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ECONOMIC ANALYSIS GROUP
DISCUSSION PAPER
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\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \= Ana\=ly\=zing \=Merg\=ers Using \\
\> \> Capacity Closures \\
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\> \> \> \> By \\
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\> \> \> Nicholas Hill* \\
\> EAG 08-8 \> \> \> \>August 2008 \\
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\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 authors and do not
represent the views of the United States Department of Justice.
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\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 e-mail address.
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\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.
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\noindent \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
\noindent * \ Economist, Economic Analysis Group, Antitrust Division, U.S. Department of
Justice. Email: nicholas.hill@usdoj.gov. Thanks are due to Beth Armington,
Dennis Carlton, Tim Daniel, Jay Ezrielev, Andy Hanssen, Ken Heyer, Greg
Leonard, Bill Nye, Russ Pittman, Joe Simons, Greg Werden, Bobby Willig, Peter
Woodward, and above all, Rene Kamita. The views expressed in this paper are
not purported to represent those of the United States Department of Justice.
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\begin{abstract}
In this paper I describe a method for analyzing mergers in industries in
which it is more cost effective to close capacity than to idle it. The
method can be used to define markets, to assess the likelihood of
competitive effects and to evaluate divestitures. I also discuss the
method's data requirements and how it can be modified to deal with the types
of issues that often arise during an antitrust investigation.
\end{abstract}
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\section{Introduction}
The goal of this paper is to describe a method for analyzing proposed mergers.
The method, which I shall call the capacity closure method, takes as its
central assumption that permanently closing capacity is the most cost
effective way to increase price.
For industries in which this assumption is justified, the capacity closure
method is a simple way to analyze competitive effects, market definition and
the impact of prospective divestitures.
The rest of this paper is organized as follows.
Section \ref{s:overview} gives an overview of how the capacity closure method
can be used to predict a transaction's competitive effect.
Section \ref{s:pofz} provides details on implementing the capacity closure
method.
Section \ref{s:variations} describes how the capacity closure method can be
used to delineate markets, analyze the but-for world and vet divestitures.
Section \ref{s:data} discusses the capacity closure method's data requirements.
Section \ref{s:bells} presents some simple extensions of the capacity closure
method.
\section{An Overview of the Capacity Closure Method} \label{s:overview}
Consider a situation in which two firms that manufacture a homogenous
product (and nothing else) propose to merge, and
assume for simplicity that the product constitutes an
antitrust product market as defined in the U.S. Horizontal Merger Guidelines.
Assume further that the product's production technology has sufficiently high
fixed costs that it is always more profitable to reduce output by closing
facilities than by reducing output across facilities.
A competition agency is charged with reviewing the merger and determining, to
the best of its ability, whether the merged firm would find it profitable to
anti-competitively raise price by closing facilities.\footnote{I discuss
analyzing the incentives of the stand-alone firms in section
\ref{s:variations}. For the moment, assume for simplicity that the stand-alone
firms have no incentive to raise the price by closing facilities.}
If the merger is approved, the combined firm would have $i = 1,\dots,n$
facilities.
Each facility $i$ can produce $q_i$ units of output and has a per unit cost
of production of $c_i(q_i)$.
Let $S$ denote the set of all possible combinations of the merged firm's $n$
facilities.
For any $s \in S$, let $\theta(s)$ denote the total capacity of the facilities
in $s$.
Finally, let $p(z)$ denote the price if $z$ units of capacity are closed,
with the current price denoted $p(0)$.
To economize on notation, I shall write $p_s$ for $p(\theta(s))$ and $p_0$ for
$p(0)$.
Let us first consider the question of whether the merged firm would find it
profitable to shut a set $s \in S$ of its facilities.
The benefit of closing the facilities in $s$ is a higher price on the merged
firm's remaining sales.
The cost of closing the facilities in $s$ is the profit that they would have
earned on the facilities in $s$.
Formally, let $c(s)$ and $q(s)$ denote the firm's cost per unit of output and
total output, respectively, if the facilities in $s$ are closed.
The price after the facilities in $s$ have been closed is $p_s$.
The merged firm's profit if it were to close the facilities in $s$ would be
\begin{equation}
\pi(s) = [p_s - c(s)]q(s)
\end{equation}
Closing the facilities in $s$ is profitable if $\pi(s)$ exceeds the firm's
current profit.
Now let us expand the firm's decision problem in a natural way.
Specifically, let us assume that the merged firm will identify the set of
facilities $s^*$ whose closure maximizes the firm's profit and close those
facilities.\footnote{Note that the optimal set of facilities to close may be
the empty set. Also, while $s^*$ need not be unique, I assume that it is for
simplicity.}
That is, the firm solves the problem
\begin{equation}
\underset{s \in S}{\operatorname{arg\,max}} \pi(s) = [p_s - c(s)]q(s)
\end{equation}
and closes the facilities in the resulting $s^*$
If the competition agency---or any other interested party---can identify the
optimal set of closures $s^*$, it can predict the price change $\Delta_p(s^*)
= p_{s^*}-p_0$ that is likely to result from the merger.
Given a functional form for $p(\cdot)$ and adequate data, it can do this---i.e.,
identify $s^*$---by calculating $\pi(s)$ for all $s \in S$.\footnote{This can
be computationally burdensome if $n$ is large. However, one can reduce
significantly the computational burden by exploiting the fact that closing
high cost plants is more profitable than closing low cost plants.}
In the following two sections, I discuss in detail how precisely this can be
done.
In particular, I present two functional forms for $p(\cdot)$ and discuss the
capacity closure method's data requirements.
\section{Implementing the Capacity Closure Method} \label{s:pofz}
The following two subsections present two functional forms for $p(\cdot)$.
The first functional form is derived under the assumption that the price
elasticity of demand is constant.
The second functional form is derived under the assumption that demand is
linear.
These two forms of demand are frequently used in merger cases and elsewhere.
There are many other functional forms that $p(\cdot)$ could take though, and
the reader is welcome to experiment with them.
\subsection{The Constant Elasticity of Demand Model}
An increase in the price of the merged firm's product will have two effects:
the demand for the product will decrease and the supply of the product will
increase.
If the merged firm wishes to raise the price of its product by $\Delta_p$, it
can do so by closing an amount of capacity sufficient to absorb the fall in
demand and increase in supply associated with $\Delta_p$.\footnote{That is,
starting from an equilibrium price $p_0$, the firm can make some $\hat{p}$
into an equilibrium if it can close enough capacity to counteract the decrease
in demand and increase in supply associated with moving from $p_0$ to
$\hat{p}$.}
Given a method for estimating the demand and supply responses associated with
any price increase $\Delta_p$, one can therefore calculate the amount of
capacity that must be closed to implement it.
Conversely, given a set of mills $s$ with capacity $\theta(s)$, one can
calculate the price increase $\Delta_p(s)$ that would result from closing
those mills.
The key to calculating an estimate of $\Delta_p(s)$ then is specifying the
relationship between price and demand and supply.
A natural specification for the change in demand given a change in price
$\Delta_p$ is
\begin{equation}
\Delta_p \eta_d Q_d(p_0)
\end{equation}
where $\eta_d$ is the price elasticity of demand for any price and $Q_d(p_0)$
is the level of demand at the current price $p_0$.
Note that the price elasticity of demand is assumed to be constant!
When specifying the supply response to an increase in price, one approach is
to disaggregate the supply response into two parts: the response of domestic
competitors and the response of importers.\footnote{For expositional clarity I
shall assume that the geographic market includes all other domestic producers
of the product and no foreign producers of the product, but that there is a
non-trivial amount of imports.}
The response of domestic competitors to a price increase can itself take
several, non-exclusive forms, but I shall assume for ease of exposition that
it takes the form of bringing excess capacity to bear.\footnote{Other possible
responses include the repatriation of exports and the expansion of productive
capacity.}
In particular, let us assume that domestic producers hold $\varepsilon$ units
of excess capacity and let $\beta(\Delta_p) \in[0,1]$ denote the proportion of
these units that are brought to bear in the event of a price increase
$\Delta_p$.
The general expression for the increase in supply by domestic competitors is:
\begin{equation}
\beta(\Delta_p) \varepsilon
\end{equation}
For ease of exposition I shall make the conservative assumption that the
domestic competitors constitute a competitive fringe so that $\beta(\cdot)$ is
equal to 1.
Note, however, that this term can be allowed to vary with $\Delta_p$ to
reflect the fact that the marginal cost of bringing excess capacity to bear is
typically not constant.
Importers of the product can respond to a price increase in a number of ways,
but I shall assume that they do so by increasing imports.
Denote the current level of imports as $I$ and the price elasticity of import
supply as $\eta_I \ge 0$.
The change in imports in response to a price increase is\footnote{Note the
implicit assumption that the price elasticity of import supply is a constant
function of the market price.}
\begin{equation}
\Delta_p \eta_I I
\end{equation}
Bringing together the demand response and supply responses and equating them
to a capacity closure $\theta(s)$ yields
\begin{equation}
\Delta_p(s) \eta_d Q_d(p_0) + \varepsilon + \Delta_p(s) \eta_I I = \theta(s)
\end{equation}
Solving this equation for $\Delta_p(s)$ yields
\begin{equation}
\Delta_p(s) = \frac{\theta(s) - \varepsilon}{\eta_d Q_d(p_0) + \eta_I I}
\end{equation}
If one can obtain estimates of $\varepsilon$, $\eta_d$, $Q_d(p_0)$, $\eta_I$
and $I$, one can calculate $\Delta_p(s)$ for any $s \in S$.
Once $\Delta_p(s)$ has been calculated, $p_s$ can be calculated as:
\begin{equation}
p_s = p_0 + \frac{\theta(s) - \varepsilon}{\eta_d Q_d(p_0) + \eta_I I}
\end{equation}
\subsection{The Linear Demand Model}
The constant elasticity of demand model assumes that the price elasticity of
demand is a constant function of the price.
An alternative assumption is to assume that demand is a linear in price:
\begin{equation} \label{eq:ldemand}
Q_d(p) = a - bp
\end{equation}
Under this assumption, the elasticity of demand is
\begin{equation} \label{eq:ldelast}
\eta_d(p) = -\frac{bp}{a-bp}
\end{equation}
It is easy to verify that demand becomes more elastic as the price increases.
Given $\eta_d(p_0)$, $Q_d(p_0)$ and $p_0$, one can calculate the values of
$a$ and $b$ using equations \ref{eq:ldemand} and \ref{eq:ldelast}.
Once these values have been calculated, the solution strategy is to specify a
method for calculating $Q_d(p_s)$ when one does not know $p_s$.
Once this has been done, $p_s$ is calculated as:
\begin{equation}
p_s = \frac{a-Q_d(p_s)}{b}
\end{equation}
The trick is calculating $Q_d(p_s)$ when one does not know
$p_s$.
One approach is to calculate $Q_s(p_s)$, the quantity supplied under $p_s$,
and assume that the market is in equilibrium (i.e., $Q_s(p_s) = Q_d(p_s)$.
One can calculate $Q_s(p_s)$ as the sum of the supply by domestic producers
and the supply of imports.
Supply by domestic producers can be estimated as the sum of
competitors' current production $q_c(p_0)$, competitors' excess capacity
$\varepsilon$ and the merged firm's current production $q_d(p_0)$ minus the
capacity closed by the merged firm $\theta(s)$.\footnote{The price change will
be negative if competitors hold more excess capacity than the merged firm
closes. This difficulty can be resolved by adopting a less conservative
functional form for $\beta(\cdot)$, though closing the facilities in $s$ is
unlikely to be profitable if $\theta(s) < \varepsilon$.}
One approach to estimating the supply of imports is to assume that they are a
linear function of the market price:
\begin{equation}
I = c + dp
\end{equation}
Provided that one can estimate the current level of imports and the price
elasticity of import supply $\eta_I$, one can estimate the values of $c$ and $d$.
Bringing all of the pieces together, the demand is
\begin{equation}
Q_d(p_s) = Q_s(p_s) = (q_c(p_0) + \varepsilon +
q_d(p_0)) - \theta(s) + (c + dp_s)
\end{equation}
Inserting this quantity into the linear demand function and solving for
$p_s$ gives
\begin{equation}
p_s = \frac{a - (q_c(p_0)+\varepsilon+q_d(p_0)-\theta(s)+c)}{b+d}
\end{equation}
Thus, given any set of facilities $s \in S$, one can calculate $p_s$.
\section{Variations on a theme: market definition, the but-for world and
divestitures} \label{s:variations}
An attractive feature of the capacity closure method
is its flexibility: it can be easily adapted to define markets,
analyze the but-for world, and evaluate divestitures.
Two simple modifications allow the method to be used to test whether
a candidate market is indeed an antitrust market.
First, $S$ is defined as the set of all facilities in the
candidate market.
In essence, this modification replaces the merged firm with a hypothetical
monopolist.
Second, any supply response by competitors in the candidate market is removed
from the calculation of $p_s$, since all competitors in the candidate market
are now part of the hypothetical monopolist.
That is, $\beta(\cdot)$ is set equal to zero.
Having made these changes, the candidate market is an antitrust market if
$p_{s^*}$ is larger than $p_0$ by some amount, say 5\%.
Once a market has been defined, one can use the capacity closure method to
test whether the merger is likely to lead to a price increase.
If the method predicts that the merger will result in a price increase, one
may wonder if the prediction is an artifact of parameter values that are
biased towards finding harm.
A crude one-sided test of the parameter values is to test whether, under these
values, the stand-alone firms have an incentive to raise the price in the
but-for world.
If they do, one may conclude that the parameter values are indeed biased
toward finding harm given that the stand-alone firms presumably find their
current facility portfolios to be profit-maximizing.
Thus, analyzing the but-for world can
act as a valuable check on one's parameter values.
To use the capacity closure method to analyze the incentives of a stand-alone
firm one proceeds as for the merged firm but defines $S$ as the facilities
owned by that firm alone.
Nothing else needs to be changed.
If a competition agency determines that a merger is likely to lead to an
anticompetitive price increase, it may wish to identify a divestiture that
will allow the firms to merge and capture any (merger-specific) efficiencies
while removing the merged firm's incentive to raise price.
Using the capacity closure method, one can, for each $s \in S$, calculate
whether a firm that owned the facilities in $s$ would find it profitable to
raise the price.
Any $s \in S$ for which the merged firm would not find it profitable to raise
price is a set of mills that can be owned by the merged firm without raising
antitrust concerns.
The competition agency can then select its preferred $s$---e.g., the $s$ that
disrupts the transaction's efficiencies the least---and require the merged
firm to divest any facilities that the stand-alone firms own that are not in
$s$.
Alternatively, the competition agency can use the capacity closure method to
test whether leaving the merged firm with only $R \subset S$ facilities
removes its incentive to raise price.
One can do this by testing whether there exists an $r \in R$ such that
$p_r > p_0$.
If no such $r$ exists, then a divestiture that leaves the merged firm with
only those facilities in $R$ is a sufficient remedy.
This approach is useful if a divestiture package has been
proposed by the merging parties, its customers or other market participants.
\section{Data Requirements} \label{s:data}
\subsection{Facility costs}
The capacity closure method requires data on the costs of all
the facilities owned by the merging parties if it is to be used to analyze
competitive effects, the but-for world and divestitures.
If market definition is also to be addressed, these data are needed for all
facilities in the proposed market.
The method's basic assumption---that capacity is reduced by closing
facilities---short circuits the usual debate about what constitutes a variable
cost, since any cost that will not be incurred once the facility is closed
should be classified as variable.\footnote{In practice, with the exception of
taxes and depreciation, all of a facility's costs generally end up being
classified as variable.}
The cost data necessary to estimate $c_i$ for each facility can therefore
generally be provided by the merging parties with little effort.
Indeed, it is often the case that the required cost data are found in
ordinary course of business documents, which can significantly shorten the
time needed to begin analyzing a merger.
\subsection{Closing costs}
Given the capacity closure method's core assumption, the costs of closing a
facility must be considered.
These closing costs often include---but are not limited to---severance
packages, environmental clean up, inventory write-off and demolition.
Once the relevant closing costs have been identified, they are most easily
incorporated into the analysis by introducing them as a lump sum loss in the
profit function.
Closing costs should be treated as facility specific whenever possible as
there is often significant variation in these costs across facilities.
In the event that one cannot form an independent evaluation
of each facility's closing cost, one may wish to apply the same generic
closing cost to each facility or to assign each facility a closing cost that
is a function of the facility's capacity.
Two final points on closing costs should be made.
First, one must verify that closing a facility is in fact costly: in some
cases the sale of the land on which a facility is located will generate enough
income to cover closing expenses.
Second, it is reasonable to treat closing costs as an annuity.
If this approach is taken, the per period cost of servicing the annuity is
deducted from the firm's per period profit.
\subsection{Capacity, output, imports and prices}
Reliable data on each in-market facility's capacity and output can usually be
gathered directly from producers or industry trade groups.
Reliable data on the current level of imports are often more difficult to
gather, but potential data sources include known importers, industry trade
groups and National governments.
Obtaining data on $p_0$ is not as always as straightforward as it sounds,
since even homogenous products are often sold in different
forms.\footnote{Newsprint, for example, is sold in different basis weights}
However, it is typically the case that some estimate of $p_0$ can be formed
from data provided by the merging firms.
If this is not possible, it may be possible to estimate $p_0$ using data
collected by an industry trade group or by speaking with customers of the
merging firms.
\subsection{Other parameters}
In addition to the basic data already mentioned, the capacity closure method
requires data on the values of a number of other variables (e.g., $\eta_d$,
$I$, etc.).
The values of these variables can typically be estimated or approximated using
data gathered from the merging parties, competitors, customers or industry
trade groups.
Since the predictions of the models depend upon these parameter values, it is
wise to undertake some sensitivity testing.
Re-running the capacity closure method using different input parameters allows
one to test the robustness of its predictions and to identify particular
variables whose values are of critical importance (so that, e.g., their true
values can be investigated more intensively).
\section{Extending the Capacity Closure Method} \label{s:bells}
One way to extend the capacity closure method is to impose different
assumptions about the demand function.
Such extensions change the calculation of the function $p(\cdot)$.
I shall not discuss this class of extensions further because, though such an
extension may be necessary in rare cases, the two demand functions
already described should generally suffice.
A different class of extensions is those that change the calculation of the
profit function, $\pi(\cdot)$, but not the price function $p(\cdot)$.
Extensions in this class are easy to implement and can address the sorts of
economic issues that frequently arise during merger cases.
In the subsections below I provide two examples of such extensions to
highlight the fact that the capacity closure method can be easily modified to
accommodate the facts of a particular merger.
\subsection{Long-term contracts}
It is often the case that some customers of the merging firms hold contracts
that insulate them from price changes for a fixed length of time.
When such contracts are long enough and numerous enough, they can
significantly affect the incentives of the merged firm.\footnote{Such
contracts also affect the incentives of the stand-alone firms, of course.}
The effects of long-term contracts can be incorporated into the capacity
closure method by adjusting the impact of a price increase on the merged
firm's profits.
Specifically, the price increase associated with a capacity closure is
applied only to the merged firm's unprotected sales.
Consider, for example, a merger in which $\alpha$ percent of the merged firm's
sales are protected from price increases by long-term contracts.
As before let $s$ denote a set of facilities to be closed, let $q(s)$ denote
the firm's sales if it closes the facilities $s$ and let $c(s)$ denote the per
unit cost of producing $q(s)$.
The price after the facilities in $s$ are closed is of course $p_s$.
The post-merger average price, denoted $\bar{p}(s)$, is
\begin{equation}
\bar{p}(s) = (1-\alpha)p_s + \alpha p_0
\end{equation}
The merged firm's profit after closing the facilities in $s$ is
\begin{equation}
\pi(s) = [\bar{p}(s) -c(s)] q(s)
\end{equation}
Notice that the presence of $\alpha$ does not affect the calculation of
$p(\cdot)$ and so adds no meaningful complication to the task of identifying
the set of optimal closures $s^*$.\footnote{As before, $s^*$ is simply the $s
\in S$ that maximizes $\pi(s)$.}
\subsection{Recapture}
Recapture can be an issue in a merger involving homogenous product A if one of
the merging parties also produces a product B that is a substitute for A but
is not a close enough substitute to be included in the antitrust market.
In such a case, an increase in the price of product A will trigger increased
demand for product B.
If the merged firm were to capture some of the increased sales of product B,
it would soften the blow of having lost some sales of product A.
The degree to which the blow would be softened will depend upon such factors
as the relative margins earned on the two products and the price elasticity of
demand for product B with respect to the price of product A.\footnote{Bear in
mind that product B cannot be \textit{too} close a substitute for product A or
it would belong in the market.}
Consider a case in which both of the merging firms produce product $A$ and one
of the merging firms also produces product $B$.
Suppose further that product $A$ is a relevant product market, that product
$B$ is product $A$'s closest substitute, and that product $B$ is a relevant
product market.
Let $d(s)$ denote that sales that will switch from product $A$ to product $B$
in the event that the facilities in $s$ are closed (and the price of product
$A$ rises from $p_0$ to $p_s$), let $\sigma_B$ denote the merged firm's
market share for product $B$ and let $m_B$ denote the merged firm's margin on
the sales of product $B$.
The effect of recapture on the merged firm's incentive to close facilities
that produce product $A$ can be incorporated by adding to $\pi(s)$ the
quantity
\begin{equation}
m_B \sigma_B d(s)
\end{equation}
Note two assumptions that are implicit in the equation above.
First, the merged firm's margin on product $B$ is assumed not to change when
the demand for product $B$ increases (i.e., the market for product $B$ is
assumed to be competitive).
This assumption can easily be relaxed to incorporate a price effect in the
market for product $B$.
Second, the merged firm is assumed to capture transferred sales in proportion
to its market share.
This may be inappropriate if customers would punish the merged firm by
purchasing product $B$ from rival suppliers.
It may also be inappropriate if some of the producers of product $B$
(including the merged firm) do not have capacity available to expand their
production of product $B$.
If this assumption is deemed inappropriate for either of these reasons, it too
can be easily modified.
\section{Conclusion}
I have described a simple method for performing merger
simulations for industries with high fixed costs and homogenous products.
The method can be used to identify antitrust markets, to perform competitive
effects analysis, and to evaluate divestitures.
The method requires only data that are typically available to antitrust
agencies.
Finally, I have shown how the method can be modified to incorporate
the types of issues that often arise in merger cases.
\end{document}