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%An Empirical Model of Spatial Competition with an Application to
%Cement
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\begin{abstract}
% beginning of the abstract
The theoretical literature of industrial organization shows that the
distances between consumers and firms have first-order implications for
competitive outcomes whenever transportation costs are large. To assess
these effects empirically, we develop a structural model of competition
among spatially differentiated firms and introduce a GMM estimator that
recovers the structural parameters with only regional-level data. We apply
the model and estimator to the portland cement industry. The estimation
fits, both in-sample and out-of-sample, demonstrate that the framework
explains well the salient features of competition. We estimate
transportation costs to be \$0.30 per tonne-mile, given diesel prices at the
2000 level, and show that these costs constrain shipping distances and
provide firms with localized market power. To demonstrate policy-relevance,
we conduct counter-factual simulations that quantify competitive harm from a
hypothetical merger. We are able to map the distribution of harm over
geographic space and identify the divestiture that best mitigates harm.
\end{abstract}

% REQUIRED

\thispagestyle{empty}

\begin{center}
ECONOMIC ANALYSIS GROUP

DISCUSSION PAPER

\vspace{2.8cm}

Competition among Spatially Differentiated Firms: An Empirical Model with an
Application to Cement

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

Nathan H. Miller* and Matthew Osborne** \\[0pt]

EAG 10-2 $\quad$ March 2010
\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 or the Bureau of Economic Analysis. \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 * \ Economic Analysis Group, Antitrust Division, U.S. Department
of Justice. Email: nathan.miller@usdoj.gov.

\noindent ** \ Bureau of Economic Analysis. Email: matthew.osborne@bea.gov.
We thank Robert Johnson, Ashley Langer, Russell Pittman, Chuck Romeo, Jim
Schmitz, Charles Taragin, Raphael Thomadsen, and seminar participants at the
Bureau of Economic Analysis, Georgetown University, and the U.S. Department
of Justice for valuable comments. Sarolta Lee, Parker Sheppard, and Vera
Zavin provided research assistance.

\newpage \thispagestyle{empty}

% end of the abstract

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

\section{Introduction}

Geography is understudied in the empirical literature of industrial
organization. Although the theoretical literature has established that the
physical distances between firms and consumers have first-order implications
for competitive outcomes whenever transportation costs are large (e.g., %
\citeN{hotelling29}, \citeN{agt79}, \citeN{salop79}, \citeN{thissevives88}, %
\citeN{econ89}, \citeN{vogel08}), the complexities associated with modeling
spatial differentiation have made it difficult to translate theoretical
insights into workable empirical models.\footnote{%
Surely it would be too strong to claim that, in empirical industrial
organization, space is the final frontier.}

Standard empirical methodologies simply sidestep spatial differentiation
through the delineation of distinct geographic markets. This simplifies
estimation but requires the dual assumptions that (1) transportation costs
are sufficiently large to preclude substantive competition across market
boundaries, and (2) transportation costs are sufficiently small that spatial
differentiation is negligible within markets. It can be difficult to meet
both conditions.\footnote{\citeN{syverson04} discusses how this tension can
compel researchers to seek compromise between markets that are ``too small''
and markets that are ``too large''. It is sometimes argued that markets that
are too large may overstate the intensity of competition while markets that
are too small may understate competition.} In practice, markets are often
based on political borders of questionable economic significance (e.g.,
state or county lines). Nonetheless, market delineation is employed
routinely in studies of industries characterized by high transportation
costs, including ready-mix concrete (e.g., \citeN{syverson04}, %
\citeN{syversonhortacsu07}, \citeN{cw09}), portland cement (e.g., %
\citeN{salvo08}, \citeN{ryan06}), and paper (e.g., \citeN{pes03}).\footnote{%
These valuable contributions focus on wide range of topics, including the
competitive impacts of horizontal and vertical mergers, heterogeneity in
plant productivity and its implications for competition, the inference of
market power, and dynamic investment decisions.}

Our purpose is to introduce an alternative empirical framework. To that end,
we develop and estimate a structural model of competition among spatially
differentiated firms that accounts for transportation costs in a realistic
and tractable manner. We focus on production and consumption within a two
dimensional Euclidean space, which we refer to informally as a geographic
space. Competition involves a discrete number of plants, each endowed with a
physical location, and a continuum of consumers that spans the space. Each
plant sets a distinct price to each consumer, taking into consideration its
proximity to the consumer and the proximities of its competitors. Thus,
plants discriminate between elastic and inelastic consumers based on the
pre-determined plant and consumer locations, and the model resembles the
theoretical work of \citeN{thissevives88}. Competitive outcomes depend on
the magnitude of transportation costs and the degree of spatial
differentiation within the geographic space.

We discretize the geographic space into small ``consumer areas'' to
operationalize the model. Since each plant may ship to each consumer area,
the model diverges starkly from more conventional approaches in which plants
do not compete across market boundaries. We employ standard
differentiated-product methods to model competition within consumer areas.
On the supply side, domestic plants compete in prices given capacity
constraints and the existence of a competitive fringe of foreign importers.
On the demand side, consumers select the plant that maximizes utility,
taking into consideration their proximity to the plants, the plant prices,
and a nested logit error term. To be clear, the plants are differentiated
primarily by price and location -- we assume that the product itself is
homogeneous. We derive an equation that characterizes the equilibrium price
and market share for each plant-area pair as a function of data and
parameters.\footnote{%
We refer to the fraction of potential demand in a consumer area that is
captured by a given plant as the ``market share'' of the plant. We select
the term purely for expositional convenience. We do not argue that consumer
areas reflect antitrust markets in any sense; indeed, a defining
characteristic of the model is that it avoids market delineation entirely.}

The central challenge for estimation is that prices and market shares are
unobserved in the data, at least at the plant-area level. We develop a
generalized method of moments (GMM) estimator that exploits variation in
data that are more often observed: regional level consumption, production,
and prices. The key insight is that each candidate parameter vector
corresponds to an equilibrium set of plant-area prices and market shares. We
compute numerical equilibrium for each candidate parameter vector using a
large-scale nonlinear equation solver developed in \citeN{dfsane06}. We then
aggregate the predictions of the model to the regional level and evaluate
the distance between the data and the predictions. The estimator can be
interpreted as having inner and outer loops: the outer loop minimizes an
objective function over the parameter space while the inner loop computes
numerical equilibrium for each candidate parameter vector. We show that the
estimator consistently recovers the structural parameters of the data
generating process in an artificial data experiment.

%\footnote{We conduct an artificial data
%experiment to evaluate the accuracy of the estimator.
%The results suggest that the estimator consistently recovers the structural parameters
%of the data generating process.}

We apply the model and the estimator to the portland cement industry in the
U.S. Southwest over the period 1983-2003. The choice of industry conveys at
least three substantive advantages to the analysis. First, transportation
costs contribute substantially to overall consumer costs because portland
cement is inexpensive relative to its weight. Second, it may be reasonable
to treat portland cement as a homogenous product because strict industry
standards govern the production process.\footnote{%
Many plants produce a number of different types of portland cement, each
with slightly different specifications and characteristics (e.g., superior
early strength or higher sulfate resistance). These products are close
substitutes for most construction projects.} This conformity matches the
simplicity of the demand system, in which spatial considerations (e.g.,
plant and consumer locations) are the main source of plant heterogeneity.
Third, high quality data on the industry are publicly available. We obtain
information on regional consumption, production, and average prices, as well
as limited information on cross-region shipments, from annual publications
of the United States Geological Survey. We pair these regional-level metrics
with information on the location and characteristics of portland cement
plants from publications of the Portland Cement Association. We exploit
variation in these data to estimate the model.

The results of estimation suggest that consumers pay roughly \$0.30 per
tonne-mile, given diesel prices at the 2000 level.\footnote{%
Strictly speaking, the model identifies consumer willingness-to-pay for
proximity to the plant, which incorporates transportation costs as well as
any other distance-related costs (e.g., reduced reliability). We refer to
this willingness-to-pay as the transportation cost, although the concepts
may not be precisely equivalent.} Given the shipping distances that arise in
numerical equilibrium, this translates into an average transportation cost
of \$24.61 per metric tonne over the sample period -- sufficient to account
for 22 percent of total consumer expenditure. Costs of this magnitude have
real effects on competition in the industry. We focus on two such effects:
First, transportation costs constrain the distance that cement can be
shipped economically. The results indicate that cement is shipped only 92
miles on average between the plant and the consumer; by contrast, a
counter-factual simulation suggests that the average shipment would be 276
miles absent transportation costs. Second, transportation costs insulate
firms from competition and provide localized market power. For instance, the
prices that characterize numerical equilibrium decrease systematically in
the distance between the plant and the consumer, as do the corresponding
market shares.\footnote{%
These patterns are precisely what economic theory would predict given that
consumers pay the costs of transportation in the portland cement industry.}

The estimation procedure produces impressive in-sample and out-of-sample
fits despite parsimonious demand and marginal cost specifications. The model
predictions explain 93 percent of the variation in regional consumption, 94
percent of the variation in regional production, and 82 percent of the
variation in regional prices. The model predictions also explain 98 percent
of the variation in cross-region shipments, even though we withhold the bulk
of these data from estimation. As we detail in an appendix, the quality of
these fits is underscored by the rich time-series variation in these data.
Together, the regression fits suggest that a small quantity of exogenous
data, properly utilized, may be sufficient to explain some of the most
salient features of competition in the portland cement industry. We
interpret this as substantial support for the power of the analytical
framework.

We suspect that our method may prove useful for future research and policy
endeavors relating to international trade, environmental economics, and
industrial organization. One such application is merger simulation, an
important tool for competition policy. We use counter-factual simulations to
evaluate the effects of a hypothetical merger between two multi-plant cement
firms in 1986. We find that the merger reduces consumer surplus by \$1.4
million if no divestitures are made, and we are able to map the distribution
of this harm over the U.S. Southwest. The overall magnitude of the effect is
modest relative to the amount of commerce -- by way of comparison, we
calculate total pre-merger consumer surplus to be more than \$239 million.
We then consider the six possible single-plant divestitures, and find that
the most powerful reduces consumer harm by 56 percent.\footnote{%
We refer to the divesture plan that offsets the greatest amount of consumer
harm among the set of single-plant divestitures as the most powerful. We do
not attempt to characterize, in any way, the appropriate course of action
for an antitrust authority.}

At least two caveats are important. First, the estimation procedure rests on
the uniqueness of equilibrium at each candidate parameter vector, but there
is no theoretical reason to expect this condition to hold generally. To
assess the issue, we conduct a Monte Carlo experiment that computes
numerical equilibrium using several different starting points for each of
6,300 randomly drawn candidate parameter vectors. The results are strongly
supportive of uniqueness, at least in our application (see the appendices
for details). Second, the promise of spatial differentiation creates
incentives for firms to locate optimally in order to secure a base of
profitable customers, provide separation from an efficient competitor,
and/or deter nearby entry. We abstract from these considerations entirely
and instead assume that firm location is pre-determined and exogenous.
Nonetheless, our framework could help define stage-game payoffs in more
dynamic models that endogenize firm location choices (e.g., as in %
\citeN{seim06} and \citeN{aguirregabiria06}).

Our work builds on recent contributions to the industrial organization
literature that model competition among spatially differentiated retail
firms (e.g., \citeN{thomadsen051}, \citeN{davis06}, \citeN{mcmanus09}).
These papers employ a framework in which each firm sets a single price to
all consumers, consistent with practice in most retail settings, and recover
the structural parameters with firm-level data and more standard estimation
techniques (e.g., \citeN{blp95}).\footnote{%
Thomadsen (2005) may be the closest antecedent to our work. Thomadsen shows
that a supply-side equilibrium condition can be substituted for firm-level
market shares in estimation. We develop the potency of equilibrium
conditions more fully: given an equilibrium condition and aggregate data,
estimation is feasible with neither firm-level prices nor firm-level market
shares. \citeN{aguirregabiria06} develop a sophisticated model of spatial
differentiation but do not take the model to data.} We make two distinct
contributions to this literature. First, our model extends the existing
framework to incorporate firms that price discriminate among consumers. In
such a setting, firm-level data are no longer sufficient to support standard
estimation techniques. Thus, our second contribution is the demonstration
that estimation is feasible with relatively aggregated data. Of course, the
regional-level data we use in our application would also support estimation
in simpler retail settings.

More generally, our results indicate that firm-level heterogeneity can
affect competitive outcomes even when the product is relatively homogenous.
Refinements to the standard toolbox available for structural research in
these settings have been outstripped in empirical industrial organization by
models of observed and (especially) unobserved product heterogeneity. We
suspect that models of competition for homogenous-product industries are
currently of heightened value, not only because of the imbalance in the
literature, but also because these industries provide fertile testing
grounds for methodological innovations regarding industry dynamics (e.g., as
in \citeN{ryan06}). We hope that the framework we introduce helps,
incrementally, to redirect the attention of researchers to these settings.

The paper proceeds as follows. In Section \ref{sec:ind}, we sketch the
relevant institutional details of the portland cement industry, focusing on
transportation costs, production technology, and trends in production and
consumption. We develop the empirical model of Bertrand-Nash competition in
Section \ref{sec:model} and discuss our data sources in Section \ref%
{sec:data}. Then, in Section \ref{sec:est}, we develop the estimator and
provide identification arguments. We present the estimation results in
Section \ref{sec:res}, discuss the merger simulations in Section \ref%
{sec:sim}, and then conclude.

\section{The Portland Cement Industry \label{sec:ind}}

Portland cement is a finely ground dust that forms concrete when mixed with
water and coarse aggregates such as sand and stone. Concrete is an essential
input to many construction and transportation projects, either as pourable
fill material or as pre-formed concrete blocks, because its local
availability and lower maintenance costs make it more economical than
substitutes such as steel, asphalt, and lumber (e.g., \citeN{vanoss02}).%
\footnote{%
We draw heavily from the publicly available documents and publications of
the United States Geological Survey and the Portland Cement Association to
support the analysis in this section. We defer detailed discussion of these
sources for expositional convenience.}

Most portland cement is shipped by truck to ready-mix concrete plants or
construction sites, in accordance with contracts negotiated between
individual purchasers and plants.\footnote{%
A smaller portion is shipped by train or barge to terminals, and only then
distributed to consumers by truck. Shipment via terminals reduces
transportation costs for more distant consumers. Roughly 23 percent of
portland cement produced in the United States was shipped through terminals
in 2003.} Transportation costs contribute substantially to overall consumer
expenditures because portland cement is inexpensive relative to its weight,
a fact that is well understood in the academic literature. For example,
Scherer et al (1975) estimates that transportation would have accounted for
roughly one-third of total consumer expenditures on a hypothetical 350-mile
route between Chicago and Cleveland, and a 1977 Census Bureau study
determines that most portland cement is consumed locally -- for example,
more than 80 percent is transported within 200 miles.\footnote{%
Scherer et al (1975) examined more than 100 commodities and determined that
the transportation costs of portland cement were second only to those of
industrial gases. Other commodities identified as having particularly high
transportation costs include concrete, petroleum refining,
alkalies/chlorine, and gypsum.} More recently, \citeN{salvo08b} presents
evidence consistent with the importance of transportation costs in the
Brazilian portland cement industry.

A recent report prepared for the Environmental Protection Agency identifies
five main variable input costs of production: raw materials, coal,
electricity, labor, and kiln maintenance (\citeN{epa09}). In the production
process itself, a feed mix composed of limestone and supplementary materials
is fed into large rotary kilns that reach temperatures of 1400-1450$^\circ$
Celsius. The combustion of coal is the most efficient way to generate this
extreme heat. Kilns generally operate at peak capacity with the exception of
an annual shutdown period for maintenance. It is possible to adjust output
by extending or shortening the maintenance period -- for example, plants may
simply forego maintenance at the risk of kiln damage and/or breakdowns. The
feed mix exits the kiln as semi-fused clinker modules. Once cooled, the
clinker is mixed with a small amount of gypsum, placed into a grinding mill,
and ground into tiny particles averaging ten micrometers in diameter. This
product -- portland cement -- is shipped to purchasers either in bulk or
packaged in smaller bags.

We focus on the production and consumption of portland cement in the U.S.
Southwest -- by which we mean California, Arizona and Nevada -- over the
period 1983-2003. This region accounts for roughly 15 percent of domestic
portland cement production and consumption during the sample period. Figure %
\ref{fig:map1} provides a map of the region based on the plant locations in
the final year of the sample. As shown, most plants are located along an
interstate highway and nearby one or more population centers. Although some
firms own more than one plant, production capacity in the area is not
particularly concentrated -- the capacity-based Herfindahl-Hirschman Index
(HHI) of 1260 is well below the threshold level that defines highly
concentrated markets in the 1992 Merger Guidelines. The plants also face
competition from foreign importers that ship portland cement through the
customs offices of San Francisco, Los Angeles, San Diego, and Nogales.
Still, transportation costs may insulate some plants from both foreign and
domestic competition.\footnote{%
We observe little entry and exit over the sample period. The sole entrant
(Royal Cement) began operations in 1994 and the only two exits occurred in
1988. This stability is consistent with substantial sunk costs of plant
construction, as documented in \citeN{ryan06}. Plant ownership is somewhat
more fluid; we observe fourteen changes in plant ownership, spread among
nine of the sixteen plants.}

\begin{figure}[t]
\centering
\includegraphics[height=4in]{map-col-hwy-2.eps}  
\caption{{\protect\footnotesize {Portland Cement Production Capacity in the
U.S. Southwest circa 2003. }}}
\label{fig:map1}
\end{figure}

% The Figure is a map of California, Arizona, and Nevada.  Cement plants are marked with circles, with
% higher capacity plants getting bigger circles.  The four import points -- San Francisco, Los Angeles,
% San Diego, and Nogales -- are marked with black squares.  See text for details.

In Figure \ref{fig:imports}, we plot total consumption and production in the
U.S. Southwest over the sample period, together with two measures of foreign
imports. Several patterns are apparent. Both consumption and production
increase over 1983-2003, and both metrics are highly cyclical. However,
consumption is more cyclical than production, so that the gap between
consumption and production increases in overall activity; foreign importers
provide additional supply whenever domestic demand outstrips domestic
capacity. Strikingly, observed foreign imports are nearly identical to
``apparent imports,'' which we define as consumption minus production,
consistent with negligible net trade between the U.S. Southwest and other
domestic areas. Finally, we note that the average free-on-board price
charged by domestic plants in the region (not shown) falls over the sample
period from \$107 per metric tonne to \$74 per metric tonne, primarily due
to lower coal and electricity costs.\footnote{%
Prices are in real 2000 dollars.}

\begin{figure}[t]
\centering
\includegraphics[height=3in]{imports.eps}  
\caption{{\protect\footnotesize {Consumption, Production, and Imports of
Portland Cement. Apparent imports are defined as consumption minus
production. Observed imports are total foreign imports shipped into San
Francisco, Los Angeles, San Diego, and Nogales.}}}
\label{fig:imports}
\end{figure}

%The figure plots consumption, production, and imports over the sample period.  It also plots the apparent imports,
%defined as consumption minus production.  The text describes the relevant data patterns in detail.

\section{Empirical Model \label{sec:model}}

\subsection{Overview}

We develop a structural model of competition that accounts for
transportation costs in a realistic and tractable manner. We focus on
production and consumption within a single geographic space. Competition
involves a discrete number of plants, each endowed with a physical location,
and a continuum of consumers that spans the space. Each plant sets a
distinct price to each consumer, taking into consideration its proximity to
the consumer and the proximities of its competitors. To operationalize the
model, we discretize the geographic space into small consumer areas. We
employ standard techniques to model competition within consumer areas:
plants charge area-specific prices subject to exogenous capacity
constraints, and consumer demand is nested logit. We derive an equation that
characterizes the equilibrium price and market share for each plant-area
pair.

\subsection{Supply}

We take as given that there are $F$ firms, each of which operates some
subset $\Im_f$ of the $J$ plants. Each plant is endowed with a set of
attributes, including a physical location. Consumers exist in $N$ different
areas that span the geographic space. Every plant can ship into every area.
Firms set prices at the plant-area level to maximize variable profits: 
\begin{equation}
\pi_{f} = \underbrace{\sum_{j \in \Im_{f}} \sum_{n} P_{jn}Q_{jn}(\boldsymbol{%
\theta_d})}_{\text{variable revenues}} - \underbrace{\sum_{j \in \Im_{f}}
\int_0^{Q_j (\boldsymbol{\theta_d})}MC_j(Q; \boldsymbol{\theta_c}) dQ}_{%
\text{variable costs}},  \label{eq:vprofs}
\end{equation}
where $Q_{jn}$ and $P_{jn}$ are the quantity and price, respectively, of
plant $j$ in area $n$, $MC_j$ is the marginal cost of production, and $Q_j$
is total plant production (i.e., $Q_j=\sum_n Q_{jn}$). The vectors $%
\boldsymbol{\theta_c}$ and $\boldsymbol{\theta_d}$ include the supply and
demand parameters, respectively, and together form the joint parameter
vector $\boldsymbol{\theta} = (\boldsymbol{\theta_c}, \boldsymbol{\theta_d})$%
.

The marginal cost function can accommodate most forms of scale economies or
diseconomies. We choose a functional form that captures the plant-level
capacity constraints that are important for portland cement production. In
particular, we let marginal costs increase flexibly in production whenever
production exceeds some threshold level. The marginal cost function is: 
\begin{equation}
MC(\boldsymbol{w}_{j},Q_{j}(\boldsymbol{\theta_d}); \boldsymbol{\theta_c}) = 
\boldsymbol{w}_{j}^\prime \boldsymbol{\alpha} + \gamma \; 1 \left\{ \frac{%
Q_j(\boldsymbol{\theta_d})}{CAP_j}>\nu \right\} \left(\frac{Q_{j}(%
\boldsymbol{\theta_d})}{CAP_{j}} - \nu \right) ^\phi ,  \label{eq:mc}
\end{equation}
where $\boldsymbol{w_j}$ is a vector that includes the relevant marginal
cost shifters and $CAP_j$ is the maximum quantity that plant $j$ can
produce. The parameter $\nu$ is the utilization threshold above which
marginal costs increase in production, the parameter $\phi$ determines the
curvature of the marginal cost function when utilization exceeds $\nu$, and
the combination $\gamma(1-\nu)^\phi$ represents the marginal cost penalty
associated with production at capacity. The marginal cost function is
continuously differentiable in production for $\phi>1$. The vector of cost
parameters is $\boldsymbol{\theta_c}= (\boldsymbol{\alpha}, \gamma, \nu,
\phi)$.

We let the domestic firms compete against a competitive fringe of foreign
importers, which we denote as plant $J+1$. We assume that the fringe is a
non-strategic actor and that import prices are exogenously set based on some
marginal cost common to all importers. The fringe is endowed with one or
more geographic locations and ships into every consumer area. It sets a
single price across all consumer areas (i.e., the fringe does not price
discriminate), consistent with perfect competition among importers.

\subsection{Demand}

We specify a nested logit demand system that captures the two most important
characteristics that differentiate portland cement plants: price and
location. To that end, we assume that each area features many potential
consumers. Each consumer either purchases cement from one of the domestic
plants or the importer, or foregoes a purchase of portland cement
altogether. We refer to the domestic plants and the importer as the inside
goods, and refer to the option to forego a purchase as the outside good. We
place the inside goods in a separate nest from the outside good.

We express the indirect utility that consumer $i$ receives from plant $j$ as
a function of the relevant plant and location observables: 
\begin{equation}
u_{ij} = \beta_0 + \boldsymbol{x}_{jn}^\prime \boldsymbol{\beta} + \zeta_i +
\lambda \epsilon_{ij},  \label{eq:util}
\end{equation}
where the vector $\boldsymbol{x}_{jn}$ includes the price of cement and the
distance between the plant and the area, as well as other plant-specific
demand shifters. The idiosyncratic portion of the indirect utility function
is composed of consumer-specific shocks to the desirability of the inside
good ($\zeta_i$) and the desirability of each plant ($\epsilon_{it}$). We
assume that the combination $\zeta_i + \lambda \epsilon_{ij}$ has an extreme
value distribution in which the parameter $\lambda$ characterizes the extent
to which valuations of the inside good are correlated across consumers.%
\footnote{\citeN{cardell97} derives the conditions under which the specified
taste shocks produce the extreme value distribution. Tastes are perfectly
correlated if $\lambda=0$ and tastes are uncorrelated if $\lambda=1$. In the
latter case the model collapses to a standard logit.} We normalize the mean
utility of the outside good to zero, so that the indirect utility associated
with foregoing cement purchase is $u_{i0}=\epsilon_{i0}$, with $\epsilon_{i0}
$ also having the extreme value distribution. The vector of demand
parameters is $\boldsymbol{\theta_d}=(\beta_0, \boldsymbol{\beta}, \lambda)$.%
\footnote{%
We exclude product characteristics from the specification because portland
cement is a homogenous good, at least to a first approximation. It may be
desirable to control for observed product characteristics in other
applications. (Though the presence of unobserved characteristics would pose
a challenge to the estimation procedure.) Also, we assume that consumers pay
the cost of transportation, consistent with practice in the cement industry.
One could alternatively incorporate distance into the marginal cost function.%
}

The nested logit structure yields an analytical expression for the market
shares captured by each plant. The market shares are specific to each
plant-area pair because the relative desirability of each plant varies
across areas: 
\begin{equation}
S_{jn}(\boldsymbol{P}_{n}; \boldsymbol{\theta_d}) = \frac{\exp(\beta_0+
\lambda I_{n})}{1+\exp(\beta_0+\lambda I_{n})} * \frac{\exp(\boldsymbol{x}%
_{jn}^\prime \boldsymbol{\beta})}{\sum_{k=1}^{J+1} \exp(\boldsymbol{x}%
_{kn}^\prime \boldsymbol{\beta})},  \label{eq:shares}
\end{equation}
where $\boldsymbol{P}_n$ is a vector of area-specific prices and $I_{n}= \ln
\left( \sum_{k=1}^{J+1} \exp(\boldsymbol{x}_{kn}^\prime \boldsymbol{\beta})
\right)$ is the inclusive value of the inside goods. The first factor in
this expression is the marginal probability that a consumer in area $n$
selects an inside good, and the second factor is the conditional probability
that the consumer purchases from plant $j$ given selection of an inside
good. Both take familiar logit forms due to the distributional assumptions
on idiosyncratic consumer tastes. The demand system maps cleanly into
supply: the quantity sold by plant $j$ to area $n$ is $Q_{jn} = S_{jn}M_n$,
where $M_n$ is the potential demand of area $n$.\footnote{%
The substitution patterns between cement plants are characterized by the
independence of irrelevant alternatives (IIA) within the inside good nest.
We argue that IIA is a reasonable approximation for our application.
Portland cement is purchased nearly exclusively by ready-mix concrete plants
and other construction companies. These firms employ similar production
technologies and compete under comparable demand conditions. We are
therefore skeptical that meaningful heterogeneity exists in consumer
preferences for plant observables (e.g., price and distance). Without such
heterogeneity, the IIA property arises quite naturally -- for example, the
random coefficient logit demand model collapses to standard logit when the
distribution of consumer preferences is degenerate.}

\subsection{Equilibrium}

The profit function yields the first-order conditions that characterize
equilibrium prices for each plant-area pair: 
\begin{equation}
Q_{jn} + \sum_{n} \sum_{k \in \Im_{f}} (P_{kn}-MC_{kn})\frac{\partial Q_{kn}%
}{\partial P_{jn}}=0.
\end{equation}
Since each plant competes in every consumer area and may price differently
across areas, there are $J\times N$ first-order conditions. For notational
convenience, we define the block-diagonal matrix $\boldsymbol{\Omega(P)}$ as
the combination of $n=1,\dots,N$ sub-matrices, each of dimension $J \times J$%
. The elements of the sub-matrices are defined as follows: 
\begin{equation}
\Omega^n_{jk}(\boldsymbol{P}) = \left\{ 
\begin{array}{ll}
\frac{\partial Q_{jn}}{\partial P_{kn}} & \text{if $j$ and $k$ have the same
owner} \\ 
0 & \text{otherwise},%
\end{array}
\right.  \label{eq:omega}
\end{equation}
where the demand derivatives take the nested logit forms. Thus, the elements
of each sub-matrix $\Omega^n_{jk}(\boldsymbol{P})$ characterize substitution
patterns within area $n$, and the matrix $\boldsymbol{\Omega(P)}$ has a
block diagonal structure because prices in one area do not affect demand in
other areas. We can now stack the first-order conditions: 
\begin{equation}
\boldsymbol{P} = \boldsymbol{MC(P)} - \boldsymbol{\Omega(P)}^{-1}\boldsymbol{%
Q(P)},  \label{eq:equil}
\end{equation}
where $\boldsymbol{P}$, $\boldsymbol{MC(P)}$, and $\boldsymbol{Q(P)}$ are
vectors of prices, marginal costs and quantities, respectively. Provided
that marginal cost parameter $\phi$ exceeds one, the mappings $\boldsymbol{%
MC(P)}$, $\boldsymbol{\Omega(P)}$, and $\boldsymbol{Q(P)}$ are each
continuously differentiable. Further, the price vector belongs to a compact
set in which each price is (1) greater or equal to the corresponding
marginal cost, and (2) smaller than or equal to the price a monopolist would
charge to the relevant area. Therefore, by Brower's fixed-point theorem,
Bertrand-Nash equilibrium exists and is characterized by the price vector $%
\boldsymbol{P^*}$ that solves Equation \ref{eq:equil}.\footnote{%
The existence proof follows \citeN{aguirregabiria06}. Multiple equilibria
may exist.}

Spatial price discrimination is at the core of the firm's pricing problem:
firms maximize profits by charging higher prices to nearby consumers and
consumers without a close alternative. Aside from price discrimination, the
firm's pricing problem follows standard intuition. For example, a firm that
contemplates a higher price for cement from plant $j$ to area $n$ must
evaluate a number of effects: (1) the tradeoff between lost sales to
marginal consumers and greater revenue from inframarginal consumers; (2)
whether the firm would recapture lost sales with its other plants; and (3)
whether the lost sales would ease capacity constraints and make the plant
more competitive in other consumer areas.

\section{Data Sources and Summary Statistics \label{sec:data}}

\subsection{Data sources}

We cull the bulk of our data from the Minerals Yearbook, an annual
publication of the U.S. Geological Survey (USGS). The Minerals Yearbook is
based on an annual census of portland cement plant and contains
regional-level information on portland cement consumption, production, and
free-on-board prices.\footnote{%
The census response rate is typically well over 90 percent (e.g., 95 percent
in 2003), and USGS staff imputes missing values for the few non-respondents
based on historical and cross-sectional information. Other academic studies
that feature these data include \citeN{mcbride83}, \citeN{rosenbaumreading88}%
, \citeN{rosenbaum94}, \citeN{jansrosenbaum96}, \citeN{ryan06}, and %
\citeN{syversonhortacsu07}. The Minerals Yearbook provides the average
free-on-board price charged the plants located in each region, rather than
the price paid by the consumers in each region.} The four relevant regions
include Northern California, Southern California, Arizona, and Nevada. We
observe annual consumption in each region over the period 1983-2003. The
USGS combines the Arizona and Nevada regions when reporting production and
prices over 1983-1991, and no usable production or price information is
available for Nevada over 1992-2003.\footnote{%
The USGS combines Nevada with Idaho, Montana and Utah starting in 1992. We
adjust the Arizona data to remove the influence of a single plant located in
New Mexico whose production is aggregated into the region. We detail this
adjustment in an appendix. Also, it is worth noting that the USGS does not
intend for its regions to approximate geographic markets. Rather, the
regions are delineated such that plant-level information cannot be
backward-engineered from the Minerals Yearbook.} The Minerals Yearbook also
includes information on the price and quantity of portland cement that is
imported into the U.S. Southwest via the customs offices in San Francisco,
Los Angeles, San Diego, and Nogales.

We also make use of more limited data on cross-region shipments from the
California Letter, a second annual publication of the USGS. The California
Letter provides information on the quantity of portland cement that is
shipped from plants in California to consumers in Northern California,
Southern California, Arizona, and Nevada. However, the level of aggregation
varies over the sample period, some data are redacted to protect sensitive
information, and no information is available before 1990. In total, we
observe 96 data points:

\begin{tabular}{p{2.7in}p{2.7in}}
& 
\end{tabular}

%  CA to N. CA over 1990-2003    S. CA to N. CA over 1990-1999
%  CA to S. CA over 2000-2003   S. CA to S. CA over 1990-1999
%  CA to AZ over 1990-2003     S. CA to AZ over 1990-1999
%  CA to NV over 2000-2003    S. CA to NV over 1990-1999
%  N. CA to N. CA over 1990-1999  N. CA to AZ over 1990-1999

\noindent We withhold the bulk of these data from the estimation procedure
and instead use the data to conduct out-of-sample checks on the model
predictions.\footnote{%
The California Letter is based on a monthly survey rather than on the annual
USGS census. As a result, the data are not always consistent with the
Minerals Yearbook. We normalize the data prior to estimation so that total
shipments equal total production in each year.}

We supplement the USGS data with basic plant-level information from the
Plant Information Survey (PIS), an annual publication of the Portland Cement
Association. The PIS provides the location of each portland cement plant in
the United States, together with its owner, the annual kiln capacity, and
various other kiln characteristics.\footnote{%
We multiply kiln capacity by 1.05 to approximate cement capacity, consistent
with the industry practice of mixing clinker with a small amount of gypsum
(typically 3 to 7 percent) in the grinding mills.} We approximate consumer
areas using counties, which meet the criterion of being small relative to
the overall geographic space -- there are 90 counties in the U.S. Southwest.
We collect county-level data from the Census Bureau on construction
employment and residential construction permits to account for county-level
heterogeneity in potential demand. Finally, we collect data on diesel, coal,
and electricity prices from the Energy Information Agency, data on average
wages of durable good manufacturing employees from the BEA, and data on
crushed stone prices from the USGS; we exploit state-level variation for all
but the diesel data.

\subsection{Summary statistics}

We provide summary statistics on consumption, production, and prices for
each of the regions in Table \ref{tab:swData}. Some patterns stand out:
First, substantial variation in each metric is available, both
inter-temporally and across regions, to support estimation. Second, Southern
California is larger than the other regions, whether measured by consumption
or production. Third, consumption exceeds production in Northern California,
Arizona, and Nevada; these shortfalls must be countered by cross-region
shipments and/or imports. The observation that plants in these regions
charge higher prices is consistent with transportation costs providing some
degree of local market power.\footnote{%
The data on cross-region shipments are also suggestive of large
transportation costs. For example, more than 90 percent of portland cement
produced in Northern California was shipped to consumers in Northern
California over 1990-1999.} Finally, imports are less expensive than
domestically produced portland cement. This discrepancy may exist in part
because the reported prices exclude duties; more speculatively, domestic
producers may be more reliable or may maintain relationships with consumers
that support higher prices.

\begin{table}[tbp]
\caption{Consumption, Production, and Prices}
\begin{center}
\begin{tabular}[h]{lcccc}
\hline\hline
\rule[0mm]{0mm}{5.5mm}Description & Mean & Std & Min & Max \\ \hline
\rule[0mm]{0mm}{5.5mm}\textit{Consumption} &  &  &  &  \\ 
$\;$ Northern California & 3,513 & 718 & 2,366 & 4,706 \\ 
$\;$ Southern California & 6,464 & 1,324 & 4,016 & 8,574 \\ 
$\;$ Arizona & 2,353 & 650 & 1,492 & 3,608 \\ 
$\;$ Nevada & 1,289 & 563 & 416 & 2,206 \\ 
\rule[0mm]{0mm}{5.5mm}\textit{Production} &  &  &  &  \\ 
$\;$ Northern California & 2,548 & 230 & 1,927 & 2,894 \\ 
$\;$ Southern California & 6,316 & 860 & 4,886 & 8,437 \\ 
$\;$ Arizona-Nevada & 1,669 & 287 & 1050 & 2,337 \\ 
\rule[0mm]{0mm}{5.5mm}\textit{Domestic Prices} &  &  &  &  \\ 
$\;$ Northern California & 85.81 & 11.71 & 67.43 & 108.68 \\ 
$\;$ Southern California & 82.81 & 16.39 & 62.21 & 114.64 \\ 
$\;$ Arizona-Nevada & 92.92 & 14.24 & 75.06 & 124.60 \\ 
\rule[0mm]{0mm}{5.5mm}\textit{Import Prices} &  &  &  &  \\ 
$\;$ U.S. Southwest & 50.78 & 9.30 & 39.39 & 79.32 \\ \hline
\multicolumn{5}{p{3.8in}}{{\footnotesize {\rule[0mm]{0mm}{0mm}Statistics are
based on observations at the region-year level over the period 1983-2003.
Production and consumption are in thousands of metric tonnes. Prices are per
metric tonne, in real 2000 dollars. Import prices exclude duties. The region
labeled ``Arizona-Nevada" incorporates information from Nevada plants only
over 1983-1991.}}}%
\end{tabular}%
\end{center}
\end{table}

In Table \ref{tab:swDistData}, we explore the spatial characteristics of the
regions in more depth, based on the plant locations of 2003. First, the
average county in Northern California is 65 miles from the nearest domestic
cement plant. Since the comparable statistics for Southern California,
Arizona, and Nevada are 72 miles, 92 miles, and 100 miles, respectively, one
may infer that average transportation costs may differ substantively across
the four regions. Second, the average \textit{additional} distance to the
second closest domestic plant is 44 miles in Northern California, 11 miles
in Southern California, 82 miles in Arizona, and 77 miles in Nevada,
suggestive that plants in Arizona and Nevada may hold more local market
power than plants elsewhere. Finally, the average distance to the nearest
customs office is 123 miles in Northern California, 110 miles in Southern
California, 181 miles in Arizona, and 281 miles in Nevada, suggestive that
imports may constrain domestic prices less severely in Arizona and Nevada.
The latter two empirical patterns are consistent with the higher cement
prices observed in Arizona and Nevada.

\begin{table}[tbp]
\caption{Distances between Counties and Plants}
\begin{center}
\begin{tabular}[h]{lcccc}
\hline\hline
\rule[0mm]{0mm}{5.5mm}Description & Mean & Std & Min & Max \\ \hline
\multicolumn{5}{l}{\rule[0mm]{0mm}{5.5mm}\textit{Miles to the closest plant}}
\\ 
$\;$ Northern California & 64.65 & 30.00 & 7.36 & 115.39 \\ 
$\;$ Southern California & 72.28 & 39.74 & 18.58 & 127.46 \\ 
$\;$ Arizona & 91.62 & 40.01 & 29.13 & 163.99 \\ 
$\;$ Nevada & 100.04 & 61.97 & 17.38 & 232.03 \\ 
\multicolumn{5}{l}{\rule[0mm]{0mm}{5.5mm}\textit{Additional miles to the
second closest plant}} \\ 
$\;$ Northern California & 43.95 & 49.84 & 0.49 & 176.07 \\ 
$\;$ Southern California & 11.22 & 8.83 & 0.49 & 31.08 \\ 
$\;$ Arizona & 81.96 & 55.65 & 0.69 & 172.38 \\ 
$\;$ Nevada & 77.38 & 43.28 & 12.82 & 177.20 \\ 
\multicolumn{5}{l}{\rule[0mm]{0mm}{5.5mm}\textit{Miles to the closest import
point}} \\ 
$\;$ Northern California & 122.65 & 67.42 & 4.33 & 283.00 \\ 
$\;$ Southern California & 110.17 & 67.43 & 30.29 & 221.28 \\ 
$\;$ Arizona & 180.80 & 90.26 & 14.07 & 314.91 \\ 
$\;$ Nevada & 281.45 & 81.93 & 170.09 & 442.02 \\ \hline
\multicolumn{5}{p{3.8in}}{{\footnotesize {\rule[0mm]{0mm}{0mm}Distances are
calculated based on plant locations in 2003. There are 46 counties in
Northern California, 12 counties in Southern California, 15 counties in
Arizona, and 17 counties in Nevada.}}}%
\end{tabular}%
\end{center}
\end{table}

\section{Estimation \label{sec:est}}

\subsection{Overview}

The challenge for estimation is that prices and market shares are unobserved
in the data, at least at the plant-area level. We develop a GMM estimator
that exploits variation in data that are more often observed: regional level
consumption, production, and prices. The key insight is that each candidate
parameter vector corresponds to an equilibrium set of plant-area prices and
market shares. We compute equilibrium numerically for each candidate
parameter vector, aggregate predictions of the model to the regional-level,
and then evaluate the ``distance'' between the data and the aggregate
predictions. The estimation routine is an iterative procedure defined by the
following steps:

\begin{enumerate}
\item Select a candidate parameter vector $\boldsymbol{\widetilde{\theta}}$.

\item Compute the equilibrium price and market share vectors.

\item Calculate regional-level metrics based on the equilibrium vectors.

\item Evaluate the regional-level metrics against the data.

\item Update $\boldsymbol{\widetilde{\theta}}$ and repeat steps 1-5 to
convergence.
\end{enumerate}

The estimation procedure can be interpreted as having both an inner loop and
an outer loop: the inner loop computes equilibrium for each candidate
parameter vector and the outer loop minimizes an objective function over the
parameter space. We discuss the inner loop and the outer loop in turn, and
then address some additional details.

\subsection{Computation of numerical equilibrium}

We use a large-scale nonlinear equation solver developed in \citeN{dfsane06}
to compute equilibrium. The equation solver employs a quasi-Newton method
and exploits simple derivative-free approximations to the Jacobian matrix;
it converges more quickly than other algorithms and does not sacrifice
precision. We define a numerical Bertrand-Nash equilibrium as a price vector
for which $\parallel \boldsymbol{\iota(P)} \parallel / \; \text{dim}(%
\boldsymbol{\iota(P)}) < \delta$, where $\parallel \cdot \parallel$ denotes
the Euclidean norm operator and 
\begin{equation}
\boldsymbol{\iota(P)} = \boldsymbol{\Omega(P)}(\boldsymbol{P}-\boldsymbol{%
MC(P)}) - \boldsymbol{Q(P)}.
\end{equation}
We denote the equilibrium prices that correspond to the candidate parameter
vector $\boldsymbol{\widetilde{\theta}}$ as $\widetilde{P}_{jnt}(\boldsymbol{%
\widetilde{\theta}},\boldsymbol{\chi})$, where $\boldsymbol{\chi}$ includes
the exogenous data. We denote the corresponding equilibrium market shares as 
$\widetilde{S}_{jnt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi})$.
From a computational standpoint, our construction of $\boldsymbol{\iota(P)}$
avoids the burden of inverting $\boldsymbol{\Omega(P)}$ that would be
required by the straight application of Equation \ref{eq:equil}. Further,
the structure of the problem permits us to compute equilibrium separately
for each period. The price vector that characterizes the equilibrium of a
given period has length $J_t \times N$ so that, for example, the equilibrium
price vector for 2003 has $14 \times 90 =1,260$ elements.\footnote{%
We set $\delta=$1e-13. Numerical error can propagate into the outer loop
when the inner loop tolerance is substantially looser (e.g., 1e-7), which
slows overall estimation time and can produce poor estimates. The inner loop
tolerance is not unit free and must be evaluated relative to the price
level. We also note that, in settings characterized by constant marginal
costs, one could ease the computational burden of the inner loop by solving
for equilibrium prices in each consumer area separately.}

The empirical analogs to the computed plant-area prices and market shares
are not observed in the data, so we aggregate the prices and market shares
to construct more useful regional metrics. For notational precision, we
define the sets $\aleph_r$ and $\jmath_r$ as the counties and plants,
respectively, located in region $r$. The aggregated region-period metrics
take the form: 
\begin{eqnarray}
\widetilde{C}_{rt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi}%
)&=&\sum_{n\in\aleph_r}\left(1-\widetilde{S}_{0nt}(\boldsymbol{\widetilde{%
\theta}},\boldsymbol{\chi})\right)M_{nt}  \notag \\
\widetilde{Q}_{rt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi}%
)&=&\sum_{j\in\jmath_r}\widetilde{S}_{jnt}(\boldsymbol{\widetilde{\theta}},%
\boldsymbol{\chi})M_{nt} \\
\widetilde{P}_{rt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi}%
)&=&\sum_{j\in\jmath_r}\sum_n\frac{\widetilde{S}_{jnt}(\boldsymbol{%
\widetilde{\theta}},\boldsymbol{\chi})M_{nt}}{\sum_{j\in\jmath_r}\sum_n%
\widetilde{S}_{jnt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi})M_{nt}}%
\widetilde{P}_{jnt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi}), 
\notag
\end{eqnarray}
where $\widetilde{C}_{rt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi})$%
, $\widetilde{Q}_{rt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi})$,
and $\widetilde{P}_{rt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi})$
are total consumption, total production, and weighted-average price,
respectively. We calculate regional consumption for each of the four regions
in the data -- Northern California, Southern California, Arizona, and
Nevada. We calculate production and prices for Northern California, Southern
California, and a combined Arizona-Nevada region. We denote the empirical
analogs $C_{rt}$, $Q_{rt}$, and $P_{rt}$.

We find that, in practice, we can better identify some of the model
parameters by exploiting information on aggregated cross-region shipments.
We denote the quantity of shipments from region $r$ to region $s$ as $%
\widetilde{Q}^s_{rt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi})$.
The shipments take the form: 
\begin{eqnarray}
\widetilde{Q}^s_{rt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi}%
)&=&\sum_{j\in\jmath_r}\sum_{n\in\aleph_s}\widetilde{S}_{jnt}(\boldsymbol{%
\widetilde{\theta}},\boldsymbol{\chi})M_{nt},
\end{eqnarray}
We calculate the quantity of portland cement produced by plants in
California (both Northern and Southern) that is shipped to consumers in
Northern California. The empirical analog, which we denote $Q^s_{rt}$, is
available over the period 1990-2003. We withhold data on other cross-region
shipments from estimation because the number of parameters that we estimate
exceeds the lengths of the available time-series. The withheld data,
however, provide natural out-of-sample tests on the model predictions.

\subsection{Objective function}

We estimate the parameters using the standard GMM framework for systems of
nonlinear regression equations (e.g., \citeN{greene03}, page 369). The
equations we use compare the metrics computed in the inner loop to their
empirical analogs: 
\begin{eqnarray}
\boldsymbol{C}_r&=&\boldsymbol{\widetilde{C}}_r(\boldsymbol{\theta},%
\boldsymbol{\chi})+\boldsymbol{e_r^1}  \notag \\
\boldsymbol{Q}_r&=&\boldsymbol{\widetilde{Q}}_r(\boldsymbol{\theta},%
\boldsymbol{\chi})+\boldsymbol{e_r^2} \\
\boldsymbol{P}_r&=&\boldsymbol{\widetilde{P}}_r(\boldsymbol{\theta},%
\boldsymbol{\chi})+\boldsymbol{e_r^3}  \notag \\
\boldsymbol{Q}^s_{r}&=&\boldsymbol{\widetilde{Q}}^s_r(\boldsymbol{\theta},%
\boldsymbol{\chi})+\boldsymbol{e_{rs}^4}.  \notag  \label{eq:moments}
\end{eqnarray}
We write the equations in vector form with one element per period. There are
four consumption moments, three production moments, three price moments, and
one cross-region shipments moment. We have 21 observations on each of the
consumption, production and price moments, and 14 observations on the
cross-region shipments moment. We interpret the disturbances as measurement
error, and assume the disturbances to have expectation zero and
contemporaneous covariance matrix $\boldsymbol{\Sigma}$.

The GMM estimator is: 
\begin{equation}
\boldsymbol{\widehat{\theta}} = \arg \min_{\boldsymbol{\Theta}} \boldsymbol{e%
}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi})^\prime \boldsymbol{A}%
^{-1} \boldsymbol{e}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi}),
\end{equation}
where $\boldsymbol{e}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi})$ is
a vector of empirical disturbances obtained by stacking the nonlinear
regression equations and $\boldsymbol{A}$ is a positive definite weighting
matrix. We employ the usual two-step procedure to obtain consistent and
efficient estimates (\citeN{hansen82}). We first minimize the objective
function using $\boldsymbol{A}=\boldsymbol{I}$. We then estimate the
contemporaneous covariance matrix and minimize the objective function a
second time using the weighting matrix $\boldsymbol{A} = \boldsymbol{%
\widehat{\Sigma}} \otimes\boldsymbol{I}.$ We compute standard errors that
are robust to both heteroscedasticity and arbitrary correlations among the
error terms of each period, using the methods of \citeN{hansen82} and %
\citeN{neweymcfadden94}.\footnote{%
Measurement error that is zero in expectation is sufficient for consistency.
Estimation of the contemporaneous covariance matrix $\boldsymbol{\Sigma}$ is
complicated by the fact that we observe consumption, production, and prices
over 1983-2003 but cross-region shipments over 1990-2003. We use methods
developed in \citeN{sz73} and \citeN{hhs90} to account for the unequal
numbers of observations.}

\subsection{Potential demand}

We normalize the potential demand of each county using two exogenous demand
predictors that we observe at the county level: the number of construction
employees and the number of new residential building permits. We regress
regional portland cement consumption on the demand predictors (aggregated to
the regional level), impute predicted consumption at the county level based
on the estimated relationships, and then scale predicted consumption by a
constant of proportionality to obtain potential demand.\footnote{%
The regression of regional portland cement consumption on the demand
predictors yields an $R^2$ of 0.9786, which foreshadows an inelastic
estimate of aggregate demand. Additional predictors, such as land area,
population, and percent change in gross domestic product, contribute little
additional explanatory power. We use a constant of proportionality of 1.4,
which is sufficient to ensure that potential demand exceeds observed
consumption in each region-year observation.} The results indicate that
potential demand is concentrated in a small number of counties. In 2003, the
largest 20 counties account for 90 percent of potential demand, the largest
10 counties account for 65 percent of potential demand, and the largest two
counties -- Maricopa County and Los Angeles County -- together account for
nearly 25 percent of potential demand.\footnote{%
The largest five counties are Maricopa County (3,259 thousand metric
tonnes), Los Angeles County (3,128 thousand metric tonnes), Clark County
(1,962 thousand metric tonnes), Riverside County (1,803 thousand metric
tonnes) and San Diego County (1,733 thousand metric tonnes).} In the
time-series, potential demand more than doubles over 1983-2003, due to
greater activity in the construction sector and the onset of the housing
bubble.

\subsection{The geographic space \label{sec:ussw}}

Our restricted geographic focus eases the computational burden of the
estimation routine. For instance, a national sample would require the
computation of more than 300 thousand equilibrium plant-county prices in
each time period, for every outer loop iteration. The geographical
restriction is valid provided that gross domestic inflows/outflows are
insubstantial. The data provide some support. Most directly, the California
Letter indicates that more than 98 percent of cement produced in Southern
California was shipped within the U.S. Southwest over the period 1990-1999,
and more than 99 percent of cement produced in California (both Northern and
Southern) was shipped within the U.S. Southwest over the period 2000-2003.%
\footnote{%
Analogous statistics for Northern California over 1990-1999 are unavailable
due to data redaction.} We consider outflows from Arizona unlikely because
the Minerals Yearbook indicates that consumption routinely exceeds
production in that state, and we consider outflows from Nevada unlikely
because production capacity is low relative to potential demand. Since 
\textit{net} domestic inflows/outflows are insubstantial (see Figure \ref%
{fig:imports}), these data patterns suggest that gross inflows are also
insubstantial.

\subsection{Identification \label{sec:ID}}

We use an artificial data experiment to test identification. We draw 40 data
sets, each with 21 time periods, using our model and a vector of ``true''
parameters as the data generating process. We then seek to recover the
parameter values with the GMM estimation procedure. The exogenous data
includes the plant capacities, the potential demand of counties, the diesel
price, the import price, and two cost shifters. We randomly draw capacity
and potential demand from the data (with replacement), and we draw the
remaining data from normal distributions.\footnote{%
Specifically, we use the following distributions: diesel price $\sim
N(1,0.28)$, import price $\sim N(50,9)$, cost shifter 1 $\sim N(60,15)$, and
cost shifter 2 $\sim N(9,2).$ We redraw data that are below zero. We also
redraw data that lead the estimator to nonsensical areas of parameter space.}
We hold plant and county locations fixed to maintain tractability, and rely
on the random draws on capacity, potential demand, and diesel prices to
create variation in the distances between production capacity and consumers.

Table \ref{tab:fakedata} shows the results of the experiment. Interpretation
is complicated somewhat because we use non-linear transformations to
constrain the price and distance coefficients below zero, constrain the
coefficients on the cost shifters and the over-utilization cost above zero,
and constrain the inclusive value and utilization threshold coefficients
between zero and one. We defer details on these transformations to Appendix %
\ref{app:estdet}. As shown, the means of the estimated coefficients are
close to (transformed) true parameters. The means of the price and distance
coefficients, which are of particular interest, are within 6 percent and 11
percent of the truth, respectively. The root mean-squared errors tend to be
between 0.45 and 0.66 -- the two exceptions that generate higher
mean-squared errors are the import dummy and the over-utilization cost,
which appear to be less cleanly identified.

\begin{table}[tbp]
\caption{Artificial Data Test for Identification}
\begin{center}
\begin{tabular}[h]{lccccc}
\hline\hline
\rule[0mm]{0mm}{6.5mm}Variable & Parameter & Truth ($\theta$) & Transformed (%
$\widetilde{\theta}$) & Mean Est & RMSE \\ \hline
\multicolumn{3}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Demand}} &  &  &  \\ 
\; Cement Price & $\beta_1$ & -0.07 & -2.66 & -2.51 & 0.66 \\ 
\; Miles$\times$Diesel Price & $\beta_2$ & -25.00 & 3.22 & 2.86 & 0.59 \\ 
\; Import Dummy & $\beta_3$ & -4.00 & -4.00 & -6.07 & 1.23 \\ 
\; Intercept & $\beta_0$ & 2.00 & 2.00 & 1.11 & 0.51 \\ 
\; Inclusive Value & $\lambda$ & 0.09 & -2.31 & -1.73 & 0.54 \\ 
\multicolumn{3}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Marginal Costs}} &  &  & 
\\ 
\; Cost Shifter 1 & $\alpha_1$ & 0.70 & -0.36 & -0.88 & 0.51 \\ 
\; Cost Shifter 2 & $\alpha_2$ & 3.00 & 1.10 & 0.54 & 0.45 \\ 
\; Utilization Threshold & $\nu$ & 0.90 & 2.19 & 1.71 & 0.59 \\ 
\; Over-Utilization Cost & $\gamma$ & 300.00 & 5.70 & 6.14 & 1.05 \\ \hline
\multicolumn{6}{p{6.3in}}{{\footnotesize {\rule[0mm]{0mm}{0mm}Results of GMM
estimation on 40 data sets that are randomly drawn based on the ``true''
parameters listed. The parameters are transformed prior to estimation to
place constraints on the parameter signs/magnitudes (see Appendix \ref%
{app:estdet}). Mean Est and RMSE are the mean of the estimated (transformed)
parameters and the root mean-squared error, respectively. }}}%
\end{tabular}%
\end{center}
\end{table}

To further build intuition, we explore some of the empirical relationships
(graphed in Figure \ref{fig:ID}) that drive parameter estimates in our
application. On the demand side, the price coefficient is primarily
identified by the relationship between the consumption and price moments. In
panel A, we plot cement prices and the ratio of consumption to potential
demand (``market coverage'') over the sample period. The two metrics have a
weak negative correlation, consistent with downward-sloping but inelastic
aggregate demand. The distance coefficient is primarily identified by (1)
the cross-region shipments moment, and (2) the relationship between the
consumption and production moments. To explore the second source of
identification, we plot the gap between production and consumption (``excess
production'') for each region over the sample period. In many years, excess
production is positive in Southern California and negative elsewhere,
consistent with inter-regional trade flows. The magnitude of these implied
trade flows helps drive the distance coefficient. Interestingly, the implied
trade flows are higher later in the sample, when the diesel fuel is less
expensive.

\begin{figure}[t]
\centering
\includegraphics[height=4in]{stats_SWsample.eps}  
\caption{{\protect\footnotesize {Empirical Relationships in the U.S.
Southwest. Panel A plots average cement prices and market coverage. Prices
are in dollars per metric tonne and market coverage is defined as the ratio
of consumption to potential demand (times 100). Panel B plots excess
production in each region, which we define as the gap between between
production and consumption. Excess production is in millions of metric
tonnes. Panel C plots average coal prices, electricity prices, durable-goods
manufacturing wages, and crushed stone prices in California. For
comparability, each time-series is converted to an index that equals one in
2000. Panel D plots the average cement price and industry-wide utilization
(times 100). } }}
\label{fig:ID}
\end{figure}

On the supply side, the parameters on the marginal cost shifters are
primarily identified by the price moments. In panel C, we plot the coal
price, the electricity price, the durable-goods manufacturing wage, and the
crushed stone price for California. Coal and electricity prices are highly
correlated with the cement price (e.g., see panel A), consistent with a
strong influence on marginal costs; inter-regional variation in input prices
helps disentangle the two effects. It is less clear that wages and crushed
stone prices are positively correlated with cement prices. The utilization
parameters are primarily identified by the relationship between the
production moments (which determine utilization) and the price moments. In
panel D, we plot cement prices and industry-wide utilization over the sample
period. The two metrics are negatively correlated over 1983-1987 and
positively correlated over 1988-2003.

\section{Estimation Results \label{sec:res}}

\subsection{Specification and fits}

We estimate the model with parsimonious specifications of the utility and
marginal cost functions. Specifically, the utility specification includes
the plant-county price, the ``distance'' between the plant and county, a
dummy for the import option, and an intercept. We proxy distance using a
diesel price index interacted with the miles between the plant and the
center of the county (in thousands). The marginal cost specification
incorporates the five variable inputs identified by \citeN{epa09}. The
constant portion of marginal costs includes shifters for the price of coal,
the price of electricity, the average wages of durable-goods manufacturing
employees, and the price of crushed stone. We let marginal costs increase in
production once utilization exceeds some (estimated) threshold value, as
written in Equation \ref{eq:mc}, and we normalize $\phi=1.5$ to ensure the
theoretical existence of equilibrium.

Before turning to the parameter estimates, we note that this specification
produces impressive in-sample and out-of-sample fits. In Figure \ref%
{fig:fitsregX}, we plot observed consumption against predicted consumption
(panel A), observed production against predicted production (panel B), and
observed prices against predicted prices (panel C). The model explains 93
percent of the variation in regional consumption, 94 percent of the
variation in regional production, and 82 percent of the variation in
regional prices. The model also generates accurate out-of-sample
predictions. In panel D, we plot observations on cross-region shipments
against the corresponding model predictions. We use only 14 of these
observations in the estimation routine -- the remaining 82 data points are
withheld from the estimation procedure and do not directly influence the
estimated parameters. Even so, the model explains 98 percent of the
variation in these data.\footnote{%
We provide additional information on the estimation fits in Appendix \ref%
{app:fits}.}

\begin{figure}[t]
\centering
\includegraphics[height=4in]{fits-regX.eps}  
\caption{{\protect\footnotesize {GMM Estimation Fits for Regional Metrics.
Consumption, production, and cross-region shipments are in millions of
metric tonnes. Prices are constructed as a weighted-average of plants in the
region, and are reported as dollars per metric tonne. The lines of best fit
and the reported $R^2$ values are based on univariate OLS regressions.}}}
\label{fig:fitsregX}
\end{figure}

%The figure shows four scatter-plots.  The first plots predicted consumption against observed consumption (panel A).  The second
%plots predicted production against observed consumption (panel B).  The third plots predicted prices against observed prices
%(panel C).  The fourth plots predicted cross-region shipments against observed shipments.  Each plot also shows the line of best fit.
%and the resulting r-squared value.  The text discusses the fits in detail.

\subsection{Demand estimates and transportation costs \label{sec:demres}}

Table \ref{tab:GMM} presents the parameter estimates of the GMM procedure.
The price and distance coefficients are the two primary objects of interest
on the demand side; both are negative and precisely estimated.\footnote{%
The other demand parameters take reasonable values and are precisely
identified. The negative coefficient on the import dummy may be due to
consumer preferences for domestic plants or to the fact that observed import
prices do not reflect the full price of imported cement (e.g., the data
exclude duties). The inclusive value coefficient suggests that consumer
tastes for the different cement providers are highly correlated,
inconsistent with the standard (non-nested) logit model.} The ratio of these
coefficients identifies consumers' willingness-to-pay for proximity,
incorporating transportation costs and any other distance-related costs
(e.g., reduced reliability). In the following discussion, we refer to the
willingness-to-pay as the transportation cost, although the two concepts may
not be strictly equivalent.

\begin{table}[tbp]
\caption{Estimation Results }
\begin{center}
\begin{tabular}[h]{lccc}
\hline\hline
\rule[0mm]{0mm}{6.5mm}Variable & Parameter & Estimate & St. Error \\ \hline
\multicolumn{3}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Demand}} &  \\ 
\; Cement Price & $\beta_1$ & -0.087 & 0.002 \\ 
\; Miles$\times$Diesel Price & $\beta_2$ & -26.42 & 1.78 \\ 
\; Import Dummy & $\beta_3$ & -3.80 & 0.06 \\ 
\; Intercept & $\beta_0$ & 1.88 & 0.08 \\ 
\; Inclusive Value & $\lambda$ & 0.10 & 0.004 \\ 
\multicolumn{3}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Marginal Costs}} &  \\ 
\; Coal Price & $\alpha_1$ & 0.64 & 0.05 \\ 
\; Electricity Price & $\alpha_2$ & 2.28 & 0.47 \\ 
\; Hourly Wages & $\alpha_3$ & 0.01 & 0.04 \\ 
\; Crushed Stone Price & $\alpha_4$ & 0.29 & 0.31 \\ 
\; Utilization Threshold & $\nu$ & 0.86 & 0.01 \\ 
\; Over-Utilization Cost & $\gamma$ & 233.91 & 38.16 \\ \hline
\multicolumn{4}{p{4.3in}}{{\footnotesize {\rule[0mm]{0mm}{0mm}GMM estimation
results. Estimation exploits variation in regional consumption, production,
and average prices over the period 1983-2003, as well as variation in
shipments from California to Northern California over the period 1990-2003.
The prices of cement, coal, and crushed stone are in dollars per metric
tonne. Miles are in thousands. The diesel price is an index that equals one
in 2000. The price of electricity is in cents per kilowatt-hour, and hourly
wages are in dollars per hour. The marginal cost parameter $\phi$ is
normalized to 1.5, which ensures the theoretical existence of equilibrium.
Standard errors are robust to heteroscedasticity and contemporaneous
correlations between moments.}}}%
\end{tabular}%
\end{center}
\end{table}

First, however, we briefly summarize the implied price elasticities. We
estimate that the aggregate elasticity is -0.16 in the median year. This
inelasticity is precisely what one should expect based on economic theory
and the fact that portland cement composes only a small fraction of total
construction expenses. Indeed, \citeN{syverson04} makes a similar argument
for ready-mix concrete, which accounts for only two percent of total
construction expenses according to the 1987 Benchmark Input-Output Tables.
The cost share of portland cement (an input to ready-mix concrete) is surely
even lower. By contrast, we estimate that the median firm-level elasticity
is -5.70, consistent with substantive competition between firms. Finally, we
estimate that the domestic elasticity -- which captures the responsiveness
of domestic demand to domestic prices, holding import prices constant -- is
-1.11 in the median year. The discrepancy between the aggregate and domestic
elasticities is suggestive that imports have a disciplining effect on
domestic prices.

Returning to transportation costs, we estimate that consumers pay roughly
\$0.30 per tonne mile, given diesel prices at the 2000 level.\footnote{%
The calculation is simply $\frac{26.42}{0.087}\frac{index}{1000}=0.3037$,
where $index=1$ in 2000.} Given the shipping distances that arise in
numerical equilibrium, this translates into an average transportation cost
of \$24.61 per metric tonne over the sample period -- sufficient to account
for 22 percent of total consumer expenditure. Transportation costs of this
magnitude have real effects on the industry. We develop two such effects
here: (1) transportation costs constrain the distance that cement can be
shipped economically; and (2) transportation costs insulate firms from
competition and provide some degree of localized market power.

In Figure \ref{fig:miles4}, we plot the estimated distribution of the
shipping distances over 1983-2003. We calculate that portland cement is
shipped an average of 92 miles, that 75 percent of portland cement is
shipped under 110 miles, and that 90 percent is shipped under 175 miles.%
\footnote{%
The average shipping distance fluctuates between a minimum of 72 miles in
1983 and a maximum of 114 miles in 1998, and is highly correlated with the
diesel price index.} To better place these numbers in context, we ask the
question: ``How far would portland cement be shipped if transportation costs
were negligible?'' We perform a counter-factual simulation in which we
normalize the distance coefficient to zero, keeping the other coefficients
at their estimated values. The results suggest that, in an average year,
portland cement would have been shipped on average 276 miles absent
transportation costs. Intriguingly, the ratio of actual miles shipped to
this simulated measure provides a unit-free measure that could enable
cross-industry comparisons. The ratio in our application is 0.33.

\begin{figure}[t]
\centering
\includegraphics[height=3in]{miles4.eps}  
\caption{{\protect\footnotesize {The Estimated Distribution of Miles Shipped
over 1983-2003.}}}
\label{fig:miles4}
\end{figure}

% The figure is a non-parametric bar chart that characterizes the distances that cement is shipped.
% See text for summary statistics and more details.

We now develop the empirical evidence regarding localized market power. We
start with an illustrative example. Figure \ref{fig:priceshare} shows the
prices (Map A) and market shares (Map B) that characterize numerical
equilibrium for the Clarksdale plant in 2003, evaluated the optimized
coefficient vector. We mark the location of the Clarksdale plant with a
star, and mark other plants with circles. The Clarksdale plant captures more
than 40 percent of the market in the central and northeastern counties of
Arizona. It charges consumers in these counties its highest prices,
typically \$80 per metric tonne or more. Both market shares and prices are
lower in more distant counties, and in many counties the plant captures less
than one percent of demand despite substantial discounts. The locations of
competitors may also influence market share and prices, though these effects
are difficult to discern on the map.

\begin{figure}[t]
\centering
\includegraphics[height=4in]{priceshare-erika.eps}  
\caption{{\protect\footnotesize {Equilibrium Prices and Market Shares for
the Clarksdale Plant in 2003. The Clarksdale plant is marked with a star,
and other plants are marked with circles.}}}
\label{fig:priceshare}
\end{figure}

%The figure is a map of California, Arizona, and Nevada.  In Map A, counties for which the Clarksdale plant has higher prices
%are shaded darker.  In Map B, counties for which the plant has higher market shares are shaded darker.  The location of the
%Clarksdale plant is marked with a star, and the locations of competitors are marked with circles.  The text provides a description
% of the spatial price and share patterns that emerge.

We explore these relationships more rigorously with regression analysis,
based on the prices and market shares that characterize equilibrium at the
optimized coefficient vector. We regress price and market share on three
independent variables: the distance between the plant and the county, the
distance between the county and the nearest other domestic plant, and the
estimated marginal cost of the plant. We define distance as miles times the
diesel index, and use a log-log specification to ease interpretation. We
focus on three data samples, composed of the plant-county pairs with
distances of 0-100, 100-200, and 200-300, respectively. Our objective is
purely descriptive and the regression coefficients should not be interpreted
as consistent estimates of any underlying structural parameters.

Table \ref{tab:disteffect} presents the results, which are consistent with
the illustrative example and demonstrate that (1) plants have higher prices
and market shares in counties that are closer; and (2) plants have lower
prices and market shares in counties that have nearby alternatives. For the
closest plants and counties, a 10 percent reduction in distance is
associated with prices and market shares that are 0.9 percent and 14 percent
higher, respectively. For the same sample, a 10 percent reduction in the
distance separating the county from its the closest alternative is
associated with prices and market shares that are 0.7 percent and 11 percent
lower, respectively. Interestingly, these price effects attenuate for plants
and counties that are somewhat more distant, whereas the market share
effects amplify.

\begin{table}[tbp]
\caption{Plant Prices and Market Shares}
\begin{center}
\begin{tabular}[h]{lcccccc}
\hline\hline
\rule[0mm]{0mm}{5.5mm}{\footnotesize {Dependent Variable:}} & {\footnotesize 
{ln(Price)}} & {\footnotesize {ln(Price)}} & {\footnotesize {ln(Price)}} & 
{\footnotesize {ln(Share)}} & {\footnotesize {ln(Share)}} & {\footnotesize {%
ln(Share)}} \\ 
\rule[0mm]{0mm}{5.5mm}{\footnotesize {Distance from Plant:}} & 
{\footnotesize {0-100}} & {\footnotesize {100-200}} & {\footnotesize {200-300%
}} & {\footnotesize {0-100}} & {\footnotesize {100-200}} & {\footnotesize {%
200-300}} \\ \hline
\rule[0mm]{0mm}{6.5mm}{\footnotesize {ln(Distance from Plant)}} & 
{\footnotesize {-0.098*}} & {\footnotesize {-0.038*}} & {\footnotesize {%
-0.003}} & {\footnotesize {-1.369*}} & {\footnotesize {-2.655*}} & 
{\footnotesize {-5.902*}} \\ 
& {\footnotesize {(0.026)}} & {\footnotesize {(0.009)}} & {\footnotesize {%
(0.009)}} & {\footnotesize {(0.102)}} & {\footnotesize {(0.186)}} & 
{\footnotesize {(0.195)}} \\ 
\rule[0mm]{0mm}{6.5mm}{\footnotesize {ln(Distance to Nearest}} & 
{\footnotesize {0.071*}} & {\footnotesize {0.018*}} & {\footnotesize {0.007*}%
} & {\footnotesize {1.073*}} & {\footnotesize {0.813*}} & {\footnotesize {%
1.279*}} \\ 
$\; \;$ {\footnotesize {Alternative)}} & {\footnotesize {(0.019)}} & 
{\footnotesize {(0.004)}} & {\footnotesize {(0.002)}} & {\footnotesize {%
(0.117)}} & {\footnotesize {(0.081)}} & {\footnotesize {(0.098)}} \\ 
\rule[0mm]{0mm}{6.5mm}{\footnotesize {ln(Marginal Cost)}} & {\footnotesize {%
0.723*}} & {\footnotesize {0.835*}} & {\footnotesize {0.841*}} & 
{\footnotesize {-1.126*}} & {\footnotesize {-2.057*}} & {\footnotesize {%
-3.212*}} \\ 
& {\footnotesize {(0.054)}} & {\footnotesize {(0.014)}} & {\footnotesize {%
(0.013)}} & {\footnotesize {(0.304)}} & {\footnotesize {(0.645)}} & 
{\footnotesize {(0.874)}} \\ 
&  &  &  &  &  &  \\ 
{\footnotesize {R}$^2$} & {\footnotesize {0.7186}} & {\footnotesize {0.9209}}
& {\footnotesize {0.9499}} & {\footnotesize {0.5278}} & {\footnotesize {%
0.4735}} & {\footnotesize {0.5407}} \\ 
{\footnotesize {\# of Obs.}} & {\footnotesize {2,840}} & {\footnotesize {%
5,460}} & {\footnotesize {6,088}} & {\footnotesize {2,840}} & {\footnotesize 
{5,460}} & {\footnotesize {6,088}} \\ \hline
\multicolumn{7}{p{5.8in}}{{\footnotesize {\rule[0mm]{0mm}{0mm}Results of OLS
regression. The units of observation are at the plant-county-year level. The
dependent variables are the natural logs of the plant-county specific prices
and market shares that characterize numerical equilibrium at the GMM
estimates. Distance from Plant is the miles between the plant and county,
times a diesel price index that equals one in 2000. Distance to Nearest
Alternative is the miles between the county and the nearest other domestic
plant, times the diesel price index. Marginal Cost is the marginal cost of
the plant implied by the GMM estimates. All regressions include an
intercept. Standard errors are robust to heteroscedasticity and correlations
among observations from the same plant. Statistical significance at the one
percent level is denoted by *.}}}%
\end{tabular}%
\end{center}
\end{table}

\subsection{Marginal cost estimates}

The marginal costs estimates shown in Table \ref{tab:GMM} correspond to a
marginal cost of \$69.40 in the mean plant-year (weighted by production). Of
these marginal costs, \$60.50 is attributable to costs related to coal,
electricity, labor and raw materials, and the remaining \$8.90 is
attributable to high utilization rates. In Figure \ref{fig:costs}, we plot
the three metrics over 1983-2003, together with the average prices that
arise in numerical equilibrium at the optimized parameter vector. The
constant portion of marginal costs declines through the sample period, due
primarily to cheaper coal and electricity, whereas the utilization portion
increases. We estimate that utilization-related expenses account for roughly
25 percent of overall marginal costs over 1997-2003, during the onset of the
housing bubble. Finally, we note that the average markup (i.e., price minus
marginal cost) is quite stable through the sample period around its mean of
\$17.20.

\begin{figure}[t]
\centering
\includegraphics[height=3in]{costs.eps}  
\caption{{\protect\footnotesize {Estimated Marginal Costs and Average Prices.%
}}}
\label{fig:costs}
\end{figure}

% The figure plots estimated marginal costs, estimated constant marginal costs, estimated marginal costs due to high
% utilization, and average prices, over the sample period.  The text discusses the relationships between these
% metrics in detail.

We calculate that the average plant-year observation has variable costs of
\$51 million by integrating the marginal cost function over the production
levels that arise in numerical equilibrium. Virtually all of these variable
costs -- 98.5 percent -- are due to coal, electricity, labor and raw
materials, rather than due to high utilization. Thus, although capacity
constraints may have substantial affects on marginal costs, the results
suggest that their cumulative contribution to plant costs can be minimal.
Taking the accounting statistics further, we calculate that the average
plant-year has variable revenues of \$73 million and that the average gross
margin (variable profits over variable revenues) is 0.32. As argued in %
\citeN{ryan06}, margins of this magnitude may be needed to rationalize entry
given the sunk costs associated with plant construction.\footnote{%
These gross margins are consistent with publicly-available accounting data.
For instance, Lafarge North America -- one of the largest domestic producers
-- reports an average gross margin of 0.33 over 2002-2004.}$^,$\footnote{%
Fixed costs are well understood to be important for production, as well. The
trade journal \textit{Rock Products} reports that high capacity portland
cement plants incurred averaged \$6.96 in maintenance costs per production
tonne in 1993 (\citeN{rock94}). Evaluated at the production levels that
correspond to numerical equilibrium in 1993, this number implies that the
average plant would have incurred \$5.7 million in maintenance costs
relative to variable profits of \$17.7 million. The GMM estimation results
suggest that the bulk of these maintenance costs are best considered fixed
rather than due to high utilization rates. Of course, the static nature of
the model precludes more direct inferences about fixed costs.}

Finally, we discuss the individual parameter estimates shown in Table \ref%
{tab:GMM}, each of which deviates somewhat from production data available
from the Minerals Yearbooks and \citeN{epa09}. To start, the coal parameter
implies that plants burn 0.64 tonnes of coal to produce one tonne of cement,
whereas in fact plants burn roughly 0.09 tonnes of coal to produce each
tonne of cement. The electricity parameter implies that plants use 228
kilowatt-hours per tonne of cement, whereas the true number is closer to
150. Each tonne of cement requires approximately 0.34 employee-hours yet the
parameter on wages is essentially zero. Lastly, the crushed stone
coefficient of 0.29 is too small, given that roughly 1.67 tonnes of raw
materials (mostly limestone) are used per tonne of cement. We suspect that
these discrepancies are due to measurement error in the data.\footnote{%
In particular, the coal prices in the data are free-on-board and do not
reflect any transportation costs paid by cement plants; cement plants may
negotiate individual contracts with electrical utilities that are not
reflected in the data; the wages of cement workers need not track the
average wages of durable-goods manufacturing employees; and cement plants
typically use limestone from a quarry adjacent to the plant, so the crushed
stone price may not proxy the cost of limestone acquisition (i.e., the
quarry production costs).}

\subsection{A comparison to standard methods}

The standard method of structural analysis for homogenous product industries
assumes independent markets and Cournot competition. In this section, we
contrast some of our results to those generated by the standard method in %
\citeN{ryan06}, a recent paper that estimates a structural model of the
portland cement industry based on data from the Minerals Yearbook and the
Plant Information Summary. We focus on two economic concepts -- the
aggregate elasticity of demand and the consequences of high utilization --
for which our model generates distinctly different estimates than the
standard method. These discrepancies do not diminish the substantial
contribution of \citeN{ryan06}, which embeds the standard method within an
innovative dynamic discrete choice game and focuses primarily on the dynamic
parameters. Rather, the discrepancies suggest two reasons that our model may
sometimes provide more reasonable results than conventional approaches.

First, we estimate the aggregate elasticity of demand to be -0.16 in the
median sample year whereas Ryan works with an aggregate elasticity of -2.96,
obtained from a constant elasticity demand system. The difference is due to
specific specification choices -- the constant elasticity demand system
produces an aggregate elasticity of -0.15 once housing permits are included
as a control.\footnote{%
See Table 3 in \citeN{ryan06}. We consider the inelastic estimate more
plausible because portland cement is a minor cost for most construction
projects (see Section \ref{sec:demres}).} Ryan cannot use the inelastic
estimate because, within the context of Cournot competition, it implies firm
elasticities that are small and inconsistent with profit maximization. This
occurs because the Cournot model restricts each firm elasticity to be
linearly related to the aggregate elasticity according the relationship $e_j
= e/s_j$, where $e_j$, $e$, and $s_j$ denote the firm elasticity, the
aggregate elasticity, and the firm market shares, respectively. This
critique is fundamental: the standard method can be inappropriate for
intermediate goods, such as portland cement, that account for only a
fraction for the total production costs of the final good. By contrast, the
nested logit demand system divorces the firm elasticities from the aggregate
elasticity and, in our case, produces inelastic aggregate demand and elastic
firm demand.

Second, the two methods produce vastly different estimates of the marginal
cost curve once utilization reaches the threshold level (which both methods
place just above 0.85). We estimate that marginal costs increase gradually
so that full utilization increases marginal costs by a total of \$12.25
relative to utilization below the threshold. By contrast, Ryan estimates
that the slope of the marginal cost curve past the threshold is nearly
infinite.\footnote{%
Our coefficient $\gamma$ is roughly analogous to Ryan's $\delta_2$
coefficient. We estimate $\gamma$ to be 233.91, while Ryan estimates $%
\delta_2$ to be $1157\times 10^7$. See Table 4 in \citeN{ryan06}.} We
suspect that the difference is data driven. The standard method requires
data on firm-level utilization. However, firm production is not available
from the publicly-available data, and Ryan imputes utilization as annual
capacity divided by annualized daily capacity.

In Figure \ref{fig:ryan}, we plot total production, total annual capacity,
and total annualized daily capacity in the U.S. Southwest over 1983-2003,
together with total consumption. Both production and consumption are
pro-cyclical, and actual utilization (i.e., production over annual capacity)
varies substantively and predictably with demand conditions. By contrast,
annual capacity simply tracks annualized daily capacity so that Ryan's
utilization proxy is uncorrelated with demand conditions. The strength of
the relationship between utilization and demand is precisely what identifies
the magnitude of utilization costs. Thus, we suspect that the lower data
requirements of our model -- estimation is feasible when some variables of
interest (e.g., firm-level production) are unobserved -- may improve
economic estimates.

\begin{figure}[t]
\centering
\includegraphics[height=3in]{ryan.eps}  
\caption{{\protect\footnotesize {Consumption, Production, and Two Capacity
Measures.}}}
\label{fig:ryan}
\end{figure}

% The figure plots production, consumption, annual utilization, and annualized daily utilization, over the sample
% period.  The text discusses the relationships between these metrics in detail.

\section{An application to competition policy \label{sec:sim}}

The model and estimator may prove useful for a variety of policy endeavors.
One potential application is merger simulation, an important tool for
competition policy. In this subsection, we use counter-factual simulations
to evaluate a hypothetical merger between Calmat and Gifford-Hill in 1986.
During that year, Calmat and Gifford-Hill operated six plants and accounted
for 43 percent of industry capacity in the U.S. Southwest.  We calculate the
loss of consumer surplus due to the unilateral effects of the merger, map
the distribution of harm over the U.S. Southwest, and evaluate six
alternative divestiture plans.\footnote{%
We follow standard practice to perform the counterfactuals. For each merger
simulation, we define a matrix $\boldsymbol{\Omega^{post}(P)}$ using
Equation \ref{eq:omega} and the post-merger structure of the industry. We
compute the equilibrium post-merger price vector as the solution to Equation %
\ref{eq:equil}, substituting $\boldsymbol{\Omega^{post}(P)}$ for $%
\boldsymbol{\Omega(P)}.$ Following \citeN{mcfadden81} and %
\citeN{smallrosen81}, the change in consumer surplus due to the merger is: 
\begin{equation}
\Delta CS =\sum_{n=1}^N \frac{\ln (1+\exp(\beta_0+ \lambda
I_{nt}^{pre}))-\ln (1+\exp(\beta_0+ \lambda I_{nt}^{post}))}{\beta_1} M_n, 
\notag
\end{equation}
where $I_{n}^{pre}$ is the inclusive value of the inside goods calculated
using equilibrium pre-merger prices, $I_n^{post}$ is the inclusive value
calculated using equilibrium post-merger prices, and $\beta_1$ is the price
coefficient.}

Table \ref{tab:mergsim} shows that the merger reduces consumer surplus by
\$1.40 million in 1986, absent a divestiture. The magnitude of the effect is
modest relative to the amount of commerce; by way of comparison, we
calculate total pre-merger consumer surplus to be more than \$239 million.
We refer to the six plants available for divestiture as Calmat1, Calmat2,
Calmat3, Gifford-Hill1, Gifford-Hill2, and Gifford-Hill3, respectively. The
single-plant divestitures mitigate between 31 percent and 56 percent of the
harm. The ``optimal'' divestiture -- that of Gifford-Hill2 -- results in
consumer harm of only \$614 thousand.

\begin{table}[tbp]
\caption{Divestitures and Consumer Surplus}
\begin{center}
\begin{tabular}[h]{lccccccc}
\hline\hline
& \multicolumn{7}{c}{\rule[0mm]{0mm}{5.5mm}\underline{Required Divestiture}}
\\ 
\rule[0mm]{0mm}{4.5mm} & {\footnotesize {None}} & {\footnotesize {Calmat1}}
& {\footnotesize {Calmat2}} & {\footnotesize {Calmat3}} & {\footnotesize {%
Gifford-Hill1}} & {\footnotesize {Gifford-Hill2}} & {\footnotesize {%
Gifford-Hill3}} \\ \hline
\rule[0mm]{0mm}{5.5mm} {\footnotesize {$\Delta$ Surplus}} & -1,397 & -964 & 
-618 & -797 & -827 & -614 & -891 \\ 
\rule[0mm]{0mm}{7.5mm} {\footnotesize {\% Mitigated}} & $\cdot$ & 31\% & 56\%
& 43\% & 41\% & 56\% & 36\% \\ \hline
\multicolumn{8}{p{6.1in}}{{\footnotesize {\rule[0mm]{0mm}{0mm}Results of
counterfactual simulations. $\Delta$ Surplus is the change in consumer
surplus due to a hypothetical merger between Calmat and Gifford-Hill in
1986, and is reported in thousands of 2000 dollars. \% Mitigated is
calculated relative to the change in consumer surplus that occurs when no
plant is divested. We consider six single-plant divestiture plans, and refer
to the different plants as Calmat1, Calmat2, Calmat3, Gifford-Hill1,
Gifford-Hill2, and Gifford-Hill3. }}}%
\end{tabular}%
\end{center}
\end{table}

We map the distribution of consumer harm over the U.S. Southwest in Figure %
\ref{fig:mapdivest}, both for the merger without divestiture (panel A) and
under the optimal divestiture plan (panel B). As shown in panel A, the
unilateral effects of the merger are concentrated in Southern California and
Arizona. Together, Maricopa County and Los Angeles County account for 60
percent of consumer harm, and more than 90 percent of consumer harm occurs
only 10 counties. The best single-plant divestiture mitigates consumer harm
in Southern California but does little to reduce harm in Maricopa County
(see panel B). The results of an additional counterfactual exercises, in
which we also divest one of the Arizona plants, suggest that a two-plant
divestiture can mitigate this harm as well (results not shown).

\begin{figure}[t]
\centering
\includegraphics[height=4in]{Consumer-Harm2.eps}  
\caption{{\protect\footnotesize {Loss of Consumer Surplus Due to a
Hypothetical Merger between Calmat and Gifford-Hill}}}
\label{fig:mapdivest}
\end{figure}

\section{Conclusion \label{sec:conc}}

We develop a structural model of competition among spatially differentiated
firms. The model accounts for transportation costs in a realistic and
tractable manner. We estimate the model with relatively disaggregated data
and recover the underlying structural parameters. We argue that the model
and estimator together provide an appealing framework with which to evaluate
competition in industries characterized by transportation costs and
relatively homogenous products. We apply the model and estimator to the
portland cement industry and demonstrate that (1) the framework explains the
salient features of competition, (2) the framework provides novel insights
regarding transportation costs and spatial differentiation, and (3) the
framework could inform merger analysis and other competition policy
endeavors. Although the model is static, it could be utilized to define
payoffs in more dynamic settings. Such extensions could examine a number of
research topics -- such as entry deterrence and product differentiation --
that have been emphasized in the theoretical literature of industrial
organization since at least \citeN{hotelling29}.

\newpage 
\bibliographystyle{chicago}
\bibliography{cement}
\clearpage

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\appendix

%\singlespace

\section{Regression fits \label{app:fits}}

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In this appendix, we develop further the quality of the estimation fits. We
focus on the ability of the model to predict the inter-temporal variation
that exist in the data.

Figure \ref{fig:fits1} aggregates the data and the model predictions across
regions, and plots the resulting time-series. Panel A shows consumption,
panel B shows production, panel C shows imports, and panel D shows average
prices (imports are defined as production minus consumption). In each case,
the model predictions mimic the inter-temporal patterns observed in the
data. Univariate regressions of the data on the prediction explain 96
percent of the variation total consumption, 75 percent of the variation in
total production, 76 percent of the variation in imports, and 91 percent of
the variation in average prices.\footnote{%
The model does not fully capture the fall in average prices over the 1980s
and early 1990s. One possible explanation is that the model, as specified,
does not incorporate potential changes to total factor productivity. %
\citeN{schmitz09} review the evidence regarding productivity and argue that
the gradual elimination of onerous clauses from labor contracts improved
productivity in the 1980s.}

\begin{figure}[t]
\centering
\includegraphics[height=4in]{fits-insample.eps}  
\caption{{\protect\footnotesize {GMM Estimation Fits for Aggregate Metrics.
The solid lines plot data and the dashed lines plot predictions.
Consumption, production, and imports are in millions of metric tonnes.
Imports are defined as production minus consumption. Prices are constructed
as a weighted-average of the plant-county prices and are reported in dollars
per metric tonne. The $R^2$ values are calculated from univariate
regressions of the observed metric on the predicted metric. }}}
\label{fig:fits1}
\end{figure}

%The figure shows four graphs that characterize the estimation fits.  Panel A is for total consumption, panel B is for
%total production, panel C is for apparent imports, and panel D is for average prices.  Each graph plots the predicted
% and observed values over the sample period.  The graphs also show the R-squared from a regression of the observed value
% on the predicted value.   See text for details.

Figure \ref{fig:outfits2} provides analogous time-series fits for eight of
the time-series for cross-region shipments. The data in panel A pertain to
shipments from California (both Northern and Southern) to Northern
California, and are used in estimation. The data in the remaining panels are
excluded from the estimation procedure, so the corresponding fits are
out-of-sample. As shown, the model predictions are close to the data in each
panel and tend to track the variation well (when variation exists).

\begin{figure}[t]
\centering
\includegraphics[height=4in]{fits-outsample.eps}  
\caption{{\protect\footnotesize {GMM Estimation Fits for Cross-Region
Shipments. The solid lines plot data and the dashed lines plot predictions.
Shipments are expressed as a percentage of production in California (panels
A-D) or Southern California (panels E-H).}}}
\label{fig:outfits2}
\end{figure}

%The figure shows eight graphs that characterize the regression fits for the cross-region shipments.
%Each graph plots both predicted and observed shipments for one particular region-to-region
%combination, over the sample period.  The region-to-region combination are, in order:
%CA to N.CA, CA to S.CA, CA to AZ, CA to NV,  S.CA to N.CA, S.CA to S.CA, S.CA to AZ, and S.CA to NV.
%See text for details.

\section{The uniqueness of equilibrium}

The estimation procedure rests on uniqueness of equilibrium at each
candidate parameter vector. The results of a Monte Carlo experiment suggest
the assumption holds, at least in our application. We consider 300 parameter
vectors for each of the 21 years in the sample, for a total of 6,300
candidate parameter vectors. For each $\theta_i \in \boldsymbol{\theta}$, we
draw from the distribution $N(\widehat{\mu}_i,\widehat{\sigma}_i^2)$, where $%
\widehat{\mu}_i$ and $\widehat{\sigma}_i$ are the coefficient and standard
error, respectively, reported in Table \ref{tab:GMM}. We then compute the
numerical equilibrium for each parameter vector, using eleven different
starting vectors. We define the elements of the starting vectors to be $%
P_{jnt}=\phi \overline{P_t},$ where $\overline{P_t}$ is the average price of
portland cement and $\phi = 0.5, 0.6, \dots, 1.4, 1.5$. Thus, we start the
equation solver at initial prices that sometimes understate and sometimes
overstate the average prices in the data. The experiment produces eleven
equilibrium price vectors for each parameter vector. We calculate the
standard deviation of each price element across the eleven observations.
Thus, we would calculate 1,260 standard deviations for a typical equilibrium
price vector with 1,260 plant-county elements. The experiment provides
support for the uniqueness condition if these standard deviations are small.%
\footnote{%
The equal-solver computes numerical equilibria for 90.3 percent of the
candidate vectors. See Appendix \ref{app:estdet} for a discussion of
non-convergence in the inner-loop.} This proves to be the case. In fact, the
maximum maximum standard deviation is \textit{zero}, considering all prices
and draws, so the experiment finds no evidence of multiple equilibria.

\section{Estimation details \label{app:estdet}}

We minimize the objective function using the Levenberg-Marquardt algorithm (%
\citeN{levenberg44}, \citeN{marquardt63}), which interpolates between the
Gauss-Newton algorithm and the method of gradient descent. We find that the
Levenberg-Marquardt algorithm outperforms simplex methods such as simulated
annealing and the Nelder-Mead algorithm, as well as quasi-Newton methods
such as BFGS. We implement the minimization procedure using the nls.lm
function in R, which is downloadable as part of the minpack.lm package.

We compute numerical equilibrium using Fortran code that builds on the
source code of the dfsane function in R. The dfsane function implements the
nonlinear equation solver developed in \citeN{dfsane06} and is downloadable
as part of the BB package. We find that Fortran reduces the computational
time of the inner loop by a factor of 30 or more, relative to the dfsane
function in R. The computation of equilibrium for each time period can be
parallelized, which further speeds the inner loop calculations. The
numerical computation of equilibrium takes between 2 and 12 seconds for most
candidate parameter vectors when run on a 2.40GHz dual core processor with
4.00GB of RAM.

We use observed prices to form the basis of the initial vector in the inner
loop computations, which limits the distance that the nonlinear equation
solver must ``walk'' to compute numerical equilibrium. In practice, the
equation solver occasionally fails to compute a numerical equilibrium at the
specified tolerance level (1e-13) within the specified maximum number of
iterations (600). The candidate parameter vectors that generate
non-convergence in the inner loop tend to be less economically reasonable,
and may be consistent with equilibria that are simply too distant from
observed prices. When this occurs, we construct regional-level metrics based
on the price vector that comes closest to satisfying our definition of
numerical equilibrium.

We constrain the signs and/or magnitudes of some parameters, based on our
understanding of economic theory and the economics of the portland cement
industry, because some parameter vectors hinder the computation of numerical
equilibrium in the inner loop. For instance, a positive price coefficient
would preclude the existence of Bertrand-Nash equilibrium. We use the
following constraints: the price and distance coefficients ($\beta_1$ and $%
\beta_2$) must be negative; the coefficients on the marginal cost shifters ($%
\boldsymbol{\alpha}$) and the over-utilization cost ($\gamma$) must be
positive; and the coefficients on the inclusive value ($\lambda$) and the
utilization threshold ($\nu$) must be between zero and one. We use nonlinear
transformations to implement the constraints. As examples, we estimate the
price coefficient using $\widetilde{\beta_1}=\log(-\beta_1)$ in the GMM
procedure, and we estimate the inclusive value coefficient using $\widetilde{%
\lambda} = \log\left(\frac{\lambda}{1-\lambda}\right)$. We calculate
standard errors with the delta method.

\section{Data details}

We make various adjustments to the data in order to improve consistency over
time and across different sources. We discuss some of these adjustments
here, in an attempt to build transparency and aid replication.

The Minerals Yearbook reports the total production and average  price of
plants in the ``Nevada-Arizona-New Mexico'' region over  1983-1991, and in
the ``Arizona-New Mexico'' region over  1992-2003. We scale the USGS
production data  downward, proportional to plant capacity, to remove for the
influence  of the single New Mexico plant. Since the two plants in Arizona
account for 89 percent of  kiln capacity in Arizona and New Mexico in 2003,
we  scale production by 0.89.

The portland cement plant in Riverside closed its kiln permanently in 1988
but continued operating its grinding  mill with purchased clinker. We
include the plant in the analysis over  1983-1987, and we adjust the USGS
production data to remove the  influence of the plant over 1988-2003 by
scaling the data  downward, proportional to plant grinding capacities. Since
the Riverside plant accounts for 7 percent of grinding  capacity in Southern
California in 1988, so we scale the  production data for that region by 0.93.

We exclude one plant in Riverside that produces white  portland cement.
White cement takes the color of dyes and is used for decorative  structures.
Production requires kiln temperatures that are roughly 50$^\circ$C  hotter
than would be needed for the production of grey cement. The resulting cost 
differential makes white cement a poor substitute for grey cement.

The PCA reports that the  California Cement Company idled one of two kilns
at its Colton plant over 1992-1993 and three of four  kilns at its Rillito
plant over 1992-1995, and that the Calaveras Cement Company idled all kilns
at  the San Andreas plant following the plant's acquisition  from Genstar
Cement in 1986. We adjust plant capacity accordingly.

The data on coal and electricity prices from the Energy Information Agency
are available at the state level starting in 1990. Only national-level data
are available in earlier years. We impute state-level data over 1983-1989 by
(1) calculating the average discrepancy between each state's price and the
national price over 1990-2000, and (2) adjusting the national-level data
upward or downward, in line with the relevant average discrepancy.

%%% Appendix Figures %%%

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