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Economic Quarterly—Volume 97, Number 4—Fourth Quarter 2011—Pages 359–387

Accounting for the
Non-Employment of U.S.
Men, 1968–2010
Marianna Kudlyak, Thomas Lubik, and Jonathan Tompkins

M

en in their prime working age, defined as men between the ages of
25 and 64, constitute 33 percent of the civilian non-institutionalized
population in the United States. At the trough of the 1969–1970
recession, 6.5 percent of this group (henceforth, “population”) were out of the
labor force (OLF), 90.8 percent were employed, and 2.7 percent were unemployed. Since then, the employment-to-population ratio has trended persistently downward, while the OLF-to-population ratio has increased substantially.1 In 2010, the aftermath of the 2007–2009 recession, the employmentto-population ratio of this same group declined to an all-time low of 76.3
percent, while the OLF-to-population ratio increased to an all-time high of
14.7 percent (see Figure 1, Panels A–C).
In this article, we investigate the extent to which the change in the
sociodemographic composition of the population (by age, educational attainment, marital status, and race) has contributed to the changes in the aggregate
labor market outcomes. Our emphasis on the compositional changes in the
sociodemographic characteristics of the population is motivated by a literature rife with correlations between sociodemographic factors and labor market
outcomes. In particular, older workers typically experience lower rates of labor force participation and, conditional on participating, older workers are
We would like to thank Huberto Ennis, Arantxa Jarque, Nadezhda Malysheva, and Alexander
Wolman for their invaluable comments. The views expressed here do not necessarily reflect
those of the Federal Reserve Bank of Richmond or the Federal Reserve System. E-mail:
marianna.kudlyak@rich.frb.org; thomas.lubik@rich.frb.org; jonathan.tompkins@rich.frb.org.
1 The focus of this article is on the employment-, unemployment- and OLF-to-population
ratios. These are defined as the proportion of individuals in the entire population with a given
labor status. They are thus distinct from rates (e.g., the unemployment rate), which are defined
as the proportion of the labor force (i.e., the sum of unemployed and employed persons) with a
given labor status outcome.

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 1 Labor Market Outcomes of 25–64-Year-Old Men
Panel A: Employment-to-Population

Proportion of Population

.90

.85

.80

.75
1970

1980

1990

2000

2010

Year
Panel B: Unemployment-to-Population

Proportion of Population

.10

.08

.06

.04

.02
1970

1980

1990

2000

2010

2000

2010

Year
Panel C: OLF-to-Population

Proportion of Population

.16

.14

.12

.10

.08

.06
1970

1980

1990

Year

Notes: Authors’ own calculations from the IPUMS-CPS data. Shaded areas on this
and subsequent graphs represent National Bureau of Economic Research (NBER)-dated
recessions.

less likely to be unemployed than younger workers (see, for example, Shimer
1999). The literature also finds that (i) more highly educated workers have
a higher opportunity cost of not working; (ii) married men are more likely
to participate in the labor force and, conditional on participation, more likely

M. Kudlyak, T. Lubik, and J. Tompkins: Non-Employment of U.S. Men

361

to be employed; and (iii) non-white persons are usually underrepresented in
the labor force and employment. Thus, one expects a strong association between labor market outcomes and the demographic composition of the labor
force, which serves as a reduced-form representation of underlying structural
relationships.
In this article, we decompose the observed changes in aggregate labor market outcomes into changes in the sociodemographic composition of the population and changes in the labor market outcomes of different sociodemographic
groups. For each year we generate two sets of counterfactual aggregate labor
market outcomes. The first set is generated by using the sociodemographic
composition of the population from all the years in the sample and holding
the labor market outcomes of different sociodemographic groups constant at
the actual level of the reference year. The second set is generated by holding the sociodemographic composition constant instead. We then use these
counterfactuals to perform the decomposition of the changes in the aggregate
labor market outcomes. Finally, we use the most recent sociodemographic
composition of the population to forecast the aggregate OLF-to-population
ratio in 2015.
Given the similarities between the 1980–1982 and 2007–2009 recessions
in terms of severity, we emphasize, throughout this article, comparisons between the labor market outcomes in 1983 and 2010. We find that the changes in
the demographic composition of the population explain much of the historical
upward trend in the OLF ratio. The OLF ratio increased from 11.1 percent in
1983 to 14.7 percent in 2010. Of this increase, 1.9 percentage points (49 percent of the total change) are attributable to changes in the sociodemographic
composition of the population. The employment-to-population ratio fell from
80.2 percent in 1983 to 76.3 percent in 2010. We find that changes in the
sociodemographic composition of the population account for 1.7 percentage
points (44 percent) of this decline. The unemployment-to-population ratio
increased from 8.7 percent in 1983 to 8.9 percent in 2010, but none of this
increase can be accounted for by changes in the demographic composition
of the population. Finally, using predicted changes in the age distribution of
the population, we estimate that the OLF-to-population ratio will increase to
more than 16 percent in 2015 as compared to 14.7 percent in 2010.
When interpreting our results we need to be wary that changes in the
sociodemographic composition might cause changes in the labor market outcomes of different sociodemographic groups. Alternatively, changes in the labor market outcomes of some sociodemographic groups might cause changes
in the sociodemographic composition of the population. For example, an increase in the employment probability for higher educated workers relative to
other education levels might contribute to an increase in the educational attainment of the population. Alternatively, an increase in educational attainment of

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Federal Reserve Bank of Richmond Economic Quarterly

the population can change the employment probabilities of different groups.
Our accounting exercise does not account for these effects.
This article is related to the existing literature that documents a secular
decline in the labor force participation of prime working age men. Autor and
Duggan (2003) document a substantial fall in labor force participation among
men. Using data from the Current Population Survey (CPS), Juhn, Murphy,
and Topel (1991, 2002) find that falling unemployment rates among men in
the 1990s greatly exaggerated the improvements of the labor outcomes for this
population because the period was also characterized by a fall in the labor force
participation rate. We update their analyses by focusing on the decomposition
of the changes in the labor outcomes into changes in the sociodemographic
characteristics of the population and by adding data from the most recent
decade. Our work is also closely related to Little and Bradley (2007), who use
a multinomial logistic model to study the sociodemographic determinants of
labor market outcomes, distinguishing employment, unemployment, peripheral inactivity (marginally attached to labor force), and OLF, in Great Britain.
Finally, our work expands on that of Fallick and Pingle (2006), who decompose movements in U.S. labor force participation into aging of the population
and labor force trends within age groups. Whereas these authors focus solely
on changes in the OLF-to-population ratio caused by aging of the population,
we consider changes in each employment status caused by changes in four
different sociodemographic factors.
The article is structured as follows. Section 1 describes the data. Section 2
summarizes the changes in the demographic composition of the population of
working age men between 1968–2010 and documents trends in labor market
outcomes by sociodemographic groups. Section 3 describes the decomposition exercise and presents results of the decomposition of changes in labor
outcomes between 2010 and the earlier years. Section 4 presents the forecast
of the 2015 OLF-to-population ratio. Section 5 concludes.

1.

DATA

We use data from the Integrated Public Use Microdata Series CPS (IPUMSCPS), which comprises data from the March Supplement of the Current
Population Survey (hereafter referred to as the March CPS). The CPS is
a monthly survey of U.S. households’ activities, conducted by the Census
Bureau and the Bureau of Labor Statistics and designed to measure unemployment. The basic survey is conducted every month; over time various supplementary surveys have been conducted to study different social
and economic questions. The March CPS contains in-depth information on
sociodemographic characteristics of the population and income. The variables
in IPUMS-CPS are coded identically or “harmonized” over the years.

M. Kudlyak, T. Lubik, and J. Tompkins: Non-Employment of U.S. Men

363

The CPS is a collection of individual-level data obtained from the interviewed households. We focus on males between the ages of 25 and 64.2
Throughout the analysis we use the March CPS sampling weights that account
for a complex survey design. The aggregate annual statistics that we report
thus correspond to March of a respective year.
It should be noted that in 1994 the CPS underwent a major redesign both
in the wording of its questions and in the methodology of the data collection
process, which led to some discrepancies between the aggregate series constructed from the microdata prior to the redesign and post-redesign. However,
it is not a concern of our analysis because the inconsistencies associated with
the aggregate labor statistics for the sample of 25–64-year-old men are minor
and not statistically significant (see Polivka and Miller [1995]).

2.

SOCIODEMOGRAPHIC COMPOSITION OF THE
POPULATION AND LABOR OUTCOMES BY
DIFFERENT GROUPS

Between 1968–2010 there have been considerable changes in the distribution
of 25–64-year-old civilian, non-institutionalized men by age and education,
and some noticeable changes by marital status and race (Figure 2). Figures
3–5 display the time series of labor outcomes by different sociodemographic
groups. In general, across different groups the employment-to-population
ratio has been trending down, while the OLF-to-population has been gradually
increasing. We now describe each figure in detail.

Sociodemographic Composition
Panel A of Figure 2 shows the changes in the shares of 25–34, 35–44, 45–
54, and 55–64-year-old men in the population. From 1968–1986, the share
of 25–34-year-old men grew steadily, reaching its largest fraction of 35.6
percent in 1986, and declined thereafter. The share of 55–64-year-old men
fell from 1968–1995, reaching its smallest fraction of 15 percent in 1995,
and has increased steadily since. From 2000 to present, the shares of older
workers (45–54 and 55–64-year-olds) have been increasing, while the shares
of younger workers (25–34 and 35–44-year-olds) have been decreasing. This
shift in the age distribution toward older workers is largely because of the
2 This article focuses on the male population. The OLF-to-population ratio for women fell
drastically from 1968 to the mid-1990s as females entered employment, while that of males trended
upward. Since the mid-1990s, the OLF-to-population ratios for males and females have experienced
similar trends, though the OLF-to-population ratio for females remains approximately 10 percentage
points higher. While studying aggregate employment outcomes for both genders would be an
interesting exercise, doing so is beyond the scope of this article.

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 2 Distribution by Age, Education, Marital Status, and Race
Panel B: Population Distribution by Years of Education

Panel A: Population Distribution by Age
25-34
45-54

35-44
55-64

.30

.25

.20

.8
Proportion of Total Population

Proportion of Total Population

.35

At Most High School
College

Some College
Higher than College

.6

.4

.2

.0

.15
1970

1980

1990

2000

1970

2010

1980

1990

2000

2010

Year

Year
Panel C: Population Distribution by Marital Status

Panel D: Population Distribution by Race

.8

.6

Proportion of Total Population

Proportion of Total Population

1

Separated/Divorced
Single, Never Married

Married
Widowed

.4

.2

.0

.8

.6

.4
White
Other

.2

Black

.0
1970

1980

1990

Year

2000

2010

1970

1980

1990

2000

2010

Year

Notes: Authors’ own calculations from the IPUMS-CPS data. In this graph, “population”
refers to the population of 25–64-year-old civilian, non-institutionalized men.

aging of the Baby Boom generation. In 2010, the shares of 25–34, 35–44,
45–54, and 55–64-year-olds were 25.8, 25.2, 27.5, and 21.5, respectively.
Panel B of Figure 2 shows the upward trend in the educational attainment
of the population and reveals that the shares of men with some college, college,
or higher than college education have been increasing at the expense of men
with at most a high school degree. The share of the latter has declined from 74.6
percent in 1969 to 44.0 percent in 2010. Panel C shows that the population
distribution by marital status has shifted toward divorced or separated and
single men at the expense of married men. In 1968, 84.8 percent of the 25–
64-year-old men were married, while only 61.1 percent were married in 2010.
Finally, Panel D shows that the share of white men in the population has been

M. Kudlyak, T. Lubik, and J. Tompkins: Non-Employment of U.S. Men

365

Figure 3 Employment-to-Population Ratio by Sociodemographic
Group, 25–64-Year-Old Males
Panel B: Proportion of Males Employed by Years of Education

Panel A: Proportion of Males Employed by Age
1

25-34
45-54

35-44
55-64

.95
.90

.9

.85
.8
.80
.7

.75
At Most High School
College

.70

.6
1970

1980

1990

2000

2010

1970

1980

Year

Some College
Higher than College

1990

2000

2010

Year

Panel C: Proportion of Males Employed by Marital Status

Panel D: Proportion of Males Employed by Race
White
Other

.9

.9

Black

.8
.8
.7

Separated/Divorced
Single, Never Married

Married
Widowed

.7
.6

.5

.6
1970

1980

1990

2000

2010

1970

1980

Year

1990

2000

2010

Year

Notes: Authors’ own calculations from the IPUMS-CPS data.

steadily falling over the last 40 years while the shares of black men and men
of other races have been increasing.

Employment-to-Population Ratio
Employment-to-population ratios by age are shown in Figure 3, Panel A. The
figure shows that the proportion of employed workers for the 25–34, 35–44,
and 45–54 age groups move in sync over the years, declining from roughly
.95 in 1968 to approximately .80 in 2010. The 55–64 age group has displayed
markedly different behavior, declining from approximately .82 in 1968 to an
all-time low of .60 in 1994 before trending back up to .65 in 2010.

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 3, Panel B displays the employment-to-population ratio by educational attainment. For each group there is a clear decline in the employmentto-population ratio over time, with the severity of this decline decreasing in
years of schooling. From 1968–2010, those with more than a college education and those with just a college degree have seen moderate declines of
.06 and .11, respectively, while those with some college experience and those
with at most a high school degree have experienced larger declines of .18 and
.21, respectively.
The employment-to-population ratio trends by marital status are displayed
in Figure 3, Panel C. Though there have been decreases in the employmentto-population ratio across marital status groups from 1968–2010, we observe
that much of these declines came prior to 1980 and after 2007. Between these
two dates, the employment-to-population ratio across marital status groups
shows little or no trend.
Panel D of Figure 3 shows the employment-to-population ratio for each
race. We observe declines in the employment-to-population ratio for whites,
blacks, and others over the sample period, with this decline being most
pronounced for blacks. The employment-to-population ratio fell by approximately .24 for this group from 1968–2010, whereas the employment-topopulation ratio for whites and others declined by .14 and .10, respectively.

Unemployment-to-Population Ratio
The unemployment-to-population ratio shows a clear cyclical pattern, rising
during economic contractions and falling during expansions. As can be seen
from Panel A of Figure 4, the unemployment-to-population ratio is decreasing
with age. The rise in the unemployment-to-population ratio across age groups
from 2007–2010 is comparable to the increase from 1980–1983, though the
increase for the 25–34 age group was more pronounced in the 1980–1983
recession, and the increases for the 35–44 and 45–54 age groups are more
pronounced in the 2007–2009 recession.
The unemployment-to-population ratios for educational attainment groups
are displayed in Figure 4, Panel B, which shows that more years of schooling are associated with lower unemployment. Figure 4, Panel C shows the
unemployment-to-population ratio by marital status and shows that single,
never married and separated/divorced individuals have consistently higher
unemployment-to-population ratios than those who are married. Finally, as
can be seen from Panel D of Figure 4, the unemployment-to-population ratio for blacks is consistently higher than the unemployment-to-population
ratio for whites and others. Between 2007 and 2010, the unemployment-topopulation ratio for blacks increased by .08, whereas the ratios of whites and
others increased by approximately .04.

M. Kudlyak, T. Lubik, and J. Tompkins: Non-Employment of U.S. Men

367

Figure 4 Unemployment-to-Population Ratio by Sociodemographic
Group, 25–64-Year-Old Males
Panel B: Proportion of Males Unemployed by Years of Education

Panel A: Proportion of Males Unemployed by Age
.12

25-34
45-54

.15

35-44
55-64

At Most High School
College

Some College
Higher than College

.10
.10
.08
.06
.05
.04
.02

.00
1970

1980

1990

2000

2010

1970

1980

Year

Panel C: Proportion of Males Unemployed by Marital Status
.15

Married
Widowed

1990

2000

2010

Year

Separated/Divorced
Single, Never Married

Panel D: Proportion of Males Unemployed by Race
.15

.10

Black

.10

.05

White
Other

.05

.00

.00
1970

1980

1990

2000

2010

1970

1980

Year

1990

2000

2010

Year

Notes: Authors’ own calculations from the IPUMS-CPS data.

OLF-to-Population Ratio
Figure 5, Panel A displays the OLF-to-population ratio for each age group. We
note that each time series has a distinct upward trend from 1968–2010, with
the upward trend for the 55–64 age group from 1968–1985 being particularly
severe. However, the OLF-to-population ratio for the 35–44 age group has
been relatively stable since 1994 and that of the 55–64 age group has actually
declined moderately since 1994. The most notable feature of this figure is the
disparity between the 55–64 age group and the other age groups. Historically,
the OLF-to-population ratio of the 55–64 age group has dwarfed that of the
younger age groups. In 2010, the OLF-to-population ratio for the 55–64 age
group was .30, whereas the ratios of the other age groups ranged from .09

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 5 OLF-to-Population Ratio by Sociodemographic Group,
25–64-Year-Old Males
Panel B: Proportion of Males OLF by Years of Education

Panel A: Proportion of Males OLF by Age
.4

.20

.3
25-34
45-54

35-44
55-64

At Most High School
College

Some College
Higher than College

.15

.2
.10
.1
.05
.0
1970

1980

1990

2000

2010

1970

1980

Year

1990

2000

2010

Year

Panel C: Proportion of Males OLF by Marital Status

Panel D: Proportion of Males OLF by Race
.25

White
Other

.4

Black

.20

.3

Married
Widowed

Separated/Divorced
Single, Never Married

.15

.2

.10

.1

.0

.05
1970

1980

1990

2000

2010

1970

1980

Year

1990

2000

2010

Year

Notes: Authors’ own calculations from the IPUMS-CPS data.

to .13. We also observe that the OLF-to-population ratio of the 45–54 group
tends to be higher than the 25–34 and 35–44 age groups by approximately .02.
The OLF-to-population ratios for different educational attainment groups
are shown in Panel B of Figure 5. There are upward trends since 1968 for
each group, with the largest increases occurring for those with at most a high
school education and those with some college education. The figure shows
that those with a college degree and those with more than a college education
have experienced similar OLF-to-population ratios across time, though since
2005 these series have diverged, with the OLF-to-population ratio for those
with a college degree continuing to trend upward while the ratio for those with
more than a college education has fallen. Finally, we observe that fewer years
of education are associated with a larger OLF-to-population ratio: In 2010,

M. Kudlyak, T. Lubik, and J. Tompkins: Non-Employment of U.S. Men

369

the OLF-to-population ratios were .19, .15, .09, and .07 for those with at most
a high school degree, some college education, a college degree, and more than
a college degree, respectively.
Panel C of Figure 5 gives the OLF-to-population ratio by marital status.
There is a notable upward trend in the OLF-to-population ratio for each marital
status group. Those in the married group have the lowest OLF-to-population
ratio throughout the observation period, while the separated/divorced and single, never married groups have an OLF-to-population ratio that is between
.05 and .1 higher. The widowed group has had the highest OLF-to-population
ratio historically, occasionally exceeding .40.
Figure 5, Panel D breaks down the OLF-to-population ratio by race. The
most notable feature of this figure is the large difference in the growth of the
OLF-to-population ratio between blacks and the other groups. The OLF-topopulation ratio for blacks has increased from .09 in 1968 to .24 in 2010,
whereas the OLF-to-population ratios for the other and white groups have
increased from .09 and .06 to .15 and .14, respectively. Thus, while each
series has trended upward, that of blacks has done so more rapidly.

3. ACCOUNTING FOR CHANGES IN AGGREGATE
LABOR OUTCOMES
Method
In this subsection we discuss the methods by which we create counterfactual
labor outcomes and decompose the changes in actual labor market outcomes.
The aggregate share of persons with labor outcome LO, where LO =
{Employed, Unemployed, OLF}, in year t can be described by the following
equation:
⎛
⎞
gi,t
LOt
⎝ LOi,t ×
⎠,
=
(1)
popt
gi,t
gi,t
i
i

where i ∈ A × E × M × R corresponds to a vector of demographic characteristics consisting of age (A), educational attainment (E), marital status (M),
and race (R); gi,t is the number of persons in group i; LOi,t is the number
of persons with the labor status LO in group i; and popt is the size of the
population.
In essence, we divide the population into mutually exclusive groups (e.g.,
married, college-educated white males between the ages of 25–34). Equation
(1) describes the aggregate proportion as the sum of the labor outcomes by
group (the first term in the equation) weighted by the size of the groups in the
population (the second term in the equation). For example, fixing either term
for all i at its 2010 level while allowing the other to take on historical values
allows us to construct counterfactual aggregate labor outcomes for 2010. By

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Federal Reserve Bank of Richmond Economic Quarterly

creating and comparing time series of these counterfactuals, we can observe
the degree to which these two terms are driving changes in aggregate labor
outcomes in 2010.
One concern when creating these counterfactuals is that the changing
sociodemographic composition of the population could affect the labor outcomes of different groups. Alternatively, changes in the labor outcomes of a
sociodemographic group could change the sociodemographic composition of
the population. For example, if college-educated individuals are more likely
to be employed relative to other educational attainment subgroups, there will
likely be an influx of individuals into the college-educated demographic group.
By simply fixing either of these terms while varying the other, we do not account for these endogeneity effects.
To analyze the change in labor outcomes, we perform the following
decomposition:
LOt2 LOt1
−
=
popt2 popt1
where si,tx ,ty =

LOi,tx
gi,tx

si,t2 ,t1 −
i

×

gi,ty
gi,ty

si,t2 ,t2 −

si,t1 ,t1 +
i

i

si,t2 ,t1 , (2)
i

.

i

The component in the first set of brackets in equation (2) measures the
effect of changes in the labor outcomes of different groups from year t1 to
t2 , given the sociodemographic composition of the population in year t1 . The
second term captures the effect of changes in the sociodemographic composition of the population, given the labor outcomes of different groups in
year t2 .
Alternatively, we can write
LOt1
LOt2
−
=
popt2 popt1

si,t1 ,t2 −
i

si,t2 ,t2 −

si,t1 ,t1 +
i

i

si,t1 ,t2 . (3)
i

The component in the first set of brackets in equation (3) measures the
effect of changes in the sociodemographic composition of the population,
given the labor outcomes of different groups in year t1 . The second term
captures the effect of changes in the labor outcomes of different groups from
year t1 to t2 , given the sociodemographic composition of the population in
year t2 . The difference between these two decompositions is the base year,
i.e., the year at which the component, other than the component of interest,
is held constant. For example, the change as a result of a change in the labor
outcomes of different groups in equation (2) is calculated using t1 as the base
year, while in equation (3) it is calculated using t2 as the base year. Because
of the endogeneity issues mentioned above, these two decompositions do not
necessarily deliver the same results. It is also unclear that one is theoretically
better than the other. We perform both decompositions and, despite some

Labo r Outco mes
Demo graphics 1
968

1
969

1
970

1
971

1
972

1
973

1
974

1
975

1
976

1
977

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978

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979

1
980

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982

1
983

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984

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985

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986

1
987

1
988

1
993

1
994

1
997

1
998

1
999 2000

2001 2002 2003 2004

2007 2008 2009

201
0

1
968

91
.9

91
.7

90.7

89.0

88.7

88.7

88.5

84.5

84.8

85.1 85.9

86.4

85.2

83.8

81
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79.5

81
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82.4

82.0

82.5

83.3

83.7

83.3

81
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80.5

80.5

79.9

81
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81
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82.4

83.0

83.4

83.3

82.8

81
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81
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81
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81
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82.7

82.5

81
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77.2

76.9

1
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91
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91
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90.8

89.0

88.8

88.8

88.6

84.6

84.8

85.2

85.9

86.4

85.3

83.9

81
.8

79.6

82.0

82.5

82.1 82.6

83.4

83.7

83.3

81
.8

80.6

80.6

80.1

81
.3

81
.8

82.5

83.1 83.4

83.4

82.9

81
.3

81
.2

81
.4

82.1 82.8

82.6

81
.8

77.3

77.0

1
970

91
.8

91
.8

90.8

89.0

88.8

88.7

88.6

84.6

84.9

85.2

85.9

86.4

85.3

83.9

81
.8

79.6

82.0

82.5

82.1 82.7

83.4

83.8

83.3

81
.8

80.6

80.6

80.1

81
.4

81
.8

82.5

83.1 83.5

83.4

82.9

81
.4

81
.2

81
.4

82.1 82.8

82.6

81
.8

77.4

1
971

91
.8

91
.7

90.7

89.1 88.7

88.6

88.5

84.5

84.8

85.2

85.9

86.4

85.3

83.9

81
.8

79.6

82.0

82.5

82.1 82.7

83.4

83.8

83.4

81
.8

80.7

80.6

80.1

81
.4

81
.9

82.6

83.2

83.5

83.5

82.9

81
.4

81
.2

81
.4

82.1 82.8

82.6

81
.8

77.4

77.1

1
972

91
.8

91
.7

90.7

89.0

88.8

88.6

88.5

84.6

84.9

85.2

85.9

86.4

85.3

84.0

81
.8

79.7

82.0

82.6

82.2

82.7

83.5

83.8

83.4

81
.9

80.7

80.7

80.2

81
.5

81
.9

82.6

83.2

83.5

83.5

83.0

81
.5

81
.3

81
.5

82.1 82.9

82.7

81
.9

77.4

77.2

1
973

91
.8

91
.7

90.7

89.1 88.8

88.7

88.6

84.7

84.9

85.4

86.1 86.5

85.4

84.1 82.0

79.8

82.2

82.8

82.4

82.9

83.7

84.0

83.6

82.0

80.9

80.9

80.4

81
.7

82.1 82.8

83.4

83.6

83.7

83.2

81
.6

81
.4

81
.7

82.3

82.9

82.0

77.6

77.3

1
974

91
.7

91
.5

90.5

88.9

88.6

88.5

88.6

84.6

84.8

85.2

86.0

86.4

84.0

81
.9

79.7

82.3

82.8

83.6

83.9

83.5

1
975

91
.6

91
.6

90.5

88.7

88.5

88.5

88.4

84.6

84.8

85.2

86.0

86.5

85.4

84.1

81
.8

79.8

82.2

82.7

82.4

82.9

83.6

83.9

1
976

91
.7

91
.7

90.6

88.9

88.6

88.6

88.5

84.7

84.9

85.3

86.1 86.6

85.5

84.3

82.1 80.0

82.4

82.9

82.6

83.1 83.8

1
977

91
.7

91
.6

90.5

88.8

88.5

88.5

88.5

84.7

84.9

85.4

86.2

86.6

85.5

84.3

82.1 80.0

82.4

83.0

82.6

1
978

91
.5

91
.5

90.4

88.6

88.4

88.4

88.3

84.5

84.7

85.2

86.1 86.5

85.4

84.2

82.1 80.0

82.3

82.9

82.5

83.7

82.1

81
.1

81
.1 80.7

1
979

91
.6

91
.5

90.4

88.7

88.4

88.3

88.4

84.6

84.8

85.3

86.2

86.6

85.5

84.3

82.1 80.0

82.4

82.9

82.6

83.1 83.7

84.1 83.8

82.3

81
.3

81
.2

80.8

1
980

91
.5

91
.4

90.3

88.6

88.3

88.3

88.3

84.5

84.7

85.3

86.1 86.6

85.5

84.3

82.1 80.0

82.4

83.0

82.6

83.1 83.8

84.1 83.9

82.3

81
.3

81
.2

80.9

82.1 82.4

85.4

82.1 82.7

1
989

1
990

1
991 1
992

1
995

1
996

2005 2006

83.0

77.1

81
.9

80.8

80.8

80.4

81
.6

82.0

82.8

83.3

83.6

83.6

83.1

81
.6

81
.3

81
.6

82.2

82.9

82.8

82.0

77.5

77.3

83.5

82.0

80.9

80.9

80.4

81
.6

82.0

82.8

83.4

83.7

83.7

83.2

81
.7

81
.4

81
.6

82.2

83.0

82.9

82.0

77.6

77.4

84.1 83.7

82.2

81
.1

81
.1 80.6

81
.8

82.2

83.0

83.6

83.9

83.9

83.3

81
.9

81
.6

81
.8

82.4

83.2

83.1 82.2

77.8

77.6

83.1 83.8

84.1 83.8

82.2

81
.2

81
.2

81
.9

82.4

83.1 83.7

83.9

84.0

83.4

81
.9

81
.6

81
.8

82.5

83.2

83.1 82.3

77.9

77.7

83.0

84.0

81
.9

82.3

83.0

83.6

83.9

83.9

83.3

81
.9

81
.5

81
.7

82.4

83.1

83.1 82.2

77.9

77.6

82.0

82.4

83.1 83.7

84.0

84.0

83.5

82.0

81
.6

81
.8

82.5

83.2

83.2

82.3

78.0

77.7

83.2

83.8

84.0

84.0

83.5

82.0

81
.7

81
.8

82.5

83.3

83.2

82.4

78.0

77.7

83.7

80.7

1
981

91
.4

91
.3

90.2

88.5

88.2

88.2

88.2

84.4

84.6

85.2

86.1 86.5

85.5

84.2

82.0

79.9

82.3

82.9

82.6

83.1 83.7

84.2

83.9

82.3

81
.3

81
.2

80.8

82.1 82.4

83.2

83.8

84.0

84.1 83.6

82.0

81
.7

81
.8

82.5

83.3

83.3

82.3

77.9

77.7

1
982

91
.5

91
.4

90.2

88.5

88.2

88.2

88.2

84.5

84.7

85.2

86.1 86.6

85.5

84.3

82.2

80.0

82.5

83.0

82.7

83.2

83.8

84.2

84.0

82.3

81
.4

81
.4

81
.0

82.2

82.5

83.3

83.9

84.2

84.3

83.7

82.2

81
.8

81
.9

82.6

83.4

83.4

82.4

78.0

77.8

1
983

91
.6

91
.5

90.2

88.6

88.3

88.3

88.3

84.7

84.8

85.3

86.2

86.7

85.6

84.4

82.3

80.2

82.6

83.2

82.9

83.3

84.0

84.4

84.2

82.5

81
.6

81
.5

81
.1 82.4

82.7

83.5

84.1 84.3

84.4

83.8

82.3

81
.9

82.0

82.8

83.5

83.6

82.6

78.2

77.9

1
984

91
.5

91
.4

90.3

88.6

88.2

88.3

88.2

84.6

84.8

85.3

86.2

86.7

85.6

84.3

82.2

80.1 82.6

83.1 82.8

83.3

84.0

84.3

84.2

82.5

81
.6

81
.5

81
.1 82.4

82.7

83.4

84.1 84.3

84.4

83.8

82.3

81
.9

82.0

82.7

83.5

83.6

82.6

78.2

77.9

1
985

91
.4

91
.3

90.1 88.5

88.2

88.2

88.1 84.5

84.8

85.3

86.2

86.7

85.5

84.3

82.2

80.1 82.6

83.1 82.8

83.3

83.9

84.3

84.2

82.5

81
.6

81
.5

81
.1 82.4

82.7

83.4

84.1 84.3

84.4

83.8

82.3

81
.9

82.0

82.7

83.5

83.6

82.6

78.2

77.9

1
986

91
.4

91
.3

90.1 88.4

88.1 88.2

88.1 84.5

84.7

85.2

86.1 86.7

85.5

84.3

82.2

80.1 82.6

83.1 82.9

83.3

83.9

84.4

84.2

82.5

81
.6

81
.5

81
.2

82.5

82.7

83.4

84.1 84.3

84.4

83.9

82.3

81
.9

82.0

82.8

83.5

83.6

82.6

78.2

77.9

1
987

91
.5

91
.4

90.1 88.4

88.1 88.3

88.2

84.6

84.8

85.3

86.2

86.8

85.7

84.4

82.4

80.2

82.7

83.2

83.0

83.5

84.1 84.5

84.4

82.7

81
.8

81
.7

81
.4

82.6

82.9

83.6

84.3

84.5

84.7

84.1 82.5

82.1 82.2

82.9

83.7

83.8

82.8

78.3

78.0

1
988

91
.4

91
.3

90.0

88.3

88.0

88.2

88.2

84.5

84.6

85.2

86.2

86.8

85.6

84.4

82.3

80.1 82.6

83.2

82.9

83.4

84.0

84.4

84.4

82.6

81
.7

81
.5

81
.3

82.5

82.8

83.5

84.2

84.4

84.6

84.0

82.4

82.0

82.1 82.8

83.6

83.6

82.6

78.2

77.9

1
989

91
.3

91
.3

90.0

88.3

88.0

88.1

88.1 84.5

84.7

85.2

86.2

86.8

85.7

84.4

82.4

80.2

82.7

83.3

83.0

83.4

84.1 84.6

84.4

82.7

81
.9

81
.6

81
.4

82.7

82.8

83.6

84.3

84.5

84.6

84.1 82.5

82.1

82.1 82.9

83.7

83.7

82.8

78.3

78.0

1
990

91
.4

91
.3

90.0

88.3

88.0

88.2

88.2

84.5

84.7

85.3

86.3

86.9

85.7

84.4

82.4

80.2

82.8

83.3

83.0

83.5

84.1 84.6

84.5

82.8

81
.9

81
.7

81
.5

82.7

82.9

83.6

84.4

84.5

84.7

84.2

82.6

82.1 82.2

83.0

83.7

83.8

82.8

78.3

78.0

1
991

91
.3

91
.2

89.9

88.2

87.9

88.0

88.0

84.5

84.6

85.1 86.2

86.7

85.6

84.2

82.2

80.0

82.6

83.1 82.8

83.3

83.8

84.3

82.6

81
.7

81
.5

81
.3

82.5

82.6

83.4

84.2

84.3

84.5

84.0

82.4

81
.9

81
.9

82.8

83.5

83.6

82.6

78.1 77.8

1
992

91
.2

91
.2

89.9

88.2

87.9

88.1 88.0

84.6

84.6

85.2

86.3

86.9

85.7

84.4

82.4

80.2

82.8

83.2

83.0

83.5

84.0

84.6

84.5

82.9

81
.9

81
.6

81
.5

82.7

82.8

83.6

84.3

84.5

84.7

84.2

82.6

82.1

82.1 82.9

83.6

83.8

82.8

78.3

77.9

1
993

91
.2

91
.2

89.9

88.2

87.9

88.2

88.0

84.7

84.7

85.2

86.3

86.9

85.8

84.4

82.5

80.3

82.9

83.3

83.0

83.5

84.0

84.6

84.5

82.9

82.0

81
.6

81
.5

82.7

82.8

83.6

84.3

84.5

84.7

84.2

82.6

82.1

82.1 82.9

83.6

83.9

82.9

78.4

78.0

1
994

91
.2

91
.2

89.9

88.1 87.9

88.1 88.0

84.8

84.8

85.3

86.4

87.0

85.9

84.5

82.5

80.3

83.0

83.3

83.1 83.5

84.1 84.7

84.6

83.0

82.1

81
.7

81
.6

82.8

82.8

83.7

84.4

84.6

84.8

84.3

82.7

82.1

82.1 83.0

83.7

83.9

83.0

78.5

78.1

1
995

91
.2

91
.3

90.0

88.2

88.0

88.3

88.0

85.0

85.0

85.5

86.6

87.2

86.0

84.6

82.7

80.5

83.1 83.5

83.2

83.7

84.2

84.8

84.8

83.2

82.3

81
.9

81
.8

83.0

83.0

83.9

84.5

84.8

84.9

84.5

82.9

82.3

82.3

83.2

83.9

84.1 83.2

78.7

78.3

1
996

91

91
.1

89.8

88.0

87.9

88.0

87.9

84.7

84.8

85.3

86.4

86.9

85.8

84.5

82.6

80.4

83.0

83.1 83.5

84.0

84.7

84.6

83.1

82.1

81
.7

81
.6

82.7

82.8

83.7

84.3

84.6

84.7

84.3

82.7

82.1

82.1 83.0

83.7

83.9

83.0

78.5

78.1

1
997

90.9

91

89.7

87.8

87.8

87.9

87.8

84.6

84.7

85.1 86.3

86.8

85.7

84.4

82.5

80.3

82.9

83.2

83.0

83.4

83.9

84.6

84.5

83.0

82.0

81
.5

81
.5

82.6

82.7

83.6

84.3

84.4

84.6

84.2

82.6

81
.9

81
.9

82.9

83.6

83.8

82.9

78.4

78.0

1
998

90.9

91
.1

89.7

87.8

87.7

87.9

87.6

84.6

84.6

85.2

86.9

85.7

84.4

82.5

80.3

82.9

83.2

83.0

83.3

83.8

84.5

84.4

82.9

81
.9

81
.5

81
.4

82.5

82.6

83.5

84.2

84.4

84.5

84.1 82.5

81
.9

81
.9

82.8

83.4

83.7

82.9

78.4

78.0

83.0

1
999

90.8

91

89.7

87.8

87.7

87.9

87.7

84.7

84.6

85.2

86.3
86.3

86.9

85.8

84.4

82.6

80.3

83.3

83.4

83.8

84.3

83.2

82.9

84.1 84.4

84.5

81
.9

81
.9

78.5

78.1

2000

90.9

91
.1

89.6

88.0

87.8

88.0

87.8

84.8

84.6

85.2

86.5

87.0

85.9

84.5

82.8

80.5

83.1 83.2

83.0

83.5

83.9

84.6

84.5

83.0

82.1

81
.5

81
.5

82.6

82.6

83.6

84.2

84.4

84.6

84.1 82.6

81
.9

82.0

82.9

83.6

83.8

83.0

78.6

78.2

2001

90.6

90.9

89.5

87.8

87.6

87.8

87.6

84.5

84.4

85.0

86.3

86.7

85.7

84.4

82.5

80.4

82.9

83.1 82.9

83.3

83.7

84.4

84.5

84.3

84.4

82.9

83.0

81
.9

81
.9

81
.3

81
.4

81
.3

81
.4

82.4

82.5

82.4

82.6

83.4

83.5

84.0

84.2

84.3

84.0

84.1 82.5

82.4

81
.8

81
.8

82.8

83.4

83.6

82.9

78.4

78.1

2002

90.5

90.8

89.3

87.6

87.5

87.6

87.4

84.5

84.3

84.8

86.1 86.6

85.5

84.1 82.3

80.2

82.6

82.9

82.7

83.1 83.5

84.1 84.0

82.6

81
.7

81
.1

81
.1

82.1 82.2

83.1 83.8

84.0

84.1 83.7

82.2

2003

90.3

90.6

89.0

87.4

87.2

87.4

87.2

84.2

84.0

84.7

85.9

86.4

85.4

84.0

82.2

80.0

82.4

82.6

82.5

82.8

83.3

83.9

83.9

82.5

81
.6

80.9

80.9

82.0

82.0

83.0

83.6

83.8

83.9

83.5

82.0

81
.4

81
.5

82.4

83.0

83.2

82.5

78.1 77.7

2004

90.3

90.5

88.9

87.2

87.1 87.3

87.0

84.0

83.8

84.5

85.7

86.2

85.3

83.8

82.0

79.7

82.1 82.4

82.2

82.6

83.0

83.7

83.7

82.2

81
.3

80.7

80.7

81
.8

81
.7

82.8

83.4

83.6

83.7

83.3

81
.8

81
.2

81
.2

82.2

82.8

83.0

82.3

78.0

77.6

81
.1 80.4

77.3

81
.5

81
.6

82.8

82.5

83.5

83.2

83.7

83.4

82.9

82.6

78.2

77.9

2005

90

90.3

88.6

86.9

86.9

87.1 86.8

83.8

83.5

84.2

85.5

86.0

85.0

83.5

81
.7

79.5

81
.9

82.2

81
.9

82.3

82.7

83.4

83.3

82.0

80.4

81
.4

81
.4

82.4

83.1 83.4

83.5

83.1

81
.6

80.9

81
.0

81
.9

82.5

82.8

82.1 77.7

2006

89.8

90.2

88.5

86.8

86.7

86.9

86.6

83.6

83.3

84.0

85.3

85.8

84.8

83.2

81
.6

79.3

81
.6

81
.9

81
.7

82.2

82.6

83.1

83.1

81
.8

80.9

80.2

80.1

81
.2

81
.2

82.2

82.9

83.1 83.2

82.8

81
.3

80.7

80.8

81
.7

82.3

82.6

81
.9

77.4

2007

89.9

90.2

88.4

86.8

86.8

86.8

86.7

83.7

83.4

84.0

85.2

85.8

84.8

83.3

81
.6

79.3

81
.6

81
.9

81
.6

82.1 82.5

83.1

83.1

81
.7

80.8

80.2

80.1

81
.2

81
.2

82.1 82.8

83.1 83.2

82.7

81
.3

80.6

80.8

81
.7

82.3

82.5

81
.8

77.4

77.1

2008

89.6

89.9

88.2

86.6

86.4

86.6

86.3

83.4

83.0

83.7

85.0

85.5

84.6

83.0

81
.3

79.1

81
.3

81
.6

81
.3

81
.8

82.2

82.8

82.8

81
.4

80.6

79.9

79.8

80.9

80.8

81
.9

82.6

82.8

82.9

82.5

81
.0

80.3

80.5

81
.5

82.0

82.3

81
.6

77.2

76.8

2009

89.3

89.7

88.0

86.3

86.1 86.4

86.1 83.2

82.8

83.6

84.9

85.4

84.4

82.8

81
.2

78.9

81
.1

81
.5

81
.2

81
.6

82.0

82.6

82.6

81
.3

80.4

79.7

79.7

80.7

80.7

81
.7

82.4

82.7

82.7

82.3

80.9

80.2

80.3

81
.3

81
.8

82.1

81
.4

77.0

76.6

201
0

89.1

89.4

87.8

86.1 85.9

86.2

85.8

82.9

82.5

83.3

84.6

85.1

84.1 82.5

80.9

78.6

80.8

81
.2

80.9

81
.4

81
.7

82.4

81
.0

80.1 79.4

79.3

80.4

80.3

81
.4

82.1 82.4

82.4

82.1 80.5

79.9

80.0

81
.0

81
.5

81
.8

81
.1 76.7

76.3

91
.9

91
.9

90.8

89.1 88.8

88.7

88.6

84.6

84.9

85.4

86.1 86.6

85.5

82.2

80.2

82.6

83.1 82.9

83.5

84.0

84.6

82.6

81
.9

81
.6

83.0

82.8

83.6

84.2

84.6

84.0

81
.4

81
.2

81
.9

82.3

82.5

81
.6

76.3

82.3

77.1

A ctual
84.2

84.5

81
.6

84.4

82.2

77.0

371

Notes: Authors’ own calculations from the IPUMS-CPS data. The columns correspond to the year of the labor outcomes
used. The rows correspond to the year of the sociodemographic composition used. Lightly shaded columns correspond to the
NBER-dated contractions (from peak to trough). Darkly shaded elements correspond to actual values for given year.

M. Kudlyak, T. Lubik, and J. Tompkins: Non-Employment of U.S. Men

Table 1 Counterfactual Predictions of the Employment-to-Population
Ratio by Percent, 25–64-Year-Old Males

372

Table 2 Counterfactual Predictions of the
Unemployment-to-Population Ratio by Percent,
25–64-Year-Old Males
Labo r Outco mes
Demo graphics 1
968

1
969

1
970

1
971

1
972

1
973

1
974

1
975

1
976

1
977

1
978

1
979

1
980

1
985

1
986

1
987

1
988

1
968

2.1

1
.7

2.7

3.8

3.5

3.0

2.9

6.0

4.9

4.6

3.7

3.5

4.3

1
981 1
982

5.2

6.9

1
983

8.8

1
984

6.1

5.3

5.5

5.2

4.5

3.9

4.0

5.7

6.5

6.2

5.1

4.2

4.5

3.8

3.5

3.0

2.8

3.3

4.8

4.7

4.3

3.9

3.3

3.6

4.3

8.2

1
969

2.0

1
.7

2.7

3.8

3.5

3.0

2.9

6.0

4.8

4.6

3.7

3.4

4.2

5.2

6.8

8.7

6.1

5.3

5.5

5.1

4.4

1
989

3.9

1
990

4.0

1
991 1
992

5.6

6.5

1
993

6.2

1
994

5.1

1
995

4.2

1
996

4.4

1
997

3.8

1
998

3.5

1
999 2000

3.0

2.8

2001 2002 2003 2004

3.3

4.8

4.7

4.3

2005 2006

3.9

3.3

2007 2008 2009

3.5

4.3

8.2

201
0

9.0

1
970

2.0

1
.7

2.7

3.8

3.5

3.0

2.9

6.0

4.9

4.6

3.7

3.4

4.3

5.1

6.8

8.7

6.0

5.3

5.5

5.1

4.4

3.9

4.0

5.6

6.5

6.2

5.1

4.1

4.4

3.8

3.5

3.0

2.7

3.3

4.8

4.7

4.3

3.9

3.3

3.5

4.3

8.1

9.0

9.1

2.0

1
.7

2.7

3.8

3.5

3.0

2.9

6.0

4.9

4.7

3.7

3.5

4.3

5.2

6.8

8.7

6.0

5.3

5.5

5.1

4.4

3.9

4.0

5.7

6.5

6.2

5.1

4.2

4.4

3.8

3.5

3.0

2.7

3.3

4.8

4.7

4.3

3.9

3.3

3.5

4.3

8.1

9.0

2.0

1
.7

2.7

3.9

3.5

3.0

2.9

6.0

4.9

4.6

3.7

3.4

4.2

5.1

6.8

8.7

6.0

5.2

5.4

5.1

4.4

3.9

4.0

5.6

6.4

6.1

5.1

4.1

4.4

3.8

3.5

3.0

2.7

3.3

4.8

4.7

4.3

3.9

3.3

3.5

4.2

8.1

9.0

1
973

2.0

1
.7

2.7

3.8

3.5

3.0

2.9

5.9

4.9

4.6

3.7

3.4

4.2

5.1

6.8

8.7

5.9

5.2

5.4

5.1

4.4

3.9

4.0

5.6

6.4

6.1

5.0

4.1

4.4

3.8

3.5

3.0

2.7

3.2

4.8

4.6

4.2

3.9

3.2

3.5

4.2

8.1

8.9

1
974

2.0

1
.7

2.7

3.9

3.6

3.0

2.9

5.9

4.9

4.7

3.7

3.4

4.2

5.1

6.8

8.6

5.9

5.2

5.4

5.0

4.4

3.9

4.0

5.6

6.4

6.1

5.0

4.1

4.4

3.8

3.5

3.0

2.7

3.3

4.8

4.6

4.3

3.9

3.3

3.5

4.2

8.1

8.9

1
975

2.0

1
.7

2.7

3.9

3.6

3.0

2.9

5.9

4.9

4.7

3.7

3.4

4.2

5.0

6.8

8.6

5.9

5.2

5.3

5.0

4.3

3.8

3.9

5.6

6.4

6.1

5.0

4.1

4.4

3.8

3.4

2.9

2.7

3.2

4.7

4.6

4.2

3.9

3.3

3.5

4.2

8.1

8.8

1
976

2.0

1
.7

2.7

3.9

3.6

3.1

2.9

5.9

4.9

4.7

3.7

3.4

4.2

5.0

6.7

8.6

5.9

5.2

5.3

5.0

4.3

3.8

3.9

5.5

6.3

6.1

5.0

4.1

4.4

3.8

3.4

2.9

2.7

3.2

4.7

4.6

4.3

3.8

3.3

3.5

4.1

8.0

8.8

1
977

2.0

1
.7

2.7

3.9

3.6

3.1

3.0

5.9

4.9

4.7

3.7

3.4

4.2

5.0

6.7

8.5

5.9

5.2

5.3

5.0

4.3

3.8

3.9

5.5

6.3

6.0

5.0

4.1

4.3

3.8

3.4

2.9

2.7

3.2

4.7

4.6

4.3

3.8

3.3

3.5

4.1

8.0

8.7

1
978

2.0

1
.7

2.7

4.0

3.6

3.1

3.0

5.9

4.9

4.7

3.7

3.4

4.2

5.0

6.7

8.5

5.9

5.2

5.3

5.0

4.3

3.9

3.9

5.5

6.3

6.0

5.0

4.1

4.4

3.8

3.4

2.9

2.8

3.2

4.7

4.6

4.3

3.8

3.3

3.5

4.1

8.0

8.7

1
979

2.0

1
.7

2.7

4.0

3.6

3.1

3.0

5.9

5.0

4.8

3.7

3.4

4.2

5.0

6.8

8.5

5.9

5.2

5.3

5.0

4.3

3.9

3.9

5.5

6.3

6.1

5.0

4.1

4.4

3.8

3.4

2.9

2.8

3.2

4.7

4.7

4.3

3.8

3.3

3.5

4.1

8.0

8.7

1
980

2.0

1
.7

2.7

4.0

3.6

3.1

3.0

5.9

5.0

4.9

3.7

3.5

4.3

5.1

6.8

8.6

5.9

5.2

5.3

5.1

4.3

3.9

4.0

5.6

6.4

6.1

5.0

4.1

4.4

3.8

3.4

2.9

2.8

3.3

4.7

4.7

4.4

3.9

3.3

3.5

4.2

8.0

8.8

1
981

2.1

1
.7

2.8

4.1

3.7

3.2

3.1

6.0

5.1

5.0

3.8

3.6

4.4

5.2

7.0

8.7

6.0

5.3

5.4

5.1

4.4

4.0

4.1

5.7

6.4

6.2

5.1

4.2

4.5

3.9

3.5

3.0

2.9

3.3

4.8

4.8

4.4

4.0

3.4

3.6

4.3

8.2

1
982

2.1

1
.7

2.8

4.1

3.7

3.2

3.1

6.1

5.1

5.0

3.8

3.6

4.4

5.2

7.1

8.8

6.1

5.3

5.4

5.2

4.4

4.0

4.1

5.7

6.4

6.2

5.1

4.2

4.5

3.9

3.5

3.0

2.9

3.3

4.9

4.8

4.5

4.0

3.4

3.6

4.3

8.2

8.9

1
983

2.1

1
.7

2.8

4.1

3.7

3.2

3.2

6.0

5.1

5.0

3.8

3.6

4.4

5.2

7.0

8.7

6.1

5.3

5.4

5.2

4.4

4.0

4.1

5.7

6.4

6.2

5.1

4.2

4.5

3.9

3.5

3.0

2.9

3.3

4.9

4.8

4.5

4.0

3.4

3.6

4.2

8.1

8.9

1
984

2.1

1
.7

2.8

4.1

3.8

3.2

3.1

6.1

5.1

5.1

3.9

3.6

4.4

5.3

7.1

8.8

6.1

5.4

5.5

5.2

4.5

4.1

4.1

5.8

6.5

6.3

5.2

4.3

4.5

4.0

3.5

3.0

2.9

3.3

4.9

4.8

4.5

4.0

3.5

3.6

4.3

8.2

9.0

1
985

2.1

1
.7

2.9

4.1

3.8

3.2

3.2

6.1

5.2

5.1

3.9

3.6

4.4

5.3

7.1

8.8

6.1

5.4

5.5

5.2

4.5

4.1

4.1

5.8

6.5

6.3

5.2

4.3

4.6

4.0

3.5

3.0

2.9

3.4

4.9

4.8

4.6

4.0

3.5

3.6

4.3

8.2

9.0

1
986

2.1

1
.7

2.9

4.2

3.8

3.3

3.2

6.1

5.2

5.2

4.0

3.6

4.5

5.3

7.2

8.9

6.2

5.5

5.5

5.3

4.6

4.1

4.2

5.8

6.5

6.3

5.2

4.3

4.6

4.1

3.6

3.1

3.0

3.4

5.0

4.9

4.6

4.1

3.5

3.7

4.4

8.3

9.1

1
987

2.1

1
.8

2.9

4.2

3.9

3.3

3.2

6.1

5.2

5.2

3.9

3.6

4.5

5.3

7.2

8.9

6.2

5.5

5.5

5.3

4.6

4.1

4.2

5.9

6.6

6.4

5.3

4.3

4.6

4.1

3.6

3.1

3.0

3.4

5.0

4.9

4.6

4.1

3.5

3.7

4.4

8.3

9.1

1
988

2.2

1
.8

3.0

4.2

3.9

3.3

3.3

6.2

5.3

5.3

4.0

3.7

4.5

5.4

7.3

9.0

6.2

5.6

5.6

5.4

4.6

4.2

4.2

5.9

6.6

6.4

5.3

4.4

4.6

4.1

3.6

3.1

3.0

3.4

5.0

4.9

4.7

4.1

3.6

3.7

4.4

8.4

1
989

2.2

1
.7

3.0

4.2

3.8

3.3

3.3

6.1

5.2

5.3

4.0

3.7

4.5

5.4

7.3

8.9

6.2

5.5

5.6

5.4

4.6

4.2

4.2

5.9

6.6

6.4

5.3

4.4

4.7

4.1

3.7

3.1

3.0

3.4

5.0

4.9

4.7

4.1

3.6

3.7

4.4

8.4

9.1

1
990

2.2

1
.7

3.0

4.3

3.9

3.3

3.3

6.1

5.2

5.3

4.0

3.7

4.5

5.4

7.3

8.9

6.2

5.5

5.6

5.4

4.6

4.2

4.2

5.9

6.6

6.5

5.3

4.4

4.7

4.2

3.7

3.1

3.0

3.4

5.0

5.0

4.7

4.1

3.6

3.7

4.4

8.4

9.2

1
991

2.3

1
.8

3.0

4.3

3.9

3.3

3.3

6.2

5.3

5.3

4.0

3.7

4.5

5.4

7.4

9.0

6.3

5.6

5.7

5.5

4.7

4.3

4.3

6.0

6.7

6.6

5.4

4.5

4.7

4.2

3.7

3.2

3.1

3.5

5.0

5.0

4.8

4.1

3.7

3.8

4.5

8.5

9.3

1
992

2.3

1
.7

3.0

4.3

3.9

3.3

3.3

6.1

5.3

5.3

4.0

3.6

4.5

5.3

7.3

8.9

6.2

5.6

5.6

5.4

4.7

4.2

4.3

5.9

6.7

6.5

5.3

4.4

4.7

4.2

3.7

3.1

3.0

3.4

5.0

5.0

4.8

4.1

3.7

3.8

4.4

8.4

9.2

1
993

2.2

1
.7

3.0

4.3

3.9

3.3

3.3

6.0

5.2

5.2

3.9

3.6

4.4

5.2

7.2

8.8

6.1

5.5

5.5

5.4

4.6

4.2

4.2

5.9

6.6

6.5

5.3

4.4

4.6

4.1

3.6

3.1

3.0

3.4

4.9

5.0

4.8

4.1

3.7

3.7

4.3

8.3

9.1

1
994

2.3

1
.7

3.0

4.3

3.9

3.3

3.3

6.0

5.2

5.2

3.9

3.6

4.4

5.3

7.2

8.8

6.1

5.6

5.6

5.5

4.6

4.2

4.2

5.9

6.6

6.5

5.4

4.4

4.7

4.2

3.7

3.2

3.0

3.4

4.9

5.1

4.8

4.1

3.7

3.7

4.4

8.4

9.2

1
995

2.2

1
.7

3.0

4.3

3.9

3.2

3.3

5.9

5.1

5.1

3.9

3.6

4.4

5.2

7.2

8.7

6.1

5.5

5.5

5.4

4.6

4.2

4.2

5.9

6.5

6.5

5.3

4.4

4.6

4.1

3.7

3.1

3.0

3.4

4.9

5.1

4.8

4.1

3.7

3.7

4.3

8.3

9.1

1
996

2.3

1
.7

3.0

4.3

3.9

3.3

3.3

5.9

5.2

5.2

3.9

3.6

4.4

5.2

7.2

8.7

6.1

5.5

5.5

5.4

4.6

4.2

4.2

5.9

6.6

6.5

5.3

4.5

4.6

4.1

3.7

3.1

3.0

3.4

4.9

5.1

4.8

4.1

3.7

3.7

4.3

8.3

9.1

1
997

2.3

1
.7

3.0

4.3

3.9

3.2

3.3

5.8

5.1

5.1

3.9

3.6

4.4

5.2

7.2

8.7

6.1

5.5

5.5

5.4

4.6

4.2

4.2

5.9

6.6

6.5

5.3

4.4

4.6

4.1

3.6

3.1

3.0

3.4

4.9

5.1

4.8

4.1

3.7

3.7

4.3

8.3

1
998

2.3

1
.7

3.0

4.3

3.9

3.2

3.3

5.8

5.1

5.1

3.8

3.6

4.3

5.1

7.1

8.6

6.0

5.4

5.5

5.4

4.6

4.1

4.2

5.8

6.6

6.5

5.3

4.4

4.6

4.1

3.6

3.1

3.0

3.4

4.9

5.1

4.7

4.1

3.7

3.7

4.3

8.3

9.1

1
999

2.3

1
.7

3.0

4.2

3.9

3.2

3.3

5.7

5.0

5.0

3.8

3.5

4.2

5.0

7.0

8.5

5.9

5.4

5.4

5.3

4.5

4.1

4.1

5.8

6.5

6.4

5.3

4.4

4.6

4.1

3.6

3.1

3.0

3.4

4.8

5.0

4.7

4.1

3.7

3.7

4.2

8.2

9.0

2000

2.2

1
.7

3.0

4.1

3.9

3.1

3.3

5.7

5.0

4.9

3.7

3.5

4.2

4.9

6.9

8.4

5.9

5.4

5.4

5.3

4.5

4.1

4.1

5.8

6.4

6.4

5.2

4.4

4.5

4.0

3.5

3.1

2.9

3.3

4.8

5.0

4.7

4.0

3.6

3.7

4.2

8.1

8.9

5.7

6.4

6.3

5.2

4.3

4.5

4.0

3.5

3.1

2.9

3.3

4.8

5.0

4.6

4.0

3.6

3.6

8.9

9.1

9.1

2001

2.2

1
.7

3.0

4.2

3.9

3.1

3.2

5.6

5.0

4.9

3.7

3.5

4.2

4.9

6.8

8.4

5.8

5.3

5.4

5.3

4.5

4.0

4.1

4.2

8.1

8.9

2002

2.2

1
.7

3.0

4.3

3.9

3.1

3.3

5.6

5.0

4.9

3.7

3.5

4.2

4.9

6.8

8.3

5.8

5.3

5.4

5.3

4.5

4.0

4.0

5.7

6.4

6.3

5.2

4.4

4.5

4.0

3.5

3.1

2.9

3.3

4.8

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

2003

2.2

1
.7

3.0

4.3

3.9

3.1

3.3

5.6

5.0

4.9

3.8

3.5

4.1

4.9

6.8

8.3

5.8

5.4

5.3

5.3

4.5

4.0

4.1

5.7

6.4

6.3

5.2

4.4

4.5

4.0

3.6

3.1

2.9

3.3

4.7

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

2004

2.2

1
.7

3.0

4.3

3.9

3.1

3.3

5.6

5.0

4.9

3.8

3.5

4.1

4.9

6.8

8.3

5.9

5.3

5.3

5.3

4.5

4.0

4.1

5.7

6.4

6.3

5.2

4.4

4.5

4.0

3.6

3.1

3.0

3.3

4.8

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

2005

2.2

1
.7

3.0

4.3

4.0

3.1

3.3

5.6

5.0

5.0

3.8

3.5

4.1

4.9

6.9

8.4

5.9

5.4

5.4

5.3

4.6

4.1

4.1

5.7

6.5

6.4

5.2

4.4

4.5

4.1

3.6

3.1

3.0

3.4

4.8

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

2006

2.2

1
.7

3.0

4.4

3.9

3.1

3.3

5.6

5.0

5.0

3.8

3.5

4.1

4.9

6.8

8.4

5.9

5.4

5.4

5.3

4.6

4.1

4.1

5.7

6.5

6.4

5.2

4.4

4.6

4.1

3.6

3.1

3.0

3.4

4.8

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

2007

2.2

1
.6

3.0

4.4

4.0

3.1

3.3

5.6

4.9

5.0

3.8

3.5

4.1

4.8

6.8

8.3

5.9

5.4

5.4

5.3

4.6

4.1

4.0

5.7

6.5

6.4

5.2

4.4

4.5

4.0

3.6

3.1

3.0

3.3

4.7

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

2008

2.2

1
.7

3.1

4.4

4.0

3.1

3.3

5.5

5.0

5.0

3.8

3.6

4.1

4.8

6.8

8.3

5.8

5.4

5.3

5.3

4.5

4.1

4.0

5.7

6.5

6.3

5.2

4.4

4.5

4.0

3.6

3.1

3.0

3.4

4.7

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

2009

2.2

1
.7

3.1

4.4

4.0

3.1

3.4

5.5

5.0

5.0

3.8

3.6

4.1

4.8

6.8

8.3

5.9

5.4

5.4

5.3

4.6

4.0

4.1

5.7

6.5

6.4

5.2

4.4

4.6

4.0

3.6

3.1

3.0

3.4

4.7

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

201
0

2.2

1
.7

3.1

4.4

4.0

3.2

3.4

5.6

5.1

5.1

3.9

3.6

4.1

4.8

6.8

8.3

5.9

5.4

5.4

5.3

4.6

4.0

4.1

5.7

6.5

6.4

5.2

4.4

4.5

4.0

3.6

3.1

3.0

3.4

4.8

5.0

4.7

4.1

3.7

3.8

4.2

8.1

8.9

6.0

6.7

6.5

5.4

4.4

4.6

4.1

3.6

3.1

2.9

3.3

4.8

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

A ctual
2.1

1
.7

2.7

3.8

3.5

3.0

2.9

5.9

4.9

4.7

3.7

3.4

4.3

5.2

7.1

8.7

6.1

5.4

5.5

5.3

4.6

4.2

4.2

Notes: Authors’ own calculations from the IPUMS-CPS data. The columns correspond to the year of the labor outcomes
used. The rows correspond to the year of the sociodemographic composition used. Lightly shaded columns correspond to the
NBER-dated contractions (from peak to trough). Darkly shaded elements correspond to actual values for given year.

Federal Reserve Bank of Richmond Economic Quarterly

1
971
1
972

Labo r Outco mes
Demo graphics 1
968

1
969

1
970

1
971

1
972

1
973

1
974

1
975

1
976

1
977

1
978

1
979

1
980

1
985

1
986

1
987

1
988

1
989

1
990

1
995

1
996

1
997

1
998

1
968

2.1

1
.7

2.7

3.8

3.5

3.0

2.9

6.0

4.9

4.6

3.7

3.5

4.3

1
981 1
982

5.2

6.9

1
983

8.8

1
984

6.1

5.3

5.5

5.2

4.5

3.9

4.0

5.7

6.5

6.2

5.1

4.2

4.5

3.8

3.5

3.0

2.8

3.3

4.8

4.7

4.3

3.9

3.3

3.6

4.3

8.2

9.1

1
969

2.0

1
.7

2.7

3.8

3.5

3.0

2.9

6.0

4.8

4.6

3.7

3.4

4.2

5.2

6.8

8.7

6.1

5.3

5.5

5.1

4.4

3.9

4.0

1
991 1
992

5.6

6.5

1
993

6.2

1
994

5.1

4.2

4.4

3.8

3.5

1
999 2000

3.0

2.8

2001 2002 2003 2004

3.3

4.8

4.7

4.3

2005 2006

3.9

3.3

2007 2008 2009

3.5

4.3

8.2

201
0

9.0

1
970

2.0

1
.7

2.7

3.8

3.5

3.0

2.9

6.0

4.9

4.6

3.7

3.4

4.3

5.1

6.8

8.7

6.0

5.3

5.5

5.1

4.4

3.9

4.0

5.6

6.5

6.2

5.1

4.1

4.4

3.8

3.5

3.0

2.7

3.3

4.8

4.7

4.3

3.9

3.3

3.5

4.3

8.1

9.0

1
971

2.0

1
.7

2.7

3.8

3.5

3.0

2.9

6.0

4.9

4.7

3.7

3.5

4.3

5.2

6.8

8.7

6.0

5.3

5.5

5.1

4.4

3.9

4.0

5.7

6.5

6.2

5.1

4.2

4.4

3.8

3.5

3.0

2.7

3.3

4.8

4.7

4.3

3.9

3.3

3.5

4.3

8.1

9.0

1
972

2.0

1
.7

2.7

3.9

3.5

3.0

2.9

6.0

4.9

4.6

3.7

3.4

4.2

5.1

6.8

8.7

6.0

5.2

5.4

5.1

4.4

3.9

4.0

5.6

6.4

6.1

5.1

4.1

4.4

3.8

3.5

3.0

2.7

3.3

4.8

4.7

4.3

3.9

3.3

3.5

4.2

8.1

9.0

1
973

2.0

1
.7

2.7

3.8

3.5

3.0

2.9

5.9

4.9

4.6

3.7

3.4

4.2

5.1

6.8

8.7

5.9

5.2

5.4

5.1

4.4

3.9

4.0

5.6

6.4

6.1

5.0

4.1

4.4

3.8

3.5

3.0

2.7

3.2

4.8

4.6

4.2

3.9

3.2

3.5

4.2

8.1

8.9

1
974

2.0

1
.7

2.7

3.9

3.6

3.0

2.9

5.9

4.9

4.7

3.7

3.4

4.2

5.1

6.8

8.6

5.9

5.2

5.4

5.0

4.4

3.9

4.0

5.6

6.4

6.1

5.0

4.1

4.4

3.8

3.5

3.0

2.7

3.3

4.8

4.6

4.3

3.9

3.3

3.5

4.2

8.1

8.9

1
975

2.0

1
.7

2.7

3.9

3.6

3.0

2.9

5.9

4.9

4.7

3.7

3.4

4.2

5.0

6.8

8.6

5.9

5.2

5.3

5.0

4.3

3.8

3.9

5.6

6.4

6.1

5.0

4.1

4.4

3.8

3.4

2.9

2.7

3.2

4.7

4.6

4.2

3.9

3.3

3.5

4.2

8.1

8.8

1
976

2.0

1
.7

2.7

3.9

3.6

3.1

2.9

5.9

4.9

4.7

3.7

3.4

4.2

5.0

6.7

8.6

5.9

5.2

5.3

5.0

4.3

3.8

3.9

5.5

6.3

6.1

5.0

4.1

4.4

3.8

3.4

2.9

2.7

3.2

4.7

4.6

4.3

3.8

3.3

3.5

4.1

8.0

8.8

1
977

2.0

1
.7

2.7

3.9

3.6

3.1

3.0

5.9

4.9

4.7

3.7

3.4

4.2

5.0

6.7

8.5

5.9

5.2

5.3

5.0

4.3

3.8

3.9

5.5

6.3

6.0

5.0

4.1

4.3

3.8

3.4

2.9

2.7

3.2

4.7

4.6

4.3

3.8

3.3

3.5

4.1

8.0

8.7

1
978

2.0

1
.7

2.7

4.0

3.6

3.1

3.0

5.9

4.9

4.7

3.7

3.4

4.2

5.0

6.7

8.5

5.9

5.2

5.3

5.0

4.3

3.9

3.9

5.5

6.3

6.0

5.0

4.1

4.4

3.8

3.4

2.9

2.8

3.2

4.7

4.6

4.3

3.8

3.3

3.5

4.1

8.0

8.7

1
979

2.0

1
.7

2.7

4.0

3.6

3.1

3.0

5.9

5.0

4.8

3.7

3.4

4.2

5.0

6.8

8.5

5.9

5.2

5.3

5.0

4.3

3.9

3.9

5.5

6.3

6.1

5.0

4.1

4.4

3.8

3.4

2.9

2.8

3.2

4.7

4.7

4.3

3.8

3.3

3.5

4.1

8.0

8.7

1
980

2.0

1
.7

2.7

4.0

3.6

3.1

3.0

5.9

5.0

4.9

3.7

3.5

4.3

5.1

6.8

8.6

5.9

5.2

5.3

5.1

4.3

3.9

4.0

5.6

6.4

6.1

5.0

4.1

4.4

3.8

3.4

2.9

2.8

3.3

4.7

4.7

4.4

3.9

3.3

3.5

4.2

8.0

8.8

1
981

2.1

1
.7

2.8

4.1

3.7

3.2

3.1

6.0

5.1

5.0

3.8

3.6

4.4

5.2

7.0

8.7

6.0

5.3

5.4

5.1

4.4

4.0

4.1

5.7

6.4

6.2

5.1

4.2

4.5

3.9

3.5

3.0

2.9

3.3

4.8

4.8

4.4

4.0

3.4

3.6

4.3

8.2

8.9

1
982

2.1

1
.7

2.8

4.1

3.7

3.2

3.1

6.1

5.1

5.0

3.8

3.6

4.4

5.2

7.1

8.8

6.1

5.3

5.4

5.2

4.4

4.0

4.1

5.7

6.4

6.2

5.1

4.2

4.5

3.9

3.5

3.0

2.9

3.3

4.9

4.8

4.5

4.0

3.4

3.6

4.3

8.2

8.9

1
983

2.1

1
.7

2.8

4.1

3.7

3.2

3.2

6.0

5.1

5.0

3.8

3.6

4.4

5.2

7.0

8.7

6.1

5.3

5.4

5.2

4.4

4.0

4.1

5.7

6.4

6.2

5.1

4.2

4.5

3.9

3.5

3.0

2.9

3.3

4.9

4.8

4.5

4.0

3.4

3.6

4.2

8.1

8.9

1
984

2.1

1
.7

2.8

4.1

3.8

3.2

3.1

6.1

5.1

5.1

3.9

3.6

4.4

5.3

7.1

8.8

6.1

5.4

5.5

5.2

4.5

4.1

4.1

5.8

6.5

6.3

5.2

4.3

4.5

4.0

3.5

3.0

2.9

3.3

4.9

4.8

4.5

4.0

3.5

3.6

4.3

8.2

9.0

1
985

2.1

1
.7

2.9

4.1

3.8

3.2

3.2

6.1

5.2

5.1

3.9

3.6

4.4

5.3

7.1

8.8

6.1

5.4

5.5

5.2

4.5

4.1

4.1

5.8

6.5

6.3

5.2

4.3

4.6

4.0

3.5

3.0

2.9

3.4

4.9

4.8

4.6

4.0

3.5

3.6

4.3

8.2

9.0

1
986

2.1

1
.7

2.9

4.2

3.8

3.3

3.2

6.1

5.2

5.2

4.0

3.6

4.5

5.3

7.2

8.9

6.2

5.5

5.5

5.3

4.6

4.1

4.2

5.8

6.5

6.3

5.2

4.3

4.6

4.1

3.6

3.1

3.0

3.4

5.0

4.9

4.6

4.1

3.5

3.7

4.4

8.3

9.1

1
987

2.1

1
.8

2.9

4.2

3.9

3.3

3.2

6.1

5.2

5.2

3.9

3.6

4.5

5.3

7.2

8.9

6.2

5.5

5.5

5.3

4.6

4.1

4.2

5.9

6.6

6.4

5.3

4.3

4.6

4.1

3.6

3.1

3.0

3.4

5.0

4.9

4.6

4.1

3.5

3.7

4.4

8.3

9.1

1
988

2.2

1
.8

3.0

4.2

3.9

3.3

3.3

6.2

5.3

5.3

4.0

3.7

4.5

5.4

7.3

9.0

6.2

5.6

5.6

5.4

4.6

4.2

4.2

5.9

6.6

6.4

5.3

4.4

4.6

4.1

3.6

3.1

3.0

3.4

5.0

4.9

4.7

4.1

3.6

3.7

4.4

8.4

1
989

2.2

1
.7

3.0

4.2

3.8

3.3

3.3

6.1

5.2

5.3

4.0

3.7

4.5

5.4

7.3

8.9

6.2

5.5

5.6

5.4

4.6

4.2

4.2

5.9

6.6

6.4

5.3

4.4

4.7

4.1

3.7

3.1

3.0

3.4

5.0

4.9

4.7

4.1

3.6

3.7

4.4

8.4

9.1

1
990

2.2

1
.7

3.0

4.3

3.9

3.3

3.3

6.1

5.2

5.3

4.0

3.7

4.5

5.4

7.3

8.9

6.2

5.5

5.6

5.4

4.6

4.2

4.2

5.9

6.6

6.5

5.3

4.4

4.7

4.2

3.7

3.1

3.0

3.4

5.0

5.0

4.7

4.1

3.6

3.7

4.4

8.4

9.2

1
991

2.3

1
.8

3.0

4.3

3.9

3.3

3.3

6.2

5.3

5.3

4.0

3.7

4.5

5.4

7.4

9.0

6.3

5.6

5.7

5.5

4.7

4.3

4.3

6.0

6.7

6.6

5.4

4.5

4.7

4.2

3.7

3.2

3.1

3.5

5.0

5.0

4.8

4.1

3.7

3.8

4.5

8.5

9.3

1
992

2.3

1
.7

3.0

4.3

3.9

3.3

3.3

6.1

5.3

5.3

4.0

3.6

4.5

5.3

7.3

8.9

6.2

5.6

5.6

5.4

4.7

4.2

4.3

5.9

6.7

6.5

5.3

4.4

4.7

3.1

3.0

3.4

5.0

5.0

4.8

4.1

3.7

3.8

4.4

8.4

9.2

1
993

2.2

1
.7

3.0

4.3

3.9

3.3

3.3

6.0

5.2

5.2

3.9

3.6

4.4

5.2

7.2

8.8

6.1

5.5

5.5

5.4

4.6

4.2

4.2

5.9

6.6

6.5

5.3

4.4

4.6

4.1

3.6

3.1

3.0

3.4

4.9

5.0

4.8

4.1

3.7

3.7

4.3

8.3

9.1

1
994

2.3

1
.7

3.0

4.3

3.9

3.3

3.3

6.0

5.2

5.2

3.9

3.6

4.4

5.3

7.2

8.8

6.1

5.6

5.6

5.5

4.6

4.2

4.2

5.9

6.6

6.5

5.4

4.4

4.7

4.2

3.7

3.2

3.0

3.4

4.9

5.1

4.8

4.1

3.7

3.7

4.4

8.4

9.2

1
995

2.2

1
.7

3.0

4.3

3.9

3.2

3.3

5.9

5.1

5.1

3.9

3.6

4.4

5.2

7.2

8.7

6.1

5.5

5.5

5.4

4.6

4.2

4.2

5.9

6.5

6.5

5.3

4.4

4.6

4.1

3.7

3.1

3.0

3.4

4.9

5.1

4.8

4.1

3.7

3.7

4.3

8.3

9.1

1
996

2.3

1
.7

3.0

4.3

3.9

3.3

3.3

5.9

5.2

5.2

3.9

3.6

4.4

5.2

7.2

8.7

6.1

5.5

5.5

5.4

4.6

4.2

4.2

5.9

6.6

6.5

5.3

4.5

4.6

4.1

3.7

3.1

3.0

3.4

4.9

5.1

4.8

4.1

3.7

3.7

4.3

8.3

9.1

1
997

2.3

1
.7

3.0

4.3

3.9

3.2

3.3

5.8

5.1

5.1

3.9

3.6

4.4

5.2

7.2

8.7

6.1

5.5

5.5

5.4

4.6

4.2

4.2

5.9

6.6

6.5

5.3

4.4

4.6

4.1

3.6

3.1

3.0

3.4

4.9

5.1

4.8

4.1

3.7

3.7

4.3

8.3

1
998

2.3

1
.7

3.0

4.3

3.9

3.2

3.3

5.8

5.1

5.1

3.8

3.6

4.3

5.1

7.1

8.6

6.0

5.4

5.5

5.4

4.6

4.1

4.2

5.8

6.6

6.5

5.3

4.4

4.6

4.1

3.6

3.1

3.0

3.4

4.9

5.1

4.7

4.1

3.7

3.7

4.3

8.3

9.1

1
999

2.3

1
.7

3.0

4.2

3.9

3.2

3.3

5.7

5.0

5.0

3.8

3.5

4.2

5.0

7.0

8.5

5.9

5.4

5.4

5.3

4.5

4.1

4.1

5.8

6.5

6.4

5.3

4.4

4.6

4.1

3.6

3.1

3.0

3.4

4.8

5.0

4.7

4.1

3.7

3.7

4.2

8.2

9.0

2000

2.2

1
.7

3.0

4.1

3.9

3.1

3.3

5.7

5.0

4.9

3.7

3.5

4.2

4.9

6.9

8.4

5.9

5.4

5.4

5.3

4.5

4.1

4.1

5.8

6.4

6.4

5.2

4.4

4.5

4.0

3.5

3.1

2.9

3.3

4.8

5.0

4.7

4.0

3.6

3.7

4.2

8.1

8.9

4.2

3.7

9.1

9.1

2001

2.2

1
.7

3.0

4.2

3.9

3.1

3.2

5.6

5.0

4.9

3.7

3.5

4.2

4.9

6.8

8.4

5.8

5.3

5.4

5.3

4.5

4.0

4.1

5.7

6.4

6.3

5.2

4.3

4.5

4.0

3.5

3.1

2.9

3.3

4.8

5.0

4.6

4.0

3.6

3.6

4.2

8.1

8.9

2002

2.2

1
.7

3.0

4.3

3.9

3.1

3.3

5.6

5.0

4.9

3.7

3.5

4.2

4.9

6.8

8.3

5.8

5.3

5.4

5.3

4.5

4.0

4.0

5.7

6.4

6.3

5.2

4.4

4.5

4.0

3.5

3.1

2.9

3.3

4.8

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

2003

2.2

1
.7

3.0

4.3

3.9

3.1

3.3

5.6

5.0

4.9

3.8

3.5

4.1

4.9

6.8

8.3

5.8

5.4

5.3

5.3

4.5

4.0

4.1

5.7

6.4

6.3

5.2

4.4

4.5

4.0

3.6

3.1

2.9

3.3

4.7

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

2004

2.2

1
.7

3.0

4.3

3.9

3.1

3.3

5.6

5.0

4.9

3.8

3.5

4.1

4.9

6.8

8.3

5.9

5.3

5.3

5.3

4.5

4.0

4.1

5.7

6.4

6.3

5.2

4.4

4.5

4.0

3.6

3.1

3.0

3.3

4.8

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

2005

2.2

1
.7

3.0

4.3

4.0

3.1

3.3

5.6

5.0

5.0

3.8

3.5

4.1

4.9

6.9

8.4

5.9

5.4

5.4

5.3

4.6

4.1

4.1

5.7

6.5

6.4

5.2

4.4

4.5

4.1

3.6

3.1

3.0

3.4

4.8

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

2006

2.2

1
.7

3.0

4.4

3.9

3.1

3.3

5.6

5.0

5.0

3.8

3.5

4.1

4.9

6.8

8.4

5.9

5.4

5.4

5.3

4.6

4.1

4.1

5.7

6.5

6.4

5.2

4.4

4.6

4.1

3.6

3.1

3.0

3.4

4.8

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

2007

2.2

1
.6

3.0

4.4

4.0

3.1

3.3

5.6

4.9

5.0

3.8

3.5

4.1

4.8

6.8

8.3

5.9

5.4

5.4

5.3

4.6

4.1

4.0

5.7

6.5

6.4

5.2

4.4

4.5

4.0

3.6

3.1

3.0

3.3

4.7

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

2008

2.2

1
.7

3.1

4.4

4.0

3.1

3.3

5.5

5.0

5.0

3.8

3.6

4.1

4.8

6.8

8.3

5.8

5.4

5.3

5.3

4.5

4.1

4.0

5.7

6.5

6.3

5.2

4.4

4.5

4.0

3.6

3.1

3.0

3.4

4.7

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

2009

2.2

1
.7

3.1

4.4

4.0

3.1

3.4

5.5

5.0

5.0

3.8

3.6

4.1

4.8

6.8

8.3

5.9

5.4

5.4

5.3

4.6

4.0

4.1

5.7

6.5

6.4

5.2

4.4

4.6

4.0

3.6

3.1

3.0

3.4

4.7

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

201
0

2.2

1
.7

3.1

4.4

4.0

3.2

3.4

5.6

5.1

5.1

3.9

3.6

4.1

4.8

6.8

8.3

5.9

5.4

5.4

5.3

4.6

4.0

4.1

5.7

6.5

6.4

5.2

4.4

4.5

4.0

3.6

3.1

3.0

3.4

4.8

5.0

4.7

4.1

3.7

3.8

4.2

8.1

8.9

6.0

6.7

6.5

5.4

4.4

4.6

4.1

3.6

3.1

2.9

3.3

4.8

5.0

4.7

4.0

3.7

3.7

4.2

8.1

8.9

A ctual
2.1

1
.7

2.7

3.8

3.5

3.0

2.9

5.9

4.9

4.7

3.7

3.4

4.3

5.2

7.1

8.7

6.1

5.4

5.5

5.3

4.6

4.2

4.2

373

Notes: Authors’ own calculations from the IPUMS-CPS data. The columns correspond to the year of the labor outcomes
used. The rows correspond to the year of the sociodemographic composition used. Lightly shaded columns correspond to the
NBER-dated contractions (from peak to trough). Darkly shaded elements correspond to actual values for given year.

M. Kudlyak, T. Lubik, and J. Tompkins: Non-Employment of U.S. Men

Table 3 Counterfactual Predictions of the OLF-to-Population Ratio
by Percent, 25–64-Year-Old Males

374

Federal Reserve Bank of Richmond Economic Quarterly

small quantitative differences, find the qualitative conclusions from the two
decompositions to be the same. For this reason, we report only the results of
the decomposition corresponding to equation (2).3

Changes in Aggregate Labor Outcomes
Tables 1–3 show the predicted labor outcomes—proportion employed, unemployed, and OLF, respectively—calculated using the labor outcomes of different groups from year t1 and the demographic composition of the population
from year t2 , where t1 , t2 ∈ [1968, 2010].
The diagonal elements in Tables 1–3 (darkly shaded) show the actual
labor outcomes for their respective year. The off-diagonal elements show
the counterfactual labor outcomes. Thus, for each year we have two sets
of counterfactual predictions. Moving along a row gives the predicted labor
outcome for a fixed demographic composition, while moving down a column
gives the predicted labor outcome for fixed labor outcomes of different groups.
For example, the (1983, 2010) entry of Table 1 gives the predicted proportion
of employed individuals in 2010 given the 1983 demographic composition
(77.9 percent), while the (2010, 1983) entry gives the predicted proportion
of employed individuals in 2010 given the 1983 labor outcomes of different
groups (78.6 percent).
Employment

The actual 2010 value of the employment-to-population ratio for 25–64-yearold men and the two series of counterfactual predictions for 2010 are shown
in Figure 6, Panel A. The dashed line shows the predicted employment-topopulation ratio from holding the demographic composition of the population constant at its 2010 level but varying the labor outcomes of different
groups. The dotted line shows the predicted employment-to-population ratio from holding the labor outcomes of different groups at their 2010 level
but varying the sociodemographic composition of the population. The actual employment-to-population ratio in 2010 is lower than any point of the
predicted counterfactual series. This implies that changes in both the sociodemographic composition and labor outcomes of different groups in 2010
contribute to the historically low employment-to-population ratio among men.
To examine the contribution of the change in the labor outcomes of
different groups and the sociodemographic composition to changes in the
employment-to-population ratio, we construct each term of equation (2)
for t1 ∈ [1968, 2010] and t2 = 2010. Table 4 reports the change in the
employment-to-population ratio, the total change as a result of change in each
3 Results of the alternative decomposition are available upon request.

M. Kudlyak, T. Lubik, and J. Tompkins: Non-Employment of U.S. Men

375

Figure 6 Counterfactual Predictions of Labor Outcomes,
25–64-Year-Old Males
Panel A: 2010 Counterfactual - Employment
90

Percentage Employed

Actual 2010 Value
Predicted, 2010 Labor
Outcome Proportions

Predicted, 2010
Demographics

85

80

75
1970

1980

1990

2000

2010

Year
Panel B: 2010 Counterfactual - Unemployment

Percentage Unemployed

10

8

6

4
Actual 2010 Value
Predicted, 2010 Labor
Outcome Proportions

2
1970

1980

Predicted, 2010
Demographics

1990

2000

2010

Year
Panel C: 2010 Counterfactual - OLF

Percentage OLF

16

14

12

10
Actual 2010 Value
Predicted, 2010 Labor
Outcome Proportions

8
1970

1980

1990

Predicted, 2010
Demographics

2000

2010

Year

Notes: Authors’ own calculations from the IPUMS-CPS data. The figure shows the
outcome-to-population ratio for each of the three labor outcomes. “Predicted, 2010
Demographics” gives the predicted 2010 labor outcome using the 2010 demographic
composition and historical labor outcome proportions. “Predicted, 2010 Labor Outcome
Proportions” gives the predicted 2010 labor outcome using the 2010 labor outcome proportions and historical demographic compositions.

376

Federal Reserve Bank of Richmond Economic Quarterly

of the two terms, and the percentage of the total change that is accounted for
by changes in each of the two terms. Figure 7, Panel A plots the change in the
employment-to-population ratio and the total change as a result of each of the
two terms. We find that the change in the labor outcomes of different groups
accounts for the majority of the change in the employment-to-population ratio. Comparing the aftermath of the 2007–2009 and 1980–1982 recessions,
we see that between 1983–2010 the employment-to-population ratio fell by
3.86 percentage points, of which 41.7 percent is a result of the change in the
demographic composition.
We conclude that the decline in the employment-to-population ratio is a
result of both changes in the sociodemographic composition and changes in
the labor outcomes of different groups.
Unemployment

Panel B of Figure 6 plots the two counterfactual series of the 2010 unemployment-to-population ratio against its actual 2010 value. We draw two key
observations from the figure: (1) the actual 2010 unemployment-to-population
ratio is higher than any point of the predicted counterfactual series that holds
the sociodemographic composition constant at its 2010 level; and (2) the
actual 2010 unemployment-to-population ratio is lower than the predicted
counterfactual for some periods when we hold the labor outcomes of different
groups constant at their 2010 level.
These observations suggest that (1) the labor outcomes of different
sociodemographic groups in 2010 contribute to a higher unemployment-topopulation ratio than the labor outcomes of different groups in all previous
years; and (2) the sociodemographic composition of the population in 2010
actually contributes to a lower unemployment-to-population ratio as compared
to the sociodemographic composition in some earlier years.
Table 5 breaks down the total change in the unemployment-to-population
ratio between a given year and 2010 into change caused by developments in
the demographic composition of the population and change caused by developments in the labor outcomes of different groups. Figure 7, Panel B plots
the total change in the unemployment-to-population ratio and the change as a
result of each of the two terms. The results of this table and graph corroborate
our above claims. Changes in the demographic composition of the population
have contributed a small, and often negative, amount of the increase in the
unemployment-to-population ratio, whereas changes in the labor outcomes
of different groups have been responsible for approximately 100 percent of
the increase. We observe that, relative to 1983, the 2010 unemployment-topopulation ratio is 0.18 percentage points higher, of which 101.0 percent of
the change is a result of a change in the labor outcomes of different groups.
Thus, the rise in the 2010 unemployment-to-population ratio relative to its

Total
Change

1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989

−15.52
−15.52
−14.45
−12.75
−12.46
−12.40
−12.22
−8.26
−8.61
−9.08
−9.79
−10.30
−9.19
−7.91
−5.83
−3.86
−6.25
−6.81
−6.56
−7.17
−7.66
−8.26

Change Because of
Labor Outcomes
Percentage Points
% of Total Change
−14.96
−14.87
−13.70
−12.03
−11.61
−11.40
−11.26
−7.23
−7.33
−7.75
−8.54
−8.90
−7.77
−6.57
−4.37
−2.25
−4.65
−5.25
−5.03
−5.50
−6.07
−6.59

96.39
95.81
94.78
94.32
93.18
91.92
92.11
87.47
85.20
85.36
87.21
86.41
84.59
83.10
74.85
58.29
74.44
77.03
76.73
76.69
79.23
79.68

Change Because of
Sociodemographics
Percentage Points
% of Total Change
−0.56
−0.65
−0.76
−0.72
−0.85
−1.00
−0.96
−1.04
−1.27
−1.33
−1.25
−1.40
−1.42
−1.34
−1.47
−1.61
−1.60
−1.56
−1.53
−1.67
−1.59
−1.68

3.61
4.19
5.22
5.68
6.82
8.08
7.89
12.53
14.80
14.64
12.79
13.59
15.41
16.90
25.15
41.71
25.56
22.97
23.27
23.31
20.77
20.32

377

Year

M. Kudlyak, T. Lubik, and J. Tompkins: Non-Employment of U.S. Men

Table 4 Decomposition of the Change in the
Employment-to-Population Ratio Between 2010 and Earlier
Years, 25–64-Year-Old Males

Total
Change

1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010

−8.20
−6.28
−5.61
−5.32
−5.28
−6.62
−6.48
−7.25
−7.82
−8.04
−8.22
−7.62
−5.89
−5.07
−4.92
−5.59
−6.00
−6.20
−5.28
−0.68
0.00

Change Because of
Labor Outcomes
Percentage Points
% of Total Change
−6.49
−4.80
−4.01
−3.61
−3.53
−4.61
−4.69
−5.56
−6.17
−6.29
−6.38
−5.90
−4.33
−3.66
−3.69
−4.62
−5.24
−5.46
−4.80
−0.39
0.00

79.15
76.45
71.44
67.81
66.78
69.59
72.39
76.65
78.87
78.27
77.57
77.39
73.61
72.14
75.03
82.75
87.39
88.10
91.00
56.79
0.00

Change Because of
Sociodemographics
Percentage Points
% of Total Change
−1.71
−1.48
−1.60
−1.71
−1.76
−2.01
−1.79
−1.69
−1.65
−1.75
−1.84
−1.72
−1.55
−1.41
−1.23
−0.96
−0.76
−0.74
−0.47
−0.29
0.00

20.85
23.55
28.56
32.19
33.22
30.41
27.61
23.35
21.13
21.73
22.43
22.61
26.39
27.86
24.97
17.25
12.61
11.90
9.00
43.21
0.00

Notes: Authors’ own calculations from the IPUMS-CPS data. Bold rows correspond to the NBER-dated contractions (from
peak to trough). Columns 3–6 correspond to the decomposition as described in (2).

Federal Reserve Bank of Richmond Economic Quarterly

Year

378

Table 4 (Continued) Decomposition of the Change in the Employment-to-Population Ratio Between
2010 and Earlier Years, 25–64-Year-Old Males

M. Kudlyak, T. Lubik, and J. Tompkins: Non-Employment of U.S. Men

379

1983 level is driven entirely by changes in the labor outcomes of different
groups.
Out of Labor Force

Panel C of Figure 6 plots the two counterfactual series of the 2010 OLFto-population ratio against its actual 2010 value. We highlight two features
of the figure: (1) The actual 2010 OLF-to-population ratio is always higher
than the counterfactual series calculated by holding the labor outcomes of
different groups at their 2010 level; and (2) prior to 1994, the actual 2010
OLF-to-population ratio is always higher than the counterfactual series that is
calculated by holding the sociodemographic composition constant at its 2010
level, although after 1994 the counterfactual is sometimes higher. Thus, we
infer that the demographic composition contributes substantially to the high
OLF-to-population ratio in 2010.
Table 6 and Figure 7, Panel C formalize this result, showing that the
total change in the incidence of the OLF-to-population ratio between any
year prior to 2009 and 2010 is positive. It also shows that the change in
the sociodemographic composition contributes to a higher OLF-to-population
ratio. The contribution from the change in the sociodemographic composition
has been increasing since 1968. The contribution from the change in the
labor outcomes of different groups has been significantly smaller, and has
even lowered the OLF-to-population ratio in more recent years. Turning our
attention once again to the 1983 and 2010 comparison, we see that the OLF-topopulation ratio increased 3.68 percentage points from 1983–2010, of which
1.66 percentage points can be attributed to the change in the demographic
composition. Thus, changes to the demographic composition of the population
have played a large role in increasing the 2010 OLF-to-population ratio relative
to its 1983 level.

4.

FORECAST OF THE OLF-TO-POPULATION RATIO

Our findings show that changes in the employment- and OLF-to-population
ratios of 25–64-year-old men during the last four decades are, to a large degree, associated with changes in the sociodemographic composition of the
population. Using the labor outcomes of different sociodemographic groups
from 2010 and a projected sociodemographic composition of the population
in 2015, we are able to create projections of the aggregate labor outcomes in
2015 using equation (1).
As Figure 1 shows, there is a large cyclical component in the employmentand unemployment-to-population ratios. Consequently, the forecasts of these
ratios depend heavily on the business cycle phase of the year of our decomposition. In addition, the changes in unemployment (and employment to a lesser
extent) are mostly dominated by changes in the labor outcomes of different

380

Federal Reserve Bank of Richmond Economic Quarterly

Figure 7 The Decomposition of the Change in Labor Outcomes
Panel A: Change Decomposition - Employment

Percentage Point Change

0

-5

-10
Total Change
Change Because
of Demographics

-15
1970

1980

1990

Change Because
of Labor Outcome
Proportions

2000

2010

Year
Panel B: Change Decomposition - Unemployment

Percentage Point Change

8

Total Change
Change Because
of Demographics

Change Because
of Labor Outcome
Proportions

6

4

2

0
1970

1980

1990

2000

2010

Year
Panel C: Change Decomposition - OLF
Total Change

Percentage Point Change

8

Change Because
of Demographics

Change Because
of Labor Outcome
Proportions

6

4

2

0
1970

1980

1990

2000

2010

Year

Notes: Authors’ own calculations from the IPUMS-CPS data.

sociodemographic groups rather than changes in the sociodemographic composition. In contrast, the OLF-to-population ratio has a much smaller cyclical
component. Consequently, we focus on forecasting the OLF-to-population
ratio.

Total
Change

1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989

6.87
7.20
6.25
5.10
5.41
5.90
6.02
3.00
3.99
4.22
5.24
5.48
4.66
3.73
1.86
0.18
2.82
3.50
3.40
3.62
4.32
4.72

Change Because of
Labor Outcomes
Percentage Points
% of Total Change
7.04
7.31
6.28
5.18
5.44
5.88
5.97
2.89
3.83
4.05
5.07
5.28
4.49
3.72
1.86
0.18
2.86
3.56
3.55
3.78
4.53
4.95

102.42
101.46
100.57
101.62
100.53
99.61
99.14
96.20
95.98
95.79
96.64
96.33
96.41
99.76
99.75
101.02
101.41
101.66
104.50
104.28
105.07
104.72

Change Because of
Sociodemographics
Percentage Points
% of Total Change
−0.17
−0.10
−0.04
−0.08
−0.03
0.02
0.05
0.11
0.16
0.18
0.18
0.20
0.17
0.01
0.00
0.00
−0.04
−0.06
−0.15
−0.15
−0.22
−0.22

−2.42
−1.46
−0.57
−1.62
−0.53
0.39
0.86
3.80
4.02
4.21
3.36
3.67
3.59
0.24
0.25
−1.02
−1.41
−1.66
−4.50
−4.28
−5.07
−4.72

381

Year

M. Kudlyak, T. Lubik, and J. Tompkins: Non-Employment of U.S. Men

Table 5 Decomposition of the Change in the
Unemployment-to-Population Ratio Between 2010 and
Earlier Years

382

Table 5 (Continued) Decomposition of the Change in the Unemployment-to-Population Ratio
Between 2010 and Earlier Years
Total
Change

1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010

4.69
2.90
2.26
2.45
3.55
4.47
4.28
4.78
5.32
5.83
5.99
5.57
4.16
3.93
4.24
4.89
5.24
5.25
4.76
0.83
0.00

Change Because of
Labor Outcomes
Percentage Points
% of Total Change
4.96
3.25
2.58
2.67
3.79
4.63
4.49
5.00
5.49
5.90
6.00
5.53
4.12
3.87
4.20
4.87
5.23
5.19
4.71
0.80
0.00

105.69
112.03
114.43
108.79
106.79
103.61
105.03
104.59
103.31
101.11
100.09
99.21
98.96
98.53
98.87
99.63
99.98
98.87
99.04
96.35
0.00

Change Because of
Sociodemographics
Percentage Points
% of Total Change
−0.27
−0.35
−0.33
−0.22
−0.24
−0.16
−0.21
−0.22
−0.18
−0.06
−0.01
0.04
0.04
0.06
0.05
0.02
0.00
0.06
0.05
0.03
0.00

−5.69
−12.03
−14.43
−8.79
−6.79
−3.61
−5.03
−4.59
−3.31
−1.11
−0.09
0.79
1.04
1.47
1.13
0.37
0.02
1.13
0.96
3.65
0.00

Notes: Authors’ own calculations from the IPUMS-CPS data. Bold rows correspond to the NBER-dated contractions (from
peak to trough). Columns 3–6 correspond to the decomposition as described in (2).

Federal Reserve Bank of Richmond Economic Quarterly

Year

M. Kudlyak, T. Lubik, and J. Tompkins: Non-Employment of U.S. Men

383

To perform this forecasting exercise, we project the sociodemographic
composition of the population of 25–64-year-old men in 2015. In projecting
this composition, we focus on the changes in age of the individuals in the 2010
sample while holding education, race, and marital characteristics constant at
their 2010 levels.
To simulate a sample of 25–64-year-old male workers in 2015, we use the
2010 sample of 20–59-year-old male workers and construct the age variable
for 2015. We use the age-specific annual male mortality rates from the Social
SecurityAdministration4 and accordingly choose which workers of a particular
age survive from 2010–2015.5 Each worker in the simulated sample is aged
five years, but has the same education, race, and marital status as in the 2010
sample. We use the projected 2015 population and the CPS sampling weights
to construct the sociodemographic composition terms in equation (1). Then we
use these forecasted demographic composition terms and the labor outcomes
of the corresponding sociodemographic groups from 2010 to construct the
predicted aggregate 2015 OLF-to-population ratio using equation (1).
Note that this exercise assumes that the mortality rates for each age remain unchanged from 2007–2015. Also, we use the sampling weights from
2010, which may not deliver a representative population for our simulated
2015 sample (for example, because we do not adjust the weights to reflect the
demographic composition of the surviving individuals). However, given the
relatively short forecast horizon, these weights provide a good approximation
for aggregation. Finally, when aging the 2010 population, we do not accordingly adjust demographic factors other than age. For example, aging a male
from 20 to 25 might alter both his educational attainment and marital status.
Our forecasting exercise does not take these effects into account.
The results of our forecast are displayed in Table 7. Panel C of Table
7 contains our forecast and the U.S. Census forecast for the age distribution
of 25–64-year-old men in 2015. As the forecast shows, the shares of 55–
64 and 25–34-year-old males are projected to increase, while the share of
35–54-year-old males is projected to decrease.
Panel A of Table 7 displays the results of the forecast of the OLF-topopulation ratio based on the labor status outcomes of different groups in
2010. For comparison, Panel B contains the results based on the labor status
outcomes of different groups in 2007, i.e., the year of a recent business cycle
peak. The results show that under both sets of labor status outcomes of different
groups, the OLF-to-population ratio is predicted to reach more than 16 percent
in 2015 as compared to the actual rate of 14.7 percent in 2010.

4 Data available at: www.ssa.gov/oact/STATS/table4c6.html#ss.
5 For example, the probability that a 20-year-old worker in 2010 survives to 2015 is (1 −

m
m
m
m
m
m
p20 )(1 − p21 )(1 − p22 )(1 − p23 )(1 − p24 ), where pa is the annual mortality rate of a worker at
age a.

384

Table 6 Decomposition of the Change in the OLF-to-Population Ratio
Between 2010 and Earlier Years
Total
Change

1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989

8.65
8.32
8.21
7.65
7.05
6.50
6.20
5.26
4.61
4.85
4.55
4.83
4.53
4.18
3.97
3.68
3.43
3.31
3.16
3.55
3.35
3.54

Change Because of
Labor Outcomes
Percentage Points
% of Total Change
7.92
7.55
7.41
6.84
6.15
5.51
5.28
4.31
3.50
3.70
3.46
3.60
3.27
2.85
2.50
2.06
1.77
1.68
1.46
1.72
1.53
1.63

91.51
90.73
90.25
89.40
87.27
84.88
85.16
81.82
75.80
76.15
76.03
74.62
72.15
68.10
62.90
55.89
51.70
50.69
46.24
48.30
45.69
46.01

Change Because of
Sociodemographics
Percentage Points
% of Total Change
0.73
0.77
0.80
0.81
0.90
0.98
0.92
0.96
1.12
1.16
1.09
1.22
1.26
1.33
1.47
1.62
1.66
1.63
1.70
1.84
1.82
1.91

8.49
9.27
9.75
10.60
12.73
15.12
14.84
18.18
24.20
23.85
23.97
25.38
27.85
31.90
37.10
44.11
48.30
49.31
53.76
51.70
54.31
53.99

Federal Reserve Bank of Richmond Economic Quarterly

Year

Year

Total
Change

1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010

3.51
3.38
3.36
2.87
1.73
2.15
2.21
2.47
2.51
2.21
2.23
2.05
1.72
1.14
0.67
0.70
0.76
0.95
0.52
−0.15
0.00

Change Because of
Labor Outcomes
Percentage Points
% of Total Change
1.51
1.54
1.42
0.93
−0.28
−0.03
0.18
0.54
0.63
0.37
0.36
0.35
0.19
−0.24
−0.53
−0.27
0.00
0.27
0.08
−0.43
0.00

43.12
45.67
42.26
32.32
−16.19
−1.41
7.96
21.98
25.06
16.76
15.93
16.92
10.93
−20.80
−78.90
−38.66
−0.16
28.02
14.69
284.63
0.00

Change Because of
Sociodemographics
Percentage Points
% of Total Change
2.00
1.84
1.94
1.94
2.02
2.18
2.03
1.93
1.88
1.84
1.88
1.70
1.53
1.38
1.20
0.97
0.76
0.69
0.44
0.28
0.00

56.88
54.33
57.74
67.68
116.19
101.41
92.04
78.02
74.94
83.24
84.07
83.08
89.07
120.80
178.90
138.66
100.16
71.98
85.31
−184.63
0.00

385

Notes: Authors’ own calculations from the IPUMS-CPS data. Bold rows correspond to the NBER-dated contractions (from
peak to trough). Columns 3–6 correspond to the decomposition as described in (2).

M. Kudlyak, T. Lubik, and J. Tompkins: Non-Employment of U.S. Men

Table 6 (Continued) Decomposition of the Change in the OLF-to-Population Ratio Between 2010
and Earlier Years

386

Federal Reserve Bank of Richmond Economic Quarterly

Table 7 Predicted Aggregate OLF-to-Population Ratio Among
25–64-Year-Old Men, 2015
Panel A: Based on 2010 Labor Outcomes
Actual 2010
Predicted 2015
Employment-to-Population
76.33
X
Unemployment-to-Population
8.92
X
OLF-to-Population
14.75
16.27
Panel B: Based on 2007 Labor Outcomes
Actual 2007
Predicted 2015
Employment-to-Population
82.53
X
Unemployment-to-Population
3.67
X
OLF-to-Population
13.79
16.04
Age
25–34
35–44
45–54
55–64

5.

Panel C: Age Composition (Percent)
Actual 2010
Simulated 2015
25.80
26.36
25.20
23.93
27.50
25.87
21.50
23.85

Census 2015
26.74
24.63
25.43
23.20

CONCLUSIONS

The OLF-to-population ratio among 25–64-year-old men has increased from
6.5 percent in 1970 to 14.7 percent in 2010. In the aftermath of the 1969–1970
recession, the employment-to-population ratio among this group was 89.1
percent, while in the aftermath of the 2007–2009 recession, the ratio is nearly
13 percentage points lower. Throughout this article we have examined the
degree to which these changes can be explained by changes in the composition
of the population by age, race, education, and marital status, and the degree
to which they can be attributed to changes in the labor market outcomes of
different sociodemographic groups.
We find that the rise in the OLF-to-population ratio since the early 1980s
is primarily a result of changes in the demographic composition of the population. Changes in the demographic composition account for about 25 percent of
the increase in the employment-to-population ratio during the same period, and
changes in the unemployment-to-population ratio are almost entirely driven by
changes in the employment status composition. Finally, simulating the 2010
sample five years forward and using labor outcomes of different sociodemographic groups from 2010, we project that the OLF-to-population ratio among
25–64-year-old men will rise to 16 percent in 2015.

M. Kudlyak, T. Lubik, and J. Tompkins: Non-Employment of U.S. Men

387

REFERENCES
Autor, David H., and Mark G. Duggan. 2003. “The Rise in the Disability
Rolls and the Decline in Unemployment.” The Quarterly Journal of
Economics 118 (February): 157–205.
Fallick, Bruce, and Jonathan Pingle. 2006. “A Cohort-Based Model of Labor
Force Participation.” Federal Reserve Board of Governors Finance and
Economics Discussion Series 2007–2009.
Juhn, Chinhui, Kevin M. Murphy, and Robert H. Topel. 1991. “Why Has the
Natural Rate of Unemployment Increased Over Time?” Brookings
Papers on Economic Activity 22 (2): 75–142.
Juhn, Chinhui, Kevin M. Murphy, and Robert H. Topel. 2002. “Current
Unemployment, Historically Contemplated.” Brookings Papers on
Economic Activity 33 (1): 79–136.
Little, Allan, and Steve Bradley. 2007. “The Determinants of Changes in the
Risk of Unemployment and Inactivity in Britain: A Decomposition
Analysis.” Labour Economics Research Group Discussion Paper No. 17.
Polivka, Anne E., and Stephen M. Miller. 1995. “The CPS After the
Redesign: Refocusing the Economic Lens.” Manuscript, Bureau of
Labor Statistics.
Shimer, Robert. 1999. “Why is the U.S. Unemployment Rate so Much
Lower?” In NBER Macroeconomics Annual 1998, Volume 13, edited by
Ben S. Bernanke and Julio Rotemburg. Chicago, The University of
Chicago Press, 11–61.
U.S. Census Bureau. 2008. “2008 National Population Projections.”
Washington, D.C.: U.S. Census Bureau. www.census.gov/population/
www/projections/2008projections.html.

Economic Quarterly—Volume 97, Number 4—Fourth Quarter 2011—Pages 389–413

Strategic Behavior in the
Tri-Party Repo Market
Huberto M. Ennis

R

epo contracts are a kind of collateralized loan that has become predominant in the United States among large cash investors. There
are several types of repo contracts, such as bilateral delivery-versuspayment repos, interdealer repos, and tri-party repos. A significant portion of
repo transactions in the United States take the form of tri-party repos, where
a third party (a clearing bank) provides collateral management and settlement
services to the borrower and the lender. The tri-party segment of the U.S. repo
market is the subject of this article.
The tri-party repo market played a significant role during the 2007–2009
global financial crisis. Tri-party repos were, for example, a major source of secured funding for Bear Sterns prior to its demise. In March 2008, repo lenders
in general, and tri-party repo counterparties in particular, lost confidence in
their ability to recoup loans to Bear Stearns and, hence, refused to renew them,
asking instead for immediate repayment (Bernanke 2008). To avoid a failure,
the Federal Reserve facilitated the acquisition of Bear Stearns by the bank J.P.
Morgan Chase. The withdrawal of tri-party repo funding also played a role in
the collapse of Lehman Brothers in September 2008. As a result of the events
during the crisis, it is now widely believed that the tri-party repo market is
subject to serious vulnerability (see, for example, Dudley [2009]). Attesting
to this is the fact that in 2009 the New York Fed asked a group of senior private
U.S. bank officials to form a task force “to address the weaknesses” in the
I would like to thank Jeff Lacker for many stimulating conversations that motivated me to
write this article, Jim Peck for answering my game theory questions patiently, and Borys
Grochulski, Bob Hetzel, Andreas Hornstein, Tim Hursey, Todd Keister, Antoine Martin, Ned
Prescott, Alex Wolman, and John Walter for comments on an earlier draft. All errors and
imprecisions are of course my exclusive responsibility. The views expressed in this article are
those of the author and do not necessarily represent the views of the Federal Reserve Bank
of Richmond or the Federal Reserve System. E-mail: huberto.ennis@rich.frb.org.

390

Federal Reserve Bank of Richmond Economic Quarterly

infrastructure of the tri-party repo market (Federal Reserve Bank of New York
2010). A broad set of reforms are currently under way.1
In this article, we study a simple model of the tri-party repo arrangement
that allows us to capture in a parsimonious way some of the strategic interactions that arise in this market. In our analysis, we use standard non-cooperative
game theory to uncover the basic mechanisms that can create some of the vulnerabilities commonly attributed to the tri-party repo market. We will show
that a change in perceptions can create a sudden coordinated withdrawal of
lenders from this market. Also, we will highlight the crucial role that the clearing bank plays in this game of “withdrawing before the rest,” which appears
to be a good representation of the situation that was present in the tri-party
repo market during the recent financial crisis.
A repo (repurchase agreement) transaction is a sale of an asset that is
combined with an agreement to repurchase the asset at a pre-specified price
on a later day. Effectively, it is equivalent to a collateralized loan, where
the loan is the amount paid for the initial sale and the asset plays the role of
collateral. Repayment of the loan takes place at the repurchase time, with
the interest rate being implicit in the repurchase price. In a tri-party repo, a
third party—the tri-party agent—facilitates the transaction between the two
main parties, the lender (a cash investor such as a money market mutual fund)
and the borrower (a securities dealer such as the broker-dealer arm of an
investment bank). The tri-party agent provides custodial and other services
to the lender and efficient collateral assignment and allocation tools to the
borrower. Settlement happens entirely in the books of the tri-party agent
where both the borrower and the lender have cash and securities accounts.
Also, in many cases, the tri-party agent (via the so-called “morning unwind”)
extends intraday credit to the borrowers to give them access, during the day,
to the securities used as collateral overnight. In the United States, the tri-party
agents are the two clearing banks, Bank of New York Mellon and J.P. Morgan
Chase.2
The volume of repo transactions in the United States is large. There are no
official data covering the entire market but Gorton and Metrick (forthcoming)
estimate that its size peaked before the crisis at a level that is in the same
order of magnitude as the value of all the assets held by U.S. commercial
banks (approximately $12 trillion). The tri-party repo segment of the market
is large as well. The value of securities financed in this way was around $1.7
trillion at the end of 2011, down from about $2.8 trillion in early 2008 (Federal
1 Implementation of the reforms have proven to be more difficult than previously expected.
On February 15, 2012, the New York Fed issue a statement indicating that the vulnerabilities in
this market still persist.
2 In what follows, for the purpose of concreteness, we will always call the lender in the triparty repo the (cash) investor, and we will call the borrower the (securities) dealer. The tri-party
agent will be called the clearing bank, or sometimes just the bank, for short.

H. M. Ennis: Strategic Behavior in the Tri-Party Repo Market

391

Reserve Bank of New York 2010). Furthermore, some large broker dealers,
arguably of systemic importance, finance large portions of their portfolios
in this market. While U.S. Treasury securities and agency mortgage-backed
securities (considered virtually riskless) are the most common class of assets
used as collateral in tri-party repos, equities and other fixed income securities
are also sometimes used. According to some estimates, at its peak in early
2008, about 30 percent of the assets used as collateral were subject to nonnegligible liquidity risk (Federal Reserve Bank of New York 2010).3
The amount of the loan in a repo transaction is often lower than the value of
the posted collateral. In other words, the value of the collateral gets discounted
and this discount is commonly referred to as a “haircut.” Haircuts are aimed
at reducing the exposure of the lending side to liquidation costs in case the
borrower defaults (see Gorton and Metrick 2010). In principle, choosing
the appropriate haircut would leave the repo transaction free of virtually any
repayment risk. This is the case because repo transactions are generally exempt
from the automatic stay that applies to debt under the U.S. Bankruptcy Code.
This implies that the lender side in a repo transaction can take possession of
the collateral immediately upon failure of the borrower.4
Data on haircuts for different types of repos is limited. However, the
available evidence suggests that the level and sensitivity of haircuts depend
on the kind of repo transaction being considered. Gorton and Metrick (2010,
forthcoming), for example, study a sample of interdealer repo transactions and
show that the average haircut increased significantly during the crisis. This
is the manifestation of what they call “the run on repos.” Repos were used to
finance portfolios of securities and, as the haircuts increased, the capacity to
borrow against those securities decreased. The owners of the securities, then,
had to find alternative sources of funding or sell the securities in the market.
This deleveraging is tantamount to the liquidation of loans that takes place in
traditional bank runs.
In contrast, Copeland, Martin, and Walker (2010) show that collateral
haircuts in the tri-party repo market did not appear to adjust in any meaningful
way to changes in the riskiness of the borrowers. The infrastructure that made
tri-party repos attractive to investors seems to have made it less convenient
for them to adjust collateral haircuts on a per-transaction basis. Instead, when
the financial conditions of a given dealer deteriorated, cash investors tended to
withdraw from dealing with such a dealer (PRC Task Force 2010). Evidence
3 Up-to-date information on the composition of collateral in the tri-party repo market can be
found at www.newyorkfed.org/tripartyrepo/.
4 The repo exemption from the stay is likely to extend to the case of the failure of a broker
dealer, as explained by Copeland, Martin, and Walker (2010, Appendix C). There is an ongoing
debate about the appropriateness of granting safe-harbor exemptions from the automatic stay to a
broad range of derivative transactions, including repos. See, for example, Roe (2009) and Lubben
(2010).

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Federal Reserve Bank of Richmond Economic Quarterly

suggests that this behavior was predominant during the events that led to the
failure of Lehman Brothers (Copeland, Martin, and Walker 2010). This way
of reacting to counterparty credit risk in the tri-party repo market is taken as a
premise in this article and plays an important role in the theoretical arguments
advanced later. In particular, we will investigate the problems that can arise
in the strategic interaction between the main players in this market given
that withdrawal from lending (and not adjustments of haircuts) constitutes the
typical reaction to a change in perceptions about the viability of the borrowing
side in the transaction.
Policymakers believe that a breakdown of the repo market can have systemic consequences. In March 2008, after the collapse of Bear Stearns, the
Federal Reserve created the Primary Dealer Credit Facility (PDCF) on the
premise that “unusual and exigent” circumstances justified the provision of
emergency (collateralized) lending to large securities dealers. The idea behind the PDCF was to provide backup liquidity to dealers to give them time
to arrange other sources of funding if repo lenders were to suddenly withdraw
from the market. The program was designed as a backstop facility, charging
a penalty rate on tri-party repo transactions in which the Fed took the lending
side (see Adrian, Burke, and McAndrews 2009). Initially, only high-quality
collateral (investment-grade securities) was accepted in the PDCF. At the time
of the failure of Lehman Brothers, the Fed expanded collateral acceptability
to a broader set of assets and usage of the PDCF soared. We will use our
model to illustrate one possible role for a lending facility such as the PDCF.
However, a more careful assessment of the suitable policy responses to the
type of vulnerabilities highlighted in this article is left for future research.
The article is organized as follows. In the rest of this section, we describe
the “morning unwind,” a feature of the tri-party repo market that is crucial
for understanding the main strategic interaction explored in this article. In
Section 1, we set up a simple model of the tri-party repo market and proceed
to study the induced strategic interaction between investors and the clearing
bank using standard tools in non-cooperative game theory. In Section 2, we
discuss some related issues that pertain to the functioning of the tri-party repo
market as presented in this article. Finally, Section 3 concludes.

The Morning Unwind
The maturity of most tri-party repo contracts is overnight, but there are also
contracts being arranged for a week, 30 days, and even longer periods of
time. A common practice in this market, however, is that the clearing bank
“unwinds” all repos, regardless of maturity, at the beginning of each day (at
around 8:00 a.m. EST).
The process of unwinding takes place as follows. Overnight, the cash
investor has the securities in its account at the clearing bank. As part of
an implicit arrangement, early in the morning (before the open of Fedwire

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securities at 8:30 a.m. EST), the clearing bank transfers the securities back
from the investor’s account to the dealer’s account, and transfers the corresponding cash to the investor (much like in a cancellation of the repo). To
finance the transfer of cash, the clearing bank (normally) extends intraday
credit to the dealer. In other words, the investor gets a credit in its cash
account at the bank and the dealer gets a debit, which usually results in an
intraday overdraft of its cash account.
There are several reasons why it is convenient for investors and dealers to
have the repos unwound in the morning. Investors benefit from having their
cash available to make various payments and to satisfy withdrawal demands
placed by their clients during the day. Dealers benefit from having access to the
securities for the purpose of trading. In fact, as a result of the trading activities
of dealers, the composition of their portfolio of securities changes during the
day. If some of the securities being used as collateral in outstanding repos
are sold, then they need to be substituted with new securities. This process
of collateral substitution is simpler if all the securities are transferred to the
dealer’s account in the morning and only reallocated back to repo contracts at
the end of the day.
With the morning unwind, the tri-party repo contract constitutes a loan
based on the combination of two sources of funding: investors covering the
night and the clearing bank covering the day. As with the overnight credit
provided by investors, the intraday credit provided by the clearing bank is
secured by the securities held by the dealer in its account at the bank.5 In
other words, if the dealer were to fail during the day, after the unwind has
occurred, then the clearing bank would get ownership of the securities as a
way to cancel the dealer’s overdraft. If, instead, the failure of the dealer were to
occur during the night, then investors would retain ownership of the securities
that served as collateral for the tri-party repo transaction.
The morning unwind, then, to the extent that it is financed with the provision of intraday credit to dealers, exposes the clearing bank to the risk of
receiving ownership of a batch of securities upon the failure of one (or more)
of those dealers.6 This unplanned increase in assets of the clearing bank
may create some extra costs associated with balance sheet capacity (capital
5 The clearing bank has a lien on the dealer’s collateral structured as a repo with broad
flexibility for collateral substitution. When the dealer sells (delivery versus payment) a security
during the day, the cash received as payment cancels out the part of the overdraft that is no
longer collateralized because of the sale of the security. When a dealer delivers a security free
of payment, the clearing bank is protected by its “right of offset” on all the securities that the
dealer has at the clearing bank, including those that were not used in tri-party repo transactions.
6 The ongoing reorganization of the market intends to reduce the predominance of the automatic “morning unwind” practice. See PRC Task Force (2010) for details. However, in the
statement issued on February 15, 2012, the New York Fed said: “the amount of intraday credit
provided by clearing banks has not yet been meaningfully reduced, and therefore, the systemic
risk associated with this market remains unchanged.”

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Federal Reserve Bank of Richmond Economic Quarterly

constraints, for example). Furthermore, it is possible that part of the overdraft
extended to the dealer by the clearing bank is, in turn, being financed by an
intraday overdraft of the clearing bank on its account at the Fed. If the dealer
fails and the clearing bank cannot resell the securities by the end of the day, it
may incur an overnight overdraft at the Fed, which is much more expensive,
or it may need to borrow at the discount window. Aside from being provided
at a penalty rate, discount window borrowing may also be associated with a
stigma effect that can make such an activity very costly for the clearing bank.7
The risk of incurring these costs is likely to be a crucial determinant of the
willingness of the clearing bank to unwind the repos every morning. The
clearing bank retains the right to refuse to unwind the repos of any particular
dealer.
At the end of the day, tri-party repos are “rewound” and cash investors
are the party exposed to the risk of failure of the dealer during the night. It
is common for cash investors in tri-party repos to accept certain securities
that they are not allowed to hold permanently in their portfolios. If the dealer
were to fail during the night, then, the cash investor would receive a batch
of securities that they would need to sell as soon as possible. Rush sales
may result in unfavorable prices (beyond the haircut applied to the collateral),
effectively exposing cash investors to financial losses.
It is important to realize here that the reason why the clearing bank is
(potentially) exposed to credit risk during the day is not because of the process
of unwinding the repos in the morning itself, but because such unwinding is
generally financed with intraday credit (an overdraft) extended by the clearing
bank to the dealer. If, every morning, the dealer were to have enough cash
in its account at the clearing bank, then the unwinding would make the repo
essentially a secured debt contract with a half-day maturity. The only exposure
in that case would be on the lending side (cash investors) and only to the extent
that the haircut on the collateral is not enough to cover any discount associated
with selling the assets.

1. A SIMPLE MODEL
The tri-party repo market in the United States is a complex system. There are
multiple participants facing diverse situations. Some of them are always there,
day after day, and some only participate occasionally. The clearing banks, the
main broker dealers, and some of the large cash investors participate every day;
one can suspect, then, that implicit relationships and reputation, for example,
play a significant role in determining outcomes (Copeland, Martin, and Walker
2010). Dealing with all these different dimensions formally is a challenging
7 For recent work on the possibility of stigma at the discount window see Ennis and Weinberg
(2010) and Armantier et al. (2011).

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395

task and it may not be the most illuminating approach. Instead, here, we will
provide a very simple environment that captures only some of the forces at
play in this market and we will use standard non-cooperative game theory to
analyze the strategic component associated with such a situation.8
The model is very simple. There are two time periods, t = 1, 2, and three
types of agents, a clearing bank, a securities dealer, and N cash investors. At
the beginning of period 1, each cash investor has an endowment of c dollars
and the dealer has the opportunity to invest 1 dollar in securities, which will
pay 1 + ρ at the end of period 2. We allow for ρ to be a random variable and
consider the natural case in which ρ has a positive expected value. We also
assume that N c > 1.
At the beginning of period 1, cash investors deposit (some of) their cash at
the clearing bank. Also at that time, the dealer can request a 1 dollar intraday
overdraft at the clearing bank to buy the securities. The clearing bank may or
may not agree to grant the dealer’s overdraft request.
At the end of period 1, the dealer needs to close the overdraft in its account at the clearing bank. We assume that overnight overdrafts are expensive
enough to give the dealer incentives to do this. In order to obtain the cash
needed to fund the overdraft position, the dealer arranges tri-party repos with
cash investors using the securities as collateral. The interest rate on the repos
is taken parametrically and denoted by r.9
If the dealer is not able to repo the securities, then it has to sell the securities
to pay back as much of the overdraft as possible. We assume that securities sold
before the end of period 2 only return a portion of what was invested. In such
a situation, then, the dealer gets no return and the clearing bank experiences a
loss equal to yB > 0.
If the dealer is able to repo the securities, it closes the overdraft at the
bank, and the next morning the bank has to decide whether or not to unwind
the repos. If the bank decides not to unwind the repos, then the dealer has
no funding for the securities, it fails, and investors take possession of the
collateral. We also assume that investors cannot hold the securities and need
to sell them at a loss at the beginning of period 2. In such a case, again, the
dealer gets no return and investors experience a loss equal to yI > 0. The
dealer stops being a customer of the bank at that point and the bank gets no
payoff from the transaction.
If the bank agrees to unwind the repos instead, the dealer gets a new
daylight overdraft in its bank account and investors get their cash and interest
8 See Duffie (2010) for a detailed description of the various activities generally undertaken
by broker dealers and the role that the repo market plays in funding those activities.
9 In the United States, most tri-party repos are arranged in the morning and settle in the
books of the clearing bank late in the afternoon, after the close of Fedwire securities. For the
formal representation of the problem, the only relevant aspect is that, each day, new repo funding
is arranged only after the morning unwind.

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Federal Reserve Bank of Richmond Economic Quarterly

back. At the end of the day, the securities pay off and the revenue is used by
the dealer to close the overdraft and pay a fee φ to the bank.
Note that the initial overdraft could be thought of as the result of the
unwinding of a (set of) pre-existing repo contract(s). In that sense, we could
think that our simple framework is able to handle two rounds of unwinding, to
the extent that the decision to unwind, in this model, will be exclusively driven
by forward-looking considerations. This interpretation of the initial overdraft
will be useful when we discuss some of the results.
Since we are assuming that N c > 1, investors’ initial endowment would
be enough to (fully) fund the investment opportunity of the dealer. The way
this funding is channeled from investors to dealers is via the clearing bank.
The clearing bank receives an initial deposit d ≤ N c from investors and then
grants a daylight overdraft to the dealer. If d > 1, then, on the books of the
clearing bank, the overdraft (loan) to the dealer is (fully) funded by the deposit
of investors. However, if investors do not deposit all of their endowment at
the bank and d < 1, then initial funding for the dealer could still be available.
At the beginning of period 1, the bank obtains daylight credit from the central
bank in the amount 1 − d. Later in the period, when (and if) the dealer secures
repo funding from investors, the corresponding cash that closes the negative
position of the dealer can be used by the bank to close its negative position with
the central bank. In this way, the bank can avoid a more expensive overnight
overdraft at the central bank.
Finally, notice that we have simplified the dealer’s side of the problem
by assuming that whenever funding is not forthcoming, the dealer fails. This
strategy allows us to concentrate our attention on the interaction between
investors and the clearing bank.10 Furthermore, when the dealer fails and the
securities need to be liquidated before the end of period 2, the proceeds from
the sale are not enough to cover the total value of the loan—the lender suffers
losses. In effect, this is a direct counterpart of postulating that insufficient
haircuts are applied to the collateral. As discussed in the introduction, the
evidence described in Copeland, Martin, and Walker (2010) suggests that this
is a reasonable approach to take.

The Non-Cooperative Game
The key strategic interaction in the model is between the clearing bank and
the set of investors. To study the outcome from this interaction we can use
the tools of non-cooperative game theory. In particular, we will concentrate
our attention here on the implied formal game played between the bank and
investors.
10 See Martin, Skeie, and von Thadden (2010) for a more fleshed out formal treatment of
the role of investors’ decisions in determining the fate of the dealer.

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Figure 1 The Game with No Uncertainty: Game 1
1

NO

O

3

2

0
0

NR
R

5

-yB
0

4

NU
U

7

0
-y

6

I

xB
xl

Let us start with the case when N = 1 and ρ = H ∈ R+ (i.e., ρ is a given
number greater than zero, not a random variable). Assume that H > φ + r.
The extensive form representation of this game, which we call Game 1, appears
in Figure 1. The game starts in node 1 (represented by an open circle in the
figure) with the move by the clearing bank, who has to decide whether to grant
the dealer a daylight overdraft (O) or not (NO). After that, if an overdraft is
granted, the investor has to decide whether to enter a repo contract with the
dealer (R) or not (NR). This is the decision presented in node 2. Finally, if a
repo contract is arranged, then the bank has to decide, in node 4, whether to
unwind the repo (U) or not (NU) the next morning. In each of the terminal
nodes (nodes 3, 5, 6, and 7) the payoffs of the players are listed in a column,
with the top element representing the payoff for the clearing bank (the first
player to move) and the bottom element representing the payoff of the investor.
We use the variables xi with i = B, I to represent the payoffs to the bank (B)
and the investor (I ) in the case where an unwinding of the repo happens on the
morning of date 2. From our description of the model above, we know that
xB = φ and xI = r. In a less stylized setup, xi could be equal to something
more complicated, but the basic results from the strategic interaction will be
the same as long as the conditions on xi and yi established below still hold.
We look for a subgame perfect Nash (SPN) equilibrium of this game.
Since Game 1 is a finite game of perfect information, an equilibrium always

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 2 Solving for the SPN Equilibrium of Game 1
1

O

1

NO

O

3

2

0
0

NR
R

5

2
xB
xl

NO

3

0
0

-yB
0

4
xB
xl

exists, and, given the payoffs, it is easy to see that the equilibrium is actually
unique (see, for example, Osborne and Rubinstein [1994]).
Proposition 1 There is a unique SPN equilibrium of Game 1 for which the
equilibrium actions are (O, R, U ).
Proof. As is standard with dynamic games, we proceed by solving backward.
First, consider the decision of the bank in the subgame that starts at node 4, that
is, after investors have agreed to repo the securities. If the bank unwinds the
repos, then it gets a payoff equal to xB , which is greater than the payoff of zero
obtained from not unwinding. Then, the bank will agree to unwind the repos.
We can now write an auxiliary game tree that takes this result into account.
This is the tree represented in the left-hand side of Figure 2. Following the
same logic, we can now solve backward in this game to find that the investor
will agree to repo the securities because xB > −yB .
Finally, we can draw an auxiliary tree that incorporates this last result (on
the right-hand side of Figure 2) and find that the bank will agree to grant an
overdraft since xB > 0. Hence, we have that the bank will always play O,
then the investor will always play R, and lastly the bank will always play U ,
which completes the proof of the proposition.
When there is no uncertainty with respect to the long-term solvency of
the dealer and there is only one cash investor (or a well-coordinated group of
them), the dealer always receives funding from the clearing bank (via daylight
overdraft) and from the investor (via repo transactions). There is no instability
associated with the tri-party repo contract in this case.

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Uncertainty over the Dealer’s Solvency
Suppose now that ρ is, in fact, a random variable that can take value H > 1
with probability ξ and −L with probability 1 − ξ . We associate the outcome
ρ = −L with a situation where the dealer experiences a solvency problem not
triggered by the actions of the participants in the tri-party repo market.11 We
will consider two cases: one where the game is played without the investor
or the bank knowing the realization of the random variable ρ, and the other
where the bank gets to know the realization of ρ before deciding whether or
not to unwind the repos the morning of date 2.
Uninformed clearing bank

In this first case, both the bank and the investor, when making decisions, share
the same degree of uncertainty about the expected performance of the dealer.
The structure of the game is almost exactly the same as in Game 1, except that
the payoff to the bank in terminal node 6 is now given by ξ xB +(1 − ξ ) (−fB ) ,
where fB is the loss to the exposed bank when the dealer fails. We call this
Game 2a. Note that the payoff to the repo investor in node 6 is still equal to
xI since the unwinding of the repos occurs as in normal circumstances in that
branch of the tree. Basically, the idea is that with some probability, the bank
finds out that the dealer is insolvent after unwinding the repos and hence is
left with a loss equal to fB = L + r in our model.12
Proposition 2 Define ξ a ≡ fB / (xB + fB ). If ξ > ξ a , then there is a unique
SPN equilibrium of Game 2a for which the equilibrium actions are (O, R, U ).
The proof of the proposition follows the same logic as the proof of Proposition 1, so we do not repeat it here. If the probability of the dealer not
experiencing a solvency problem is high enough (i.e., if ξ is high enough),
then the dealer will get funding from the bank and from the cash investor.
However, if the probability ξ is below the threshold value ξ a , then the unique
SPN equilibrium has the bank playing NO in node 1 and the dealer does not
obtain funding in such a situation.13 We could summarize this result as saying
that those dealers who are perceived as “fragile” will not get funded.
11 See Duffie (2010) for a thorough description of the various factors that can contribute to
the failure of a dealer bank.
12 The bank, at the time of unwinding the repos, grants an overdraft to the dealer of size
1 + r. After the dealer fails, the securities pay 1 − L and the bank gets the proceeds. Hence, the
net loss for the bank is equal to L + r.
13 Recall that in game theory, an equilibrium is a property of a profile of strategies. A
strategy is a complete contingent plan of play for all possible circumstances in the game, not just
the ones that occur in equilibrium. For example, when ξ < ξ a , the equilibrium strategy of the
bank is {NO, NU if the investor plays R} and the equilibrium strategy of the investor is {NR if
the bank plays O}. In this article, we sometimes loosely describe equilibrium play by the actions
taken in equilibrium, just to keep the presentation simple.

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It is interesting to note that the bank plays NO when ξ < ξ a because it
anticipates that the investor will not be willing to enter into a repo agreement
at the end of the day to finance the dealer. The investor, in turn, does not agree
to participate in the repo because it anticipates that the bank will not be willing
to unwind the repos the next morning if the repos were outstanding.14 This
anticipation game makes the tri-party repo market very sensitive to changes
in perceptions, not just about actual weaknesses of the dealer being funded,
but also about the perceptions of other players about those weaknesses.
If we interpret the initial overdraft as (possibly) the result of an unwinding
of previously arranged repos, then the model says that if the clearing bank
places a high probability on the eventual failure of the dealer the next day, the
refusal to unwind will take place immediately. This result suggests that the
situation can potentially unravel long before the actual failure of the dealer is
expected to occur, even if such failure is only regarded as a possibility (and
not a certainty).
A crucial issue left unexplored here is how the perception of the probability
of failure gets determined and how it changes over time. What the theory here
makes clear is that, once such probability has crossed a certain threshold, the
whole tri-party repo arrangement is bound to immediately collapse.
Informed clearing bank

The second case we would like to consider in this section is the case when
the bank gets to know the realization of ρ before deciding whether or not to
unwind the repos on the morning of date 2. We refer to this game as Game 2b.
The extensive form representation of this game is provided in Figure 3 where
nature moves at node 4. We denote by NF the situation when the realized
state of nature is such that ρ = H , and by F the situation when ρ = −L.15
The other new piece of notation in Figure 3 is the payoff fI , which is the
loss experienced by the repo investor when the repo is not unwound by the
bank and ρ = −L. In principle, fI could be different than yI because the
liquidation value of the securities may depend on the state of nature.
Proposition 3 Define ξ b ≡ fI / (xI + fI ). If ξ > ξ b , then there is a unique
SPN equilibrium of Game 2b for which the equilibrium actions are (O, R, U
if ρ = R, NU if ρ = −L).
Proof. First note that in the proper subgame that starts at node 6, the bank
should agree to unwind the repos, and in the one that starts in node 7, the
14 Copeland, Martin, and Walker (2010) call this strategic interaction “the hand-off of risk
between investors and clearing banks.”
15 Osborne and Rubinstein (1994, 101) call games with this structure extensive games with
perfect information and chance moves.

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Figure 3 The Game with Uncertainty: Game 2b
1

O

NO

3

2

0
0

NR
R

5

4

NF
6
U

xB
xl

F

ξ

1-ξ

NU

8

U
9

0
-yl

-yB
0

10
-fB
xl

7
NU
11
0
-f l

bank should not unwind the repos. Now, using backward induction, we can
construct the reduced game where nodes 6 and 7 are terminal nodes and the
payoffs are the ones associated with nodes 8 and 11 in the full game. Given
that nature moves according to the probability ξ , we have that the payoff
for the investor from playing R is equal to ξ xI + (1 − ξ ) (−fI ). Also, the
payoff for the bank after playing O and given that the investor is playing R
is ξ xB . Now, again, using backward induction, we can construct a reduced
game with node 4 as a terminal node and the associated payoffs being {ξ xB ,
ξ xI + (1 − ξ ) (−fI )}. Clearly, if ξ > ξ b , the investor wants to play R and,
given this, the bank wants to play O (since ξ xB > 0).
If ξ < ξ b , the investor will want to play NR when node 2 is reached and,
anticipating this, the bank will want to play NO. Thus, if ξ < ξ b , the dealer
will not obtain the initial overdraft funding from the bank and no repo will be
ultimately arranged.
The logic behind these results is clear. The cash investor anticipates that
the bank will be able to infer somehow, before the unwinding of the repos,
the future performance of the dealer. If the investor believes that it is very
likely that the bank will find out that the dealer is bound to fail (and hence
that the bank will not unwind the repos), then the investor will not be willing
to agree to the repo transaction. In turn, anticipating this, the bank will not

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grant an initial overdraft to the dealer and the whole tri-party repo arrangement
collapses.
Here, again, we can loosely interpret the initial overdraft as the result of
unwinding previously arranged tri-party repos. In this informal interpretation,
the crucial element for such a story to work is that there must have been a
change in perceptions about the situation of the dealer after repo contracts
were arranged prior to the beginning of Game 2b. In particular, right at the
beginning of Game 2b, it must be the case that all the participants in the triparty repo arrangement realized that the dealer actually has a probability of
success (the next day) smaller than the threshold ξ b and that the bank will be
able to find out whether or not the dealer will fail before the unwinding takes
place the following day. If this is the case, then the tri-party repo arrangement
immediately collapses, not at the time when the failure of the dealer is expected
to occur but when the perceptions about that failure actually change (which
could very well be much sooner, as the game illustrates).
Discussion

It is interesting to compare the results in Propositions 2 and 3. Note that the
thresholds are increasing in the size of the loss if the dealer fails, and they are
decreasing in the size of the gain if funding is granted and the dealer does not
fail. This is true for both thresholds, although in Proposition 2 the relevant
payoffs are those of the bank and in Proposition 3, those of the cash investor.
The reason for this difference is the fact that in Game 2a the bank is playing
the role of creditor at the time when the dealer fails, while in the case of Game
2b the bank finds out whether or not the dealer will fail before unwinding the
repos, and if the dealer is actually expected to fail, then investors will be the
party exposed to losses.
This difference in the threshold values has implications for the relationship
between fragility and information flows in the tri-party repo market. We can
interpret a situation with a lower threshold value as a situation where the
tri-party repo arrangement is more likely to survive shifts in participants’
perceptions. The idea is that the creditor will accept to stay in the transaction
even after larger increases in the perceived probability of failure 1 − ξ when
the threshold value is lower. Then, if we think that cash investors have less
to gain from the repo contract and more to lose relative to the bank—so that
the threshold ξ b > ξ a —a situation where everybody anticipates that the bank
will be able to obtain information about the solvency conditions of the dealer
before the morning unwind (as in Proposition 3) would result in a more fragile
tri-party repo market. In such a situation, it is worth noticing, increasing the
haircuts applied to the collateral will tend to reduce the loss fI , reduce the
threshold value ξ b , and, in this way, improve the stability of the repo market.
In the simple formal game we have studied in this section, the initial
perceptions about the probability ξ are shared by all participants and are correct

H. M. Ennis: Strategic Behavior in the Tri-Party Repo Market

403

in the sense of being equal to the actual objective probability associated with
the random variable ρ. This stark information structure hides the fact that the
crucial driver of behavior in this strategic situation is the perception that the
bank has about the perception of investors about the probability of failure of
the dealer. Notice that, in fact, the bank would be willing to grant the initial
overdraft to the dealer regardless of the bank’s perception of the probability ξ ,
as long as the bank expects that investors will be willing to repo the securities
later in the day. Whether or not investors will be willing to repo the securities
depends only on the perception that those investors (and not the bank) have
about ξ . So, if the bank thinks that investors are optimistic about the dealer,
then, even if the bank is not, the bank will be willing to grant the initial
overdraft. This is the case because the bank will get to know whether or not
the dealer will fail before unwinding the repos in the morning of the second
date and, hence, can effectively get out of the deal without experiencing any
losses.
We have considered here the case of only one cash investor with no interim
information. However, it would be more realistic to have many investors, each
getting some partial information about the solvency condition of the dealer.
Because the clearing bank observes the actions of investors in the tri-party
repo market, it has a vantage point to aggregate all the dispersed information
available to investors and hence, to some degree, anticipate the potential failure
of the particular dealer. In other words, after the round of repos during the
day, the bank is likely to become better informed about the situation of the
dealer. The structure of Game 2b attempts to capture the gist of this situation
by having the bank become perfectly informed before deciding whether or not
to undertake the morning unwind.
Having more than one investor makes the game more complicated and
can produce other interesting insights. In particular, the issue of coordination among multiple investors is key to understanding the sources of possible
fragility in the tri-party repo market. We discuss some of those issues in the
following sections. The analysis in this section applies to a situation where investors can (somehow) perfectly coordinate their actions and play R whenever
such a move benefits all of them.
Before we move on to discuss potential coordination issues, it is worth
mentioning an interesting implication coming out of the structure of Game 2b.
In situations such as the one captured by the timing in that game, any measure
aimed at reducing the potential losses of a clearing bank will not change the
resiliency of the tri-party repo market. If the clearing bank (by obtaining
independent information or by inferring information from the behavior of
investors) can (fully) anticipate the failure of any particular dealer before the
morning unwind, then the bank is effectively not exposed to actual losses
(i.e., the value of fB is irrelevant for equilibrium, as long as it is positive).

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Figure 4 Coordination in the Repo Market: Game 3
1

O

NO

3

2

R

NR
5

4

R

NR
7

6

U
10
xB
xl
xl

0
0
0

R

-yB
-z l
0

NU

NR
9

8

-yB
0
-z l

-yB
0
0

11
0
-yl
-y
l

Hence, any attempt at reducing a clearing bank’s potential losses will not have
a material effect on the behavior of the market.

Coordination in the Repo Market
Suppose that there are N = 2 cash investors and that, at the beginning of date
2, these investors play a simultaneous move game to decide whether or not
to agree to enter repo contracts with the dealer. Also assume that if only one
of the two investors agrees to a repo, then the dealer stops operations and the
investor that entered the repo agreement experiences a loss equal to zI . The
extensive form representation of this game, which we call Game 3, is given in
Figure 4.16
The encircled decision nodes 4 and 5 constitute a single information set
for the investor moving in those nodes. This is the result of the fact that
investors play simultaneously and, hence, each investor does not know if the
other investor has played R or NR at the time that he has to decide what to play
(that is, the investor does not know if he is in node 4 or in node 5, respectively).
As before, we look for a SPN equilibrium of Game 3.
16 Osborne and Rubinstein (1994, 102) call games with this structure extensive games with
perfect information and simultaneous moves.

H. M. Ennis: Strategic Behavior in the Tri-Party Repo Market

405

Figure 5 Normal Form Representation of the Coordination Game in
the Repo Market
Investor 2

R

NR

R

(x l ,x l )

(-z l ,0)

NR

(0,-z l )

(0,0)

Investor 1

Proposition 4 There are two pure-strategy SPN equilibria of Game 3; in one
the dealer gets funded and in the other it does not.
Proof. Note that the branch of the game tree that starts at node 6 is indeed a
proper subgame of this game. Clearly, if play reaches node 6, then the bank
should agree to unwind the repos (i.e., play U ) at that point. Using backward
induction, we can substitute the payoff from node 10 at node 6 and consider the
reduced game that results after this first iteration. In this reduced game (and in
the complete game), there is one proper subgame that starts at node 2. In fact,
this subgame has the structure of a coordination game between investors and
has two pure-strategy Nash equilibria: (R, R) and (NR,NR) (Figure 5 depicts
the normal-form representation of this coordination game).
As a result of this multiplicity, the full game actually has two pure-strategy
SPN equilibria, one where investors play (R, R) if the proper subgame starting at node 2 is reached, and another where investors play (NR,NR) if this
subgame is reached. In the first case, when both investors agree to enter repo
transactions, the bank will be willing to grant an overdraft (i.e., play O) in node
1. The equilibrium actions will then be (O, {R, R}, U ) and the equilibrium
payoffs will be (xB , xI , xI ).
In the other case, when investors play (NR,NR), we have that the bank will
not agree to initially grant the overdraft and the equilibrium payoffs are equal
to zero for all players since the dealer does not get funded from the outset.
The equilibrium in which the bank does not agree to grant the dealer an
overdraft in node 1 captures in a stylized way a source of potential fragility in
the tri-party repo market. If the clearing bank expects that, because of what
amounts to a coordination failure, cash investors in the afternoon will not be
willing to fund the securities dealer via repo transactions, then the bank will
not be willing to grant an overdraft to the dealer in the morning. Recall that,
for all practical purposes, the overdraft could originate on an initial request

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Federal Reserve Bank of Richmond Economic Quarterly

for funding by a dealer or as the result of the unwinding of outstanding repo
transactions. In this sense, then, the model underscores the fragility associated
with the daily unwinding of repo transactions that are financed with daylight
overdrafts on the accounts that securities dealers have at their clearing banks.
Note here that all agents in the model prefer that the equilibrium in which
the dealer gets funded be played at all times. However, because of the possibility of a coordination failure among investors, it is consistent with rational
play and equilibrium that the dealer not be funded. Martin, Skeie, and von
Thadden (2010) call such a situation a repo run. One way to deal with this
problem would be to have the central bank provide backstop liquidity in the
repo market, as the Federal Reserve did with the PDCF. In such a situation,
investors would get payoff xI from choosing R, independent of what the other
investor is choosing. This change in the structure of payoffs makes (R, R) the
unique equilibrium of the game, and the dealer always gets funded. The key
to this result is that the policy intervention changes the game among investors
so that it is no longer a coordination game.17 Interestingly, in the model, the
PDCF would not be tapped by investors in equilibrium, even though it is essential for ruling out the possibility of coordination failures and, in this way,
stabilizing the market.
Martin, Skeie, and von Thadden (2010) (see, also, Copeland, Martin, and
Walker [2010]) consider the game played by investors in the case when there
is no “morning unwind.” In the context of their model, they show that the
investors’ game is no longer a coordination game and, hence, runs can no
longer happen. Their model is different, yet related to the model presented
here. In particular, they consider the case where there are old and new investors
playing the game. Then, the result relies on the assumption that, without the
unwind, the dealer gets to observe whether or not it will fail before making any
payments to existing (old) investors. This removes the incentives of existing
investors to run, even if no new investor is willing to fund the dealer. But,
when existing investors do not run, the dealer can withstand a run by new
investors, which removes the incentives for new investors to run.
One way to obtain a similar result in our setup is by assuming that, barring
daylight credit from the clearing bank, the dealer needs to arrange repo funding
before making any investments. Also, let us assume that the dealer goes ahead
with the investment only if it is able to convince both investors to fund the
operation. In this situation, the payoff to an investor that agrees to enter a repo
17 This role of the PDCF is highlighted by Adrian, Burke, and McAndrews (2009) when
they say: “The PDCF has the potential to benefit trading in the repo market beyond the direct
injection of funding. The very existence of the facility is a source of reassurance to the primary
dealers and their customers.” Dudley (2009) also says that “the PDCF essentially placed the Fed
in the role of the tri-party repo investor of last resort thereby significantly reducing the risk to the
clearing banks that they might be stuck with the collateral. As a consequence, the PDCF reassured
end investors that they could safely keep investing. This, in turn, significantly reduced the risk
that a dealer would not be able to obtain short-term funding through the tri-party repo system.”

H. M. Ennis: Strategic Behavior in the Tri-Party Repo Market

407

contract, when the other investor does not, is the same as the payoff from not
entering a repo contract; i.e., it is equal to zero. Assuming, as Martin, Skeie,
and von Thadden (2010) do, that in case of indifference an investor agrees to
repo, we have that the “unique” equilibrium in the investors’ game is to play
(R, R), and the dealer always gets funded.

Correlated Equilibrium
In the SPN equilibria of Proposition 4, the clearing bank in the morning has no
doubts about the events that will take place during the afternoon when the cash
investors have to decide whether or not to fund the securities dealer: Either
the bank anticipates that funding from cash investors will be broadly available
or it anticipates that no investor will be accepting repo requests. In principle,
however, the bank may not be sure about the availability of funding in the
afternoon. A simple representation of this uncertainty can be accomplished
by using the alternative equilibrium concept of correlated equilibrium.18
In particular, suppose that at the time when investors have to decide
whether or not to fund the dealer in the afternoon of the first date, they observe
a public signal that can take two possible values: α with probability π , and
β with probability 1 − π . Suppose also that, when investors observe α, they
play the equilibrium with actions (R, R), and when they observe β, they play
the equilibrium with actions (NR,NR). The bank, instead, does not observe
the public signal at the time when it has to decide whether or not to allow the
dealer to incur an overdraft on its account at the bank.
Proposition 5 Define π ≡ yB / (xB + yB ). If π ≥ π , then there is a correlated equilibrium in which the bank plays O in node 1 of Game 3. If π < π ,
then there is a correlated equilibrium in which the bank plays NO in node 1.
The proof of the proposition is very similar to the other proofs and is
not included here. We can interpret π as the clearing bank’s perception of
the likelihood that the dealer will obtain funding in the afternoon. If the
probability is high enough, above the threshold π , then the bank will agree
to grant an overdraft. Note that, after the bank allows for the overdraft, with
probability 1 − π , investors do not agree to fund the dealer in the afternoon
and the clearing bank is stuck with the securities that served as collateral for
the overdraft. In such case, the bank suffers a loss given by yB . Note that,
as the loss increases, the threshold value π increases and gets closer to unity.
18 There is also a mixed-strategy SPN equilibrium of Game 3 in which investors randomize
over actions R and N R, playing R with probability zI /(xI + zI ). In such an equilibrium, the
bank also faces uncertainty about the ability of the dealer to get funding at the end of period 1.
However, we find the interpretation of this equilibrium less appealing and, for this reason, we do
not discuss it here.

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Federal Reserve Bank of Richmond Economic Quarterly

In other words, as the loss for the clearing bank becomes larger, the bank
needs to be more and more certain that investors will fund the dealer in the
afternoon if an overdraft is to be granted in the morning. We can think that a
lower π represents a situation where confidence in the ability of the dealer to
participate in the repo market decreases. If the situation deteriorates enough,
to the point when π gets below the threshold π , then the clearing bank will not
agree to grant an overdraft (or unwind previously arranged repo transactions
by granting the dealer daylight credit).
Note that, in contrast to the situation described in the previous subsection,
here the payoff of the bank in case the dealer defaults after the morning unwind
is relevant for the outcomes of the game. In the equilibrium of Proposition
5, the clearing bank retains some uncertainty about the ability of the dealer
to obtain repo funding in the afternoon of date 1. The key to this result is
that the public signal is only observed after the morning unwind and, hence,
it creates the potential for a sudden shift in the behavior of investors in the
afternoon repo market. Coordination failures are, perhaps, more likely to
happen abruptly since they are based only on changes in the beliefs of market
participants about the behavior of other market participants. Instead, changes
in behavior driven by fundamentals, such as the ones studied in Propositions
2 and 3, are more likely to happen gradually over time, allowing the clearing
bank to potentially exploit its informational advantage.
For concreteness, we have considered here a situation with only two investors. However, in general, there could be many more cash investors.19 An
alternative formalization would be to have a continuum of investors deciding
at the end of date 1 whether or not to fund the dealer via repo transactions.
In such case, it is clear that the decision of any one investor will not have
a material consequence on the overall ability of the dealer to fund itself. In
other words, if an investor enters a repo contract with a dealer when all the
other investors do not, then the dealer will indeed fail and the investor with
the repo contract will be stuck with the securities. The structure of payoffs
that implies a coordination game arises more naturally in this case, relative to
the case where there are only two investors. However, given our assumptions
on payoffs, the results would be basically the same in both cases.

2.

DISCUSSION

From the perspective of cash investors, the tri-party repo contract is almost
equivalent to an interest-bearing demand deposit. Because of the daily unwind,
investors have access to their cash during the day (on demand). During the
19 Copeland, Martin, and Walker (2010) consider a coordination game similar to the one

studied here but where there are three investors in the game.
von Thadden (2010).

See also Martin, Skeie, and

H. M. Ennis: Strategic Behavior in the Tri-Party Repo Market

409

night, the cash is locked in with the repo transaction. The next morning, the
contract entitles the investor to a positive interest payment. In an uninsured
demand deposit contract, investors are exposed to counterparty credit risk. In
contrast, the tri-party repo contract could be considered, in principle, a form
of secured lending since there is collateral pledged to address default risk.
Haircuts on the collateral could be set so as to leave the lender with virtually
no exposure to credit risk. However, in reality, evidence suggests that cash
investors still perceive themselves as being exposed to some risk of losses
when the borrower defaults (see, for example, Copeland, Martin, and Walker
[2010] and PRC Task Force [2010]). We have taken the possibility of losses
as a premise for our model, without trying to explain the fundamental reasons
for under-collateralization. Understanding how this arrangement could arise
optimally is not an easy task. Lacker (2001) provides a framework to think
about collateralized debt that could be used to address these kinds of issues
(see, also, Dang, Gorton, and Holmstr¨ m [2010]). More work is clearly
o
needed in this area.
In the United States, paying interest on demand deposits was not allowed
until very recently. This restriction was especially binding for businesses.
However, the financial system has developed some alternatives that constitute close substitutes of interest-bearing demand deposits. The tri-party repo
arrangement could be considered one such alternative. The newly enacted
Dodd-Frank financial reform legislation includes a provision that repeals the
prohibition of paying interest on demand deposits and, starting on July 21,
2011, banks are now allowed to pay interest on these accounts. It is an open
question how this will impact the tri-party repo market in the long run. It
seems plausible that some cash investors looking for a way to earn interest on
their cash holdings overnight may now turn to demand deposits at banks for
this purpose. But, of course, there is a demand as well as a supply side in the
tri-party repo market. On the demand side, securities dealers will still need
to fund their portfolios of securities. Some form of repo contract is likely to
play a role in satisfying that demand.
As we have explained, the source of funding for tri-party repos is twofold: during the night, cash investors provide the funding and, during the day,
daylight overdrafts granted by the clearing banks provide (most of) the funding. Some (if not most) of the cash owned by cash investors does not leave
the books of the clearing bank during the day. Those funds are effectively
demand deposits held by cash investors in their accounts at the clearing bank.
These deposits, then, can be used by the clearing bank to fund the daylight
credit provided to the dealers as part of the tri-party repo contract. But, to
the extent that some of the cash owned by investors is used during the day to
make payments and other transfers, the clearing bank needs to obtain daylight
funding for the overdraft granted to the dealers. Of course, one readily available source of daylight funding for clearing banks is their daylight overdraft

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Federal Reserve Bank of Richmond Economic Quarterly

capabilities with the Federal Reserve. If we think that the rate charged by the
Fed for daylight credit is intentionally kept low (“to ensure the smooth functioning of payment and settlement systems”), then we could conclude that, to
a certain extent, the tri-party repo arrangement is an indirect way for dealers
to access subsidized funding during the day.20
With its simplified treatment of the events associated with a dealer’s default, our formal analysis could not be used to address some significant issues
being discussed in policy circles (see, for example, Copeland et al. [2011]).
For example, the possibility that the liquidation of a dealer’s portfolio could
result in fire sale prices and externalities to other dealers (and to market participants in general) was left unexplored.21 Another important issue that was
not examined here is the possibility of “a loss of confidence” in the solvency
of a clearing bank. This was a major concern for policymakers during the
crisis and has been a salient point in the discussions about possible reforms to
the infrastructure in the tri-party repo market (Bernanke 2008). Each clearing
bank in the United States provides services to multiple dealers and to a large
number of investors. To some extent, dealers need the clearing bank for their
daily operation. It seems plausible, then, that problems at a clearing bank
could spread to its client dealers if, for example, those dealers were relying on
daylight credit to stay in business. Furthermore, cash investors usually have
large unsecured exposures to their clearing bank during the day that could
also destabilize them if that cash were no longer readily available. These are
important issues that deserve careful consideration and are certainly related to
the subject of this article. Here, however, we chose to keep the model simple
on these dimensions to be able to sharpen our understanding of the strategic
interaction between the clearing bank and investors, which may play a crucial
role in the functioning of this complex market during a crisis.
In May 2010, the Tri-Party Repo Infrastructure Reform Task Force issued
a set of recommendations to increase the stability of this market (PRC Task
Force 2010). Their main proposal was to reform the system in order to reduce
as much as possible the reliance of market participants on large amounts of
intraday credit provided by clearing banks. In short, the proposal calls for
an elimination of the indiscriminate daily unwind of all tri-party repo trades.
Evidently, reducing the credit exposure of the clearing banks will limit the
power of some of the strategic interactions highlighted in this article. However,
20 Currently, the Fed provides daylight credit to depository institutions using a two-tiered
fee schedule. Those institutions that pledge enough acceptable collateral with their Reserve Bank
receive daylight credit (up to a cap) at no charge. Uncollateralized daylight credit is charged
a fee that is calculated per minute using an annual rate of 50 basis points. This system was
only recently introduced. During the crisis, the Fed charged a uniform rate of 36 basis points
for intraday credit and this credit was all uncollateralized. For more information on the current
system see www.federalreserve.gov/paymentsystems/psr policy.htm.
21 For a model that is useful to address some of these issues, see Acharya and Viswanathan
(2011).

H. M. Ennis: Strategic Behavior in the Tri-Party Repo Market

411

if the morning unwind creates some valuable operational advantages that make
the tri-party repo contract especially attractive to dealers and investors, then an
obvious tradeoff arises between stability and effectiveness.22 In such a case,
fragility is not to be combated at all costs. As in many other situations where
a risk-return frontier results in a tradeoff, the optimal arrangement could very
well involve actually tolerating some positive risk.
There are also other alternatives that have been considered to limit this
source of fragility in the tri-party repo market. For example, a system of
capital requirements and risk charges that penalizes the intraday exposure of
the clearing banks may give the appropriate incentives to participants, inducing
them to move away from their over-reliance on intraday credit from the clearing
banks (Tuckman 2010). Similarly, changes in the treatment of repos under
bankruptcy law, such as removing them from the exception to the automatic
stay (Roe 2009), could make these contracts less attractive and, hence, reduce
the size of this potentially destabilizing market.
As the process of evaluating possible reforms continues, it is important to
keep in mind that many of the features of the tri-party repo contract that we
observe in the data are contingent on a set of rules (and common practices) that
existed when the data was collected. If some of those rules are changed (by fiat
or by newly built consensus among major participants), then some prevalent
characteristics of the existing contract may also change. A case in point is the
distribution of maturity terms in the market. Currently, term trades represent
10 percent to 40 percent of the market (PRC Task Force 2010). To the extent
that participants stop perceiving the morning unwind as an automatic event for
repos of longer maturities, it seems plausible that an even higher proportion of
the outstanding repos will become overnight contracts. This may seem a fairly
obvious point, yet it clearly highlights the limitations of evaluating the effects
of possible changes in policies using only historical data. To complement our
data analysis, we need to develop better models of the tri-party repo market
that can allow us to conduct policy evaluations in a more meaningful way. The
alternative is a costly process of trial and error purely based on experience in
the actual market. Considering the current importance of this market, pushing
forward a model-based agenda for studying this market seems worthwhile.
The model introduced in this article is an attempt to take a preliminary step in
this direction.
22 For example, changing to a system in which repos get unwound only later in the day (or,
not unwound at all, in the case of term repos) will make those contracts less comparable with
a demand deposit from the perspective of cash investors. While it is true that during the day
investors are unlikely to need all the cash used in tri-party repos, the option to have that cash
available presumably has some value for investors.

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Federal Reserve Bank of Richmond Economic Quarterly

CONCLUSION

In this article, we study a simple model of the strategic interaction between
investors and the clearing bank in the tri-party repo market. In order to be
able to apply simple game theory techniques to the problem, we abstract from
many important features of this complex market. We mention several of them
along the way in the presentation. Clearly, a lot more work is needed to extend
the formal analysis in ways that would allow us to evaluate the role, and the
relative importance, of those various features left unexplored here.
Perhaps the aspect most clearly highlighted by the model in this article
is the role in the inception of a crisis played by participants’ anticipation
of each others’ perceptions and actions. In particular, the model eloquently
illustrates how changes in expectations about future events and actions can
make a crisis happen abruptly before the fundamental factors behind it visibly
manifest themselves. We conclude, then, that swings in perceptions (about
fundamentals or about market confidence) can, in principle, trigger sudden
crises in the tri-party repo market.

REFERENCES
Acharya, Viral V., and S. Viswanathan. 2011. “Leverage, Moral Hazard, and
Liquidity.” Journal of Finance 66 (February): 99–138.
Adrian, Tobias, Christopher R. Burke, and James J. McAndrews. 2009. “The
Federal Reserve’s Primary Dealer Credit Facility.” Federal Reserve Bank
of New York Current Issues in Economics and Finance 15 (August).
Armantier, Olivier, Eric Ghysels, Asani Sarkar, and Jeffrey Shrader. 2011.
“Stigma in Financial Markets: Evidence from Liquidity Auctions and
Discount Window Borrowing during the Crisis.” Federal Reserve Bank
of New York Staff Report 483 (January).
Bernanke, Ben S. 2008. “Financial Regulation and Financial Stability.”
Speech at the FDIC Forum on Mortgage Lending for Low and Moderate
Income Households. Arlington, Va., July 8.
Copeland, Adam, Antoine Martin, and Michael Walker. 2010. “The Tri-Party
Repo Market before the 2010 Reforms.” Federal Reserve Bank of New
York Staff Report 477 (November).
Copeland, Adam, Darrell Duffie, Antoine Martin, and Susan McLaughlin.
2011. “Policy Issues in the Design of Tri-Party Repo Markets.”
Unpublished manuscript, Federal Reserve Bank of New York.

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Dang, Tri Vi, Gary Gorton, and Bengt Holmstr¨ m. 2010. “Financial Crises
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and the Optimality of Debt for Liquidity Provision.” Unpublished
manuscript, Yale University.
Dudley, William C. 2009. “More Lessons from the Crisis.” Remarks at the
Center for Economic Policy Studies (CEPS) Symposium. Princeton, N.
J., November 13.
Duffie, Darrell. 2010. “The Failure Mechanics of Dealer Banks.” Journal of
Economic Perspectives 24 (Winter): 51–72.
Ennis, Huberto M., and John A. Weinberg. 2010. “Over-the-Counter Loans,
Adverse Selection, and Stigma in the Interbank Market.” Federal
Reserve Bank of Richmond Working Paper 10-07 (April).
Federal Reserve Bank of New York. 2010. “Tri-Party Repo Infrastructure
Reform.” Federal Reserve Bank of New York White Paper (May 17).
Gorton, Gary, and Andrew Metrick. 2010. “Haircuts.” Federal Reserve Bank
of St. Louis Review 92 (November/December): 507–19.
Gorton, Gary B., and Andrew Metrick. Forthcoming. “Securitized Banking
and the Run on Repo.” Journal of Financial Economics.
Lacker, Jeffrey M. 2001. “Collateralized Debt as the Optimal Contract.”
Review of Economic Dynamics 4 (October): 842–59.
Lubben, Stephen J. 2010. “The Bankruptcy Code Without Safe Harbors.”
American Bankruptcy Law Journal 84 (Spring): 123.
Martin, Antoine, David Skeie, and Ernst-Ludwig von Thadden. 2010. “Repo
Runs.” Federal Reserve Bank of New York Staff Report 444. (April).
Osborne, Martin J., and Ariel Rubinstein. 1994. A Course in Game Theory.
Cambridge, Mass.: The MIT Press.
PRC Task Force. 2010. Report of the Payments Risk Committee Task Force
on Tri-party Repo Infrastructure. (May 17)
Roe, Mark. 2009. “End Bankruptcy Priority for Derivatives, Repos and
Swaps.” Financial Times, December 16.
Tuckman, Bruce. 2010. “Systemic Risk and the Tri-Party Repo Clearing
Banks.” CFS Policy Paper (February 2).

Economic Quarterly—Volume 97, Number 4—Fourth Quarter 2011—Pages 415–430

K-Core Inflation
Alexander L. Wolman

S

tandard measures of inflation (for example, personal consumption expenditure [PCE] or consumer price index [CPI]) are constructed in order
to accurately describe the behavior of consumption prices as a whole.
However, to the extent that the inflation rate in a given period is accounted
for by large relative price changes for particular goods and services, it may be
desirable to have additional measures of inflation that adjust for those large relative price changes. These alternatives would be useful if large relative price
changes are a source of noise, obscuring underlying patterns in the economy.
Any such alternative inflation measure could never be the best measure of
overall price changes, but it might provide valuable information about the
behavior of future inflation, or more generally about the “state of the world”
relevant for conducting monetary policy. This article describes a new class of
measures of underlying inflation called “k-core inflation.”
The term “core inflation” came into use in the 1970s, when large price
increases for food and energy coincided with high overall CPI inflation and, in
some years, with weak economic activity. Researchers using a Phillips curve
framework at that time sought a notion of inflation that was consistent with a
positive association between inflation and real activity. For example, Robert
Gordon (1975) referred to “underlying ‘hard-core’ inflation” as distinct from
the contributions made by food and energy, dollar devaluation, and the end of
price and wage controls.1 The Bureau of Labor Statistics responded to these
conditions in 1977 by beginning to publish a measure of the CPI that omitted
food and energy components (“the index for all items less food and energy”).2
Today that subindex of the CPI is widely referred to as the core CPI, and, more
The author thanks Todd Clark, Marianna Kudlyak, Nika Lazaryan, Thomas Lubik, and Ned
Prescott for helpful comments. The views in this paper are the author’s and do not represent
the views of the Federal Bank of Richmond, the Federal Reserve Board of Governors, or the
Federal Reserve System. E-mail: alexander.wolman@rich.frb.org.
1 Beryl Sprinkel (1975) seems to have used the same term (“hard-core” inflation) a few
months earlier than Gordon.
2 See the Bureau of Labor Statistics (2011).

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Federal Reserve Bank of Richmond Economic Quarterly

generally, “core inflation” is understood to refer to some broad price index that
excludes food and energy contributions.
Although both the term “core inflation” and the CPI measure originated
in the 1970s, it was not until around 1990 (see Ball and Cecchetti [1990]) that
the two became essentially synonymous. In recent years, economists have
proposed many alternative measures of core inflation. One of the more prominent alternatives is trimmed mean inflation, which removes from the inflation
calculation those price changes above and below specified percentiles in the
distribution. K-core inflation, the measure proposed in this article, is a close
cousin both to the standard core inflation measure (the index for all items less
food and energy) and to trimmed mean inflation. Instead of removing food and
energy prices—as core does—and instead of removing prices beyond specified percentiles in the distribution—as a trimmed mean does—k-core inflation
removes items whose relative prices change by more than a specified threshold. If one’s objective is to construct a measure of inflation on which large
relative price changes have a limited effect, then k-core inflation seems clearly
preferable to both inflation ex-food and energy and trimmed mean inflation. In
periods during which food and energy prices move with the overall price level,
whereas other categories experience large relative price changes, inflation exfood and energy will exclude small relative price changes and include large
relative price changes. In contrast, k-core inflation will always exclude—and
only exclude—the large relative price changes. Likewise, in periods during
which the distribution of relative price changes is unusually concentrated but
asymmetric, trimmed mean inflation would exclude many small relative price
changes, and could produce a measure that deviates markedly from overall
inflation. In contrast, k-core inflation would simply replicate overall PCE
inflation if there were no large relative price changes.
Section 1 provides some background information on the construction of
PCE inflation and the behavior of the category price changes that go into
constructing PCE inflation. Section 2 describes k-core inflation in general
terms. Whereas the measure in Section 2 is a parametric family, in Section 3
we show how the properties of k-core inflation vary with that parameter (k,
the criterion for a large relative price change). We specify a value for k and
compare k-core inflation to core inflation and trimmed mean inflation. Section
4 suggests areas for future research and concludes.

1.

INFLATION AND THE DISTRIBUTION OF
PRICE CHANGES

The two most commonly discussed measures of inflation in the United States
are PCE inflation and CPI inflation. PCE inflation is an index of the rate of
price change for a broad array of consumption goods and services—technically
the entirety of consumption in the national income and product accounts.
CPI inflation is an index of the rate of price change for “out-of-pocket”

A. L. Wolman: K-Core Inflation

417

consumption expenditures. As such, there are a number of differences between the components of PCE inflation and those of CPI inflation. Most
importantly, PCE inflation puts a significantly higher weight on health care
spending, and CPI inflation puts a significantly higher weight on housing services. There are also differences in the way the indexes are calculated; for
details, see Clark (1999). Because the PCE index is a more comprehensive
measure and is generally believed to be a more accurate measure of overall price changes, in the remainder of this article all references to inflation
(without other qualifiers) will be to PCE inflation.
PCE inflation (π t ) is a Fisher ideal index of price changes for a large
number (N ) of categories of consumption goods; it is the geometric mean of
the Paasche and Laspeyres indexes of price change. The Paasche index in
period t, denoted π P , is the rate of price change from period t − 1 to period
t
t for the consumption basket purchased in period t:
πP =
t

N
n=1 pn,t qn,t
N
n=1 pn,t−1 qn,t

(1)

.

In this expression, pn,t and qn,t are the price and quantity purchased of category
n in period t. The Laspeyres index in period t, denoted π L , is the rate of price
t
change from period t − 1 to period t of the consumption basket purchased in
period t − 1:
πL =
t

N
n=1 pn,t qn,t−1
.
N
n=1 pn,t−1 qn,t−1

(2)

Thus, the PCE inflation rate is given by the following formula:
πt =

N
n=1 pn,t qn,t
N
n=1 pn,t−1 qn,t

N
n=1 pn,t qn,t−1
N
n=1 pn,t−1 qn,t−1

.

(3)

Note that both the Paasche and Laspeyres indexes can be written as weighted
averages of price changes for each category, thus
N

πt =

N

ωP π n,t
n,t−1
n=1

ωL π n,t ,
n,t−1

(4)

n=1

where π n,t is the rate of price change for consumption category n in period t
(that is, π n,t = pn,t /pn,t−1 ), and where the weights are given by
pn,t−1 qn,t−1
(5)
ωL
n,t−1 =
N
j =1 pj,t−1 qj,t−1
and
ωP
n,t−1 =

pn,t−1 qn,t
N
j =1 pj,t−1 qj,t

.

(6)

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 1 PCE Inflation: Constructed and from the BEA

5

12-Month Constructed PCE Inflation Rate
12-Month Published BEA PCE Inflation Rate

Percent

4
3
2
1
0
-1
1988 Jan 1991 Jan 1994 Jan 1997 Jan 2000 Jan 2003 Jan 2006 Jan 2009 Jan

The Laspeyres weight, ωL , is the period t − 1 expenditure share for caten,t−1
gory n, and the Paasche weight, ωP , is the hypothetical expenditure share
n,t−1
associated with evaluating the period t consumption basket at period t − 1
prices.
Hundreds of consumption categories comprise PCE inflation, which is
compiled by the Bureau of Economic Analysis of the U.S. Department of
Commerce (BEA). We aggregate some of those categories in order to have a
consistent panel going back to January 1987, and are left with 240 categories,
covering 100 percent of personal consumption expenditure, for the period
from January 1987–October 2011. Figure 1 plots the behavior of official 12month PCE inflation over this period (solid line), together with the series we
constructed using (4) with 240 categories (open circles). A careful look at the
figure reveals slight differences between the two measures in some periods.
Overall however, the two series are close enough that it appears legitimate to
proceed using the constructed PCE measure instead of the BEA’s measure.
If the component price changes that aggregate up to PCE inflation were
all close to each other, and thus close to PCE inflation, then there would be
no reason to consider inflation measures that control for large relative price
changes. The black line in Figure 2 displays the distribution of relative price
changes for all categories across all periods in the sample, where the relative
price change for category n in period t is simply the difference between the
rate of price change for that category and the rate of PCE price change:
rn,t = π n,t − π t .

A. L. Wolman: K-Core Inflation

419

Figure 2 CDF of Monthly Relative Price Changes, Jan. 1987–Oct. 2011

To construct the distribution, each rn,t is weighted by the corresponding expenditure share ωL . The distribution of monthly relative price changes is indeed
n,t
concentrated around zero, with an interquartile range of ( −0.23 percent, 0.25
percent). However, there are also many large relative price changes: For example, 12.1 percent of (weighted) relative price changes are greater than 1
percent per month in absolute value. Figure 2 also displays the distribution
of relative price changes for the 28 food and energy categories (dark gray)
and for the 212 non-food and energy categories (light gray). Food and energy
relative prices vary much more than their complement: The interquartile range
for food and energy categories is (−0.53 percent, 0.55 percent) compared to
(−0.19 percent, 0.24 percent) for other categories.
In sum, from Figure 2 it is clear that (i) there is nontrivial variation in the
relative prices of different categories of consumption, and (ii) the variation
is especially large for food and energy categories. We take those facts as
motivation for constructing measures of inflation that attempt to control for

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Federal Reserve Bank of Richmond Economic Quarterly

the contributions of large relative price changes.3 We refer to any such measure
below as a measure of underlying inflation.

2.

OLD AND NEW MEASURES OF UNDERLYING INFLATION

Because food and energy prices are so much more volatile than the prices of
other consumption categories (see Figure 2), a natural underlying inflation
measure is one that simply removes food and energy prices from the inflation
calculation. This measure, so-called “ex-food and energy” PCE inflation,
has the virtue of simplicity. However, always and only removing food and
energy prices does not mean always and only removing categories with the
largest relative price changes. Of the top 10 price increases and the top 10
price decreases each period, on average less than one quarter of those largest
price changes were from food and energy categories. And of the 20 smallest
relative price changes each period (measured by absolute value), more than
8 percent were from food and energy categories.4 Thus, removing only food
and energy price changes means not removing most of the large relative price
changes, and it means removing a significant number of very small relative
price changes.
An alternative to ex-food and energy inflation that does remove only the
largest price changes each period is trimmed mean inflation. Trimmed mean
inflation begins with the weighted cumulative distribution function (CDF) of
monthly price changes each period, and removes those price changes that lie
outside upper and lower percentile cutoffs. If the upper and lower cutoffs are
the 50th percentile, then trimmed mean inflation is simply the rate of price
change for the median category. Bryan and Cecchetti (1994) and Dolmas
(2005) provide detailed discussions of trimmed mean inflation, with the former
focusing on the CPI and the latter on PCE inflation. They suggest various
methods of choosing the specific percentile cutoffs for trimmed mean inflation.
The Federal Reserve Bank of Dallas maintains a trimmed mean inflation series
(Federal Reserve Bank of Dallas 2012)—currently, their preferred cutoffs are
24 percent from the bottom of the distribution and 31 percent from the top (see
Section 3 for further discussion). From the data behind Figure 2, on average
these criteria remove relative price decreases greater than 0.25 percent per
month, and relative price increases greater than 0.18 percent per month.
If the goal is to construct a measure of underlying inflation by removing
large relative price changes, then a trimmed mean has an obvious advantage
3 From the definition of PCE inflation, it is tautological that all relative prices together account
completely for the behavior of inflation.
4 These statements refer to unweighted price changes—meaning that each category is weighted
equally. There are 28 food and energy categories out of 240 total categories in our sample, so
that if price change distributions were identical across categories then 11.6 percent of any range
of price changes would be from food and energy categories.

A. L. Wolman: K-Core Inflation

421

Figure 3 How the Distribution (Across Time) of k-Core Inflation Varies
with k
6

Max
5

4

75th Percentile

3

Mean
Median

2

25th Percentile
1

Min

0

-1
0.00

0.05

0.10

0.15

0.20

k

relative to ex-food and energy inflation: It removes categories with the largest
price changes, regardless of whether they are food and energy categories.
However, the fact that a trimmed mean removes fixed percentiles of each
period’s distribution of price changes has an important implication. Depending on how the distribution of price changes behaves in a given period, price
changes of different sizes will be removed. That is, once one specifies the percentile cutoffs, the largest price changes are removed each period, regardless
of the size of those price changes. But if the goal is to remove large relative
price changes, it seems preferable to specify the size of relative price changes
that will be removed and hold that size fixed each period. In the remainder of
the article we consider such a measure, which we call k-core inflation.5
K-core inflation specifies a cut-off value, k, for the size of relative price
changes. If the relative price change for category n is less than k in absolute
5 Researchers such as Bryan and Cechetti (1994) and Dolmas (2005) motivate trimmed mean

inflation partly on statistical grounds and partly on theoretical grounds. In the conclusion, we
suggest a theoretical grounding for soft-core inflation.

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 4 Benchmark k-Core Inflation (k = 2 Percent Annual Rate)

12-Month PCE Inflation Rate
12-Month K-Core Inflation Rate

5

Percent

4
3
2
1
0
-1
1988 Jan 1991 Jan 1994 Jan 1997 Jan 2000 Jan 2003 Jan 2006 Jan 2009 Jan

value, then the price change for category n is included without modification. If
the relative price change for category n in period t is greater than k in absolute
value, then the price change for category n is truncated at k. Formally, for a
given k, k-core inflation (π sc ) is defined as follows:
t
N

π kc
t

(k) =

N

ωP π kc
n,t−1 n,t
n=1

ωL π kc (k) ,
n,t−1 n,t

(k)

(7)

n=1

where
π kc (k) =
n,t

π n,t , if π n,t − π t < k
.
π t 1 + k · sign π n,t − π t , if π n,t − π t ≥ k

(8)

Three assumptions embodied in this definition require some discussion. First,
large price changes are truncated rather than being omitted. This choice is
based on the facts that there is uncertainty about the proper value of k, and about
whether or not every relative price change greater than k should be omitted
from underlying inflation. An appealing implication of this assumption is that
varying k between zero and infinity makes π kc (k) a smooth function that starts
t
and ends at π t . For low k all price changes are replaced with actual inflation,
and for high k all price changes are admitted, which returns actual inflation.
The second important assumption is that the criterion for truncating price
changes is the size of relative price change, rather than the size of nominal
price change. This choice simply reflects the view that it is large relative price
changes that we want to control for. Third, the criterion (k) does not vary with
the level of inflation. There is a large literature on the relationship between

A. L. Wolman: K-Core Inflation

423

Table 1 Summary Statistics for One-Month Inflation
(Annualized Percent)

PCE
k-core
XFE
Trimmed Mean

Mean
2.51
2.43
2.56
2.47

Min
−13.55
−1.31
−5.35
0.30

25th and 75th Percentiles
1.40
3.74
1.43
3.37
1.59
3.37
1.87
2.99

Max
13.66
6.28
8.30
5.68

Std. Dev.
2.41
1.38
1.55
0.95

relative price variability and inflation (see Hartman [1991], for example).
Based on that literature, one might argue that k should be an increasing function
of the inflation rate. Because the data used in this article is from a period of
relatively low and stable inflation, we assume that such considerations are not
quantitatively important.
From Figure 2, one can see how the choice of k maps into the fraction of
price changes that will be truncated: k ≥ 0.04 (4 percent monthly in Figure
2) would mean truncating a tiny fraction of price changes, whereas k = 0.005
(one-half percent monthly) would mean truncating 13.9 percent of weighted
price changes because of relative price decreases, and 12.7 percent because
of relative price increases. Of course, these are averages, and the fraction of
expenditures (equivalently, price changes) affected in a given period would
depend on the distribution of price changes in that period.

3.

BEHAVIOR OF K-CORE INFLATION

Figure 3 plots summary statistics for 12-month k-core inflation as a function
of k, using the entire sample. For each value of k, we compute the time series
for k-core inflation and plot the summary statistics, mean, median, maximum,
minimum, and 25th and 75th percentiles. The figure shows how these summary statistics of the time series vary with k. For low and high values of k,
the statistics are similar, reflecting the fact that k-core inflation converges to
overall PCE inflation as k approaches zero or infinity. The properties of k-core
inflation are sensitive to k for values around 0.02 (2 percent monthly relative
price change). The range (maximum minus minimum) of k-core inflation
shrinks from almost six percentage points (the range for PCE inflation) for
high and low k to less than four percentage points when k is around 0.02.
Because it is a round number and comes close to minimizing the range of
k-core inflation, we will use k = 0.02 as our benchmark.
Figure 4 plots the time series for benchmark k-core inflation, together
with overall PCE inflation (the constructed measure from Figure 1). Although
we construct k-core inflation as a monthly measure, using (7), the time series
plotted in Figure 4 and in subsequent figures display the 12-month cumulative

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Federal Reserve Bank of Richmond Economic Quarterly

Table 2 Summary Statistics for 12-Month Inflation (Percent)

PCE
k-core
XFE
Trimmed Mean

Mean
2.46
2.41
2.55
2.45

Min
−0.91
0.88
0.98
0.80

25th and 75th Percentiles
1.86
3.02
1.81
2.88
1.84
2.67
2.05
2.66

Max
5.42
4.59
5.19
4.28

Std. Dev.
1.11
0.88
1.10
0.74

k-core inflation rate.6 As expected from Figure 3, k-core inflation is notably
less volatile than PCE inflation. The behavior of inflation in the depths of the
Great Recession illustrates this point well: From mid-2008 to mid-2009, PCE
inflation fell by more than five percentage points, whereas k-core inflation fell
by less than three percentage points. However, it is not always the case that
k-core inflation is a smoother version of PCE inflation. For example, in the
second half of 2010, PCE inflation was relatively low (generally below 1.5
percent), yet k-core inflation was below PCE inflation.

K-Core Inflation and Ex-Food and Energy Inflation
Having motivated k-core inflation as an appealing alternative to ex-food and
energy inflation and trimmed mean inflation, we now compare the behavior of
k-core to inflation ex-food and energy (henceforth XFE), and in the next section, to trimmed mean inflation (henceforth TMI). The top three rows of Tables
1 and 2 display summary statistics for one-month and 12-month PCE inflation,
k-core inflation, and XFE.7 For monthly price changes, both k-core and XFE
are much less volatile than PCE inflation. This statement holds whether one
measures volatility by max-min, standard deviation, or interquartile range.
K-core inflation is less volatile than XFE, apart from the interquartile range
measure. Moving from one-month to 12-month inflation, the comparisons become more muddied. Because each of these series has a substantial transitory
component, the volatility of 12-month inflation is lower in every case than the
volatility of the one-month measure. The transitory component is strongest in
PCE inflation, so that the standard deviation of that series is cut by more than
half when comparing one-month and 12-month changes. In contrast, the standard deviation of XFE inflation falls by just 29 percent, leaving the standard
deviations of 12-month PCE and XFE inflation essentially identical. K-core’s
standard deviation is 36 percent lower for 12-month than one-month changes,
6 The only reason for doing this is that one-month inflation is quite volatile. Some of the
tables, as well as Figure 2, refer to one-month price changes.
7 Note that the version of XFE analyzed here is not the version reported by the BEA. Instead,
we calculate our own version by removing the 28 food and energy categories (and adjusting the
other weights accordingly) in equation (3). The resulting series is close to the one reported by
the BEA.

A. L. Wolman: K-Core Inflation

425

Figure 5 K-Core Inflation and Ex-Food and Energy Inflation

5

12-Month K-Core Inflation Rate (k = 2 percent annually)
12-Month Ex-Food and Energy Inflation Rate
12-Month PCE Inflation Rate

Percent

4
3
2
1
0
-1
1988 Jan 1991 Jan 1994 Jan 1997 Jan 2000 Jan 2003 Jan 2006 Jan 2009 Jan

leaving it well below XFE (0.88 versus 1.10). However, the interquartile range
for 12-month k-core inflation is well above that for XFE.
Figure 5 plots the time series for 12-month k-core inflation and XFE.
Although Tables 1 and 2 indicate that k-core inflation is generally less volatile
than XFE, Figure 5 shows that this volatility ranking is heavily influenced
by the first few years of the sample, when PCE inflation was often above
5 percent. During that time, k-core inflation was well below XFE. In the
last several years, by contrast, XFE has been markedly less volatile than kcore. The recent period has involved dramatic swings in energy prices. In
September 2008 for example, 12-month inflation was 4.03 percent, whereas
XFE was 2.52 percent. During this period, Figure 6 shows that there were
many large relative price decreases that k-core inflation adjusted for, whereas
XFE did not. As a result, k-core inflation was much higher than XFE, 3.37
percent, in the 12 months preceding September 2008.
From Figure 2, it is already clear that k-core inflation with k = 0.02 does
not always truncate food and inflation categories, and sometimes truncates
categories other than food and inflation. Table 3 lists the 15 categories whose
price changes are truncated most frequently when k = 0.02, restricting to
categories representing more than 0.01 percent of expenditure on average
over the sample period.8 Seven of the 15 categories are either food or energy
categories (they are indicated in bold in the table). The 15 categories together
8 The restriction based on expenditure shares meant that two categories were eliminated and
replaced with other categories.

426

Federal Reserve Bank of Richmond Economic Quarterly

Table 3 Categories Whose Relative Price Changes Most Frequently
Exceed k = 0.02 in Absolute Value
Category
Eggs
Fuel Oil
Gasoline & Other Motor Fuel
Fresh Vegetables
Indirect Securities Commissions
Mutual Fund Sales Charges
Air Transportation
Direct Securities Commissions
Used Truck Margin
Natural Gas
Fresh Fruit
Other Fuels
Luggage & Similar Personal Items
Tobacco
Commissions for Trust, Fiduciary, & Custody Activities

Freq. Exceed k
81
80
79
64
64
63
52
44
42
41
39
39
35
32
31

Avg. Share
0.0007
0.0023
0.0269
0.0040
0.0013
0.0011
0.0060
0.0032
0.0017
0.0065
0.0027
0.0002
0.0024
0.0096
0.0011

Notes: Food and energy categories are listed in bold.

represent 7 percent of expenditures, and the seven food and energy categories
represent 4.3 percent of expenditures.

K-Core Inflation and Trimmed Mean Inflation
Next, we compare our k-core inflation measure to TMI. To generate TMI
we use a lower cutoff of 20 percent of expenditures, and an upper cutoff
of 23 percent. Dolmas (2005) proposes three different criteria for choosing
the upper and lower cutoffs. One of the criteria he uses is to minimize the
squared deviations from a centered 36-month moving average of overall PCE
inflation. Applying that criterion to our sample generates the 20 percent and 23
percent cutoffs. Note that for k-core inflation, our choice of k = 0.02 nearly
represents the value of k that would minimize the deviation of k-core inflation
from the 36-month moving average of overall PCE inflation; that value is
˜
k = 0.018. However, even this “optimized” version of k-core is considerably
less successful than the optimized TMI at matching the moving average. The
sum of squared deviations for the TMI is 7.5 × 10−5 , whereas the sum of
squared deviations for k-core inflation is 1.2 × 10−4 .
Tables 1 and 2 contain summary statistics for TMI, in the bottom row,
and Figure 7 plots TMI along with k-core inflation and PCE inflation. TMI is
less volatile than either XFE or k-core inflation. The difference is especially
striking for one-month inflation, where the standard deviation of TMI is at least
30 percent lower than that of the other measures, and the difference between
the maximum and minimum values is 5.4 percent for TMI, compared to 7.6

A. L. Wolman: K-Core Inflation

427

Figure 6 Pooled Distribution of Monthly Relative Price Changes from
Sept. 2007–Sept. 2008

percent for k-core and 13.7 percent for XFE. The relative stability of TMI
compared to k-core can be partly understood by referring back to Figure 2,
the distribution of relative price changes. Although k-core inflation is not a
trimmed mean, we can think of it as a “truncated mean,” where the percentile
cutoffs for truncation (at 0.02) change each period. From Figure 2, on average
both the lower and upper cutoffs for truncation are close to 0.25 percent of
expenditure-weighted price changes. Thus, TMI with cutoffs at about 20
percent results in a price index that differs much more dramatically from
PCE inflation than does our k-core inflation measure. If we were to exclude
categories instead of truncating their price changes, the resulting series would
be precisely a trimmed mean with time-varying cutoffs. From the numbers
reported in the previous section we know that relatively little trimming would
be implied by k = 0.02. Figure 8 displays the somewhat smoother series
generated by eliminating categories with k = 0.02 instead of truncating their
price changes.

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 7 K-Core and Trimmed Mean Inflation

12-Month K-Core Inflation Rate (k = 2 percent annually)
12-Month Trimmed Mean Inflation Rate (23 percent from
top, 20 percent from bottom)
12-Month PCE Inflation Rate

5

Percent

4
3
2
1
0
-1
1988 Jan 1991 Jan 1994 Jan 1997 Jan 2000 Jan 2003 Jan 2006 Jan 2009 Jan

4.

CONCLUSION

We have proposed a new measure of underlying inflation, referred to as k-core
inflation. All such measures are motivated to some degree by the idea that large
relative price changes may represent noise, which the monetary authority ought
to filter out for the purpose of forecasting or for inferring the current stance of
monetary policy. K-core inflation does this filtering by specifying a threshold
for a large relative price change. Relative price increases or decreases beyond
that threshold are truncated to be equal to the threshold. In contrast, inflation
ex-food and energy excludes food and energy prices regardless of how much
those prices change, and trimmed mean inflation excludes fixed percentiles of
the price change distribution, regardless of the size of price changes to which
those percentiles correspond.
The figures and tables in the article illustrate how k-core inflation behaves,
and how it compares to inflation ex-food and energy and to trimmed mean
inflation. Mid-2008 was a period in which the differences between k-core inflation and these other measures were particularly large and persistent. K-core
inflation indicated significantly higher underlying inflation in mid-2008 than
either ex-food and energy inflation or trimmed mean inflation. The situation
looks somewhat similar today, when energy price increases are again in the
headlines: In the 12 months preceding October 2011, k-core inflation was

A. L. Wolman: K-Core Inflation

429

Figure 8 K-Core Inflation when Large Price Changes are Eliminated
Rather than Truncated

5

12-Month K-Core Inflation Rate (k = 2 percent annually)
Version of K-Core that Eliminates Categories
12-Month PCE Inflation Rate

Percent

4
3
2
1
0
-1
1988 Jan 1991 Jan 1994 Jan 1997 Jan 2000 Jan 2003 Jan 2006 Jan 2009 Jan

2.3 percent, compared to 2.7 percent for overall PCE inflation, 1.6 percent for
PCE inflation ex-food and energy, and 1.9 percent for trimmed mean inflation.
This article is exploratory in nature. It would be interesting to investigate
k-core inflation further in at least three dimensions. First, measures of underlying inflation are often evaluated on the basis of their ability to forecast PCE
inflation. How does k-core inflation fare according to this criterion? Second,
the definition of k-core inflation used here has maintained that PCE inflation
is the correct inflation rate against which to measure relative price changes.
Perhaps k-core inflation is instead the correct inflation rate against which to
measure relative price changes. Applying this change to our definition would
require solving a fixed-point problem to compute k-core inflation. Finally,
and most importantly, it would be interesting to pursue possible theoretical
underpinnings of k-core inflation. If there are large sector-specific shocks (as
suggested by much research on price adjustment, such as Golosov and Lucas
[2007]) and if the structure of the economy and the behavior of monetary
policy are such that monetary policy does not generate large relative price
changes, then something like k-core inflation might be a useful indicator of
monetary conditions. It would be straightforward to study this issue in a multisector equilibrium model. Of course, it is also possible that large relative price
changes could actually signal loose monetary policy. That would go against
the spirit of this article, but it cannot be ruled out a priori. Whether or not such
a possibility is empirically relevant would seem to depend on the nature of

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Federal Reserve Bank of Richmond Economic Quarterly

cross-sectoral variation in price stickiness and demand and supply elasticities.
These issues could be studied in the context of a calibrated equilibrium model.

REFERENCES
Ball, Laurence, and Stephen G. Cecchetti. 1990. “Inflation and Uncertainty
at Long and Short Horizons.” Brookings Papers on Economic Activity 21
(1): 215–54.
Bryan, Michael, and Stephen Cecchetti. 1994. “Measuring Core Inflation.”
In Monetary Policy, edited by N. Gregory Mankiw. Chicago: University
of Chicago Press, 195–215.
Bureau of Labor Statistics. 2011. “The So-called ‘Core’ Index: History and
Uses of the Index for All Items Less Food and Energy.” In Focus on
Prices and Spending. Washington, D.C.: Bureau of Labor Statistics
(February).
Clark, Todd E. 1999. “A Comparison of the CPI and the PCE Price Index.”
Federal Reserve Bank of Kansas City Economic Review Q3: 15–29.
Dolmas, Jim. 2005. “Trimmed Mean PCE Inflation.” Federal Reserve Bank
of Dallas Working Paper 0506 (July).
Federal Reserve Bank of Dallas. 2012. “Trimmed Mean PCE Inflation Rate.”
Available at www.dallasfed.org/research/pce/index.cfm.
Golosov, Mikhail, and Robert E. Lucas, Jr. 2007. “Menu Costs and Phillips
Curves.” Journal of Political Economy 115: 171–99.
Gordon, Robert J. 1975. “Alternative Responses of Policy to External Supply
Shocks.” Brookings Papers on Economic Activity 6 (1): 183–206.
Hartman, Richard. 1991. “Relative Price Variability and Inflation.” Journal
of Money, Credit and Banking 23 (May): 185–205.
Sprinkel, Beryl W. 1975. “1975: A Year of Recession, Recovery, and
Decelerating Inflation.” Journal of Business 48 (January): 1–4.

Economic Quarterly—Volume 97, Number 4—Fourth Quarter 2011—Pages 431–450

The Cost of Unanticipated
Household Financial
Shocks: Two Examples
Kartik Athreya and Urvi Neelakantan

H

ouseholds sometimes experience unexpected negative changes to
their financial circumstances. In this article, we quantify the consequences of two representative types of unanticipated financial
shocks. By “unanticipated,” we mean that households in our experiments
are modeled as ignoring even the possibility that the shock could occur. We
are thus interested in the cost of an event that comes as such a surprise to the
household that its previous consumptions-savings decisions in no way prepared it for such an eventuality. Our analysis is therefore exactly analogous
to a standard form of experiment in business cycle contexts, e.g., the impulse
response of an unanticipated fiscal or monetary policy shock that agents know
is permanent as soon as it occurs (see, for example, Baxter and King [1993]).
For each shock, our calculations tell us how costly it is for households
to live in a world where the shock occurs compared to a world in which it
does not. Why might such costs be useful to study? If the household (or a
policymaker) could pay—say through investment in financial education—for
information that would enable it to avoid the shock or mitigate its effect, our
calculations may provide an upper bound on how much it might be willing to
pay. The reason is that the cost of the shock depends not just on its magnitude
but also its likelihood. For a shock of a given size, households will be less
willing to pay to avoid it as its likelihood falls. We assume that the shock is
We thank Juan Carlos Hatchondo, Marianna Kudlyak, Ned Prescott, Pierre Sarte, Max
Schmeiser, Jonathan Tompkins, John Weinberg, Kim Zeuli, seminar participants at the Federal Reserve Bank of Richmond and the University of Illinois at Urbana-Champaign, and
participants at the Georgetown Center for Economic Research and the Southern Economics
Association conferences for helpful comments and suggestions. We are solely responsible for
any errors. The views expressed in this paper are those of the authors and do not necessarily
reflect the views of the Federal Reserve Bank of Richmond or the Federal Reserve System.
E-mail: kartik.athreya@rich.frb.org; urvi.neelakantan@rich.frb.org.

432

Federal Reserve Bank of Richmond Economic Quarterly

completely unanticipated or seen as one with zero likelihood. If its likelihood
is truly close to zero, this makes it not worth doing much about, all else
equal. Moreover, if the household is incorrect in assigning zero or nearzero likelihood to the shock, that is a belief that maximizes the amount by
which households “underestimate” the risks. If the household instead knew
that the shock could occur with positive probability, it would take actions (to
the extent warranted by the magnitude and likelihood of the shocks, and the
household’s aversion to risk) to reduce its severity. By contrast, our model
features households that will, by their unawareness, have made no provisions
at all at the onset of either of the shocks we consider. Our cost calculations
will also allow us to compare shocks, that is, to point out which shocks are
costlier (assuming equal likelihood) and therefore worthy of greater attention.
The two types of shocks we consider here are (1) an unanticipated drop in
net worth and (2) an unpredicted increase in borrowing costs for all forms of
unsecured debt. Each is meant to represent the occurrence of an empirically
plausible scenario. The first provides insight into the cost borne by those who
are surprised by declines in the value of assets in their portfolio. Consider,
for example, a household that has a net worth that is largely composed of
equity in its home, and for which the recent decline in U.S. house prices came
as a shock. It is evident that many commentators and experts placed little
probability on a widespread decline in home prices.1 The second case is that
of a sudden, widespread increase in the cost of rolling over debt and captures
the effects of general credit market tightening as might occur in the midst of
a severe recession that was a priori assigned zero probability. Note that both
shocks are fully persistent.
The size of a shock is an inadequate measure of its importance to a household, in particular because the cost is likely to vary across households. Thus,
quantifying the cost requires a model of household financial decision making.
Households make consumption-savings decisions with the goal of smoothing
consumption over their lifetime. A consequence of this hypothesis is that
households’ financial positions (and, hence, the cost of the shock to them)
will differ by age. Moreover, to the extent that households face other, more
predictable forms of risk throughout their lives, they will also differ from each
other at any given age. In turn, the cost of a shock will vary across households
of any given age as well. The economic model we use is a fairly standard version of a life-cycle model of consumption and savings, and follows Athreya
(2008). We use the model and, in particular, the optimal value function of
the household, to quantify the effects of the shocks. Specifically, we use the
1 Freddie Mac’s “Rent or Buy” calculator provides anecdotal evidence of the lack of concern
with house price declines. The calculator did not allow users to analyze the effects of negative
realizations for home prices, even though the same device allowed for increases in house prices
of up to 100 percent (Joffe-Walt and Davidson 2010).

Athreya and Neelakantan: Unanticipated Household Financial Shocks 433
model to determine the amount of annual consumption that a household would
be willing to give up to avoid facing the shock.
The reader will no doubt see that our article is highly stylized. Importantly, it abstracts from portfolio choice and focuses instead on a simple scalar
measure of net worth. In its current form it therefore cannot speak directly to
particular kinds of financial decisions, such as house purchases or any other
leveraged purchase of risky assets. In particular, our focus on net worth effectively precludes us from being able to assess the impact of decisions whose
effects derive primarily from their impact on the gross financial positions of
households—as well as on any attendant changes in the periodic payment
obligations—while leaving net worth essentially unchanged. Our model also
abstracts from the labor supply decision, which could mitigate the cost of the
shock by allowing households to simply “work their way” out of a reduction
in net worth. However, this is not wholly unreasonable because the shocks
we consider are most relevant to recessionary settings, in which labor markets
could plausibly preclude such adjustments.
Finally, our model embodies a strong assumption with respect to the information that households possess: We assume that the shocks that the household
faces are completely unanticipated. It is possible, instead, that households are
aware of the existence of the kinds of shocks we analyze in this article, but
wrong about the exact probabilities with which they could occur. Nonetheless,
while strong, this assumption allows us to determine what are likely to be upper bounds on the consequences of such shocks. After all, any information
received in advance about the likelihood of such events can only make the eventual shock, if it occurs, easier to deal with, as households will have consumed
and saved in anticipation of such possible outcomes. In addition, the current
work is simply a small first step, and we have indeed begun to incorporate
each of these features in ongoing work (Athreya, Ionescu, and Neelakantan
2011) that we hope will shed greater light on the questions addressed here.
With the preceding in mind, we describe the model in Section 1. Section 2
describes how each shock is introduced within this framework. Section 3
reports the results in terms of the costs of each shock. Section 4 concludes.

1. A LIFE-CYCLE MODEL OF CONSUMPTION AND
NET WORTH
The economy is that of Athreya (2008), and consists of a continuum of J overlapping generations of working households. Households value consumption,
do not value leisure, and therefore supply labor inelastically.

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Table 1 Model Parametrization
Parameter
α

Value
2

Parameter
Rf

Value
1.01

Parameter
γ

Value
0.99

Parameter
τ

Value
$7,600

β

0.96

ψ

3.4%

σ2
u

0.063

τR

$8,600

x1

0

λ

0.9

σ2
η

0.0275

σ2
η1

0.22

Preferences
The household chooses consumption, {cj }J=1 , and retirement wealth, xR , to
j
solve
J

sup

({cj },xR )∈ (

βj

E0
0)

j =1

1−α
cj

1−α

+

1−α
xR
.
1−α

(1)

Here, ( 0 ) denotes the space of all feasible combinations ({cj }, xR )
given initial state 0 , α denotes risk aversion, and β is the discount factor.
In the calibration, risk aversion and the discount factor are set at the standard
values of α = 2 and β = 0.96. (See Table 1 for all model parameter values,
which follows Athreya [2008].)

Income
Households have three potential sources of income: labor income, meanstested transfer income, and retirement income, with labor income being subject
to shocks drawn from a probability structure that is known perfectly by the
agent.
Labor Income

The model period is one calendar year. Households begin working life at age
21 and retire at age 65. Households face uncertainty in their labor income
because of stochastic productivity shocks to their labor supply. Following
the literature (e.g., Hubbard, Skinner, and Zeldes 1995; Huggett and Ventura
2000; Storesletten, Telmer, and Yaron 2004), the evolution of log income,
ln yj , is modeled as
ln yj = μj + zj + uj ,

(2)

where μj is an age-specific mean of log income, zj is the persistent shock,
and uj is the transitory shock.

Athreya and Neelakantan: Unanticipated Household Financial Shocks 435
The profile {μj }J=1 is parameterized using data on the median earnings
j
of U.S. males from the 2000 Census.2
The persistent shock, zj , is given by
zj = γ zj −1 + ηj , γ ≤ 1, j ≥ 2, ηj ∼ i.i.d. N (0, σ 2 ).
η

(3)

We set γ = 0.99 and
= 0.0275 to capture the facts that, in the data, the
cross-sectional variance in log income increases substantially, and roughly
linearly, over the life cycle; it is roughly 0.28 among 21-year-olds and roughly
0.90 among new retirees. The transitory shock, uj , is distributed as uj ∼
i.i.d N (0, σ 2 ) and is independent of ηj .
u
To capture initial heterogeneity across households, it is assumed that they
draw their first realization of the persistent shock from a distribution with a
different variance than at all other ages. That is,
σ2
η

z0 = 0, and η1 ∼ N (0, σ 21 ).
η

(4)

In the above, σ 21 = 0.22.
η
Note that the assumption that households supply labor inelastically restricts them from using a smoothing mechanism that could be particularly
useful in the face of unanticipated shocks. However, not only does this assumption keep the model parsimonious, it is in keeping with the usefulness of
providing an upper bound on the costs of the shocks we study.
Means-Tested Transfer Income

Following Hubbard, Skinner, and Zeldes (1995), means-tested transfers τ (·)
are specified as a function of age, j , net worth, xj , and income, yj , as follows:
τ (j, xj , yj ) = max{0, τ − (max(0, xj ) + yj )}.

(5)

Social insurance in the United States aims to provide a floor on consumption and the specification in equation (5) captures this feature. The transfer
scheme provides households with a minimum of τ units of the consumption
good at the beginning of the period. In the calibration, τ ∼ $7,600 to match
=
data on the asset accumulation of households in the lower percentiles of the
wealth distribution.
Retirement Income
x 1−α

R
Household utility at retirement is evaluated as 1−α . Retirement wealth, xR ,
is the sum of household personal savings, xJ +1 , and a baseline retirement
benefit, xτ R :

xR = xJ +1 + xτ R .

(6)

2 Since income is lognormally distributed, the mean of log income equals the log of median
income. Therefore, the log of median earnings is used to generate the profile.

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Federal Reserve Bank of Richmond Economic Quarterly

The baseline retirement benefit, xτ R , yields an annual income of τ R when
annuitized using discount rate R f . That is, the baseline retirement benefit
solves
K

k=1

τR
= xτ R .
(R f )k

(7)

Here, τ R represents the societal minimum amount of consumption at retirement. This amount is not means tested and is intended to represent the sum of
welfare programs, Social Security, and Medicare.3 The interest rate, R f > 0,
is the risk-free rate of return on savings and is exogenously given.
In the calibration, the minimum amount of consumption at retirement, τ R ,
is set equal to $8,600 and R f = 1.01.

Technology and Market Arrangement
At each age j , households choose whether to save (xj +1 > 0) or borrow
(xj +1 < 0). Savings earn the exogenous risk-free rate of return R f > 0.
The interest rate on borrowing is R(·), which incorporates credit risk (because
households can default on the debt next period) and transaction costs, ψ,
arising from resources used in intermediation. Default is costly and reduces
household utility by λ in the period in which debts are repudiated. This cost
includes, but is not limited to, the cost of legal representation and court fees.
It is meant to capture all costs deemed relevant by households, and will be
calibrated to help the model match default-related behavior.

Recursive Formulation
The household’s problem is recursive in a state vector that includes age, j ,
beginning-of-period net worth, xj , current-period realization of the persistent
shock, zj , and current-period realization of transitory income, uj .
Value Functions

Households that enter a period with debt must decide whether or not to default.
The value function when repaying debts is W R (·), which solves
W R (j, xj , zj , uj ) = sup
xj +1

1−α
cj

1−α

+ βEzj +1 |zj V (j + 1, xj +1 , zj +1 , uj +1 ) ,
(8)

3 This approach follows Huggett (1996) and Gourinchas and Parker (2002).

Athreya and Neelakantan: Unanticipated Household Financial Shocks 437
subject to
cj +

xj +1
≤ yj + τ (j, xj , yj ) + xj ,
R(j, xj +1 , zj )

(9)

where R(j, xj +1 , zj ) is the interest rate associated with the level of savings or
borrowing, xj +1 , chosen by the household of age j and current realization of
the persistent shock zj .
The value of defaulting is given by W D (·), which solves
W (j, xj , yj , uj ) = sup
D

xj +1

1−α
cj

1−α

− λ +βEzj +1 |zj V (j + 1, xj +1 , zj +1 , uj +1 ) ,
(10)

subject to
cj +

xj +1
≤ yj + τ (j, xj , yj ),
Rf
xj +1 ≥ 0.

(11)
(12)

The debt obligation in the right-hand side of (9) does not appear in (11) because
the household defaults. The household pays the utility cost, λ, associated
with defaulting. In the parametrization, λ is set to 0.9. Along with the other
parameters in the model, this targets the Chapter 7 filing rate of 0.5 percent
and the mean net worth of Chapter 7 bankruptcy filers of $16,815.4
The household is not allowed to borrow in the period in which it defaults.
Thus, net worth chosen in the current period must be non-negative and earns
the risk-free rate of interest, R f . The household can borrow in all subsequent
periods.
The beginning-of-period value function must therefore satisfy
V (j, xj , zj , uj ) = max W R (j, xj , zj , uj ), W D (j, xj , zj , uj ) .

(13)

Once borrowing or savings is chosen, the period ends.

Loan Pricing
In the market for loans, creditors are assumed to be competitive and to hold
a sufficiently large number of loans of any given size for the law of large
numbers to guarantee them a deterministic rate of return on loans of that size.
They pay transactions costs, ψ, in exchange for which they can observe all
factors needed to forecast the risk of default one period ahead. In the model,
these factors are age, j , the persistent shock, zj , and household debt, xj .
Creditors expect to break even on each loan by pricing contingent on these
factors. Let π D (j, xj +1 , zj ) denote the probability of default on a loan of
4 This data is as of 1991 in order to be consistent with the timing of the income and
consumption data.

438

Federal Reserve Bank of Richmond Economic Quarterly

size xj +1 , made to a household of age j , with persistent income shock zj .
Let I (j + 1, xj +1 , zj +1 , uj +1 ) be the indicator function over whether or not
a household with debt xj +1 and shocks zj +1 and uj +1 will choose to default.
That is,
I (j + 1, xj +1 , zj +1 , uj +1 ) = 1,
if and only if
W D (j + 1, xj +1 , zj +1 , uj +1 ) > W R (j + 1, xj +1 , zj +1 , uj +1 ).
Therefore, π D (·) is calculated at each age j as follows:
π D (j, xj +1 , zj ) =

I (j +1, xj +1 , zj +1 , uj +1 )f (zj +1 , uj +1 |zj )dzj +1 duj +1 .

(14)
Given π D (·), the interest rate function, R(j, xj +1 , zj ), is determined as
follows:
Rf + ψ
R(j, xj +1 , zj ) =
.
(15)
(1 − π D (j, xj +1 , zj ))

2.

UNANTICIPATED SHOCKS AND THEIR SIZES

We now introduce unanticipated shocks to households in the above framework.
We capture the effect of the shock on the “representative” household of any
given age, as described by the age-specific median value of wealth.
As mentioned earlier, we quantify the effect of such shocks in terms of annual consumption. There are several ways in which we could do this. For ease
of interpretation, we express all quantities in terms of constant consumption
levels. We now describe the two scenarios under study and detail the particular calculations needed to derive the costs in terms of equivalent constant
consumption levels under each.

Case 1: An Unanticipated Drop in Net Worth
The first case analyzes the consequences of an unanticipated drop in net worth.
The empirical parallels we have in mind are unexpected decreases in house
prices or stock prices. Since wealthier households are likely to have more
expensive homes and larger stock portfolios, we assume that the shock is
proportional to net worth.
The cost of this shock is calculated as follows. Let V (j, x k , zj , uj ) denote
the value of arriving in a given period k with wealth x k . Let V (j, xk , zj , uj )
denote the value of arriving in a given period k with wealth xk , where, for
0 < θ < 1,
xk = (1 − θ )x k if x k ≥ 0
xk = (1 + θ )x k if x k < 0.

Athreya and Neelakantan: Unanticipated Household Financial Shocks 439
The latter is the value associated with the discounted expected utility that a
household can obtain from behaving optimally after the occurrence of the
shock to net worth. Given this, we define c and c as the constant values for
¯
˜
consumption that, when received over an entire lifetime, generate discounted
utility equal to V (j, x k , zj , uj ) and V (j, xk , zj , uj ), respectively. Thus, c and
c solve, respectively,
J +25

V (j, x k , zj , uj ) =

β j −k

c1−α
¯
,
1−α

β j −k

c1−α
˜
.
1−α

k=j
J +25

V (j, xk , zj , uj ) =
k=j

The difference c − c represents the number of units of consumption that would
¯ ˜
make the household indifferent to facing the shock or not. The difference thus
represents the cost of the shock in units of consumption, or, alternately, the
amount the household would pay in units of consumption to avoid facing the
shock. (See the Appendix for calculation details.)
What are empirically plausible sizes of the net worth shocks? We may
arrive at upper bounds by assuming that the household’s entire net worth is
composed of a single asset and attribute the shock to a drop in the value of
that asset. This asset might be the household’s equity in its home or its stock
portfolio. To obtain an approximation of the upper bound on the size of shocks
to house and stock prices, we use available data. The recent house price bust
serves as a case in point. The largest annual decline of nearly 19 percent in the
S&P/Case-Shiller Home Price Index of U.S. National Values since 1987 came
between the first quarter of 2008 and the corresponding quarter of 2009.5 If we
look at stocks, the shock could correspond to the worst one-year performance
of diversified mutual funds such as Vanguard’s S&P 500 Index or Total Stock
Market Index, in which their value fell by roughly 37 percent. Different values
may be appropriate for households with different profiles, but we carry out
the exercise for a 40 percent drop in asset values, i.e., θ = 0.4, to serve as an
upper bound.

Case 2: An Unexpected Tightening of Credit Markets
Our second scenario aims to measure the cost imposed on a household by a
sudden change in borrowing premia, and so intends to be reflective of dysfunction in credit markets very generally. We capture this in the model by
comparing the maximal value that is attainable to an agent under the initial
5 See
www.standardandpoors.com/indices/sp-case-shiller-home-price-indices/en/us/?indexId=spusacashpidff–p-us—-.

440

Federal Reserve Bank of Richmond Economic Quarterly

interest rate function with that attainable from living in an environment where
loans are costlier than before. Specifically, we model the shock as raising the
interest rate on credit to R(·) + ι, ι > 0. Importantly, we assume that agents
will be faced with the tighter credit conditions for the rest of their lives. As
a result, our calculations will likely represent an upper bound on the cost of
such credit market tightening.
The net worth shock did not change household value functions. This is
because the shock did not alter the subsequent uncertainty or costs in the
household’s environment. In this case, by changing the interest rate faced
by households for the rest of their lives, the shock does change the maximal value of attainable utility coming from any given wealth position. Let
V (j, xk , zj , uj ) be the value function before the shock, when the interest on
credit is R(·). Let V (j, xk , zj , uj ) be the value function after the shock, derived from solving the household’s optimization problem over its remaining
life under the new credit-pricing function.
In this case, c and c solve
¯
˜
J +25

V (j, xk , zj , uj ) =

β j −k

c1−α
¯
,
1−α

β j −k

c1−α
˜
.
1−α

k=j
J +25

V (j, xk , zj , uj ) =
k=j

As before, cost of shock is c − c.
¯ ˜
We increase borrowing costs at all debt levels by 300 basis points, which
corresponds to among the largest spreads observed between mortgage market
interest rates and 10-year Treasury securities.

3.

RESULTS

The following section presents results for the two cases. In both scenarios,
we impose the shock on households at one period and calculate the cost of
the shock for households of various ages. The household does not expect, nor
does it receive, any further shocks aside from the age at which we study them.
Notice that in our model, the presence of uninsurable risk will lead households to vary not only in income, but importantly, in consumption and wealth.
There is, therefore, no “representative” household. This raises the issue of
whose well-being and costs we are studying. A natural candidate, which we
use, is the household with the median level of wealth and the median level of
income shocks for its age group. Figure 1 shows median wealth for households
at each age.
We find in general that the cost depends on the household’s initial level of
assets, the size of the shock, and the time period in which it occurred. Each
case is discussed in detail.

Athreya and Neelakantan: Unanticipated Household Financial Shocks 441

Figure 1 Median Wealth by Age
400

Median Wealth by Age (Thousands of Dollars)

350

300

250

200

150

100
50

0
-50
20

25

30

35

45

40

50

55

60

65

Age

Case 1: Net Worth Shock
We calculate the cost of a 40 percent decrease in net worth to a household with
the median level of assets and income shocks for its age. Figure 2 shows the
cost of the shock to the household in dollars of constant annual consumption,
which is calculated as c − c. Figure 3 shows the cost as a fraction of constant
˜ ¯
˜ ¯
annual consumption, calculated as c−c .
c
¯
The cost as a function of age displays a U-shape and then a steep increase.
The U-shape corresponds to ages at which the household has negative net
worth. The proportional shock pushes the household deeper into debt, and
is costliest when the household has the most debt (at age 26). Subsequently,
the cost of the shock is small for a while. This is because the absolute value
of wealth is low, so a proportional decline amounts to a very small number.
However, the cost rapidly increases with age. As age increases, so does median
wealth and the same percentage drop in net worth represents a much larger
absolute loss. For example, the cost to a household that faces the shock at
age 40 is $481 in constant annual consumption for the rest of its life.6 For
6 All costs in this article are reported in 2010 dollars.

442

Federal Reserve Bank of Richmond Economic Quarterly

Figure 2 Difference in Consumption by Age at Shock: Case 1
Net Worth Shock

Difference in Constant Consumption Streams ($)

0

-2,000

-4,000

-6,000

-8,000

-10,000

-12,000
20

25

30

35

40

45

50

55

60

65

Age at Shock

a household at age 60, the cost of an unforeseen 40 percent reduction in net
worth is $6,870 per year for the rest of its life. Figure 3 represents the same
cost in percentage terms. The cost to the 40-year-old household is 1 percent of
annual consumption without the shock, while for the 60-year-old household,
it is nearly 15 percent.
While older households face greater costs in terms of annual consumption,
they also face them for fewer periods. It is therefore useful to compare the
present value of the sequence {c − c}J=k for various k, where k is the date of
¯ ˜ j +25
the shock. Calculating the present value in the first period of the model gives
the perspective of one household looking ahead, while calculating the present
value at date k compares the relative cost of the shock to older and younger
households living in date k. Figure 4 shows the results of both calculations.7
Even in present value terms, the cost of the shock is highest for the oldest
households.
7 See the Appendix for calculation details.

Athreya and Neelakantan: Unanticipated Household Financial Shocks 443

Figure 3 Percentage Difference in Consumption by Age at Shock:
Case 1

Net Worth Shock
0.00

Ratio of Difference in Consumption (

~ −
c−c
______
−
c

)

-0.05

-0.10

-0.15

-0.20

-0.25

-0.30-

-0.35
20

25

30

35

40

45

50

55

60

65

Age at Shock

Case 2: Interest Rate Shock
We now consider the shock to a household that comes from facing unexpectedly higher interest rates on credit for the rest of its working life. Recall that
in this case, we measure the cost as arising from a 300-basis-point increase
in the interest rate on all debt levels that the household might choose. The
corresponding costs in terms of annual consumption, fraction of original consumption, and present value of annual consumption are shown in Figures 5–7.
The cost of this shock is very small relative to the net worth shock; it does
not exceed $300 of annual consumption for agents of any age. An important
part of why this cost remains small is that households can adjust savings in
the interim fairly effectively to nullify the effects of such an increase in costs.
Moreover, as long as such a shock does not occur at the time when a household
is holding peak debt (lowest net worth), the size of the shock itself is not large.
Lastly, once households leave the first 15 years or so of their working life cycle,
they are typically not in debt (have non-negative net worth), and, moreover,
do not typically expect to return to such a state. Thus, contractions in credit
markets will not hurt them.

444

Federal Reserve Bank of Richmond Economic Quarterly

0.0

x 10

5

Net Worth Shock
PV of Difference in Constant Consumption Streams at Birth ($)

PV of Difference in Constant Consumption Streams at Date k ($)

Figure 4 Present Value of Difference in Consumption: Case 1

-0.5

-1.0

-1.5

-2.0

-2.5

x 10

0

4

Net Worth Shock

-2

-4

-6

-8

-10

-12

-14

-16
-18

-3.0
20

30

40

50

60

70

20

Age at Shock

30

40

50

60

70

Age at Shock

The costs we report have all been calculated under the assumption that
households can declare bankruptcy and remove all unsecured debt obligations
subject to a penalty. As a result, debt in the model is priced to reflect this
possibility. Household net worth over the life cycle is, of course, different
than what it would be in a setting where households did not have this option
but instead had access to risk-free borrowing. As a check for robustness, we
have shut down this option in the model by making the utility cost, λ, infinite,
which effectively precludes bankruptcy.8 We find that this has little or no effect
on the size of the costs. This is because the option to declare bankruptcy is
relevant only to a subset of households—those with negative net worth (that
do not have sufficient assets to pay off their debts). In our model, these are
younger households. Because the debt they hold is not large on average, the
proportional net worth shock translates into a small shock for them in absolute
8 Results are available from the authors upon request.

Athreya and Neelakantan: Unanticipated Household Financial Shocks 445

Figure 5 Difference in Consumption by Age at Shock: Case 2
Interest Rate Shock

Difference in Constant Consumption Streams ($)

0

-50

-100

-150

-200

-250

-300
20

25

30

35

40

45

50

55

60

65

Age at Shock

terms. Our results have shown that the cost of the interest rate shock is also
small in absolute terms for these households. As a result, neither shock is
large enough to make bankruptcy an important consideration for the examples
we study.
We note here that the focus of the model on net worth is likely very
important for the small role it assigns to the effects of an interest rate shock.
In a richer setting, the fact that households are often engaged in very heavily
leveraged investment (taking out mortgages to finance a home purchase) means
that credit market costs could likely affect people well into late middle-age.
This is simply because, while they might have positive net worth by middleage (indeed will, in most instances), they may also owe substantial amounts on
a mortgage, and the size of these obligations may be quite large and difficult
to deal with. In ongoing work (Athreya, Ionescu, and Neelakantan 2011), we
are considering precisely this.

4.

CONCLUSION

In this article, we take a first step in measuring the cost of two particular and, we
think, representative types of financial shocks. The results yield some general

446

Federal Reserve Bank of Richmond Economic Quarterly

Figure 6 Percentage Difference in Consumption by Age at Shock:
Case 2
Interest Rate Shock

Ratio of Difference in Consumption (

~ −
c−c
______
− )
c

0.00

-0.01

-0.02

-0.03

-0.04

-0.05

-0.06
20

25

30

35

40

45

50

55

60

65

Age at Shock

insights about such shocks and their costs. Comparing across households of
various ages, shocks that are proportional to net worth are costliest to the
oldest households for which the proportional shock translates into the largest
absolute drop in net worth. Interest rate shocks, in the form of an unanticipated
tightening of the credit market, are much less costly.
As mentioned at the outset, this article is stylized along several dimensions, and thus represents only a small first step in the important task of
assessing the power of financial shocks to compromise household well-being.
In particular, the model abstracts from portfolio choice and focuses instead
on a simple scalar measure of net worth. This in turn prevents us from fully
analyzing particular kinds of financial decisions, such as house purchases or
any other leveraged purchase of risky assets, which can greatly change gross
financial positions and periodic payment obligations while leaving net worth
essentially unchanged. The model also abstracts from the labor supply decision, which could mitigate the cost of the shocks. Finally, the shocks that the
household faces are completely unanticipated, something that is likely not as
stark in reality. Households may well be aware of the existence of the kinds
of shocks we analyze in this article, but incorrect about or unable to assess

Athreya and Neelakantan: Unanticipated Household Financial Shocks 447

Figure 7 Present Value of Difference in Consumption: Case 2
Interest Rate Shock
0
PV of Difference in Constant Consumption Streams at Birth ($)

PV of Difference in Constant Consumption Streams at Date k ($)

Interest Rate Shock
0

-2,000

-4,000

-6,000

-8,000

-10,000

-12,000

-14,000

-2,000

-4,000

-6,000

-8,000

-10,000

-12,000

-14,000
20

30

40

50

60

70

20

30

Age at Shock

40

50

60

70

Age at Shock

“true” probabilities for such events. In ongoing work (Athreya, Ionescu, and
Neelakantan 2011), we consider these issues in a richer model of household
portfolio choice.

APPENDIX
Solving for c and c
¯
˜
J +25

V (j, x k , zj , uj ) =
j =k

β j −k

c1−α
¯
.
1−α

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Federal Reserve Bank of Richmond Economic Quarterly

First, we change the index of summation. Let i = j − k. Then
J +25−k

V (j, x k , zj , uj ) =

βi
i=0

c1−α
¯
.
1−α

We use the following rule for the sum of a finite series:
n

ai =
i=0

1 − a n+1
,
1−a

to obtain
V (j, x k , zj , uj ) =

c1−α 1 − β J +25−k+1
¯
.
1−α
1−β

Let
k

=

1 − β J +25−k+1
.
1−β

Then
V (j, x k , zj , uj ) =

c1−α
¯
1−α

k.

Solving for c yields
¯
c=
¯

V (j, x k , zj , uj )

(1 − α)

1
1−α

.

(16)

.

(17)

k

Similarly,
c=
˜

V (j, xk , zj , uj )

(1 − α)

1
1−α

k

Finding the Present Value of c − c
¯ ˜
To allow us to easily compare the cost of shocks at various ages, we now
compute two types of present values. First, we begin by discounting to age 0,
not just back the date at which the shock occurred (date k). The present value
at date zero of a shock occurring at date k, given that the constant consumption
equivalents are c and c, is
¯
˜
c−c
¯ ˜
c−c
¯ ˜
c−c
¯ ˜
+
+ ... +
,
(1 + r)k
(1 + r)k+1
(1 + r)J +25
where r is the interest rate on savings. To be clear, notice that the first dis1
counting term (1+r)k shows that we are discounting to age 0 events that begin
at age k.
1
ˆ
Let 1+r = β. Next, we’ll use the known formula for the finite sum of a
geometric series. We want the sum from age k to death (age J + 25). We
P V0 (k) =

Athreya and Neelakantan: Unanticipated Household Financial Shocks 449

therefore first take the series from 0 to J + 25,
this the sum going from 0 to k − 1,

J +25+1

ˆ
1−β
ˆ
1−β

and subtract from

k

ˆ
1−β
ˆ
1−β

.

ˆ
ˆ J +25+1 1 − β k
1−β
−
¯ ˜
P V0 (k) = (c − c)
ˆ
ˆ
1−β
1−β
ˆ k ˆ J +25+1
β −β
= (c − c)
¯ ˜
ˆ
1−β
ˆ J +25−k+1
1−β
ˆk
= (c − c)
¯ ˜
β
ˆ
1−β
= (c − c)
¯ ˜

diff at birth ,

letting
diff at birth

=

ˆ J +25−k+1
1−β
ˆk
β ,
ˆ
1−β

we have
¯ ˜
P V0 (k) = (c − c)

diff at birth .

Similarly, the present value of a shock occurring at date k, discounted
back to date k is
¯ ˜
P Vk (k) = (c − c)

diff at k ,

where we define
diff at k

=

ˆ
1−β

J +25−k+1

ˆ
1−β

.

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