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Economic Quarterly—Volume 95, Number 4—Fall 2009—Pages 335–355
Heterogeneity in Sectoral
Employment and the
Business Cycle
Nadezhda Malysheva and Pierre-Daniel G. Sarte
T
his paper uses a factor analytic framework to assess the degree to which
agents working in different sectors of the U.S. economy are affected
by common rather than idiosyncratic shocks. Using Bureau of Labor
Statistics (BLS) employment data covering 544 sectors from 1990–2008, we
first document that, at the aggregate level, employment is well explained by
a relatively small number of factors that are common to all sectors. In particular, these factors account for nearly 95 percent of the variation in aggregate
employment growth. This finding is robust across different levels of disaggregation and accords well with Quah and Sargent (1993), who perform a similar
analysis using 60 sectors over the period 1948–1989 (but whose methodology
differs from ours), as well as with Foerster, Sarte, and Watson (2008), who
carry out a similar exercise using data on industrial production.1
Interestingly, while common shocks represent the leading source of variation in aggregate employment, the analysis also suggests that this is typically
not the case at the individual sector level. In particular, our results indicate
that across all goods and services, common shocks explain on average only 31
percent of the variation in sectoral employment. In other words, employment
at the sectoral level is driven mostly by idiosyncratic shocks, rather than common shocks, to the different sectors. Put another way, it is not the case that “a
rising tide lifts all boats.” Moreover, it can be easy to overlook the influence
of idiosyncratic shocks since these tend to average out in aggregation.
We wish to thank Kartik Athreya, Sam Henly, Andreas Hornstein, and Thomas Lubik for
helpful comments. The views expressed in this article do not necessarily represent those
of the Federal Reserve Bank of Richmond, the Board of Governors of the Federal Reserve
System, or the Federal Reserve System. All errors are our own.
1 See also Forni and Reichlin (1998) for an analysis of ouput and productivity in the United
States between 1958 and 1986.
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Federal Reserve Bank of Richmond Economic Quarterly
Despite the general importance of idiosyncractic shocks in explaining
movements in sectoral employment, we nevertheless further document substantial differences in the way that sectoral employment is tied to these shocks.
Specifically, we identify sectors where up to 85 percent of the variation in
employment is driven by the common shocks associated with aggregate employment variations. Employment in these sectors, therefore, is particularly
vulnerable to the business cycle with little in the way of idiosyncratic shocks
that might be diversified away. These sectors are typically concentrated in
construction and include, for example, residential building.
More generally, our empirical analysis indicates that employment in
goods-producing industries tends to more tightly reflect changes in aggregate
conditions relative to service-providing industries. However, even within the
goods-producing industries, substantial heterogeneity exists in the way that
sectoral employment responds to common shocks. For instance, the durable
goods and construction industries are significantly more influenced by common shocks than the nondurable goods and mining industries. Among the
sectors where employment is least related to aggregate conditions are government, transportation, and the information industry.
Finally, we present evidence that the factors uncovered in our empirical
work play substantially different roles in explaining aggregate and sectoral
variations in employment. Although the findings we present are based on
a three-factor model, our analysis suggests that one factor is enough to explain roughly 94 percent of the variation in aggregate employment. At the
same time, however, that factor appears almost entirely unrelated to employment movements in specific sectors such as in natural resources and mining
or education and health services. Interestingly, the reverse is also true in the
sense that the analysis identifies factors that significantly help track employment movements in these particular sectors but that play virtually no role in
explaining aggregate employment fluctuations.
This article is organized as follows. Section 1 provides an overview of
the data. Section 2 describes the factor analysis and discusses key summary
statistics used in this article. Section 3 summarizes our findings and Section
4 offers concluding remarks.
1.
OVERVIEW OF THE DATA
Our analysis uses data on sectoral employment obtained from the BLS covering the period 1990–2008. The data are available monthly, seasonally adjusted, and disaggregated by sectors according to the North American Industry
Classification System (NAICS). Our data cover the period since 1990, the
date at which this classification system was introduced. Prior to 1990, BLS
employment data were disaggregated using Standard Industry Classification
codes, which involve a lower degree of disaggregation and were discontinued
Malysheva and Sarte: Sectoral Employment and the Business Cycle
337
Figure 1 Breakdown of Sectoral Employment Data
Level 4 Disaggregation (Five-digit NAICS)
Total Nonfarm (544)
Goods-producing Industries (186)
23 Construction (28)
31-33 Manufacturing (150) 1133, 21 Natural
Resources & Mining (8)
Service-providing Industries (358)
Private Industries (346)
91 Government (12)
22 Utilities (4)
321, 327, 33 Durable Goods (85)
31, 322-326 Nondurable Goods (65)
42, 44-45 Wholesale & Retail Trade (108)
48-49 Transportation & Warehousing (33)
321 Wood Products (5)
311 Food Mfg (16)
327 Nonmetallic Mineral Product Mfg (6)
312 Beverage & Tobacco Products (3)
331 Primary Metal Mfg (9)
313 Textile Mills (4)
332 Fabricated Metal Product Mfg (12)
314 Textile Product Mills (4)
333 Machinery Mfg (11)
315 Apparel Mfg (6)
334 Computer & Electronic Product Mfg (8)
316 Leather & Allied Products (2)
335 Electrical Equipment &
Appliance Mfg (8)
322 Paper Mfg (6)
336 Transportation Equipment Mfg (15)
323 Printing & Related Support
Activities (1)
337 Furniture & Related Product Mfg (4)
324 Petroleum & Coal Products (2)
339 Miscellaneous Mfg (7)
325 Chemical Mfg (13)
326 Plastics & Rubber Products (8)
51 Information (18)
52, 53 Financial Activities (39)
54-56 Prof. & Business Services (54)
61, 62 Education & Health Services (38)
71, 72 Leisure & Hospitality (27)
81 Other Services (25)
as of 2002. For most of the article, we use a five-digit level of disaggregation
that corresponds to 544 sectors, although our findings generally apply to other
levels of disaggregation as well. The raw data measure the number of employees in different sectors, from which we compute sectoral employment growth
rates as well as the relative importance (or shares) of industries in aggregate
employment.
When aggregated, the data measure total nonfarm employment. Nonfarm
employment is further subdivided into two main groups: goods-producing
sectors, comprising 186 sectors at the five-digit level, and service-providing
sectors, comprising 358 sectors. The goods-producing sectors are further subdivided into three main categories: construction, with 28 sectors; manufacturing, with 150 sectors; and natural resource and mining, with eight sectors. The
manufacturing component of the goods sector contains two main categories:
durable goods, comprising 85 sectors, and nondurable goods, with 65 sectors.
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 2 Monthly and Quarterly Employment, All Goods and Services
Panel A: Monthly Data
5
σ
4
1990--2008 = 1.7
3
2
1
0
-1
-2
-3
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
Panel B: Quarterly Data
5
σ 1990--2008 = 1.5
4
3
2
1
0
-1
-2
-3
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
The service-providing sectors employ more than four times as many workers
as the goods-producing sectors. They are made up of two main components:
government, with 12 sectors, and a variety of private industries that include
346 sectors spanning wholesale and retail trade, information, financial activities, education and health, as well as many other services. Figure 1 illustrates
a breakdown of our sectoral data, along with the number of industries within
each broad category of sectors in parenthesis, as well as their corresponding
NAICS codes.
Let et denote aggregate employment across all goods- and servicesproducing industries at date t, and let eit denote employment in the i th industry. We construct quarterly values for employment as averages of the
months in the quarter. We further denote aggregate employment growth by
et and employment growth in industry, i, by eit . At the monthly frequency,
we compute eit as 1, 200 × ln(eit /eit−1 ) and, at the quarterly frequency, as
Malysheva and Sarte: Sectoral Employment and the Business Cycle
339
Figure 3 Distribution of Standard Deviations of Sectoral Growth Rates
(1990–2008)
Panel A: Monthly Growth Rates
5.0
4.5
25th Percentile = 7.4
Median = 10.6
75th Percentile = 14.5
4.0
3.5
Percent
3.0
2.5
2.0
1.5
1.0
0.5
0.0
10
20
30
40
50
60
Sectoral Standard Deviation
Panel B: Quarterly Growth Rates
2.8
25th Percentile = 4.0
Median = 5.5
75th Percentile = 7.4
2.4
Percent
2.0
1.6
1.2
0.8
0.4
0.0
10
20
30
40
50
60
Sectoral Standard Deviation
400 × ln(eit /eit−1 ). Aggregate employment growth is computed similarly.
Finally, we represent the N × 1 vector of sectoral employment growth rates,
where N is the number of sectors under consideration, by et .
Figures 2A and 2B illustrate the behavior of aggregate employment growth
at the monthly and quarterly frequencies, respectively, over our sample period.
Monthly aggregate employment growth is somewhat more volatile than quarterly employment growth, but in either case the recessions of 1991 and 2001
stand out markedly. At a more disaggregated level, Figures 3A and 3B show
the distributions of standard deviations of both monthly and quarterly sectoral
employment growth across all 544 sectors. As with aggregate data, quarterly
averaging reduces the volatility of sectoral employment. More importantly, it
is clear that there exists substantial heterogeneity across sectors in the sense
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Federal Reserve Bank of Richmond Economic Quarterly
Table 1 Standard Deviation of Employment Growth Rates
Full Covariance Matrix
Diagonal Covariance Matrix
Monthly Growth Rates
1.8
0.7
Quarterly Growth Rates
1.5
0.4
Notes: The table reflects percentage points at an annual rate.
that fluctuations in employment are unequivocally more pronounced in some
sectors than others.
Let si denote the (constant mean) share of sector i’s employment in aggregate employment and the corresponding N × 1 vector of sectoral shares
be denoted by s. Then, we can express aggregate employment growth as
et = s et . Furthermore, it follows that the volatility of aggregate employment growth in Figure 2, denoted σ 2e , is linked to individual sectoral
employment growth volatility in Figure 3 through the following equation,
σ 2e = s ee s,
(1)
where ee is the variance-covariance matrix of sectoral employment growth.
Thus, we can think of the variation in aggregate employment as driven by
two main forces—individual variation in sectoral employment growth (the
diagonal elements of ee ) and the covariation in employment growth across
sectors (the off-diagonal elements of ee ).2
Table 1 presents the standard deviation of aggregate employment, σ 2e ,
computed using the full variance-covariance matrix ee in the first row, and
using only its diagonal elements in the second row. As stressed in earlier
work involving sectoral data, notably by Shea (2002), it emerges distinctly
that the bulk of the variation in aggregate employment is associated with
the covariance of sectoral employment growth rates rather than individual
sector variations in employment. The average pairwise correlation in sectoral
employment is positive at approximately 0.10 in quarterly data and 0.04 in
monthly data. Moreover, if one assumed that the co-movement in sectoral
employment growth is driven primarily by aggregate shocks, then Table 1
would immediately imply that these shocks represent the principal source
of variation in aggregate employment. For example, focusing on quarterly
growth rates, the fraction of aggregate employment variability explained by
aggregate shocks would amount roughly to 1 − (0.42 /1.52 ) or 0.93. This
calculation, of course, represents only an approximation in the sense that the
diagonal elements of ee would themselves partly reflect the effects of changes
2 As in Foerster, Sarte, and Watson (2008), time variation in the shares turns out to be
immaterial for the results we discuss in this article.
Malysheva and Sarte: Sectoral Employment and the Business Cycle
341
in aggregate conditions. That said, it does suggest, however, that despite clear
differences in employment growth variability at the individual sector level,
these differences, for the most part, vanish in aggregation and so become
easily overlooked.
2. A FACTOR ANALYSIS OF SECTORAL EMPLOYMENT
As discussed in Stock and Watson (2002), the approximate factor model provides a convenient means by which to capture the covariability of a large
number of time series using a relatively few number of factors. In terms
of our employment data, this model represents the N × 1 vector of sectoral
employment growth rates as
et = λFt + ut ,
(2)
where Ft is a k × 1 vector of unobserved factors common to all sectors, λ
is an N × k matrix of coefficients referred to as factor loadings, and ut is an
N × 1 vector of sector-specific idiosyncratic shocks that have mean zero. We
denote the number of time series observations in this article by T . Using (1),
the variance-covariance matrix of sectoral employment growth is now simply
given by
ee = λF F λ + uu ,
(3)
where F F and uu are the variance-covariance matrices of Ft and ut ,
respectively.
In classical factor analysis, uu is diagonal so that the idiosyncratic shocks
are uncorrelated across sectors. Stock and Watson (2002) weaken this assumption and show that consistent estimation of the factors is robust to weak
cross-sectional and temporal dependence in these shocks. Equation (2) can
be interpreted as the reduced form solution emerging from a more structural
framework (see Foerster, Sarte, and Watson 2008). Given this, features of
the economic environment that might cause the “uniquenesses,” ut , to violate
the weak cross-sectional dependence assumption include technological considerations, such as input-output (IO) linkages between sectors or production
externalities across sectors. In either case, idiosyncratic shocks to one sector may propagate to other sectors via these linkages and give rise to internal
co-movement that is ignored in factor analysis. Using sectoral data on U.S. industrial production, Foerster, Sarte, and Watson (2008) show that the internal
co-movement stemming from IO linkages in a canonical multisector growth
model is, in fact, relatively small. Hence, the factors in that case capture mostly
aggregate shocks rather than the propagation of idiosyncratic shocks by way
of IO linkages. Thus, for the remainder of this article, we shall interpret Ft as
capturing aggregate sources of variation in sectoral employment.
When N and T are large, as they are in this article, the approximate
factor model has proved useful because the factors can simply be estimated by
342
Federal Reserve Bank of Richmond Economic Quarterly
principle components (Stock and Watson 2002). By way of illustration, the
Appendix provides a brief description of the principle component problem and
its relationship to the approximate factor model (2). Bai and Ng (2002) further
show that penalized least-square criteria can be used to consistently estimate
the number of factors, and the estimation error in the estimated factors is
sufficiently small that it need not be taken into account in carrying out variance
decomposition exercises (Stock and Watson 2002).
Key Summary Statistics
Given equation (2), we shall summarize our findings in mainly two ways.
First, we compute the fraction of aggregate employment variability explained
by aggregate or common shocks, which we denote by R 2 (F). In particular,
since et = s et = s λFt + s ut , we have that
R 2 (F) =
s λF F λ s
.
σ 2e
(4)
For the 544 sectors that make up all goods and services at the five-digit level,
R 2 (F) then captures the degree to which fluctuations in aggregate employment
growth are driven by aggregate rather than sector-specific shocks. Second, we
also assess the extent to which aggregate shocks explain employment growth
variability in individual sectors. More specifically, denoting a typical equation
for sector i in (2) by
eit = λi Ft + uit ,
(5)
where λi represents the 1 × k vector of factor loadings specific to sector i and
uit denotes sector i’s idiosyncratic shocks, we compute
Ri2 (F) =
λi F F λi
,
σ 2ei
(6)
where σ 2ei is the variance of employment growth in sector i.
Note that the analysis yields an entire distribution of Ri2 (F) statistics, one
for each sector. Consider the degenerate case where Ri2 (F) = 1 for each
i. In this case, employment variations in each sector are completely driven
by the shocks common to all sectors and idiosyncratic shocks play no role.
Put another way, variations in aggregate employment reflect only aggregate
shocks and the fate of each sector is completely tied to these shocks. A direct
economic implication, therefore, is that the issue of market incompleteness
or insurance considerations (at the sectoral level) tend to become irrelevant
as there is no scope for diversifying away idiosyncratic shocks. To the extent
that factor loadings differ across sectors, aggregate shocks still affect sectoral employment differentially so that there may remain some opportunity to
complete markets. However, in the limit where λi = λj ∀i, j , the standard
Malysheva and Sarte: Sectoral Employment and the Business Cycle
343
Table 2 Decomposition of Variance from the Approximate Factor
Model
Monthly Growth Rates
Std. Dev. of et
Implied by Factor
Model
R 2 (F)
Quarterly Growth Rates
1
Factor
2
Factors
3
Factors
1
Factor
2
Factors
3
Factors
1.80
0.77
1.80
0.80
1.80
0.80
1.53
0.94
1.53
0.95
1.53
0.95
representative agent setup becomes a sufficient framework with which to study
business cycles (i.e., without loss of generality). In contrast, when Ri2 (F) < 1
for a subset of sectors, it is no longer true that the fortunes of individual sectors
are dictated only by aggregate shocks. Sector-specific shocks help determine
sectoral employment outcomes, and the degree of market completeness potentially plays an important part in determining the welfare implications of
business cycles.
3.
EMPIRICAL FINDINGS
Tables 2 through 4, as well as Figures 4 and 5, summarize the results from computing these key summary statistics using our data on sectoral employment
growth rates. We estimated the number of factors using the Bai and Ng (2002)
ICP1 and ICP2 estimators, both of which yielded three factors over the full
sample period. For robustness, Table 2 shows the factor model’s implied standard deviation of aggregate employment (computed using constant shares),
as well as the fraction of aggregate employment variability explained by the
common factors, R 2 (F), using either one, two, or three factors. Most of our
discussion will focus on the three-factor model. Two important observations
stand out in Table 2. First, the common factors explain essentially all of the
variability in quarterly employment growth rates. These common shocks also
explain the bulk, or more specifically 80 percent, of fluctuations in monthly
growth rates. Second, note that for both monthly and quarterly growth rates,
the first factor almost exclusively drives aggregate employment growth, with
the second and third factors contributing little additional variability to the aggregate series in relative terms. That is not to say that the absolute variance
of the latter factors is small, and we shall see below that these are essential in
helping track subsets of the sectors that make up total nonfarm employment.
At a more disaggregated level, Figure 4 illustrates the fraction of quarterly
employment growth variability in individual sectors that is attributable to common shocks or, alternatively, the distribution of Ri2 (F). As the figure makes
344
Federal Reserve Bank of Richmond Economic Quarterly
Figure 4 Contribution of Sector-Specific Shocks to Sectoral
Employment
2
5
Distribution of R i (f) for Sectoral Employment Growth (1990--2008), 3 Factors
25th Percentile = 0.12
Median = 0.27
Mean = 0.31
75th Percentile = 0.48
4
Percent
3
2
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2
R i (f)
clear, sector-specific shocks play a key role in accounting for employment
variations at the sectoral level, with common shocks explaining, on average,
only 31 percent of the variability in sectoral employment. In addition, observe that there exists substantial heterogeneity in the way that employment is
driven by aggregate and idiosyncratic shocks across sectors. Specifically, the
interquartile range suggests Ri2 (F) statistics that are between 0.12 to 0.48, or
a 0.36 point gap.
It may seem counterintuitive at first that R 2 (F) is close to 1 in Table 2 while
the mean or median Ri2 (F) statistic is considerably less than 1 in Figure 4. To
see the intuition underlying this result, consider equation (2) when aggregated
across sectors:
s et = s λFt + s ut .
(7)
When the number of sectors under consideration is large, as in this article,
the “uniquenesses” will tend to average out by the law of large numbers. Put
another way, since the
correlated across sectors and have
Nuit s are weakly
p
mean zero, s ut =
s
u
→
0
as N becomes large. This result
i=1 i it
Malysheva and Sarte: Sectoral Employment and the Business Cycle
345
Table 3 Fraction of Variability in Sectoral Employment Growth
Explained by Common Shocks
Sector
Residential Building Construction
Electrical Equipment Manufacturing
Wood Kitchen Cabinet and Countertop
Plumbing and HVAC Contractors
Printing and Related Support Activities
Other Building Material Dealers
Wireless Telecommunications Carriers
Construction Equipment
Plywood and Engineered Wood Products
Semiconductors and Electronic Components
Management of Companies and Enterprises
Electrical Contractors
Lumber and Wood
Metalworking Machinery Manufacturing
Electric Appliance and Other Electronic Parts
Ri2 (F)
0.85
0.85
0.84
0.84
0.80
0.80
0.78
0.78
0.77
0.77
0.77
0.77
0.77
0.76
0.76
holds provided that the distribution of sectoral shares is not too skewed so
that a few sectors
have very large weights (see Gabaix 2005). In contrast,
s λFt = Ft N
s
i=1 i λi does not necessarily go to zero with N since the λi s
are fixed parameters.3 Therefore, whatever the importance of idiosyncratic
shocks in driving individual sectors (i.e., whatever the distribution of Ri2 (F)),
R 2 (F) will generally tend towards 1 in large panels. The rate at which R 2 (F)
approaches 1 will depend on the particulars of the data-generating process. In
this case, with 544 sectors, we find that R 2 (F) is around 0.8 in monthly data
and 0.95 in quarterly data.
Interestingly, Figure 4 suggests that at the high end of the cross-sector
distribution of Ri2 (F) statistics, there exist individual sectors whose variation
in employment growth is almost entirely driven by the common shocks that
explain aggregate employment, and, thus, that are particularly vulnerable to
the business cycle. Table 3 lists the top 15 sectors in which idiosyncratic shocks
play the least role in relative terms. Note that all of the sectors listed in Table
3 are goods-producing sectors. In other words, even though service-providing
sectors employ more than four times as many workers as the goods-producing
sectors, it turns out that it is the latter sectors that are most informative about the
state of aggregate employment. In essence, because employment variations
in the sectors listed in Table 3 reflect mainly the effects of common shocks,
and because movements in aggregate employment growth are associated with
3 In Foerster, Sarte, and Watson (2008), the factor loadings correspond to reduced-form pa-
rameters that can be explicitly tied to the structural parameters of a canonical multi-sector growth
model.
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Federal Reserve Bank of Richmond Economic Quarterly
Table 4 Sectoral Information Content of Aggregate Employment
Selected Sectors Ranked by Ri2 (F)
Top 5 Sectors
Top 10 Sectors
Top 20 Sectors
Top 30 Sectors
Fraction of et Explained by Selected Sectors
0.88
0.92
0.94
0.96
these shocks (Table 2), information regarding aggregate employment tends to
be concentrated in these sectors.
This notion of sectoral concentration of information regarding aggregate
employment can be formalized further as follows. Consider the problem of
tracking movements in aggregate employment using only a subset, M, of the
available sectors, say the the five highest ranked sectors in Table 3. This
problem pertains, for example, to the design of surveys that are meant to track
aggregate employment in real time such as those carried out by the Institute for
Supply Management, as well as by various Federal Reserve Banks including
the Federal Reserve Bank of Richmond.4 In particular, the question is: Which
sectors are the most informative about the state of aggregate employment and
should be included in the surveys? To make some headway toward answering
t denote the vector of employment growth rates associated
this question, let e
t = met , where m is an M × N selection
with the M sectors such that e
matrix. To help track aggregate employment growth, s et , we compute the
M × 1 vector of weights, w, attached to the different employment growth
t as the orthogonal projection of s et on e
t . That is to say, the
series in e
weights are optimal in the sense of solving a standard least-square problem,
w = (mee m )−1 mee s.
Table 4 reports the fraction of aggregate employment growth explained by
the (optimally weighted) employment series related to various sector selections
t . Strikingly, using only the sectors associated with the
in our data set, w e
2
highest five Ri (F) statistics in Table 3, this particular filtering already helps
us explain 88 percent of the variability in aggregate employment growth.
Moreover, virtually all of the variability in aggregate employment growth is
accounted for by only considering the 30 highest ranked sectors, according to
Ri2 (F), out of 544 sectors. It is apparent, therefore, that information concerning
movements in aggregate employment growth is concentrated in a small number
of sectors. Contrary to conventional wisdom, these sectors are not necessarily
those that have the largest weights in aggregate employment nor the most
volatile employment growth series. Because aggregate employment growth
4 Employment numbers are typically released with a one-month lag and revised up to three
months after their initial release. In addition, a revision is carried out annually in March.
Malysheva and Sarte: Sectoral Employment and the Business Cycle
347
Figure 5 Distribution of Ri2 (F ) in Goods-Producing and
Service-Providing Sectors
Panel A: Goods-Producing Industries
12.5
25th Percentile = 0.20
Median = 0.43
Mean = 0.40
75th Percentile = 0.59
10.0
Percent
7.5
5.0
2.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2
R i (f)
Panel B: Service-Providing Industries
12.5
25th Percentile = 0.09
Median = 0.22
Mean = 0.26
75th Percentile = 0.40
10.0
Percent
7.5
5.0
2.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
R 2i(f)
is almost exclusively driven by common shocks, the factor analysis proves
useful precisely because it allows us to identify the individual sectors whose
employment growth also moves most closely with these shocks.
From the exercise we have just carried out, it should be clear that there is
much heterogeneity in the way that individual sector employment growth compares to aggregate employment growth over the business cycle. To underscore
this point, Figure 5 depicts the breakdown of Ri2 (F) statistics across the main
sectors that make up total goods and services separately. Differentiating between goods-producing and service-providing industries, Figure 5 shows that
aggregate shocks play a lesser role in driving employment variations in the
service sectors relative to the goods-producing sectors. In particular, both the
mean and median Ri2 (F) statistics are notably lower in the service-providing
348
Federal Reserve Bank of Richmond Economic Quarterly
industries than in the goods sectors. That said, it is also the case that there isn’t
much uniformity within the goods-producing sectors. In particular, we find
that employment variations in the durable goods sectors are significantly more
subject to common shocks than in the nondurable goods sectors. The median
Ri2 (F) statistic is 0.54 in durable goods but only 0.20 in the nondurable goods
sectors. In service-providing industries, we find that sector-specific shocks
generally play a much greater role in determining employment growth variations. Moreover, the distributions of Ri2 (F) tend to be more similar across
service sectors than they are across goods-producing industries. The smallest
median Ri2 (F) value across private industries is 0.19, in financial activities,
while the largest value is relatively close at 0.29, in the information sector. As
indicated above, although employment variations in individual sectors tend to
be dominated by sector-specific shocks, these shocks tend to lose their importance in aggregation. To further illustrate this notion, let sj denote a vector
comprising either the shares corresponding to a particular subsector j of total
goods and services, say goods-producing sectors, or zero otherwise. In other
words, sj effectively selects out employment growth in the different industries
making up subsector j . It follows that employment growth in that subsector
is given by sj et , and the corresponding factor component in that subsector
is sj λFt . Note that to the degree s λFt successfully captures the business
cycle as it relates to movements in aggregate employment, sj λFt captures the
analogous concept at a more disaggregated level.
Figures 6 and 7 depict the behavior of sj et and sj λFt for the various
sectoral components of our data. Despite the heterogeneity in sectoral employment across sectors as captured by Ri2 (F), the figures suggest that employment
growth generally follows movements in the factor component not only at the
aggregate level but in subsectors of the economy as well. Of course, at the
aggregate level, we have argued that this is to be expected given the results in
Table 2 and confirmed in Figure 6. However, we also find that employment
growth and the factor component generally move together in goods-producing
and service-providing industries separately (Figure 7). In fact, this finding is
also true of the main subsectors that make up total goods and services, with the
notable exception of government. Perhaps not surprisingly, the latter finding
simply reflects the lack of a business cycle component in government services
relative to other sectors. Consistent with our earlier findings, our work additionally suggests that employment growth moves less closely with the factor
component in service-providing industries than in goods-producing sectors,
notably in financial services for instance. On the whole, however, the factor analysis appears to provide a helpful way to track the business cycle as it
relates to employment in the broad sectoral components of goods and services.
Finally, we note that the factors uncovered in this analysis play substantially different roles in explaining aggregate and sectoral variations in employment. Specifically, even though the first factor alone explains roughly 94
Malysheva and Sarte: Sectoral Employment and the Business Cycle
349
Figure 6 Aggregate Employment Growth and Factor Component
3
Aggregate Employment = solid
Factor Component = dashed
2
Percent
1
0
-1
-2
-3
-4
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
Date
percent of the variation in aggregate employment growth (Table 2), this factor
does very little to explain employment growth in particular sectoral components of goods and services. To see this, Figure 8 shows plots of employment
growth in natural resources and mining, as well as education and health services, against the factor component using one, two, and three factors. In the
first row of Figure 8, we see unambiguously that, despite accounting for the
bulk of the variations in aggregate employment, the first factor does very little to capture employment variations in either of the sectors. The correlation
between the factor component and employment growth is virtually nil at 0.03
in natural resources and mining and 0.08 in education and health services. In
sharp contrast, this correlation jumps to 0.57 in education and health once the
second factor is included, and to 0.77 in natural resources and mining once the
third factor is included. Note, in particular, that the second factor does little
to capture employment growth in natural resources and mining, and it is the
third factor alone that helps capture business cycle movements in employment
in that sector. In that sense, the Bai and Ng (2002) ICP1 and ICP2 estimators
help identify factors that not only explain aggregate employment variations
but also account for employment movements at a more disaggregated level.
350
Federal Reserve Bank of Richmond Economic Quarterly
Figure 7 Employment Growth and Factor Component in Goods and
Services
Panel A: Goods-Producing Industries
1.2
Employment = solid
Factor Component = dashed
0.8
Percent
0.4
0.0
-0.4
-0.8
-1.2
-1.6
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
Date
Panel B: Service-Providing Industries
2.0
Employment = solid
Factor Component = dashed
1.5
1.0
Percent
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
Date
4.
CONCLUSIONS
In the standard neoclassical one-sector growth model, fluctuations in the representative agent’s circumstances are largely determined by shocks to aggregate
total factor productivity. This notion is developed, for example, in work going
as far back as King, Plosser, and Rebelo (1988). The assumption of a representative agent stands in for a potentially more complicated world populated
by heterogenous agents, but where homothetic preferences and complete markets justify focusing on the average agent. Alternatively, we can also think
of the representative agent framework as approximating a world in which all
agents are essentially identical and affected in the same way by shocks to the
economic environment. Under the latter interpretation, a boom in the course
of a business cycle characterizes a situation in which “a rising tide lifts all
Malysheva and Sarte: Sectoral Employment and the Business Cycle
351
Figure 8 Contribution of Individual Factors in Explaining Sectoral
Employment Growth
Education and Health Services: 1 Factor
Natural Resources and Mining: 1 Factor
0.4
0.06
0.3
0.02
0.2
Percent
Percent
Correlation = 0.03
0.04
0.00
-0.02
0.1
0.0
-0.04
-0.1
-0.06
-0.2
-0.08
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
Correlation = 0.08
-0.3
1990
1992
1994
1996
1998
Date
Natural Resources and Mining: 2 Factors
Correlation = 0.09
Percent
Percent
2008
2010
0.2
-0.02
0.1
0.0
-0.04
-0.1
-0.06
-0.2
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
-0.3
1990
1992
1994
1996
1998
Date
2000
2002
2004
2006
2008
2010
Date
Natural Resources and Mining: 3 Factors
Education and Health Services: 3 Factors
0.4
0.06
Correlation = 0.77
0.04
0.3
0.02
0.2
Percent
Percent
2006
Correlation = 0.57
0.3
0.00
0.00
-0.02
0.1
-0.1
-0.06
-0.2
1992
1994
1996
1998
2000
Date
2002
2004
2006
2008
2010
Correlation = 0.74
0.0
-0.04
-0.08
1990
2004
Education and Health Services: 2 Factors
0.02
-0.08
1990
2002
0.4
0.06
0.04
2000
Date
-0.3
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
Date
boats,” and vice versa in the case of a recession. Put another way, idiosyncratic shocks play no role in determining agents’ outcomes. More importantly,
when individual agents’ fortunes are driven mainly by common shocks, the
significance of market incompleteness and the importance of insurance considerations tend to vanish since there is no scope for diversifying idiosyncratic
shocks away.
352
Federal Reserve Bank of Richmond Economic Quarterly
Using factor analytic methods, this article documents instead significant
differences in employment variations across sectors. In some industries, notably in goods production, variations in employment growth are dominated by
aggregate shocks so that these sectors are particularly sensitive to the business
cycle. In other industries, in particular some service-providing industries, employment movements are virtually unrelated to aggregate shocks and instead
result almost exclusively from sector-specific shocks. The analysis, therefore, suggests that agents working in different sectors of the U.S. economy
are affected in very different ways by shocks to the economic environment.
Moreover, it underscores the potential importance of market incompleteness
and mitigates the usefulness of representative agent models in determining the
welfare costs of business cycles.
APPENDIX
This Appendix gives a brief description of the Principle Component (PC)
problem based on the discussion in Johnston (1984). See that reference for a
more detailed presentation of the problem and its implications.
As described in the main text, suppose we have (demeaned) employment
growth observations across N sectors over T time periods summarized in an
N × T matrix, X. In that way, et in the text is a typical column of X.
The nature of the PC problem is to capture the degree of co-movement across
these N sectors in a simple and convenient way. To this end, the PC problem
transforms the Xs into a new set of variables that will be pairwise uncorrelated
and of which the first will have maximum possible variance, the second the
maximum possible variance among those uncorrelated with the first, and so
on.
Let
F1 = X λ1
denote the first such variable where λ1 and F1 are N × 1 and T × 1 vectors,
respectively. In other words, F1 is a linear combination of the elements of X
across sectors. The sum of squares of F1 is
F1 F1 = λ1 XX λ1 ,
(8)
where XX = XX represents the variance-covariance matrix (when divided
by T ) of employment growth rates across sectors. We wish to choose the
weights λ1 to maximize F1 F1 , but some constraint must evidently be imposed
on λ1 to prevent the sum of squares from being made infinitely large. Thus, a
Malysheva and Sarte: Sectoral Employment and the Business Cycle
353
convenient normalization is to set
λ1 λ1 = 1.
The PC problem may now be stated as
max λ1 XX λ1 + μ1 (1 − λ1 λ1 ),
λ1
where μ1 is a Lagrange multiplier. Using the fact that XX is a symmetric
matrix, the first-order condition associated with this problem is
2xx λ1 − 2μ1 λ1 = 0.
Thus, it follows that
xx λ1 = μ1 λ1 .
In other words, the weights λ1 are given by an eigenvector of xx with corresponding eigenvalue μ1 . Observe that when λ1 is chosen in this way, the sum
of squares in (8) reduces to
λ1 XX λ1 = λ1 μ1 λ1 = μ1 .
Therefore, our choice of λ1 must be the eigenvector associated with the largest
eigenvalue of XX . The first principle component of X is then F1 .
Now, let us define the next principle component of X as F2 = X λ2 .
Similar to the choice of λ1 we have just described, the problem is to choose
the weights λ2 so as to maximize λ2 XX λ2 subject to λ2 λ2 = 1. In addition,
however, because we want the second principle component to capture comovement that is not already reflected in the first principle component, we
impose the further restriction λ2 λ1 = 0. This last restriction ensures that F2
will be uncorrelated with F1 .
The problem associated with the second principle component may then
be stated as
max λ2 XX λ2 + μ2 (1 − λ2 λ2 ) + φλ2 λ1 .
λ2
The corresponding first-order condition is
2xx λ2 − 2μ2 λ2 + φλ1 = 0.
Pre-multiplying this last equation by λ1 gives
2λ1 XX λ2 − 2μ2 λ1 λ2 + φλ1 λ1 = 0,
or
φ = 0,
λ1 λ1
λ1 XX
since
= 1,
=
weights λ2 must satisfy
μ1 λ1 ,
and λ1 λ2 = 0. Therefore, we have that the
xx λ2 = μ2 λ2 ,
354
Federal Reserve Bank of Richmond Economic Quarterly
and, in particular, should be chosen as the eigenvector associated with the
second largest eigenvalue of XX .
Proceeding in this way, suppose we find the first k principle components
of X. We can arrange the weights λ1 , λ2 ,...,λk in the N × k orthogonal matrix
k = [λ1 , λ2 , ..., λk ].
Furthermore, the general PC problem may then be described as finding the
T × k matrix of components, F = X k , such that k solves
max k XX k subject to k k = Ik .
k
(9)
Now, consider the approximate factor model (2) in the text written in matrix
form,
X = k F + u,
where X is N × T , k is a N × k matrix of factor loadings, F is a k × T
matrix of latent factors, and u is N × T . One can then show that solving the
constrained least-square problem,
min
{F1 }Tt=1 ,...,{Fk }Tt=1 ,k
T
(Xt − k Ft ) (Xt − k Ft ) subject to k k = Ik ,
t=1
is equivalent to solving the general principle component problem (9) we have
just described (see Stock and Watson 2002).
REFERENCES
Bai, Jushan, and Serena Ng. 2002. “Determining the Number of Factors in
Approximate Factor Models.” Econometrica 70 (January): 191–221.
Foerster, Andrew, Pierre-Daniel Sarte, and Mark W. Watson. 2008. “Sectoral
vs. Aggregate Shocks: A Structural Factor Analysis of Industrial
Production,” Working Paper 14389. Cambridge, Mass.: National Bureau
of Research (October).
Forni, Mario, and Lucrezia Reichlin. 1998. “Let’s Get Real: A Factor
Analytical Approach to Disaggregated Business Cycle Dynamics.”
Review of Economic Studies 65 (July): 453–73.
Gabaix, Xavier. 2005. “The Granular Origins of Aggregate Fluctuations.”
Manuscript, Massachusetts Institute of Technology.
Johnston, Jack. 1984. Econometric Methods, third edition. New York:
McGraw-Hill Book Company.
Malysheva and Sarte: Sectoral Employment and the Business Cycle
355
King, Robert, Charles Plosser, and Sergio Rebelo. 1988. “Production,
Growth and Business Cycles: The Basic Neoclassical Model.” Journal
of Monetary Economics 21: 195–232.
Quah, Danny, and Thomas J. Sargent. 1993. “A Dynamic Index Model for
Large Cross Sections.” In Business Cycles, Indicators and Forecasting,
edited by J. H. Stock and M. W. Watson. Chicago: University of Chicago
Press for the NBER, 285–310.
Shea, John. 2002. “Complementarities and Comovements.” Journal of
Money, Credit, and Banking 34 (May): 412–34.
Stock, J. H., and M. W. Watson. 2002. “Forecasting Using Principal
Components from a Large Number of Predictors.” Journal of the
American Statistical Association 97 (December): 1,167–79.
Economic Quarterly—Volume 95, Number 4—Fall 2009—Pages 357–382
Inventories and Optimal
Monetary Policy
Thomas A. Lubik and Wing Leong Teo
I
t has long been recognized that inventory investment plays a large role in
explaining fluctuations in real gross domestic product (GDP), although it
makes up only a small fraction of it. Blinder and Maccini (1991) document
that in a typical recession in the United States, the fall in inventory investment
accounts for 87 percent of the decline in output despite being only one half of
1 percent of real GDP. A lot of research has been trying to explain how this
seemingly insignificant component of GDP has such a disproportionate role in
business cycle fluctuations.1 However, surprisingly few studies have focused
on the conduct of monetary policy when firms can invest in inventories. In this
article we attempt to fill this gap by investigating how inventory investment
affects the design of optimal monetary policy.
We employ the simple New Keynesian model that has become the benchmark for analyzing monetary policy from both a normative and a positive
perspective. We introduce inventories into the model by assuming that the inventory stock facilitates sales, as suggested in Bils and Kahn (2000). We first
establish that the dynamics, and therefore the monetary transmission mechanism, differ between the models with and without inventories for a given
behavior of the monetary authority. Monetary policy is then endogenized by
assuming that policymakers solve an optimal monetary policy problem.
First, we compute the optimal Ramsey policy. A Ramsey planner maximizes the welfare of the agents in the economy by taking into account the
We are grateful to Andreas Hornstein, Pierre Sarte, Alex Wolman, and Nadezhda Malysheva,
whose comments greatly improved the paper. Lubik is a senior economist at the Federal
Reserve Bank of Richmond. Teo is an assistant professor at National Taiwan University.
Lubik wishes to thank the Department of Economics at the University of Adelaide, where
parts of this research were conducted, for their hospitality. The views expressed in this paper
are those of the authors and should not necessarily be interpreted as those of the Federal Reserve Bank of Richmond or the Federal Reserve System. E-mails: thomas.lubik@rich.frb.org;
wlteo@ntu.edu.tw.
1 See Ramey and West (1999) and Khan (2003) for extensive surveys of the literature.
358
Federal Reserve Bank of Richmond Economic Quarterly
private sector’s optimality conditions. In doing so, the planner chooses a socially optimal allocation. While this does not necessarily bear any relationship
to the typical conduct of monetary policymakers, it provides a useful benchmark. Subsequently, we study optimal policy when the planner is constrained
to implement simple rules. That is, we specify a set of rules that lets the policy
instrument (the nominal interest rate) respond to target variables such as the
inflation rate and output. The policymaker chooses the respective response
coefficients that maximize welfare. Optimal rules of this kind may be preferable to Ramsey plans from an actual policymaker’s perspective since they can
be operationalized and are easier to communicate to the public.
Our most interesting but surprising finding is that Ramsey-optimal monetary policy deviates from full inflation stabilization in our model with inventories. This stands in contrast to the standard New Keynesian model. In the
New Keynesian model, perfectly stable inflation is optimal since movements
in prices represent deadweight costs to the economy. Introducing inventories
potentially modifies that basic calculus for the following reasons. First, we
assume that a firm’s inventory holdings are relevant for its sales only in relative terms, that is, when they deviate from the aggregate inventory stock. This
presents an externality, which a Ramsey planner may want to address. Second,
inventories change the economy’s propagation mechanism as they allow firms
to smooth sales over time with concomitant effects on consumption; that is,
output and consumption need no longer coincide, which has a similar effect
as capital in that it provides future consumption opportunities. Changes in
prices serve as the equilibrating mechanism for the competing goals of reducing consumption volatility and avoiding price adjustment costs. The inventory
specification therefore contains something akin to an inflation-output tradeoff. Consequently, the optimal policy no longer fully stabilizes inflation. The
second important finding concerns the efficacy of implementing simple rules.
Similar to most of the optimal policy literature, we show that simple rules
can come exceedingly close to the socially optimal Ramsey policy in welfare
terms.
Our article relates to two literatures. First, the amount of research on optimal monetary policy in the New Keynesian framework is very large already,
and we do not have much to contribute conceptually to the modeling of optimal
policy. Schmitt-Grohé and Uribe (2007) is a recent important and comprehensive contribution. A main conclusion from this literature is that optimal
monetary policy will choose to almost perfectly stabilize inflation. In environments with various nominal and real distortions, this policy prescription
becomes slightly modified, but nevertheless perseveres. We thus contribute
to the optimal policy literature by demonstrating that the results carry over to
a framework with another, previously unconsidered modification to the basic
framework in the form of inventories.
T. A. Lubik and W. L. Teo: Inventories and Optimal Monetary Policy
359
The study of inventory investment has a long pedigree, to which we cannot
do full justice here. Much of the earlier literature, as surveyed in Blinder and
Maccini (1991), was concerned with identifying the determinants of inventory investment, such as aggregate demand and expectations thereof, or the
opportunity costs of holding inventories. Most work in this area was largely
empirical using semi-structural economic models, with West (1986) being a
prime example.2 Almost in parallel to this more explicitly empirical literature,
inventories were introduced into real business cycle models. The seminal article by Kydland and Prescott (1982) introduces inventories directly into the
production function. More recent contributions include Christiano (1988),
Fisher and Hornstein (2000), and Khan and Thomas (2007). The latter two
articles especially build a theory of a firm’s inventory behavior on the microfoundation of an S-s environment. The focus of these articles is on the business
cycle properties of inventories, in particular the high volatility of inventory
investment relative to GDP and the countercylicality of the inventory-sales
ratio, both of which are difficult to match in typical inventory models. In an
important article, Bils and Kahn (2000) demonstrate that time-varying and
countercyclical markups are crucial for capturing this co-movement pattern.
This insight lends itself to considering inventory investment within a New
Keynesian framework since it features interplay between marginal cost, inflation, and monetary policy, which might therefore be a source of inventory
fluctuations.3 Recently, several articles have introduced inventories into New
Keynesian models. Jung and Yun (2005) and Boileau and Letendre (2008)
both study the effects of monetary policy from a positive perspective. The
former combines Calvo-type price setting in a monopolistically competitive
environment with the approach to inventories as introduced by Bils and Kahn
(2000). The use of the Calvo approach to modeling nominal rigidity allows
these authors to discuss the importance of strategic complementarities in price
setting. Boileau and Letendre (2008), on the other hand, compare various approaches to introducing inventories in a sticky-price model. This article is
differentiated from those contributions by its focus on the implications of
inventories as a transmission mechanism for optimal monetary policy.
The rest of the article is organized as follows. In the next section we
develop our New Keynesian model with inventories. Section 2 analyzes the
differences between the standard New Keynesian model and our specification
with inventories. We calibrate both models and compare their implications for
business cycle fluctuations. We present the results of our policy exercises in
2 A more recent example of applying structural econometric techniques to partial equilibrium
inventory models is Maccini and Pagan (2008).
3 Incidentally, Maccini, Moore, and Schaller (2004) find that an inventory model with regime
switches in interest rates is quite successful in explaining inventory behavior despite much previous
empirical evidence to the contrary. The key to this result is the exogenous shift in interest rate
regimes, which lines up with breaks in U.S. monetary policy.
360
Federal Reserve Bank of Richmond Economic Quarterly
Section 3, which also includes a robustness analysis with respect to changes
in the parameterization. Section 4 concludes with a brief discussion of the
main results and suggestions for future research.
1. THE MODEL
We model inventories in the manner of Bils and Kahn (2000) as a mechanism
for facilitating sales. When firms face unexpected demand, they can simply
draw down their stock of previously produced goods and do not have to engage in potentially more costly production. This inventory specification is
embedded in an otherwise standard New Keynesian environment. There are
three types of agents: monopolistically competitive firms, a representative
household, and the government. Firms face price adjustment costs and use
labor for the production of finished goods, which can be sold to households or
added to the inventory. Households provide labor services to the firms and engage in intertemporal consumption smoothing. The government implements
monetary policy.
Firms
The production side of the model consists of a continuum of monopolistically
competitive firms, indexed by i ∈ [0, 1]. The production function of a firm i
is given by
yt (i) = zt ht (i) ,
(1)
where yt (i) is output of firm i, ht (i) is labor hours used by firm i, and zt is
aggregate productivity. We assume that it evolves according to the exogenous
stochastic process
ln zt = ρ z ln zt−1 + ε zt ,
(2)
where εzt is an i.i.d. innovation.
We introduce inventories into the model by assuming that they facilitate sales as suggested by Bils and Kahn (2000).4 In their partial equilibrium
framework, they posit a downward-sloping demand function for a firm’s product that shifts with the level of inventory available. As shown by Jung and
Yun (2005), this idea can be captured in a New Keynesian setting with monopolistically competitive firms by introducing inventories directly into the
4 This approach is consistent with a stockout avoidance motive.
Wen (2005) shows that
it explains the fluctuations of inventories at different cyclical frequencies better than alternative
theories.
T. A. Lubik and W. L. Teo: Inventories and Optimal Monetary Policy
Dixit-Stiglitz aggregator of differentiated products:
μ
1
at (i) θ
st (i)(θ−1)/θ di
st =
at
0
361
θ/(θ−1)
,
(3)
where st are aggregate sales; st (i) are firm-specific sales; at and at (i) are,
respectively, the aggregate and firm-specific stocks of goods available for
sales; θ > 1 is the elasticity of substitution between differentiated goods; and
μ > 0 is the elasticity of demand with respect to the relative stock of goods.
Holding inventories helps firms to generate greater sales at a given price since
they can rely on the stock of previously produced goods when, say, demand
increases. Note, however, that a firm’s inventory matters only to the extent that
it exceeds the aggregate level. In a symmetric equilibrium, having inventories
does not help a firm to make more sales, but it affects the firm’s optimality
condition for inventory smoothing.
Cost minimization implies the following demand function for sales of
good i:
at (i) μ Pt (i) −θ
st (i) =
st ,
(4)
at
Pt
where Pt (i) is the price of good i, and Pt is the price index for aggregate sales
st :
1
1/(1−θ)
at (i) μ
Pt =
Pt (i)1−θ di
.
(5)
at
0
A firm’s sales are thus increasing in its relative inventory holdings and decreasing in its relative price. The inventory term can alternatively be interpreted
as a taste shifter, which firms invest in to capture additional demand (see
Kryvtsov and Midrigan 2009). Finally, the stock of goods available for sales
at (i) evolves according to
at (i) = yt (i) + (1 − δ) (at−1 (i) − st−1 (i)) ,
(6)
where δ ∈ (0, 1) is the rate of depreciation of the inventory stock. It can also
be interpreted as the cost of carrying the inventory over the period.
Each firm faces quadratic costs for adjusting its price relative to the steady
2
t (i)
state gross inflation rate π : φ2 πPPt−1
− 1 st , with φ > 0, and π ≥ 1, the
(i)
steady state gross inflation rate. Note that the costs are measured in units of
aggregate sales instead of output since st is the relevant demand variable in the
model with inventories. Firm i’s intertemporal profit function is then given
by
2
∞
Pt+τ (i) st+τ (i) Wt+τ ht+τ (i) φ
Pt+τ (i)
Et
ρ t,t+τ
−
−
− 1 st+τ ,
Pt+τ
Pt+τ
2 π Pt+τ −1 (i)
τ =0
(7)
362
Federal Reserve Bank of Richmond Economic Quarterly
where Wt is the nominal wage and ρ t,t+τ is the aggregate discount factor that
a firm uses to evaluate profit streams.
Firm i chooses its price, Pt (i), labor input, ht (i), and stock of goods
available for sales, at (i), to maximize its expected intertemporal profit (7),
subject to the production function (1), the demand function (4), and the law
of motion for at (i) (6). The first order conditions are
Pt (i)
st
st (i)
φ
−1
= (1 − θ)
π Pt−1 (i)
π Pt−1 (i)
Pt
st (i)
st+1 Pt+1 (i)
Pt+1 (i)
+
−
δ)
θ
+Et ρ t,t+1 φ
−1
mc
(i)
(8)
(1
t+1
π Pt (i)
Pt (i)
π Pt2 (i)
Wt
= zt mct (i),
Pt
(9)
and
Pt (i) st (i)
st (i)
mct (i) = μ
+ (1 − δ) 1 − μ
Et ρ t,t+1 mct+1 (i),
Pt at (i)
at (i)
(10)
where mct (i) is the Lagrange multiplier associated with the demand constraint
(4). It can also be interpreted as real marginal cost.
Equation (8) is the optimal price-setting condition in our model with inventories. It resembles the typical optimal price-setting condition in a New
Keynesian model with convex costs for price adjustment (e.g., Krause and
Lubik 2007), except that marginal cost now enters the optimal pricing condition in expectations because of the presence of inventories. In this model,
the behavior of marginal cost, mc, can be interpreted from two different directions. As captured by Equation (9), it is the ratio of the real wage to the
marginal product of labor, which in the standard model is equal to the cost of
producing an additional unit of output. Alternatively, it is the cost of generating an additional unit of goods available for sale, which can either come out of
current production or out of (previously) foregone sales. This in turn reduces
the stock of goods available for sales in future periods, which would eventually
have to be replenished through future production. This intertemporal tradeoff
between current and future marginal cost is captured by Equation (10).
Household
We assume that there is a representative household in the economy. It
maximizes expected intertemporal utility, which is defined over aggregate
T. A. Lubik and W. L. Teo: Inventories and Optimal Monetary Policy
363
consumption,5 ct , and labor hours, ht :
E0
∞
1+η
β
t
t=0
h
ζ t ln ct − t
,
1+η
(11)
where η ≥ 0 is the inverse of the Frisch labor supply elasticity.
ζ t is a preference shock and is assumed to follow the exogenous AR(1)
process
ln ζ t = ρ ζ ln ζ t−1 + ε ζ ,t ,
(12)
where 0 < ρ ζ < 1 and εζ ,t is an i.i.d. innovation.
The household supplies labor hours to firms at the nominal wage rate,
Wt , and earns dividend income, Dt , (which is paid out of firms’ profits) from
owning the firms. It can purchase one-period discount bonds, Bt , at a price of
1/Rt , where Rt is the gross nominal interest rate. Its budget constraint is
Pt ct + Bt /Rt ≤ Bt−1 + Wt ht + Dt .
(13)
The first-order conditions for the representative household’s utility maximization problem are
η
ht =
ζ t Wt
, and
ct Pt
ζt
= βRt Et
ct
ζ t+1 Pt
.
ct+1 Pt+1
(14)
(15)
Equation (14) equates the real wage, valued in terms of the marginal utility of consumption, to the disutility of labor hours. Equation (15) is the
consumption-based Euler equation for bond holdings.
Government and Market Clearing
In order to close the model, we also need to specify the behavior of the monetary authority. The main focus of the paper is the optimal monetary policy in
the New Keynesian model with inventories. In the next section, however, we
briefly compare our specification to the standard model without inventories
in order to assess whether introducing inventories significantly changes the
model dynamics. We do this conditional for a simple, exogenous interest rate
feedback rule that has been used extensively in the literature:
t = ρ R
t−1 + ψ 1
R
π t + ψ 2
yt + ε R,t ,
(16)
5 Consumption can be thought of as a Dixit-Stiglitz aggregate, as is typical in New Keynesian
models. We abstract from this here for ease of exposition.
364
Federal Reserve Bank of Richmond Economic Quarterly
where a tilde over a variable denotes its log deviation from its deterministic
steady state. ψ 1 and ψ 2 are monetary policy coefficients and 0 < ρ < 1 is the
interest smoothing parameter. ε R,t is a zero mean innovation with constant
variance; it is often interpreted as a monetary policy implementation error.
Finally, we impose a symmetric equilibrium, so that the firm-specific indices,
i, can be dropped. In addition, we assume that bonds are in zero net supply,
Bt = 0. Market clearing in the goods market requires that consumption,
together with the cost for price adjustment, equals aggregate sales:
st = ct +
φ πt
−1
2 π
2
st .
(17)
2. ANALYZING THE EFFECTS OF MONETARY POLICY
The main focus of this article is how the introduction of inventories into an
otherwise standard New Keynesian framework changes the optimal design of
monetary policy. However, we begin by briefly comparing the behavior of
the model with and without inventories to assess the changes in the dynamic
behavior of output and inflation, given the exogenous policy rule (16). The
standard New Keynesian model differs from our model with inventories in
the following respects. First, there is no explicit intertemporal tradeoff in
terms of marginal cost as in equation (10). This implies, secondly, that the
driving term in the Phillips curve (8) is current marginal cost, as defined by
equation (9). Finally, in the standard model, consumption, output, sales, and
goods available of sales are first-order equivalent. We note, however, that
the standard specification is not nested in the model with inventories; that is,
the equation system for the latter does not reduce to the former for a specific
parameterization.
Calibration
The time period corresponds to a quarter. We set the discount factor, β,
to 0.99. Since price adjustment costs are incurred only for deviations from
steady-state inflation, its value is irrelevant for first-order approximations of
the model’s equation system but plays a role when we perform the optimal
policy analysis. We therefore set π = 1.0086 to be consistent with the average
post-war, quarter-over-quarter inflation rate. In the baseline calibration, we
choose a fairly elastic labor supply and set η = 1, which is a common value
in the literature and corresponds to quadratic disutility of hours worked. We
impose a steady-state markup of 10 percent, which implies θ = 11. The
price adjustment cost parameter is then calibrated so that η(θ − 1)/φ = 0.1,
as in Ireland (2004). This is a typical value for the coefficient on marginal
T. A. Lubik and W. L. Teo: Inventories and Optimal Monetary Policy
365
cost in the standard New Keynesian Phillips curve.6 The parameters of the
monetary policy rule are chosen to be broadly consistent with the empirical
Taylor rule literature for a unique equilibrium. That is, ψ 1 and ψ 2 are set to
0.45 and 0, respectively, while the smoothing parameter is set to ρ = 0.7. This
choice corresponds to an inflation coefficient of 0.45/0.3 = 1.5 that obeys the
Taylor principle. We specify the policy rule in this manner since it allows us
to analyze later the effects of inertial and super-inertial rules with ρ ≥ 1.
The persistence of the technology shock and the preference shock are
both set to ρ z = ρ ζ = 0.95. The standard deviation of the productivity
innovation is then chosen so as to match the standard deviation of HP-filtered
U.S. GDP of 1.61 percent. This yields a value of σ z = 0.005. We set the
standard deviation of the preference shocks at three times the value of the
former, which is consistent with empirical estimates from a variety of studies
(e.g., Ireland 2004). In the same manner, we choose a standard deviation of
the monetary policy shock of 0.003. The parameters related to inventories, μ
and δ, are calibrated following Jung and Yun (2005); specifically the elasticity
of demand with respect to the stock of goods available for sales is μ = 0.37,
while the depreciation rate of the inventory stock is δ = 0.01.
Do Inventories Make a Difference?
To get an idea how the introduction of inventories changes the model dynamics, we compare the responses of some key variables to technology, preference,
and monetary policy shocks for the specification with and without inventories.
The impulse responses are found in Figures 1–3, respectively. In the figures,
the label “Base” refers to the responses under the specification without inventories, while “Inv” indicates the inventory specification. The key qualitative
difference between the two models is the behavior of labor hours. In response
to a persistent technology shock, labor increases in the model with inventories,
while it falls in the standard New Keynesian model before quickly returning
to the steady state.7 In the New Keynesian model, firms can increase production even when economizing on labor because of the higher productivity
level. There is further downward pressure on labor since the productivity
shock raises the real wage. Higher output is reflected in a drop in prices,
which are drawn out over time due to the adjustment costs, and marginal cost
falls strongly.
The presence of inventories, however, changes this basic calculus as firms
can use inventories to take advantage of current low marginal cost. With
6 This value is also consistent with an average price duration of about four quarters in the
Calvo model of staggered price adjustment.
7 Chang, Hornstein, and Sarte (2009) also emphasize that in the presence of nominal rigidities
labor hours can increase in response to a persistent technology shock when firms hold inventories.
366
Federal Reserve Bank of Richmond Economic Quarterly
Figure 1 Impulse Response Functions to Productivity Shock
Sales and Sales-Stock Ratio
1.0
Output
1.5
y Inv
y Base
0.0
% Deviation
% Deviation
0.5
s
γ
-0.5
1.0
0.5
-1.0
0.0
-1.5
0
5
10
15
20
25
0
5
Labor Hours
20
25
-0.02
h Inv
h Base
-0.04
% Deviation
0.2
% Deviation
15
Interest Rate
0.3
0.1
0.0
-0.1
-0.2
-0.06
-0.08
R Inv
R Base
-0.10
0
5
10
15
20
0
25
5
10
15
20
25
Marginal Cost
Inflation Rate
0.00
0.0
-0.05
-0.1
% Deviation
% Deviation
10
-0.10
-0.15
π Inv
π Base
-0.2
-0.3
mc Inv
mc Base
-0.20
-0.4
0
5
10
15
20
25
0
5
10
15
20
25
inventory accumulation firms need not sell the additional output immediately,
which prompts them to increase labor input. Consequently, output rises by
more than in the standard model and the excess production is put in inventory.
The stock of goods available for sales thus rises, whereas the sales-to-stock
ratio, γ t ≡ st /at , falls. This is also reflected in the (albeit small) fall in
marginal cost, which is, however, persistent and drawn out. In other words,
firms use inventories to take advantage of current and future low marginal cost.
Inflation moves in the same direction as in the standard model, but is much
smoother, as the increased output does not have to be priced immediately.
This behavior is just the flip side of the smoothing of marginal cost.
T. A. Lubik and W. L. Teo: Inventories and Optimal Monetary Policy
367
Figure 2 Impulse Response Functions to Preference Shock
Output
Sales and Sales-Stock Ratio
0.8
0.6
% Deviation
% Deviation
0.4
s
0.2
γ
0.4
0.3
0.2
0.0
0.1
0
5
10
15
20
25
0
5
10
% Deviation
h Inv
h Base
0.4
0.3
25
R Inv
0.04
R Base
0.03
0.02
0.2
0.1
0.01
0
5
10
15
20
25
0
5
10
15
20
25
Marginal Cost
Inflation Rate
0.08
0.20
π Inv
π Base
mc Inv
0.15
% Deviation
0.06
% Deviation
20
0.05
0.5
0.04
0.02
0.00
15
Interest Rate
Labor Hours
0.6
% Deviation
y Inv
y Base
0.5
0.6
mc Base
0.10
0.05
0.00
0
5
10
15
20
25
0
5
10
15
20
25
In response to a preference shock, hours move in the same direction in
both models. However, the response with inventories is smaller since firms can
satisfy the additional demand out of their inventory holdings, which therefore
does not drive up marginal cost as much. Compared to the standard model,
firms do not have to resort to increases in price or labor input to satisfy the
additional demand. Inventories are thus a way of smoothing revenue over time,
which is also consistent with a smoother response of inflation. The dynamics
following a contractionary policy shock are qualitatively similar to those of
technology shocks in terms of co-movement. Sales in the inventory model
fall, but output and hours increase to take advantage of the falling marginal
368
Federal Reserve Bank of Richmond Economic Quarterly
Figure 3 Impulse Response Functions to Monetary Policy Shock
Output
Sales and Sales-Stock Ratio
1.0
1
0.5
% Deviation
% Deviation
0
-1
s
-2
γ
0.0
-0.5
-1.0
y Inv
-1.5
-3
y Base
-2.0
0
5
10
15
20
0
25
5
0.8
0.5
0.6
0.0
-0.5
-1.0
h Inv
h Base
-1.5
-2.0
0
25
R Inv
0.4
0.2
0.0
-0.2
5
10
15
20
0
25
5
10
15
20
25
Marginal Cost
Inflation Rate
0
-0.2
-1
% Deviation
% Deviation
20
R Base
0.0
-0.4
π Inv
π Base
-0.6
-0.8
15
Interest Rate
1.0
% Deviation
% Deviation
Labor Hours
10
-2
mc Inv
-3
mc Base
-4
0
5
10
15
20
25
0
5
10
15
20
25
cost. All series are again noticeably smoother when compared to the standard
model.
We now briefly discuss some business cycle implications of the inventory model.8 Table 1 shows selected statistics for key variables. A notable
stylized fact in U.S. data is that production is more volatile than sales. We
find that our inventory model replicates this observation in the case of productivity shocks, that is, output is 30 percent more volatile. This implies that
8 This aspect is discussed more extensively in Boileau and Letendre (2008) and Lubik and
Teo (2009).
T. A. Lubik and W. L. Teo: Inventories and Optimal Monetary Policy
369
Table 1 Business Cycle Statistics
Moments
Standard Deviation (%)
Output
Sales
Hours
Correlation
Sales )
(Sales, Inventory
Sales , Marginal Cost
( Stock
Technology
Preference
Policy
All Shocks
1.61
1.18
0.25
1.93
2.37
1.93
0.23
0.74
0.23
2.52
2.80
2.02
−0.85
0.87
0.51
0.49
0.95
0.90
0.72
0.49
consumption, which is equal to sales in our linearized setting, is also less
volatile than GDP. The introduction of inventories is thus akin to the modeling of capital and investment in breaking the tight link between output and
consumption embedded in the standard New Keynesian model. However,
the model has counterfactual implications for the co-movement of inventory
variables. Sales are highly negatively correlated with the sales-inventory ratio, whereas in the data the two series co-move slightly positively and are
at best close to uncorrelated. This finding can be overturned when either
preference or policy shocks are used, both of which imply a strong positive
co-movement. However, in the case of policy shocks, sales are counterfactually more volatile than output. When all shocks are considered together, we
find that co-movement between the inventory variables are positive, but not
unreasonably so, while sales are slightly more volatile than output.
The model also has implications for inflation dynamics. Most notably,
inflation is less volatile in the inventory specification than in the standard
model. In the New Keynesian model, inflation is driven by marginal cost;
hence, the standard model predicts that the two variables are highly correlated.
In the data, however, proxies for marginal cost, such as unit labor cost or the
labor share, co-move only weakly with inflation. This has been a challenge
for empirical studies of the New Keynesian Phillips curve. Our model with
inventories may, however, improve the performance of the Phillips curve in
two aspects. First, marginal cost smoothing translates into a smoother and thus
more persistent inflation path; second, the form and the nature of the driving
process in the Phillips curve equation changes, as is evident from equations
(8) and (10). The latter equation predicts a relationship between marginal cost
and the sales-to-stock ratio, γ , which changes the channel by which marginal
cost affects inflation dynamics.9
9 This is further and more formally empirically investigated in Lubik and Teo (2009), who
suggest that the inventory channel does not contribute much to explain observed inflation behavior.
370
Federal Reserve Bank of Richmond Economic Quarterly
We can tentatively conclude that a New Keynesian model with inventories
presents a modified set of tradeoffs for an optimizing policymaker. In the
standard model optimal policy is such that both consumption and the labor
supply should be smoothed and price adjustment costs minimized. In the
inventory model, these objectives are still relevant since they affect utility in the
same manner, but the channel through which this can be achieved is different.
Inventories allow for a smoother adjustment path of inflation, which should
help contain the effects of price stickiness, while the consumption behavior
depends on the nature of the shocks. We now turn to an analysis of optimal
policy with inventories.
3.
OPTIMAL MONETARY POLICY
The goal of an optimizing policymaker is to maximize a welfare function subject to the constraints imposed by the economic environment and subject to
assumptions about whether the policymaker can commit or not to the chosen action. In this article, we assume that the optimizing monetary authority
maximizes the intertemporal utility function of the household subject to the
optimal behavior chosen by the private sector and the economy’s feasibility
constraints. Furthermore, we assume that the policymaker can credibly commit to the chosen path of action and does not re-optimize along the way. We
consider two cases. For our benchmark, we assume that the monetary authority implements the Ramsey-optimal policy.10 We then contrast the Ramsey
policy with an optimal policy that is chosen for a generic set of linear rules of
the type used in the simulation analysis above.
We can alternatively interpret the policymaker’s actions as minimizing
the distortions in the model economy. In a typical New Keynesian setup like
ours, there are two distortions. The first is the suboptimal level of output
generated by the presence of monopolistically competitive firms. The second
distortion arises from the presence of nominal price stickiness, as captured by
the quadratic price adjustment cost function, which is a deadweight loss to the
economy. In the standard model, the optimal policy perfectly stabilizes inflation at the steady-state level. Introducing inventories can change this basic
calculus in our model, as the sales-relevant terms are relative inventory holdings that present an externality for a Ramsey planner. We will now investigate
whether this additional wedge matters quantitatively for optimal policy.
Welfare Criterion
We use expected lifetime utility of the representative household at time
zero, V0a , as the welfare measure to evaluate a particular monetary policy
10 See Khan, King, and Wolman (2003), Levin et al. (2006), and Schmitt-Grohé and Uribe
(2007) for wide-ranging and detailed discussions of this concept in New Keynesian models.
T. A. Lubik and W. L. Teo: Inventories and Optimal Monetary Policy
371
regime, a:
V0a
≡ E0
∞
t=0
β
t
ζ t ln Cta
a 1+η
h
− t
.
1+η
(18)
As in Schmitt-Grohé and Uribe (2007), we compute the expected lifetime
utility conditional on the initial state being the deterministic steady state for
given sequences of optimal choices of the endogenous variables and exogenous
shocks. Our welfare measure is in the spirit of Lucas (1987) and expresses
welfare as a percentage of steady-state consumption that the household
is willing to forgo to be as well off under the steady state as under a given
monetary policy regime, a. can then be computed implicitly from
∞
h1+η
c −
= V0a ,
β t ζ ln 1 −
(19)
100
1
+
η
t=0
where variables without time subscripts denote the steady state of the corresponding variables.11 Note that a higher value of corresponds to lower
welfare. That is, the household would be willing to give up percent of
steady-state consumption to implement a policy that delivers the same level
of welfare as the economy in the absence of any shocks. This also captures
the notion that business cycles are costly because they imply fluctuations that
a consumption-smoothing and risk-averse agent would prefer not to have.
Optimal Policy
We compute the Ramsey policy by formulating a Lagrangian problem in which
the government maximizes the welfare function (18) of the representative
household subject to the private sector’s first-order conditions and the marketclearing conditions of the economy. The optimality conditions of this Ramsey
policy problem can then be obtained by differentiating the Lagrangian problem
with respect to each of the endogenous variables and setting the derivatives
to zero. This is done numerically by using the Matlab procedures developed
by Levin and Lopez-Salido (2004). The welfare function is then approximated around the distorted, non-Pareto-optimal steady state. The source of
steady-state distortion is the inefficient level of output due to the presence of
monopolistically competitive firms.
In our second optimal policy case, we follow Schmitt-Grohé and Uribe
(2007) and consider optimal, simple, and implementable interest rate rules.
11 We assume that the policymaker chooses the same steady-state inflation rate for all mon-
etary policies that we consider. The steady state of all variables will thus be the same for all
policies.
372
Federal Reserve Bank of Richmond Economic Quarterly
Specifically, we consider rules of the following type:
t = ρ R
t−1 + ψ 1 Et
R
π t+i + ψ 2 Et
yt+i , i = −1, 0, 1.
(20)
The subscript i indicates that we consider forward-looking (i = 1), contemporaneous (i = 0), and backward-looking rules (i = −1). Following the
suggestion in Schmitt-Grohé and Uribe (2007), we focus on values of the
policy parameters ρ, ψ 1 , and ψ 2 that are in the interval [0, 3]. Note that this
rule also allows for the possibility that the interest rate is super-inertial; that
is, we assume ρ can be larger than 1. In order to find the constrained-optimal
interest rate rule, we search for combinations of the policy coefficients that
maximize the welfare criterion. As in Schmitt-Grohé and Uribe (2007), we
impose two additional restrictions on the interest rate rule: (i) the rule has
to be consistent with a locally unique rational expectations equilibrium; (ii)
the interest rate rule cannot violate 2σ R < R, where σ R is the unconditional
standard deviation of the gross interest rate while R is its steady-state value.
The second restriction is meant to approximate the zero bound constraint on
the nominal interest rate.12
Ramsey-Optimal Policy
A key feature of the standard New Keynesian setup is that Ramsey-optimal
policy completely stabilizes inflation. Price movements represent a deadweight loss to the economy because of the existence of adjustment costs.13
An optimizing planner would, therefore, attempt to remove this distortion.
This insight is borne out by the impulse response functions for the standard
model without inventories in Figure 4. Inflation does not respond to the technology shock, nor do labor hours or marginal cost as per the New Keynesian
Phillips curve. The path of output simply reflects the effect of increased and
persistent productivity. The Ramsey planner takes advantage of the temporarily high productivity and allocates it straight to consumption without feedback
to higher labor input or prices. The planner could have reduced labor supply
to smooth the time path of consumption. However, this would have a level effect on utility due to lower consumption, positive price adjustment cost via the
feedback from lower wages to marginal cost, and increased volatility in hours.
The solution to this tradeoff is thus to bear the brunt of higher consumption
volatility.
The possibility of inventory investment, however, changes this rationale
(see Figure 4). In response to a technology shock, output increases by more
12 If R is normally distributed, 2σ < R implies that there is a 95 percent chance that R
R
will not hit the zero bound.
13 In a framework with Calvo price setting, the deadweight loss comes in the form of relative
price distortions across firms, which lead to the misallocation of resources.
T. A. Lubik and W. L. Teo: Inventories and Optimal Monetary Policy
373
Figure 4 Impulse Response Functions to Productivity Shock: Ramsey
Policy
Output
Sales and Sales-Stock Ratio
1.0
1.5
y Inv
y Base
0.0
% Deviation
% Deviation
0.5
s
γ
-0.5
1.0
0.5
-1.0
-1.5
0.0
0
5
10
15
20
25
0
5
Labor Hours
15
-0.01
h Inv
h Base
% Deviation
0.1
0.0
-0.02
-0.03
-0.04
R Inv
-0.05
2
0
x 10
5
-3
10
15
20
R Base
-0.06
25
0
5
Inflation Rate
10
15
20
25
Marginal Cost
0.005
0.000
% Deviation
0
% Deviation
25
0.00
0.2
-0.1
20
Interest Rate
0.3
% Deviation
10
-2
-4
π Inv
π Base
-6
-0.005
mc Inv
-0.010
mc Base
-0.015
-0.020
-8
-0.025
0
5
10
15
20
25
0
5
10
15
20
25
compared to the model without inventories, while consumption, which is firstorder equivalent to sales, rises less. Ramsey-optimal policy can induce a
smoother consumption profile by allowing firms to accumulate inventories.
Similarly, the planner takes advantage of higher productivity in that he induces the household to supply more labor hours. Inflation is now no longer
completely stabilized as the lower increase in consumption leads to an initial
decline in inflation. Inventories thus serve as a savings vehicle that allows
the planner to smooth out the impact of shocks. The planner incurs price
adjustment costs and disutility from initially high labor input. The benefit
is a smoother and more prolonged consumption path than would be possible
374
Federal Reserve Bank of Richmond Economic Quarterly
Table 2 Welfare Costs and Standard Deviations under
Ramsey-Optimal Policy
Technology
Welfare Cost ()
Standard Deviation (%)
Output
Inflation
Consumption
Labor
Welfare Cost ()
Standard Deviation (%)
Output
Inflation
Consumption
Labor
Welfare Cost ()
Standard Deviation (%)
Output
Inflation
Consumption
Labor
Preference
All Shocks
Panel A: Model without Inventories
0.0000
−0.0521
−0.0521
1.60
0.00
1.60
0.00
0.000
1.73
0.02
1.45
0.24
2.40
0.00
2.40
2.40
Panel B: Model with Inventories
−0.0529
2.28
0.04
2.60
2.28
2.89
0.00
2.89
2.40
−0.0529
2.86
0.04
2.97
2.29
Panel C: Full Inflation Stabilization
0.000
−0.0528
−0.0528
1.73
0.00
1.45
0.24
2.29
0.00
2.61
2.29
2.87
0.00
2.99
2.30
without inventories. The model with inventories therefore restores something
akin to an output-inflation tradeoff in the New Keynesian framework.
The quantitative differences between the two specifications are small,
however. Table 2 reports the welfare costs and standard deviations of selected
variables for the two versions of the model under Ramsey-optimal policy. The
welfare costs of business cycles in the standard model are vanishingly small
when only technology shocks are considered and undistinguishable from the
specification with inventories. The standard deviation of inflation is zero for
the model without inventories while it is slightly higher for the model with
inventories. This is consistent with the evidence from the impulse responses
and highlights the differences between the two model specifications. Note
also that consumption is less volatile in the model with inventories than in the
standard model, which reflects the increased degree of consumption smoothing
in the former.14
14 This is consistent with the simulation results reported in Schmitt-Grohé and Uribe (2007)
in a model with capital. They also find that full inflation stabilization is no longer optimal since
T. A. Lubik and W. L. Teo: Inventories and Optimal Monetary Policy
375
Figure 5 Impulse Response Functions to Preference Shock: Ramsey
Policy
Sales and Sales-Stock Ratio
Output
0.5
0.8
y Inv
y Base
s
γ
0.4
% Deviation
% Deviation
0.6
0.4
0.2
0.0
0.3
0.2
0.1
0
5
10
15
20
0
25
5
Labor Hours
15
20
25
Interest Rate
0.05
0.5
h Inv
h Base
0.4
R Inv
0.04
% Deviation
% Deviation
10
0.3
0.2
R Base
0.03
0.02
0.01
0.00
0.1
0
5
-3
15
20
0
25
5
Inflation Rate
x 10
2
10
10
15
20
25
Marginal Cost
0.02
0.01
% Deviation
% Deviation
0
-2
π Inv
π Base
-4
-6
0.00
-0.01
-0.02
mc Inv
-0.03
mc Base
-0.04
0
5
10
15
20
25
0
5
10
15
20
25
Figure 5 depicts the impulse responses to the preference shock under
Ramsey-optimal policy. Inflation and marginal cost are fully stabilized in
the standard model, which the planner achieves through a higher nominal
interest rate that reduces consumption demand in the face of the preference
shock. At the same time, the planner lets labor input go up to meet some of
the additional demand. In contrast, Ramsey policy for the inventory model
investment in capital provides a mechanism for smoothing consumption, just as inventory holdings
do in our model.
376
Federal Reserve Bank of Richmond Economic Quarterly
can allow consumption to increase by more since firms can draw on their
stock of goods for sale. Consequently, output and labor increase by less for
the inventory model. Similarly to the case of the technology shock, optimal
policy does not induce complete inflation stabilization as it uses the inventory
channel to smooth consumption. This is confirmed by the simulation results
in Table 2, which show the Ramsey planner trading off volatility between
inflation, consumption, and labor when compared to the standard model.
Interestingly, eliminating business cycles and imposing the steady-state
allocation is costly for the planner in the presence of preference shocks that
multiply consumption. This is evidenced by the negative entries for the welfare
cost in both model specifications. In other words, agents would be willing to
pay the planner 0.05 percent of their steady-state consumption not to eliminate
preference-driven fluctuations. This stems from the fact that, although fluctuations per se are costly in welfare terms for risk-averse agents, they can also
induce co-movement between the shocks and other variables that have a level
effect on utility. Specifically, preference shocks co-move positively with consumption due to an increase in demand. This positive co-movement is reflected
in a positive covariance between these two variables. In our second-order approximation to the welfare functions, this overturns the negative contribution
to welfare from consumption volatility.
When we consider both shocks together, the differences between the two
specifications are not large in welfare terms and with respect to the implications for second moments. Inflation and consumption are more volatile in the
inventory version, while labor is less volatile compared to standard specification. We also compare Ramsey-optimal policy with inventories to a policy of
fully stabilizing inflation only (as opposed to using the utility-based welfare
criterion from above). Panel C of Table 2 shows that the latter is very close to
the Ramsey policy. The welfare difference between the two policies is small—
less than 0.001 percentage points of steady-state consumption. The effects of
inventories can be seen in the slightly higher volatility of consumption and
labor under the full inflation stabilization policy. Inventory investment allows
the planner to smooth consumption more compared to the standard model,
and the mechanism is a change in prices. Although price stability is feasible,
the planner chooses to incur an adjustment cost to reduce the volatility of
consumption and labor.
Optimal Policy with a Simple and Implementable
Rule
Ramsey-optimal policy provides a convenient benchmark for welfare analysis
in economic models. However, from the point of view of a policymaker, pursuing a Ramsey policy may be difficult to communicate to the public. It may
also not be operational in the sense that the instruments used to implement
T. A. Lubik and W. L. Teo: Inventories and Optimal Monetary Policy
377
the Ramsey policy may not be available to the policymaker. For instance, in a
market economy the government cannot simply choose allocations as a Ramsey plan might imply. The literature has therefore focused on finding simple
and implementable rules that come close to the welfare outcomes implied by
Ramsey policies (see Schmitt-Grohé and Uribe 2007).
Therefore, we investigate the implications for optimal policy conditional
on the simple rule (20). Panel A of Table 3 shows the constrained-optimal interest rate rules for the model without inventories with all shocks considered
simultaneously. The rule that delivers the highest welfare is a contemporaneous rule, with a smoothing parameter ρ = 1 and reaction coefficients on
inflation ψ 1 and output ψ 2 of 3 and 0, respectively.15 This is broadly consistent
with the results of Schmitt-Grohé and Uribe (2007), where the constrainedoptimal interest rate rule also features interest smoothing and a muted response
to output. Without interest rate smoothing the welfare cost of implementing
this policy increases, which is exclusively due to a higher volatility of inflation.
On the other hand, the difference between the constrained-optimal contemporaneous rule and the Ramsey policy is small—less than 0.001 percentage
points. This confirms the general consensus in the literature that simple rules
can come extremely close to Ramsey-optimal policies in welfare terms. The
characteristics of constrained-optimal backward-looking and forward-looking
rules are similar to the contemporaneous rule, i.e., they also feature full interest smoothing and no output response. The welfare difference between the
constrained-optimal contemporaneous rule and the other two rules are also
small.
Turning to the model with inventories, we report the results for the constrained-optimal rules in Panel B of Table 3. All rules with interest smoothing deliver virtually identical results but strictly dominate any rule without
smoothing. As before, the coefficient on output is zero, while the policymakers implement a strong inflation response. The main difference to the Ramsey
outcome is that inflation is slightly less volatile, while output is more volatile.
This again confirms the findings in other articles that a policy rule with a
fully inertial interest rate and a hawkish inflation response delivers almost
Ramsey-optimal outcomes.
Sensitivity Analysis
We now investigate the robustness of our optimal monetary policy results
to alternative parameter values. The results of alternative calibrations are
reported in Table 4, where we only document results for the rule that comes
closest to the Ramsey benchmark. In the robustness analysis, we change
15 The reader may recall that we restricted the policy coefficients to lie within the
interval [0, 3].
378
Federal Reserve Bank of Richmond Economic Quarterly
Table 3 Optimal Policy with a Simple Rule
ρ
Ramsey Policy
Optimized Rules
Contemporaneous (i = 0)
Smoothing
No Smoothing
Backward (i = −1)
Smoothing
No Smoothing
Forward (i = 1)
Smoothing
No Smoothing
Ramsey Policy
Optimized Rules
Contemporaneous (i = 0)
Smoothing
No Smoothing
Backward (i = −1)
Smoothing
No Smoothing
Forward (i = 1)
Smoothing
No Smoothing
ψ1
ψ2
Welfare
Cost ()
σπ
Panel A: Model without Inventories
−0.0521
0.00
σy
2.89
1.00
0.00
3.00
3.00
0.00
0.00
−0.0520
−0.0499
0.04
0.28
2.89
2.89
1.00
0.00
3.00
3.00
0.00
0.00
−0.0520
−0.0501
0.05
0.27
2.89
2.90
1.00
0.00
3.00
3.00
0.00
0.00
−0.0518
−0.0496
0.08
0.30
2.90
2.90
Panel B: Model with Inventories
−0.0529
0.04
2.86
1.00
0.00
3.00
3.00
0.00
0.00
−0.0528
−0.0518
0.01
0.20
2.87
2.87
1.00
0.00
3.00
3.00
0.00
0.00
−0.0528
−0.0518
0.02
0.19
2.87
2.87
1.00
0.00
3.00
3.00
0.00
0.00
−0.0528
−0.0517
0.02
0.20
2.87
2.87
one parameter at a time while holding all other parameters at their benchmark
values. The overall impression is that in all alternative calibrations the optimal
simple rule comes close to the Ramsey policy, and that the relative welfare
rankings for the individual rules established in the benchmark calibration are
unaffected. Specifically, inertial rules tend to dominate rules with a lower
degree of smoothing.
We first look at the implications of alternative values for the two parameters
related to inventories: the elasticity of demand with respect to the stock of
goods available for sale, μ, and the depreciation rate of the inventory stock, δ.
As in Jung and Yun (2005), we consider the alternative value μ = 0.8. Since
sales now respond more elastically to the stock of goods available for sale, the
inventory channel becomes more valuable as a consumption-smoothing device
and inflation becomes more volatile under a Ramsey policy. The best simple
rule has contemporaneous timing and comes very close to the Ramsey policy
in terms of welfare. The optimal rule is inertial and strongly reacts to inflation
only. The volatility of inflation is lower than under the Ramsey policy and
closer to that of the optimally simple rule with the benchmark calibration. This
T. A. Lubik and W. L. Teo: Inventories and Optimal Monetary Policy
379
Table 4 Optimal Policy for the Model with Inventories: Alternative
Calibration
ρ
ψ1
Ramsey Policy
Contemporaneous (i = 0)
1.0
3.0
Ramsey Policy
Contemporaneous (i = 0)
1.0
3.0
Ramsey Policy
Contemporaneous (i = 0)
1.0
3.0
Ramsey Policy
Contemporaneous (i = 0)
1.0
3.0
Welfare
Cost ()
Panel A: μ = 0.8
−0.0508
0.0
−0.0507
Panel B: δ = 0.05
−0.0557
0.0
−0.0553
Panel C: η = 5
−0.0193
0.0
−0.0190
Panel D: θ = 21
−0.0539
0.0
−0.0537
ψ2
σπ
σy
0.05
0.01
2.88
2.89
0.09
0.02
2.85
2.86
0.03
0.01
1.79
1.80
0.05
0.02
2.85
2.86
suggests that the response coefficients of the optimal rule are insensitive to
changes in elasticity parameter μ, and that the Ramsey planner can exploit the
changes in the transmission mechanism in a way that the simple rule misses.
The quantitative differences are small, however.
In the next experiment, we increase the depreciation rate of the inventory
stock to δ = 0.05. It is at this value that Lubik and Teo (2009) find that the
inclusion of inventories has a marked effect on inflation dynamics in the New
Keynesian Phillips curve. Panel B of Table 4 shows that the preferred rule is
again contemporaneous, but the differences between the alternatives are very
small. Interestingly, Ramsey policy leads to a volatility of inflation that is
almost an order of magnitude higher than in the benchmark case, which is
consistent with the findings in Lubik and Teo (2009).
The benchmark calibration imposed a very elastic labor supply with η = 1.
The results of making the labor supply much more inelastic by setting η = 5
are depicted in Panel C of the table. For this value, the differences to the
benchmark are most pronounced. In particular, the volatility of output declines
substantially across the board, which is explained by the difficulty with which
firms change their labor input. The best simple rule is contemporaneous, but
the differences to the other rules are vanishingly small. Optimal policy again
puts strong weight on inflation, with the optimal rule being inertial. Another
difference to the benchmark parameterization is that the welfare cost of no
interest smoothing is also much bigger for η = 5.16 Finally, we also report
results for calibration with a lower steady-state markup of 5 percent, which
16 The welfare cost of no interest smoothing is 0.0088 for η = 5, while it is 0.0021 for the
benchmark parameterization.
380
Federal Reserve Bank of Richmond Economic Quarterly
corresponds to a value of θ = 21. The qualitative and quantitative results are
mostly similar to the benchmark results.
In summary, the results from the benchmark calibration are broadly robust. Under a Ramsey policy full inflation stabilization is not optimal, while
the best optimal simple rule exhibits inertial behavior on interest smoothing
and a strong inflation response. The welfare differences between alternative
calibrations are very small, with the exception of changes in the labor supply
elasticity. A less elastic labor supply reduces the importance of the inventory channel to smooth consumption by making it more difficult to adjust
employment and output in the face of exogenous shocks.
4.
CONCLUSION
We introduce inventories into an otherwise standard New Keynesian model
that is commonly used for monetary policy analysis. Inventories are motivated
as a way to generate sales for firms. This changes the transmission mechanism
of the model, which has implications for the conduct of optimal monetary policy. We emphasize two main findings in the article. First, we show that full
inflation stabilization is no longer the Ramsey-optimal policy in the simple
New Keynesian model with inventories. While the optimal planner still attempts to reduce inflation volatility to zero since it is a deadweight loss for the
economy, the possibility of inventory investment opens up a tradeoff. In our
model, production no longer needs to be consumed immediately, but can be
put into inventory to satisfy future demand. An optimizing policymaker therefore has an additional channel for welfare-improving consumption smoothing,
which comes at the cost of changing prices and deviations from full inflation
stabilization. Our second finding confirms the general impression from the literature that simple and implementable optimal rules come close to replicating
Ramsey policies in welfare terms.
This article contributes to a growing literature on inventories within the
broader New Keynesian framework. However, evidence on the usefulness
of including inventories to improve the model’s business cycle transmission
mechanism is mixed, as we have shown above. Future research may therefore
delve deeper into the empirical performance of the New Keynesian inventory
model, in particular on how modeling inventories affect inflation dynamics.
Jung and Yun (2005) and Lubik and Teo (2009) proceed along these lines.
A second issue concerns the way inventories are introduced into the model.
An alternative to our setup is to add inventories to the production structure so
that instead of smoothing sales, firms can smooth output. Finally, it would be
interesting to estimate both model specifications with structural methods and
compare their overall fit more formally.
T. A. Lubik and W. L. Teo: Inventories and Optimal Monetary Policy
381
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Economic Quarterly—Volume 95, Number 4—Fall 2009—Pages 383–418
Dynamic Provisioning: A
Countercyclical Tool for
Loan Loss Reserves
Eliana Balla and Andrew McKenna
T
he methodology to recognize loan losses set forth by the Financial Accounting Standards Board (FASB) and the International Accounting
Standards Board (IASB) is referred to as the incurred loss model and
defined as the identification of inherent losses in a loan or portfolio of loans.
Inherent credit losses, under current accounting standards in countries following FASB and IASB, are event driven and should only be recognized upon an
event’s occurrence.1 This has tended to mean that reserves for loan losses on
a bank’s balance sheet need to grow significantly during an economic downturn, a time associated with increased credit impairment and default events.
Critics of the incurred loss model have pointed to it as one of the causes
of the severity of strain many financial institutions experienced at the onset
of the financial crisis of 2007–2009. As rapid provisioning to increase loan
loss reserves made headlines, discussions of international regulatory banking
reform included the method of dynamic provisioning as a potential alternative to the incurred loss approach (see, for example, Cohen 2009). Dynamic
provisioning is a statistical method for loan loss provisioning that relies on
historical data for various asset classes to determine the level of provisioning
that should occur on a quarterly basis in addition to any provisions that are
The authors would like to thank Teresita Obermann, Jesús Saurina, and Tricia Squillante
for their help in researching the details of the Spanish provisioning system, and Stacy
Coleman, Borys Grochulski, Sabrina Pellerin, Mike Riddle, Diane Rose, David Schwartz,
and John Walter for their helpful comments. We are also grateful to David Gearhart and
Susan Maxey for their research assistance. Any errors are our own. The views expressed
in this article are those of the authors and do not necessarily reflect those of the Federal
Reserve Bank of Richmond or the Federal Reserve System. The authors can be reached at
Eliana.Balla@rich.frb.org and Andrew.McKenna@rich.frb.org.
1 FASB and IASB, March 2009 meeting. Information for Observers. Project: Loan Loss
Provisioning.
384
Federal Reserve Bank of Richmond Economic Quarterly
event driven.2 The primary goal of dynamic provisioning is the incremental
building of reserves during good economic times to be used to absorb losses
experienced during economic downturns.
We begin this paper with a discussion of the current approach to loan loss
reserves (LLR) in the United States. We argue that, to a social planner who
cares both about avoiding bank failures and the efficiency of bank lending, the
current accounting and regulatory approach for LLR may be suboptimal on
both fronts. First, by taking provisions after the economic downturn has set in,
a bank faces higher insolvency risk. When a banker or regulator determines
that a bank has inadequate LLR, the bank will have to build the reserves in
an unfavorable economic environment. Also, inadequate reserves imply that
regulatory capital ratios have been overstated, placing the bank at a higher risk
for resolution by the Federal Deposit Insurance Corporation (FDIC). Second,
as most banks tend to increase LLR during the economic downturn, the current
approach may be procyclical; that is, it may amplify the business cycle. We
aim to highlight some of the potential inefficiencies under the incurred loss
approach by contrasting it to dynamic provisioning. Dynamic provisioning
was instituted in Spain in 2000 in response to some of the same problems we
highlight in the United States. We present a conceptual framework to compare
loan loss provisioning under the incurred loss framework and dynamic provisioning. Then we simulate dynamic provisioning with U.S. data to present
an empirical comparison. In the remainder of this section, we offer a brief
summary of our main arguments and the conclusions from the simulation
exercise.
In accounting terms, the LLR account, also known as the allowance for
loan and lease losses (ALLL), is a contra-asset account used to reduce the value
of total loans and leases on a bank’s balance sheet by the amount of losses that
bank managers anticipate in the most likely future state of the world.3 LLR
incorporate both statistical estimates and subjective assessments. Provisioning
is the act of building the LLR account through a provision expense item on the
income statement. While we present the intuition behind LLR in this section,
the Appendix to the paper describes their important accounting features in
basic terms.
Interest margin income from loans is a smooth flow whereas a loan default
or impairment event causes a lumpy drop in the stock of bank assets. This
introduces volatility to banks’ balance sheets. By themselves, a large number
2 Dynamic provisioning is also known as statistical provisioning and countercyclical
provisioning.
3 See, for example, Ahmed, Takeda, and Thomas (1999). See Benston and Wall (2005) for
a treatment of fair value accounting as it pertains to loan losses. The key to Benston and Wall’s
arguments is that if loans could be reported reliably at fair value, where fair value is value in use,
there would be no need for a loan loss provision or reserves. But a market for the full transfer
of credit risk does not exist and loans cannot be reported reliably at fair value.
E. Balla and A. McKenna: Dynamic Provisioning
385
of loans may be insufficient to smooth these fluctuations out due to the correlation between the risks in the portfolio of bank loans. Some defaults are
to be expected in a typical portfolio of bank loans. In order to avoid excess
volatility of bank capital levels, banks can build a buffer stock of reserves
against expected losses. Intuitively, LLR should serve to absorb expected
loan losses while bank capital serves to absorb unexpected losses.4 The key
difference between a conventional economic definition of expected losses and
incurred losses is that, unlike expected losses, incurred losses cannot incorporate information from expected future changes into economic variables that
affect credit defaults. Incurred losses are entirely based on historical information.5 If expected losses are greater than the loan loss reserve, regulatory
capital ratios overstate the bank capital available to protect against insolvency
risk.
To understand the argument that current loan loss accounting standards
may have procyclical effects, we have to think about the LLR through the
economic cycle. U.S. banking data show that LLR tend to be much lower
during good economic times relative to bad economic times. An event-driven
approach to LLR does not account for a booming economy resulting in banks
relaxing their underwriting standards and taking greater risks.6 Most bad
loans will only reveal themselves in a recession. In that sense, the current
approach may magnify the economic boom. By delaying provisioning for
loan losses until the economic downturn has set in, the current approach may
also magnify the bust. Reserves have to be built at a time when bank funds
are otherwise strained, potentially furthering the credit crunch.7 Therefore,
even though banks should want to build “excess” LLR voluntarily during the
boom years (it is efficient to do so from their perspective and it would have the
benefit of offsetting cyclicality), the accounting guidelines pose a constraint.
4 See Laeven and Majnoni (2003, Appendix A) for a detailed description of the conceptual
relationship between LLR, provisions, capital, and earnings.
5 Typically, expected loss is the mean of a loss distribution measured over a one-year horizon
(expected loss is loss given default times the probability of default times the exposure at default);
see Davis and Williams (2004). One way to separate the two concepts is by stating that no
expected economic impacts are taken into account in LLR methodology. A bank manager cannot,
for example, consider the increases in default risk due to future increases in unemployment.
6 Independently of any LLR effects, stylized facts and a burgeoning literature suggest that
bank lending behavior is procyclical. Many explanations have been presented. The classical
principal-agent problem between shareholders and managers may lead to procyclical banking if
managers’ objectives are related to credit growth. Two of the more recent theories are “herd
behavior” and “institutional memory hypothesis.” Rajan (1994) suggests that credit mistakes are
judged more leniently if they are common to the whole industry (herd behavior). Berger and
Udell (2003) suggest that, as the time between the current period and the last crisis increases,
experienced loan officers retire or genuinely forget about the lending errors of the last crisis and
become more likely to make “bad” loans (the institutional memory hypothesis). Our argument here
is that LLR effects may add to this otherwise present procyclicality of bank lending.
7 See Hancock and Wilcox (1998) and the sources cited therein for a discussion of the literature on the credit crunch. Also see Eisenbeis (1998) for a critique of Hancock and Wilcox
(1998).
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Federal Reserve Bank of Richmond Economic Quarterly
“Excess” reserves are associated with managing earnings, which is viewed
as undesirable by the accounting profession. Wall and Koch (2000) offer a
review of the theoretical and empirical evidence on earnings management via
loan loss accounting. The evidence they summarize suggests that banks both
have an incentive to and, in general, are using loan loss accounting to manage
reported earnings. From the perspective of the accounting profession, using
LLR to manage reported earnings is in conflict with the goals of transparency
of a bank’s balance sheet as of the date of the financial statement. We take it
as a given that the goals and concerns of the accounting standard setters are
valid. We simply highlight the resulting tradeoffs.
We illustrate the tradeoffs by pointing to an alternative system of reserving for loan losses—dynamic provisioning. Dynamic provisioning is, at its
core, a deliberate method to build LLR in good economic times to absorb loan
losses during an economic downturn, without putting undue pressure on earnings and capital. Spanish regulators instituted dynamic provisioning in 2000
explicitly to combat their banking system’s procyclicality.8 In maintaining a
focus on the use of historical data in its approach to loan loss provisioning,
the Bank of Spain (the regulator of Spanish banks) has been able to adopt
dynamic provisioning in compliance with IASB standards. We describe the
Spanish method in some detail and present data on Spanish reserves (relative to
contemporaneous credit quality) against the United States and other Western
European economies in 2006, before the beginning of the crisis. According
to these data, the Spanish policy was effective in building relatively higher
reserves and thus worthy of further study.
We compare the incurred loss and dynamic provisioning approaches.
Through a basic example we illustrate that the key difference is not the level
of provisioning but the timing of the provisioning. By taking provisions early
when economic conditions are good, banks will avoid using capital in an economic downturn when it is more expensive, thereby reducing the probability
of failure from capital deficiencies. Moreover, a goal of dynamic provisioning is to ensure that the balance sheet accurately reflects the true value of
assets to banks. If income is not reduced to provision for assets that are not
collectable, then managers may be pressured to provide greater dividends to
investors based on the income that is reported in the period.
As a next step in our analysis, we conduct an empirical simulation to
illustrate that a dynamic provisioning framework (akin to the one implemented
in Spain) could have allowed for a build-up of reserves during the boom years
in the United States. The results demonstrate that the alternate framework
would have smoothed bank income through the cycle and provisioning levels
8 In the current provisioning system, outside of Spain, loan loss provisions are generally
countercyclical but their effect is thought to be procyclical. We refer to “procyclicality” as the
amplification of otherwise normal business fluctuations.
E. Balla and A. McKenna: Dynamic Provisioning
387
would have been significantly lower during the financial crisis of 2007–2009.
Note that, in contrast to accountants, bank regulators would not take issue with
LLR resulting in income smoothing because the regulators’ primary concern
is the adequacy of the reserves to sustain loan losses.
The remainder of the article proceeds as follows. Section 1 describes
current rules for LLR in the United States, as well as the issues confronting
the current system, particularly as identified during the financial crisis of 2007–
2009. Section 2 provides a conceptual framework for comparing the incurred
loss and the dynamic provisioning approaches to LLR. Section 3 describes the
approach as implemented by the Bank of Spain. Section 4 builds a simulation
of dynamic provisioning with historical U.S. data. Section 5 concludes.
1. THE CURRENT ACCOUNTING AND REGULATORY
FRAMEWORK FOR LOAN LOSS PROVISIONING IN THE
UNITED STATES
Bank regulators view adequate LLR as a “safety and soundness” issue because
a deficit in LLR implies that a bank’s capital ratios overstate its ability to
absorb unexpected losses. As a result of their important relationship to bank
capital and financial reporting transparency, rules governing LLR have been
revisited many times by bank regulators and accounting standard setters. Two
crucial revision points relate to the new regulatory capital rules in the Basel
Capital Accord (signed in 1988) as enacted by the Federal Deposit Insurance
Corporation Improvement Act of 1991 (FDICIA)9 and the landmark case of
the SunTrust Bank earnings restatement that occured in 1998.10 Changes in
capital rules may have reduced bank manager incentives to keep large reserve
buffers, while the implementation of accounting rules following the SunTrust
case may have made it more difficult to justify building a reserve buffer during
good economic times. This section documents current rules that govern LLR,
the U.S. data from the last three cycles, and the importance of LLR both for
bank solvency and the procyclicality of bank lending.
Incurred Loss Accounting
Provisioning for loan losses in the United States is accounted for under Financial Accounting Standard (FAS) Statement 5, Accounting for Contingencies,
and FAS 114, Accounting by Creditors for Impairment of a Loan—an amendment of FAS Statements 5 and 15. Impaired loans evaluated under FAS 114,
9 See Walter (1991) for extensive coverage of LLR leading to the 1991 changes. Ahmed,
Taekda, and Thomas (1999) study how FDICIA (1991) changes affected the relationship between
loan loss provisioning, capital, and earnings.
10 See Wall and Koch (2000) for an extensive summary of the theoretical and empirical
evidence on bank loan loss accounting and LLR philosophies.
388
Federal Reserve Bank of Richmond Economic Quarterly
which provides guidance on estimating losses on loans individually evaluated, must be valued based on the present value of cash flows discounted at
the loan’s effective interest rate, the loan’s observable market price, or the fair
value of the loan’s collateral if it is collateral-dependent.11 Loans individually
evaluated under FAS 114 that are not found to be impaired are transferred to
homogenous groups of loans that share common risk characteristics, which
are evaluated under FAS 5. FAS 5 provides for accrual of losses by a charge
to the income statement based on estimated losses if two conditions are met:
(1) information available prior to the issuance of the financial statements indicates that it is probable that an asset has been impaired or a liability has been
incurred at the date of the financial statement, and (2) the amount of the loss
can be reasonably estimated.12
Both FAS 114 and 5 allow banks to include environmental or qualitative
factors in consideration of loan impairment analysis. Examples of these factors
include, but are not limited to, underwriting standards, credit concentration,
staff experience, local and national economic trends, and business conditions.
In addition, FAS 5 allows for the use of loss history in impairment analysis.13
These elements provide bankers with flexibility in determining the level of
provisions taken against incurred losses when they are well substantiated by
relevant data and/or documentation required by supervisors and accountants.
This paper includes illustrative examples to support our explanation of the
technical aspects of accounting for loan losses. For consistency, when we
refer to identified loan losses, we mean the accounting conditions for taking
a provision were met. In practice, banks identify losses by categorizing loans
based on their payment status (i.e., current, 30 days past due, 60 days past
due, etc.) and the severity of delinquency (which can vary by asset class) and
assess whether a provision should be taken on loans they expect to experience
a loss, if the loss is probable and estimable.14
The U. S. Data
The adequacy of LLR to cover loan losses is generally measured against the
level of non-performing loans (the ratio of the two is known as the coverage
ratio), meaning loans that are seriously delinquent by being 90 or more days
past due or in non-accrual status. Figure 1 shows LLR and non-performing
11 Financial Accounting Standards Board, Summary of Statement No. 114: Accounting by
Creditors for Impairment of a Loan—An amendment of FASB Statements No. 5 and 15 (Issued
5/93).
12 Financial Accounting Standards Board, Summary of Statement No. 5: Accounting for
Contingencies (Issued 3/75).
13 SR 06-17: Interagency Policy Statement on the ALLL, December 13, 2006. SR 01-17:
Policy Statement on ALLL Methodologies and Documentation for Banks and Savings Institutions,
July 2, 2001. SR 99-22: Joint Interagency Letter on the Loan Loss Allowance, July 26, 1999
IPS.
14 See Walter (1991) for an explanation of how banks identify and categorize defaults.
E. Balla and A. McKenna: Dynamic Provisioning
389
Figure 1 Loan Loss Reserves Versus Non-Performing Loans Ratios:
U. S. Bank Aggregates in Levels, 1983–2008
4.0
3.5
Recession
NPTL as a % of TL
LLR as a % of TL
3.0
Percent
2.5
2.0
1.5
1.0
0.5
0.0
1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007
Notes: LLR = loan loss reserves; NPTL = non-performing total loans; TL = total loans.
Source: Call Reports.
loans, both scaled by total loans, between 1983 and 2008. The data come from
the Commercial Bank Consolidated Reports of Condition and Income Reports
(Call Reports) and they are constructed by combining all U.S. banks into an
“aggregate” balance sheet. We show the aggregate level of reserves in the U.S.
banking system at any point in time, the aggregate level of non-performing
loans, and so on.
Figure 1 depicts the cyclicality of LLR. The first cycle is different from
the subsequent two. Reserves are lower than non-performing loans during
the banking crisis of the late 1980s and early 1990s. Because of the major
regulatory changes that took full effect in 1992, we look more closely at the
last two cycles. At the height of the boom, in 2005, we saw some of the lowest
reserves relative to total loans on record. Note that this is not surprising given
that non-performing loans and reserves move together. At any point between
1992 and 2006, there were more reserves in the system than there were nonperforming loans. In 2005, banks had historically high coverage ratios, not
390
Federal Reserve Bank of Richmond Economic Quarterly
Figure 2 Loan Loss Reserves Versus Non-Performing Loans Ratios:
U. S. Bank Aggregates Year Over Year Percentage Change,
1983–2008
125
100
Recession
LLR to TL
NPTL to TL
Percent
75
50
25
0
1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007
-25
-50
Notes: LLR = loan loss reserves; NPTL = non-performing total loans; TL = total loans.
Source: Call Reports.
because reserves were high (indeed they were at a historical low), but because
non-performing loans were so low. The period 2006–2008 demonstrated how
reserves needed to be built in an economic downturn to keep pace with the
rapid deterioration in credit quality. Like credit problems, reserves grow just
ahead of a recession and continue to grow after the recession has ended. In
the first quarter of 2006, overall U.S. reserves were 1.1 percent of total loans.
In the first quarter of 2009, they were built up to 2.7 percent of total loans.
Trends in reserve adjustments against changes in non-performing loans,
shown in Figure 2, illustrate that modifications to LLR tend to lag credit problems, and reserves increase more slowly than non-performing loans during
economic busts and fall more slowly in booms. Both Figures 1 and 2 indicate
that some build-up of reserves relative to non-performing loans existed in the
E. Balla and A. McKenna: Dynamic Provisioning
391
U.S. banking system but the cushion shrank in the 2000s boom relative to the
1990s boom.15
Loan Loss Provisioning and Bank Solvency
The Basel Accord set current rules for LLR that prescribe the use of impairment and estimated loss methodology.16 FDICIA enacted these changes into
law. LLR were no longer counted as a component of Tier 1 capital but were
counted toward Tier 2 capital, up to 1.25 percent of the bank’s risk-weighted
assets. Laeven and Majnoni (2003) have argued that “. . . from the perspective
of compliance with regulatory capital requirements, it became much more
effective for U.S. banks to allocate income to retained earnings (entirely included in Tier 1 capital) than to loan loss reserves (only partially included in
Tier 2 capital)” (Laeven and Majnoni 2003, 194).
In the new regulatory regime of Basel I, banking regulators remained
concerned with the roles that the loan losses and banks’ reserve for losses play
in insolvency risk. Comptroller of the Currency John Dugan, the regulator of
U.S. national banks, has stated that “. . . banking supervisors love the loan loss
reserve. When used as intended, it allows banks to recognize an estimated loss
on a loan or portfolio of loans when the loss becomes likely, well before the
amount of the loss can be determined with precision and is actually charged off.
That means banks can be realistic about recognizing and dealing with credit
problems early, when times are good, by building up a large ‘war chest’ of loan
loss reserves. Later, when the loan losses crystallize, the fortified reserve can
absorb the losses without impairing capital, keeping the bank safe, sound, and
able to continue extending credit” (Dugan 2009). But accounting guidelines,
as enforced in the late 1990s and 2000s, may have limited the ability of LLR
to function in the way summarized by Comptroller Dugan.
In the mid-1990s, the Securities and Exchange Commission (SEC) was
increasingly concerned that U.S. banks may be overstating their LLR, potentially using this account to manage reported earnings. In 1998, following
an SEC inquiry, SunTrust Bank agreed to restate prior years’ financial statements, reducing its provisions in each of the years 1994–1996 and resulting in
a cumulative reduction of $100 million to its LLR (see Wall and Koch 2000).
Analysts of the U.S. banking industry viewed the SunTrust restatement as
a permanent strengthening of the existing accounting constraint on a bank’s
LLR policy.
15 Note that non-performing loans are only shown as a limited approximation of incurred
losses. There is no regulatory guidance that advocates a 100 percent reserve coverage for nonperforming loans. Nonetheless, it is a helpful standard simplification to present the data in this
way.
16 Basel II: International Convergence of Capital Measurement and Capital Standards: A
Revised Framework—Comprehensive Version. http://www.bis.org/publ/bcbs128.htm.
392
Federal Reserve Bank of Richmond Economic Quarterly
Figure 3 Loan Loss Provisions as a Percentage of Net Operating
Revenue: U. S. Bank Aggregates, 1984–2009
45
Recession
40
35
Percent
30
25
20
15
10
5
3/
31
/
6/ 198
30
4
/1
9
9/
30 85
/
12 198
6
/3
1/
19
87
3/
31
/1
9
6/
30 89
/1
9
90
9/
30
/
12 199
1
/3
1/
19
3/
9
2
31
/1
99
6/
4
30
/1
99
9/
5
30
12 /199
/3
1/ 6
19
3/
97
31
/1
99
6/
9
30
/2
0
9/
30 00
/
12 200
1
/3
1/
20
02
3/
31
/2
00
6/
4
30
/2
0
9/
30 05
/
12 200
6
/3
1/
20
07
3/
31
/2
00
9
0
Source: Call Reports.
Bankers desire flexibility in recognition of the subjective aspects in determining appropriate reserves. Bank regulators desire flexibility in recognition
of the importance of LLR for bank safety and soundness. Accounting standard setters stress the need for transparency and comparability across banks’
financial statements. We take it as a given that the goals of the accounting
standard setters and their concern over earnings management are valid. We
simply highlight the resulting tradeoffs.
Figure 3 illustrates the importance of provision expense relative to bank
income. First, the size of provisions relative to earnings helps us understand
their importance to bank managers, accountants, and bank regulators. Second, the period 2007–2009 illustrates nicely the inverse relationship between
earnings and provisions in a recession. Banks had to sharply increase provisions in recognition of pending losses, which for many banks more than offset
earnings and reduced capital.
E. Balla and A. McKenna: Dynamic Provisioning
393
Loan Loss Provisioning and Procyclicality
By entering the current economic downturn with low LLR, the banking sector
may have unintentionally exacerbated the cycle. In a speech in March 2009,
Ben Bernanke, the Chairman of the Board of Governors of the Federal Reserve stated that there is “considerable uncertainty regarding the appropriate
levels of loan loss reserves over the cycle. As a result, further review of accounting standards governing. . . loan loss provisioning would be useful, and
might result in modifications to the accounting rules that reduce their procyclical effects without compromising the goals of disclosure and transparency”
(Bernanke 2009).
The cyclicality of loan loss provisioning is well documented with crosscountry data. During periods of economic expansion, provisions fall (as a
percentage of loans) and, conversely, they rise during downturns. Figure 1
illustrated the cyclicality of loan loss provisioning with U.S. data. As with bank
regulatory capital, the concern is that with an approach in which banks have to
rapidly raise reserves during bad times, the bad times could get prolonged.17
Laeven and Majnoni (2003) and Bouvatier and Lepetit (2008) document the
procyclicality of loan loss provisions with cross-country data. Banks delay
provisioning for bad loans until economic downturns have already begun,
amplifying the impact of the economic cycle on banks’ income and capital.
Section 1 documented the current framework around LLR in the United
States, the U.S. data from the last three cycles, and the importance of LLR both
for bank solvency and the procyclicality of bank lending. In response to the
recent experience where many banks had to increase their LLR abruptly and
drastically, various U.S. and international regulators have expressed the desire
to revisit LLR policies. The Financial Stability Forum’s Working Group on
Provisioning (2009) has recommended that accounting standard setters give
due consideration to alternative approaches to recognizing and measuring loan
losses. One approach that has garnered attention is dynamic provisioning.
2.
INCURRED LOSS ACCOUNTING VERSUS DYNAMIC
PROVISIONING: A CONCEPTUAL FRAMEWORK
In this section, we discuss and compare dynamic provisioning with the incurred
loss methodology using two simplified examples that will set the stage for a
more complicated simulation completed in a subsequent section. And, while
17 For simplicity, we are not addressing in this article all the links between LLR and regulatory capital, nor the similarities between the cyclical effects of LLR and regulatory bank capital.
On the latter, we refer the reader to a large literature ranging from Bernanke and Lown (1991)
to Peek and Rosengren (1995) to Pennacchi (2005), who use U.S. data to analyze the effects of
capital requirements on banks and the economy.
394
Federal Reserve Bank of Richmond Economic Quarterly
we will review at a high level the technical nuances of accounting, a more
detailed discussion of accounting basics is included in the Appendix.
The IASB and the British Bankers Association (BBA) provide more flexibility than the FASB for firms to include forward-looking elements based on
expectations of total losses over the life of loans, rather than losses already
realized, but are still similar to the U.S. standards. The BBA Statement of
Recommended Practice on advances indicates specific provisions should be
made to cover the difference between the carrying value and the ultimate realizable value, made when a firm determines that information suggests impairment. General provisions18 take into account past experience and assumptions
about economic conditions, but are generally small because the Basel Accord
of 1988 limits general provisions to 1.25 percent of risk-weighted assets and
such provisions are not tax deductible (Mann and Michael 2002).
The IASB, in standard 39, requires assessment at each balance sheet date
to determine whether there is objective evidence that an asset or group of
assets is impaired—the difference between the asset’s carrying amount and
the present value of estimated cash flows discounted at the original interest
rate—based on criteria such as the financial condition of the issuer, breach
of contract, probability of bankruptcy, and historical pattern of collections of
accounts receivable.19
The incurred loss model for accounting for loan losses is divorced from
prudential goals of maintaining the safety and soundness of financial institutions in that the FASB and IASB frameworks do not aim to influence the
decisions investors make toward a specific objective, such as financial stability.20 Instead, the goal of financial reporting is to provide financial statement
users with the most accurate information about identified losses in the loans
and leases portfolio. The dynamic provisioning model, conversely, has as its
primary objective the enhancement of the safety and soundness of banks. Its
fundamental premise is that when loans are made the probability that default
will occur on a loan is greater than zero. In accordance with this philosophy,
the dynamic provisioning approach provides a model-based mechanism for a
bank to build a stock of provisions in good times so it will not face insolvency
due to charge-offs and provisions in bad times. In a simple exercise, presented
in Tables 1 and 2, we demonstrate how the application of the incurred loss
model would differ from that of dynamic provisioning.21
We will make a number of simplifications to help facilitate an understanding of this topic. These assumptions, if not applied, wouldn’t substantively
18 The Bank of Spain refers to statistical provisions as general provisions. See Saurina
(2009a).
19 Deloitte:
Summaries
of
International
Financial
Reporting
Standards.
http://www.iasplus.com/standard/ias39.htm.
20 The Financial Crisis Advisory Group. Public Advisory Meeting Agenda, February 13, 2009.
21 The illustration builds on an example provided by Mann and Michael (2002).
E. Balla and A. McKenna: Dynamic Provisioning
395
Table 1 Incurred Loss Model
Assets
Total Loans and Leases
Expected Losses
Stock of Specific Provisions
Yearly Charge-offs
Total Charge-offs
Total Stock of Provisions Net
of Stock of Charge-offs
Loans Net of Charge-offs
Year 1
1,000
40
0
0
0
Year 2
1,000
40
0
0
0
Year 3
1,000
70
30
5
5
Year 4
1,000
100
100
60
65
Year 5
1,000
100
100
35
100
0
1,000
0
1,000
25
995
35
935
0
900
Income Statement
Profit before Provision
Specific Provision
Profit after Provision
30
0
30
30
0
30
30
30
0
30
70
−40
30
0
30
Shareholders’ Equity
Shareholders Equity, Beginning
Other Expenses and Dividends
Retained Earnings
Shareholders’ Equity, End
44
24
6
50
50
24
6
56
56
0
0
56
56
0
−40
16
16
24
6
22
5.00%
Solvent
5.60%
Solvent
5.63%
Solvent
1.71%
Insolvent
2.44%
Solvent
Equity to Assets
Solvency: Equity to Assets ≥ 2%
Notes: Expected losses represent bank managers’ expectations for losses over the fiveyear period in the listed year. Specific provisions are those identified as probable and
estimable. Yearly charge-offs are those taken in the listed year, while the total is the
sum of charge-offs over the five-year period. Equity to assets is shareholders’ equity,
end divided by loans net of charge-offs.
change the results of our example, but instead, on average, only reinforce our
conclusions. First, the example bank in question will make all its loans inYear
1 and will not grow its portfolio. This prevents us from having to account for
changes in bank managers’ preference for varying risks of loans over time as
the economic environment changes. Second, we assume that, over the fiveyear cycle, the bank’s pre-provision profits don’t vary. If we introduced profit
declines as conditions worsened, which would likely occur since the bank’s
net interest margin (its only source of income in the example) would decline
due to charge-offs, the result would be reduced pre-provision profits and the
reduction of profit due to provisions would be greater. Third, we assume we
know the ex post level of risk in the portfolio of loans. In our example, losses
at the end of the five-year cycle will equal 10 percent of the loan portfolio.
For illustrative purposes we make the assumption that bank managers believe,
based on the bank’s historical loss experience, total ex post losses will equal
4 percent in Year 1 when loans are issued. As a loan approaches maturity in
Year 5, managers can, with greater certainty, estimate the true extent of ex
396
Federal Reserve Bank of Richmond Economic Quarterly
Table 2 Dynamic Provisioning Model
Assets
Total Loans and Leases
Expected Losses
Stock of Specific Provisions
Stock of Statistical Provisions
Total Stock of Provisions
Yearly Charge-offs
Total Charge-offs
Total Stock of Provisions Net
of Stock of Charge-offs
Loans Net of Charge-offs
Year 1
1,000
40
0
20
20
0
0
Year 2
1,000
40
0
40
40
0
0
Year 3
1,000
70
30
40
70
5
5
Year 4
1,000
100
100
0
100
60
65
Year 5
1,000
100
100
0
100
35
100
0
1,000
0
1,000
65
995
35
935
0
900
Income Statement
Profit before Provision
Provisions
Profit after Provision
30
20
10
30
20
10
30
30
0
30
30
0
30
0
30
Shareholders’ Equity
Shareholders’ Equity, Beginning
Other Expenses and Dividends
Retained Earnings
Shareholders’ Equity, End
44
8
2
46
46
8
2
48
48
0
0
48
48
0
0
48
48
24
6
54
4.60%
Solvent
4.80%
Solvent
4.82%
Solvent
5.13%
Solvent
6.00%
Solvent
Equity to Assets
Solvency: Equity to Assets ≥ 2%
Notes: Expected losses represent bank managers’ expectations for losses over the fiveyear period in the listed year. Specific provisions are those identified as probable and
estimable. Statistical provisions are those taken under the dynamic provisioning model
based on the historical data used to estimate the annual statistical provision. Yearly
charge-offs are those taken in the listed year, while the total is the sum of charge-offs
over the five-year period. Equity to assets is shareholders’ equity, end divided by loans
net of charge-offs.
post losses. To show how provisioning levels are determined over time, we
allowed managers’ expectations about expected losses to change in each year
to justify provisions each period (the expected losses entry in Tables 1 and 2).
Lastly, since the primary intent of dynamic provisioning is to better prepare
financial institutions to absorb loan losses, we make assumptions about shareholders’ equity. First, we assume shareholders’ equity is all tangible equity
and total loans and leases equal total assets. Accordingly, shareholders’ equity
is determined by adding retained earnings, after dividends and expenses, to
shareholders’ equity. To determine yearly retained earnings we assume a constant ratio of dividend and other expenses equal to 80 percent of after-provision
profit. Under current prompt corrective action (PCA) standards used by banking regulators, the Tier I capital ratio, total capital ratio, and leverage ratio
are used to determine when banks need to be resolved by the FDIC. In this
E. Balla and A. McKenna: Dynamic Provisioning
397
Table 3 Capital Guidelines
Well Capitalized
Adequately Capitalized
Undercapitalized
Significantly Undercapitalized
Critically Undercapitalized
Leverage Ratio
5 percent
4 percent
< 4 percent
< 3 percent
< 2 percent
example, since shareholders’ equity most closely resembles tangible equity,
we will focus on the leverage ratio, which in the example is equal to equity
divided by assets, as the primary measure of solvency. In Year 1 we assume
the bank starts with $44 shareholders’ equity and $1,000 total assets, leaving
it adequately capitalized at 4.4 percent. The PCA guidelines listed in Table 3
apply to the leverage ratio.
We begin our discussion with the incurred loss model. As previously
mentioned, bank managers expect that, given the average risk and performance
of loans and leases issued, the banks’total losses will amount to 4 percent of the
face value of loans at the end ofYear 5; this is reflected inYear 1, the first column
in Tables 1 and 2, listed in the row entitled Expected Losses. However, since
data do not exist that support identification of losses—evidence suggesting
the 4 percent of losses is probable and estimable—bank managers cannot take
provisions until such data exist. In Year 3, the first losses of $30 are identified
in the portfolio and managers’ expectation about total losses increases to 7
percent (or $70) based on new data. However, since only $30 in losses have
been identified, only a provision of that amount can be taken. Additionally,
since the $30 pre-provision profit was wiped out by the provision, the bank’s
managers can’t modify other expenses and dividends to increase capital to
prepare for the increase in losses. In Year 4, charge-offs increase to $60 for
the year and bank managers believe losses will amount to $100, or 10 percent
of loans and leases. The bank’s profit of $30 is eliminated by a provision of
$70 required by accountants, and a net loss of $40 (shown as negative retained
earnings in Table 1) requires the bank to reduce its capital by that amount.
When a net loss occurs, banks must use capital to equate assets with liabilities
and shareholders’ equity on the balance sheet. The reduction in capital leaves
the bank with only $16, resulting in a leverage ratio of 1.71 percent. By
PCA standards, this leaves the bank critically undercapitalized and it will be
resolved by banking regulators.
Table 2 presents the example bank under dynamic provisioning. Under the
dynamic provisioning model, bank managers would take a different approach
to provision for loan losses focusing on incrementally building up a fund
(referred to as the statistical fund) to protect the bank against losses expected
but not yet identified in the loans and leases portfolio. The model driving
398
Federal Reserve Bank of Richmond Economic Quarterly
the building process for the statistical fund would rely on historical default
data for the types of loans and leases issued by the bank rather than models
estimating expected losses; this is an important distinction between the dynamic provisioning approach and a strict expected loss approach. In Year 1,
when loans and leases of $1,000 were made, bank managers expected a total
of 4 percent of the loans and leases portfolio to default based on its historical
data, as reflected in the stock of statistical provisions built. Accordingly, the
managers establish a plan to build a fund for the bank to absorb those losses
over a two-year period. In Years 1 and 2 the bank provisions $20 to reach
the $40 fund desired. This has the effect of reducing profits in those years
to $10 instead of $30, which in turn reduces the amount of other expenses
and dividends the bank’s managers have and the amount of retained earnings
available to build capital. However, since the fund is capital set aside to absorb
losses the bank is anticipating based on historical data, the Year 2 capital ratio
including the fund is 8.8 percent, compared to 5.6 percent under the incurred
loss model in the same year. In Year 3, charge-offs increase to 0.5 percent and
the bank’s managers modify their expectation of total losses expected on the
loans and leases portfolio to 7.0 percent.
As with the incurred loss model, the $30 identified in specific provisions
would also be taken under the dynamic provisioning model. Bank managers,
in segmenting specific from statistical provisions, are not only preparing for
losses they expect to incur at some point over the life of the loan, but they are
also signaling to users of financial statements the difference between the expectation for losses as suggested by the historical data on defaults for the assets
held and those actually identified in the portfolio as probable and estimable.
The combined fund of $40 and the specific provision of $30 allow the bank to
have total provisions equal to $70, preparing the bank to fully absorb the expected losses. And while this reduces profit to $0, the bank’s solvency is well
protected. In Year 4, charge-offs increase sharply to $60 and the bank’s managers expect losses to increase to $100. Bank managers shift the $40 statistical
fund to specific provisions and add $30. After annual charge-offs of $60, the
bank has $35 remaining at the end of Year 4 in its stock of provisions to absorb
the remaining $35 in identified losses. Again, the bank’s provisioning reduces
its profit to $0, but in a difficult economic environment when bank losses are
sharply increasing, dividends are being halted throughout the industry and the
bank’s managers are actively attempting to reduce expenses, the primary concern is solvency. The bank under the incurred loss model becomes insolvent
in Year 4, while the bank under the dynamic provisioning model is actually
able to increase its leverage ratio, although due to a reduction in balance sheet
assets from charge-offs. In Year 5, under the assumption of no recoveries, the
remaining charge-offs of $35 occur and the stock of provisions is depleted.
The remaining loans and leases are paid down and the bank’s leverage ratio
increases again.
E. Balla and A. McKenna: Dynamic Provisioning
399
This basic example illustrates one of the primary points of importance for
dynamic provisioning: the key difference is not the level of provisioning but
the timing of the provisioning. By taking provisions early when economic
conditions are good, banks may be able to avoid using their capital in an economic downturn when it is more expensive, thereby reducing the probability
of failure from capital deficiencies. Moreover, an objective of dynamic provisioning is to ensure that the balance sheet accurately reflects the true value
of assets to banks. If income is not reduced to provision for assets that are not
collectable, then managers may be pressured to provide greater dividends to
investors based on the income that is reported in the period.
3.
DYNAMIC PROVISIONING AND THE SPANISH
EXPERIENCE
In the wake of the financial crisis of 2007–2009, as various banking policymakers revisit loan loss provisioning rules, there have been calls to study
the Spanish experience. The Spanish provisioning system has been credited
by most market observers as positioning its banks to avoid the strain that
their international peers experienced in 2007. This section reviews the historical developments that led to Spain’s adoption of dynamic provisioning.
Subsequently, we provide methodological details of the original dynamic provisioning approach implemented in 2000 and the modifications made in 2004.
Lastly, we briefly discuss the position of Spanish banks at the end of 2006, just
prior to the onset of the financial crisis, which will allow readers to gauge the
efficacy of the policy; 2006 is the year of emphasis for our analysis because the
primary intent of dynamic provisioning is to prepare banks to absorb losses,
which, if effective, should have been accomplished by year’s end in 2006.
Why is Spain a Relevant Example?
Spain makes for an interesting case in reference to an international financial
crisis precipitated by a widespread housing boom.22 Spain’s housing sector
boom greatly outpaced the United States’, Japan’s, and the United Kingdom’s,
as seen in Figure 4. Spanish loan loss provisions historically demonstrated
high procyclicality with the business cycle. From 1991–1999, Spain’s correlation between loan loss provisioning levels and GDP was −0.97, the highest
in the Organisation for Economic Co-operation and Development (OECD).
Similarly, in 1999, Spain had the lowest level of loan loss provisions (to total
loans) of any OECD country (Saurina 2009a). In response to pronounced
procyclicality, the Bank of Spain in 2000 implemented a countercyclical
22 For a discussion of the housing boom in Spain, see Garcia-Herrero and Fernández de Lis
(2008).
400
Federal Reserve Bank of Richmond Economic Quarterly
Figure 4 Gross Fixed Capital Formation in Housing as a Percentage of
GDP, 1977–2006
16
14
12
Ireland
Japan
Spain
United Kingdom
United States
Percent
10
8
6
4
2
0
1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005
Source: OECD data.
method of loan loss provisioning that allowed banks to build a reserve in
good times to cover losses in bad times.23, 24
The Original Model
In 1989, the Bank of Spain was authorized to establish the accounting
practices of the banks it supervises, helping to resolve the conflict between
the objectives of accountants and regulators over issues such as loan loss provisioning. The Bank of Spain has historically viewed loan loss provisioning
as a necessary policy to accomplish its goals of prudential supervision. In a
23 Subsequent to the Asian Financial Crisis, many emerging Asian economies instituted loan
loss provisioning policies (some used discretionary measures) that increased provisioning in good
times, leaving the banking system well prepared to absorb losses associated with an economic
downturn. For a detailed description of these measures, see Angklomkliew, George, and Packer
(2009).
24 For a brief summary of the Spanish method, see Saurina (2009b).
E. Balla and A. McKenna: Dynamic Provisioning
401
Table 4 Risk Categories Under Standard Approach to Statistical
Provisioning: The 2000 Methodology
Category
Without Risk (0.0%)
Low Risk (0.1%)
Medium-Low Risk (0.4%)
Medium Risk (0.6%)
Medium-High Risk (1.0%)
High Risk (1.5%)
Description
Risks involving the public sector
Mortgages with outstanding risk below 80 percent of the
property value, as well as risks with firms whose
long-term debts are rated at least “A”
Financial leases and other collateralized risks (different
from the former in point 2)
Risks not mentioned in other points
Personal credits to financial purchases of durable consumer
goods
Credit card balances, current account overdrafts, and credit
account excesses
regulation adopted in 2000 upon a foundation set out in Circular 4/1991, the
Bank of Spain instituted dynamic provisioning (Poveta 2000). The Bank of
Spain places significant emphasis on the growth years in a credit cycle and,
given the enhanced procyclicality recognized in Spain (see Fernández de Lis,
Pagés, and Saurina 2000), the goal of statistical provisioning is to compensate
for the underpricing of risk that takes place during those years. Statistical provisioning intends to anticipate the next economic cycle, although it is not meant
to be a pure expected loss model because it is backward-looking (Fernández
de Lis, Pagés, and Saurina 2000). The statistical fund, built by quarterly
provisions recognized on the income statement, was meant to complement
the insolvency fund, but, instead of covering incurred losses, was built from
estimates of latent losses on homogenous asset groups.
The regulation established two methods for computing the quarterly statistical provision. Banks can create their own internal model using their own
loss experience, provided that the data used spans at least one economic cycle
and is verified by the bank supervisor. Conversely, banks can use the standard
approach outlined by the Bank of Spain, based on a parameter measuring the
risk of institutions’ portfolio of loans and leases. For this analysis, our focus
will be on the standard approach for two reasons. First, the majority of institutions in Spain use the standard method and, second, the internal models are
not available publicly and therefore cannot be the subject of analysis.
The standard approach was developed on the assumption that asset risk
is homogenous. The Bank of Spain, in adopting the statistical provision in
July 2000, created six risk categories of coefficients by assets for banks to
use to take a quarterly provision. The coefficients, shown in Table 4, are
based on historical data of the average net specific provision over the period
402
Federal Reserve Bank of Richmond Economic Quarterly
1986–1998, meant to reflect one economic cycle in Spain.25 Similar to loan
loss provision methods under International Financial Reporting Standards
(IFRS) and generally accepted accounting principles (GAAP), the statistical provision is recognized on the income statement and thus has the effect
of reducing profits when the difference between the statistical and specific
provision is positive. When the statistical provision is negative it reduces the
statistical fund, which cannot be negative, and increases profits. The statistical
provision is not a tax-deductible expense. To limit the size of the statistical
fund, which is a function of the type of assets a bank holds and the duration of
economic growth, the fund was capped at 300 percent of the coefficient times
the exposure.
The Current Model
In 2004, the Bank of Spain made several modifications to its approach to the
statistical provision to conform to the IFRS guidance adopted by the European
Union, becoming compulsory for Spanish banks in 2005.26 The following
equation sets out the new approach:
6
6
Specific provisionit
General provisiont =
α i Cit +
βi −
Cit .
Cit
i=1
i=1
The new model retains a general risk parameter, α, meant to capture the
different risks across homogenous categories of assets. For each asset class, α
is the average estimate of credit impairment in a cyclically neutral year; and,
although it is meant to anticipate the next cycle, it is not intended to predict it.
Two components of the α parameter—that it is backward-looking and unbiased
(cyclically neutral)—are crucial in differentiating the policy from an expected
loss approach, which would attempt to gauge the characteristics of the next
economic cycle through forecasting methods or incorporating expectations
about future economic performance. Rather, the use of cyclically unbiased
historical data allows provisions to be taken based on the past experience
of assets. This feature removes the potential for conflict between banking
regulators and banks because there is no discretion over the expectations of
25 Fernández de Lis, Pagés, and Saurina (2000) indicate that the data incorporate banks’ credit
risk measurement and management improvements without specifically providing details.
26 The Spanish experience could be considered unique in that the central bank is also the
accounting standard setter for banks. Safety and soundness prudential decisions are not made in
an accounting vacuum as illustrated by the need for changes to the Spanish model in 2004. The
ongoing debate in many countries sets those who argue that a countercyclical reserve account
could be made transparent in financial reporting (and sophisticated investors would differentiate
the statistical build-up) against the accounting standard setters’ reluctance to turn bank financial
reports into prudential regulatory reports. Any LLR reform would involve compromise between
bank regulators and accounting standard setters.
E. Balla and A. McKenna: Dynamic Provisioning
403
Table 5 Risk Categories Under Standard Approach to Statistical
Provisioning: The 2004 Methodology
Category
Negligible Risk (α = 0%, β = 0%)
Low Risk (α = 0.6%, β = 0.11%)
Medium-Low Risk (α = 1.5%, β = 0.44%)
Medium Risk (α = 1.8%, β = 0.65%)
Medium-High Risk (α = 2.0%, β = 1.1%)
High Risk (α = 2.5%, β = 1.64%)
Description
Cash and public sector exposures (both
loans and securities)
Mortgages with a loan-to-value ratio
below 80 percent and exposure to
corporations with a rating of “A” or higher
Mortgages with a loan-to-value ratio above
80 percent and other collateralized loans
not previously mentioned
Other loans, including corporate exposures
that are non-rated or have a rating below
“A” and exposures to small- and mediumsize firms
Consumer durables financing
Credit card exposures and overdrafts
economic performance that may lead to over- or underprovisioning. It is also
beneficial for investors because it provides transparency of provisioning.
In contrast to the original model, the 2004 approach includes a β parameter
that interacts with the specific provision (see Table 5). β is the historical
average specific provision for each of the homogenous groups. The interaction
between β and the specific provision measures the speed with which nonspecific provisions become specific for each asset class. Cit represents the
stock of asset i at time t. The limit of the statistical fund was modified to 1.25
times latent exposure.27
Spanish Banks in 2006
What difference did dynamic provisioning make for Spain?28 In 2006, the
Spanish banking system had by far the highest coverage ratio (the ability of
the LLR to cover non-performing loans) among Western European countries
at 255 percent. At the same time, the U.S. aggregate coverage ratio was 176
percent.29 Figure 5 shows the coverage ratios for Spain and the United States
and also Spain’s European peers as of 2006. To reiterate, 2006 was important
27 The Bank of Spain defines latent loss generally as the probability of default times loss
given default. See Saurina (2009a).
28 As part of the Financial Sector Assessment Program, the International Monetary Fund
(2006) states that the Central Bank of Spain has pioneered a rigorous provisioning system that
enhances the safety and soundness of Spanish banks. While we will briefly discuss the condition
of Spanish banks in 2006, see Saurina (2009d) for a more detailed review.
29 The data were obtained from Banco de España (2008), U.S. Call Reports, Saurina (2009c)
and IMF cited in Catan and House (2008).
404
Federal Reserve Bank of Richmond Economic Quarterly
Figure 5 Loan Loss Reserves as a Percentage of Non-Performing Assets
300
250
Percent
200
150
100
50
ly
Ita
ga
l
Sw
ed
en
Au
str
ia
Fr
an
ce
Gr
ee
ce
Ire
lan
Ne
d
th
er
Un
lan
ite
ds
d
Ki
ng
do
m
No
rw
ay
Be
lgi
um
Po
r
tu
d
d
lan
Ice
lan
at
itz
er
St
Sw
Un
ite
d
Sp
ain
es
*
0
Notes: *The coverage ratio for the United States is the aggregate LLR for all commercial
banks as a percentage of the aggregate non-performing loans. It was computed using Call
Report data. All other countries’ data come from the International Monetary Fund.
for measurement purposes because the primary intent of dynamic provisioning
is the timing of provisions. At the onset of the financial crisis and global
recession, dynamic provisioning worked as expected in Spain, allowing large
Spanish banks like Santander and BBVA to enter the crisis with substantial
reserve cushions relative to non-Spanish peers. Many observers, including
G20 Finance Ministers, have singled out the Spanish dynamic provisioning
system as a contributor to that banking sector’s soundness entering the financial
crisis of 2007–2009. It is beyond the scope of this article to assess what, if
any, fraction of banking sector stability in Spain is attributable to dynamic
provisioning versus other Spain-specific factors. Additional factors unique to
Spain may have a bearing were this policy to be adopted more widely.
E. Balla and A. McKenna: Dynamic Provisioning
4.
405
DYNAMIC PROVISIONING: A SIMULATION WITH U. S.
DATA
One way to illustrate potential inefficiencies of the current U.S. loan loss
provisioning framework is to simulate U.S. bank loan loss provisioning over
the last two business cycles under an alternate provisioning framework. This
section illustrates that a dynamic provisioning framework (akin to that implemented in Spain) could have allowed for a build-up of reserves during the
boom years. The results demonstrate that the alternate framework would have
smoothed provisions and bank income through the cycle. The severe drop in
bank income associated with the actual steep rise in loan loss provisioning
during the financial crisis of 2007–2009 would have been substantially reduced. With positive net income in its place, banks could have increased their
capital through internal means and thus reduced the need for assistance from
the U.S. government. Note that no precise conclusions can be reached as to the
magnitude of these effects from the exercise. Any conclusions based on the
simulation are subject to the Lucas Critique in that a change in LLR methodology is likely to influence bank lending behavior (Lucas 1976). Lending and
other related variables would therefore take on values different from those
actually observed and used in the simulation. Nonetheless, the simulation is a
useful illustration of the relationship between loan loss provisioning and bank
income under the two different frameworks.
Using aggregate U.S. data from the year-end quarterly FDIC banking reports (key actual data presented in Figure 6),30 we created an example bank.
We have populated the example bank’s financial report with aggregates from
a hypothetically consolidated U.S. commercial banking industry so as to observe the interaction of a dynamic provisioning approach with historical U.S.
banking data. We apply the 2004 Spanish provisioning methodology to the
financials of the example bank so as to display and describe the technical nuances of dynamic provisioning. The purpose of this exercise is to demonstrate
a well-functioning variant of the policy, that is, how the policy should work,
not how it would or would have worked. Accordingly, several simplifying
assumptions were required to complete the simulation.
First, we selected representative α and β parameters to compute the statistical provision on a yearly basis. We selected 1.8 percent and 0.65 percent,
respectively. While an attempt could have been made to approximate the parameters for the example bank, the available banking data does not permit that
level of analysis.31 Further, these parameters allowed us to show the results
under a case where the policy is effective and, by examining the results under
30 FDIC Quarterly Banking Profile: 1998–2008
31 The coefficients used in the Spanish standard approach are estimated from a large credit
database maintained by the Bank of Spain. No equivalent consolidated credit database exists in
the United States as of the publication of this article.
406
Federal Reserve Bank of Richmond Economic Quarterly
Figure 6 Aggregate U. S. Bank Loan Loss Provisions and Net Income as
a Percentage of Total Loans, 1993–2008
2.50
Recession
Loan Loss Provisions
Net Income
2.00
Percent
1.50
1.00
0.50
0.00
1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008
Source: FDIC data.
this example, readers should be able to also grasp how the results would vary
under the case where the parameters were either too high or low.
Second, we had to select a time period for illustrative purposes. The
simulation for the example bank begins in 1993 with five years devoted to
building the statistical fund. We targeted a build-up of about 2 percent of
total loans by 1999 (close to the largest statistical fund possible of 1.25 α).
Subsequent to 1999, the statistical fund was built using only the statistical
provision, shown in line 6 of Table 6 (all line references for the simulation
are from Table 6). Line 13 is the actual allowance for loan and lease losses
(ALLL),32 while line 14 is the ALLL plus the statistical fund, equal to the total
stock of provisions. Line 16, the actual loan loss provision for the year divided
by total loans and leases (LLP/TLL), and line 17, the five-year moving
32 ALLL is the formal accounting term for loan loss reserves.
1999
3,491
7.81%
2000
3,820
9.40%
2001
3,889
1.83%
2002
4,156
6.86%
2003
4,429
6.55%
2004
4,904
10.74%
2005
5,380
9.70%
2006
5,981
11.17%
2007
6,626
10.80%
2008
6,840
3.22%
69,395
4,411
73,238
5,714
74,494
976
62,229
4,492
54,873
4,677
59,550
8,392
67,942
8,387
76,329
10,646
86,976
11,249
82,077
2,862
3,843
1,256
0
0
4,677
8,392
8,387
10,646
0
0
0
0
12,265
7,356
0
0
0
0
4,899
86,404
69,395
78,554
73,238
85,940
62,229
87,513
54,873
93,519
54,873
99,649
59,550
110,351
67,942
121,051
76,329
134,567
82,077
149,094
0
153,900
0
73,238
58,770
132,008
0
74,494
64,137
138,631
0
62,229
72,323
134,552
0
54,873
76,999
131,872
0
59,550
77,152
136,702
0
67,942
70,990
138,932
0
76,329
68,688
145,017
0
86,976
68,984
155,960
0
82,077
89,004
171,081
0
0
156,152
156,152
21,814
0.62%
30,001
0.79%
43,433
1.12%
48,196
1.16%
34,837
0.79%
26,098
0.53%
26,610
0.49%
25,583
0.43%
57,310
0.86%
151,244
2.21%
0.61%
0.67%
0.78%
0.87%
0.89%
0.88%
0.82%
0.68%
0.62%
0.91%
569
4,458
13,242
11,847
0
0
0
0
16,147
89,266
407
Balance Sheet Accounts
1 TLL, in billions
2 Actual Rate of Growth
Parameter Scenarios
3
α = 1.8%
4
β = 0.65%
5
Statistical Fund,
Beginning
6
Statistical Provision
7
Statistical Provision
after Shift
8
Amount Needed from
Statistical Fund
9
Statistical Fund
after Shift
10 Statistical Fund Limit
11 Statistical Fund Limit
Reversal
12 Statistical Fund, End
13 ALLL
14 Total Stock of Provisions
Income Statement Accounts
15 LLP
16 LLP/TLL
17 Five-Year Moving
Average
18 Amount Needed from
Statistical Fund
E. Balla and A. McKenna: Dynamic Provisioning
Table 6 Dynamic Provisioning Simulation: Example Bank, 1999–2008
(in millions, except TLL)
408
Table 6 (Continued) Dynamic Provisioning Simulation: Example Bank, 1999–2008 (in millions. except TLL)
19
22
23
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
569
21,245
4,458
25,543
13,242
30,191
11,847
36,349
0
34,837
0
26,098
0
26,610
0
25,583
16,147
41,163
84,939
66,305
71,556
71,009
73,967
89,861
102,440
104,172
114,016
128,217
97,630
18,726
3,843
1,256
−12,265
−7,356
4,677
8,392
8,387
10,646
−4,899
−82,077
67,713
69,753
86,232
97,217
97,763
95,780
105,629
117,571
102,529
100,803
Federal Reserve Bank of Richmond Economic Quarterly
20
21
Amount Taken from
Statistical Fund
New Loan Loss Provision
Profits Without
Statistical Provision
Changes in Provisions
under Dynamic
Provisioning
Profits With Statistical
Provision
E. Balla and A. McKenna: Dynamic Provisioning
409
Figure 7 Aggregate U. S. Bank Reserves Compared to Total and
Statistical Reserves as a Percentage of Total Loans, 1993–2008
4.50
Recession
Statistical Fund with Dynamic Provisioning
Allowance for Loan and Lease Losses
Total Stock of Provisions
4.00
3.50
Percent
3.00
2.50
2.00
1.50
1.00
0.50
0.00
1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008
Source: FDIC data.
average of the same ratio, were used to objectively determine if funds were
needed from the statistical fund; the difference between a given year’s LLP/TLL
ratio and the five-year moving average determine exactly how much should
be drawn from the statistical fund. The statistical provision in a given year
was used to cover as much of the amount needed as possible, and then, if
necessary, the remaining amount would be taken from the statistical fund; the
amount needed from the statistical fund is listed in line 18, while the actual
amount taken is listed in line 19. If funds from the statistical provision and
statistical fund were used to cover specific provisions, then the loan loss provision in a given year would change. The new loan loss provision reflecting the
changes is shown in line 20, and the changes in provisions under the dynamic
provisioning simulation, which has an effect on net income, are shown in line
22. The new net income (profit) under the dynamic provisioning simulation
is listed in line 23.
The simulation has several interesting results. First, by allowing the bank
to use five years to build the fund (during good times from 1993–1998), the
410
Federal Reserve Bank of Richmond Economic Quarterly
Figure 8 Aggregate U. S. Bank Net Income Compared to Net Income as
a Percentage of Total Loans, 1993–2008
2.50
2.00
Percent
1.50
1.00
0.50
Recession
Profit without Dynamic Provisioning
Profit with Dynamic Provisioning
0.00
1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008
Source: FDIC data.
example bank was able to build a total stock of provisions, both statistical
and specific, of 3.9 percent of total loans. When compared to the actual (U.S.
aggregate) total allowance for loan and lease losses of 1.7 percent, the bank
is clearly better positioned to withstand losses associated with a recession
in 1999. The statistical fund was set to be constrained, if necessary, by the
aforementioned limit of 1.25 times the latent losses, which was not reached
in the simulation.
The second result to note is the impact of a recession, or increase in loan
losses, on the statistical fund. Figure 7 shows total provisions and the statistical
fund compared to the actual allowance for loan and lease losses. Both the
2001 and current recessions result in declines in the statistical fund, driving
down total provisions, compared to large increases in the actual allowance
for loan and lease losses. Since the statistical provision is recorded against
the income statement, it reduces profits, as demonstrated in the first example
E. Balla and A. McKenna: Dynamic Provisioning
411
that contrasted dynamic provisioning to the incurred loss model.33 Figure
8 shows how profits under statistical provisioning compare to actual profits,
emphasizing the impact this has on earnings.
The simulation’s primary purpose was to depict the results of dynamic
provisioning under one possible scenario with representative parameters selected to demonstrate how the policy should work. The selected parameters,
combined with the other simplifying assumptions, allow a sizeable fund of
statistical provisions to be built in good times and absorb losses in bad times.
However, if representative parameters were selected that were too low, higher
levels of specific provisioning would have prevented the appropriate fund from
being built. Conversely, if higher parameters were selected, then the fund built
would have been larger, which could have resulted in inefficiently used capital given the size of realized losses over the time horizon for which the bank
provisioned under the policy. This is an important point because the standard
approach in the Spanish system is dependent on the parameters estimated by
the Bank of Spain.34 The cost of underestimating the risk weights relates to
bank-specific solvency, whereas an overestimation of the risk weights is a tax
to the bank and could reduce the overall supply of credit.
The simulation illustrates how dynamic provisioning prepared the example bank to weather the economic downturn while remaining profitable and
leaving its capital levels in periods of economic growth accurately stated and
allowing managers to further build capital if deemed necessary.
5.
CONCLUSIONS
Current accounting guidelines require banks to recognize losses prompted by
events that make the losses probable and estimable. But this method may be
at odds with the bank regulators’ desire for banks to build a “war chest” of
reserves in good times to be depleted in bad times. In 1998, the SEC required
SunTrust Bank to restate its LLR by $100 million, reflecting the SEC’s aversion
to what it considered to be an overstated reserve. Remarkably, the action taken
by the SEC wasn’t long after the banking crises of the 1980s and early 1990s,
which, in 1990 alone, resulted in a total provision for loan losses almost twice
bank profits (Walter 1991). By the same measure, the financial crisis that
began in 2007 was far more severe, resulting in total loan loss provisions
greater than eight times bank profits in 2008.35 Following the current episode,
33 Note that both examples treat loan growth rate as independent of reserve rules. Dynamic
provisioning is likely to smooth out loan growth expansion and contraction. One could argue that
the rules could impose a high enough cost as to result in lower loan volume through the credit
cycle.
34 Even banks that use internal models are subject to approval by the Bank of Spain. Presumably, the approval process could involve some reference to the standard risk weights.
35 FDIC Quarterly Banking Profile, December 31, 2008.
412
Federal Reserve Bank of Richmond Economic Quarterly
Table 7 Aggregate Condition and Income Data: All Commercial
Banks, 2008 (in millions)
Assets
Total Loans and Leases
Less: Reserve for Loss
Net Loans and Leases
Securities
All Other Assets
Total Assets
$6,839,998
156,152
6,683,846
1,746,539
3,882,529
12,312,914
Liabilities and Capital
Deposits
$8,082,104
Other Borrowed Funds
2,165,821
Subordinated Debt
182,987
All Other Liabilities
724,353
Equity Capital
1,157,648
Total Liabilities and Capital
12,312,914
Source: FDIC Quarterly Banking Profiles, 2008.
many have called for revisiting the loan loss provisioning method so as to
allow for the “war chest” approach and to mitigate the current approach’s
procyclical effects.
We compared the incurred loss (as implemented in the United States)
and dynamic provisioning (as implemented in Spain) approaches. Through a
basic example we illustrated that the key difference is not the aggregate level
of provisioning but the timing of the provisioning. We conducted an empirical
simulation to illustrate that a dynamic provisioning framework, like the one
implemented in Spain, could have allowed for a build-up of reserves during
the boom years in the United States. The results demonstrate that the alternate
framework would have smoothed bank income through the cycle and the need
to provision for loan losses would have been significantly lower during the
financial crisis of 2007–2009.
Both in the context of a conceptual framework and through an empirical
simulation, we highlighted that dynamic provisioning could mitigate some
of the problems associated with the current U.S. system. Even so, it would
be premature to advocate the adoption of dynamic provisioning in the United
States. We have three reasons for this opinion. First, the distortions embedded
in the current U.S. system need to be more fully understood and quantified.
Second, although 2006 is a relevant point in time for measuring the efficacy of
the Spanish policies, Spanish banking needs to be studied through this cycle
and the lessons learned need to be included in potential reform. Third, and
most important, the U.S. approach to loan loss reserves should not be reformed
independently of other bank capital regulation reform.36
36 To allow for a focus on LLR issues, this article abstracted away from any discussion of
broader reform of regulatory capital.
E. Balla and A. McKenna: Dynamic Provisioning
413
Figure 9 The Balance Sheet
Assets = Liabilities
+
Assets = Liabilities
+
Contributed
Capital
+
Assets = Liabilities
+
Contributed
Capital
+ Retained Earnings, + Net Income - Dividends
Beginning of Period
for Period
for Period
Assets = Liabilities
+
Contributed
Capital
+ Retained Earnings, + Rev. - Exp. - Dividends
for Period
Beginning of Period
for Period
Shareholders' Equity
Retained Earnings
Source: Stickney and Weil 2007.
APPENDIX:
ACCOUNTING FOR LOAN LOSSES
The primary function of banking is to collect depositors’ funds to provide
them with transactional support and savings. Banks use depositors’ funds to
invest in loans and securities that provide the yields necessary to conduct their
operations. As shown in Table 7, as of the second quarter of 2009 loans and
securities represented 54.3 and 14.2 percent of total assets for all commercial
banks, respectively, while deposits represented 65.6 percent of total liabilities
and capital.
As discussed in the subsection entitled “Incurred Loss Accounting,” accounting for loan losses is handled under FAS 114 and 5. Under FAS 114,
bank managers recognize individually identified losses—that is, losses that
managers believe are probable and can be reasonably estimated. FAS 5 provides for assessment of losses of homogeneous groups of loans and, similar
to FAS 114, should be probable and estimable.37 In Table 7, listed below
37 Provided in GAAP guidance in March 1975.
414
Federal Reserve Bank of Richmond Economic Quarterly
Figure 10 Balance Sheet Accounts
Any Asset Account
Beginning
Balance
Increases Decreases
+
Debit
Credit
Ending
Balance
Beginning
Balance
Any Shareholder's
Equity Account
Beginning
Balance
Decreases Increases
+
Debit
Credit
Decreases Increases
+
Debit
Credit
Any Liability Account
Ending
Balance
Ending
Balance
the total loans and leases, is the reserve for losses, an account that represents
the total value of loans and leases that managers expect is probable to not be
collected as of the balance sheet data.
To fully understand the reserve for losses, a brief review of accounting
basics is helpful. Figure 9 helps provide a simplified understanding of the
balance sheet accounts: assets, liabilities, and shareholders’ equity. The most
important concept is the relationship between the income statement accounts
(revenue and expenses) and shareholders’ equity. Net income for a firm is
computed as revenues for the period minus expenses. Net income flows to
shareholders’ equity, dividends are paid, and the remaining funds are added
to the accumulated retained earnings. And, since retained earnings and contributed capital make up shareholders’ equity, negative retained earnings will
reduce total capital.
Another important concept is that of credits and debits. As shown in Figure
10, a debit serves to increase an asset account, while it reduces a liability and
shareholders’ equity account. That is, referencing Table 7, loans and leases,
securities, and all other assets are increased with a debit to their respective
accounts and therefore, at yearend, should have a remaining debit balance.
A credit to those accounts would serve to reduce the value of assets on the
balance sheet.
In accounting, there are contra-accounts—assets, liabilities, or shareholders’ equity accounts that have a negative balance for the account; the opposite
of where Figure 10 has the beginning balances shown. The reserve for loan
losses is an example of a contra-asset. As previously mentioned, the purpose of
E. Balla and A. McKenna: Dynamic Provisioning
415
Table 8 Aggregate Condition and Income Data: All Commercial
Banks, 2008 (in millions)
Income Data
Total Interest Income
Total Interest Expense
Net Interest Expense
Provision for Loan and Lease Losses
Total Noninterest Income
Total Noninterest Expense
Securities Gains (Losses)
Applicable Income Taxes
Extraordinary Gains, Net
Net Income
Net Charge-offs
Cash Dividend
Retained Earnings
$530,513
210,569
319,944
151,244
193,853
329,050
−14,066
6,163
5,452
18,726
87,990
42,724
22,355
the reserve for losses is to accurately reflect the value of total loans and leases
on the balance sheet. As a contra-asset—meaning it has a negative (credit)
asset balance—it serves to reduce the value of assets by the amount that bank
managers believe it will not collect; hence, it reflects the true value of loans
and leases to the bank. The reserve for losses is increased on a quarterly basis
when losses are recognized and charged against income through an account
called the provision for loan and lease losses. The income statement results
for all FDIC-insured commercial banks (Table 8) show that the total provision
for loan and lease losses was $151,244, greater than eight times bank profits
(net income).
The provision for loan and lease losses is an expense account. All income
statement accounts (i.e., revenues and expenses) affect shareholders’ equity.
An expense account is a debit to shareholders’ equity and reduces its value.
Therefore, the provision for loan and lease losses reduces net income and
shareholders’ equity.
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Economic Quarterly—Volume 95, Number 4—Fall 2009—Pages 419–454
The U.S. Establishment-Size
Distribution: Secular
Changes and Sectoral
Decomposition
Samuel E. Henly and Juan M. Sánchez
E
stablishment heterogeneity has been modeled in economics at least
since the seminal work of Lucas (1978). More recently, this feature
has been incorporated into calibrated models to provide quantitative
evaluations of different mechanisms. This article aims to contribute to this
literature by providing a set of facts about the establishment-size distribution
since the 1970s that may be used to calibrate and test the predictions of these
models.
First, this article analyzes establishment data from 1974–2006.1 During
this period, the number of workers (size) of a “representative establishment” is
relatively constant. Next, the analysis turns to the dispersion of establishment
sizes. The size distribution of establishments has become slightly more even.
The same analysis is then applied at the sector level. Service establishments
became larger and service labor became more concentrated in large establishments while opposite trends were observed in manufactures. Although these
intrasector shifts played an important role in explaining aggregate movements, intersector changes were also found to be important. Finally, this
article considers whether trends in the firm-size distribution resemble those
We gratefully acknowledge comments from Anne Stilwell, Devin Reilly, Kartik Athreya,
Marianna Kudlyak, and Ned Prescott. All remaining errors are our own. The views expressed
in this paper are those of the authors and do not necessarily reflect those of the Federal
Reserve Bank of Richmond or the Federal Reserve System. E-mail: sam.henly@rich.frb.org;
juan.m.sanchez@rich.frb.org.
1 Two alternative production units will be considered—firms and establishments. A firm may
be a collection of establishments. For instance, Walmart is one firm but it has more than 4,000
establishments in the United States.
420
Federal Reserve Bank of Richmond Economic Quarterly
found in establishments. They are similar, although labor became slightly
more concentrated in large firms.
Davis and Haltiwanger (1989) also analyze secular trends at the establishment level.2 In particular, they study changes in the establishment-size
distribution during the period 1962–1985. First, they study how workers are
distributed across establishments; they find that the “representative” worker
was working in a larger establishment in 1962 than in 1985. Second, they
consider the establishment-size distribution; conversely, they find that the
“representative” establishment was smaller in 1962 than in 1985.3 The opposite behavior of these series reveals a decline in the dispersion of establishment
size. Davis and Haltiwanger also decompose these changes by sector. They
find that “changes in the industry distribution of employment and movements
in the employee size distribution within the average two-digit industry make
roughly equal contributions to the secular shift towards mid-size establishments in the aggregate economy.” This article extends part of their work
through 2006 and complements it with an analysis of firm data and alternative
statistics, figures, and decompositions. The earlier change in the first moments contrasts with the finding in this article, while the downward trend in
the dispersion of establishment size continued after 1985.
Buera and Kaboski (2008) also study the evolution of the scale of production and sectoral reallocation. They emphasize the difference between the
size distribution for manufactures and services establishments. Additionally,
they present evidence of the rise in the size of service establishments and the
reallocation of resources from manufacturing to services.4 Our article extends
their analysis by studying changes in the size distribution of manufacturing
and service establishments over time.
Several studies take an interest in which distribution best fits the firmsize distribution. Gibrat (1931) finds that the log-normal distribution effectively described French industrial firms. This distribution is a consequence of
the “law of proportional effect,” also known as Gibrat’s Law, whereby firm
growth is treated as a random process and growth rates are independent of
firm size (Sutton 1997). As noticed by Axtell (2001), census data display
monotonically increasing numbers of progressively smaller firms, a shape the
log-normal distribution cannot reproduce. Using data from the U.S. Census
Bureau from 1988–1997, Axtell (2001) shows that firm size is approximately
Zipf-distributed. Although we find that the aggregate distribution is relatively
2 See also Davis and Haltiwanger (1990) and Davis, Haltiwanger, and Schuh (1996).
3 Our article also considers the distribution of employees by establishment size and the dis-
tribution of establishments by size. Notice that while the latter describes which proportion of the
establishments is of a given size, the former studies which proportion of employees work in an
establishment of a given size.
4 They also show evidence of sectoral reallocation for 30 countries.
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
421
stable, results for manufacturing and services suggest that it would be interesting to extend Axtell’s analysis to the sectoral level.
Recent articles use establishments data to study economic development.
They argue that the misallocation of resources among heterogeneous establishments may be a key determinant of cross-country income differences.
Banerjee and Duflo (2005) conclude that “the microeconomic evidence indeed suggests that there are some sources of misallocation of capital, including
credit constraints, institutional failures, and others.” Restuccia and Rogerson
(2008) illustrate this mechanism using a model with establishment heterogeneity similar to Hopenhayn and Rogerson (1993). In a similar framework,
Hsieh and Klenow (2007) find that productivity would increase by 30–50
percent in China and 40–60 percent in India “if capital and labor were reallocated to equalize marginal products across plants to the extent observed
in the U.S.” Similarly, Greenwood, Sánchez, and Wang (2008) study the role
of informational frictions for economic development in a model with establishments heterogeneity.5 All the theories above analyze mechanisms that
may contribute to an explanation of differences in income across countries.
The calibrations of these and similar models generally use targets from the
size distribution. For instance, Restuccia and Rogerson (2008) use the 2000
establishment size distribution and Greenwood, Sánchez, and Wang (2008)
use the Lorenz curve for the distribution of employment by establishment size
for 1974. The subsequent sections of this article present evidence for size
distributions of establishments and firms and supply a set of stylized facts that
new theories in this strand of literature may find useful as calibration targets.
Perhaps more importantly, these sections analyze secular changes in the size
distribution that could be used to test the predictions of these models. For
example, we find that the average size of establishments is fairly constant (or
slightly decreasing) over the last 30 years. This finding supports models in
which the average size is constant on the balanced-growth path.
The remainder of the article is organized as follows. Section 1 introduces
and summarizes our findings. Section 2 describes the secular changes in the
establishment-size distribution. The decomposition of secular changes into
changes in the sectoral composition (intersector) and distribution changes
within each sector (intrasector) is undertaken in Section 3. A description of
the data on firms, as an alternative to establishments, is presented in Section
4. Finally, Section 5 concludes. An Appendix presents detailed information
about data sources, formulae used to compute the statistics, and some figures
and tables.
5 See also Caselli and Gennaioli (2003); Amaral and Quintin (2007); Alfaro, Charlton, and
Kanczuk (2008); Bartelsman, Haltiwanger, and Scarpetta (2008); Buera and Shin (2008); Guner,
Ventura, and Yi (2008); and Castro, Clementi, and McDonald (2009).
422
1.
Federal Reserve Bank of Richmond Economic Quarterly
PRODUCTION UNIT SIZE TRENDS, 1970–2006
In the sections below, several statistics are defined and used to evaluate the
distributions of productive units and their workers from the 1970s to 2006.
The aggregate economy, as well as two component sectors (manufacturing
and services), are considered in each analysis.
Section 2 develops statistics and functions that are used in the analysis of
trends in establishment size and shifts in the dispersion of establishments and
workers. We find that the aggregate establishment size changes negligibly.
Manufacturing establishments are very large and shrink over time, while service establishments are initially smaller than average but become much larger
by 2006. Variation of establishment size does not change significantly apart
from a small increase in the service sector. The distribution of employees
across establishments becomes slightly more even. This trend is driven by
the decline of large manufacturing firms and dampened by increased labor
concentration in services.
Section 3 decomposes, by sector, several statistics introduced in Section
2. The results are used to disentangle changes in aggregate statistics caused by
intrasector distribution movements from those caused by shifts in the sectoral
composition of the aggregate (intersector changes). We find that both intraand intersector movements are important, but the importance of each varies
by statistic.
Section 4 examines the question of whether and when firm distribution
patterns should resemble those found in establishments. We argue that movements in establishment distributions should be more similar to those in firms
when large firms are composed of relatively large establishments, and present
evidence is consistent with this hypothesis. Trends in the aggregate and sectoral distributions of firms and employees across firms generally conform to
trends at the establishment level.
2.
SECULAR CHANGES IN THE SIZE DISTRIBUTION OF
ESTABLISHMENTS
The U.S. Census Bureau (USCB) publishes annual data on establishments
in their County Business Patterns series. This section presents a variety of
statistics derived from these data. The statistics describe the size distribution
of establishments and the dispersion of labor and establishments across establishments. Major trends in these statistics since 1974 are noted and depicted
in Figures 1–8.
County Business Patterns Data
County Business Patterns (CBP), released by the USCB annually since 1964,
contains tables listing establishment quantity, worker quantity, and payroll by
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
423
Table 1 Example Establishment Data
Size Group
1–2 Workers (Small)
3–4 Workers (Large)
Establishment Size
1
2
3
4
Number of Establishments
5
2
2
1
establishment size groups. For example, CBP tables in any given year list the
number of establishments employing 20–49 workers, the number of people
employed by those establishments, and other data (like payroll) not used in
this article. Similar data are provided for other establishment size groups (1–4
workers, 5–9 workers, etc.). This information is given for the aggregate and
also by SIC (1997 and earlier) or NAICS (1998 onward) industry category.
We use data for years 1974 and later due to a significant methodological shift
taking place between 1973 and 1974.6
A caveat is in order. In the service sector, data for years before and after
1997 are not directly comparable: After 1997, an establishment’s sector was
determined by the North American Industrial Classification System (NAICS),
which is not easily reconciled with the Standard Industrial Classification (SIC)
system used for the same purpose in previous years.7 Consequently, analysis
of labor concentration across service sector establishments treats SIC years
(1974–1997) and NAICS years (1998–2006) separately. The composition
of the manufacturing sector also changes with NAICS, but a single series is
available under each system and differences are minimal.
Mean Establishment Size and Coworker Mean Size
Two different measures of mean size will be considered to describe the size of
a “representative” establishment. Given data restrictions, the comparison of
these two measures will be used later to study the dispersion of establishments
by size.
It may be useful to consider the world described in Table 1, where establishments have between one and four employees (inclusive) and are separated
into two size groups: two or fewer workers and three or more workers. In
6 Some data were retrieved from the National Historical Geographic Information System, an
online database operated by the Minnesota Population Center (Ruggles et al. 2009).
7 Under the SIC system, a single series summing all portions of a “service” sector was
available. NAICS split the sector into numerous constituents (educational services; health care and
social assistance; professional, technical, and scientific services; and so on). A composite service
sector was constructed from these NAICS service subsectors (see Appendix) but it was not possible
to precisely recreate the SIC service sector’s composition.
424
Federal Reserve Bank of Richmond Economic Quarterly
Employees per Establishment
Figure 1 Mean Size of Establishments, 1974–2006
Manufacturing
Services
Aggregate
60
45
30
15
0
1975
1980
1985
1990
1995
2000
2005
this world, a “small” group is comprised of seven establishments employing a
total of nine workers; the remaining three establishments form a “large” group
employing 10 workers.
Establishment mean size
We begin by asking: What is the average establishment size across establishments? The answer is the mean of the distribution of establishments by
establishment size, referred to hereafter as the mean size of establishments
(or simply as the establishment mean) and denoted E(esize). Denote index
establishment size groups by i. Then, we obtain the establishment mean by
taking a weighted sum of the expected size of establishments within each size
group i:
E(esize) =
E(esize | egroup = i) ∗ P (egroup = i).
(1)
i
Here, egroup = i is the condition in which an establishment is a member of
size group i.8 Considering our example world, we find that
E(esize) = [9/7] ∗ (7/10) + [10/3] ∗ (3/10) = 1.9.
8 Calculations of expected values and probabilities are detailed in the Appendix.
(2)
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
425
Figure 1 displays the mean size of establishments between 1974 and 2006.
Across the period, this mean changes negligibly: In 1974, the average establishment employed about 15 workers, a figure that ranged between 14 and 16
workers in subsequent years through 2006. This constancy in the aggregate
masks significant shifts at the sector level. The average manufacturing establishment size fell from almost 70 employees in the late 1970s to about 41
employees in 2006. The greatest decline occurred between 1979 and 1983,
when the average size dropped from 67 employees to 52 employees. In spite of
this decline, manufacturing establishments tend to be much larger than other
establishments in all years. For instance, in 1974 the average manufacturing
establishment employed about 50 more workers than the aggregate economy’s
average establishment; this gap was halved by 2006. Contemporaneously, the
average service sector establishment increased in size, from about 11 workers
in 1974 to 14.7 workers in 1997 and from 14.8 workers in 1998 to 16 workers
in 2006.
Coworker mean size
What is the average number of coworkers across workers? The answer is the
mean of the distribution of workers by establishment size, referred to hereafter
as the coworker mean size of establishments or simply the coworker mean,
denoted E(wsize). This statistic is interesting because it may vary even when
the mean size of establishments is constant.9 The following formula can be
used to compute this measure:
E(wsize) =
E(wsize | wgroup = i) ∗ P (wgroup = i),
(3)
i
where wgroup = i denotes a worker who is employed by an establishment in
size group i. In our example, we have data that allow us to compute E(wsize)
directly:
((1 ∗ 5) + (2 ∗ 4))
9
((3 ∗ 6) + (4 ∗ 4))
10
]∗( )+[
]∗( )
9
19
10
19
≈ 2.47.
(4)
E(wsize) = [
Unfortunately, E(wsize) cannot be computed directly from public CBP data
because we are unable to obtain E(wsize | wgroup = i) without information
about the distribution of workers within size groups. We use an alternative
method of computation that employs an assumption about the distribution of
establishments within size groups.10
9 This was actually the case for the time period studied by Davis and Haltiwanger (1989).
10 See details in the Appendix.
426
Federal Reserve Bank of Richmond Economic Quarterly
Figure 2 Coworker Establishments Mean Size, 1974–2006
Coworker Mean of Establishments
1,600
Manufacturing
Services
Aggregate
1,400
1,200
1,000
800
600
400
200
0
1975
1980
1985
1990
1995
2000
2005
Figure 2 shows the coworker mean size of establishments. As expected,
worker mean size is much greater than establishment mean size. In 1974, the
worker mean stands around 830 at the aggregate level, 1,560 for manufactures,
and 480 for services. Subsequent trends resemble those for the mean size of
establishments. The aggregate worker mean remains fairly flat through 2006,
dropping 11 percent. Simultaneously, the coworker mean in manufactures is
halved (falling from 1,560 to 760) even as the services coworker mean doubles
(480 to 970).
Establishment Size Dispersion and Employment
Concentration
Coefficient of variation
The statistic used to analyze the dispersion of establishment size is the coefficient of variation (CV ). It measures the dispersion of establishment size
relative to the mean size.11
11 This statistic is computed from equation (18) in the Appendix.
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
427
Figure 3 Coefficient of Variation of Establishments Size, 1974–2006
10
Coefficient of Variation
8
6
4
Manufacturing
Services
Aggregate
2
0
1975
1980
1985
1990
1995
2000
2005
The coefficients of variation for the aggregate and for industries are displayed in Figure 3. In the aggregate, this measure fell about 8 percent from
1974 to 2006 (7.2 to 6.1). The coefficient also fell slightly in the manufacturing sector, from 4.7 to 4.2; note that this figure indicates a much lower
variation in establishment size than is present in services or the aggregate. Service establishments actually saw their coefficient increase about 21 percent
(6.3 to 7.8).
Large establishment employment share
The fraction of workers employed by very large establishments (those with
more than 1,000 workers) serves as a simple measure of labor concentration
(Figure 4). In the aggregate this figure decreased slightly. Very large establishments employed about 16 percent of all workers in 1974. By 2006, they
were responsible for only 13 percent of employment, although this number
had earlier dipped to a 1987 nadir of 12.5 percent. In the manufacturing sector, a decline in large establishment employment share was observed. Large
establishments employed 29 percent of manufacturing workers in 1974; in
2006, they employed only 16 percent. Finally, the large establishment share
of service labor moved erratically upward. In this sector the employment share
428
Federal Reserve Bank of Richmond Economic Quarterly
Figure 4 Employment Share of Large Establishments, 1974–2006
Manufacturing
Services
Aggregate
Employee Share
0.30
0.25
NAICS
adopted
0.20
0.15
0.10
1970
1975
1980
1985
1990
1995
2000
2005
2010
increased from 12.5 percent in 1974 to about 18 percent between 1990–1997;
from 1998–2006, the share increased from 14 percent to 17 percent.
Lorenz curve
One frequently employed instrument for the analysis of inequality is the
Lorenz curve. This measure of the distribution of labor across establishments
is independent of the absolute size of establishments. Thus, if all establishments grow or shrink proportionally, there are no changes in the Lorenz curve.
Here, a Lorenz curve represents the fraction y of total workers employed
by the fraction x of total establishments employing the smallest number of
workers. A 45◦ line means that all establishments employ the same number
of workers; the further a curve is below this line, the greater the unevenness in
worker distribution across establishments. Given the data restriction, we have
values for the Lorenz function, L, at the upper bound of each size group i:
L(P (egroup ≤ i)) = P (wgroup ≤ i).
(5)
The function is linearly interpolated elsewhere.
Panel A of Figure 5 shows the Lorenz curve for the distribution of labor
across establishments. This curve shifted slightly upward over time, suggesting a decrease in labor concentration. This movement is minor: In 1974, the
largest 5 percent establishment employed about 60 percent of the country’s
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
429
Figure 5 Establishment-Size Distribution; Aggregate Economy,
1974–2006
Panel A: Employee-Establishment Lorenz Curve, Aggregate
Cumulative Fraction Employees
0.8
1974
2006
0.6
0.4
0.2
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Cumulative Fraction Establishments
Panel B: Cumulative Employee Share, Aggregate
Cumulative Employee Share
1.0
0.8
1974
2006
0.6
0.4
0.2
0.0
10
100
1,000
Establishment Size by Employees
workers. In 2006, the same icosile employed about 57 percent of the work
force.
The manufacturing sector’s Lorenz curve is found in Panel A of Figure
6. The curve shows a clear shift upward near the top of the scale from 1974
to 2006, as the employee share of the top 5 percent establishments fell from
58.2 percent to 51.7 percent. Workers, then, became more evenly distributed
among manufacturing establishments.
430
Federal Reserve Bank of Richmond Economic Quarterly
Figure 6 Establishment-Size Distribution; Manufacturing Sector,
1974–2006
Panel A: Employee-Establishment Lorenz Curve, Manufacturing
Cumulative Fraction Employees
0.8
1974
2006
0.6
0.4
0.2
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Cumulative Fraction Establishments
Panel B: Cumulative Employee Share, Manufacturing
Cumulative Employee Share
1.0
1974
2006
0.8
0.6
0.4
0.2
0.0
10
100
1,000
Establishment Size by Employees
Service-sector Lorenz curves are located in Panels A and C of Figure
7. Over the SIC years (Panel A) the employee-establishment Lorenz curve
shifted downward: The top 5 percent establishments employed about 58 percent of all service workers in 1974 and 62 percent in 1997, reflecting a greater
concentration of employment in the largest service establishments. Service
labor also became more concentrated in large establishments in the NAICS
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
431
Figure 7 Establishment-Size Distribution; Service Sector, 1974–2006
Panel A: Employee-Establishment Lorenz, Services (SIC)
Cumulative Fraction Employees
Cumulative Fraction Employees
0.8
1974
1997
0.6
0.4
0.2
0.8
1998
2006
0.6
0.4
0.2
0.0
0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
0.0
0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Cumulative Fraction Establishments
Cumulative Fraction Establishments
Panel B: Cumulative Employee Share, Services (SIC)
1.0
0.8
1.0
1974
1997
Cumulative Employee Share
Cumulative Employee Share
Panel C: Employee-Establishment Lorenz, Services
(NAICS Composite)
0.6
0.4
0.2
0.0
0.8
Panel D: Cumulative Employee Share, Services
(NAICS Composite)
1998
2006
0.6
0.4
0.2
0.0
10
100
1,000
Establishment Size by Employees
10
100
1,000
Establishment Size by Employees
period (Panel C) when the largest 5 percent establishment employment share
rose from 1998 (56.6 percent) to 2006 (57.6 percent).
Cumulative employee distributions
To consider the distribution of workers across establishments without explicit
disregard for the absolute size of establishments (in contrast to the Lorenz
curve), we construct the cumulative distribution function (CDF). This function
provides the share of employment held by establishments of or less than a
432
Federal Reserve Bank of Richmond Economic Quarterly
particular size and is computed at the upper bound of each size group, maxi :
CDF (maxi ) = P (wgroup ≤ i),
(6)
and then linearly interpolated elsewhere.
Panel B of Figure 5 plots the CDF for the aggregate. This graph shows that
the distribution of labor across establishments shifted toward mid-size firms
between 1974 and 2006. In 1974, small establishments (10 or fewer employees) and larger establishments (more than 500 employees) are responsible for
larger shares of total employment than in 2006. This change is visible as the
2006 curve begins below the 1974 curve but rises more quickly through the
mid-size establishments. In both years, employment is nearly evenly divided
between establishments with more than and fewer than 100 workers: Establishments with 99 or fewer workers employed 53 percent of the work force in
1974 and 54 percent in 2006.
The cumulative employment curve in Panel B of Figure 6 shows that every
size group of manufacturing establishments below 500 workers increased its
employee share from 1974 to 2006. Manufacturing establishments employing
fewer than 250 workers held 56 percent of the manufacturing employment
share in 2006, up from only 42 percent in 1974.
Conversely, in both SIC and NAICS periods, cumulative employment
share curves for services (Figure 7, Panels B and D) moved to the right,
implying a broad increase in the size of service establishments (recall data in
Figures 1 and 2). Establishments employing fewer than 1,000 workers saw
their employee share drop from 88 percent to 82 percent between 1974 and
1997 and from 85 percent to 82 percent between 1998 and 2006.
Histograms
While the CDF is useful for revealing shifts in the distribution of labor across
establishments, simple histograms of the distribution of labor across establishments are helpful to identify which size groups are actually responsible
for those shifts. This function is computed as
f ((mini − 1, maxi ]) = P (wgroup = i).
(7)
where mini and maxi are, respectively, the lower and upper establishment size
bounds for size group i. The histogram for distribution of labor among size
categories at the aggregate level is depicted in the top row of Figure 8. These
histograms show movement of worker share from the smallest and largest
establishments into establishments of intermediate size. The employee share
of the smallest establishment size group decreases (1–9 workers, 15.5 percent
to 13.7 percent) while intermediate size categories see their employee share
increase. Establishments with 10–249 workers employed 50.6 percent of the
labor force in 1974, and their share increased to 56.7 percent by 2006. Larger
establishments (250–999 employees) lose employment share (18 percent to
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
433
Figure 8 Histograms for the Distribution of Labor Across
Establishments
Aggregate
1974
1991
0.20
0.20
0.15
0.15
0.15
0.10
0.10
0.10
0.05
0.05
0.05
0.00
Manufacturing
1
Services
2006
0.20
10
100
1,000
0.00
1
10
100
1,000
0.00
0.30
0.30
0.30
0.25
0.25
0.25
0.20
0.20
0.20
0.15
0.15
0.15
0.10
0.10
0.10
0.05
0.05
0.05
0.00
0.00
1
10
100
1,000
1
10
100
1,000
0.00
0.20
0.20
0.20
0.15
0.15
0.15
0.10
0.10
0.10
0.05
0.05
0.05
0.00
10
100
1,000
10
100
1,000
1
10
100
1,000
1
10
100
1,000
0.00
0.00
1
1
1
10
100
1,000
Establishment Size by Employees
16.2 percent) as do the largest establishments (1,000 or more employees; 16
percent to 13.4 percent). Large establishments lost the most share before 1991,
while small establishments lost the most after 1991.
Figure 8 also contains histograms illustrating the labor distribution across
manufacturing establishments. As in previous figures, it is apparent that manufacturing sector employment was less concentrated in large establishments
in 2006 than in 1974. Every establishment size group of 499 employees or
fewer saw significant increases in its employment share from 1974 to 1991 and
again from 1991 to 2006. Establishments employing 100–249 workers saw
434
Federal Reserve Bank of Richmond Economic Quarterly
the greatest increase over the entire period, employing about 17.5 percent of
manufacturing workers in 1974 but 21.8 percent in 2006. By contrast, the size
group 500–999 workers saw its employment share decrease from an initial
13.7 percent to 12.0 percent over the same period. This movement is in the
same direction as the 13-percentage-point decline in the employment share of
manufacturing establishments with more than 1,000 workers.
As noted earlier, the service sector is more difficult to probe due to differences in its composition before and after 1997. The last row of histograms in
Figure 8 show that between 1974 and 1991, both years using the SIC service
sector, the smallest service establishments (1–19 workers) saw their employee
share drop from 32 percent to 27 percent. Intermediate size categories (20–
249 workers) increased their employee share slightly, from 38 percent to 39
percent, and the largest size categories depicted (250–999 workers) lost 1 percentage point of total employee share (17 percent to 16 percent). The largest
size group (1,000 or more employees) accounted for most of the balance as
between 1974 and 1991 its share increased from 12 percent to about 18 percent. A histogram for 2006 shows further erosion in the employment share
of the smallest and largest establishments depicted, but these data cannot be
directly compared with data from 1974 or 1991.
3.
SECTORAL DECOMPOSITION OF SECULAR CHANGES
Changes in the Sectoral Composition
Previous sections demonstrated that, broadly speaking, manufacturing establishments have become smaller and service establishments have become larger
since the mid-1970s. The distribution of workers became more even across
manufacturing establishments and less even across service establishments.
These sector level trends offset one another in the aggregate economy. However, to better understand the cause of the slight decline in overall establishment
size and labor concentration, it is also necessary to consider changes in the
relative share of the service and manufacturing sectors over time.
Two types of effects can be cited as contributors to observed trends in
the aggregate distribution of labor across establishments. First are intrasector
movements of labor; these are described for manufacturing and service sector establishments in the previous section. Intrasector movements of labor
include shifts of employment share of different establishment size categories
and changes in the dispersion of labor across establishments. The aggregate
can also be affected by intersector forces as the relative labor and establishment share of different sectors change.
Figure 9 displays the sector shares of total employment from 1974 to
2006, and Figure 10 shows the sector share of establishments for the same
period. The pattern is similar in both figures. The participation of other sectors
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
435
Figure 9 Employment Share by Sector, 1974–2006
0.6
Sector Employee Shares
NAICS Adopted
0.5
0.4
0.3
0.2
Manufacturing
Services
All Other Sectors
0.1
0.0
1975
1980
1985
1990
1995
2000
2005
Year
is relatively constant,12 only decreasing slightly in establishments; service
sector participation rose and manufactures participation fell. Changes are
more notable in terms of worker shares: manufacturing had 32 percent in 1974
and 11 percent in 2006, while services had 19 percent in 1974 and 46 percent
in 2006. During the same period, the establishment share of manufacturing
dropped from 8 percent to 4 percent while the services establishment share
rose from 27 percent to 47 percent.
Computation
Any aggregate statistic is a weighted average of the sectoral values of that
statistic. Therefore, it can be decomposed into its sectoral constituents. As an
example, consider the mean size of establishments, the first statistic that will
be decomposed. It can be written as
E(esize) =
E(esize | esector = s) ∗ P (esector = s),
(8)
s
12 The main change seems to be in 1997, when a new sector classification system was
adopted (NAICS). Of course, this implies that this change does not have economic meaning. These
data were derived from County Business Patterns figures.
436
Federal Reserve Bank of Richmond Economic Quarterly
Figure 10 Establishment Share by Sector, 1974–2006
1.0
Manufacturing
Services
All Other Sectors
Sector Establishment Shares
0.9
0.8
NAICS Adopted
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
1970
1975
1980
1985
1990
1995
2000
2005
Year
where s is a sector index and esector = s denotes that an establishment
operates in sector s. By separating services, manufacturing, and the combined
other sectors, and simplifying the notation, the mean size of establishments
can be written
E(esize) = nserv E serv + nmanuf E manuf + nother E other ,
(9)
where E s = E(esize | esector = s) and ns is the establishment-share of
each sector, ns = P (esector = s).
This decomposition may be used to answer two questions: (1) What would
the value of a statistic (the establishment mean in this example) be if the
intersector weights had stayed at their 1974 values? and (2) what value would
the statistic have taken if the intrasector value of the statistic had stayed the
same as in 1974? The first question is answered by computing a counterfactual
statistic,
t = nserv E serv + nmanuf E manuf + nother E other .
E (esize)
(10)
1974
t
1974
t
1974
t
Similarly, the second question is answered by computing another counterfactual statistic,
serv
manuf manuf
other
t = nserv
E1974 + nother
E1974
.
(11)
E(esize)
t E1974 + nt
t
Other statistics can be decomposed in a similar manner. The only difference
is that some of them require a different weight, the sector employment share,
Statistic
Mean Size
Year 1974
Year 2006
Intrasector
Intersector
Coworker Mean
Year 1974
Year 2006
Intrasector
Intersector
Coefficients of Variation*
Year 1974
Year 2006
Intrasector
Intersector
Aggregate
Value
Manufactures
Weight
Value
Services
Weight
Value
Other Sectors
Weight
Value
15.447
15.776
16.263
13.690
=
=
=
=
0.076
0.044
0.076
0.044
65.6
41.2
41.2
65.6
0.268
0.466
0.268
0.466
11.3
15.6
15.6
11.3
0.656
0.491
0.656
0.491
11.4
13.6
13.6
11.4
845.095
754.655
690.784
618.081
=
=
=
=
0.321
0.114
0.321
0.114
1,563.4
793.9
793.9
1,563.4
0.196
0.462
0.196
0.462
479.1
970.8
970.8
479.1
0.483
0.424
0.483
0.424
516.2
508.8
508.8
516.2
7.329
6.844
6.401
7.186
=
=
=
=
0.076
0.044
0.076
0.044
102,598.9
32,687.4
32,687.4
102,598.9
0.268
0.466
0.268
0.466
5,400.1
15,185.9
15,185.9
5,400.1
0.656
0.491
0.656
0.491
5,871.8
6,944.3
6,944.3
5,871.8
Notes: *Aggregate coefficients of variation are calculated here as the square root of the sum of the products of sector weights
and variances, all over the mean establishment size.
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
Table 2 Sectoral Decomposition of Changes Between 1974–2006
437
438
Federal Reserve Bank of Richmond Economic Quarterly
defined as es = P (wsector = s), where wsector = s is the condition that a
worker is employed at an establishment in sector s. Notice that es and ns are
the shares presented in Figures 9 and 10, respectively.
Decomposition Results
Table 2 presents the decomposition of trends in intra- and intersectoral changes.
It shows how each statistic can be constructed as a weighted average of sectoral values. It also illustrates the computation of the counterfactual statistics
used for the decomposition following the logic of equations (10) and (11).
Considering only intrasector changes, the mean size of establishments would
have increased 5 percent. Only the establishment mean of the manufacturing sector fell during this period, and its weight is relatively small. Keeping
intrasector changes constant, the mean size would have dropped 12 percent.
This is clearly because services, a sector with relatively small establishments
in 1974, nearly doubled its share during this period.
Coworker mean results are substantially different. The main reason is
that when labor shares are used instead of establishment shares, manufacturing is far more important than services. Consequently, when only intrasector changes are permitted, the drop in the coworker mean of manufacturing
dominates the rise in services, and the coworker mean drops by 20 percent.
Similarly, considering only intersector changes, the coworker mean size would
have dropped 31 percent.13 Finally, Table 2 presents the decomposition of the
coefficient of variation of the establishment size distribution. The drop at the
aggregate level is 7 percent. The decomposition shows that this drop is mainly
due to intrasector changes. Keeping the share constant at 1974 levels, the drop
would have been −14 percent; if one allows only changes in the share a fall
of −2 percent is observed.
Figures 11 and 12 further resolve changes in the concentration of labor
across establishments. Notice that these figures describe the distribution of
workers across establishments, while the coefficient of variation mentioned
earlier describes the distribution of establishments across establishment sizes.
The results of this decomposition are different than those of the decomposition
of the coefficient of variation. Allowing only intrasector changes, there would
be a less equal distribution of labor across establishments in 2006 (see Figure
11). In contrast, intersector changes imply a greater shift toward a more even
distribution than the one observed during this period.
13 It is surprising in this case that with inter- or intrasector changes alone the coworker mean
would have decreased more than when both changes occurred. This happens because the coworker
mean size of services is higher than that of manufacturing in 2006, while the reverse is true in
1974. Thus, when the shares are allowed to change (not just the sectoral means), the aggregate
coworker mean size increases.
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
439
Figure 11 Intrasectorial Changes in the Establishment-Size
Distribution, 1974–2006
Panel A: Employee-Establishment Lorenz Curve, Intrasector
Cumulative Fraction Employees
0.8
0.6
1974
2006
Intrasector
0.4
0.2
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Cumulative Fraction Establishments
Panel B: Cumulative Employee Share, Intrasector
Cumulative Employee Share
1.0
0.8
1974
2006
Intrasector
0.6
0.4
0.2
0.0
10
100
1,000
Establishment Size by Employees
4.
FIRMS VERSUS ESTABLISHMENTS
Although the establishment is usually used as the production unit in models with heterogeneity in productivity, it is conceivable that the firm might
also serve in that role. Because production units in these models vary in
productivity or in their managers’ ability, one could argue that they resemble establishments. However, since financial decisions are also made at the
440
Federal Reserve Bank of Richmond Economic Quarterly
Figure 12 Intersectorial Changes in the Establishment-Size
Distribution, 1974–2006
Panel A: Employee-Establishment Lorenz Curve, Intersector
Cumulative Fraction Employees
0.8
1974
2006
Intersector
0.6
0.4
0.2
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Cumulative Fraction Establishments
Panel B: Cumulative Employee Share, Intersector
Cumulative Employee Share
1.0
0.8
1974
2006
Intersector
0.6
0.4
0.2
0.0
10
100
1,000
Establishment Size by Employees
production unit level, it might also be argued that firm data is more appropriate. If it could be shown that the distribution of labor across firms tracks labor
patterns across establishments, however, this distinction might be irrelevant.
It might be expected that small firms and small establishments, and large
firms and large establishments, will see their labor distributions move together.
Trivially, all small firms are composed entirely of small establishments, and all
large establishments are constituent parts of large firms. If large firms contain
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
441
Average Establishment Size
Figure 13 Establishment Size by Firm Size Group, 1991
350
Aggregate
Manufacturing
Services
300
250
200
150
100
50
1-4
5-9
10-19
20-49
50-99 100-249 250-499 500-999 1k-2.5k 2.5k-5k 5k-10k >10k
Firm Size Category
few small establishments, then the employment share of small establishments
will correlate strongly with the employment share of small firms; the same will
be true of large establishments and large firms. However, one may imagine a
world in which large firms are mostly composed of many small establishments,
and in this case movements in the distribution of labor across establishments
might not be clearly reflected in movements of workers among firms. Consequently, it might be expected that co-movement in labor across establishments
and across firms tends to be greater when large firms are composed of larger
establishments.
Firm Data Sources
Firm data were obtained from three Census Bureau series: Enterprise Statistics, Statistics of U.S. Businesses (SUSB), and Business Dynamics Statistics
(BDS). All series contain tallies of establishments and employees by firm size;
Enterprise Statistics and SUSB also contain a count of firms in each firm size
group. Enterprise Statistics was published consistently every five years from
1967 to 1992; SUSB was published in 1992 and annually after 1997. BDS
was constructed retrospectively from several internal census databases and is
available annually from 1977.
442
Federal Reserve Bank of Richmond Economic Quarterly
Figure 14 Firm Size and the Share of Large Firms in Total Employment
Panel A: Average Firm Size
Employees per Firm
80
60
SUSB Introduced
Manufacturing
Services
Aggregate
40
NAICS
Adopted
20
0
1970
1975
1980
1985
1990
1995
2000
2005
Panel B: Employee Share of Firms Employing 5,000 or More Workers
0.5
Manufacturing
Services
Aggregate
Employee Share
0.4
0.3
0.2
0.1
1975
1980
1985
1990
1995
2000
2005
Whenever possible, BDS data are utilized. The publication is consistent
in scope and methodology over the entire period of study. SUSB and especially Enterprise Statistics suffer from shifting definitions and sector coverage.
These deviations, and the methods used in this article to mitigate their effects,
are discussed in the Appendix.
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
443
Figure 15 Firm-Size Distribution; Aggregate Economy, 1977–2006
Panel A: Employee-Firm Lorenz Curve, Aggregate
Cumulative Fraction Employees
0.8
1972
2005
0.6
0.4
0.2
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Cumulative Fraction Firms
Panel B: Cumulative Employee Share, Aggregate
Cumulative Employee Share
1.0
1977
2005
0.8
0.6
0.4
0.2
0.0
10
100
1,000
10,000
Firm Size by Employees
Comparison Results
Figure 13 shows the average size of establishments for firms in 12 size categories in 1991; the data in this figure are typical for the sectors depicted and
for the years 1979–2005. These data were obtained from BDS. Large firms,
unlike small firms, do seem to be composed of larger establishments, and
this is even more true in the manufacturing sector than in the rest of the economy. Movements in labor distribution should be similar across establishments
444
Federal Reserve Bank of Richmond Economic Quarterly
Figure 16 Histograms for the Distribution of Labor Across Firms
Manufacturing
Aggregate
1977
2005
0.30
0.30
0.30
0.25
0.25
0.25
0.20
0.20
0.20
0.15
0.15
0.15
0.10
0.10
0.10
0.05
0.05
0.05
0.00
0.00
1
10
100
1,000 10,000
0.00
1
10
100
1,000 10,000
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
Services
1987
1
10
100
1,000 10,000
0.00
1
10
100
1,000 10,000
0.00
0.20
0.20
0.20
0.15
0.15
0.15
0.10
0.10
0.10
0.05
0.05
0.05
0.00
0.00
1
10
100
1,000 10,000
1
10
100
1,000 10,000
1
10
100
1,000 10,000
1
10
100
1,000 10,000
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
1
10
100
1,000 10,000
Firm Size by Employees
and firms, then, especially within the manufacturing sector. Indeed, evidence
presented below generally confirms firm-establishment labor co-movement in
these sectors, and to a degree in the aggregate economy, at least in the period
under examination.
Figures 14 through 18 display firm data analogous to the establishment
data. Data used in the creation of Lorenz curves (Panel A in Figures 15, 17,
and 18) and mean firm size series (Figure 14, Panel A) were obtained through
Enterprise Statistics and SUSB. Other firm figures (Panel B in Figures 14, 15,
17, and 18, as well as all of Figures 13 and 16) were derived from the BDS
series.
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
445
Figure 17 Firm-Size Distribution; Manufacturing Sector, 1974–2006
Panel A: Employee-Firm Lorenz, Manufacturing
Cumulative Fraction Employees
0.8
1972
2005
0.6
0.4
0.2
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
Cumulative Fraction Firms
0.95
1.00
Panel B: Cumulative Employee Share, Manufacturing
Cumulative Employee Share
1.0
1977
2005
0.8
0.6
0.4
0.2
0.0
10
100
1,000
10,000
Firm Size by Employees
It is clear that labor distribution movements across establishments track
those in firms. Both the aggregate and the sectoral mean size series display
the same patterns between the early 1970s and mid-2000s that are seen at the
establishment level. Intrasector changes in the distribution of employment by
firm size resemble those in establishment data: labor in the manufacturing
sector became less concentrated (more clearly for firms than establishments),
while service sector labor grew slightly more concentrated. Perhaps the
only qualitative departure from establishment trends is a decrease in the
446
Federal Reserve Bank of Richmond Economic Quarterly
Figure 18 Firm-Size Distribution; Service Sector, 1977–2006
Panel A: Employee-Firm Lorenz, Services
0.8
Cumulative Fraction Employees
1977
2005
0.6
0.4
0.2
0.0
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Cumulative Fraction Firms
Panel B: Cumulative Employee Share, Services
Cumulative Employee Share
1.0
1977
2005
0.8
0.6
0.4
0.2
0.0
10
100
1,000
10,000
Firm Size by Employees
evenness of the aggregate labor distribution across firms that occurred between 1972 and 2005.
5.
CONCLUSIONS
This article collects and analyzes publicly available data from the 1970s onward to obtain a set of statistics that can be used to calibrate and evaluate models with establishment heterogeneity. Recently, these models have
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
447
become widely used in economics to explain phenomena as important as
economic development.
At the aggregate level, there is a minor shift of labor to mid-size establishments and away from the smallest and largest establishments. This change
is partially explained by intrasector changes. The largest manufacturing
establishments have consistently lost employee share since 1974, and manufacturing establishments smaller than 500 employees have uniformly seen
their employee share increase. Trends in the distribution of labor across service sector establishments are complicated by inconsistencies in the definition
of the service sector, but service establishments seem to have become larger
since 1974, and the largest service establishments have grown at a disproportionately fast rate. Thus, the distribution of labor across service establishments
has become less even, with most change occurring before 1997. Changes in
the aggregate distributions of establishments and labor across establishments
are also the result of changes in the share of sectors. Between 1974 and 2006
the worker share of manufacturing, a sector with large establishments and
concentrated labor, decreased as the employment share of the services sector, characterized by smaller establishments, increased. In combination with
movements in intrasector distributions, this trend explains observed changes in
the aggregate distributions of establishments and labor across establishments.
Labor movements across firms should, hypothetically, resemble the movement of labor across establishments. This will be true to a greater degree when
large firms contain fewer small establishments. This hypothesis is not contradicted by the data presented in this article.
APPENDIX
Data Sources
Enterprise Statistics
The Enterprise Statistics (ES) data set was first published in 1954; later publications came in 1958, 1963, 1967, and every five years after 1967 until the
series was discontinued after 1992. The primary virtue of ES for this article is the provision of tables detailing quantities of firms, establishments,
and employment; these values are provided for firms in different employment
size groups similar to establishment size groups in CBP. These size groups
are available for the aggregate economy as well as for sectors that generally
replicate SIC definitions.
Unfortunately, ES’s coverage and content changes significantly from publication to publication. The number of SIC sectors covered varies wildly;
448
Federal Reserve Bank of Richmond Economic Quarterly
using sector-level data we were able to homogenize the aggregate data, but
the adjusted series lacks coverage of entire sectors (transportation and communication; finance and real estate; and most services). Moreover, the 1972 publication inflates its count of small firms by including certain non-employers;
this can be corrected for the aggregate using a table found in that publication’s
appendix. The manufacturing sector from this year is still usable because
there are no manufacturing firms in the small size group affected by the 1972
methodology, but the sector-level data for service firms must be set aside.
Adjustment of ES data to obtain a homogenous aggregate composition
requires the subtraction of some sectors from each year’s aggregate. This is
a simple arithmetic task complicated in some cases by the lack of subsector
data: The Census Bureau occasionally withholds employment information
for certain firm size groups if its publication might result in the disclosure of
private information. These missing values are estimated by multiplying the
number of firms in the size group with the missing data by the mean number
of employees per firm for the size group at the aggregate level. An example
adjustment is displayed in Table 3. There, the original employee count for each
aggregate firm size group was reduced by the deduction of employees in public
warehousing, travel agencies, and dental laboratories—three small sectors not
present in the ES aggregates in all years. Values in bold were missing from the
original publication and estimated using the procedure previously described.
Similar exercises were also carried for firm and establishment series and in all
ES years.
The composition of the services sector also varied from publication to
publication. Unfortunately, homogenization was not a feasible solution: very
few firms would remain in an intertemporally consistent services sector. Consequently, the service sector is presented for each ES year unaltered with the
caveat that it is inconsistent.
Statistics of United States Businesses
Statistics of U.S. Businesses (SUSB) replaced ES in 1992; it was published
in 1992, and annually from 1997 onward. Although SUSB provides data
similar to those found in ES, there are several important differences. First,
SUSB covers many sectors not covered by ES. This leaves aggregate data
somewhat incomparable across the two publication series, especially after
this article’s sectoral homogenization of aggregate ES data. Second, SUSB
uses enterprise size groups rather than firm size groups. In ES these terms were
interchangeable and each enterprise was assigned a single industry code; in
SUSB an enterprise is composed of many firms, each of which represents the
enterprise’s production in a given industry. With this convention, it is possible
to find a 5,000–9,999 employee size group containing three firms employing
2,000 workers between them. This data is not well-suited for the creation of
Lorenz curves because it does not permit the sorting of firms by size.
Firm Size
Group
0
1–4
5–9
10–19
20–49
50–99
100–249
250–499
500–999
1,000–2,499
2,500–4,999
5,000–9,999
= 10,000
Total
Column Error
Original Total
Total
Employees
0
2,938,355
3,209,609
3,945,190
5,372,937
3,446,571
3,459,628
2,126,488
1,837,286
2,330,673
1,981,793
2,376,041
12,786,233
45,810,804
0
(Subtracted)
Public
Warehousing (42A)
0
5,235
8,471
14,670
27,508
13,739
14,281
5,833
688
4,618
0
0
0
94,464
−579
(Subtracted)
Travel
Agencies (47)
0
8,898
10,447
7,869
5,997
3,100
1,967
2,100
688
3,079
0
0
0
44,888
743
(Subtracted)
Dental
Laboratories (80)
0
6,355
5,107
5,284
6,072
1,670
1,526
686
0
1,539
0
0
0
27,744
−496
Final Figure
Adjusted Total
Employees
0
2,917,867
3,185,584
3,917,367
5,333,360
3,428,062
3,441,854
2,117,869
1,835,910
2,321,437
1,981,793
2,376,041
12,786,233
45,643,708
331
Values in bold were missing from the original publication and are estimated using the procedure described in the text of this
article.
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
Table 3 Adjustment to ES Sectoral Composition; Example
449
450
Federal Reserve Bank of Richmond Economic Quarterly
Table 4 Services Sector Assembled from NAICS
NAICS Number
54
56
61
62
71
72
81
NAICS Service Sector Component
Professional, scientific, and technical services
Administrative and support and waste management and
remediation services
Educational services
Health care and social assistance
Arts, entertainment, and recreation
Accommodation and food services
Other services (except public administration)
Moreover, it prevents any adjustment of the SUSB aggregate by the subtraction
of sector data, because too many firms would be dropped. For example, if
the construction and mining sectors are subtracted from the aggregate, and a
single enterprise has constituent firms in each sector, then two firms will be
removed from the aggregate despite the fact that the enterprise is represented
in the aggregate by a single firm. Consequently, sectoral and aggregate data
are only marginally comparable between the two series.
The utility of SUSB is further reduced by the switch to the NAICS classification system from the SIC system after 1997; it is difficult to compare sectors
between systems, and, as with CBP, it was necessary to construct a composite
service sector from several NAICS subsectors (see Table 4). Because of the
SUSB definition of a firm, the number of service firms in large size groups is
probably overstated in NAICS.
Business Dynamics Statistics
BDS is consistent in methodology and coverage; derived from a number of
internal USCB databases, it has annual data on employment for firm size
groups reaching back to 1977. For the purposes of this article, BDS has one
major shortcoming: For each firm size group, only data on establishments and
employment are provided. When firm quantities are required for a calculation,
ES and SUSB are used.
Because the series was assembled from microdata retrospectively, BDS
industry classifications are internally comparable for all years. These classifications are based on the SIC system, and so the comparability of BDS
sector data with CBP and SUSB sector series from 1998 on is somewhat
compromised.
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
451
Computing Establishment and Coworker Means and
Probabilities
We compute the expected establishment mean for a size group by dividing the
total number of workers in a size group (workersi ) by the total number of
establishments in the size group (establishmentsi ):
E(esize | egroup = i) =
workersi
.
establishmentsi
(12)
Obtaining the expected coworker mean for a size group is more involved and
the next subsection is devoted to this effort. Meanwhile, the probabilities
P (egroup = i) and P (wgroup = i) are obtained by dividing the establishments or workers (respectively) in i by the total number of establishments or
workers over all size groups j :
establishmentsi
P (egroup = i) =
, and
j establishmentsj
(13)
workersi
.
P (wgroup = i) =
j workersj
(14)
Probabilities P (egroup ≤ i) and P (wgroup ≤ i) are calculated in a similar
manner by summing the probabilities for each size group j less than or equal
to i:
i
establishmentsj
P (egroup = i) = 1
, and
(15)
j establishmentsj
i
workersj
.
P (wgroup = i) = 1
j workersj
(16)
Computing the Size-Group Coworker Mean
For each size group i, the available information is
• the minimum and maximum size in the group, mini and maxi ,
respectively;
• the total number of workers, workersi ; and
• the total number of establishments, establishmentsi .
With this information it is simple to compute the mean size of the group,
E(esize | egroup = i) =
workersi
.
establishmentsi
(17)
452
Federal Reserve Bank of Richmond Economic Quarterly
Figure 19 Triangular Distribution; Example
f(x)
min
mode
max
x
Unfortunately, it is not possible to compute the coworker mean of this group.
Davis and Haltiwanger (1989) show that the coworker mean can also be written
as
E(wsize | wgroup = i) =
V (esize | egroup = i)
,
E(esize | egroup = i) +
E(esize | egroup = i)
(18)
where V (esize | egroup = i) is the variance of the establishment size for
the size group i. Equation (18) indicates that once E(esize | egroup = i)
is known, only an estimate of V (esize | egroup = i) is needed to obtain
an estimate of E(wsize | wgroup = i). With a distributional assumption
for the distribution of establishments inside each size group, this statistic can
be recovered. A useful assumption is that this distribution is triangular. This
distribution has three parameters: the lower bound, min; the upper bound,
max; and the mode, mode. The probability density function increases linearly
from min to mode and decreases linearly from mode to max (see Figure 19
for an example). With this assumption, the mean size can be written as
E(esize | egroup = i) =
mini + maxi + modei
.
3
(19)
S. E. Henly and J. M. Sánchez: U.S. Establishments Size Trends
453
Since E(esize | egroup = i), mini , and maxi are available, one can use the
equation above to solve for modei . Then, it is simple to compute the variance
using the formula for the triangular distribution,
V (esize | egroup = i) =
min2i + maxi2 + modei2 − mini ∗ maxi − mini ∗ modei − maxi ∗ modei
.
18
(20)
Finally, equation (18) can be used to compute the coworker mean of size
group i.
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