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%An Empirical Model of Spatial Competition with an Application to
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\begin{center}
ECONOMIC ANALYSIS GROUP

DISCUSSION PAPER


\vspace{3.8cm}


Forward Contracting and the Welfare Effects of Mergers

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

\newcommand*\samethanks[1][\value{footnote}]{\footnotemark[#1]}



Nathan H. Miller*  \\

EAG 13-1  $\quad$ May 2013

\end{center}

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\small

\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 analysis and conclusions expressed herein are solely those of the author and do not represent the views 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, BICN 10-000, 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 address.

\vspace{0.25cm}

\noindent Recent EAG Discussion 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.  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 *Economic Analysis Group, Antitrust Division, U.S. Department of Justice. Author email address: nathan.miller@usdoj.gov.  The views expressed herein are entirely those of the authors and should not be purported to reflect those of the U.S. Department of Justice.

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\begin{abstract}            % beginning of the abstract
I extend the oligopoly model of Allaz and Vila (1993) to explore how forward contracting affects the adverse welfare consequences of horizontal mergers.    I derive a welfare statistic that, within the context of the model, is free of structural parameters.  The statistic allows for conclusions that generalize across different cost and demand conditions.  I then show that exogenous forward contracting mitigates welfare loss but that  endogenous forward contracting exacerbates welfare loss provided the relevant industry is sufficiently concentrated.
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\section{Introduction}

Firms in some industries engage in forward contracting, prior to competition in a spot market, through which they commit to supply some amount of output to the spot market at a locked-in price.  A long-standing result in the theoretical literature, due to Allaz and Vila (1993), is that such forward contracting leads to greater output and lower prices in the spot market.  In this paper, I explore the related topic of whether (and when) forward contracting exacerbates or mitigates the adverse welfare effects of horizontal mergers.

To that end, I extend the oligopoly model used in Allaz and Vila (1993) and, more recently, in Bushnell (2007).  The model features firms that supply undifferentiated output to a spot market and that can commit, via forward contracting, to at least some level of output.  I derive a welfare statistic that, within the context of the model, is free of structural parameters.  The statistic allows for conclusions that generalize across different cost and demand conditions.   I then show that the welfare loss caused by horizontal mergers is diminished by forward contracting, provided that forward contracting is treated as exogenous and fixed.

The model provides more nuanced results when forward contracting is endogenized.  Indeed, it is derived in Bushnell (2007) that the level of forward contracting consistent with sub-game perfect equilibrium decreases in the level of industry concentration.  In the context of the model, then, horizontal mergers reduce forward contracting and lessen an economic inducement for spot market production.  I show that when industries are sufficiently concentrated this effect dominates and forward contracting exacerbates rather than mitigates the adverse welfare effects of horizontal mergers.

In particular, mergers-to-monopoly have greater adverse welfare consequences with endogenous forward contracting than with no forward contracting at all.  Likewise, both mergers-to-monopoly and mergers-to-duopoly have greater adverse welfare consequences with endogenous forward contracting than with exogenous forward contracting fixed at 50 percent of spot market output.  While these thresholds are specific to the model, and in particular to the mechanism that generates forward contracting as an equilibrium strategy, I anticipate that the results generalize qualitatively to other settings. If a merger makes the exercise of market power more profitable but for some limiting constraint then it also introduces an incentive for firms to relax the constraint.

The results have obvious implications for policy-makers in the merger review process.  They also advance the large literature on competition in the electricity generation industry.  The closest paper to this one is Bushnell, Mansur and Wolak (2013), which estimates an empirical model of forward contracting similar in some respects to theoretical model employed here, and simulates the effects of a recent electricity merger cleared by the U.S. Department of Justice.\footnote{The merger is that of Exelon Corporation (Exelon) and Constellation Energy Group (CEG), which closed on March 12, 2012.}  Other research develops theoretically how forward contracting affects market outcomes (e.g., Powell (1993), Newberry (1998), Green (1999)),  establishes empirically the importance of forward contracting (e.g., Wolak (2000), Bushnell (2007), Bushnell, Mansur and Savaria (2008)) or incorporates forward contracting into  empirical models of oligopoly competition (e.g., Horta\c{c}su and Puller (2008)).  I refer readers to Bushnell (2007) for a more complete review of the extant literature.

The paper proceeds in three parts.  Section 2.1 develops the economics of the spot market and how forward contracting affects equilibrium output.  Section 2.2 examines the forward market and shows the level of forward contracting that is consistent with sub-game perfect equilibrium, within the context of the model.  Section 2.3 derives the welfare measures and examines the welfare effects of horizontal mergers.


\section{Theoretical Model and Results}


\subsection{The spot market}

Let $n=1\dots N$ firms supply undifferentiated output to a spot market.  The firms face a constant marginal cost $c$.  Denote the output of firm $n$ as $q_n$. Price is determined by total market output according to the inverse demand curve $P(Q)=\frac{\beta_0-Q}{\beta_1}$, where $P(\cdot)$ is the market price, $Q=\sum_n q_n$ is total production, and the demand parameters are $\beta_0$ and $\beta_1$.  Permit the firms to engage in forward contracting, through which each firm can commit to supply at least some level of production $q_n^f$ in the spot market.  Let the forward position of each firm be common knowledge and taken as given in the spot market.  The profit that firm $n$ obtains in the spot market equals the obtained markup times production in excess of that already committed:
\begin{equation}
\pi^s_n(q_n, q_{-n},q_n^f) = (P (Q)-c) (q_n - q_n^f) .
\end{equation}
The equilibrium concept is Nash-Cournot.  Each firm acts in the spot market to maximize profit taking as given its forward position and the output of its competitors.

Bushnell (2007) derives that total production in this spot market, conditional on the forward positions, is given by
\begin{equation} Q^*(N,q^f) = (\beta_0-\beta_1 c + q^f)\frac{N }{N+1}. \label{eq:qstar1} \end{equation}
where $q^f=q^f_n$, for all $n$, due to the symmetry of the model.  For the calculations that follow it will be useful to define the ``hedge rate,'' $h \equiv q^f / q$,  to be the proportion of output in the spot market equilibrium that is committed via forward contracting.  Then equation \eqref{eq:qstar1} can be re-expressed as
\begin{equation}
Q^*(N,q^f) = (\beta_0-\beta_1 c)  \frac{N }{N+1-h} . \label{eq:qstar2}
\end{equation}
The derivation is provided in an appendix.  Total output increases with the hedge rate. If $h=1$ the perfectly competitive level of output, $\beta_0-\beta_1 c$, is obtained regardless of the number of firms.  If instead $h=0$ then the standard Nash-Cournot oligopoly solution is obtained.

\subsection{The contract market}

The following contract market provides a tractable mechanism through which mergers can affect the incentive to engage in forward contracting.  Suppose that each firm $n$ can engage in forward contracting.  This entails a commitment to produce some level of output $q_n^f$ in the spot market at a fixed forward price.  Impose that the forward price equals the subsequently realized spot price, which eliminates arbitrage opportunities and follows from  rational expectations and risk neutrality assumptions.  Bushnell (2007) derives the hedge rate that is consistent with sub-game perfect equilibrium:
\begin{equation}
h^*(N) = \frac{N-1}{N}.
\label{eq:hstar}
\end{equation}
Monopolists do not participate in the contract market.  Provided $N>1$, forward contracts serve to partially deter spot market production on the part of competitors.  In the limit, as the number of firms grows large, all spot market production is pre-committed in the contract market at the perfectly competitive price.

\subsection{Forward Contracting, Welfare and Mergers}

I now extend the findings of Bushnell (2007) to examine the welfare implications of horizontal mergers.  First, welfare can be expressed as a function of the number of firms and the hedge rate:
\begin{align}
  W(N,h) &=  \int_0^{Q^*(N, \;q^f)} \left( \frac{\beta_0-x}{\beta_1}-c \right) dx   \nonumber \\
\nonumber  \\
&=  \frac{(\beta_0-\beta_1 c)^2}{\beta_1} \left[   \frac{N}{N+1-h} - \frac{1}{2} \left(\frac{N}{N+1-h}\right) ^2 \right]. \\
\intertext{With $h=1$ this reduces to the perfectly competitive level of welfare}
 W(N,1) &=  \frac{1}{2}\frac{(\beta_0-\beta_1 c)^2}{\beta_1}, \\
\intertext{and with $h=h^*(N)$ welfare equals}
 W(N,h^*) &=  \frac{(\beta_0-\beta_1 c)^2}{\beta_1} \left(   \frac{N^2}{N^2+1} - \frac{1}{2} \left(\frac{N^2}{N^2+1}\right) ^2 \right).
\end{align}
The ratio of welfare, for a given hedge rate, to the perfectly competitive level of welfare is a useful metric that is free of demand and cost parameters and thereby facilitates generalizations across different industry conditions:
\begin{equation}
\frac{W(N,h)}{W(N,1)} = 2 \left[   \frac{N}{N+1-h} - \frac{1}{2} \left(\frac{N}{N+1-h}\right) ^2 \right]
\end{equation}

Figure \ref{fig:welf3} plots this ``welfare ratio'' for the strategic hedge rate given by $h^*(N)$, as well as for four different fixed hedge rates, across industry structures with $N=1,2,3,4$.  Several relationships are apparent: (i) the welfare ratio increases with the magnitude of the fixed hedge rate for any given industry structure; (ii) the welfare ratio increases with the number of firms faster with smaller fixed hedge rates; (iii) relative to fixed hedge rates, the strategic hedge rate adds ``curvature'' to the welfare-concentration relationship such that the welfare ratio increases faster with $N$ for small $N$ but slower for large $N$.  This final result follows directly from Equation \eqref{eq:hstar}.

\begin{figure}[t]
        \centering
                \includegraphics[height=5in]{welfare3.eps}
        \caption{Welfare with Forward Contracting}
        \label{fig:welf3}
\end{figure}
% Figure is described in the text.  Welfare is shown to decrease in concentration, to a greater degree with less fixed hedging, and especially with endogenous hedging.

A horizontal merger can be modeled as a reduction in the number of firms by one.\footnote{Whether endogenous forward contracting is sufficient to make such mergers profitable is an unexplored topic, to my knowledge.}  Figure \ref{fig:welf4} plots the change in the welfare ratio (``merger harm'') due to horizontal mergers with initial $N=2,3,4,5$, for the strategic hedge as well as for two different fixed hedge rates.  Again the relevant results are apparent: (i) merger harm decreases in the fixed hedge rate; (ii) merger harm is greater in more concentrated industries; (iii) merger harm is greater with the strategic hedge than with the fixed hedges for small $N$ but smaller than with the fixed hedges for large $N$.  The last result arises because horizontal mergers reduce the optimal level of the strategic hedge (by Equation \eqref{eq:hstar}) and thereby lessen an economic inducement for spot market production.  When $N$ is small this effect dominates the counter-force that merger harm is lessened for any fixed hedge rate.   Inspection of the figure show that mergers-to-monopoly have greater adverse welfare consequences with endogenous forward contracting than with no forward contracting at all.  Likewise, both mergers-to-monopoly and mergers-to-duopoly have greater adverse welfare consequences with endogenous forward contracting than with exogenous forward contracting fixed at 50 percent of spot market output.

\begin{figure}[t]
        \centering
                \includegraphics[height=5in]{harm3.eps}
        \caption{Loss of Welfare due to Horizontal Mergers}
        \label{fig:welf4}
\end{figure}
% Figure is described in the text.  The welfare loss of mergers is shown to increase in the initial concentration, to a greater degree with less fixed hedging, and especially with endogenous hedging.

This demonstrates the main result of the paper: forward contracting diminishes welfare loss if it is exogenous but can exacerbate welfare loss provided that it is endogenous and  the relevant industry is sufficiently concentrated.  I anticipate that the result, developed within the context of specific modeling choices, generalizes qualitatively to other economic environments.  For instance, suppose that the primary motivation for hedging is to lessen price risk rather than to deter competitor production, that firms maximize profit, and that hedging is partial (i.e., spot market output exceeds forward commitments).  Then the pre-merger hedge rate reflects a balancing of risk and profit, the latter obtained from the exercise of market power. Horizontal mergers make the exercise of market power more profitable and create an incentive for firms to accept more risk.  Specific results in this alternative setting are less tractable, however, and I leave them to future research.


\appendix

\newpage

\section{Derivation of Equation \eqref{eq:qstar2} }

Bushnell (2007) derives that
\begin{align}
Q^* &=\frac{ (\beta_0-\beta_1 \alpha_0 + q^f)N }{N+1+ \beta_1 \alpha_1}. \nonumber \\
\intertext{Substituting $h q^*$ for $q^f$ yields}
Q^* &= \frac{(\beta_0-\beta_1 \alpha_0 + h q^*)N }{N+1+ \beta_1 \alpha_1}, \nonumber \\
\intertext{which further obtains}
Q^* &= \frac{(\beta_0-\beta_1 \alpha_0 )N + h Q^* }{N+1+ \beta_1 \alpha_1}.  \nonumber
\end{align}
Here the $Q^*$ appears on both sides of the equation.  Collecting the terms and solving yields Equation \eqref{eq:qstar2}.



\newpage \singlespace

\begin{center} \textbf{References} \end{center}


\bigbreak \noindent Allaz, Blaise and Jean-Luc Vila.  1993.  ``Cournot Competition, Forward Markets and Efficiency.''  \textit{Journal of Economic Theory}, 59: 1-16.

\bigbreak \noindent Bushnell, James.  2007.  ``Oligopoly Equilibria in Electricity Contract Markets.'' \textit{Journal of Regulatory Economics}, 32(3): 225-245.

\bigbreak \noindent Bushnell, James, Erin Mansur and C Saravia. 2008.  ``Vertical Relationships, Market Structure, and Competition: An Analysis of the U.S. Restructured Electricity Markets.'' \textit{American Economic Review}, 99(1), pages 237-266.

\bigbreak \noindent Bushnell, James, Erin Mansur and Frank Wolak.  2013.  ``Vertical Commitments and the Price Effects of Mergers: Evidence from Electricity Markets.''  Draft in progress.

\bigbreak \noindent Green, Richard. 1999.  ``The Electricity Contract Market in England and Wales.'' \textit{Journal of Industrial Economics}, 48(1): 107-124.

\bigbreak \noindent Horta\c{c}su, Ali and Steven Puller.  2008.  ``Understanding Strategic Bidding in Multi-Unit Auctions: A Case Study of the Texas Electricity Spot Market.''  \textit{RAND Journal of Economics}, 39(1): 86-114.

\bigbreak \noindent Newberry, David. 1998.  ``Competition, Contracts and Entry in the Electricity Spot Market.''  \textit{RAND Journal of Economics}, 29, 726-749.

\bigbreak \noindent Powell, Andrew.  1993.  ``Trading Forward in an Imperfect Market: The Case of Electricity in Britain.''  \textit{Economic Journal}, 103: 444-453.

\bigbreak \noindent Wolak, Frank. 2000.  ``An Empirical Analysis of the Impact on Hedge Contracts on Bidding Behavior in a Competitive Electricity Market.''  \textit{International Economic Journal}, 14(2): 1-39.








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