical relationship
have an incentive to introduce competition into the others' markets,
and that the elimination of this incentive (e.g., through
coordination/merger) can be anticompetitive. We employ the simplest
possible modeling framework because the intuition itself is
straight-forward.  Extensions to the model could generate additional
results -- one could incorporate more general functional forms, a
tradeoff between entry costs and entrant efficiency, and/or
first-mover advantages. We leave these extensions to future work.



\begin{thebibliography}{99}

\singlespace

\bibitem{BG99}\textbf{Bresnahan, Timothy, and Shane Greenstein.}
1999.  Tecnological Competition and the Structure of the Computer
Industry.  \textit{Journal of Industrial Economics}, 47(1): 1-40.

\bibitem{CMNY07}\textbf{Casadesus-Masanell, Ramon, Barry Nalebuff,
and David Yoffie.} 2007.  Competing Complements.  Unplublished.

\bibitem{CNY06}\textbf{Chen, Keith and Barry Nalebuff.} 2006.
One-Way Essential Complements.  Cowles Foundation Discussion Paper
No. 1588.

\bibitem{LN07}\textbf{Chen, Leonard and Jae Nahm.} 2007.
Product Boundary, Vertical Competition, and the Double-Mark-up
Problem. \textit{RAND Journal of Economics}, 38(2): 447-466.

\bibitem{FK00}\textbf{Farrell, Joseph and Michael Katz.} 2000.
Innovation, Rent Extraction, and Integration in Systems Markets.
\textit{Journal of Industrial Economics}, 48(4): 413-432.

\bibitem{P09}\textbf{Packalen, Mikko} 2009.  Complements and Potential
Competition.  Unplublished.

%\bibitem{N50}\textbf{Nash, John F.} 1950. Equilibrium Points in n-Person Games. \textit{Proceedings of the National Academy of Sciences of the United States of America}, 36(1): 48-49.
%\bibitem{N51}\textbf{Nash, John F.} 1951. Non-Cooperative Games. \textit{The Annals of Mathematics}, 54(2): 286-295.

\end{thebibliography}








\end{document}
