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\begin{center}
ECONOMIC ANALYSIS GROUP
DISCUSSION PAPER
\vspace{3.8cm}
Competition when Consumers Value Firm Scope
\vspace{0.25cm} By \vspace{0.25cm}
Nathan H. Miller* \\
EAG 08-8 $\quad$ August 2008
\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 analysis and
conclusions expressed herein are solely those of the authors and do
not represent the views of the United States Department of Justice.
\vspace{0.25cm}
\noindent Information on the EAG research program and discussion
paper series may be obtained from Russell Pittman, Director of
Economic Research, Economic Analysis Group, Antitrust Division, U.S.
Department of Justice, BICN 10-000, Washington DC 20530, or by
e-mail at russell.pittman@usdoj.gov. Comments on specific papers
may be addressed directly to the authors at the same mailing address
or at their email address.
\vspace{0.25cm}
\noindent Recent EAG Discussion Paper titles are listed at the end
of this paper. To obtain a complete list of titles or to request
single copies of individual papers, please write to Janet Ficco at
the above mailing address or at janet.ficco@usdoj.gov. In addition,
recent papers are now available on the Department of Justice website
at http://www.usdoj.gov/atr/public/eag/discussion\_papers.htm.
Beginning with papers issued in 1999, copies of individual papers
are also available from the Social Science Research Network at
www.ssrn.com.
\vspace{0.25cm}
\noindent\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
\noindent * \ Economist, Economic Analysis Group, Antitrust
Division, U.S. Department of Justice. Email:
nathan.miller@usdoj.gov. I thank Allen Berger, Joseph Farrell,
Richard Gilbert, Robert Johnson, Ashley Langer, Juan Lleras, Kevin
Stange, John Sutton, Kenneth Train, Sofia Villas-Boas, Catherine
Wolfram, and seminar participants at the University of California,
Berkeley for valuable comments. The views expressed do not
necessarily reflect those of the U.S. Department of Justice.
\newpage
\thispagestyle{empty}
\begin{abstract} % beginning of the abstract
I model multimarket competition when consumers value firm scope
across markets. Such competition is surprisingly common -- consumers
in many industries prefer firms that operate in more geographic
and/or product markets. I show that these preferences permit firms
of differing scopes to coexist in equilibrium. Within markets, firms
of greater scope have higher prices and market shares. I turn to the
commercial banking industry for the empirical implementation.
Structural estimation of the model firmly supports the assumptions
on consumer preferences, and empirical predictions specific to the
model hold in the data. The results suggest that theoretical model
is empirically relevant.
\end{abstract} % end of the abstract
\newpage
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\section{Introduction}
A central goal of industrial organization is to understand the
determinants of firm conduct and market structure. I study
competition when consumers value firm scope across markets. Such
competition is surprisingly common. Consumers have preferences for
scope across geographic markets in the commercial banking industry,
the cell phone industry, the video rental industry, the fitness club
industry, and elsewhere. Consumers also have preferences for scope
across some product markets. For example, consumers prefer to
purchase cable television together with internet service, automobile
insurance with homeowners/rental insurance, and cellular phones with
portable music players (e.g., the iPhone). Nonetheless, the effect
of these preferences on competition has yet to receive substantial
attention from the academic literature. The primary objective of
this paper is to help fill this gap.
I start with a theoretical model of multimarket competition. I
consider technologically identical firms that determine their scope
across markets and then compete in prices within markets. Consumers
prefer firms of greater scope but differ in their willingness-to-pay
for scope. I solve for the unique subgame perfect equilibrium and
show that these preferences have implications for competition both
across and within markets. Across markets, the preferences create
opportunities for scope differentiation and permit firms of greater
and lesser scope to coexist in equilibrium; the preferences are
therefore capable of generating the distribution of scopes observed
in the commercial banking industry, the cellular phone industry, and
elsewhere. Within markets, firms of greater scope have both higher
prices and higher market shares. Firms of lesser scope, meanwhile,
occupy a (relatively) unprofitable niche position in which they
attract more price-sensitive consumers.
I tailor the theoretical model to fit the institutional details of
deposit competition among commercial banks. The specific setting
engenders little loss of generality -- the model easily accommodates
consumer preferences for geographic scope in other industries, as
well as consumer preferences for scope across product markets.
Further, the focus on the commercial banking industry conveys a
substantial advantage to the empirical analysis. Due to the
industry's history of regulation, comprehensive panel data exist on
bank balance sheets and income statements, as well as on the
geographic location of bank branches and deposits. In the empirical
analysis, I estimate the theoretical model structurally and test the
first-order assumption on consumer preferences. I then examine a
number of distinct empirical predictions for firm conduct and market
structure. Thus, I am able to provide evidence, for at least one
specific setting, that consumers prefer firms of greater scope but
differ in willingness-to-pay, and that these preferences have real
implications for competition within and across markets.
I estimate the model structurally using the simulated generalized
method of moments. Estimation exploits nearly 40,000
bank-market-year observations over the period 2001-2006, as well as
individual-level data from 2000 Consumer Population Survey (CPS)
March Supplement. The results suggest that the mean depositor
values a unit increase in bank scope at 42.90 cents annually, so
that the scope of Bank of America (which operates in 207
metropolitan markets) is 88.41 dollars more valuable than the scope
of a single-market bank. The number is quite plausible. A typical
deposit account earns 33.86 fewer dollars at Bank of America than at
the average single-market bank, yet Bank of America averages higher
market shares. The data also suggest that scope valuations increase
in depositor income, and a statistical test firmly rejects the null
hypothesis of no depositor heterogeneity. Consistent with the
theoretical model, the elasticity of deposit demand decreases in
scope -- single-market banks face a median elasticity of demand that
is more than double the median elasticity faced by banks that
operate in more than twenty metropolitan markets.
I then evaluate three empirical predictions of the theoretical model
that are not easily generated under (realistic) alternative
assumptions. Each prediction relates the assumption on depositor
preferences to competition through the same mechanism -- if
depositors prefer banks of greater scope but differ in their
willingness-to-pay then banks should differentiate in scope. The
predictions are as follows: First, banks should be less likely to
enter an outside market if the original market already features
banks of greater scope. Second, banks should be less likely to
enter a specific outside market (conditional on entry to some
outside market) if other banks of similar scope already exist in
both the original market and the specific outside market. Third, the
number of banks within a given geographic market should increase
with that market's nearness to other markets. The final prediction
follows the theoretical result that the number of banks within a
market is limited by the number of relevant outside markets. I test
these predictions using standard reduced-form econometric
techniques. Evaluated together, the results suggest that depositor
preferences for bank scope affect firm conduct and market structure
in the commercial banking industry -- and therefore that the
theoretical model is empirically relevant, at least in one specific
setting.
To my knowledge, this paper is the first to formally model
competition when consumers value firm scope. Nonetheless, the paper
relates to existing work in at least three other areas. First, the
theoretical model extends the substantial literature on the
boundaries of the firm. The assumption on consumer preferences
limits firms both across and within markets. Across markets, the
scopes of some firms are limited because expansion would intensify
price competition with competitors of greater scope. Within markets,
firm market shares are bounded above because revenue gains from
monopolization are outweighed by revenue losses on inframarginal
consumers. Of course, the mechanism by which the theoretical model
bounds firm size differs from the standard supply-side mechanisms
predominately featured in the literature (e.g., transaction costs in
Williamson [1965, 1979, 1985] and Klein, Crawford and Alchian
[1978], and control rights in Grossman and Hart [1986], Hart and
Moore [1990] and Hart [1995]). The assumption on depositor
preferences is more closely related to the vertical differentiation
models of Shaked and Sutton (1982, 1983).
Second, the paper offers a new explanation for the stylized fact
that commercial banks differ greatly in their scope. The differences
are difficult to understate: for example, the Bank of America (which
operates in more than 200 metropolitan markets) competes with more
the 3,000 single-market commercial banks. In the extant literature,
the most prominent explanation for this diversity invokes
technological factors, in particular the notion that small banks
have comparative advantages evaluating ``opaque'' small business
loans but comparative disadvantages evaluating other loans (e.g.,
Stein 2002; Cole, Goldberg and White 2004; Berger, Miller, Petersen,
Rajan and Stein 2005). The argument that depositor heterogeneity is
an important determinant of the scope distribution is compatible
with this and other alternative explanations, in the sense that the
results presented here to not rule out other determinants of bank
scope. I leave exploration of the relative importance of these
determinants to future research.
Third, the paper relates to a growing empirical literature on
competition among depository institutions. Although a full review
of this literature is beyond the scope of this paper, the work of
Adams, Brevoort and Kiser (2007) and Cohen and Mazzeo (2007) is of
particular relevance.\footnote{Other recent empirical work on the
relationship between bank scope and deposit rate competition
includes Beihl (2002), Hannan and Prager (2004), Park and Pennacchi
(2004), Berger, Dick, Goldberg and White (2007), and Dick
(forthcoming).} These authors estimate structural models that
distinguish between single-market and multimarket commercial banks
and show that competition within partitions is more pronounced than
competition between partitions. Cohen and Mazzeo argue informally
that the results are due to product differentiation across
partitions. The theoretical model developed here formalizes and
generalizes their argument. Of course, depository institutions may
be of substantive general interest because they are ubiquitous
(e.g., nearly 90 percent of U.S. households maintain a checking
account [Bucks, Kennickell and Moore 2006]) and important for
monetary policy transmission (e.g., Kashyap and Stein 2000).
The paper proceeds as follows. Section \ref{sec:facts} discusses
two stylized facts of the commercial banking industry that are
consistent with the theoretical model, namely that a bank scope
distribution exists and that banks of greater scope offer lower
deposit rates (i.e., higher prices) yet capture higher deposit
market shares within markets. Section \ref{sec:theory} develops the
theoretical model and derives empirical predictions. Section
\ref{sec:empirics} describes the data, estimates the theoretical
model structurally, and tests a number of distinct empirical
predictions using standard reduced-form techniques. Section
\ref{sec:conc} concludes.
\section{Two Stylized Facts \label{sec:facts}}
The theoretical model conforms to two stylized facts of the
commercial banking industry: 1) a distribution of bank scope exists
and 2) banks of greater scope offer lower deposit rates yet capture
greater deposit market shares within markets. I develop these facts
here and discuss the relevant literature. I defer detailed
discussion of the data for expositional convenience.
Panel A of Table \ref{tab:fact1} shows the total number of
commercial banks with branches in at least one metropolitan Core
Based Statistical Area (CBSA) and Panel B shows the mean number of
commercial banks per CBSA.\footnote{The Office of Management and
Budget defines a metropolitan CBSA to be a geographic areas that
contains at least one urban area of 50,000 or more inhabitants. A
CBSA also includes surrounding counties that meet specific commuting
requirements.} Each panel is tabulated by bank scope over the period
2001-2006. Most banks operate within a single CBSA (e.g., nearly 80
percent in 2006) and the vast majority operate in fewer than six
CBSAs (e.g., 97 percent in 2006). The diversity of bank scope is
pronounced within individual CBSAs -- nearly 75 percent of the
CBSA-year combinations over the sample period include at least one
bank in each scope category, and the average CBSA-year combination
features 11.21, 5.20, 2.22, and 3.80 banks that operate in 1, 2-5,
5-20 and more than 20 CBSAs, respectively. The distribution is
relatively stable through the sample period. Although banks of
greater scope are more common in 2006 than 2001, the changes appear
to be of only second-order magnitude. It may therefore be reasonable
to conjecture that bank scope diversity is real and not simply part
of an out-of-equilibrium transition.\footnote{Deregulation and
technological advances increased the efficient scope of banking
activity during the 1980s and 1990s. The result was a protracted
period of consolidation: for example, Berger (2003) reports that the
number of banks, inclusive of rural banks, decreased from 14,392 to
8,016 over the period 1984-2001. Berger, Kashyap and Scalise (1995)
and Berger (2003) provide excellent reviews of the relevant banking
literature.}
\begin{table}[t]
\begin{center}
\caption{The Bank Scope Distribution, 2001-2006 \label{tab:fact1}}
\begin{tabular}{c c c c c c c }
\hline \hline
\multicolumn{7}{c}{\rule[0mm]{0mm}{6.5mm}Panel A: The total number of banks} \\
\rule[0mm]{0mm}{4.5mm}\# of CBSAs & 2001 & 2002 & 2003 & 2004 & 2005 & 2006 \\
\hline
\rule[0mm]{0mm}{4.5mm}1 & 3,806 & 3,694 & 3,645 & 3,558 & 3,510 & 3,480 \\
\rule[0mm]{0mm}{4.5mm}2-5 & 700 & 725 & 758 & 753 & 781 & 831 \\
\rule[0mm]{0mm}{4.5mm}6-20 & 74 & 77 & 82 & 89 & 91 & 95 \\
\rule[0mm]{0mm}{4.5mm}$\geq$21 & 23 & 26 & 27 & 28 & 24 & 28 \\
\hline
\multicolumn{7}{c}{\rule[0mm]{0mm}{6.5mm}Panel B: The mean number of banks per CBSA} \\
\rule[0mm]{0mm}{4.5mm}\# of CBSAs & 2001 & 2002 & 2003 & 2004 & 2005 & 2006 \\
\hline
\rule[0mm]{0mm}{4.5mm}1 & 11.29 & 11.09 & 10.98 & 10.68 & 10.64 & 10.71 \\
\rule[0mm]{0mm}{4.5mm}2-5 & 4.94 & 5.07 & 5.19 & 5.21 & 5.45 & 5.83 \\
\rule[0mm]{0mm}{4.5mm}6-20 & 2.41 & 2.46 & 2.62 & 2.72 & 2.88 & 2.89 \\
\rule[0mm]{0mm}{4.5mm}$\geq$21 & 3.49 & 3.69 & 3.82 & 3.96 & 3.90 & 4.22 \\
\hline
\multicolumn{7}{p{4.00in}}{\footnotesize{The data include all
commercial banks with branches in at least one CBSA, over the period
2001-2006.}}
\end{tabular}
\end{center}
\end{table}
The extant literature explains the coexistence of large and small
banks as a product of comparative advantages in loan underwriting
technologies. In particular, larger banks may exploit economies to
scale in processing easily quantifiable information (i.e., ``hard
information''). By contrast, smaller banks may better handle
qualitative information that is difficult to transmit and/or verify
across layers of bureaucracy (i.e., ``soft information''), because
they can properly align loan officer research incentives (Stein
2002) and/or mitigate agency problems (Berger and Udell 2002). Some
empirical evidence supports this hypothesis. Smaller banks typically
allocate a far greater proportion of their assets to small business
loans, which may be more difficult to evaluate via hard information
(e.g., Berger, Kashyap, and Scalise 1995). More directly, the
quantitative financial statements of loan applicants are less
predictive of subsequent underwriting decisions at smaller banks
(Cole, Goldberg and White 2004), and smaller banks also tend to have
closer, more personal, more exclusive, and longer relationships with
their borrowers (Berger et al. 2005).\footnote{Similarly, Liberti
and Mian (2006) show that the underwriting decisions of loan
officers may weight soft information more heavily than those of bank
managers.}
The extent to which comparative advantages in loan underwriting
technologies fully explain the bank scope distribution is not clear.
From a theoretical standpoint, the framework predicts a bimodal
scope distribution characterized by distinctly large and small banks
that specialize in the analysis of hard and soft information,
respectively. The observed distribution, by contrast, suggests a
prominent role for ``medium-sized'' banks. Further, recent empirical
evidence suggests that only 20 percent of small business loans
issued by banks with assets under \$1 billion are evaluated with
soft information underwriting technologies (Berger and Black 2007).
The market for soft information loans, by itself, may not be
sufficient to support the operation of small banks.
I now turn to the second stylized fact. Table 2 shows mean deposit
interest rates and market shares, tabulated by bank scope. The means
are based on 45,785 bank-CBSA-year observations over the period
2001-2006. I calculate the deposit interest rates as interest
expenses over total deposits, and the market shares as a proportion
of all commercial bank deposits in the CBSA-year. As shown, deposit
rates decrease with scope and market shares increase in scope. The
differences are dramatic: for example, the average single-market
bank offers a deposit rate that is 58 percent higher than the
average bank with branches in more than twenty CBSAs, yet it
captures less than one-fifth the market share.
\begin{table}[t]
\begin{center}
\caption{Deposit Rates and Market Shares \label{tab:fact2}}
\begin{tabular}{c c c }
\hline \hline
\rule[0mm]{0mm}{4.5mm} \# of CBSAs & Deposit Rate & Market Share \\
\hline
\rule[0mm]{0mm}{4.5mm}1 & 0.019 & 0.020 \\
\rule[0mm]{0mm}{4.5mm}2-5 & 0.018 & 0.039 \\
\rule[0mm]{0mm}{4.5mm}6-20 & 0.014 & 0.075 \\
\rule[0mm]{0mm}{4.5mm}$\geq$21 & 0.012 & 0.114 \\
\hline \multicolumn{3}{p{3.0in}}{\footnotesize{The table shows mean
deposit rates and market shares by bank scope. The data include all
commercial banks with branches in at least one CBSA over the period
2001-2006, for a total of 45,785 bank-CBSA-year observations.}}
\end{tabular}
\end{center}
\end{table}
One common explanation for the negative relationship between deposit
rates and bank scope is that larger banks have superior access to
wholesale funds and substitute away from deposits (e.g., Kiser 2004;
Hannan and Prager 2004; Park and Pennacchi 2004). However, the
wholesale funds hypothesis predicts that deposit shares decrease
with scope, and is therefore incomplete at best. Potentially closer
is the observation of Bassett and Brady (2003) that the deposit
growth of small banks exceeded that of large banks over the period
1990-2001. Indeed, the combination of sticky deposit supply -- due
to switching costs and/or other factors -- and appropriate initial
conditions could generate the patterns shown in Table
\ref{tab:fact2}. Although there is some empirical evidence
supporting the existence of deposit supply stickiness (e.g., Sharpe
1997; Kiser 2002a, 2002b), to my knowledge no work has attempted to
quantify its importance for the banking industry structurally.
\section{Theoretical Model \label{sec:theory}}
\subsection{The equilibrium concept}
The theoretical model is based on a three stage non-cooperative
game. The timing of the game is as follows: In the first stage,
banks decide whether to enter a single geographic market, which I
label the inside market. Entry decisions become common knowledge at
the end of the first stage. In the second stage, banks that enter
the inside market choose whether to establish a presence in each of
$M$ outside markets. Finally, banks set deposit rates given the
second stage actions and compete for deposits within the inside
market.
Strategies consist of actions to be taken in each of the three
stages. Strategies are therefore of the form: do not enter the
inside market; or enter the inside market, establish a presence in
$0,1,\dots,$ or $M$ outside markets (conditional on the first stage
actions), and set a deposit interest rate (conditional on the first
and second stage actions).
Banks that enter the inside market receive payoffs (profits) based
on a depositor choice model introduced below. Banks that do not
enter the inside market receive payoffs of zero.
The solution concept is that of subgame perfect equilibrium (e.g.,
Selton 1975). An $n$-tuple of strategies forms a subgame perfect
equilibrium if it forms a Nash equilibrium in every stage-game. To
solve the game, I first analyze deposit rate competition in the
third stage, and then turn to the second and first stages.
\subsection{Deposit rate competition}
Suppose that $j=1,2,\dots,J$ banks enter the inside market. Each
bank is characterized by the number of outside markets in which it
is present ($m_j$), and chooses a deposit rate $r^D_j$ to maximize
profits:
\begin{equation}
r^{D}_j = \arg \max (r^L-r^D_j)N s_j(r^D_.,m_.),
\label{eq:tprofs}
\end{equation}
where $r^L$ is the fixed interest rate obtained from investments in
a competitive lending market, $N$ is the size of the inside market,
$s_j$ is the share of deposits obtained from the inside market, and
$r^D_.$ and $m_.$ are vectors of the deposit rates and bank scopes,
respectively. Without loss of generality, I normalize the size of
the inside market to one.
Deposit shares are determined by a continuum of depositors that
differ only in the probability with which they travel to outside
markets. This probability of travel, which I denote as $\alpha$, is
distributed according to some cumulative distribution function
$F(\alpha)$ with support between zero and one. Depositors split the
probability of travel evenly across the outside markets, so that the
probability of travel to each outside market is simply $\alpha
/M$.\footnote{Alternatively, it is possible model depositors that
differ in their market-specific travel probabilities. For example,
one New Yorker may be more likely to visit Chicago than Boston while
another may be more likely to visit Boston than Chicago. The
results presented here extend naturally.} Depositors that travel to
an outside market from which their bank is absent pay a cost
$\gamma$ to participate in a competitive ATM market. Thus, the
expected utility that depositor $i$ receives from bank $j$ takes the
form
\begin{equation}
\text{E}[u(\alpha,j)] = (1-\alpha) r^D_j + \alpha \frac{m_j}{M}
r^D_j + \alpha \left(1-\frac{m_j}{M}\right)(r^D_j-\gamma),
\end{equation}
where the term $\frac{m_j}{M}$ is the conditional probability of
travel to an outside market in which bank $j$ has a presence, given
that travel occurs. Combining terms yields the tractable expression
\begin{equation}
\text{E}[u(\alpha,j)] = r^D_j + \gamma \alpha \left(\frac{m_j}{M}-1
\right). \label{eq:tutil}
\end{equation}
The utility representation has the interpretation that depositors
that never travel (i.e., $\alpha=0$) consider only the deposit
interest rate. Importantly, all depositors at least weakly prefer
banks of greater scope for a given deposit interest rate. The set-up
therefore fits within the class of vertical differentiation models
first analyzed by Shaked and Sutton (1982, 1983) and Tirole (1988,
Section 2.1).\footnote{Vertical differentiation models typically
analyze firms that choose their ``quality'' and then compete in
prices for consumers that differ in their willingness-to-pay. In
these models, the support of consumer preferences tends to bound the
number of profitable firms (e.g., Shaked and Sutton 1983, 1984;
Motta 1993) and prices, market shares, and profits generally
increase in quality (e.g., Shaked and Sutton 1982; Choi and Shin
1992; Donnenfeld and Weber 1992; Wauthy 1996; Lehmann-Grube 1997).}
Depositors select the bank that provides the greatest expected
utility. This implicitly defines the set of travel probabilities
that corresponds to the selection of bank $j$, and integrating over
the set yields an expression for the market share of bank $j$:
\begin{equation}
s_{j} = \int_{[\alpha | \; u_{ij} \geq \;u_{ik} \forall \; k=1,
\dots, \; J]} d F(\alpha). \label{eq:tshares}
\end{equation}
Ranking banks in increasing order of scope, such that
$m_11$, these indifference travel probabilities have the
expression:
\begin{equation}
\alpha_j = \frac{r^D_{j-1}-r^D_j}{\gamma \left( \frac{m_j -
m_{j-1}}{M}\right) }. \label{eq:tcutoff}
\end{equation}
Depositors with the travel probability $\alpha<\alpha_j$ strictly
prefer bank $j-1$ to bank $j$, whereas depositors with
$\alpha>\alpha_j$ strictly prefer bank $j$ to bank $j-1$. The
indifference travel probability for the bank of least scope has the
special form $\alpha_1 = \frac{r^D_1}{\gamma(1- m_1/M)},$ which
reflects the possibility that not all depositors select a bank.
Further progress requires an evaluation of the market shares, and I
let $\alpha$ have uniform density in order to facilitate an analytic
solution. The uniform density simplifies the market share
equations: $s_j=\alpha_{j+1}-\alpha_j$ for $0\leq jm_k$). If bank $j$ sets a deposit
rate at least as large as bank $k$ ($r^D_j \geq r^D_k$) then bank
$k$ has zero market share and does not earn positive
profits.}\end{quote}
\noindent Lemma 2 helps deliver the key result that deposit interest
rates decrease in scope and market shares increase in scope.
\begin{quote} \textbf{Proposition 1.} \textit{Any stage game
Nash equilibrium (if it exists) features deposit rates that decrease
strictly in scope, i.e., $r^D_j>r^D_{j+1}$ for any $jM+1$:
\begin{quote} \textbf{Lemma 4.} \textit{If $J \leq M+1$ then
the two stage subgame has a unique class of subgame perfect
equilibria in which each bank differs in scope.}
\end{quote}
\begin{quote} \textbf{Lemma 5.} \textit{If $J>M+1$ then the two stage
subgame has a unique class of subgame perfect equilibria in which
$m_j=m_k$ for some $j$ and $k$ and there is at least one bank of
each scope over $m_{j+1}, \; m_{j+2},\; \dots,M$.}
\end{quote}
\noindent The results follow naturally from the third stage deposit
rate competition. So long as the number of entrants is no greater
than the upward bound established in Corollary 1, any bank that is
not differentiated in scope has a profitable deviation in the second
stage. On the other hand, if the number of entrants exceeds the
upward bound then at least one bank is undifferentiated and earns no
profits in the third stage, yet has no profitable deviation in the
second stage.
To complete the analysis, in the first stage some number of banks
choose whether to enter the inside market given the number of
outside markets ($M$) and with full knowledge of competition in the
subsequent stages. In order to eliminate equilibria in which banks
enter and then fail to earn positive profits in the third stage, I
introduce an arbitrarily small entry cost $\epsilon>0$. The main
result of the theoretical model follows immediately.
\begin{quote} \textbf{Proposition 2.} \textit{For any sufficiently small
$\epsilon>0$ and any number of potential entrants $N>M$, there
exists a unique subgame perfect equilibrium in which exactly $M+1$
banks enter the inside market. Each bank enters a different number
of outside markets and earns positive profits.} \end{quote}
\noindent Proposition 2 establishes that the unique subgame perfect
equilibrium is consistent the first stylized fact presented in
Section \ref{sec:facts}, namely that a diversity of bank scopes
exists. This diversity exists despite the fact that every depositor
at least weakly prefers banks of greater scope for a given interest
rate. The presence of depositor heterogeneity allows larger banks
to reduce their deposit rates and still attract depositors with high
scope valuations. Smaller banks, meanwhile, maintain a (relatively)
unprofitable niche position in which they attract depositors with
low scope valuations.\footnote{The result that some smaller firms
may prefer their niche position has empirical parallels. For
example, Bajari, Fox and Ryan (2006) note that small cellular
carriers often do not agree to charge roamers low per-minute rates.}
\subsection{Testable empirical predictions}
To the extent that differences in depositor willingness-to-pay exist
and are important, the theoretical model has a number of empirical
predictions:
\begin{quote} \textbf{Prediction 1.} A diversity of bank scopes
should exist within markets. \end{quote}
\begin{quote} \textbf{Prediction 2a.} Banks of greater scope should
have lower deposit interest rates and higher market shares.
\end{quote}
\begin{quote} \textbf{Prediction 2b.} Banks of greater scope should
have less elastic demand.
\end{quote}
\noindent Section \ref{sec:facts} shows that Predictions 1 and 2a
hold in the banking data over the period 2001-2006, and results from
the structural model (Section \ref{sec:logit}) are consistent with
Prediction 2b. Although a number of alternative theories can
together generate these predictions, the theoretic model presented
here may retain its appeal because it provides a single intuitive
explanation. The next predictions further differentiate the
theoretical model:
\begin{quote} \textbf{Prediction 3.} A bank in market
$a$ should be less likely to enter an outside market if banks of
greater scope are already in market $a$.
\end{quote}
\begin{quote} \textbf{Prediction 4.} Conditional on entry to some
outside market, a bank in market $a$ should be less likely to enter
market $b$ if another bank of similar scope already exists in
markets $a$ and $b$.
\end{quote}
\begin{quote} \textbf{Prediction 5.} The number of banks within a
market should increase with the market's nearness to other markets.
\end{quote}
\noindent The third and fourth predictions follow the theoretical
result that scope differentiation improves profitability (e.g.,
Lemmas 1 and 4). The fifth prediction follows the intuition that
depositors are more likely travel to nearby markets than distant
ones. Markets that are near others have more relevant outside
markets, and thus offer superior opportunities for scope
differentiation (e.g., Corollary 1). I use standard reduced-form
econometric techniques to evaluate these predictions in Section
\ref{sec:entry}.
\section{Data and Empirical Implementation \label{sec:empirics}}
\subsection{Data \label{sec:data}}
The bulk of the data used in this study comes from the Summary of
Deposits and the December and June Call Reports. The Summary of
Deposits tracks the location of all commercial bank branches and
deposits and is maintained by the Federal Deposit Insurance
Corporation (FDIC). The Call Reports contain the balance sheets and
income statements of commercial banks and are maintained by the
Federal Financial Institutions Examination Council (FFIEC). I
compile data from these sources over the period 2001-2006. The data
yield 26,905 observations on the bank-year level and 45,785
observations on the bank-CBSA-year level; each observation is in
June of its respective year.
Table \ref{tab:sumstat1} presents summary statistics at the
bank-year and bank-CBSA-year level. The main quantities of interest
are the deposit interest rates, the market shares, and the bank
scopes. I calculate the deposit rates as interest expenses (incurred
over the previous year) over deposits (averaged over the previous
year), the market shares as deposits over the sum of all commercial
bank, thrift, and credit union deposits, and the scopes as the
numbers of CBSAs in which the banks have branches. To be clear, the
deposit interest rates and the scopes vary on the bank-year level,
and the market shares vary on the bank-CBSA-year level.\footnote{The
lack of CBSA-specific deposit rates may not hinder empirical
analysis. More than 80 percent of the bank-year observations have
branches in a single CBSA, and larger banks tend to set uniform
deposit rates across CBSAs (e.g., Radecki 1998; Heitfield 1999).
Although the regulatory reports do not report the location of ATMs,
Dick (forthcoming) reports a correlation coefficient between the
number of branches and the number of ATMs of 0.80, based on data
obtained from a large ATM network.} The mean bank-year observation
has a deposit rate of 0.019, an average market share of 0.025, and
scope of 1.70. The means of the bank-year-CBSA observations more
heavily weight banks that operate in many CBSAs. The lower deposit
rate mean (0.017) and higher market share mean (0.047) are
consistent with the stylized fact that banks of greater scope offer
lower deposit rates and have higher market shares.
Turning to the remaining commercial bank variables, the mean
bank-year observation has 1.235 billion dollars in gross total
assets. The assets are funded, in part, by an average of 0.76
billion dollars in deposits and invested in an average of 0.72
billion dollars worth of loans. Finally, the mean bank-year
observation has 2.94 branches in each of its markets, employs 23.41
people per branch, and charged off roughly five million dollars of
loans over the previous year. The mean bank-CBSA-year observation
has much higher gross total assets, loans, deposits, and branch
density because the mean more heavily weights the larger banks.
\begin{table}[t]
\begin{center}
\caption{Commercial Bank Summary Statistics \label{tab:sumstat1}}
\begin{tabular}{l l c c c c }
\hline \hline
\multicolumn{2}{l}{\rule[0mm]{0mm}{5.5mm}Units of Observation:} & \multicolumn{2}{c}{Bank-Year} & \multicolumn{2}{c}{Bank-CBSA-Year} \\
\rule[0mm]{0mm}{4.5mm}Variable & Description & Mean & St. Dev. & Mean & St. Dev. \\
\hline
\multicolumn{3}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Deposit pricing and market share}} \\
\rule[0mm]{0mm}{4.5mm}$\; \; r^D_{jt}$ & Deposit rate & 0.019 & (0.011) & 0.017 & (0.011) \\
\rule[0mm]{0mm}{4.5mm}$\; \; s_{jmt}$ & Market share & 0.025 & (0.053) & 0.047 & (0.078) \\
\multicolumn{3}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Bank scope}} \\
\rule[0mm]{0mm}{4.5mm}$\; \; m_{jt}$ & \# of CBSAs & 1.702 & (5.167) & 17.389 & (39.024) \\
\rule[0mm]{0mm}{4.5mm}$\; \; \left(\frac{m_{jt}}{M}-1\right)$ & Normalized scope & -0.998 & (0.014) & -0.954 & (0.109) \\
\multicolumn{3}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Other bank variables}} \\
\rule[0mm]{0mm}{4.5mm}$\; \;$ GTA$_{jt}$ & Gross total assets & 1.235 & (17.017) & 38.269 & (124.512) \\
\rule[0mm]{0mm}{4.5mm}$\; \;$ LOANS$_{jt}$ & Loans & 0.721 & (9.217) & 22.211 & (68.094) \\
\rule[0mm]{0mm}{4.5mm}$\; \;$ DEPS$_{jt}$ & Deposits & 0.762 & (9.366) & 24.197 & (78.223) \\
\rule[0mm]{0mm}{4.5mm}$\; \;$ BRDEN$_{jmt}$ & Branch density & 3.935 & (8.190) & 7.176 & (18.933) \\
\rule[0mm]{0mm}{4.5mm}$\; \;$ NEMP$_{jt}$ & \# of employees & 23.412 & (32.002) & 24.404 & (29.377) \\
\rule[0mm]{0mm}{4.5mm}$\; \;$ CHG$_{jt}$ & Charge-offs & 0.005 & (0.107) & 0.152 & (0.542) \\
\hline
\multicolumn{6}{p{5.1in}}{\footnotesize{\rule[0mm]{0mm}{0mm}Summary
statistics for 26,905 bank-year observations and 45,785
bank-CBSA-year observations over the period 2001-2006. Gross total
assets, deposits, loans, and charge-offs are in billions of 2000
dollars.}}
\end{tabular}
\end{center}
\end{table}
The statistical tests of the empirical predictions require some
aggregation to the CBSA-year level, and Table \ref{tab:sumstat2}
presents summary statistics for the 2,160 CBSA-year observations in
the data over the period 2001-2006. As shown, the mean CBSA-year
contains 9.49 billion dollars in deposits, spread among 21.02 banks
and 152.11 bank branches. The mean CBSA population, median
household income (in dollars), and land area are 645.47, 48.03, and
22.11 thousand, respectively. Implicit in the data construction is
the notion that metropolitan CBSAs approximate well the relevant
geographic markets for depository services. Recent research provides
some support: Amel and Starr-McCluer (2001) show that the median
household travels only three miles to its depository institution and
that roughly 90 percent of checking and savings accounts are held by
local depository institutions. Kwast, Starr-McCluer and Wolken
(1997) report similar numbers for small businesses. Finally,
Heitfield (1999) and Heitfield and Prager (2004) show that average
deposit rates vary across cities, and interpret the results as
consistent with local market competition.
\begin{table}[t]
\begin{center}
\caption{CBSA Summary Statistics \label{tab:sumstat2}}
\begin{tabular}{l l c c }
\hline \hline
\multicolumn{2}{l}{\rule[0mm]{0mm}{5.5mm}Units of Observation:} & \multicolumn{2}{c}{CBSA-Year} \\
\rule[0mm]{0mm}{4.5mm}Variable & Description & Mean & St. Dev. \\
\hline
\multicolumn{4}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Number of commercial banks, branches, and deposits}} \\
\rule[0mm]{0mm}{4.5mm}$\; \;$ BANK$_{mt}$ & Total banks & 21.202 & (24.156) \\
\rule[0mm]{0mm}{4.5mm}$\; \;$ BRANCH$_{mt}$ & Total branches & 152.114 & (296.166) \\
\rule[0mm]{0mm}{4.5mm}$\; \;$ DEP$_{mt}$ & Total deposits & 9.491 & (33.771) \\
\multicolumn{4}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Other CBSA variables}} \\
\rule[0mm]{0mm}{4.5mm}$\; \;$ POP$_{m}$ & Population & 645.468 & (1487.606) \\
\rule[0mm]{0mm}{4.5mm}$\; \;$ INC$_{m}$ & Median HH income & 48.033 & (7.692) \\
\rule[0mm]{0mm}{4.5mm}$\; \;$ MIL$_{m}$ & Land area & 22.111 & (22.545) \\
\hline
\multicolumn{4}{p{4in}}{\footnotesize{\rule[0mm]{0mm}{0mm}Summary
statistics for 2,160 CBSA-year observations over the period
2001-2006. Total deposits are in billions, and population, median
household income, and land area are in thousands.}}
\end{tabular}
\end{center}
\end{table}
\subsection{The estimation of depositor willingness-to-pay \label{sec:logit}}
In this section, I develop and estimate a structural model of
depositor choice and test the first-order assumption of the
theoretical model, namely that depositors prefer banks of greater
scope but differ in their willingness-to-pay.\footnote{The
structural model requires its own set of assumptions, and I evaluate
throughout the extent to which estimation is robust to different
specifications and identification strategies.} To improve the
empirical relevancy of the theoretical model, I augment the expected
utility specification of Equation \ref{eq:tutil} such that it more
flexibly accommodates bank differentiation and depositor
heterogeneity:
\begin{equation} \text{E}[u_{ijmt}] = \beta_i^D r^D_{jt} +
\beta^M \left(\frac{m_{jt}}{M}-1 \right) + x_{jmt}^\prime \beta^X +
\xi_{jmt} + \epsilon_{ijmt}, \label{eq:sutil}
\end{equation}
where $i$, $j$, $m$, and $t$ index depositors, banks, markets, and
years, respectively. The vector $x_{jmt}$ includes observable bank
characteristics, the scalar $\xi_{jmt}$ is the mean depositor
valuation of bank $j$, and the scalar $\epsilon_{ijmt}$ allows for
depositor-specific taste shocks. Within the context of the
theoretical model, the parameter $\beta^D_i$ is the inverse travel
probability (i.e., $\beta^D_i = 1/\alpha$) and the parameter
$\beta^M$ is the annualized cost of participating in the competitive
ATM market (i.e., $\beta^M=\gamma$.) I combine these parameters to
calculate willingness-to-pay for a unit increase in scope, given by
$\beta^M / (M\beta^D_i)$.\footnote{I convert willingness-to-pay into
dollar terms by multiplying by 3,800, the median dollar amount held
in transaction accounts (Bucks, Kennickell and Moore 2006).} The
final parameter vector, $\beta^X$, captures the contribution of bank
observables to expected utility.
The depositor-specific parameter $\beta^D_i$ allows for
heterogeneity in the willingness-to-pay for bank scope. I model the
parameter as having the multivariate normal distribution conditional
on depositor income:
\begin{equation}
\beta_i^D = \beta^D + \pi y_i + \sigma \nu_i, \qquad \nu_i \sim
\text{N}(0,1),\label{eq:srpar}
\end{equation}
where $y_i$ is income, $\nu_i$ captures unobserved demographics, the
parameter $\beta^D$ is the mean depositor interest rate valuation,
the parameter $\pi$ allows this mean valuation to vary across
incomes, and $\sigma$ is a scaling vector. This formulation permits
a simple statistical test for the presence of heterogeneity in
willingness-to-pay: under the null hypothesis of no heterogeneity,
the demographic parameters $\pi$ and $\sigma$ are jointly zero.
As in the theoretical model, I permit depositors to select the
outside option, i.e., elect not to choose a commercial bank. I let
the outside option be thrifts and credit unions, and use thrift and
credit union balance sheets to determine the outside option market
shares directly.\footnote{The use of thrifts and credit unions as
the outside good is somewhat compelling. Amel and Starr-McCluer
(2001) show that commercial banks, thrifts and credit unions
together account for 98 percent of all checking and savings
accounts. Further, lumping the individual thrift and credit union
institutions together may not unduly restrict the substitution
patterns of interest: Adams, Brevoort and Kiser (2007) estimate a
median cross interest rate elasticity between commercial banks and
thrifts and only -0.002.} The use of alternative outside option
shares, calculated as population (times a constant of
proportionality) less commercial bank deposits, returns similar
parameter estimates. I denote the outside option as $j=0$. Since the
mean valuations of the outside option are not separably
identifiable, I let the expected utility obtainable from the outside
option be $\text{E}[u_{i0mt}] = \epsilon_{i0mt}.$
I again maintain the assumption that depositors select the bank that
provides the greatest expected utility. This defines the set of
depositor attributes \{$y_{im}$, $\nu_{im}$, $\epsilon_{i.mt}$\}
that correspond to the selection of bank $j$ in market $m$ and year
$t$. Integrating over the set yields an expression for the deposit
market shares:
\begin{equation}
s_{jmt} = \int_{[(y_i,\; \nu_i,\; \epsilon_{ijmt}) | \; u_{ijmt}>
\;u_{ikmt} \forall \; k=0, \dots, \; J]} d F(\epsilon) d F(\nu) d
F(y), \label{eq:sshar}
\end{equation}
where $F(\cdot)$ denotes the relevant population distribution
functions. Specific distributional assumptions on $F(\cdot)$ enable
evaluation of the integral via analytic and/or numerical methods.
Throughout, I let $\epsilon_{ijmt}$ be distributed $iid$ with the
extreme value type I density. The distributional assumption on
$\epsilon$ is itself not restrictive, as mixed logit models can
approximate any random utility model, to any degree of accuracy,
given appropriate choices of regressors and random parameter
distributions (McFadden and Train 2000).\footnote{Somewhat more
troublesome, given the empirical evidence regarding depositor
switching costs (e.g., Sharpe 1997; Kiser 2002a, 2002b), is the
assumption that the taste shocks are independent over time.}
As a prelude to the mixed logit estimation, I first impose the
restriction that depositors have homogenous tastes for observables
(i.e., I impose $\pi=\sigma=0$). The restriction makes integration
over the demographics $y$ and $\nu$ unnecessary. The resulting logit
demand system follows from analytical integration over the
depositor-specific taste shocks:
\begin{equation}
\log(s_{jmt}) - \log(s_{0mt})=\beta^D r^D_{jt} + \beta^M
\left(\frac{m_{jt}}{M}-1 \right) + x_{jmt}^\prime \beta^X +
\xi_{jmt}. \label{eq:slogit}
\end{equation}
The equation can be estimated by treating the mean valuation
($\xi_{jmt}$) as an unobserved error term. The logit demand system
is problematic because it specifies unrealistic elasticities (e.g.,
Berry 1994; Berry, Levinsohn and Pakes 1995; Nevo 2001) and does not
permit the test of depositor heterogeneity that is of interest here.
Nonetheless, its estimation is much less computationally burdensome
than that of the mixed logit model, and may also provide substantial
intuition regarding the validity of the identification
strategy.\footnote{The mixed logit estimation requires demographic
information from the Consumer Population Survey (CPS). The
information is available for a subset of the CBSAs in the full
sample. For consistency, I use the subset to estimate the logit
model. The results are robust to the use of the full sample.}
Table \ref{tab:logit} shows the logit results. The Column 1 results
are computed with OLS. The coefficients have the expected signs:
depositors appear to prefer to prefer banks with higher deposit
rates, greater scopes and branch densities, and more employees per
branch. The deposit rate coefficient of 12.78 corresponds to a
median deposit rate elasticity of only 0.18. One might expect these
numbers to understate the true depositor responsiveness to deposit
interest rate changes. If higher quality banks (i.e., those with
high mean valuations) tend to offer lower deposit rates, then the
assumed orthogonality between the deposit rates and mean valuations
fails and the estimated coefficient should be too
small.\footnote{Adams, Brevoort and Kiser (2007), Knittel and Stango
(2007), and Dick (forthcoming), estimate median interest rate
elasticities for the commercial banking industry of 2.31, 1.20, 1.70
respectively. Elasticities below one in magnitude are often
considered inconsistent with profit maximization.}
\begin{table}
\begin{center}
\caption{Logit Regression Results \label{tab:logit}}
\begin{tabular}[t]{l c c c c c c }
\hline \hline
\rule[0mm]{0mm}{5.5mm}Variables & (1) & (2) & (3) & (4) & (5) & (6) \\
\hline
\multicolumn{6}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Deposit interest rate}} \\
\rule[0mm]{0mm}{4.5mm} \; $r^D_{jt}$ & \footnotesize{12.780***} & \footnotesize{105.060***} & \footnotesize{115.269***} & \footnotesize{115.846***} & \footnotesize{98.576***} & \footnotesize{100.821***} \\
\; &\footnotesize{(1.436)} & \footnotesize{(6.983) } & \footnotesize{(13.591) } & \footnotesize{(7.721) } & \footnotesize{(5.704) } & \footnotesize{(5.603) } \\
\multicolumn{6}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Bank scope}} \\
\rule[0mm]{0mm}{4.5mm} \; $\left(\frac{m_{jt}}{M}-1 \right)$ & \footnotesize{2.455***} & \footnotesize{4.347***} & \footnotesize{4.557***} & \footnotesize{4.568***} & \footnotesize{2.313 } & \footnotesize{4.387**} \\
\; & \footnotesize{(0.333) } & \footnotesize{(0.572) } & \footnotesize{(0.632) } & \footnotesize{(0.605) } & \footnotesize{(2.091) } & \footnotesize{(2.134)} \\
\multicolumn{6}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Control variables}} \\
\rule[0mm]{0mm}{4.5mm} \; BRDEN$_{jmt}$ & \footnotesize{1.434***} & \footnotesize{1.463***} & \footnotesize{1.466***} & \footnotesize{1.466***} & \footnotesize{1.377***} & \footnotesize{1.015***} \\
\; & \footnotesize{(0.048) } & \footnotesize{(0.045) } & \footnotesize{(0.045) } & \footnotesize{(0.044) } & \footnotesize{(0.060) } & \footnotesize{(0.053)} \\
\rule[0mm]{0mm}{4.5mm} \; NEMP$_{jt}$ & \footnotesize{0.062* } & \footnotesize{ 0.073* } & \footnotesize{0.074* } & \footnotesize{0.074* } & \footnotesize{-0.002 } & \footnotesize{-0.040} \\
\; & \footnotesize{(0.033) } & \footnotesize{(0.041) } & \footnotesize{(0.043) } & \footnotesize{(0.043) } & \footnotesize{(0.040) } & \footnotesize{(0.038)} \\
\rule[0mm]{0mm}{4.5mm} \; lag($s_{jmt}$) & & & & & & \footnotesize{8.824***} \\
\; & & & & & & \footnotesize{(0.693)} \\
\multicolumn{7}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Imputed
willingness-to-pay for scope}} \\
\rule[0mm]{0mm}{4.5mm} WTP & \footnotesize{2.033} & \footnotesize{0.437 } & \footnotesize{0.418} & \footnotesize{0.418} & \footnotesize{0.248} & \footnotesize{0.464} \\
\\
\rule[0mm]{0mm}{4.5mm} $R^2$ & \footnotesize{0.479} & \footnotesize{0.286} & \footnotesize{0.240} & \footnotesize{0.238} & \footnotesize{0.437} & \footnotesize{0.443} \\
\rule[0mm]{0mm}{4.5mm} 1st stage $F$-test & \footnotesize{$\cdot$} & \footnotesize{226.96***} & \footnotesize{71.73***} & \footnotesize{318.56***}& \footnotesize{214.67***}& \footnotesize{287.16***} \\
\rule[0mm]{0mm}{4.5mm} Instruments & $\cdot$& \footnotesize{Loans+CC}&\footnotesize{Loans}& \footnotesize{CC}& \footnotesize{Loans+CC} & \footnotesize{Loans+CC} \\
\rule[0mm]{0mm}{4.5mm} Fixed effects & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & \footnotesize{Bank} & \footnotesize{Bank} \\
\hline
\multicolumn{7}{p{6.2in}}{\footnotesize{\rule[0mm]{0mm}{0mm}Results
from logit estimation. The data include 38,993 bank-CBSA-year
observations. The dependent variable is $\log(s_{jmt}) -
\log(s_{0mt})$, where $s_{jmt}$ is the market share of bank $j$ and
$s_{0mt}$ is the outside good market share. All regressions include
a constant. Standard errors are clustered at the bank level and
shown in parenthesis. Significance at the 10\%, 5\%, and 1\% levels
is denoted by *, **, and ***, respectively.}}
\end{tabular}
\end{center}
\end{table}
I employ two sets of instruments to help mitigate this potential
endogeneity problem. The first set relies on the notion that
loan-side conditions may affect deposit pricing (e.g., Kiser 2004)
but have no direct effect on depositor valuations. In particular, I
proxy the demand for funds and loan cost:
\begin{equation}
z_{1,jt} = \frac{\text{LOANS}_{jt}}{\text{GTA}_{jt}}, \quad
\text{and} \quad z_{2,jt} =
\frac{\text{CHG}_{jt}}{\text{LOANS}_{jt}}.
\end{equation}
Banks with higher loans-to-assets ratios may be more likely seek
additional deposits with which to fund loans, and banks with higher
charge-offs may have greater lending costs and less incentive to
attract additional deposits. Turning to the second set of
instruments, under the maintained assumption that the non-deposit
rate characteristics are exogenous, competitor non-price
characteristics provide natural instruments. Similarly to Berry,
Levinsohn and Pakes (1995), I calculate the sum of competitor
characteristics and then average this sum across each bank's
markets:
\begin{equation}
z_{3,jt}= \frac{1}{m_{jt}} \sum_{l \in M_{jt}} \left( \sum_{k \neq
j, \; k \in J_{lt}} w_{klt} \right),
\end{equation}
where $w_{ijt}$ is a vector of bank observables that includes
$\left(\frac{m_{jt}}{M}-1 \right)$ and $x_{jmt}$, $M_{jt}$ is the
set of markets in which bank $j$ has branches, and $J_{mt}$ is the
set of banks with branches in market $m$. In Table \ref{tab:logit},
I refer to the two sets of instruments as ``Loans'' and ``CC,''
respectively.\footnote{The model does not provide intuition
regarding the form of the relationship (if any) between the deposit
interest rate and the instruments. I use third-order polynomials in
each instrument to flexibility estimate the first stage.}
I show the baseline 2SLS logit results in Column 2. As expected,
the estimated deposit rate coefficient of 105.06 is much larger, and
the implied median deposit rate elasticity is a more plausible 1.47.
A statistical comparison to the OLS results, ala Hausman (1978),
easily rejects the null that the deposit rate is exogenous to
depositor mean valuations. The bank scope coefficient of 4.35
remains statistically different than zero and is consistent with
depositors that prefer banks of greater scope. The coefficients
imply an annual willingness-to-pay for a unit increase in bank scope
of 43.70 cents. The number may be substantial. For example, it
implies that depositors value the scope of Bank of America (branches
in 207 CBSAs in 2006) at roughly 89.97 dollars more than that of a
single-market bank. By way of comparison, the 2006 data suggest
that the median account of 3,800 dollars earns 33.86 fewer dollars
at Bank of America than at the average single-market
bank.\footnote{The results are robust to a number of alternative
specifications. For example, the inclusion of market or year fixed
effects (which limits identification derived from outside good
market shares), the inclusion of the deposit fee rate as an
additional endogenous regressor (i.e., fee income over deposits),
and the use of the alternative outside good (calculated as
population times a constant of proportionality less commercial bank
deposits), do not substantially alter the results.}
The consistency of the logit estimation depends on 1) the validity
of the instruments and 2) the exogeneity of the non-deposit rate
characteristics. I examine these assumptions in the remaining
columns. To start, Column 3 estimates the model using only the
loan-side instruments. The results are quite similar to the
baseline. If one is willing to assume that the loan-side instruments
are valid, then a comparison to the baseline results yields a
statistical test for the validity of the competitor characteristic
instruments (e.g., Hausman 1978; Ruud 2000). Intuitively, the Column
3 estimates are consistent given the exogeneity of the loan-side
instruments. Under the null hypothesis that the competitor
characteristics are also valid, the baseline coefficients should be
similar. The data do not reject the null ($p$-value$=0.983$).
Column 4 estimates the model using only the competitor
characteristic instruments. The results are again similar to the
baseline, and the data fail to reject the null that the loan-side
instruments are valid, given the validity of the competitor
characteristic characteristics ($p$-value$=0.764$). Together, the
results provide some evidence that the instruments may be valid.
Columns 5 and 6 help evaluate the exogeneity of bank scope. One
might expect the baseline estimated scope coefficient to overstate
the true effect. If higher quality banks (i.e., banks for which
depositors have higher mean valuations) tend to enter more markets,
then the assumed orthogonality between scope and the mean valuations
fails and the estimated coefficient should be too large. To help
address this concern, I decompose the mean valuations into bank
fixed effects and market-year specific valuations:
\begin{equation}
\xi_{jmt} = \xi_j + \Delta \xi_{jmt},
\end{equation}
and estimate the bank fixed effects directly.\footnote{The procedure
yields consistent estimates provided that a bank's quality is fixed
across markets. In the implementation, I estimate separate effects
for the 1,212 banks that have branches in at least two CBSAs at some
point during the sample, and estimate a shared effect for the
remaining banks. The restriction greatly eases the computational
burden, and the procedure should still mitigate bias due to scope
endogeneity. The parameters of interest are not well-identified
empirically when a separate fixed effect is estimated for each
single-market bank because the effects overtax the data (there are
six observations per single-market bank).} The scope coefficient is
then identified from changes in scope, i.e. off of bank entry. For
illustration, consider a bank $j$ that operates in CBSA $m$ during
period $t$ and in both CBSAs $m$ and $n$ during period $t+1$. Two
comparisons identify the scope coefficient. The first comparison is
that of bank $j$'s market share in CBSA $m$ across periods. An
increase in market share would suggest that scope provides value to
depositors. The second comparison is that of bank $j$'s market share
in CBSA $m$ during period $t$ and bank $j$'s market share in CBSA
$n$ during period $t+1$. A market share that is higher in the
latter instance would again suggest that depositors value scope. The
second comparison is confounding empirically, however, because bank
entrants tend to have small market shares initially (e.g., Berger
and Dick 2007), due to switching costs and/or other factors.
Column 5 presents the results of the bank fixed effects
specification. The bank scope coefficient of 2.31 is more than 45
percent smaller than the baseline coefficient and is no longer
statistically different than zero.\footnote{The data reject the null
that the bank fixed effects are jointly zero ($p$-value$=0.00$).}
However, it is not clear whether the reduction is due to the
correction of endogeneity bias or due to the confounding market
shares of recent entrants. To address the matter, I add lagged
market share to the specification, which controls for the
out-of-equilibrium effects associated with entry. Column 6 presents
the results. The bank scope coefficient of 4.39 is larger, close to
the baseline coefficient, and statistically significant. Together,
the Column 5 and 6 results suggest that bank scope endogeneity may
be unimportant in the baseline specification. More generally, the
results shown in Columns 3 through 6 provide empirical support for
the identification strategy.
I now return to the mixed logit case, in which depositors are
permitted to have heterogenous tastes for observables. Estimation
requires numerical integration over demographic characteristics (as
in Equation \ref{eq:sshar}). I let income have the lognormal
distribution within CBSAs and estimate the income distributions
using individual-level data from the 2000 CPS March Supplement. I
then draw 200 quasi-random incomes for each CBSA from the
appropriate estimated distribution. I proxy the unobserved
demographics with 200 quasi-random draws from a standard normal
distribution. To ease interpretation, I normalize income to have
mean zero and variance one across all CBSAs, following Nakamura
(2006).\footnote{CPS data are available for 259 CBSAs. Estimation
uses 38,993 of 45,785 bank-CBSA-year observations. I use Halton
numbers to capture income and unobserved demographics. Train (1999)
and Bhat (2001) find that the simulation variance caused by 100
Halton quasi-random numbers is smaller than the simulation variance
caused by 1,000 random draws. In estimating the CBSA-specific income
distributions, I consider the household income of individuals over
18 years in age. I trim the bottom 1.5 percent of the sample to
eliminate negative and zero incomes, which are incompatible with the
lognormal assumption.}
With the numerical integration in hand, I am able to obtain the
vector of mean depositor valuations, $\xi$, for any set of proposed
parameters $\theta = (\beta^D, \beta^M, \beta^X, \pi, \sigma)$ via a
contraction mapping algorithm (e.g., Berry 1994; Berry, Levinsohn
and Pakes 1995, Nevo 2001). The parameters are then identified by
the usual assumption, namely that $ \text{E}[Z^\prime \xi(\theta^*)]
= 0,$ where $Z$ includes the instruments and $\theta^*$ includes the
true population parameters. The simulated generalized method of
moments (SGMM) estimator takes the form:
\begin{equation}
\widehat{\theta}_{SGMM} = \arg \min_{\theta \in \Theta} \xi(\theta)
^\prime Z \Phi^{-1} Z^\prime \xi(\theta),
\end{equation}
where $\Phi^{-1}$ is a consistent estimate of $\text{E}[Z^\prime
\xi(\theta^*) \xi(\theta^*)^\prime Z]$. I compute the standard
errors with the usual formulas (e.g., Hansen 1982; Newey and
McFadden 1994), and apply a clustering correction that allows for
the consistent estimation of the standard errors in the presence of
arbitrary correlation patterns between observations from the same
bank.\footnote{I follow the procedure outlined in Nevo (2000) to
perform the contraction mapping. I use a gradient algorithm
supplied by the Mathworks optimization package to select the
nonlinear parameters.}
Table \ref{tab:mlogit} presents the results of the mixed logit
regression. The mean depositor valuations ($\beta$'s), shown in the
first column, are similar to those of the baseline logit results.
The mean valuations for the deposit interest rate and bank scope are
115.92 and 4.70, respectively, and the implied mean
willingness-to-pay for a unit increase in bank scope is 42.92 cents.
Again, this number may be substantial, as it implies that the mean
depositor values the scope of Bank of America (branches in 207 CBSAs
in 2006) at roughly 88.41 dollars more than that of a single-market
bank. The results also suggest that depositors may prefer banks with
greater branch densities and more employees per branch, though the
first coefficient is smaller in magnitude (vis-a-vis the logit
results) and the second is no longer statistically significant.
\begin{table}[t]
\begin{center}
\caption{Mixed Logit Regression Results \label{tab:mlogit}}
\begin{tabular}{l c c c }
\hline \hline
\rule[0mm]{0mm}{4.5mm} & Means & St. Dev & Income \\
\rule[0mm]{0mm}{4.5mm}Variable & ($\beta$'s) & ($\sigma$) & ($\pi$) \\
\hline
\multicolumn{4}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Deposit interest rate}} \\
\rule[0mm]{0mm}{4.5mm} \; $r^D_{jt}$ & 115.923*** & 2.263 & -65.580*** \\
\rule[0mm]{0mm}{4.5mm} & (8.178) & (41.228) & (18.997) \\
\multicolumn{4}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Bank scope}} \\
\rule[0mm]{0mm}{4.5mm}\; $\left(\frac{m_{jt}}{M}-1 \right)$ & 4.701*** \\
\rule[0mm]{0mm}{4.5mm} & (0.627) \\
\multicolumn{4}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Control variables}} \\
\rule[0mm]{0mm}{4.5mm}\; BRDEN$_{jmt}$ & 1.070*** \\
\rule[0mm]{0mm}{4.5mm} & (0.044) \\
\rule[0mm]{0mm}{4.5mm}\; NEMP$_{jt}$ & 0.070 \\
\rule[0mm]{0mm}{4.5mm} & (0.045) \\
\rule[0mm]{0mm}{4.5mm}GMM objective & & 14,268.73 \\
\rule[0mm]{0mm}{4.5mm}\% of $r^D_{jt}$ coefficients$<$0 & & 5.46 \\
\hline
\multicolumn{4}{p{4.2in}}{\footnotesize{\rule[0mm]{0mm}{0mm}Results
from mixed logit estimation. The data include 38,993 bank-CBSA-year
observations. The regression also includes an intercept. The
instruments are loan-side measures and competitor characteristics.
Standard errors are clustered at the bank level and shown in
parenthesis. Significance at the 10\%, 5\%, and 1\% levels is
denoted by *, **, and ***, respectively.}}
\end{tabular}
\end{center}
\end{table}
The next columns show estimates of depositor heterogeneity around
the mean deposit rate valuation. The estimated deposit rate
parameter standard deviation ($\sigma$) is small and not
statistically significant. By contrast, the coefficient on the
interaction with depositor income ($\pi$) is large and statistically
different than zero. The coefficient suggests that a one standard
deviation increase in income lowers the deposit rate valuation by
65.58, so that higher income depositors are less price-sensitive. A
joint statistical test rejects the null of no heterogeneity (i.e.,
$\sigma=\pi=0$) at any conventional level. Figure \ref{fig:wtp1}
graphs the estimated willingness-to-pay
distribution.\footnote{Roughly five percent of depositors have
deposit rate valuations near or below zero. The corresponding
willingness-to-pay for these depositor is quite large (or quite
small) and is not shown.} The skew of the distribution reflects the
result that the lognormally distributed depositor incomes matter
more to deposit rate valuations than the normally distributed
unobserved demographics. The 25th, 50th, and 75th percentiles of the
empirical distribution are 31.30, 35.22, and 44.56 cents,
respectively, so that a depositor at the 75th percentile has a
willingness-to-pay that is 42.34 percent greater than a depositor at
the 25th percentile. Overall, the results are strongly consistent
with the notion that depositor heterogeneity exists and is
substantial.
\begin{figure}
\centering
\includegraphics[height=3in]{wtp1.eps}
\caption{The frequency distribution of the
willingness-to-pay for a unit increase in bank scope
(based on Table \ref{tab:mlogit}). Willingness-to-pay is
measured in cents per year.}
\label{fig:wtp1}
\end{figure}
% The figure is a bar graph that plots the frequency distribution of willingness-to-pay
% for a unit increase in bank scope. More than sixty percent of the distribution for
% willingness-to-pay is around thirty to fifty cents. Slightly less than twenty percent
% of the distribution is around fifty to seventy cents. Higher willingness-to-pay is less
% frequent.
The theoretical model predicts that banks of greater scope should
have less elastic deposit demand. Intuitively, these banks attract
depositors that value scope more relative to the deposit interest
rate. The median elasticity of demand in the sample is 1.84;
consistent with the theoretical model, demand elasticities decrease
substantially with bank scope.\footnote{The demand elasticities in
the mixed logit take the form \[ \frac{\partial s_{jmt}}{\partial
r^D_{jt}} \frac{r^D_{jt}}{s_{jmt}} = \frac{r^D_{jt}}{s_{jmt}}\int
\beta_i^D s_{ijmt}(1-s_{ijmt}) dF(\nu) dF(y),\] where the term
$s_{ijmt}$ represents the depositor-specific probability (based on
the logit formulas) of choosing bank $j$ during period $t$ given the
demographic characteristics $y$ and $\nu$.} The median elasticity
faced by banks with branches in 1, 2-5, 6-20, and more than 20
markets is 2.20, 1.95, 1.47, and 1.01, respectively, so that the
median elasticity faced by a single-market bank is more than double
that of the largest banks. The results underscores the empirical
relevance of the theoretical model to competition for depositors
within metropolitan markets.
\subsection{Further tests of the empirical predictions \label{sec:entry}}
\subsubsection{The decision to enter an outside market}
The theoretical model predicts that a bank should be less likely to
enter an outside market if its original markets already feature
banks of greater scope. Intuitively, the presence of larger banks
creates an environment in which entry would lessen scope
differentiation and intensify deposit rate competition. I examine
the empirical prediction using data on the entry decisions of
single-market banks. Within this specific context, the empirical
prediction can be precisely formulated. Suppose a single-market
bank operates in market $a$. Then the single-market bank should be
less likely to enter any second market if more two-market banks
already operate in market $a$.
To test the empirical prediction, I construct a regression sample of
single-market banks over the period 2001-2005. The 14,469 bank-year
observations in the sample include 3,658 single-market banks that,
combined, operate in 191 CBSAs.\footnote{The remaining CBSAs never
have a single-market bank enter a second market. I exclude the
corresponding 3,744 bank-year observations from the regression
sample. The use of market fixed effects makes the
inclusion/exclusion of these observations irrelevant for estimation
because the fixed effect parameters perfectly predict observed
entry.} Of these single-market banks, 351 enter a second market
during the sample period. Banks that enter a second market drop out
of the sample after entry. The estimation procedure is standard
logit model. Let $j=1,2,\dots,J$ single-market banks, each based in
an original market $a_j$, determine whether to enter a second market
in period $t+1$. I represent the latent utility of entry and the
observed entry choice as
\begin{equation}
v^*_{j,t+1} = \text{NBANKS}_{a_jt}^\prime \lambda_1 + w_{jt}^\prime \lambda_2 + \gamma_{at} + \eta_{jt} \quad \text{and} \quad v_{j,t+1} = 1\{v_{j,t+1}^*>0 \}, \\
\end{equation}
respectively. The vector NBANKS$_{a_jt}$ captures the number of
banks in market $a_j$ during period $t$ that have branches in
exactly one, exactly two, three through ten, and more than ten
CBSAs. I refer to these four variables as NBK1$_{a_jt},$
NBK2$_{a_jt}$, NBK3$_{a_jt}$ and NBK4$_{a_jt}$, respectively. The
primary variable of interest is NBK2$_{a_jt}$; the theoretical model
implies that its coefficient should be negative. The vector $w_{jt}$
includes controls at the bank-year level and the vector
$\gamma_{at}$ includes market and year fixed effects. The parameter
vectors $\lambda_1$ and $\lambda_2$ can be consistently estimated
with standard logit regression under the assumption that the error
term $\eta_{jt}$ has the extreme value type I density.\footnote{Two
econometric points may be of interest. First, the estimation of
market and year fixed effects within the limited dependent variable
framework does not introduce incidental parameters bias because the
ratio of observations to parameters converges to infinity with the
number of banks per market-year. Overall, I estimate 201 parameters
using 14,469 observations. Second, I cluster the standard errors at
the market level to account for arbitrary correlation patterns among
bank-year observations in the same market.}
Before turning to the results, some discussion of the specification
may be fruitful. First, market fixed effects solve the basic
identification problem that markets may differ in their ability to
support banks of greater scope (potentially for unobservable
reasons). Such market heterogeneity, if unaccounted for in the
regression specification, would bias the NBK2$_{a_jt}$ coefficient
upwards and against the empirical prediction. The upward bias
occurs because markets that better support banks of greater scope
are likely to have more banks of greater scope; further,
single-market banks in these markets may find entry into an outside
market more profitable. The inclusion of market fixed effects
controls directly for this confounding influence. Second, the
bank-year control variables include the deposit and branch market
shares of the single-market branch in its original market. One
might expect banks that have more substantial market shares in their
original markets to be more likely to enter a second market.
The first column of Table 7 presents the baseline results. The
NBK2$_{a_jt}$ coefficient is negative and statistically different
than zero, consistent with the theoretical model. The coefficient
is also substantial in magnitude. For example, a hypothetical one
standard deviation increase in NBK2$_{a_jt}$ reduces the probability
that a single-market bank enters a second market by an average of
76.54 percent.\footnote{The mean number of two-market banks in the
regression sample is 8.14. The standard deviation is 6.39. To
measure the average percent change in entry probabilities, I first
calculate the probability of entry for each bank-year observation.
This probability, call it $p_{jt1}$, has the simple logit expression
$p_{jt1} = \exp(x_{jt}^\prime \widehat{\lambda}) /
\left(1+\exp(x_{jt}^\prime \widehat{\lambda}) \right),$ where the vectors $x_{jt}$
and $\widehat{\lambda}$ include the regressors and estimated
coefficients, respectively. I then calculate the probability of
entry for each observation given a hypothetical increase in the
number of two-markets banks present in the original market. This
probability, call it $p_{jt2}$, has an expression analogous to that
of $p_{jt1}$. The average percent change is then simply
$\frac{1}{JT} \sum_{j=1}^{J} \sum_{t=1}^{T} \left(
p_{jt2}-p_{jt1}\right) / p_{jt1}.$} Overall, the regression results
strongly support the prediction that banks should be less likely to
enter an outside market if their original markets already feature
banks of greater scope. The results also underscore the empirical
importance of this prediction. Turning quickly to the control
variables, single-market banks that have larger deposit and branch
market shares in their original markets are more likely to enter a
second market, though only the branch market share coefficient is
statistically significant (results not shown).
Interestingly, the entry choices of single-market banks appear
uncorrelated with the number of banks in the original market that
operate in \textit{more} than two markets. The NBK3$_{a_jt}$ and
NBK4$_{a_jt}$ coefficients of 0.034 and -0.069 are small in
magnitude and not statistically different than zero. This empirical
result dovetails nicely with the theoretical model, in which banks
compete for market share only with their immediate neighbors in
scope-space (e.g., Equations \ref{eq:tcutoff} and \ref{eq:tfoc}).
One might infer from the results that competition in the commercial
banking industry is predominately ``local'' in geographic
scope.\footnote{Recent theoretical and empirical work in industrial
organization has examined the extent to which competition in
differentiated markets is local versus global (e.g., Anderson, de
Palma, and Thisse 1989, Pinske, Slade and Brett 2002, Vogel 2008),
and shown that the distinction is important from a policy standpoint
(e.g., Dineckere and Rothschild 1992).}
\begin{table}
\begin{center}
\caption{The Decision to Enter an Outside Market}
\begin{tabular}[t]{l c c c c}
\hline \hline
\rule[0mm]{0mm}{5.5mm}Variables & (1) & (2) & (3) & (4) \\
\hline
\multicolumn{5}{l}{\rule[0mm]{0mm}{5mm}\textit{The number of banks in the original market, by bank scope}} \\
\rule[0mm]{0mm}{5mm} $\quad$ NBK1$_{a_jt}$ & 0.032 &-0.018*** & 0.029 &-0.021***\\
& (0.041) & (0.006) & (0.039) & (0.007) \\
\rule[0mm]{0mm}{5mm} $\quad$ NBK2$_{a_jt}$ &-0.232*** & 0.001 &-0.236*** & -0.003 \\
& (0.074) & (0.023) & (0.075) & 0.025) \\
\rule[0mm]{0mm}{5mm} $\quad$ NBK3$_{a_jt}$ & 0.034 & 0.027* & 0.031 & 0.021 \\
& (0.069) & (0.015) & (0.068) & (0.015) \\
\rule[0mm]{0mm}{5mm} $\quad$ NBK4$_{a_jt}$ &-0.069 & 0.061*** &-0.072 & 0.027 \\
& (0.080) & (0.022) & (0.080) & (0.022) \\
\\
Market fixed effects & yes & no & yes & no \\
Bank controls & yes & yes & no & no \\
\\
\rule[0mm]{0mm}{4.5mm}Pseudo $R^2$ & 0.131 & 0.069 & 0.121 & 0.048 \\
\hline
\multicolumn{5}{p{4.7in}}{\footnotesize{\rule[0mm]{0mm}{0mm}Results
from logit regressions. The data include 14,469 bank-year
observations from 3,658 single-market banks over the period
2001-2005. The dependent variable equals one if the single-market
bank enters a second market in the subsequent year, and zero
otherwise. The variables NBK1$_{a_jt},$ NBK2$_{a_jt}$, NBK3$_{a_jt}$
and NBK4$_{a_jt}$ are the number of banks in the original market
with branches in exactly one, exactly two, three through ten, and
more than ten CBSAs, respectively. The bank controls include the
single-market bank deposit and branch market shares within the
original market. The regressions also include year fixed effects.
Standard errors are clustered at the market level and shown in
parenthesis. Statistical significance at the 10\%, 5\%, and 1\%
levels is denoted by *, **, and ***, respectively.}}
\end{tabular}
\end{center}
\label{tab:entry1}
\end{table}
At the risk of digression, Columns 2 through 4 show the results when
the market fixed effects and/or the bank control variables are
excluded from the specification. As discussed above, one might
expect the estimated NBK2$_{a_jt}$ coefficient to be less negative
(or even positive) when market fixed effects are excluded, due to
the confounding influence of unobserved market heterogeneity. This
is precisely what happens. Column 2 omits the market fixed effects;
the specification is otherwise identical to that of Column 1. The
resulting NBK2$_{a_jt}$ coefficient is nearly zero. The coefficients
on NBK1$_{a_jt}$, NBK3$_{a_jt}$, and NBK4$_{a_jt}$ also change in
the predicted directions. Finally, Column 3 omits only the bank
control variables and Column 4 omits both the market fixed effects
and the bank control variables. In each case, the exclusion of the
bank controls has little effect on the estimated regression
coefficients. One might conclude that bank heterogeneity has little
effect on the main results.
\subsubsection{The choice of which outside market to enter}
Any bank that enters an outside market must decide which outside
market to enter. The second empirical prediction deals with this
choice between outside markets. In particular, the theoretical model
predicts that a bank with branches in market $a$ should be less
likely to enter market $b$ (conditional on entry somewhere) if
another bank of similar scope exists with branches in both market
$a$ and market $b$. I again turn to the entry decisions of
single-market banks to provide a specific context in which to test
the empirical prediction.
To motivate the empirical strategy, consider the following
simplified setting: a single-market bank (bank $j$) is based in an
original market ($a_j$) and enters one of two outside markets ($b_1$
or $b_2$). Suppose that a two-market bank already exists with
branches in $a_j$ and $b_1$. Then the theoretical model suggests
that bank $j$ is more likely to enter $b_2$ because it offers
greater scope differentiation. One could test the hypothesis by
regressing the observed entry decision on an indicator variable,
call it 2CBSA$_{jb}$, that equals one if a two-market bank exists in
markets $a_j$ and $b$, and 0 otherwise. Provided that the
two-market bank is randomly assigned between $b_1$ and $b_2$, the
regression coefficient provides a consistent test of the empirical
prediction. Of course, this condition is unlikely to hold in
practice. The two-market bank may operate in $a_j$ and $b_1$ because
$b_1$ is more profitable than $b_2$, and/or because greater
synergies exist between $a_j$ and $b_1$ than between $a_j$ and
$b_2$. Both possibilities threaten to bias the regression
coefficient upwards, i.e., against the empirical prediction, and I
control directly for these potentially confounding factors.
The actual empirical model generalizes the simplified setting to
allow for many banks and many outside markets. Let $j=1,2,\dots,J$
single-market banks, based in the original markets $a_j$, determine
which of $b=1,2,\dots,B$ outside markets to enter. Each
single-market banks enters the outside market that provides the
greatest value:
\begin{eqnarray}
v_{jb}^* &=& \lambda_1 \text{2CBSA}_{jb} + \lambda_2 \text{SYNERGY}_{jb} + \text{PROFITS}_{b}^\prime \lambda_3 + \eta_{jb}, \nonumber \\
v_{jb} &=& 1\{v_{jb}^* > v_{jc} \; \forall \; c \neq b \},
\label{eq:condlogit}
\end{eqnarray}
where the scalar 2CBSA$_{jb}$ equals one if a two-market bank
already exists with branches in markets $a_j$ and $b$, and zero
otherwise, the scalar SYNERGY$_{jb}$ represents the synergies
between the markets $b_j$ and $b$, and the vector PROFITS$_{b}$
captures the profitability of market $b$. The parameters
$\lambda_1$, $\lambda_2$, and $\lambda_3$ can be consistently
estimated with conditional logit regression (e.g., Greene 2003,
Section 19.7) under the assumption that the bank-specific error
$\eta_{jn}$ has the extreme value type I density.\footnote{One
advantage of the conditional logit framework is that it implicitly
controls for unobservable bank characteristics. To see this, note
that adding a bank fixed effect to Equation \ref{eq:condlogit} does
not affect the \textit{relative} value of the outside markets.} The
theoretical model implies that $\lambda_1<0$.
I take the empirical model to the 351 instances in which a
single-market bank entered an additional CBSA over the period
2001-2005. The regression observations are combinations of
single-market banks and outside markets: the 351 single-market banks
and 359 outside CBSAs form 126,009 regression observations. The
dependent variable equals one if the single-market bank entered the
outside market, and zero otherwise. The independent variable of
interest, 2CBSA$_{jb}$ is directly observable. I proxy the synergies
between the home and outside markets using the proportion of all
banks in the original market that also have branches in the
respective outside market. To proxy the profitability of the outside
markets, I include the number of outside market banks with branches
in exactly one, exactly two, three through ten, and more than ten
CBSAs. I also include second-order polynomials in median income,
population, and land area. I lag the synergy and profitability
controls to mitigate any potential endogeneity concerns.
The first column of Table 8 presents the baseline results. The
2CBSA$_{jb}$ coefficient is negative and statistically different
than zero, consistent with the empirical prediction of the
theoretical model. The coefficient is sizable in magnitude and
suggests that a single-market bank is on average 35 percent less
likely to enter market $b$ if a two-market bank already exists in
$a_j$ and $b$.\footnote{To be clear, I calculate the probability
that bank $j$ enters each outside market $b$, alternately toggling
2CBSA$_{jb}$ to be 0 and 1, and holding the remaining regressors
constant. The percent change in probability that bank $j$ enters
the outside market $b$ due to the presence of a two-market bank in
markets $a_j$ and $b$ is \[ \frac{Pr(b| \; j, \text{2CBSA}_{jb}=1) -
Pr(b|\; j, \text{2CBSA}_{jb}=0)}{Pr(b| \; j, \text{2CBSA}_{jb}=0)}.
\] The probabilities have the familiar logit closed form solutions.
I report the average percent change among the regression
observations.} This may actually understate the true effect: to the
extent that the control variables imperfectly measure synergies and
profits, the true 2CBSA$_{jb}$ parameter is likely more negative
than the estimated coefficient. Overall, the regression result
strongly supports the empirical prediction of the theoretical model.
\begin{table}
\begin{center}
\caption{The Choice of which Outside Market to Enter}
\begin{tabular}[t]{l c c c c}
\hline \hline
\rule[0mm]{0mm}{5.5mm}Variables & (1) & (2) & (3) & (4) \\
\hline
\rule[0mm]{0mm}{5mm}2CBSA$_{jb}$ &-0.442** &-0.241 & 4.460*** & 4.625*** \\
\; & (0.226) & (0.218) & (0.139) & (0.132) \\
\\
Synergy Control & yes & yes & no & no \\
Profit Controls & yes & no & yes & no \\
\\
\rule[0mm]{0mm}{4.5mm}Pseudo $R^2$ & 0.463 & 0.450 & 0.151 & 0.126 \\
\hline
\multicolumn{5}{p{4.2in}}{\footnotesize{\rule[0mm]{0mm}{0mm}Results
from conditional logit regressions. The observations are
combinations of the 351 single-market banks that enter an outside
market over the sample period and the 359 outside CBSAs. The
dependent variable equals one if the single-market bank entered the
outside market, and zero otherwise. The variable 2CBSA$_{jb}$ equals
one if a two-market bank already exists with branches in the home
and outside market, and zero otherwise. The synergy control is the
proportion of banks in the original market that also have branches
in the outside market. The profit controls include the number of
outside market banks with branches in exactly one, exactly two,
three through ten, and more than ten CBSAs, as well as second-order
polynomials in the outside market median income, population, and
land area. Statistical significance at the 10\%, 5\%, and 1\% levels
is denoted by *, **, and ***, respectively.}}
\end{tabular}
\end{center}
\label{tab:entry2}
\end{table}
Again at the risk of digression, Columns 2 through 4 show the
results when the synergy control and/or the profit controls are
excluded from the specification. As discussed above, one might
expect the estimated 2CBSA$_{jb}$ coefficient to be less negative
(or even positive) due to omitted variables bias in each of these
regressions. That is precisely what happens. Column 2 omits the
controls that proxy the outside market profitability. The resulting
2CBSA$_{jb}$ coefficient remains negative but is smaller in
magnitude (-0.241) and not statistically different than zero. Column
3 omits the synergy controls, and Column 4 omits both sets of
controls. The resulting 2CBSA$_{jb}$ coefficient is positive and
large in both cases. These alternative regressions may demonstrate
the importance of controlling for market synergies and other factors
when testing or estimating scope effects.
\subsubsection{Market nearness and the number of banks}
Finally, the theoretical model generates the empirical prediction
that the number of banks within a given market should increase with
its ``nearness'' to other markets. To examine the prediction
empirically, I proxy nearness inversely with the mean distance in
miles between a CBSA and all other CBSAs.\footnote{Formally, the
proxy is $\frac{1}{M}\sum_{n \neq m} \text{DIST}_{mn}$, where
$\text{DIST}_{mn}$ is the miles between CBSA $m$ and CBSA $n$.} The
average CBSA has a mean distance of 1060 miles, and the sample
standard deviation is 349 miles.
The empirical prediction holds in the data. Figure \ref{fig:miles}
plots the univariate relationship between CBSA nearness and the
number of banks for all 2,160 CBSA-year observations over the period
2001-2006. The three CBSAs with the greatest mean distance
(Honolulu, Anchorage, and Fairbanks) also have very few banks (e.g.,
8, 5, and 5 banks, respectively, in 2006). The CBSA with the most
banks (Chicago-Nashville-Jolie, with 231 banks in 2006) has one of
the shortest mean distances. Among all CBSAs, the correlation
coefficient between the number of banks and mean distance is -0.11.
Further, an OLS regression of the number of commercial banks per
CBSA-year observation on mean distance and second-order control
polynomials in CBSA median household income, population, and land
area yields a mean distance coefficient of -0.014 that is
statistically different than zero at standard levels (standard error
$=0.003$).\footnote{I cluster the standard errors at the CBSA level
to account for heteroscedasticity and arbitrary correlation patterns
between observations from the same CBSA.} The regression
coefficient is also substantial in magnitude: a one standard
deviation increase in mean distance corresponds to a 24 percent
reduction in the number of banks when evaluated at the mean.
\begin{figure}
\centering
\includegraphics[height=3in]{miles.eps}
\caption{Market nearness and the number of banks.
Each point represents a single CBSA-year observation. The vertical
axis is the average distance in miles between the CBSA and all
other CBSAs. The horizontal axis is the number of banks in the
CBSA-year. }
\label{fig:miles}
\end{figure}
%Figure 2 is a scatterplot of raw data. Each point represents a single CBSA-year observation.
%The horizontal axis is the number of banks in the CBSA-year and the vertical axis is the mean
%distance between the CBSA and all other CBSAs. A downward trend is apparent in the data, i.e.
%CBSA-years that have more banks tend to be closer to other CBSAs. These data support regression
%analysis. See text for details.
More detailed investigation calls into question the extent to which
the relationship between CBSA nearness and the number of banks
supports the theoretical model, however. The empirical prediction
is generated by the theoretical model because markets near many
others may offer greater opportunities for scope differentiation.
One would naturally expect the relationship between nearness and the
number of banks to be driven by differences in the number of
multimarket banks rather than the number of single-market banks
because, by definition, single-market banks do not exploit
opportunities for scope differentiation. The opposite holds in the
data -- the relationship between CBSA nearness and the number of
banks appears to be driven primarily by differences in the number of
single-market banks.
Figure \ref{fig:miles3} shows a scatterplot of CBSA nearness and the
number of single-market banks for the 2,160 CBSA-year observations
over the period 2001-2006. It is clear that, on average, CBSAs that
are near others support more single-market banks. By way of
contrast, Figure \ref{fig:miles2} shows the scatterplot of CBSA
nearness and the number of multimarket banks. No relationship is
apparent. Further, an OLS regression of the number of multimarket
banks on mean distance and second-order control polynomials in
median household income, population, and land area yields a mean
distance coefficient of -0.00003 that is small in magnitude and not
statistically different than zero (standard error $=0.0007$). Thus,
while the empirical prediction of the theoretical model holds in the
data -- CBSAs near many others do support more banks -- it is not
apparent that the prediction holds because CBSAs in close proximity
to many others offer greater opportunities for scope
differentiation. It is therefore difficult to conclude that the
relationship between CBSA nearness and the number of banks provides
substantive support for the theoretical model.
\begin{figure}
\centering
\includegraphics[height=3in]{miles3.eps}
\caption{Market nearness and the number of
banks with branches in only one CBSA.
Each point represents a single CBSA-year observation. The vertical
axis is the average distance in miles between the CBSA and all
other CBSAs. The horizontal axis is the number of banks in the
CBSA-year that have branches only one CBSA. }
\label{fig:miles3}
\end{figure}
%Figure 3 is a scatterplot of raw data. Each point represents a single CBSA-year observation.
%The horizontal axis is the number of single-market banks in the CBSA-year and the vertical axis is the mean
%distance between the CBSA and all other CBSAs. Again, a downward trend is apparent in the data, i.e.
%CBSA-years that have more single-market banks tend to be closer to other CBSAs.
\begin{figure}
\centering
\includegraphics[height=3in]{miles2.eps}
\caption{Market nearness and the number of
banks with branches in more than one CBSA.
Each point represents a single CBSA-year observation. The vertical
axis is the average distance in miles between the CBSA and all
other CBSAs. The horizontal axis is the number of banks in the
CBSA-year that have branches in more than one CBSA. }
\label{fig:miles2}
\end{figure}
%Figure 3 is a scatterplot of raw data. Each point represents a single CBSA-year observation.
%The horizontal axis is the number of multimarket banks in the CBSA-year and the vertical axis is the mean
%distance between the CBSA and all other CBSAs. Again, a downward trend is apparent in the data, i.e.
%CBSA-years that have more multimarket banks tend to be closer to other CBSAs. These data support regression
%analysis. See text for details.
\section{Conclusion \label{sec:conc}}
I model multimarket competition when consumers value firm scope
across markets. I show that these consumer preferences have
implications for firm conduct and market structure, and provide
evidence that these implications are empirically relevant. Although
the paper fills a gap in the academic literature, much remains to be
done. First, the model makes strong assumptions to isolate the
influence of consumer preferences; these assumptions could be
relaxed in future work. For example, one could examine preferences
for scope in the presence of scale economies. Second, although the
empirical implementation builds the case that consumer preferences
for scope have real effects on competition, it does less to evaluate
the magnitude of these effects. The estimation of structural
supply-side models may help fill this gap. The work of Ishii (2004)
on ATM network supply may represent a first step towards a suitable
modeling framework. Finally, the theoretical model has implications
for merger review and other policy concerns; future research may
explore these implications.
\newpage
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\newpage
\appendix
\singlespace
\section{Proofs \label{sec:proofs}}
\noindent \textbf{Proof of Lemma 1.} The first order conditions for
bank $j$ and bank $k$ simplify to $r^D_j = (1/2)*(r^L + r^D_k)$ and
$r^D_k = (1/2)*(r^L + r^D_j)$, respectively, when $m_j=m_k$ for any
$j$. Solving the conditions yields $r^D_j = r^D_k = r^L$. It follows
that $\pi_j = \pi_k = 0$. Finally, by Equation \ref{eq:tutil},
depositors strictly prefer banks $j$ and $k$ to any bank $l$
characterized by $m_lr^D_{j+1}$ for any $j(1/2)*(\frac{m_{j+1}-m_{j}}{m_{j+1}-m_{j-1}} r^D_{j-1}+
\frac{m_{j}-m_{j-1}}{m_{j+1}-m_{j-1}} r^D_{j+1})$ for $1r^D_2>\dots>R^D_J$ or
by deposit rates that increase strictly in scope, i.e.
$r^D_1r^D_2>\dots>R^D_J$ in any equilibrium.
I now turn to the claim that market shares and profits increase in
scope, i.e. $s_js_{j+1}$. Then, by the definition of market share,
$\alpha_{j+1}-\alpha_{j}>\alpha_{j}-\alpha_{j-1},$ and noting that
$m_k-m_{k-1}=1$ for any $k$,
\[ 2r^D_{j}-r^D_{j-1}-r^D_{j+1} > 2r^D_{j+1}-r^D_{j}-r^D_{j+2}, \]
from Equation \ref{eq:tcutoff}. Rearranging terms yields
\[ r^D_j - r^D_{j+1} > \frac{1}{2}(r^D_{j-1}+r^D_{j+1}) -
\frac{1}{2}(r^D_{j}+r^D_{j+2}), \] and, substituting the first order
conditions specified in Equation \ref{eq:tfoc} into the left hand
side it must be that
\[ \frac{1}{2}(r^L + \frac{1}{2}(r^D_{j-1}+r^D_{j+1})) - \frac{1}{2}(r^L +
\frac{1}{2}(r^D_{j}+r^D_{j+2})) > \frac{1}{2}(r^D_{j-1}+r^D_{j+1}) -
\frac{1}{2}(r^D_{j}+r^D_{j+2}). \] Rearranging and canceling yields
the simple condition $r_j+r_{j+2} > r_{j-1}+r_{j+1},$ which
contradicts the finding presented above that $r_{j-1}>r_j$ and
$r_{j+1}>r_{j+2}$. The claim that profits increase in scope is
trivially true from Equation \ref{eq:tprofs}, given that deposit
rates decrease in scope and market shares increase in scope. $\Box$
\bigbreak \noindent \textbf{Proof of Corollary 2.} The first-order
condition for profit maximization, based on the profit function
specified in Equation \ref{eq:tprofs}, can be recast as a Lerner
index: \[ \frac{r^L - r^D_j}{r^D_j} = \frac{s_j(r^D_.,m_.)}{r^D_j}
\frac{1}{\frac{\partial s_j(r^D_.,m_.)}{\partial r^D_j}}, \] where
the term on the left is the deposit ``markup'' and the term on the
right is the inverse deposit demand elasticity. By Proposition 1,
deposit rates decrease strictly in scope. It follows that markup
increases in scope and that demand elasticities decrease in scope.
$\Box$
\bigbreak \noindent \textbf{Proof of Lemma 3.} Since in equilibrium
$r^D_1>r^D_2>\dots>R^D_J$ (Lemma 3) and
$r^D_j>(1/2)*(\frac{m_{j+1}-m_{j}}{m_{j+1}-m_{j-1}} r^D_{j-1}+
\frac{m_{j}-m_{j-1}}{m_{j+1}-m_{j-1}} r^D_{j+1})$ for $1\alpha_j$ by Equation \ref{eq:tcutoff},
and the market share of bank $j$ is positive. Turning to banks 1 and
$J$, the equilibrium characteristics $r^D_1>r^D_2$ and
$r^D_{J-1}>r^D_J$ imply that $\alpha_2>0$ and $\alpha_J<1$, so banks
1 and $J$ have positive market shares. The condition $r^D_1>r^D_2$
necessarily implies $r^L>r^D_1$, by Equation \ref{eq:tfoc}.
Therefore, $r^L>r^D_j$ and $s_j>0$ for all $j$, and there exists at
least one stage game Nash equilibrium in which all banks have
positive profits. Finally, rearranging the first order conditions
such that $Ax=b$, where $A$ is a matrix of coefficients, $x$ is a
vector of deposit rates, and $b$ is a vector of solutions, it is
apparent by inspection that $A$ is nonsingular for any $J$. Applying
Cramer's Rule, the first order conditions generate a unique stage
game Nash equilibrium. $\Box$
\bigbreak \noindent \textbf{Proof of Lemma 4.} If $J=M+1$ then the
two stage subgame has a unique subgame perfect equilibrium in which
each bank differs in scope. Suppose that each bank enters a
different number of outside markets. By Lemma 3, the third stage
features a unique Nash equilibrium in which each bank earns positive
profits. If a bank deviates in the second stage then it has a scope
that is equal to that of exactly one other bank and, by Lemma 1,
earns zero profits in the third stage. The strategy profile in which
each bank differs in scope is therefore subgame perfect. Next,
suppose that some banks choose to be of equal scope. These banks
earn zero profits by Lemma 1, but have a profitable deviation
available in the second stage. The strategy profile in which each
bank differs in scope is therefore the unique subgame perfect
equilibrium.
If instead $JM+1$,
there must exist at least one bank $j$ such that $m_j=m_k$ for some
bank $k$. By Lemma 1, banks $j$ and $k$, as well as any banks of
lesser scope, earn zero profits in the third stage Nash equilibrium.
If the scope space is covered over $m_{j+1}, \; m_{j+2},\; \dots,M$
then bank $j$ has no profitable deviation in the second stage. The
class of equilibria is unique: if the scope space is not covered
above $m_j$ then bank $j$ has a profitable deviation in which it
selects the unoccupied scope in the second stage. $\Box$
\bigbreak \noindent \textbf{Proof of Proposition 2.} Suppose that
$M+1$ banks enter the inside market. Then, by Proposition 1 and
Lemma 2, these banks subsequently enter different numbers of outside
markets and earn positive profits in the third stage. The outcome
is subgame perfect because no bank has a profitable deviation in any
subgame. Next, suppose that $J>M+1$ banks enter the inside market.
By Lemma 4, at least one bank must earn zero profits in third stage;
and, given the entry cost $\epsilon$, this bank prefers not to enter
the inside market. Lastly, if $J