Duopoly Conduct and the Panzar-Rosse Revenue Test
Sherrill Shaffer
Guthrie Professor of Banking and Financial Services. Primary affiliation: University of Wyoming,
Department of Economics and Finance (Dept. 3985), 1000 East University Ave., Laramie, WY
82071, USA. Phone: 1-307-766-2173. Email: [email protected]. Secondary affiliation: Centre
for Applied Macroeconomic Analysis (CAMA), Australian National University, Australia.
Laura Spierdijk (corresponding author)
Professor of Econometrics. University of Groningen, Faculty of Economics and Business, Department of Economics, Econometrics and Finance, P.O. Box 800, NL-9700 AV Groningen, The
Netherlands. Phone: +31 50 363 5929. Email: [email protected].
Laura Spierdijk gratefully acknowledges financial support by a Vidi grant (452.11.007) in the ‘Vernieuwingsimpuls’ program of the Netherlands Organization for Scientific Research (NWO).
Duopoly Conduct and the Panzar-Rosse Revenue Test
Sherrill Shaffer| and Laura Spierdijk}
February 1, 2013
Abstract
The theoretical part of this study provides two examples of duopoly in which the PanzarRosse revenue test (also known as the H statistic) cannot reliably identify the degree of market
power. The empirical part of this paper analyzes the banking duopoly in Dewey County (South
Dakota, USA). In this highly concentrated market with two banks of inefficient size we expect
to find market power, which is confirmed by the Lerner index and the conjectural variation.
The H statistic indicates a highly competitive market, consistent with our theoretical result
that the revenue test may indicate competitive conduct in noncompetitive markets.
Keywords: reduced-form revenue test; Panzar-Rosse; competition; duopoly
JEL codes: D4, L1
| Guthrie Professor of Banking and Financial Services. Primary affiliation: University of Wyoming, Department of
Economics and Finance (Dept. 3985), 1000 East University Ave., Laramie, WY 82071, USA. Phone: 1-307-766-2173.
Email: [email protected]. Secondary affiliation: Centre for Applied Macroeconomic Analysis (CAMA), Australian
National University, Australia.
} Corresponding author. Professor of Econometrics. Address: University of Groningen, Faculty of Economics &
Business, Department of Economics, Econometrics, & Business, P.O. Box 800, NL-9700 AV Groningen, The Netherlands. Email: [email protected]. Phone: +31 50 363 5929. Fax: +31 50 363 7337.
1 Introduction
The response of firms’ revenue to changes in marginal cost has long been a topic of intrinsic interest, as well as affording an empirical test for competitive conduct (Rosse and Panzar, 1977; Panzar
and Rosse, 1987; Vesala, 1995). The linear homogeneity of marginal cost in factor prices suggests
a parsimonious implementation of this model with modest data requirements, an approach that
has recently seen growing popularity. Bikker et al. (2012) count 31 published applications of the
Panzar-Rosse model to the banking industry up to 2009, including 15 between 2005 and 2009.
The same method has also been applied to non-banking samples. For some recent banking applications of the Panzar-Rosse model, see Claessens and Laeven (2004), Coccorese (2009), Schaeck
et al. (2009), Podpiera and Rakova (2009), Kasman (2010), Li (2010), Masood and Sergi, (2011),
Maudos and Solis (2011), Olivero et al. (2011), Weill (2011), Bikker et al. (2012), and Coccorese
(2012). For non-banking applications, see Rosse and Panzar (1977), Sullivan (1985), Ashenfelter
and Sullivan (1987), Wong (1996), Fischer and Kamerschen (2003), Tsutsui and Kamasaki (2005),
Podpiera and Rakova (2009), Li (2010), and Coccorese (2012), among others.
The test statistic in the standard reduced-form revenue model, often denoted H , is the sum of
elasticities of total revenue with respect to each input price. A sample of firm-level observations
in long-run competitive equilibrium would exhibit H D 1, while a sample of observations from a
profit-maximizing monopoly yields H < 0 (Panzar and Rosse, 1987). For intermediate cases, the
literature varies widely and remains incomplete.
Theoretical studies report H as alternately an increasing (Shaffer, 1983) or decreasing (Panzar
and Rosse, 1987) function of the Lerner index for firms facing constant elasticity of demand. Several empirical studies interpret smaller values of H as indicating greater monopoly power, such
as De Bandt and Davis (2000), Bikker and Haaf (2002), Claessens and Laeven (2004), Coccorese
(2009), Schaeck et al. (2009), Masood and Sergi (2011), and Olivero et al. (2011). Many empirical studies interpret positive estimates of H as reflecting Chamberlinian monopolist competition,
though Panzar and Rosse (1987, p. 454) state that positive H could also result from conjectural
variation oligopoly if firms face inelastic market demand. Some studies report empirical estimates
of H > 1, which is not consistent with any previously analyzed theoretical equilibrium (De Bandt
and Davis, 2000; Bikker and Haaf, 2002; Coccorese, 2009).
1
The theoretical part of this study explores the equilibrium response of revenue to changes
in marginal cost in two previously overlooked market settings. Our findings broaden the known
causes of positive H to include two new, highly noncompetitive scenarios, thereby demonstrating
that positive H cannot rule out a high degree of market power, contrary to a common perception in
prior studies (e.g., Bikker et al., 2012). We also provide the first theoretical justification of H > 1,
thus establishing that such estimates can be a valid outcome and not just an artifact of econometric
problems such as model misspecification and estimation uncertainty.
In the empirical part of our study we illustrate the implications of our theoretical results for
the empirical assessment of banking competition. We select a sample from Dewey County (South
Dakota), which appears to be an actual duopoly banking market in the US. Exactly two banks
have operated in this county for each year since at least 1976. We therefore analyze this market
during the 1976 – 2010 period. Located in the sparsely developed north-central part of the state,
Dewey County is at least three counties away from the nearest metropolitan areas. According to the
conventional interpretation of the estimated H statistic we find long-run competitive equilibrium
(i.e., H D 1) for one of the two banks and monopolistic competition or short-run competition
(i.e., 0 < H < 1) for the other one. This is an unlikely result given the inefficient scale of the two
banks and the highly concentrated market they operate in. To verify the outcomes of the PanzarRosse revenue test, we consider two alternative measures of competition. The first alternative is
the Lerner index as proposed by Lerner (1934), and recently applied in several banking studies,
such as Angelini and Cetorelli (2003), Fernández et al. (2005), Carbó et al. (2009), and Koetter et
al. (2012). The second alternative is the conjectural variation correlation test proposed by Shaffer
(2000). To our best knowledge, we are the first to apply the latter test to a banking sample. The two
competition measures reveal significant market power for the banks in Dewey County, consistent
with a priori expectations. Our empirical findings are consistent with our theoretical result that a
positive value of H may arise under noncompetitive conditions.
While other recent work has noted additional weaknesses in the Panzar-Rosse model as widely
implemented (Bikker et al., 2012), that analysis nevertheless suggests that a properly specified
reduced-form revenue equation can yield a one-tail test of conduct based on the sign of H . By
contrast, both our theoretical and empirical findings establish that, contrary to widespread belief,
neither the sign nor the magnitude of H alone can reliably identify the degree of market power.
2
The remainder of this paper is organized as follows. Section 2 illustrates that a positive value
of H can arise in a Stackelberg duopoly. We additionally show that H > 1 may arise in a homogeneous Cournot duopoly with asymmetric costs. Section 3 describes the banking duopoly of
Dewey County. The Panzar-Rosse revenue equation is estimated for the banks of Dewey County in
Section 4. The latter estimates are verified in Section 5 using the Lerner index and the conjectural
variation correlation test. Finally, Section 6 concludes.
2 Theoretical results
Whereas the empirical P-R literature has focused almost exclusively on a simple notion that negative values of the H statistic denote monopoly power while positive values denote more competitive conditions, a growing body of theoretical analysis has pointed out important exceptions to
this pattern. For example, Shaffer (1983) and Bikker et al. (2012) showed that H < 0 can occur
even in highly competitive conditions – the former in the case of blockaded entry (fixed numbers
of firms), and the latter in the case of constant average cost. In this section, we extend the known
deviations of H from its commonly assumed properties by analyzing the implications of two standard market structures for the equilibrium response of firm revenue to marginal cost. In both cases,
we show for the first time that H > 0 can arise in highly noncompetitive settings. These findings
complement prior theoretical analysis, disprove the notion that H can provide a reliable one-tail
test of competition, and further expand the recognition that H is related to market structure in
complex ways that cannot be accurately represented as a simple sign test.
We derive our results in the context of duopoly for clarity and brevity, as well as to emphasize
that positive H can result even with a high degree of oligopoly power, contrary to common perceptions. Moreover, duopoly is often an empirically relevant case, as documented by Bresnahan and
Reiss (1991). Similar calculations establish corresponding generalizations to arbitrary numbers of
firms. Both proofs are in Appendix A.
2.1 Stackelberg and Cournot duopoly
We begin by analyzing a standard Stackelberg duopoly, and show that it provides the first direct
example supporting a claim of Panzar and Rosse (1987) and Vesala (1995, p. 58) that H > 0 is
3
possible in a conventional static oligopoly. We show:
Proposition 2.1 H can take either sign for a Stackelberg duopoly facing linear cost and demand.
Proposition 2.1 broadens the known cases of H > 0 under imperfect competition. In this
setting, both firms have the same sign of H , and the leader’s value of H is twice as far from zero
as that of the follower. As shown in Appendix A, the follower’s Lerner index (the relative markup
of price over marginal cost) is positive, confirming the existence of market power.
Next we turn to the possibility of H exceeding 1 and find the following:
Proposition 2.2 H > 1 is possible for the low-cost firm in a homogeneous Cournot duopoly with
asymmetric costs and linear demand.
Of course, the possibility of H > 1 in this case also encompasses the possibility of H > 0,
as shown in Appendix A. Thus, like Stackelberg duopoly, an asymmetric Cournot duopoly can
generate positive values of H . The appendix confirms that the Lerner index is positive for the
low-cost firm in this case, denoting market power.
2.2 Implications
Propositions 2.1 and 2.2 have the following implications. An empirical finding of H D 1 is
commonly associated with long-run competitive equilibrium, whereas 0 < H < 1 is generally
interpreted as monopolistic competition or short-run competition; see Bikker et al. (2012). Propositions 2.1 and 2.2 make clear that an empirical finding of H > 0 may also occur with a high
degree of oligopoly power. Although these theoretical results literally apply only to Stackelberg
and asymmetric Cournot cases, the broader point is that there are conditions under which positive
values of H may be observed despite substantial market power. The propositions identify two sets
of sufficient conditions, not necessary conditions, for this outcome. Establishing empirical relevance of the broader conclusion does not require or imply a showing of either Stackelberg conduct
or asymmetric Cournot conduct in particular, as there may also be other imperfectly competitive
cases that generate the same outcome.
Consequently, the H statistic alone is not enough to assess the degree of competition in a market. Additional information is required to distinguish between competitive and non-competitive
4
outcomes. Together with previously published theoretical analyses of H , these results dispel the
notion that the sign or magnitude of H alone can support reliable conclusions about market power.
In the remainder of this paper we will illustrate the implications of our results by means of an
empirical example. In particular, we will estimate H in a highly concentrated banking market that
may satisfy theoretical conditions to generate H > 0 despite significant actual market power. We
then estimate market power in this market by an alternate means, finding empirical evidence of
market power that contrasts with the common interpretation of the estimated H statistics.
3 The sample
It is possible to explore the empirical ramifications of our theory in any highly concentrated banking market, including countries such as Canada and Australia. However, to enhance the correspondence between our theoretical and empirical analyses, we select instead a sample from what
appears to be an actual duopoly banking market in the US. While cross-country studies have commonly assumed nationwide banking markets, it is widely recognized that the situation is different
in the US due to its large land area, with a much larger number of banks, and historical restrictions
on individual banks’ geographic span of operation. The long-standing practice in both academic
research and regulatory enforcement has been to regard non-metropolitan banking markets in the
US as individual counties, consistently supported by a variety of empirical evidence (Whitehead,
1980; Jackson, 1992; Kwast et al., 1997; Huang, 2008; Ho and Ishii, 2011). We follow that practice
here.
3.1 The Dewey County duopoly
Over the past 30 years, between 20 and 25 percent of US counties have had only one or two
commercial banks operating in them (Calem et al., 2000; and the FDIC’s Summary of Deposits
database), demonstrating strong empirical relevance of such high levels of concentration. For our
application, we must find such a county that has not experienced entry or exit during a sufficiently
long time period to provide an adequate sample size.1 In addition, we need to rule out counties
1 An alternate research strategy might be to examine a market that has experienced entry or exit, and test for evidence
of competitive shifts in response to the structural change. However, small numbers of observations in each regime would
make this approach statistically less feasible.
5
adjacent to metropolitan areas, where competition from banks in the core city might spill over the
county boundary, as well as suggesting a credible threat of entry.
Because detailed financial data are available only at the bank level, rather than for individual
bank offices, we further need to identify a county in which the only banks present are headquartered in that county and operate only within that county. This restriction eliminates the great majority of duopoly banking markets from consideration, due solely to data limitations and not due
to any theoretical unsuitability. Dewey County, South Dakota satisfies all these requirements.
Located in the sparsely developed north-central part of the state, Dewey County is at least three
counties away from the nearest metropolitan areas (Bismarck, North Dakota; and Rapid City, near
the western edge of South Dakota). Exactly two banks have operated in the county for each year
since at least 1976.2 We analyze the Dewey County duopoly during the 1976 – 2010 period.
The two banks in Dewey County are the Western Dakota Bank (FDIC certificate number 492)
and the State Bank of Eagle Butte (FDIC certificate number 18145).3 The Western Dakota Bank’s
main office is in the county seat Timber Lake (2010 population: 443). It has branches in Eagle
Butte (2010 population: 1,318) since 1997 and in Isabel (2010 population: 135) since 1960. The
State Bank of Eagle Butte has only one office, in Eagle Butte. Neither of the two banks was a
subsidiary of a holding company during the sample period. In 2007 a small credit union opened
in Dewey County, with a market share of only 1%. Because credit unions are legally restricted in
their ability to issue commercial loans, the presence of the small credit union does not distort our
findings and interpretations. There was no further entry or exit during the sample period.
Potential entry was very limited. Few major roads connect Dewey County to its neighbors, impeding potential competition from outside the county. Already in the early eighties, South Dakota
viewed limited deregulation of interstate branching as incentive for economic development. Interstate banking into South Dakota was fully deregulated in 1988, several years before the RiegleNeal Interstate Branching Act of 1994. However, Amel (1988) has shown that entry is much less
likely in rural than in urban banking markets. Also the Financial Modernization Act of 1999 had
2 The
FDIC’s Summary of Deposits database reports all active bank branches in each county, regardless of the
location of the bank’s headquarters.
3 The Western Dakota Bank was originated in Glencross, South Dakota on July 1, 1919 and opened under the name
of Glencross State Bank. The Bank of Timber Lake merged with Glencross State Bank in 1935. The bank name was
changed to Dewey County Bank. The bank name was changed to Western Dakota Bank in 1997 (but retained the same
bank certificate number after the name change). Source: http://www.westerndakotabank.com/about-us.html.
6
less effect on small rural US banking markets than on large banks, because community banks
operating in rural markets generally have less capability and competitive incentive to offer an expanded array of non-banking financial products than do larger banks in more densely populated
areas. Some competitive pressure may arise from consumers and firms who shop around for low
loan rates in other counties or via internet.
The population in Dewey County grew about 2% between 1976 – 2010 (from 5,227 to 5,335),
which is much lower than the growth of South Dakota as a whole (1976 – 2011: 19%). Nominal
per capita income in Dewey County rose from $ 4,155 in 1976 to $ 34,986 in 2010 and doubled
about every ten years.4 These modest growth figures ensure that our empirical results will not be
disturbed by extreme economic conditions. Furthermore, Dewey County’s small population allows
us to assume exogeneity of the regulatory environment.
The Western Dakota Bank was founded in 1919; the State Bank of Eagle Butte in 1960. Both
banks had many years to attain a state of long-run equilibrium by the start of our sample period.
We already argued that the deregulation of interstate branching and The Financial Modernization
Act of 1999 likely had little impact on rural banking markets. Furthermore, the 2001 recession
was brief and mild. Our sample also contains the years of the financial crisis, 2008 – 2010. We
will verify whether and to what extent these two observations distort the results.
3.2 Banking competition in Dewey County
Based on the typical characteristics of the banking duopoly in Dewey County, we expect the degree
of competition in this market to be limited. To see this, we observe that both banks are smaller
than the minimum efficient size as estimated in the empirical cost literature. While the literature
has historically provided mixed evidence regarding returns to scale among large banks, smaller
banks (particularly those with total nominal assets less than $ 100 million, such as the two banks
operating in Dewey County) have always been found to operate in a range of increasing returns.
Recent studies have also found evidence that increasing returns to scale extend to larger banks
as well (Hughes and Mester, 2011; Wheelock and Wilson, 2012). Both banks in Dewey County
had total nominal assets no larger than around $ 35 million throughout the sample period. In
the presence of economies of scale, marginal costs are lower than average costs. Consequently,
4 The
income and population figures in this section come from the United States Census Bureau.
7
sustainable equilibrium prices P satisfy P AC > M C (where AC denotes average costs
and MC marginal costs), implying a positive Lerner index and, consequently, a certain degree of
market power, even if the banks do not pursue an objective of profit maximization.5 Furthermore,
the small size of the banks operating in Dewey County also affects new entry into this market.
Because both banks operate in a region of strong economies of scale, entry into the market is
unattractive until the market is large enough to permit an additional bank or branch of efficient
size. When the market contains only two banks, any entrant must begin at an even smaller scale
than either incumbent, and its unit costs will therefore be higher than those of the incumbents if
all those banks are operating in a region of increasing returns to scale. The entrant will therefore
be at a competitive disadvantage because of its higher costs. As a result, the competitive pressure
in the Dewey County banking market is limited.
3.3 Data
Although all US banks have reported financial data on a quarterly basis since the mid-1980s, we
use annual data for four reasons. First, during the initial years of our sample period, quarterly data
are not available for these two banks. Second, the quarterly salary data for the two banks show a
pronounced ‘bonus effect’ in the fourth quarter, which may distort the regression result. It turns out
difficult to fully remove this effect from the data using the usual techniques to correct for seasonal
effects. Third, the quarterly data seem inaccurate sometimes (certain series have strong outliers)
and also give rise to implausible results. Fourth, banks operating in rural areas often face strong
seasonal demand patterns from the agricultural cycle that may distort quarterly estimates.
We obtain annual bank-level data for the two banks in Dewey County from the Reports of
Condition and Income (or Call Reports) of the Federal Reserve System, covering the period 1976
– 2010.6 This results in 35 annual observations for each bank, constituting a panel data set of 70
observations. For the sake of simplicity, we will refer in the sequel to the Western Dakota Bank as
bank 1 and the State Bank of Eagle Butte as bank 2. Our sample size is comparable to those used
in other studies of duopoly such as Shaffer and DiSalvo (1994).
5 An
example of such a case is when one or more banks seek to maximize size or growth subject to a break-even
constraint.
6 The Call Reports data can be downloaded from
http://chicagofed.org/webpages/banking/financial institution reports/commercial bank data.cfm.
8
4 Panzar-Rosse model
The Panzar-Rosse (P-R) revenue test is based on a reduced-form equation relating gross revenues
to a vector of input prices and other firm-specific control variables. Following the intermediation
model for banks (Klein, 1971; Monti, 1972; Sealey and Lindley, 1977), an n-input single-output
production function is assumed. The empirical reduced-form equation of the P-R model is written
as
log TR D ˛ C
n
X
iD1
ˇi log wi C
J
X
j D1
j log CFj C ";
(1)
where TR denotes total revenue, wi the price of the i-th input factor, and CFj the j -th firm-specific
control factor. Panzar and Rosse (1977) show that the sum of input price elasticities,
H D
n
X
ˇi ;
(2)
iD1
reflects the competitive structure of the market. As shown by Bikker et al. (2012), the control
variables should not contain any level variables controlling for scale, such as total assets or equity.
If such scale variables are included in the revenue equation, the H statistic will generally be
positively biased.
While many studies have estimated H using aggregate data, the only previous study of a banking duopoly estimates H separately for each bank (Shaffer and DiSalvo, 1994). We follow their
approach here. Moreover, our theoretical analysis above indicates that the two banks should exhibit
unequal values of H if there is asymmetry resulting from either sequential choice (Stackelberg)
or unequal marginal costs. In these cases it would be important to estimate H separately for each
bank.
4.1 Model specification
In line with previous studies such as Koetter et al. (2012), we consider three inputs (funding, labor,
and physical capital) and one output (loans and other earning assets). Because we have a limited
number of observations, we only consider the ratio of total loans to total assets (TLNS/TA) and
the ratio of total equity to total assets (EQ/TA) as control variables. These variables capture credit
risk and leverage, respectively; see Bikker et al. (2012). This results in the following P-R revenue
9
equation:
log TRit
D ˛i C ˇ1i log Fit C ˇ2i log Wit C ˇ3i log Ri t
C 1i log .TLNS/TAi t / C 2i log .EQ/TAit / C "it ;
(3)
where TRit denotes total revenue of bank i D 1; 2 in year t D 1; : : : ; 35, Fit the deposit interest
rate of bank i in year t , Wit the wage rate of bank i in year t , and Ri t the price of physical capital
of bank i in year t . The corresponding H statistic equals H D ˇ1 C ˇ2 C ˇ3 .
To estimate Equation (3), we have to proxy total revenue and the input prices using the data
from the Call Reports. We take total operating income (TOI) as total revenue, which is defined
as the sum of interest and non-interest revenue. The use of non-interest revenue accounts for the
increase in revenue coming from fee-based products and off-balance sheet activities, particularly
in recent years. A practical reason for using operating income instead of interest income is that the
latter is only available as a separate variable in the Call Reports beginning in 1983. In line with
other banking studies (e.g., Koetter et al., 2012), we measure the deposit interest rate (F ) as the
ratio of interest expenses to total deposits, the wage rate (W ) as the ratio of personnel expenses
to the number of fulltime employees, and the price of physical capital (R) as total expenses on
fixed assets divided by the dollar value of fixed assets. We use the Consumer Price Index (CPI)
to deflate the level variables.7 Appendix B explains in detail how the model variables have been
calculated from the Call Reports Data.
4.2 Sample statistics
Table 1 provides sample statistics for the two banks during the 1976 – 2010 period. In terms of
average total assets over the years, bank 2 is the larger one.8 Table 1 highlights certain differences
in input prices between the two banks. Labor is cheaper for bank 2, but physical capital and funding
is cheaper for bank 2. Moreover, bank 1 has a higher loan to asset ratio, whereas the equity ratios
7 The
CPI for All Urban Consumers: All Items (CPIAUCSL) can be downloaded from
http://research.stlouisfed.org/fred2/series/CPIAUCSL/downloaddata?cid=9. We use the CPI in levels, with the price
index in 1982 – 1984 normalized to 100.
8 Between 1979 and 1987, bank 1 was the larger bank in terms of total assets. As of 1988, bank 2 has been the larger
one, but the difference in total assets between the two banks has become smaller over the years. In 2012 total nominal
assets equaled $ 29,674,000 for bank 1 (13,070,347 in real dollars) and $ 34,888,000 for bank 2 (15,366,929 in real
dollars).
10
are almost the same for the two banks. Table 1 also provides the profitability statistics net income
and return on assets (ROA), which show that the variation in profitability is much larger for bank 1
than for bank 2. In terms of ROA, the two banks seem relatively unaffected by the financial crisis
that started in 2008.
Differences in scale, input prices and profitability are very common in banking samples and
do not threaten the applicability or the interpretation of the P-R model. For example, Shaffer and
DiSalvo (1994), in a related study of a US banking duopoly that included a P-R test, indicate that
the average wage rate was 22 percent higher at one bank than at the other, while its price of fixed
assets was less than half that of the other. Similarly, Bikker et al. (2012) report that the average
funding cost varied across a range of 6.5:1 from the 5% quantile to the 95% quantile in their
sample of banks, while the imputed wage rate spanned a range of more than 4:1.
4.3 Estimation results
We estimate the H statistic separately for each bank using OLS. Table 2 reports the estimation
results for the Panzar-Rosse revenue equation and the corresponding values of the H statistic, together with heteroskedasticity-consistent standard errors (White, 1980). Following Shaffer (2004),
we also estimate a short-run revenue equation for each bank, in which the price of physical capital
is omitted (yielding H D ˇ1 C ˇ2 ). The short-run H statistic acknowledges that physical capital
may be fixed in the short run, precluding an immediate effect of its price on equilibrium income.
Throughout, the H statistic is assumed to be constant over time, thus representing the average
value during the sample period over which the revenue equation is estimated. Because the equity
ratio (EQ/TA) turns out insignificant in each of the estimated revenue equations and does not affect
the value of the estimated H statistic, it has been left out in the final specifications.
From Table 2 it follows that both the short-run and the long-run H statistics for bank 1 are
not significantly different from 1. The H statistics for bank 2 are substantially lower. For the
latter bank, we cannot reject the null hypothesis that 0 < H < 1. These results are in line with
Proposition 2.2 in the following sense. From Table 1 it becomes clear that bank 1 is the low-cost
bank in terms of average costs. It will later also turn out as the bank with the lowest marginal
costs. To the extent that the conduct of these banks is sufficiently similar to Cournot conduct,
11
Proposition 2.2 would therefore imply that the H statistic of the low-cost bank is higher than for
the other bank. In particular, we might reasonably expect a positive H statistic for the low-cost
bank. Throughout, we find qualitatively the same results if we leave out the observations for 2008
– 2010, which are possibly affected by the financial crisis.
We have also performed an ROA test by estimating H roa (Shaffer, 1982). We have done this by
replacing the dependent variable in Equation (3) by ROA. As shown in Bikker et al. (2012), H roa
can be used as a test of long-run competitive equilibrium. Whenever H D 1 and H roa D 0, both
the revenue test and the ROA test provide results consistent with long-run competitive equilibrium.
For both banks in Dewey County, H roa is not significantly different from 0. For bank 1, this finding
is consistent with long-run competitive equilibrium. For bank 2 the ROA test is more difficult to
interpret.
4.4 Implications
As explained in Section 3.2, we expect to find low banking competition in Dewey County because
of the inefficient size of the two banks. Consequently, the finding of H D 1 for bank 1 seems
highly implausible, given that this value of H is commonly associated with long-run competitive
equilibrium.9 However, Propositions 2.1 and 2.2 in Section 2 emphasize that H > 0 may also
occur with a high degree of oligopoly power. As a result, we need additional information to interpret the estimated value of the H statistic; for example, about the sequence of the banks’ actions.
Because it is not straightforward how to obtain such information, we we will further examine the
competitiveness of particularly bank 1 by estimating several alternative competition measures. We
will do the same for bank 2, for which the finding 0 < H < 1 indicates monopolistic competition
or short-run competition.
5 Alternative measures for market power
The New Empirical Industrial Organization literature provides several methods for assessing banking competition, such as the conjectural variation approach of Iwata (1974), Gollop and Roberts
(1979), Appelbaum (1982), Bresnahan (1982, 1989), Lau (1982), and Genesove and Mullin (1998).
9 This statement assumes that the banks are pursuing an objective of profit maximization; Shaffer (1982) showed
that H D 1 under an objective of sales maxmization subject to a breakeven constraint.
12
Another measure for market power is the price-cost margin or Lerner index, which is defined as
the difference between output price and marginal costs, divided by the output price (Lerner, 1934).
Other methods for assessing competition can be found in Boone (2008), Bolt and Humphrey
(2010), Kumbhakar, Baardsen, and Lien (2012), and Kutlu and Sickles (2012).
The nature of our data sample puts a restriction on the choice of the alternative competition
measures. We consider only two banks observed during a 35-year period, resulting in a relatively
modest panel data set of 70 observations. With this data set it is not possible to estimate models
involving a large number of parameters. We therefore limit the subsequent analysis to the Lerner
index and a version of the conjectural variation approach that requires relatively little data.
5.1 Lerner index
The Lerner index reflects the social loss arising from deviations from marginal cost pricing. It
equals zero in a situation of perfect competition (when prices equal marginal costs), thus providing
a natural benchmark. Following e.g. Koetter et al. (2012), we specify a translog cost function
to estimate the marginal costs component of the Lerner index. Total operating costs (TOC ) are
defined as the sum of personnel expenses, interest expenses, and total expenses on fixed assets and
output (Q) is measured as total assets (TA). The last rows of Table 1 provide sample statistics for
total operating costs and average costs. We assume a common three-input one-output production
technology as before and impose linear homogeneity in input prices by normalization with the
price of fixed assets. This results in the following specification for bank i in year t :
A
log T OC it
eit C ˇ2 log W
fit C ˇ3 .log F
ei t /2
D ˛i C ˇ1 log F
fit /2 C ˇ5 log F
eit log W
fit
C ˇ4 .log W
eit log Qit C 2 log W
fit log Qit
C 1 log F
C ı1 log Qit C ı2 .log Qi t /2 C 1 t C 2 t 2 C 3 t log Qit ;
A
(4)
eit D Fi t =Rit , and W
fit D Wit =Rit . Marginal costs then follow
where T OC it D TOCit =Rit , F
as
M Cit D
Cit eit C 2 log W
fit C ı1 C 2ı2 log Qi t C 3 t :
1 log F
Qit
13
(5)
The Lerner index for bank i in year t then follows as
Lit D
Pt
M Cit
:
Pt
(6)
Following Koetter et al. (2012), we proxy the output price as the average revenue.
We estimate Equation (4) using the within estimator and clustered standard errors.10 The estimation results are displayed in Table 3. The estimated coefficients are used to calculate marginal
costs following Equation (5). The resulting average marginal costs are 3.8% for bank 1 and 4.2%
for bank 2. On average, marginal costs are lower than average costs for both banks (see Table 1),
confirming the expected economies of scale mentioned in Section 3. The average Lerner index
equals 50.6% for bank 1 and 55.3% for bank 2, which is substantially larger than the average
Lerner indices established in Koetter et al. (2012) for the US as a whole. Figure 1 displays the
Lerner indices for both banks as a function of time, including 90% confidence intervals for parameter uncertainty.11 From Figure 1 it becomes clear that both Lerner indices are significantly
positive in most of the years.
As an additional robustness check, we estimate the Lerner index using the Hall-Roeger method
(Hall, 1988; Domowitz, 1988; Roeger, 1995; Rezitis, 2010). The attractive feature of this method
is that the underlying model is relatively parsimonious, which is advantageous in our setting with
a limited data sample. Full details of the Hall-Roeger method are provided in Appendix C. The resulting bank-specific estimates of the Lerner index are 47.7% for bank 1 (with a heteroskedasticityconsistent standard error of 3.7%) and 27.1% for bank 1 (3.0%).
The Lerner estimates based on the Hall-Roeger approach are smaller than those based on the
translog cost function. The critical assumption underlying the original Hall-Roeger method is that
of constant returns to scale (CRS). As shown by Hyde and Perloff (1995), the assumption of CRS
results in an underestimation of the Lerner index in the presence of increasing returns to scale.
As mentioned in Section 3, increasing returns to scale are the most likely deviation from the CRS
assumption in the case of Dewey County, which was confirmed by our estimates of average and
marginal costs. This explains why the Hall-Roeger estimates of the Lerner index turn out smaller
10 Clustered standard errors may have poor finite-sample properties if the number of clusters is small (Cameron and
Trivedi, 2005). Because we have only N D 2 banks, we therefore cluster over time (for which we have T D 35
clusters).
11 The confidence interval accounts for the parameter uncertainty in the estimated marginal costs.
14
than those based on the translog cost function. Yet both estimation methods consistently reveal a
significantly positive Lerner index for both banks.
5.2 Conjectural variation correlation test
The role of the conjectural variation is to quantify the gap between price and marginal costs (Shaffer, 2000; Shaffer, 2004). Estimation of the conjectural variation provides an empirical test of the
degree of monopoly power. Any form of oligopoly behavior, whether arising from a static or a
dynamic game, can be represented by this parameterization.12 Our limited data sample makes
it difficult to reliably estimate Bresnahan’s (1982) conduct parameter as done by e.g. Shaffer
(2004) and Uchida and Tsutsui (2005). Instead, we apply the correlation test proposed by Shaffer
(2000), which requires much less data. The latter test uses only output price, output quantities,
and marginal costs and does not impose any parametric assumptions. Throughout, we assume that
the conjectural variation is constant over time, which means that we basically look at the average
conjectural variation during the sample period. To our best knowledge, we are the first to apply
Shaffer’s (2000) conjectural variation correlation test to bank data.
The conjectural variation is derived from the first-order conditions for a profit-maximizing
firm and equals, for bank i D 1; 2,
bi D e.M Ci =P
1/Q=Qi
1;
(7)
where e denotes the market elasticity of demand and Q is defined as aggregate output (Q D
Q1 C Q2 ). Equation (7) implies that
eQ.M Ci =P
Consequently, bi D
1/ D .1 C bi /Qi :
(8)
1 (perfect competition) implies zero correlation between Qi and the left
hand side of Equation (8) in a time-series sample, whereas bi >
1 (imperfect competition)
implies positive correlation. Shaffer (2000, Prop. 1a) shows that zero correlation between Qi and
Q.M Ci =P
1/ is both a necessary and a sufficient condition for competitive behavior for any
form of the demand elasticity.13 Only in case of a non-competitive outcome, changes in the demand
12 See
Tirole (1988), p. 245, footnote 12; Worthington (1990); Shaffer (2000, 2004).
correlation is not a sufficient condition if either the market demand curve is horizontal or if bi D 1 C
PP 0 Q=.QP 02 PP 0 PP 00 Q/Qi . Because these exceptions occur on a set of measure zero, there is sufficiency in
practice.
13 Zero
15
elasticity over time may bias the results. In a time series sample, also demand shifts may induce
spurious correlation between the two terms in Equation (8), causing a bias towards rejection of the
competitive hypothesis. Shaffer (2000) proposes first differencing to make the correlation test more
robust for time-series samples with periods of fluctuating prices and market demand. Whenever
the elasticity of demand is constant, imperfect competition implies a negative correlation between
Qit and
Q.M Ci =P
1/ C QM Ci =P
Q.M Ci =P 2 /P;
(9)
whereas perfect competition implies zero correlation between these terms.
We run two versions of the correlation test, with and without first differencing. With two
banks, this results in four tests. In all cases, we test the sign of the correlation using a Wilcoxon
non-parametric rank correlation test. The test results are displayed in Table 4. As expected, firstdifferencing leads to a slightly less negative correlation. All four tests reject the null hypothesis of
perfect competition at each reasonable confidence level.
5.3 Robustness checks
It is important to verify that the Lerner index and the conjectural variation correlation test truly
reveal market power. Although we have a modest data sample covering 35 years, both measures
point in the same direction, indicating robustness. Furthermore, we have estimated the Lerner
index in two different ways (using a translog cost function and the more parsimonious Hall-Roeger
method), resulting in qualitatively similar outcomes.
An issue of consideration is cost and profit inefficiency of banks. When banks are cost inefficient, their potential marginal costs will be lower than the ones implicitly used in estimating the
Lerner index, resulting in underestimation of the Lerner index. To our best knowledge, the impact
of inefficiencies on the H statistics has not yet been addressed in the theoretical or empirical literature. Given our modest data sample, we cannot estimate inefficiency-corrected Lerner indices
as in e.g. Koetter et al. (2012). The latter study finds, on average, a Lerner index that is about one
third larger when profit and cost inefficiencies are accounted for. This finding suggests that our
estimates of the Lerner index might be even larger if we could similarly control for inefficiencies,
further strengthening the contrast with the apparent implications of our H estimates.
16
5.4 Evaluation
According to the conventional interpretation of the P-R revenue test, the estimated H statistics
for bank 1 indicate long-run competitive equilibrium. This is highly unlikely given the highly
concentrated market the two banks operate in and their inefficient scale; see the discussion in
Section 3.2. But From Propositions 2.1 and 2.2 we know that a positive value of H may also occur
with a high degree of oligopoly power. Indeed, the Lerner index and the conjectural variation
correlation test both indicate that bank 1 possesses market power, consistent with our a priori
expectations. These findings suggest that we have indeed identified an empirical case in which
H D 1 does not correspond with perfect competition, thus emphasizing the practical relevance of
the problem identified by Propositions 2.1 and 2.2. That is, we do not claim to have shown that
these banks operate as either Stackelberg or asymmetric Cournot duopolists but, more generally,
that they exhibit measurable oligopoly power that does not show up in the H statistic.
Also for bank 2 the Lerner index and the conjectural variation correlation test reveal market
power, but this does not necessarily contradict the empirical finding of 0 < H < 1. A value
of 0 < H < 1 is consistent with both monopolistic competition and short-run competition. With
short-run competition, the Lerner index equals zero, but under monopolistic competition a positive
Lerner index is possible (Benassy, 1991).
6 Conclusions
Motivated by a recent proliferation of the Panzar-Rosse method’s use in banking studies, we have
analyzed the response of equilibrium revenue to changes in marginal cost under two market structure scenarios not previously explored. We identified two imperfectly competitive cases in which
the sum of revenue elasticities with respect to input prices, H , can take a positive value, providing
the first formal proof of a property claimed by Panzar and Rosse (1987) and Vesala (1995) and
contrary to a pattern previously established for specific forms of oligopoly. We further showed
that one of those cases can even generate H > 1, potentially explaining some previous empirical
results and clarifying that the perfectly competitive value of 1 is not a theoretical upper bound on
H as was previously thought.
Together with prior theoretical studies, these two results indicate that neither the sign nor
17
the magnitude of H can unambiguously identify the degree of market power without additional
information on the characteristics of cost and the sequence of actions by firms, contrary to prior
belief. This conclusion calls into question the standard interpretations of dozens of existing PanzarRosse studies and prescribes additional procedures for future estimation.
The results of our empirical analysis of the banking duopoly in Dewey County (South Dakota,
USA) are consistent with our theoretical result that the H statistic may indicate competitive conduct in noncompetitive markets. The value of the H statistic for one of the banks in this duopoly
does not significantly differ from 1 and corresponds to long-run competitive equilibrium according
to the conventional interpretation of the H statistic. This is unlikely given the inefficient scale of
the two banks active in Dewey County and the highly concentrated market they operate in. Indeed,
the Lerner index and the conjectural variation correlation test reveal significant market power, in
accordance with the predictions of conventional oligopoly theory.
Overall, the implications of the analysis here are not sanguine for the extensive and growing
body of empirical reduced-form revenue studies. Far from being the convenient, robust measure
of market conduct as originally portrayed, the H statistic must be augmented by considerable
additional information from independent sources to take account of a wide variety of alternative
outcomes.
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23
Table 1: Sample statistics for the Dewey county duopoly (Panzar-Rosse method)
Level variables are deflated (using the Consumer Price Index) and in thousands of dollars. The
other variables are in percentage. Bank 1 is the Western Dakota Bank and bank 2 the State Bank
of Eagle Butte. Abbreviations: TA: total assets, TOI : operating income, F : deposit rate, W :
wage rate, R: price of physical capital, EQ: total equity, LNS: total loans, NI : net income,
ROA: return on equity (ROA D NI=TA), TOC : total operating costs, AC : average costs.
TA
TOI
F
W
R
EQ
LNS
EQ=TA
LNS =TA
Bank 1
mean std. error
10484.0
2519.9
867.2
452.4
3.6%
2.1%
20.0
6.2
72.6%
44.6%
1082.2
432.5
4930.7
2879.3
10.3%
2.9%
44.7%
16.3%
Bank 2
mean std. error
12479.8
2036.0
1107.2
214.5
3.3%
1.8%
32.2
6.5
30.5%
16.0%
1317.4
428.7
4508.4
2009.2
10.4%
2.1%
35.0%
11.8%
NI
ROA
63.1
0.5%
404.4
2.9%
158.3
0.8%
98.3
0.3%
TOC
AC
597.9
5.5%
338.6
2.1%
793.4
6.5%
155.8
1.4%
24
Table 2: Estimation results for the Panzar-Rosse revenue equation
The revenue equation has been estimated separately for each bank, using OLS with
heteroskedasticity-consistent standard errors. Because the equity ratio (EQ=TA) turned out
insignificant in each equation and did not affect the value of the H statistic, it has been left out in
the final specifications. Abbreviations: F : deposit rate, W : wage rate, R: price of physical
capital, LNS =TA: total loans/total assets.
coeff.
std. error
t -value
p-value
Bank 1
short run
Intercept
log F
log W
log .LNS =TA/
adj. R2
H statistic
6.063
0.272
0.642
0.413
0.899
0.914
long run
Intercept
log F
log W
log R
log .LNS =TA/
adj. R2
H statistic
6.072
0.303
0.628
-0.116
0.319
0.922
0.814
0.649
0.077
0.157
0.106
9.341
3.524
4.100
3.897
0.000
0.001
0.000
0.000
0.132
6.912
0.000
0.535
0.062
0.137
0.030
0.110
11.340
4.894
4.564
-3.918
2.886
0.000
0.000
0.000
0.000
0.007
0.140
5.806
0.000
Bank 2
short run
Intercept
log F
log W
log .LNS =TA/
adj. R2
H statistic
6.698
0.136
0.353
0.392
0.680
0.489
0.415
0.033
0.107
0.081
16.122
4.096
3.302
4.846
0.000
0.000
0.002
0.000
0.121
4.027
0.000
long run
Intercept
log F
log W
log R
log .LNS =TA/
adj. R2
H statistic
6.872
0.160
0.345
0.025
0.414
0.674
0.531
0.408
0.042
0.100
0.031
0.075
16.851
3.796
3.456
0.825
5.515
0.000
0.001
0.002
0.416
0.000
0.131
4.038
0.000
Table 3: Estimation results for the translog cost function
The translog cost function has been estimated using the within estimator, with clustered standard
e: deposit rate, W
f: wage rate, TA: total
errors. Abbreviations: T OC : total operating costs, F
assets, t : time. All variables with a tilde have been normalized with the price of fixed assets R to
ensure linear homogeneity in input prices.
A
e
log F
f
log W
e/2
.log F
f/2
.log W
e log W
f
log F
e log TA
log F
f
log W log TA
log TA
.log TA/2
t
t2
t log TA
coeff.
std. error
t -value
p-value
1.331
-2.060
0.171
0.209
-0.384
0.181
-0.020
13.687
-0.687
-0.176
0.000
0.020
0.736
0.923
0.013
0.024
0.035
0.087
0.108
2.993
0.174
0.072
0.000
0.008
1.809
-2.231
13.117
8.684
-11.078
2.091
-0.182
4.573
-3.959
-2.426
-0.999
2.501
0.076
0.030
0.000
0.000
0.000
0.041
0.856
0.000
0.000
0.018
0.322
0.015
Table 4: Wilcoxon rank correlation for conjectural variation
The rank correlation based on level variables is in the column captioned ‘levels’, whereas the
rank correlation based on first-differenced variables is in the column captioned ‘differences’. The
p-values corresponding to the Wilcoxon’s rank correlation tests are within parentheses.
bank 1
bank 2
levels
-0.85
(0.000)
-0.86
(0.000)
26
differences
-0.76
(0.000)
-0.80
(0.000)
Figure 1: Lerner index over time
1.0
0.6
0.2
−0.2
Lerner index bank 1
The dashed lines constitute a 90% pointwise confidence interval accounting for parameter
uncertainty. The estimates of the Lerner index are significantly different from zero as long as the
confidence interval stays above the zero line.
1975
1980
1985
1990
1995
2000
2005
2010
1995
2000
2005
2010
1.0
0.6
0.2
−0.2
Lerner index bank 2
year
1975
1980
1985
1990
year
27
Appendix A Proofs of propositions
Proposition 2.1: H can take either sign for a Stackelberg duopoly facing linear cost and demand.
Proof: Because marginal cost is homogeneous of degree 1 in all input prices, H equals the elasticity form of (and has the same sign as) @TR=@M C . Thus, it suffices to solve for @TR=@M C .
Let P D a
bq1
profit i D qi .P
bq2 for firms producing qi .i D 1; 2/ at constant marginal cost c. Firm i earns
c/ D qi .a
c
bq1
bq2 /. The Stackelberg follower (firm 1) chooses output
q1 to maximize its profit, given the output selected by the other firm, with first-order condition
a
c
2bq1 D 0, so q1 D .a
bq2
c
bq2 /=2b. The leader (firm 2) chooses output q2
to maximize its profit, conditional on the reaction function of firm 1. Its first-order condition is
.a
c/=2
bq2 D 0 or q2 D .a
c/=2b. Then q1 D .a
c/=4b and P D .a C 3c/=4.14
If both firms maintain the same leader-follower sequence, we can solve directly for the impact of
a change in c on each firm’s output, equilibrium price, and revenue: @q1 =@c D
1=4b, @q2 =@c D
1=2b, and @P =@c D 3=4. In general, @TRi =@c D P @qi =@c C qi @P =@c C .@qi =@c/.@P =@c/.
Thus, @TR1 =@c D .2a
the sign of 2a
6c
6c
3/=16b and @TR2 =@c D .2a
6c
3/=8b. Both expressions have
3, which depends on the parameter values.
For a sustainable equilibrium, the demand price must cover the cost of production, so a > c.
Within this range, we can have either c < a < 3c C 3=2 (where @TR=@c < 0 for both firms) or
a > 3c C 3=2 (where @TR=@c > 0 for both firms). By the linear homogeneity of marginal cost
with respect to all input prices, this means that the Panzar-Rosse H statistic can take either sign
for Stackelberg duopolists, depending on the relative magnitudes of a and c.
The follower’s Lerner index (relative markup of price over marginal cost) is .a
which is positive for a > c, confirming the existence of market power.
c/=.a C 3c/,
2
Proposition 2.2: H > 1 is possible for the low-cost firm in a homogeneous Cournot duopoly with
asymmetric costs and linear demand.
Proof: Let P D a bx by where x is the output quantity of one firm and y is the output quantity
of the other firm. Let the total cost of x equal cx and total cost of y equal ˛cy. Assume a > c
to ensure non-negative profits in equilibrium. Assume ˛ > 1 so the first firm has lower marginal
14 The
second-order condition is satisfied for all b > 0.
28
cost. We define TRx as the first firm’s total revenue and Hx as its H statistic. By standard firstorder conditions for profit maximization, x D Œa
c.2
Then P D Œa C c.˛ C 1/=3, TRx D Œa2 C ac.˛
1/ C c2.˛ 2
Œa.˛
1/ C 2c.˛ 2
1/=Œa2 C ac.˛
˛
˛/=3b and y D Œa
˛
˛
1/=3b.
1/=9b, and @TRx =@c D
1/=9b. Thus Hx D .c=TRx /@TRx =@c D Œac.˛
1/ C c 2 .˛ 2
c.2˛
1/ C 2c 2 .˛ 2
1/. Because a > c and ˛ > 1, a2 > c 2 .1 C ˛
˛
˛ 2 / and so
Hx > 1.
Similar calculations show Hx > 0 for all ˛ > 2c
a C Œ.2c C a/2 C 16c 2 1=2 =4c. This is an-
other example of how we can get H > 0 with imperfect competition. For instance, if a D 2c, then
p
p
Hx > 0 for all ˛ > 2 and Hx > 1 for all ˛ > .1 C ˛ 21/=2. The Lerner index for the low-cost
firm is Lx D Œa C c.˛
2/=Œa C c.˛ C 1/, which is positive for a > c and ˛ > 1, confirming
market power. Moreover, @Lx =@˛ D 3c 2 =Œa C c.˛ C 1/2 > 0, so greater inequality in marginal
cost results in a higher value of the Lerner index for the low-cost firm.
29
2
Appendix B
Definition of variables
Below we explain how the variables in this study have been calculated from the data available in
the Call Reports. Throughout, level variables are deflated using the Consumer Price Index.
model variable
Call Reports variable(s)
operating income (TOI )
RIAD4000
(=RIAD4107+RIAD4079)
RCFD1400
RCFD3210
RCFD2170
TA
RIAD4130
NI/TA
total loans (T LNS )
total equity (EQ)
total assets (TA)
output (Q)
net income (NI )
return on assets (ROA)
salaries (S)
interest expenses on deposits
and interest expenses on fed funds (IE)
expenditures on fixed assets (EFA)
total operating costs (TOC )
RIAD4135
RIAD4170+RIAD4180
RIAD4217
S C IE C EFA
number of full-time employees (L)
deposits and fed funds purchased (D)
physical capital (K)
RIAD4150
RCFD2200+ RCFD2800
RCFD2145
output price (P )
deposit interest rate (F )
wage rate (W )
price of physical capital (R)
TOI=TA
IE=D
S=L
EFA=K
30
Appendix C Hall-Roeger method
The Hall method is based on the insight of Solow (1957) that productivity shocks can be measured
from the data if price equals marginal cost and the marginal product of labor equals the real wage
rate. The contribution of Hall (1988) is the extension of this relation to the case of market power;
i.e., when price exceeds marginal cost. In this way, the Hall method provides a method of estimating the Lerner index without having to estimate marginal cost. In his seminal article, Hall (1988)
develops his method for the case of two input factors, capital and labor. Domowitz et al. (1988)
include materials as a third input factor. While estimation of the original Hall model requires an
instrumental variable, the extension of Roeger (1995) can do without. We will explain this in more
detail below. A recent application of the Hall-Roeger method to the Greek banking sector can be
found in Rezitis (2010).
We continue to use the intermediation model for banks and adopt the common notation for the
Hall method. In period t the production function of bank i uses labor (Li t ) and physical capital
(Kit ) to attract deposits (Dit ). The deposits are then used to fund loans and other earning assets
(Qt ):
Qit D Ait e i f .Kit ; Lit ; Dit /;
(C.1)
where i represents the rate of Hicks-neutral technical progress, Ait a productivity shock, and
f ./ a constant returns to scale (CRS) production function.
As shown by Domowitz et al. (1988), the Solow residual ait D logAit satisfies
qit
kit
˛Lit .`it
i .1
ˇi / C ˇi .qit
kit /
˛Di t .dit
kit / C .1
kit / D
ˇi /ait :
(C.2)
Here qit D log.Qit /, kit D log.Kit /, `i t D log.Li t /, dit D log.Di t /, ˛Li t D Wit Lit =.Pit Qit /
(labor share in the value of output), and ˛Di t D Fit Dit =.Pi t Qi t / (deposit share in the value of
output). Moreover, Wit is the wage rate, Fit the deposit interest rate, Pit the output price, and ˇi
the Lerner index. In the absence of market power (ˇi D 0), the productivity shock ai t equals
the dependent variable minus i ; an expression known as the Solow residual.
To estimate the Lerner index, we calculate the dependent variable in Equation (C.2) from the data
31
and regress it on the growth rate of the output-capital ratio qit
ki t . However, the growth
rate is correlated with the productivity shock ai t , causing a potential endogeneity problem. We
could deal with this problem by means of an instrumental variable (IV) approach (Hall, 1988).
This means that we have to find an exogenous variable that is strongly correlated with inputs
and output, but that does not cause movements in productivity and does not respond to random
fluctuations in productivity growth. Because finding valid instruments is generally difficult, we
prefer the extension of Roeger (1995).
Roeger (1995) avoids the endogeneity problem by using the dual (or price-based) Solow residual
in addition to the primal Solow residual. Roeger (1995) shows that the difference between the two
residuals does not depend on the productivity shock ai t . This results in the following equation
from which the Lerner index ˇi can be estimated:
qit C pit
.1
˛Dit
˛Dit .dit C fit /
˛Li t .`it C wi t /
˛Lit /.kit C rit / D ˇi .qi t
ki t C pi t
rit /;
(C.3)
where pit ; wit ; fit and rit denote the growth rates of the output price, the wage rate, the deposit
interest rate, and the price of physical capital, respectively.
32