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Economic Quarterly— Volume 99, Number 1— First Quarter 2013— Pages 1–23

Why Labor Force
Participation (Usually)
Increases when
Unemployment Declines
Andreas Hornstein

uring the Great Recession, the unemployment rate increased
rapidly within two years from about 4 percent in 2007 to about
10 percent in 2009. Yet over the ensuing recovery, the unemployment rate has declined only gradually and, more than four years
after the end of the recession, it now stands at about 7 percent. At
the same time, the labor force participation rate has declined steadily
over this time period and now stands at about 63 percent, a level comparable to the early 1980s. Many observers view the decline in the
labor force participation rate as an indication that further declines in
the unemployment rate will come only slowly. The expectation is that
if the labor market improves, many participants who have left the labor market will return and contribute to the pool of unemployed, and
many unemployed participants will no longer exit the labor force but
continue to search for work.1
Past business cycles have indeed been characterized by a negative
correlation between the unemployment rate and the labor force participation (LFP) rate, that is, as the unemployment rate declines, the
LFP rate increases. In this article we use observations on gross ‡ows
This is a revised version of an article previously titled “The Cyclicality of the
Labor Force Participation Rate.” I would like to thank Marianna Kudlyak, John
Muth, Felipe Schwartzman, and Alex Wolman for helpful comments. Any opinions expressed are those of the author and do not necessarily re‡ect those of
the Federal Reserve Bank of Richmond or the Federal Reserve System. E-mail:
andreas.hornstein@rich.frb.org.
1

For example, see Daly et al.
Tankersley (2013).

(2012), Hatzius (2012), Davidson (2013), or

Federal Reserve Bank of Richmond Economic Quarterly

between labor market states to provide a more detailed analysis of why
the unemployment rate and the LFP rate are negatively correlated over
the business cycle. For our analysis, the total potential workforce is decomposed into three groups: the employed (E), the unemployed (U),
and the out-of-the-labor-force group, or inactive (I) for short. The LFP
rate is the share of employed and unemployed in the potential workforce, and the unemployment rate is the share of the unemployed in the
labor force. We think of labor market participants as transitioning between these three states. Figure 1 provides a stylized representation of
these transitions. The arrows connecting the circles represent the gross
‡ows between the three labor market states. For our analysis we look
at a gross ‡ow as the product of two terms: the total number of participants that could potentially make a transition and the rate at which
the participants make the transition. For example, the total number
of unemployed who become employed is the product of the number of
unemployed and the probability at which an unemployed worker will
become employed. The transition probabilities re‡ect the opportunities faced and choices made by labor market participants. For example,
the probability of an unemployed worker becoming employed depends,
among other things, on the number of available jobs (vacancies) and
the search e¤ort while unemployed. Given the size of the potential
workforce, the transition rates between labor market states determine
the LFP rate and the unemployment rate.
We have marked three groups among the transitions in Figure 1:
EU, IU, and IE. The …rst group involves transitions within the labor
force, between employment and unemployment, and these transitions
have been the focus of much recent research on the determination of
the unemployment rate.2 The working assumption of this research has
been that, for an analysis of the unemployment rate, a …xed LFP rate is
a reasonable …rst approximation. The second and third group involve
transitions between the labor force and out-of-the-labor-force, that is,
they potentially generate changes of the LFP rate. The second group,
which involves transitions between inactivity and unemployment, is at
the heart of the above mentioned concern that further reductions in the
unemployment rate will come only slowly. This concern is based on the
assumption that, as the labor market improves, unemployed workers
become less likely to exit the labor force and inactive workers become
more likely to join the labor force as unemployed; we call this the IU
hypothesis.
2

For example, see Shimer (2012) and other research mentioned below.

A. Hornstein: Unemployment and Labor Force Participation

Figure 1 Labor Market State Transitions

In this article we argue that observations on transition probabilities
obtained from gross ‡ow data are inconsistent with the IU hypothesis.
In fact, the opposite is true: As the labor market improves, unemployed
workers become more likely to exit the labor force and inactive workers
become less likely to join the labor force as unemployed. This pattern
for IU transitions would result in a positive correlation between the unemployment rate and the LFP rate. The observed negative correlation
between unemployment and LFP must then result from patterns in the
EU and IE group transition rates. We calculate the contributions of
cylical variations in the transition rates for the three groups— IU, IE,
and EU— and indeed …nd that the variations in the IE and EU group
transition rates generate a negative co-movement of the unemployment
and LFP rates that dominates the positive co-movement generated by
the IU group transition rates. This suggests that an increasing LFP
rate is more the by-product of an improving labor market rather than
a brake on the declining unemployment rate.
This article is based on a line of research that accounts for changes
in labor market ratios through changes in the rates at which labor market participants transition between labor market states. Early work in
this literature mostly ignored variations in the LFP rate and focused
on variations in transition rates between the two labor market states—
employment and unemployment— for example, Elsby, Michaels, and

Federal Reserve Bank of Richmond Economic Quarterly

Solon (2009), Fujita and Ramey (2009), and Shimer (2012). This work
…nds that variations in unemployment exit rates contribute relatively
more to unemployment rate volatility than do variations in employment exit rates. Recently, a similar approach has been applied to a
more general accounting framework that adds a third labor market
state, out-of-the-labor-force, and allows for variations in the LFP rate,
for example, Barnichon and Figura (2010) and Elsby, Hobijn, and Şahin
(2013).3 Our work is closest to Elsby, Hobijn, and Şahin (2013), but
their main focus is on accounting for the relative contributions of transition rate volatility to unemployment rate volatility.4 Nevertheless,
they also point out that the cyclical behavior of measured transition
rates between unemployment and inactivity is at odds with common
preconceptions about that behavior, and they also note that the observed cyclical behavior of these transition rates would induce a positive
correlation between the unemployment rate and the LFP rate.
The article is organized as follows. Section 1 documents the negative correlation between the detrended unemployment rate and LFP
rate for the total working age population, and men and women separately. Section 2 documents the co-movements between the unemployment rate and transition probabilities between labor market states.
Section 3 demonstrates how variations in transition rates contribute
to the co-movement of the unemployment rate and the LFP rate. In
conclusion, Section 4 speculates on the implications of the recent “unusual” co-movement of unemployment and LFP in the recovery since
2010.

UNEMPLOYMENT AND LFP

The U.S. Bureau of Labor Statistics (BLS) publishes monthly data
on the labor market status of U.S. households that are based on the
Current Population Survey (CPS). The CPS surveys about 60,000
households every month with about 110,000 household members, a
representative sample of the U.S. working age population. Household
respondents are asked if the household members are employed, and if
3
Shimer (2012) also develops tools for the analysis of a multi-state labor market
model and studies the role of variations in the LFP rate, but the focus of the article
is on the two-state model of the labor market.
4
An important part of Elsby, Hobijn, and Şahin (2013) is their analysis of a measurement issue for gross ‡ows. Since gross ‡ows are derived from survey samples, it
is always possible that survey respondents are misclassi…ed with respect to their labor
market state. Past research has demonstrated that misclassi…cation is a signi…cant issue.
Elsby, Hobijn, and Şahin (2013) argue that allowing for the possibility of misclassi…cation does not substantially a¤ect the conclusions drawn from measured gross ‡ows for
the issue studied in this article.

A. Hornstein: Unemployment and Labor Force Participation

they are not employed, whether they want to work and are actively
looking for work. The latter are considered to be unemployed, and employed and unemployed household members constitute the labor force.
Household members that are not employed and that are not actively
looking for work are considered to be not part of the labor force, or
inactive for short. The unemployment rate is the share of unemployed
workers in the labor force, and the LFP rate is the share of the labor
force in the working age population.5
The unemployment rate tends to be more volatile than the LFP
rate in the short run, but changes in the LFP rate tend to be more persistent over the long run. Figure 2, panels A and B, display quarterly
averages of monthly unemployment and LFP rates for the period from
1948 to 2012. The unemployment rate increases sharply in a recession,
and then declines gradually during the recovery. Shaded areas in Figure 2 indicate periods when the unemployment rate is increasing, and
these periods match periods of National Bureau of Economic Research
(NBER) recessions quite well.6 Even though the average unemployment rate appears to be somewhat higher than usual in the 1970s,
considering the magnitude of short-run ‡uctuations in the unemployment rate, the average unemployment rate does not change much over
subsamples of the period. The 2007–09 Great Recession stands apart
by the magnitude of the increase of the unemployment rate and the
rather slow decline of the unemployment rate from its peak.
The LFP rate does not display much short-run volatility, rather it
is dominated by long-run demographic trends. Starting in the mid1960s, the LFP rate increased gradually from values slightly below 60
percent to reach a peak of 67 percent in 2000. This slow but persistent
increase of the LFP rate can be accounted for by the increasing LFP
rate of women and early on by the baby boomer generation entering the
labor force. Since 2000, the LFP rate has declined, …rst gradually, then
at an accelerated rate since the Great Recession and is now at about
63 percent. The gradual decline in the LFP rate can be attributed to
the aging of the baby boomer generation and declining LFP rates for
women and the young (less than 25 years of age).7 In general, there is
not much short-run volatility in the LFP rate, the recent accelerated
5

Households are asked about other features of their labor market status, but the
questions about employment and active search for work when not employed are the main
questions of interest for determining the unemployment rate and the LFP rate. For a
detailed description of the survey and the methods used, see Bureau of Labor Statistics
(2012).
6
The business cycle dates provided by the NBER are a widely accepted measure
of the peaks and troughs of U.S. economic activity.
7
For example, see Aaronson et al. (2006).

Federal Reserve Bank of Richmond Economic Quarterly

Figure 2 Unemployment and Labor Force Participation,
1948{2013

Notes: The unemployment and LFP rates displayed in panels A and B are
quarterly averages of monthly values. Shaded (white) areas are periods when the
unemployment rate is increasing (declining). The dashed lines are the trend calculated using a Baxter and King (1999) bandpass …lter series with periodicity more
than 12 years for the trend. Panel C displays the di¤erence between actual and
trend values of the unemployment rate and the LFP rate.

decline following the Great Recession being the exception. This accelerated decline in the LFP rate after the Great Recession shows up in
the declining LFP rates of mature workers between 25 and 55 years
of age, especially men, and also in declining participation rates of the
young.

Sample

1952:Q1–2007:Q4
1952:Q1–1991:Q4
1992:Q1–2007:Q4
1992:Q1–2013:Q1

0.89
0.93
0.79
0.98

0.29
0.31
0.21
0.33

0.09
0.09
0.08
0.08

0.20
0.19
0.21
0.07

1952:Q1–2007:Q4
1952:Q1–1991:Q4
1992:Q1–2007:Q4
1992:Q1–2013:Q1

1.01
1.04
0.92
1.19

0.28
0.28
0.27
0.41

0.03
0.09
0.14
0.07

0.18
0.22
0.03
0.09

1952:Q1–2007:Q4
1952:Q1–1991:Q4
1992:Q1–2007:Q4
1992:Q1–2013:Q1

0.77
0.81
0.65
0.77

0.36
0.40
0.23
0.32

0.16
0.13
0.26
0.07

0.22
0.20
0.30
0.04

2
Total
0.30
0.29
0.39
0.24
Men
0.30
0.34
0.27
0.27
Women
0.28
0.25
0.38
0.17

Corr(u(t),l(t + s) ) for s=
1
0
1

0.38
0.37
0.55
0.41

0.45
0.43
0.65
0.53

0.52
0.49
0.71
0.63

0.55
0.53
0.70
0.70

0.54
0.51
0.69
0.75

0.48
0.44
0.68
0.75

0.39
0.41
0.48
0.45

0.45
0.46
0.61
0.57

0.52
0.52
0.70
0.67

0.55
0.55
0.74
0.73

0.55
0.53
0.77
0.78

0.48
0.44
0.77
0.78

0.34
0.32
0.43
0.29

0.37
0.35
0.43
0.39

0.42
0.41
0.46
0.49

0.45
0.45
0.38
0.58

0.43
0.43
0.35
0.63

0.37
0.36
0.31
0.64

Notes: Standard deviations and cross-correlations of detrended unemployment, u, and labor force participation rate,
l, for total, men, and women. The trend for each variable is calculated as a Baxter and King (1999) bandpass …lter
with periodicity more than 12 years for monthly data, from January 1948 to March 2013. Unemployment and LFP
rate are in percent, and detrended values are the di¤erence between actual values and trend. Statistics are calculated
for quarterly averages of monthly data for the indicated subsamples.

A. Hornstein: Unemployment and Labor Force Participation

Table 1 Cyclicality of Unemployment and Labor Force
Participation

Federal Reserve Bank of Richmond Economic Quarterly

The average unemployment rate in the 1960s, when the LFP rate
was low, does not appear to be much di¤erent from the average unemployment rate in the 1990s when the LFP rate was high. In other
words, the unemployment rate and the LFP rate do not appear to be
correlated over the long run. Over the short run, the unemployment
rate and the LFP rate are, however, negatively correlated, that is, the
LFP rate increases as the unemployment rate declines.
We de…ne short-run movements of the unemployment rate and the
LFP rate as deviations from trend, and we de…ne the trend of a time
series as a smooth line drawn through the actual time series. To be
precise, we construct the trend using a bandpass …lter that extracts
movements with a periodicity of more than 12 years.8 The dashed
lines in Figure 2, panels A and B, display the trends for the unemployment rate and the LFP rate.9 In panel C of Figure 2 we display the
deviations from trend, that is, the di¤erence between the actual and
trend values, for the LFP rate and the unemployment rate. Clearly,
deviations from trend are more volatile for the unemployment rate than
for the LFP rate. Furthermore, the LFP rate tends to be above trend
whenever the unemployment rate is below trend and vice versa. In
Table 1 we display the standard deviations and cross-correlations between the detrended unemployment rate and the LFP rate for the total
working age population, and for men and women separately.
The unemployment rate is about three times as volatile as the LFP
rate, and the LFP rate increases as the unemployment rate declines,
with the LFP rate lagging about half a year.10 When we split the
sample in the early 1990s, we can see that both the unemployment
rate and the LFP rate are less volatile since the 1990s, but they remain negatively correlated.11 Including the Great Recession and its
8
We use the method of Baxter and King (1999) to construct the trend. This is just
one of several alternative methods to calculate trends. The results do not di¤er much if
instead we use a Hodrick and Prescott (1997) …lter, or a random walk bandpass …lter
as described in Christiano and Fitzgerald (2003).
9
At the beginning and end of the sample, our procedure delivers an ill-de…ned measure of the trend. Essentially, the trend of a series is a symmetric moving average of
the series. Thus, at the beginning and end of the sample, we do not have enough data
points to calculate the trend. For these truncated periods we simply choose to truncate
the moving average …lter and reweigh the available data points. This procedure is arbitrary, and it implies that current data points receive much more weight in determining
the trend, which explains the high trend value for the unemployment rate in 2012. For
the statistical analysis below we therefore discard some observations at the beginning
and end of sample, and start the sample in 1952:Q1 and end the sample in 2007:Q4.
10
We de…ne the length of the lead/lag by the correlation that is largest in absolute
value.
11
This is consistent with the period being part of the “Great Moderation” in the
United States, which indicates an economy-wide decline in volatility starting in the mid1980s. We choose to split the sample in 1992 because in the next section we study
how changes in labor market transition rates contribute to the co-movement of the

A. Hornstein: Unemployment and Labor Force Participation

aftermath signi…cantly increases the measured volatility of the unemployment rate and LFP rate, but, again, it does not much a¤ect the
measured negative correlation between the two variables.12 Finally,
the cyclical co-movement between unemployment and LFP is similar
for men and women, but the unemployment rate is relatively more
volatile for men, the LFP rate is relatively more volatile for women,
and the LFP rate is lagging the unemployment rate more for men than
for women.
We now study if this negative correlation between the unemployment rate and the LFP rate can be accounted for by inactive workers
becoming more likely to enter the labor force and unemployed workers
becoming less likely to exit the labor force.

TRANSITIONS BETWEEN LABOR
MARKET STATES

The CPS household survey not only contains information on how many
people are employed, unemployed, and inactive in any month, but it
also contains information on how many people switch labor market
states from one month to the next. We can use these gross ‡ows between labor market states to calculate the probabilities that any one
household member will, within a month, transition from one labor market state to a di¤erent state. This information can be used to see if, for
example, variations in the transition rates between inactivity and unemployment are consistent with the usual interpretation of the negative
co-movement of the unemployment rate and the LFP rate.
Households are surveyed repeatedly in the CPS. In particular, the
survey consists of a rotation sample, that is, once a household enters
the sample it is surveyed for four consecutive months, then it leaves
the sample for eight months, after which it reenters the sample and is
once more surveyed for four consecutive months. Thus, in any month,
for three-fourths of the household members in the sample, we potentially have observations on their current labor market state and their
state in the previous month. We can use this information to calculate
the gross ‡ows between labor market states from one month to the
unemployment rate and the LFP rate. We calculate transition rates from data on gross
‡ows for the period after 1990, and again we discard some of the beginning and end
of sample data on deviations from trend to minimize the problems arising from an illde…ned trend.
12
Related to the discussion in footnote 9, we should note that if the unemployment rate continues to decline, then future measures of the trend unemployment rate
that include these data points will indicate a lower trend unemployment rate than do
our current measures. Thus, our current measure very likely understates the cyclical
deviations from trend for the unemployment rate.

Federal Reserve Bank of Richmond Economic Quarterly

next. The measurement of gross ‡ows su¤ers from two problems, missing data points and misclassi…ed data points. We will use data series
for gross ‡ows that have been adjusted for missing data but not for
misclassi…cation.13
Data points are missing because the actual unit of observation in
the CPS is not a particular household, but the household that is residing at a particular address. Thus, even for those addresses that have
entered the sample in the previous month, we may not have observations on the previous month’s labor market states for the members of
the current resident household. This might happen for various reasons. The household could have a new member who did not live at the
current address in the previous month, for example, a dependent returning to the family household after a longer absence. Alternatively,
the household previously residing at the address moved away and a
new household moved in. About 15 percent of the potential observations cannot be matched across months, and these observations are
not missing at random (Abowd and Zellner 1985). One can use “margin adjustment”procedures to generate gross ‡ow data consistent with
unconditional marginal distributions, and these procedures take into
account the possibility that observations are not missing at random.
In the following, we use the BLS-provided margin adjusted research
series on labor force status ‡ows from the CPS.14
Gross ‡ows from one labor market state to another can be interpreted as the product of two terms: the total number of participants in
the initial state and the probability that any one of these participants
makes the transition from the initial state to another state. For example, more people might make the transition from unemployment to
inactivity because there are more unemployed people, or because each
unemployed worker is more likely to make the transition. In Figure 3
we display the transition probabilities between employment (E), unemployment (U), and inactivity (I) that are implied by the observed gross
‡ows between labor market states for the period from 1990 to 2012. A
panel labeled AB denotes the probability that a participant who is in
labor market state A will transition to state B within a month. For
example, the center panel in the bottom row, labeled IU, denotes the
probability that a participant who is inactive in the current month will
13
The evidence for misclassi…cation in the BLS, that is, that a participant is assigned the wrong labor market state in the survey, has been discussed for a long time,
see, for example, Poterba and Summers (1986). There is currently no generally accepted
procedure to adjust CPS data on labor market states for misclassi…cation. Recently,
Elsby, Hobijn, and Şahin (2013) and Feng and Hu (2013) have worked on possible corrections for misclassi…cation.
14
The research series is available at www.bls.gov/cps/cps_‡ows.htm. Frazis et al.
(2005) describe the BLS procedure used to construct the series.

A. Hornstein: Unemployment and Labor Force Participation

Figure 3 Transition Probabilities, 1990:Q2{2013:Q1

Notes: Panel AB denotes the probability of making the transition from labor market state A to labor market state B. The dashed lines are the trend calculated using a Baxter and King (1999) bandpass …lter series with periodicity more than 12
years for the trend. The probabilities displayed are quarterly averages of monthly
values. Shaded (white) areas are periods when the unemployment rate is increasing (declining).

be unemployed in the next month. Regions that are (not) shaded denote periods when the unemployment rate increases (declines). The
trend for each transition probability is calculated using the same bandpass …lter as in the previous section, and it is displayed as a dashed line
in Figure 3. In Table 2, we display the average transition probabilities,
the standard deviations of the detrended transition probabilities, and
their cross-correlations with the detrended unemployment rate for the
total working age population, and for men and women separately.
An increase in the unemployment rate is associated with more
churning in the labor market: Employed workers are more likely to

Federal Reserve Bank of Richmond Economic Quarterly

Table 2 Cyclicality of Transition Probabilities
pij

4
EU 1.4 0.10
UE 27.5 2.35
IU
2.6 0.21
UI 22.4 1.39
IE
4.9 0.21
EI
2.7 0.09

0.70
0.48
0.36
0.59
0.24
0.02

EU 1.5 0.13
UE 29.0 2.54
IU
3.2 0.30
UI 18.9 1.47
IE
5.7 0.27
EI
2.2 0.07

0.73
0.46
0.47
0.54
0.20
0.03

EU 1.2 0.07
UE 25.8 2.31
IU
2.3 0.18
UI 26.7 1.30
IE
4.5 0.21
EI
3.4 0.14

0.39
0.50
0.21
0.54
0.21
0.03

Corr( u(t); pij (t + s) ) for s=
3
2
1
0
1
2
Total, u = 5:3, u = 0:76
0.83
0.88
0.88
0.85
0.72
0.62
0.64
0.78
0.89
0.95
0.94
0.88
0.49
0.61
0.71
0.79
0.78
0.77
0.68
0.75
0.79
0.77
0.68
0.55
0.35
0.50
0.57
0.65
0.66
0.60
0.02
0.10
0.24
0.32
0.45
0.48
Men, u = 5:4, u = 0:88
0.85
0.89
0.90
0.86
0.73
0.63
0.62
0.76
0.86
0.92
0.91
0.85
0.56
0.66
0.76
0.84
0.79
0.76
0.62
0.70
0.77
0.77
0.71
0.59
0.33
0.45
0.53
0.58
0.62
0.58
0.08
0.09
0.03
0.00
0.16
0.19
Women, u = 5:3; u = 0:63
0.57
0.67
0.68
0.70
0.57
0.48
0.62
0.77
0.86
0.91
0.90
0.84
0.35
0.48
0.60
0.71
0.68
0.69
0.62
0.68
0.68
0.66
0.53
0.40
0.32
0.46
0.48
0.61
0.60
0.53
0.08
0.18
0.34
0.43
0.53
0.54

0.51
0.78
0.75
0.36
0.55
0.45

0.42
0.65
0.70
0.16
0.45
0.36

0.53
0.77
0.72
0.41
0.50
0.23

0.43
0.65
0.68
0.17
0.43
0.20

0.40
0.73
0.68
0.22
0.51
0.47

0.34
0.59
0.61
0.08
0.39
0.36

Notes: The …rst column lists the sample average for transition probabilities from
labor market state i to j, pij , with labor market states being employed (E), unemployed (U), and out-of-the-labor-force/inactive (I). The second column lists standard deviations of detrended transition probabilities, and the remaining columns
list cross-correlations of detrended transition probabilities with the detrended unemployment rate. The trend for each variable is calculated as a Baxter and King
(1999) bandpass …lter with periodicity of more than 12 years for monthly data,
from January 1990 to March 2013. Transition probabilities and the unemployment rate are in percent, and detrended values are the di¤erence between actual
and trend values. Statistics are calculated for quarterly averages of monthly data
for the sample 1992:Q1 to 2007:Q4.

lose their jobs, and unemployed workers are less likely to return to
work, with job loss (…nding) rates slightly leading (lagging) the unemployment rate; see the panels labeled EU and UE in Figure 3 and
the corresponding correlations in Table 2.15 Considering the magnitude and volatility of the job …nding rate for unemployed workers, the
transition rate UE, it is apparent that variations in this rate are a
15
In fact, when unemployment is high, gross ‡ows between unemployment and employment are both high. Despite the lower probability of the unemployed …nding employment, gross ‡ows from unemployment to employment are high because there are
more unemployed.

A. Hornstein: Unemployment and Labor Force Participation

major source of unemployment volatility. Looking at panels IU and
UI, we can see that as the unemployment rate declines, it becomes
more likely that an unemployed worker exits the labor force and less
likely that an inactive worker joins the labor force as unemployed. This
pattern is con…rmed by the cross-correlations for the detrended rates
in Table 2. Thus, the cyclical pattern of the transition rates between
inactivity and unemployment is exactly the opposite of what the IU
hypothesis proposes as an explanation of the negative correlation between the LFP rate and the unemployment rate. However, the transition probabilities between inactivity and employment do have a cyclical
pattern that supports a negative co-movement between the unemployment rate and the LFP rate. As the unemployment rate increases it
becomes less likely that people make the transition from inactivity to
employment. It also becomes less likely that employed workers leave
the labor force, but this probability is always quite low and it is not
very volatile over the cycle. The cyclical properties of the transition
probabilities for all three groups, EU, IU, and IE, are roughly the same
for men and women. The only exception is that transition probabilities
for women tend to be somewhat less volatile overall, and that men’s
transition probabilities from employment to inactivity appear to be
acyclical.
So far we have shown that the direct evidence on labor market
transitions does not support the IU hypothesis of why the LFP rate
increases as the unemployment rate declines. In particular, as the labor
market improves and the unemployment rate declines, participants become less likely to make the transition from inactivity to unemployment
and they become more likely to make the transition from unemployment to inactivity. So what accounts for the negative correlation of
unemployment and the LFP rate?

SOURCES OF CO-MOVEMENT

Recent research on labor markets using the stock-‡ow approach points
to the importance of variations in the job …nding rate and job loss rate
for the determination of the unemployment rate. We now argue that
variations in the job …nding and job loss rates are also important for
the cyclical co-movement between the unemployment and LFP rates.
As a …rst step, note that the exit rate from the labor force is an order
of magnitude smaller for employed workers than it is for unemployed
workers (see Table 2). This means that as the unemployment rate
declines, the average exit rate from the labor force declines, and the
LFP rate increases. Furthermore, as we have just seen, when the unemployment rate declines, more people join the labor force without

Federal Reserve Bank of Richmond Economic Quarterly

Figure 4 Counterfactuals for Unemployment Rate and LFP
Rate

an intervening unemployment spell. This suggests that cyclical movements of the transition rates in the UE and IE group account for the
negative co-movement of unemployment and LFP over the business cycle. We now formalize this argument by constructing counterfactuals
for the unemployment rate and the LFP rate.
Consider the trend paths for the transition probabilities that we
have calculated for Figure 3 and Table 2. We can interpret the deviations of the unemployment rate and the LFP rate from their respective
trends as arising from deviations of the transition probabilities from
their respective trends. In the Appendix, we describe a procedure that
allows us to decompose the cyclical movements of the unemployment
and LFP rates into parts that originate from the cyclical movements of
the various transition probabilities.16 In Figure 4, we graph the contributions to trend deviations of the unemployment rate and LFP rate
(black lines) coming from variations in the transition probabilities between (1) employment and unemployment (red lines), (2) inactivity and
unemployment (blue lines), and (3) inactivity and employment (green
16

The procedure used to derive the contributions coming from variations in monthto-month transition probabilities is actually based on a model that allows for continuous
transitions between labor market states in between the monthly survey dates.

A. Hornstein: Unemployment and Labor Force Participation

Table 3 Cross-Correlations between Unemployment Rate
and LFP Rate for Counterfactuals, Deviations from
Trend, 1992:Q1{2007:Q4

UE and EU
IU and UI
UE, EU, UI,
and IU
IE and EI
Actual

4
0.20
0.15

3
0.40
0.31

Corr( u(t), l(t+s) ) for s=
2
1
0
1
2
0.58
0.74
0.87
0.95
0.99
0.48
0.64
0.82
0.89
0.92

3
0.97
0.90

4
0.91
0.84

0.41
0.33
0.10

0.37
0.50
0.22

0.32
0.66
0.40

0.02
0.55
0.69

0.07
0.43
0.68

0.24
0.86
0.55

0.23
0.99
0.65

0.13
0.83
0.71

0.04
0.65
0.70

Notes: Cross-correlations of trend deviations for the unemployment rate, u, and
the LFP rate, l. The …rst four rows represent counterfactuals for u and l, and
the last row represents actual values for u and l. For a counterfactual all monthly
transition rates, except for the ones listed in the counterfactual column, are kept
at their trend values. Statistics are calculated for quarterly averages of counterfactual monthly time series. Detrended unemployment rate and LFP rate are level
deviations from trend.

lines).17 These are the three counterfactuals for the trend deviations of
the unemployment rate and LFP rate, and they approximately add up
to the overall trend deviation of the two rates. In Table 3, we calculate
the cross-correlations between the counterfactual unemployment and
LFP rates implied by these experiments.
Past research has shown that variations in the transition probabilities between employment and unemployment are a major determinant
of the unemployment rate, e.g., Shimer (2012) or Elsby, Hobijn, and
Şahin (2013). This observation is con…rmed by Figure 4, panel A, in
that variations in these probabilities account for a substantial part of
the unemployment rate variation. Figure 4, panel B, demonstrates
that these variations also introduce substantial volatility into the LFP
rate. In fact, the counterfactual LFP rate is more volatile than the
actual LFP rate. Furthermore, variations in the transition probabilities between employment and unemployment generate a strong negative
17
Since our trend is a symmetric moving average …lter, we face a problem at the
beginning and end of our sample period (see footnote 9). If for this part of the sample
the deviations from a presumed trend are very large, such as is the case for the years
2007–12, then this problem is even more pronounced and our adjustment to the …lter
will understate deviations from trend. For this reason, we replace the calculated trend
values from 2008 on with the trend values in the fourth quarter of 2007. This essentially
keeps the trend unemployment rate …xed at 6.2 percent and the trend LFP rate …xed
at 65.5 percent from 2008 on. Thus, our procedure is likely to overstate deviations from
trend from 2008 on, especially for the LFP rate.

Federal Reserve Bank of Richmond Economic Quarterly

co-movement between the unemployment rate and the LFP rate (Table
3, …rst row).
The co-movement of the actual unemployment rate, with the transition probabilities between inactivity and unemployment, is such that
people are more likely to join the labor force as unemployed and less
likely to exit the labor force from unemployment when the unemployment rate is high. Thus, these movements simultaneously increase the
unemployment rate and the LFP rate. In other words, the observed
variations in transition probabilities between inactivity and unemployment contribute to the volatility of the unemployment rate, and they
introduce a positive co-movement between the unemployment rate and
the LFP rate (see the blue lines in Figure 4 and the second row in Table
3).
For the LFP rate, the variations of transition probabilities between
employment and unemployment on the one hand, and between inactivity and unemployment on the other hand, tend to almost o¤set each
other. This means that the joint e¤ect of the variations in these transition probabilities is a weak positive correlation between the unemployment rate and the LFP rate (see the third row of Table 3). The
stronger negative actual correlation between the unemployment rate
and the LFP rate is then determined by the pattern of transition probabilities between inactivity and employment. As the unemployment
rate increases, the probability of making a direct transition from inactivity to employment and vice versa declines. The e¤ect of the reduced
transition rate from inactivity tends to dominate, and the LFP rate
declines. Adding this feature is enough to generate a signi…cant negative correlation between the unemployment rate and the LFP rate (last
row of Table 3).
We can interpret these results using a simpli…ed version of the
dynamics between labor market states described in the Appendix. Suppose that participants make the transition from labor market state i to
labor market state j at rate j . The transition rates between employment and unemployment are EU and U E , and the transition rates between unemployment and inactivity are U I and IU . Assume also that
participants can make the transition between in- and out-of-the-laborforce only by going through unemployment, that is, there are no direct
transitions between employment and inactivity, EI = IE = 0.18 For
…xed transition rates, the unemployment rate and LFP rate converge
18
In part, we can look at this as the limiting case for the observation that U I
.
EI It is, however, also true that transitions from inactivity to employment are actually
more likely than transitions from inactivity to unemployment, IE > IU .

A. Hornstein: Unemployment and Labor Force Participation

to their steady-state values, u respectively l ,
EU

u =
EU

and l = 1 +
UE

In the data, monthly unemployment and LFP rates tend to be close to
the steady-state values implied by their monthly transition rates.
This special case illustrates three points. First, the unemployment
rate would be independent of transitions between the labor force and
inactivity, if it was not for transitions between inactivity and employment. Similar to a simple two-state model of the labor market that
ignores variations in the LFP rate, the unemployment rate would be
determined by the transition rates between employment and unemployment. Second, even with an unemployment rate that is “exogenous”
to the LFP rate, the LFP rate does depend on the unemployment rate
and transition rates between unemployment and inactivity. In particular, a lower unemployment rate implies a higher LFP rate, which helps
generate the observed negative correlation between the unemployment
rate and the LFP rate. Third, the observed cyclical movements in the
transition rates between unemployment and inactivity imply that the
ratio of U I to IU is decreasing as the unemployment rate u increases,
thereby introducing a positive correlation between the unemployment
rate and the LFP rate and dampening the co-movement. Thus, transitions between employment and inactivity have to be considered if
one wants to account for the co-movement between unemployment and
LFP.

CONCLUSION

Many observers of the U.S. labor market perceive the LFP rate to be
below its long-run trend and the unemployment rate to be above its
long-run trend. In fact, the low cyclical LFP rate is seen as keeping
the cyclical unemployment rate from being even higher, because poor
employment prospects have induced discouraged unemployed workers
to leave the labor force and have prevented marginally attached inactive participants from a return to the job search. In this article,
we have documented that direct observations on transition rates between unemployment and out-of-the-labor-force are inconsistent with
this perception. It turns out that at times of high unemployment,
unemployed workers are less likely to exit the labor force and inactive
workers are more likely to return to the labor force as unemployed. This
pattern would have introduced a positive correlation between cyclical
movements of the unemployment rate and the LFP rate. Yet we have
observed a negative correlation between the two rates. We have shown

Federal Reserve Bank of Richmond Economic Quarterly

that the negative co-movement is induced by movements in the unemployment rate itself, and by a procyclical transition rate from inactivity
to employment without an intervening unemployment spell. To summarize, a low cyclical LFP rate to some extent simply seems to re‡ect
a high current unemployment rate rather than to indicate an elevated
future unemployment rate.
We have just described the “usual” co-movements between labor
market transition rates, the unemployment rate, and the LFP rate
over the business cycle. Since 2010, the unemployment rate has been
declining gradually, and if we had observed the usual co-movement
pattern, we should have seen the LFP rate increasing with at most
a one-year lag, say, starting in 2011. We have not seen that. The
LFP rate has been on a long-run declining trend since 2000, with an
acceleration of that decline during the Great Recession. It is generally
agreed that part of the decline in the LFP rate since 2000 re‡ects a
demographic change that will persist over time. Current forecasts call
for a further decline of the LFP rate over the next 10 years (see, for
example, Toossi [2012]). But it is also argued that the more recent
decline in the LFP rate re‡ects temporary cyclical e¤ects that will be
reversed over time (see, for example, Erceg and Levin [2013]). The
recent “unusual” co-movement between the unemployment rate and
LFP rate does speak to this issue. In particular, the recent observations
on co-movement would appear to be less unusual if we were to attribute
more of the decline in the LFP rate to a change in its long-run trend
than to short-run cyclical e¤ects.
This interpretation has implications for the medium-run forecast
for gross domestic product (GDP). A falling LFP rate will dampen any
increase in employment and corresponding increase in per capita GDP,
even as the unemployment rate continues to decline. Thus, whereas
the increasing trend for the LFP rate contributed to per capita GDP
growth before 2000, the declining trend from 2000 will reduce the trend
growth rate of per capita GDP. Depending how much the LFP rate is
currently below trend, a return to trend might dampen this negative
e¤ect for per capita GDP growth in the near term.

APPENDIX:

SOME MATH

Let fij;t denote the gross ‡ow between labor market state i in period
t 1 and state j in period t, with i; j 2 fE; U; Ig. Disregarding in‡ows
to and out‡ows from the working age population, the total number of

A. Hornstein: Unemployment and Labor Force Participation

people in labor market state i at time t 1 is
X
X
si;t 1 =
fik;t =
fki;t

(1)

The probability that a participant makes the transition from state i in
period t 1 to state j in period t is simply
pij;t = fij;t =si;t

The unemployment rate and LFP rate are
sU;t + sE;t
sU;t
and lt =
:
ut =
sU;t + sE;t
sU;t + sE;t + sI;t

(2)

(3)

Conditional on initial values for the stocks, si0 , we can obtain the
sequence of future stocks from the sequence of transition probabilities
by iterating on the equation
X
si;t =
pji;t sj;t 1 :
(4)
j

This de…nes a mapping from the sequence of transition probabilities,
p, to the sequence of stocks, s,
s = G (p; s0 ) ;

(5)

conditional on initial stocks s0 . Suppose we have a series for the trend
transition probabilities, pTij;t : Then we can use the above mapping to
construct the implied trend values for stocks
sT = G pT ; s0 ;

(6)

and we calculate the implied trend values for the unemployment rate
and LFP rate, uT and lT .
In order to evaluate the contribution of a group of transition probabilities to the overall variation of the unemployment rate and LFP
rate, we simply construct a counterfactual path for the stocks where
we keep all but the probabilities of interest at their trend values and
set the probabilities of interest to their actual values. For example, in
order to evaluate the contribution of variations in the k-th transition
probability, we construct the series
sCF
= G pk ; pT k ; s0
k

(7)

with implied series for the unemployment rate and LFP rate, uCF
and
k
lkCF . The contribution of the k-th probability to unemployment rate
variations is then de…ned as uCF
uT :
k
The actual implementation of the procedure in Section 3 is slightly
more complicated in that we allow for in‡ows and out‡ows to the working age population, and we replace the discrete time month-to-month

Federal Reserve Bank of Richmond Economic Quarterly

transition probabilities with a continuous time process as described in
Shimer (2012).
Modeling labor market transitions as a continuous time process
deals with issues of time aggregation in the data. For example, if the
exit rate from unemployment is relatively high, as it is most of the time,
our estimates of entry probabilities to unemployment from month-tomonth gross ‡ow data might be biased since we are missing the people
who do become re-employed within the month. In fact, the monthto-month transition probabilities between two particular labor market
states, for example, employment and unemployment, will be an amalgam of the continuous time transition rates between all labor market
states. The procedure of Shimer (2012) simply provides a way to recover the continuous time transition rates between labor market states
that give rise to the observed discrete time transition probabilities.
The continuous time representation of labor market transitions also
provides a convenient tool to interpret the role of transitions between
unemployment and inactivity for the path of the unemployment rate
and the LFP rate. The continuous time analog for the discrete time
transition equation for labor market states (4) is given by
s_ E
s_ U
s_ I
1

=
( EU + EI ) sE + U E sU + IE sI
= EU sE ( U E + U I ) sU + IU sI
= EI sE + U I sU ( IE + IU ) sI
= sE + sU + sI ;

(8)

where a dot denotes the time derivative of a variable, ij denotes the
continuous time transition rate from state i to state j, and we have
normalized the size of the working age population to one. For example,
on the one hand, employment declines because employed workers make
the transition to unemployment at the rate EU and exit the labor force
at the rate EI . On the other hand, employment increases because
unemployed workers …nd employment at the rate U E and inactive
participants join the labor force and immediately …nd employment at
the rate IE . Subtracting out‡ows from in‡ows yields the net change
of employment.
The continuous time representation of the monthly transition probabilities assumes that the transition rates remain …xed for a month.
The observed transitions rates between labor market states tend to
be su¢ ciently large such that the steady state of the system (8) for
the given monthly transition rates is a good approximation of the actual stock values. The steady state of the system for …xed transition
rates is an allocation of the population over labor market states such
that in‡ows and out‡ows cancel and the stock values do not change,
s_ = 0. Solving equations (8) for steady-state stocks and the implied

A. Hornstein: Unemployment and Labor Force Participation

steady-state unemployment rate and LFP rate is a bit messy, but it
simpli…es considerably if we assume that transitions between in- and
out-of-the-labor-force have to proceed through unemployment, that is,
EI = IE = 0. For this case we …nd that the steady-state unemployment rate and LFP rate are

u =
EU

and l = 1 +
UE

For this special case, the unemployment rate is independent of transitions between the labor force and inactivity. Similar to a simple twostate model of the labor market that ignores variations in the LFP rate,
the unemployment rate is determined by the transition rates between
employment and unemployment. On the other hand, the LFP rate
does depend on the unemployment rate and transition rates between
unemployment and inactivity. In particular, a lower unemployment
rate implies a higher LFP rate, which helps generate the observed negative correlation between the unemployment rate and the LFP rate.
From Section 2 we have that the transition rates from unemployment
to inactivity (inactivity to unemployment) are negatively (positively)
correlated with the unemployment rate. This would imply that the
LFP rate increases as the unemployment rate increases. Thus, the
movements in the transition rates between in- and out-of-the-laborforce alone would yield a counterfactual positive correlation between
the unemployment rate and the LFP rate.

REFERENCES

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Federal Reserve Bank of Richmond Economic Quarterly

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Economic Quarterly— Volume 99, Number 1— First Quarter 2013— Pages 25–43

A Cohort Model of Labor
Force Participation
Marianna Kudlyak

The aggregate labor force participation (LFP) rate measures the
share of the civilian noninstitutionalized population who are either employed or unemployed (i.e., actively searching for work). From 1963 to
2000, the LFP rate was rising, reaching its peak at 67.1 percent. The
LFP rate has been declining ever since, with the decline accelerating
after 2007. Between December 2007 and December 2012, the LFP rate
declined from 66 percent to 63.6 percent. Prior to 2012, the last year
when the LFP rate was below 65 percent was 1986.
The decline in the LFP rate, which coincided with the Great Recession, raises the question: Is the LFP rate at the end of 2012 close to or
below its long-run trend? The question is important to policymakers
and economists. If a large portion of the workers who are currently out
of the labor force represents workers who are temporarily out of the
labor force, then the unemployment rate by itself might not be a good
measure of the slack in the economy.
In this article, we discuss the change in the aggregate LFP rate
from 2000 to 2012, with an emphasis on the changes in the age-gender
composition of the population and changes in the LFP rates of di¤erent demographic groups. We then estimate a cohort-based model of
the LFP rates of di¤erent age-gender groups and construct the aggregate LFP rate using the model estimates. The model is a parsimonious
version of the model studied in Aaronson et al. (2006). It contains agegender e¤ects, birth-year cohort e¤ects, and the estimated deviations
of employment from its long-run trend as the cyclical indicator. We
The author is grateful, without implicating them in any way, to Bob Hetzel,
Andreas Hornstein, Marios Karabarbounis, Steven Sabol, and Alex Wolman for their
comments. The author thanks Peter Debbaut and Samuel Marshall for excellent research assistance. The views expressed here are those of the author and do not
necessarily re‡ect those of the Federal Reserve Bank of Richmond or the Federal
Reserve System. E-mail: marianna.kudlyak@rich.frb.org.

Federal Reserve Bank of Richmond Economic Quarterly

estimate the model on the 1976–2007 data and then predict the aggregate LFP rate for 2008–12.
We …nd that in 2008–11, the actual LFP rate closely follows the
LFP rate predicted from the model that takes into account the estimated cyclical deviation of employment from its trend. In 2012, the
actual LFP rate is in fact above the estimated value from the model.
The actual LFP rate in 2012 is close to the estimated trend constructed
from the actual age-gender composition of the population and the agegender and cohort e¤ects estimated from the model.
What are the factors behind the LFP rate in 2012 being above
the value predicted from the model with the cyclical indicator? In
the model, we use estimated deviations of employment from its longrun trend as a cyclical indicator. While it is true that the decline in
employment during the Great Recession contributed to lowering labor
force participation in 2008–12, it also appears that other factors during
the 2007–09 recession worked to counteract this e¤ect in 2012. Our
model is silent about these factors. One can speculate that the increase
in the duration of unemployment insurance bene…ts, or the decline in
household wealth (due to the collapse of stock and housing markets),
might have contributed to workers remaining in the labor force at a
larger rate than predicted by the cyclical component of employment.
This article is related to an active debate in the recent academic
and policy circles. The theoretical models are studied in Veracierto
(2008), Krusell et al. (2012), and Shimer (2013). The empirical discussions are provided in Kudlyak, Lubik, and Tompkins (2011); Aaronson,
Davis, and Hu (2012); Daly et al. (2012); Hotchkiss, Pitts, and RiosAvila (2012); Canon, Kudlyak, and Debbaut (2013); and Schweitzer
and Tasci (2013). The cohort model employed in the modeling labor force participation rate was originally proposed by Aaronson et al.
(2006). Fallick and Pingle (2006) and Balleer, Gómez-Salvador, and
Turunen (2009) provide extensions to the model.
The …ndings in the article are consistent with the …ndings in
Aaronson et al. (2006), whose 2006 projection of the LFP rate in
2012 is 63.7 percent, the number that coincides with the actual rate in
2012. Other studies …nd that the LFP rate in 2012 is below its trend
(Aaronson, Davis, and Hu [2012]; Bengali, Daly, and Valletta [2013];
Erceg and Levin [2013]; Hotchkiss and Rios-Avila [2013]).
The rest of the article is structured as follows. The …rst section
reviews the behavior of the aggregate LFP rate during 2000–12 and
presents counterfactual exercises using an age-gender decomposition of
the aggregate LFP rate. Section 2 describes the cohort model and
presents the empirical results. Section 3 concludes.

M. Kudlyak: A Cohort Model of the LFP Rate

Figure 1 LFP Rate and Unemployment Rate

Notes: Quarterly averages of seasonally adjusted (SA) monthly series, January
1949–December 2012. Author’s calculations using series from HAVER.

WHAT COMPONENTS DRIVE THE CHANGES IN
THE AGGREGATE LFP RATE DURING 2000{12?

After reaching its peak of 67.3 percent in the …rst half of 2000, the
aggregate LFP rate declined from 2000 to Q2:2004, stabilized for a
few years, and then started falling again in 2008.1 Figure 1 shows the
aggregate LFP rate and the aggregate unemployment rate.
The aggregate LFP rate can be decomposed into the weighted sum
of the LFP rates of di¤erent demographic groups, i.e.,
X
LF Pt =
sit LF Pti ;
(1)
i

LF Pti

where
is the labor force participation rate of group i, sit is the
P opi
population share of group i, i.e., sit P optt , and P opit is the population
of group i.
1
The data reported in the article are from HAVER (SA), unless stated otherwise.
The last data point at the time of the analysis: December 2012.

Federal Reserve Bank of Richmond Economic Quarterly

Figure 2 Actual and Forecasted Population Shares

Notes: Quarterly averages of monthly series, January 1969–December 2012. Annual series for estimates. All series from HAVER. Dotted lines denote shares
based on the HAVER-estimated forecast of resident; population adjusted by the
author to represent civilian noninstitutionalized population (using 2012).

To understand what forces drove the decline of the LFP rate since
2008, we …rst examine the change in the demographic composition of
the population and the change in the LFP rates of di¤erent age-gender
groups. Figure 2 shows the population shares by age-gender group.

M. Kudlyak: A Cohort Model of the LFP Rate

Figure 3 LFP Rates

Notes: Quarterly averages of monthly series, January 1969–December 2012. All
series from HAVER. Dotted lines denote shares based on the HAVER-estimated
forecast of resident; population adjusted by the author to represent civilian noninstitutionalized population (using 2012). Author’s calculations from the population
and labor force series, SA.

Figure 3 shows the LFP rates of di¤erent age-gender groups. As can
be seen from the …gures, the developments that took place between
Q4:2007 and Q4:2012 are a continuation of the developments that have

Federal Reserve Bank of Richmond Economic Quarterly

been taking place since 2000, when the aggregate LFP rate reached its
peak:2
The composition of the population has been shifting toward older
workers who typically have lower labor force attachment. This is
in part due to the population of baby boomers gradually moving
from the prime working age group with a high LFP rate to older
age groups with lower LFP rates. Also note that the share of
older women is larger than the share of older men, and women
typically have lower labor force attachment than men.
The LFP rate of 25- to 54-year-old workers, a group with the
highest LFP rate, has been declining. From Q4:2007 to Q4:2012,
the rate declined from 82.9 percent to 81.3 percent.
The LFP rate of teenagers and young adults has been declining.
The LFP rate of women has started to decline after increasing
prior to 1999.
The LFP rate of men has continued its decline, which started in
the 1940s.

How Much Change Is Driven by the LFP
Rates of Di erent Demographic Groups?
To understand the importance of the compositional changes and of the
changes in the labor force participation rates of di¤erent demographic
groups, we …rst present counterfactual exercises to quantify the impact
of these changes on the aggregate labor force participation rate.
In the exercises, we keep the LFP rate of speci…c demographic
groups …xed at their Q4:2007 level and allow the LFP rates of all other
groups and the demographic composition of the population to follow
their actual path. We consider four such counterfactual exercises: (1)
…xing the LFP rate of 55+ year-old workers, (2) …xing the LFP rate
of 16- to 24-year-old workers, (3) …xing the LFP rate of women, and
(4) …xing the LFP rate of men. These exercises demonstrate the importance of changes in the LFP rates of di¤erent demographic groups
for changes in the aggregate LFP rate. In our …fth counterfactual exercise, we …x the population shares of age-demographic groups at their
Q4:2007 levels and allow the groups’LFP rates to follow their actual
path. The results of these exercises are shown in Figure 4.
2
See Toossi (2012a, 2012b) for a description of the trends and Canon, Kudlyak,
and Debbaut (2013) for a summary of the Bureau of Labor Statistics projections.

M. Kudlyak: A Cohort Model of the LFP Rate

Figure 4 Labor Force Participation Rate, Actual and
Counterfactual

Notes: Author’s calculations using data from HAVER, quarterly averages of
monthly series.

As can be seen from the …gure, the experiment with holding the
LFP rates of 55–64 and 65+ year-old workers …xed (the dashed blue
line) delivers the largest discrepancy between the actual aggregate LFP
(the solid black line) and the counterfactual one. Since the LFP rate
of older workers has increased, the counterfactual rate lies below the
actual LFP rate, and in Q4:2012 stands at 61.7 percent.
The second largest discrepancy (in absolute value) between the actual aggregate LFP and the counterfactual one is obtained from holding
the population shares …xed at their 2007 levels (the dashed red line).
In this case, the counterfactual LFP rate exceeds the actual one and
stands at almost 65 percent in Q4:2012. We see that between 2007
and 2012 the population composition has shifted toward a composition
with lower labor force attachment.
The results also show that the counterfactual based on the …xed
LFP of 16- to 24-year-old workers (the dashed green line) and the
counterfactual based on the …xed LFP of men (the yellow dashed line)

Federal Reserve Bank of Richmond Economic Quarterly

line up almost perfectly and are both above the actual aggregate LFP
rate.
Finally, the …gure shows that the counterfactual LFP rate based
on the …xed LFP by women (the dashed pink line) has declined more
than the one based on the …xed LFP by men (the dashed yellow line),
while both counterfactuals lie above the actual LFP rate.

An Alternative Decomposition of LFP
As an alternative way of gauging how much of the change in the LFP
rate was driven by the change in the population shares of di¤erent
demographic groups, we perform the following counterfactual. We …x
the LFP rates of 14 age-gender groups at their respective levels at
time t0 and construct the counterfactual LFP rate usingPthe actual
population shares of the respective groups, i.e., LF Ptt0 = i sit LF Pti0 .
In the analysis, we consider the following seven age groups for each
gender: 16–19, 20–24, 25–34, 35–44, 45–54, 55–64, and 65 and older.
The blue lines in Figures 5 and 6 show the counterfactual LFP for t0
equal to Q4:2007 and t0 equal to Q4:2000, respectively.
As can be seen from Figure 6, in Q4:2012, the counterfactual LFP
rate constructed from the groups’ LFP rates …xed at their levels in
Q4:2000 is 65:5 percent, while from 2000 to 2012 the actual LFP rate
declined from 67 percent to 63:6 percent. The counterfactual LFP
rate constructed from the age-gender LFP rates …xed at their levels in
Q4:2007 is 65 percent, while from 2007 to 2012 the actual LFP rate
declined from 66 percent to 63:6 percent (Figure 5). Thus, the results
suggest that the demographic change of the population is associated
with approximately 40 percent of the decline of the aggregate LFP
rate between 2000 and 2012 and 37 percent of the decline between
2007 and 2012.
For such demographic counterfactuals it is important to consider as
…ne a group classi…cation as possible, especially if there are substantial
di¤erences in the LFP rates of workers of di¤erent ages combined into a
group. For example, the red lines in Figures 5 and 6 show the counterfactual LFP rate when we consider only six age groups for each gender
(16–19, 20–24, 25–34, 35–44, 45–54, and 55 and older), i.e., combining ages 55–64 and 65+ into one group, 55+. As can be seen from the
…gures, in this case LF Pt2000 has declined more than the counterfactual
rate in the seven-age-group exercise (64:4 percent). This is because the
share of 55- to 64-year-old workers in the 55+ group, who have much
higher labor force attachment than 65+, has increased between 2000
and 2012 (see Figure 6 for the shares).

M. Kudlyak: A Cohort Model of the LFP Rate

Figure 5 The Counterfactual LFP Rate based on the Change
in the Demographic Composition of the Population,
Q4:2007

Notes: Population forecast is based on residential population forecast from
HAVER, scaled by 2012 relationship between residential and civilian noninstitutionalized population by age and gender.

The observations above show that the demographic composition of
the population and the changes in the LFP rates of di¤erent groups
have played an important role in the change of the aggregate LFP rate.
We now proceed to examine the age-gender and cohort e¤ects in the
LFP rates of di¤erent demographic groups on the aggregate LFP rate.

A COHORT-BASED MODEL OF LABOR
FORCE PARTICIPATION

The results in Section 1 show that the time-variation in the LFP rates
of di¤erent demographic groups are important for the variation in the
aggregate LFP rate. In this section, we propose a model for the trend
in the LFP rates of di¤erent demographic groups. We then estimate
the trend in the aggregate LFP rate using the estimated trends in the

Federal Reserve Bank of Richmond Economic Quarterly

Figure 6 The Counterfactual LFP Rate based on the Change
in the Demographic Composition of the Population,
Q4:2000

Notes: Population forecast is based on residential population forecast from
HAVER, scaled by 2012 relationship between residential and civilian noninstitutionalized population by age and gender.

LFP rates of di¤erent demographic groups and the actual demographic
composition of the population.

Model
Life-Cycle and Cohort E¤ ects in the LFP
Rates of Age-Gender Groups
The LFP rates of di¤erent demographic groups re‡ect life-cycle and
gender e¤ects. In addition to these e¤ects, the year-of-birth cohort
e¤ects can be an important determinant of the labor force attachment
of a demographic group in a particular period. For example, as noted
earlier, the baby boomers typically have higher labor force attachment.

M. Kudlyak: A Cohort Model of the LFP Rate

As this cohort ages and moves through the age distribution, its stronger
labor force attachment carries over to the respective age group.
We think of the demographic and the cohort e¤ects in the LFP
rates of di¤erent demographic groups as the determinants of the longrun labor force participation trend. To estimate this trend, we specify
the following model:
ln LF Pti = +ln

P
1 1996
C f ln
n b=1917 b;i;t

f 1
b +n

1996
P

b=1917

m
Cb;i;t
ln

m
b +"i;t ;

(2)

where LF Pti is the labor force participation rate of age-gender group i,
f
i is the …xed e¤ect of age-gender group i, Cb;i;t is the dummy variable
that takes value 1 if age-gender group i in period t includes women
m is the dummy variable that takes value 1 if ageborn in year b, Cb;i;t
gender group i in period t includes men born in year b, and n denotes
the number of ages in group i. We specify separate cohort e¤ects for
men and women, i.e., fb ( m
b ) is the cohort-speci…c …xed e¤ect of a
cohort of women (men) born in year b. We assume that each cohort
has equal importance in the corresponding age group conditional on
the number of cohorts in the group. For the oldest group, 65+, we set
n = 20 (setting n = 30 does not have a substantial e¤ect on the results).
To identify age-gender and cohort e¤ects, we normalize ln 1 = 0 and
ln f1969 = 0. The model is estimated using pooled quarterly data on
the LFP rates of 14 age-gender groups.
The model in equation (2) is a simpli…ed version of a model in
Aaronson et al. (2006). Using the estimates from equation (2), we
obtain the time series of ln LF P i for the 14 age-gender groups, ln\
LF P i ,
t

2
\
and calculate LF
Pti = exp ln\
LF Pti + 2" , where 2" is the variance of
"c
i;t . We then construct the estimated aggregate LFP rate as
X
\
\
LF
Pt =
sit LF
Pti ;
(3)

where sit denotes the actual population share of group i in quarter t.
Thus, the population shares capture the e¤ect of the change in the
\
demographic composition of the labor force, while LF
Pti re‡ects the
age-gender and cohort e¤ects of the di¤erent demographic groups. We
\
refer to LF
Pt from model (2) as the estimated trend in the aggregate
LFP rate.
Life-Cycle, Cohort, and Cyclical E¤ ects
To further understand the behavior of the aggregate LFP rate, we also
estimate a model similar to the one in equation (2) with a cyclical indi-

Federal Reserve Bank of Richmond Economic Quarterly

cator. The cyclical indicator is the percentage deviation of employment
from its trend. The idea behind the indicator is that when the labor
market is weak, the labor force participation declines.3
The cohort model with the cyclical indicator is
ln LF Pti =
14
P

g=1

+ ln

P
1 1996
f
Cb;i;t
ln
n b=1917

I(i = g) d ln Et ln

0
g

+ d ln Et

1 ln

f
b

+
1
g

P
1 1996
C m ln
n b=1917 b;i;t

+ d ln Et

2 ln

2
g

m
b +

+ "i;t ;
(4)

where I( ) is the indicator function, and d ln Et is the percentage deviation of the employment series from its Hodrick-Prescott (HP)-…ltered
trend with a smoothing parameter = 105 applied to the quarterly
data.
In the estimation, we use the contemporaneous percentage deviation from employment as well as the …rst and second lag of the deviation. Note that we allow the cyclical e¤ects to vary by demographic
group i. Because of the end-of-sample issues associated with HP…ltering the series, we experiment with using a counterfactual cyclical
series, d^
ln Et ; obtained by calculating the deviations from the employment series simulated to grow at the 2 percent year-over-year quarterly
rate after Q4:2012. While the cyclical components from the actual and
simulated employment series di¤er after 2009, the model-based aggregate LFP rates from the two alternative series are very similar.
The model is estimated on quarterly data. After estimating equation (4), we construct the aggregate LFP rate as described in equation
(3).
The error term in equation (4), "i;t , captures the residual between
the actual LFP rate of group i in period t and the one explained by the
historical relationship between age-gender, cohort, and cyclical e¤ects
and the LFP rates by group. Thus, the residual captures two main
e¤ects. First, it captures the factors that a¤ect the LFP of group i
that are not modeled explicitly in equation (4). These include some
structural factors (for example, changes in taxes or disability bene…ts)
and some cyclical factors that are not fully captured by the changes in
aggregate employment (for example, changes in the duration of unemployment bene…ts, house prices, and stock prices). Second, the residual
captures potential changes in individuals’behavior (i.e., changes in responses of the LFP rates to di¤erent structural and cyclical factors).
3

See recent evidence in Hotchkiss, Pitts, and Rios-Avila (2012); Kudlyak and
Schwartzman (2012); Elsby, Hobijn, and Şahin (2013); and Hornstein (2013).

M. Kudlyak: A Cohort Model of the LFP Rate

Empirical Results

One way to obtain the predictions from the models described in equations (2) and (4) is to estimate the models using the 1976–2012 data,
obtain the trend in the aggregate LFP rate (from equation [3]) and the
model-predicted aggregate LFP rate from the model with a cyclical
indicator, and compare the estimates with the actual LFP rate during
2008–12. Another way is to estimate the model on the 1976–2007 data
and then use the estimates together with the assumptions on cohort
e¤ects and predict the aggregate LFP rate for 2008–12. The cohort
model is sensitive to which approach is used.
One of the concerns associated with cohort models is the end-ofthe-sample e¤ect. In particular, the young cohorts observed in the
1976–2012 sample (i.e., those born in 1985–1996) are observed only
during the period of the declining aggregate LFP rate. Thus, the model
identi…es these cohorts’ propensity to participate from the period of
overall low participation, attributing low LFP to these young cohorts
rather than to the model’s residual. Given the severity and the length
of the Great Recession, the e¤ects of the cohorts born prior to 1985 are
also, to a large extent, identi…ed from their labor force participation
rates during 2008–2012, the period of the overall low LFP. This is the
case for cohorts for which, for example, at least half of the observations
come from the 2008–12 period.
To avoid the end-of-sample e¤ect on the estimates, we estimate the
models in equations (2) and (4) using the data from 1976–2007. To
construct the prediction of the aggregate LFP rate for 2008–12, we
assign, for cohorts born after 1991, the average cohort e¤ect of the
last 20 cohorts. Figure 7 shows the following series: (1) the actual
aggregate LFP rate, (2) the LFP rate constructed from the model with
only age-gender e¤ects, (3) the LFP rate constructed from the model
with age-gender and cohort e¤ects estimated on 1976–2007 data, and
(4) the LFP rate constructed from the model with age-gender, cohort,
and cyclical e¤ects estimated on 1976–2007 data.4
As can be seen from the …gure, the aggregate LFP rate estimated
from the model with only age-gender and cohort e¤ects on the 1976–
2007 sample exceeds the actual aggregate LFP rate after 2008, and the
two lines coincide at the end of 2012. This measure constitutes our
preferred measure of the trend in the LFP rate. The aggregate LFP
rate estimated from the model with age-gender, cohort, and cyclical
4

The estimates are available from the author.

Federal Reserve Bank of Richmond Economic Quarterly

Figure 7 Actual and Model-Based Aggregate LFP Rate,
Age-Gender and Cohorts E ects

Notes: To construct the LFP from the model estimated on the 1976–2007 data,
we estimate unrestricted cohort e¤ects for birth years from 1917 to 1991 and then
assign the average cohort e¤ect of the last 20 cohorts to cohorts born in 1992–96.

e¤ects on the 1976–2007 data closely tracks the actual aggregate LFP
rate during 2008–11 and is slightly below it in the last quarter of 2012.
For comparison, Figure 7 also shows the aggregate LFP rate estimated from the models using the 1976–2012 data. As can be seen from
the …gure, during 2008–12, the aggregate LFP rate predicted from the
model estimated using the 1976–2007 data exceeds the aggregate LFP
rate predicted from the model estimated using the 1976–2012 data.
This is true for the predictions from the model with age-gender and
cohort e¤ects and for the predictions from the model with age-gender,
cohort, and cyclical e¤ects. It appears that the model estimated using
the 1976–2012 data attributes the cyclical e¤ects of the 2008–12 period to cohort e¤ects. To minimize the end-of-sample e¤ect, we also
estimated the models employing a restriction on cohorts as described
in Aaronson et al. (2006). In particular, we constrain the evolution of
the …xed e¤ects for consecutive pairs of the cohorts born in 1985–96 so
that the di¤erence in the average propensity to participate between one

M. Kudlyak: A Cohort Model of the LFP Rate

cohort and the next is the same as for a set of cohorts observed over the
last full business cycle. The aggregate LFP rate based on the models
with restricted and unrestricted cohorts are similar, so the …gure shows
only the results without restrictions.5

Discussion
In the model, the cohort e¤ect stands for an average e¤ect of all nonmodeled factors (beyond life-cycle, gender, and cyclical e¤ects) that
a¤ect the labor force participation of a cohort (i.e., the workers born
in a particular year) throughout the period the cohort is observed in
the sample. These factors can include both structural and cyclical variables. For example, the availability of and the rules that govern Social
Security bene…ts and disability insurance might in‡uence the decision
to look for work versus drop out of the labor force. The wage premium from higher educational attainment might in‡uence the decision
of younger workers to go to school rather than participate in the labor
force. The availability and cost of child care can in‡uence the decision
of mothers to join the labor force.
Consequently, the cohort e¤ects constitute a black box that aggregates these in‡uences and serve as a useful device for accounting
exercises. The cohort model, however, might not be the best laboratory for long-term forecasts. In our estimation, we recognize explicitly
that the e¤ect of young cohorts is to a large degree identi…ed from the
few years during which we observe these cohorts in the data. In particular, for the youngest cohorts, a low cohort e¤ect can be due to the
true low propensity of these cohorts to participate or due to the model
attributing low cyclical LFP to the cohort e¤ect. In our exercise, we
control for these e¤ects. A forecasting exercise would inevitably involve
assumptions about the cohort e¤ects going forward. It is possible that,
for example, the youngest cohorts who are not participating currently
due to schooling will, in fact, increase their LFP as they grow older.
The cohort model does not provide information to support or reject
such scenarios.
5
The result with restrictions is available from the author. This result motivates
estimation of the benchmark model (i.e., using the 1976–2007 data) without restrictions
on cohorts.

40
3.

Federal Reserve Bank of Richmond Economic Quarterly

CONCLUSION

We …nd that in the aftermath of the Great Recession, the aggregate
LFP rate closely tracks the one predicted by the historical relationship
between the changes in employment and the labor force participation
rates of di¤erent age-gender groups in a cohort-based model. In 2012,
the actual LFP rate is slightly higher than the one predicted by the
model. In 2009–11, the trend component of the labor force participation
rate, which is based entirely on the life-cycle and cohort e¤ects of the
LFP rates of di¤erent age-gender groups and the actual age-gender
composition of the population, exceeds the actual LFP rate.
The result that the LFP rate in 2012 is above the level that is
predicted by the historical relationship between labor force participation and the cyclical indicator is consistent with the recent …ndings
by Hotchkiss and Rios-Avila (2013), who provide direct evidence that
some changes in behavior took place.6 What other factors could have
contributed to the estimated deviation of the actual LFP rate from
its model-based prediction? We speculate that the Great Recession
was characterized by unusually wild swings in some economic indicators that could have a¤ected labor force participation. First, the
unemployment bene…ts in some states were extended to unusually high
levels. The bene…ts extension might have kept some workers in the
labor force for up to two years to enable them to collect bene…ts rather
than dropping out of the labor force. In particular, Farber and Valletta
(2013) …nd that the e¤ect of the unemployment insurance extensions
on unemployment exits and duration is primarily due to a reduction in
exits from the labor force.7 Second, the collapse of the stock market led
to a decline in retirement savings, which might have led older workers
to stay in the labor force longer. Third, the collapse of the housing
market lowered the ability of households to borrow against their home
equity, which also might have caused individuals to join and/or remain
in the labor force at higher rates than historically predicted by age,
gender, cohort, and cyclical employment e¤ects. Finally, to understand
the behavior of labor force participation and its trend, more research
is needed that would explicitly model and account for the factors that
6
In particular, Hotchkiss and Rios-Avila (2013) use microdata from the Current
Population Survey and estimate the probability of an individual participating in the
labor force as a function of age, education, and other socioeconomic and demographic
characteristics of the individual as well the aggregate labor market conditions. They
…nd that the coe¢ cients on the socioeconomic and demographic characteristics estimated
from the post-2008–09 period di¤er from the coe¢ cients estimated from the pre-recession
period in such a way as to increase the aggregate LFP rate.
7
See also Fujita (2010, 2011) and Rothstein (2011).

M. Kudlyak: A Cohort Model of the LFP Rate

in‡uence the labor force participation decision of di¤erent demographic
groups.

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Federal Reserve Bank of Richmond Economic Quarterly

Farber, Henry S., and Robert G. Valletta. 2013. “Do Extended
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Economic Quarterly— Volume 99, Number 1— First Quarter 2013— Pages 45–81

Saving for Retirement with
Job Loss Risk
Borys Grochulski and Yuzhe Zhang

n this article, we study optimal saving and consumption decisions.
The optimal saving problem is among the most basic questions in
economics and …nance. How does one best decide on what portion
of income they should consume now and what portion they should
save for their future consumption needs? One important aspect of this
question concerns saving for retirement. What is an optimal plan for
saving enough to be able to retire? In particular, how does this plan
depend on the risk of losing one’s job? How much more should one
save if the risk of becoming jobless increases?
Our primary objective in this article is to review several important
results from the general theory of optimal consumption and saving decisions, as well as provide some novel analysis of the problem of saving for
retirement in particular. The problem of optimal timing of retirement
is most conveniently studied in a continuous-time framework, which we
employ for our analysis. Our secondary objective is to provide an accessible exposition of the techniques useful in solving continuous-time
models of the type we examine.
The basic framework economists have used to study the intertemporal tradeo¤ between current and future consumption has the following
structure. An economic agent earns a stream of labor income that can
change stochastically over time. At each point in time, the agent allocates his labor income to either current consumption or to savings.
The agent’s preferences over consumption streams are represented by
a concave utility function, i.e., the agent is averse to ‡uctuations in his
Grochulski is a senior economist with the Richmond Fed. Zhang is an assistant
professor with Texas A&M University. The authors would like to thank Arantxa
Jarque, Marianna Kudlyak, Andrew Owens, and Ned Prescott for their helpful comments. The views expressed in this article are those of the authors and not necessarily those of the Federal Reserve Bank of Richmond or the Federal Reserve System.
E-mail: borys.grochulski@rich.frb.org; yuzhe-zhang@econmail.tamu.edu.

Federal Reserve Bank of Richmond Economic Quarterly

consumption. The duration of the agent’s career is in the basic model
approximated by in…nity, i.e., the agent earns labor income and consumes inde…nitely into the future. The portion of his labor income that
the agent does not immediately consume adds to his …nancial wealth.
In the basic model, there is only one asset in which all of the agent’s
…nancial wealth is invested. The asset pays o¤ a riskless rate of return
equal to the agent’s intertemporal rate of time preference— the rate
of return with which the agent’s optimal consumption path absent all
uncertainty would be constant forever.
The model we study in this article extends this basic framework by
adding to the optimal consumption and saving decision a labor supply
decision operating on the extensive margin, meaning we allow the agent
to stop working. If the agent quits, he loses his labor income but gains
leisure. The decision to quit is irreversible, so quitting means retiring.
In retirement, the agent lives o¤ of his savings and enjoys leisure. As
in the basic model, the agent remains in…nitely lived in our analysis.
For tractability and ease of exposition, we assume in our model
a particularly simple stochastic structure for the agent’s labor income
process. The agent earns a constant stream of labor income for as long
as he is not …red. If he is …red, he earns nothing and cannot go back
to working ever again. Thus, being …red is in our model equivalent
to being sent to involuntary retirement. The observed time path of
the agent’s labor income in our model is thus constant, at some positive level, until the agent either is …red or quits. Afterward, it is also
constant at the level of zero.
Ljungqvist and Sargent (2004, Ch. 16) review the solution to the
optimal consumption and saving problem in the basic framework with
income ‡uctuating stochastically but without retirement. The main
property of the optimal consumption plan is unbounded growth of
…nancial wealth and consumption: Provided that the labor income
process does not settle down in the long run (rather, it remains suf…ciently stochastic), in almost all possible resolutions of uncertainty,
the amount of …nancial wealth the agent holds and the amount the
agent consumes grow over time without bound. When we allow for endogenous retirement, this property of the optimal wealth accumulation
and consumption plan no longer holds. In all possible resolutions of
uncertainty, wealth and consumption converge to a …nite limit.
The intuition behind this result is simple. We show that the agent’s
optimal retirement plan takes the form of a wealth threshold rule: The
agent retires as soon his accumulated …nancial wealth reaches a certain
threshold. With this rule, wealth will not grow without bound prior to
retirement. With …nite wealth and no labor income after retirement,
the agent’s optimal consumption also remains bounded in the long run.

Grochulski and Zhang: Saving for Retirement

In fact, consumption is constant and equal to the amount of interest
income generated by the agent’s wealth in retirement.
The dynamics of consumption and savings are in our model as
follows. Wealth and consumption increase monotonically over time for
as long as the agent does not involuntarily lose his job. If the agent is
…red, his wealth accumulation is stopped and his consumption jumps
downward. If the agent reaches the voluntary retirement threshold,
his wealth and consumption reach their permanent, retirement levels
smoothly. For any level of …nancial wealth the agent starts out with,
we compute the planned duration of the agent’s career, i.e., his time to
planned retirement. Agents with lower initial wealth retire later.
We provide several comparative statics results. We show how the
agent’s optimal path of wealth accumulation and consumption prior to
retirement depends on the risk of losing his job, on the value of leisure
he obtains in retirement, and on the level of the rate of return paid
by the asset in which the agent invests his savings. Higher job loss
risk implies the agent saves more, consumes less, and retires faster.
Lower utility of leisure implies the agent saves less, consumes more,
and retires later. When the interest rate is higher, the agent retires
with lower wealth and generally consumes more prior to retirement. In
solving for the agent’s optimal retirement rule, we discuss the option
value of postponing retirement.
In addition, we discuss, in the context of our model, two standard
properties of the solution to the optimal consumption and saving problem. We show that in the model with retirement, like in the standard
model without retirement, the agent’s marginal utility of consumption
is a martingale, which means the conditional expected change in its
value is always zero. We also review the result known as the permanent income hypothesis (PIH). De…ned narrowly, PIH states that the
agent chooses to consume exactly the income from his total wealth at
all times. Total wealth consists of both …nancial and human wealth,
where human wealth is de…ned as the expected present value of all
labor income the agent is to earn in the future. With quadratic preferences, PIH holds in the standard model without retirement. We show
that adding an endogenous retirement decision to the model does not
overturn PIH.
We provide an elementary-level discussion of all dynamic optimization techniques involved in the analysis of our continuous-time model,
thus making it accessible to a broad audience.

Federal Reserve Bank of Richmond Economic Quarterly

Related Literature
Our study is related to the literature on optimal consumption and
saving decisions with ‡uctuating income and incomplete markets, and
to the literature on the optimal timing of retirement.
The vast literature on the optimal saving problem with ‡uctuating income is summarized in Ljungqvist and Sargent (2004, Ch. 16).
Classic studies of this problem, which include Friedman (1957), Bewley
(1977), and Hall (1978), take the agent’s stochastic income process as
exogenous, which means they abstract from retirement. Chamberlain
and Wilson (2000) allow for stochastic changes to the interest rate and
show under weak conditions that optimal consumption diverges with
probability one. Marcet, Obiols-Homs, and Weil (2007) extend the
classic framework by including the agent’s labor supply decision along
the intensive margin. They show that with endogenous labor income
the result of divergence of almost all consumption paths does not hold
due to a wealth e¤ect suppressing the agent’s labor supply and thus
eventually eliminating ‡uctuations in the agent’s income. Our analysis
is similar but allows for changes in labor supply along the extensive
margin, i.e., it incorporates the retirement decision.
Similar to our analysis, Ljungqvist and Sargent (forthcoming) study
an optimal consumption and saving problem with endogenous retirement. They focus on the impact of the curvature of the life cycle income
pro…le on savings and the timing of retirement in a …nite-horizon model
in which all income shocks are unanticipated. Our model assumes a
‡at income pro…le in an in…nite-horizon model in which the agent anticipates the risk in his income and responds to it.
Kingston (2000) and Farhi and Panageas (2007) study the optimal
retirement timing decision combined with the problem of optimal saving and asset allocation prior to retirement, where available assets are
one risky and one riskless asset, as in Merton (1971). They show that
the option to delay retirement lets agents take on more risk than they
would have chosen otherwise. In particular, Farhi and Panageas (2007)
show that investors close to retirement may …nd it optimal to invest
more heavily in stocks than those whose retirement is far o¤ in the
future. Our analysis is di¤erent as we do not consider a portfolio allocation problem in this article. Rather, we assume an incomplete market
structure in which the riskless asset is the only vehicle for saving and
wealth accumulation, as in the classic models of optimal consumption
and saving decisions.
Our article is organized as follows. Section 1 presents our model.
Section 2 discusses the optimal consumption pattern after retirement.
Section 3 describes the optimal timing of retirement. Sections 4 and
5 study consumption and wealth accumulation prior to retirement.

Grochulski and Zhang: Saving for Retirement

Sections 6 and 7 provide comparative statics results with respect to
several parameters of the model, with particular attention given to the
job loss hazard parameter. Section 8 concludes. Appendix A contains
proofs. Appendixes B and C discuss two extensions of the model.

MODEL

We will study the following partial equilibrium model in continuous
time with a single agent. The agent consumes a single consumption
good and leisure. The agent is initially employed. When employed, the
agent earns a ‡ow of labor income of y > 0 units of the consumption
good per unit of time. The agent also consumes a ‡ow of leisure of
lW > 0 units per unit of time. If the agent is not working, his labor
income is zero but his ‡ow of leisure is lR > lW . The agent’s preferences
over deterministic paths of consumption and leisure are represented by
a standard utility function
Z 1
e rt U (ct ; lt )dt;
0

where ct is consumption, lt 2 flW ; lR g is leisure, and r > 0 is the agent’s
intertemporal rate of time preference.
While employed, the agent faces the risk of losing his job. If he
loses his job, he never works again, which e¤ectively means that losing
one’s job represents in our model involuntary retirement. The job loss
shock arrives stochastically with a constant hazard rate > 0. That
is, for any date t at which the agent is employed and for any s > 0,
the probability that the agent will have not lost his job by date t + s
is e s .
In addition to losing his job involuntarily, the agent can quit. In
this case, as well, the separation from employment is permanent, i.e.,
quitting means retiring. If he retires, the agent gives up the ‡ow of
labor income y and gains the ‡ow of extra leisure lR lW > 0.
At each point in time, the agent decides how much of his current
income to consume and how much to save. There is only one asset
in which the agent can invest his savings. It is a riskless asset with a
constant rate of return equal to the agent’s rate of time preference r.
Denote the amount of the riskless asset held by the agent at date t,
i.e., the agent’s …nancial wealth at t, by Wt .
With these assumptions, the law of motion for the agent’s …nancial
wealth Wt is as follows. While working, the …nancial wealth changes
according to
dWt = (rWt + y

ct )dt:

(1)

Federal Reserve Bank of Richmond Economic Quarterly

Thus, for example, if the agent were to consume exactly his labor income while working, i.e., if ct = y, then his …nancial wealth would
grow exponentially at the rate of interest r. When not working (i.e., in
retirement), the agent’s wealth follows
dWt = (rWt

ct )dt:

The agent maximizes
"Z
Z
minf ; f g
E
e rt U (ct ; lW )dt +
0

minf ;

(2)

U (ct ; lR )dt ;

(3)

where is the agent’s planned, voluntary retirement time, f is the time
he is forced into involuntary retirement, and the expectation E is taken
over the realizations of the involuntary job loss shock. In particular,
we will take the utility function to be separable in consumption and
leisure:
U (c; lW ) = u(c);
U (c; lR ) = u(c) + ;
where u is strictly increasing and a strictly concave utility of consumption and
0 is the utility of the extra leisure the agent enjoys in
retirement. In this speci…cation, the agent’s lifetime utility (3) can be
more simply written as
Z 1
:
E
e rt u(ct )dt + e r minf ; f g
r
0

OPTIMAL SAVING AND CONSUMPTION
IN RETIREMENT

We start by discussing the agent’s optimal use of savings in retirement.
Because the return on the …nancial wealth held by the agent is equal to
the agent’s rate of time preference and the agent faces no uncertainty
in retirement, it is natural to guess that in retirement the agent will
keep assets constant, dWt = 0, and consume his capital income, i.e., the
return rWt at all t. Thus, the natural guess is that if the agent retires
with assets Wt , the maximum present value of total lifetime utility he
can obtain after retirement, denoted by V (Wt ), is
1
V (Wt ) = u(rWt ) + :
(4)
r
r
In the remainder of this section, we will use a standard dynamic
programming argument to con…rm that this guess is correct. In the
process, we will derive an optimality condition on the value function

Grochulski and Zhang: Saving for Retirement

V — known as the Bellman equation— that will be useful when we discuss the agent’s optimal consumption and saving behavior prior to
retirement in the next section.
Following the dynamic programming approach, we take a small
time interval [t; t + h) and assume that from time t + h onward the
agent will apply the optimal saving and consumption policy, which is
not known to us as of now. Given this assumption, we seek an optimal
consumption rate c within the time interval [t; t+h). Because h is small,
we can consider c to be constant over the interval [t; t + h). The true
optimal consumption rate to be applied at time t, ct , will be obtained
by taking the limit as h goes to zero.
Because the agent follows an optimal consumption plan after t + h,
the total discounted value he will obtain as of time t+h will be V (Wt+h ),
where Wt+h is the amount of …nancial wealth the agent holds at t + h.
For a given consumption rate c to be applied in [t; t + h), the total
discounted utility value the agent obtains as of time t is
Z h
e rs (u(c) + ) ds + e rh V (Wt+h ):
(5)
0

Because this plan is a feasible consumption plan for an agent with
nonnegative wealth, the maximal utility value V (Wt ) must be at least
as large as the value of this plan, so for any c it is true that
Z h
V (Wt )
e rs (u(c) + ) ds + e rh V (Wt+h ):
0

When h becomes arbitrarily small, the maximized value of the righthand side of this expression approaches the value on the left-hand side,
which we can write as
Z h
V (Wt ) = max
e rs (u(c) + ) ds + e rh V (Wt+h )
(6)
c

with h approaching zero. Since h is very small, we can replace the
expression on the right-hand side of (6) with its …rst-order approximation. For a function f di¤erentiable at some point t, for small h, we
can approximate f (t + h) with f (t) + f 0 (t)h. In this approximation,
the …rst of the two terms in (5) equals
0 + (u(c) + ) h;
and the second term equals
V (Wt ) +

rV (Wt ) + V 0 (Wt )

dWt
dt

The value in (5) is therefore approximated by
V (Wt ) + u(c) +

rV (Wt ) + V 0 (Wt )(rWt

c) h;

Federal Reserve Bank of Richmond Economic Quarterly

where we have used the law of motion for assets in retirement (2). With
this approximation, we can thus write (6) as
V (Wt ) = max V (Wt ) + u(c) +
c

rV (Wt ) + V 0 (Wt )(rWt

c) h :

(7)
Dividing by h and simplifying terms, we obtain the following condition
for the value function V :
rV (Wt ) = max u(c) +
c

+ V 0 (Wt )(rWt

c) :

(8)

We will refer to this condition as the Bellman equation for the value
function V . This equation shows how the agent’s total utility V (Wt )
(converted to ‡ow units by multiplying it by r) depends on current
utility and the change in …nancial wealth. Higher c will increase the
current utility ‡ow u(c) + at the cost of lower saving rWt c. The
marginal value of wealth V 0 (Wt ) shows how costly a change in saving
is to the agent in utility terms. In choosing the consumption rate c the
agent optimally balances this tradeo¤ between his utility from current
consumption and his utility from future wealth.
Next, by di¤erentiating the Bellman equation (8), we will obtain the
optimal consumption policy function. Note that the Envelope Theorem
lets us treat c as a constant in this di¤erentiation. Indeed, di¤erentiation gives us
rV 0 (Wt ) = V 00 (Wt )(rWt

ct ) + V 0 (Wt )r;

which simpli…es to
0 = V 00 (Wt )(rWt

ct ):

(9)

Assuming the second derivative V 00 is nonzero, we divide both sides by
V 00 (Wt ) to obtain
ct = rWt :

(10)

This con…rms our guess that the optimal consumption policy for the
agent in retirement is to consume the interest income from his …nancial assets at all t . Using this policy in the law of motion for wealth in
retirement, (2), we con…rm that dWt = 0 and so assets and consumption remain constant in retirement. Substituting constant consumption
ct+s = rWt into the agent’s utility function at all times t + s following
the retirement date t leads to the value function (4), con…rming the
guess we made at the beginning of this section.
We will also note that the …rst-order condition for the maximum
on the right-hand side of (8) is
u0 (ct ) = V 0 (Wt ):

Grochulski and Zhang: Saving for Retirement

This condition, along with the policy function (10), lets us determine
the marginal value of wealth in retirement as
V 0 (Wt ) = u0 (rWt ):

(11)

Clearly, the same result can be obtained by di¤erentiating (4) directly.1

OPTIMAL RETIREMENT DECISION

In this section, we show that the optimal retirement policy for the
agent is a threshold policy: The agent retires when his wealth reaches
a speci…c threshold level. At this threshold, the marginal value of
income the agent can earn if he works is exactly matched by the value
of the extra leisure the agent can get if he retires.
In our analysis of the optimal voluntary retirement rule, we will use
one intuitive property of the optimal pre-retirement wealth accumulation path. Namely, that the optimal wealth accumulation path is nondecreasing, i.e., the agent actually does save for retirement. That the
agent will choose an increasing wealth accumulation path fWt ; t 0g
prior to retirement is very intuitive in our model because the agent’s
labor income process is non-increasing and the return on savings is
equal to the agent’s rate of time preference. It is clear from (1) that
Wt decreases only if ct > rWt + y, i.e., when the agent consumes more
than his capital income rWt and labor income y combined. Doing so
clearly cannot be optimal for the agent given the labor income process
the agent faces. The agent earns constant labor income y > 0 when he
works and has no labor income after he quits or loses his job. In order
to smooth consumption, the agent will want to save at least a part of
his labor income for as long as he works, i.e., will choose ct rWt + y
prior to retirement. In Section 5, we will characterize precisely what
portion of y will be saved at each point in time. For now, we will just
state that ct > rWt + y is never optimal for the agent, and thus Wt is
at least weakly increasing over time.
We now move on to the agent’s optimal retirement decision. We will
analyze this decision in two steps. First, we will compare the agent’s
value from retiring now, i.e., at some given time t, with the value from
retiring a little later, i.e., at t + h, for a small h > 0. Then, we will
argue that if the agent prefers to retire at t rather than retire at t + h
for a small h, then he also prefers to retire at t over retiring at any
future date, which means the agent’s overall optimal retirement time
is t.
1
Further, di¤erentiating (4) twice, we have V 00 (Wt ) = ru00 (rWt ) < 0, which justi…es
the assumption of nonzero V 00 we made when we divided (9) by V 00 (Wt ).

Federal Reserve Bank of Richmond Economic Quarterly

As before, we will use the …rst-order approximation for payo¤s at
t + h. In addition, we will discretize the involuntary job loss shock by
assuming that if the agent loses his job by time t + h, this loss will
occur only at t + h and not earlier. With h approaching zero, these
approximations will be su¢ ciently precise.
Suppose then that the agent is employed and has …nancial wealth
Wt as of some time t. As we know from the previous section, the agents’
value of retiring now is V (Wt ). The value of postponing retirement by
a small amount of time h, denoted here by V h (Wt ), is
Z h
e rs u(c)ds + e rh V (Wt+h )
V h (Wt ) = max
c

with wealth following (1) between t and t+h, as the agent keeps working
between t and t + h. Note that it does not matter if at t + h the job
loss shock happens or does not happen, because the agent is retiring
at t + h anyway. Since h is small, we use the …rst-order approximation
and express V h (Wt ) as
V h (Wt ) = max V (Wt ) + u(c)
c

= max V (Wt )+ u(c)
c

dWt
h
dt
rV (Wt )+V 0 (Wt )(rWt + y c) h ;
rV (Wt ) + V 0 (Wt )

where the second line uses (1).
Using the …rst-order approximation (7) for the value of retiring at
t, V (Wt ), we have that postponing retirement by h is strictly preferred
to retiring immediately, i.e., V h (Wt ) > V (Wt ), if and only if
max V (Wt ) + u(c)
c

> max V (Wt ) + u(c) +
c

rV (Wt ) + V 0 (Wt )(rWt + y
rV (Wt ) + V 0 (Wt )(rWt

c) h
c) h :

Dividing by h, simplifying, and taking terms that do not depend on c
out of the maximization on each side, we have
max u(c)
c

V 0 (Wt )c + V 0 (Wt )y > max u(c)
c

V 0 (Wt )c + :

Since the maximization problems on both sides of this inequality are
the same, we simplify the above condition further to obtain
V 0 (Wt )y > :

(12)

This says that whenever the utility ‡ow from the additional leisure
the agent can obtain by retiring is smaller than the utility he draws
from the ‡ow of his labor income, the agent will prefer to postpone
retirement. From (11) we know that the agent’s marginal value of
wealth in retirement is V 0 (Wt ) = u0 (rWt ). Thus, inequality (12) is

Grochulski and Zhang: Saving for Retirement

equivalent to
y

< u0 (rWt ):

(13)

Let u0 1 denote the inverse function of u0 . Since the right-hand side of
(13) is strictly decreasing in Wt , it is true that this inequality holds for
all Wt < W , where the threshold value W is given by
1
W = u0
r

(14)

This means that postponing retirement (by at least a small instant) is
preferred at all wealth levels Wt strictly smaller than W . The agent
thus will not retire voluntarily with wealth Wt < W . Intuitively, for as
long as his wealth is below W , by continuing to postpone retirement,
the agent obtains a larger current ‡ow return (his labor income is more
valuable than the leisure forgone to obtain it) and retains the option
to retire later.
Now that we know the agent will not retire with wealth smaller than
W ; we should ask if the agent will choose to retire as soon as his wealth
reaches W . We know already that the agent with wealth Wt equal to
or larger than W prefers to retire at t over retiring a bit later. But
what about the possibility of retiring much later? Does Wt W also
mean that the agent prefers to retire at date t rather than at any future
date T > t? The answer is yes because, as we argued earlier in this
section, the time path of wealth the agent chooses is never decreasing.
Indeed, suppose the agent’s wealth as of t satis…es Wt
W , but he
does not retire until some later date T > t. Because the path of wealth
is non-decreasing, Ws W at all dates s in t s T . In particular,
for a small h > 0, at date s = T h, the agent’s wealth is greater
than or equal to W , so, by our previous argument, the agent prefers
to retire at T h rather than wait until T . Because his wealth is not
smaller than W at T 2h, as well, the agent will prefer to retire at
T 2h rather than at T h. Extending this reasoning backward in
time all the way back to date t shows that the agent’s overall preferred
retirement rule is to retire as soon as his wealth reaches W .
In sum, the optimal retirement rule takes on a threshold form.
The agent chooses to postpone retirement for as long as his wealth is
below the threshold W and retire immediately when his wealth reaches
W . It is worth noting in (14) that the optimal wealth threshold W
increases in labor income y, decreases in the value of leisure , and
does not depend on the intensity of the job loss risk . If = 0, i.e., if
working is not costly to the agent in terms of forgone leisure at all, then
W = 1. In this case, the agent never chooses to retire voluntarily.

Federal Reserve Bank of Richmond Economic Quarterly

The Option Value of Postponing Retirement
Because retirement is permanent in our model, when the agent retires
he loses the option of working at later dates. The threshold retirement
rule we derived tells us, however, that the value of this option is zero
for the agent in the problem we study.
In general, a one-time, irreversible action has a positive option value
for an agent if he is willing to forgo an immediate bene…t that the action
can produce in order to retain the option of taking the action in the
future.2 In our model, the agent retires as soon as the current ‡ow
return from doing so turns positive, i.e., when the value of the ‡ow of
leisure, , becomes as large as the value of the ‡ow of labor income y,
V 0 (Wt )y. The agent is not willing to delay retirement beyond that point
because once wealth reaches W the agent will continue to prefer the
‡ow of leisure over the ‡ow of his labor income y at all future times in
all possible realizations of uncertainty he faces. In fact, once the agent
retires with wealth Wt W , there is no realization of uncertainty in
which he might want to go back to working, even if he could return.
The value of having the option to work in the future that the agent
gives up by retiring would in our model be positive if the parameters
determining the threshold wealth level W could change in a way that
increases the value of working relative to the value of consuming leisure.
In particular, the value of this option would be positive if the agent
could receive a positive income shock increasing the level of his labor
income y, or a taste shock decreasing the utility of leisure , or a taste
shock increasing the agent’s marginal utility of consumption u0 , or a
shock destroying a part of the agent’s …nancial wealth Wt . In Appendix
C, we discuss this point in more detail, focusing on the possibility of
an increase in labor income y.

CONSUMPTION, SAVING, AND WEALTH
ACCUMULATION PRIOR TO RETIREMENT

In this section, we study the agent’s optimal saving and consumption
decisions prior to retirement, i.e., when his wealth is strictly less than
W . The guess-and-verify method we used earlier to solve for optimal consumption in retirement will not work here because wealth and
consumption have nontrivial dynamics prior to retirement. In order to
2
For example, it is often optimal for a business owner to keep her business open
for some time after it begins to make a loss (a ‡ow of negative pro…t). The option
for the pro…ts that the business may generate in the future has a value that keeps the
owner from shutting the business down as soon as the current pro…t ‡ow turns negative
(see Leland [1994]). Pindyck (1991) discusses the option value of undertaking a one-time
investment in a stochastic environment.

Grochulski and Zhang: Saving for Retirement

study these dynamics, we will derive intertemporal optimality conditions leading to a dynamic system in wealth and consumption. We will
then use standard methods to analyze this system.

Bellman Equation
Let us denote by J(Wt ) the maximal discounted expected utility value
a working agent can obtain given his wealth Wt . Since retiring immediately is optimal when Wt
W , we have J(Wt ) = V (Wt ) for all
Wt
W . Since not retiring is strictly preferred by the agent when
Wt < W , we have J(Wt ) > V (Wt ) for all Wt < W . We look now
to learn more about J(Wt ) for Wt < W . We proceed by deriving the
Bellman equation for J analogous to the Bellman equation for V we
derived earlier.
Take a small h > 0 and assume that an agent who works at t and
holds …nancial wealth Wt chooses to consume at some constant rate c
inside the time interval [t; t + h). In addition, assume that if the agent
wants to quit inside (t; t + h), he will do so only at t + h. Likewise,
assume that if the agent loses his job involuntarily during this short
period of time, this will happen only at the end of the period, i.e., at
date t+h. As before, these assumptions will be innocuous when we take
the limit with h going to zero. Following the dynamic programming
approach, we suppose that from time t + h onward the agent applies an
optimal (to us yet unknown) consumption and saving policy. The total
utility value the agent obtains by following this strategy with some
…xed consumption rate c is
Z t+h
h
i
e rs u(c)ds + e rh e h J(Wt+h ) + (1 e h )V (Wt+h ) : (15)
t

The term in square brackets represents the expectation of the value the
agent will draw at time t + h. With probability e h he does not lose
his job as of t + h and J(Wt+h ) represents the continuation value he
obtains at that time. With probability 1 e h he loses his job, and
thus the continuation value he obtains at t + h is the retirement value
V (Wt+h ).
With the optimal choice of c, the value in (15) approaches the
overall maximal value the agent can obtain, J(Wt ), which we write as
nR
t+h
J(Wt ) = max t e rs u(c)ds
c
h
io
+e rh e h J(Wt+h )+ (1 e h )V (Wt+h )
(16)

Federal Reserve Bank of Richmond Economic Quarterly

with h approaching zero. Since h is very small, we can apply the …rstorder approximation to the value in (15) and write it as
J(Wt ) + u(c)

(r + )J(Wt ) + J 0 (Wt )(rWt + y

c) + V (Wt ) h:

Using this approximation in (16), we have
J(Wt ) = max J(Wt ) + u(c)

(r + )J(Wt )+

+J 0 (Wt )(rWt + y

c) + V (Wt ) h :

Dividing by h and simplifying terms, we get the Bellman equation for
J:
(r + )J(Wt ) = max u(c) + J 0 (Wt )(rWt + y
c

c) + V (Wt ) : (17)

To compare it with the Bellman equation for V , (8), let us rewrite (17)
as
rJ(Wt ) = max u(c) + J 0 (Wt )(rWt + y

(J(Wt )

V (Wt )) :

(18)
Bellman equations (8) and (18) di¤er in three ways. First, the tradeo¤ between consumption and saving is di¤erent, as prior to retirement
the agent earns the stream of income y. Second, the level of J is
also in‡uenced by the lower ‡ow of leisure prior to retirement. These
two di¤erences are re‡ected in the expression inside the maximization with respect to c in (18). Third, (18) contains an extra term,
(J(Wt ) V (Wt )), that re‡ects the possibility of the agent’s involuntarily losing his job. In this term, is the intensity with which the
agent loses his job and J(Wt ) V (Wt ) is the loss of value that occurs
in that event.

Euler Equation
As before, we use the envelope and …rst-order conditions associated
with the Bellman equation. Using the Envelope Theorem in di¤erentiation of the Bellman equation (17) yields
(r + )J 0 (Wt ) = J 00 (Wt )(rWt + y

ct ) + J 0 (Wt )r + V 0 (Wt ):

Simplifying terms and rearranging, we get
J 0 (Wt )

V 0 (Wt ) = J 00 (Wt ) (rWt + y

ct ) :

(19)

Unlike in the post-retirement problem we studied earlier, in the preretirement problem the envelope condition (19) does not by itself determine the optimal consumption rule. However, it gives us an important
intertemporal optimality condition for consumption known as the Euler

Grochulski and Zhang: Saving for Retirement

equation. To derive it, we use the chain rule to express the time derivative of J 0 (Wt ) as
dJ 0 (Wt )
dt

dWt
dt
= J 00 (Wt )(rWt + y
= J 00 (Wt )

ct );

where the second equality uses (1). This lets us write (19) as
dJ 0 (Wt )
= J 0 (Wt ) V 0 (Wt ) :
dt
Next, we use the …rst-order condition in the maximization problem in
the Bellman equation (17),
u0 (ct ) = J 0 (Wt );

(20)

to write the above as
du0 (ct )
= u0 (ct ) V 0 (Wt ) :
dt
Finally, we use (11) to eliminate V 0 from the above equation and express
it purely in terms of the marginal utility of consumption:
du0 (ct )
= u0 (ct ) u0 (rWt ) :
(21)
dt
This is the Euler equation for consumption prior to retirement. It shows
how the marginal utility of consumption changes along an optimal path
of consumption and …nancial wealth accumulation prior to retirement.3

Martingale Property
Before we use the Euler equation to study optimal consumption and
asset accumulation, let us discuss an implication of the Euler equation
known as the martingale property of marginal utility. As studied by
Hall (1978) and many others, (21) implies that at all times prior to
retirement the expected change in marginal utility of consumption is
zero, i.e., marginal utility of consumption is a so-called martingale.4 In
discrete-time models that are most commonly used in the literature,
the Euler equation takes the familiar form of u0 (ct ) = Et [u0 (ct+1 )] at
all t, where Et [ ] is the conditional expectation operator. In discrete
time, it is thus easy to see that the expected change in u0 is zero. In
continuous time, the martingale property is slightly less self-evident
but can still be seen as follows.
3

Note that, trivially, the Euler equation also holds after retirement.
Because consumption is constant in retirement, marginal utility of consumption is
trivially also a martingale after the agent retires, voluntarily or not.
4

Federal Reserve Bank of Richmond Economic Quarterly

Take a small h > 0 and a date t at which the agent is not retired.
In the time interval [t; t + h), the agent will be hit with the job loss
shock with probability 1 e h , which will cause his marginal utility at
t + h to change (jump) by u0 (rWt+h ) u0 (ct+h ). With probability e h ,
the agent will not lose his job, in which case the change in his marginal
utility over the time interval [t; t + h) will be simply u0 (ct+h ) u0 (ct ).
Marginal utility is a martingale when the average (i.e., expected) value
of these two changes is zero, i.e., when
(1

) u0 (rWt+h )

u0 (ct+h ) + e

u0 (ct+h )

u0 (ct ) = 0:

Rearranging this condition, we have
u0 (ct+h )

u0 (ct ) = (e
=

1) u0 (ct+h )

h u0 (ct+h )

u0 (rWt+h )

u0 (rWt+h ) ;

where the second equality uses the linear approximation e h = 1 + h.
Dividing by h and taking formally the limit as h ! 0; we get the Euler
equation (21). Thus, the Euler equation (21) says exactly that the time
trend du0 (ct )=dt in marginal utility along the path that consumption
follows conditional on the job loss shock not occurring is the negative
of the jump in marginal utility that occurs if the agent loses his job,
u0 (rWt ) u0 (ct ), times the intensity of the job loss . This trend exactly
o¤sets the jump-induced change in marginal utility, making the overall
expected change in marginal utility zero, i.e., marginal utility indeed is
a martingale.5

Dynamic Analysis
As we saw earlier, consumption and …nancial wealth have trivial dynamics in retirement: Both remain constant over time. Prior to retirement, however, wealth and consumption do change over time. We will
now use the Euler equation (21) and the law of motion for wealth (1)
to study the dynamics of wealth and consumption prior to retirement.
To do this, we use the chain rule
dct
du0 (ct )
= u00 (ct )
dt
dt
and the strict concavity of u, implying u00 6= 0, to express the Euler
equation (21) as
dct
u0 (ct ) u0 (rWt )
=
:
dt
u00 (ct )

(22)

5
In this respect, the marginal utility process is in our model similar to a compensated Poisson process. See Problem 1.3.4 in Karatzas and Shreve (1997).

Grochulski and Zhang: Saving for Retirement

Figure 1 Phase Diagram for (W,c)

Together with the law of motion for …nancial wealth Wt , (1), this
gives us a dynamic system describing the evolution of consumption and
wealth prior to retirement. In the rest of this subsection, we will study
qualitative properties of this system. We will use a phase diagram to
describe the shape of the time paths in the plane (W; c) that satisfy the
di¤erential equations (1) and (22). Any such path is called a solution
to the system (1), (22), and there are an in…nite number of them (every
point in the domain for (W; c) belongs to one solution). The solutions
represent all paths of consumption and wealth accumulation the agent
might want to follow while working that are consistent with intertemporal optimization. That is, any path that is not a solution to (1), (22)
is not optimal for the agent. In order to select the optimal path from
among all solutions to this system, a boundary condition is needed. In
standard in…nite-horizon analysis, the transversality condition serves
this role. In our model with endogenous retirement, this condition will
be provided by the optimal voluntary retirement rule we obtained in
the previous section.
The phase diagram for the system of di¤erential equations (1), (22)
is shown in Figure 1. It provides a graphical representation of the

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directions in which the system (Wt ; ct ) moves along all possible solution
paths. In Figure 1, these directions are marked by horizontal and
vertical arrows. These arrows are determined as follows.
From (22) we see that along a solution path consumption will increase over time, i.e., dct =dt > 0, if and only if u0 (ct ) u0 (rWt ) < 0 , i.e.,
if and only if ct > rWt . The line c = rW therefore divides the state
space (W; c) into two regions: one in which consumption grows over
time (the region above this line), and one in which it decreases over
time (the region below it). Similarly, we have from (1) that Wt grows
over time if and only if ct < rWt + y. Therefore, the line c = rW + y
divides the state space (W; c) into a region of wealth growth (below this
line) and a region of wealth decline (above it). Since the two lines are
parallel, we see that there are three regions in the state space (W; c)
di¤erentiated by distinct dynamic properties of the system (Wt ; ct ).
Above the line c = rW + y, wealth declines and consumption grows.
In the band rW < c < rW + y, both wealth and consumption grow.
Below the line c = rW , wealth grows and consumption declines.
The qualitative conclusions we can obtain from the phase diagram
are as follows. Inside the band rW < c < rW + y, solution paths
increase in both the c and the W direction and fall into one of the
following three types. Paths of the …rst type will reach the upper
straight line c = rW + y, where they bend backward as Wt begins
to decrease while ct continues to increase. Paths of the second type
will reach the lower straight line c = rW , where they bend downward
with ct declining and Wt continuing to increase. Note that none of the
paths of the …rst or second type return to the band rW < c < rW + y
once they leave it. Paths of the third type will stay inside the band
rW < c < rW + y forever.
Further characterization of the solution paths can be obtained analytically in the special case in which the Euler equation (22) is linear
or numerically in other cases. In the remainder of this article, we will
focus on the case with a linear Euler equation and discuss analytical
solutions. As we will see in the next section, the Euler equation is linear when the utility function u is quadratic. In Appendix B, we brie‡y
discuss how the results change for other utility functions, in particular
for preferences exhibiting constant relative risk aversion (CRRA).

EXACT SOLUTION WITH LINEAR
EULER EQUATION

We specialize the utility function to
u(c) =

1
(c
2

B)2

Grochulski and Zhang: Saving for Retirement

and restrict its domain to c
B. Under this speci…cation, marginal utility is linear and therefore the Euler equation (22) is linear in
consumption as well as in wealth:
dct
= (ct rWt ) :
(23)
dt
The system of di¤erential equations (1), (23) is now linear and can
be solved in closed form. In particular, we have the following lemma
providing analytical expressions for all solution paths of the system (1),
(23).
Lemma 1 Let f(Wt ; ct ); t 0g be a solution path. If there exists
such that c = rW + y, then
ry
ct = rWt +
1 e (r+ )( t) + ye (r+ )( t) :
(24)
r+
If there exists such that c = rW , then
ry
ct = rWt +
1 e (r+ )( t) :
(25)
r+
Otherwise,
ry
ct = rWt +
:
(26)
r+
Proof. In Appendix A.
Figure 2 plots several sample solution paths f(Wt ; ct ); t 0g of the
three types given in the above lemma. Solution paths (24) bend backward with wealth declining over time at all dates t > , where is such
that c = rW + y. None of these solution paths will be optimal for the
agent because, as we saw earlier, it is never optimal for the agent to see
his …nancial wealth decrease while he is saving for retirement. Along
all solution paths (25) and (26) wealth is increasing. These paths,
therefore, are our candidates for the optimal path of consumption and
saving prior to retirement.

The Optimal Accumulation Path
As we saw in Section 3, the agent’s retirement decision is determined by
a simple wealth threshold rule. The agent retires as soon as his wealth
reaches the level W . The threshold W depends on the parameters
r, , and y, as shown in (14). At retirement (and forever after), the
agent’s optimal level of consumption is ct = rW . The Euler equation
(23) and the wealth accumulation equation (1) tell us that prior to
retirement the agent follows one of the non-backward-bending paths
depicted in Figure 2. For a given value of W , which one of these paths
will the agent follow?

Federal Reserve Bank of Richmond Economic Quarterly

Figure 2 Solution Paths with Quadratic Utility

Notes: Parameters used in this plot: y = 1, r = 0:04,

= 0:02.

Since the agent’s wealth and consumption remain constant in retirement, in Figure 2 the evolution of wealth and consumption after
voluntary retirement is represented by a single point for each threshold
value W . That point is (W ; rW ). Thus, once the agent retires, the
time path of his wealth and consumption is absorbed at (W ; rW ).
It is easy to see in Figure 2 that for each value W
0 there is a
unique solution path f(Wt ; ct ); t 0g leading to the point (W ; rW ).
That path is the optimal path for the agent whose retirement wealth
threshold is W . Why this path? Because all other paths would imply
a jump in consumption at retirement, which the agent wants to avoid.
Because his utility function is concave, the agent prefers a smooth consumption path with no jump at retirement. The level of consumption
in voluntary retirement, rW , thus determines the optimal accumulation path the agent follows prior to retirement. It is the single path
that intersects the line c = rW at W = W .

Grochulski and Zhang: Saving for Retirement

For example, the solution path labelled A in Figure 2 crosses the
line c = rW at W = 8. Thus, this solution path is optimal for the agent
whose desired retirement wealth is W = 8. Likewise, the solution path
B is optimal for the agent whose desired retirement wealth is W = 15.
The solution path C follows a straight line parallel to the line c = rW
and therefore never crosses it. This solution path is optimal for the
agent whose retirement threshold is W = 1, which means the agent
plans to never retire voluntarily. From the formula (14) we see that
W = 1 when
= 0, i.e., when the agent does not at all value
the extra leisure he can obtain by retiring. Since the value of the extra
leisure is zero for this agent, it is natural that he never chooses to retire.
This is the case studied in standard in…nite-horizon models of optimal
saving and consumption decisions, e.g., Ljungqvist and Sargent (2004,
Ch. 16).
Note that the argument implying that the agent’s preferred path of
wealth and consumption is the one that leads to the point (W ; rW )
does not use the assumption of quadratic preferences. Rather, this
argument is based on the agent’s preference for smooth consumption,
and so it applies to any concave utility function u. Thus, although the
shape of the optimal path of wealth accumulation and consumption
f(Wt ; ct ); t 0g will in general not be the same as that presented in
Figure 2, it will be true for any concave utility function that the optimal
accumulation path is the unique solution path that leads to the point
(W ; rW ).6
Also, the phase diagram in Figure 1, which works for any concave
u, shows that the only way for a solution path to approach (W ; rW )
is through the middle band rW < c < rW + y of the state space
(W; c), where Wt and ct are both strictly increasing over time. This
con…rms the validity of the assumption we made in Section 2 about
wealth following an increasing path prior to retirement.

Planned Retirement and Optimal
Saving Rate
Figure 2 provides a clear illustration of the following point. When the
option to retire is added to the standard, in…nite-lived-agent model of
optimal consumption and saving, the model’s prediction on the optimal
amount of saving unambiguously increases.
In its textbook version (see Ljungqvist and Sargent [2004]), the
standard model of optimal consumption and saving decisions abstracts
6

See Appendix B for the case of the CRRA utility function.

Federal Reserve Bank of Richmond Economic Quarterly

from retirement. Labor income ‡uctuates stochastically, but the agent
does not have an option to retire and end his ‡ow of labor income
altogether. The model we consider in this article assumes a particularly
simple form of stochastic income ‡uctuations (labor income is a step
function initially positive and jumping down to zero at a random date
f ), but allows for endogenous retirement.
In the special case with = 0, our model is a version of the standard model with no retirement. As we saw earlier, when
= 0 the
agent never retires voluntarily, so his labor income e¤ectively follows
an exogenous process, as in the standard model. From (26) we have in
that case that the optimal fraction of labor income y to be saved by
the agent is
dWt =dt
y

rWt + y
y
r
= 1
r+
=

Thus, the standard model without retirement would predict r+ as the
agent’s optimal rate of saving out of labor income. With positive utility
of leisure, > 0 , our model predicts voluntary retirement in …nite time
as well as a higher optimal rate of saving prior to retirement. From
(25) we have
dWt =dt
y

rWt + y
y
r
= 1
r+

ct
1

(r+ )(

;

where the strict inequality follows from the agent’s time to retirement
being …nite, i.e.,
t < 1. Given that people do save for retirement
in reality, models that disregard retirement underpredict the optimal
rate of saving. Figure 2 shows this very clearly: The in…nite-horizon
solution path that runs parallel to the line c = rW is everywhere above
all solution paths that cross this line.
That the optimal saving rate should be higher when agents save for
retirement is of course very intuitive. With retirement, the agent’s labor
income is more front-loaded relative to the case without retirement. To

Grochulski and Zhang: Saving for Retirement

smooth this front-loading out, the agent saves more. Our analysis lets
us see this point clearly in Figure 2.7

Time to Retirement
As we see in (25), the agent’s optimal consumption at time t depends
on the agent’s current wealth Wt and the amount of time left before his
planned retirement,
t. The agent’s target retirement wealth level
W is given in (14). But how do we …nd the agent’s target retirement
time ?
From the law of motion for wealth prior to retirement, (1), we have
Z t
Z t
(y (ct+s rWt+s ))ds:
dWt+s = Wt +
W = Wt +
0

Using the retirement condition W = W and the consumption rule
(25) we have
Z t
r
W = Wt +
1
(1 e (r+ )s ) yds
r
+
0
r
= Wt +
(
t) y +
1 e (r+ )( t) y: (27)
r+
(r + )2
For any given values for r, , y, W , and Wt , this condition can be
solved for the agent’s planned time to retirement
t. Because the
right-hand side of (27) is increasing in both
t and Wt , the time to
retirement is decreasing in current wealth.8
In sum, the dynamics of consumption and wealth accumulation are
as follows. The agent determines his target retirement wealth level W ,
as in (14). Then the agent follows the unique wealth accumulation and
consumption path f(Wt ; ct ); t 0g in Figure 2 that leads to the point
(W ; rW ). How far away from the retirement point the agent starts on
this path depends on his initial wealth W0 . Unless he loses his job before
reaching wealth W , the agent retires voluntarily as soon as his wealth
attains W . After retirement, he consumes at the constant rate rW
and his …nancial wealth remains constant at W . Thus, the solution
path the agent follows in Figure 2 is absorbed at the point (W ; rW ).
If the agent is forced into involuntary retirement at some date f < ,
i.e., when his wealth is W f < W , his consumption jumps down at f
from his preferred accumulation path to the point (W f ; rW f ), and is
absorbed there. That is, consumption stays constant in retirement at
7
The same is true in the case of CRRA preferences we discuss in Appendix B. See
Figure 7.
8
This also con…rms that wealth grows over time while the agent is working.

68
the level rW
W .

Federal Reserve Bank of Richmond Economic Quarterly

< rW and …nancial wealth stays constant at W

Permanent Income Hypothesis
It is well known, see Ljungqvist and Sargent (2004, Ch. 16), that with
quadratic preferences the optimal saving and consumption rule satis…es
the permanent income hypothesis (PIH). Under PIH, it is optimal for
the agent at each point in time to consume simply the income from his
total wealth, where total wealth includes …nancial wealth and human
capital. Human capital of an agent is de…ned as the present value of
all the labor income that the agent is yet to earn. Thus, permanent
income has two components: the income from currently held …nancial
wealth and the income from currently held human capital.
That PIH holds in our model is most clearly evident in (26), i.e.,
in the case with = 0 in which the agent never retires voluntarily. If
the agent’s stock of …nancial wealth is Wt , his permanent income from
it is rWt because, as we saw earlier, if the agent consumes rWt , he
never depletes his …nancial wealth and therefore is able to maintain
this consumption forever. If the agent is working at t, the expected
present value of his future labor income is
Z f
y
:
E
e rs yds =
r+
0
Permanent income from human capital y=(r + ) is ry=(r + ) because
this is the perpetual ‡ow equivalent of stock y=(r + ). According to
PIH, with …nancial wealth Wt and with human capital y=(r + ), the
agent’s consumption at t should be r(Wt + r+y ), which it is, as we see
in (26).
The agent’s optimal rule for consumption and saving obeys PIH
also when he chooses to voluntarily retire at a future date . In this
case, the agent’s human capital as of t < is
"Z
#
minf f ; g
y
rs
E
e yds =
1 e (r+ )( t) :
(28)
r
+
0
Thus, (25) is consistent with PIH because the agent in this case as well
consumes exactly the return on his …nancial and human capital at all
times. Note that the value in (28) is less than y=(r + ) because a part
of expected future income is lost due to the agent’s planned retirement
at . The closer t is to , the lower the agent’s human capital. Because
the agent saves at all t < , however, his …nancial wealth Wt grows as
t gets closer to . It fact, …nancial wealth grows faster than human
capital declines, and so the agent’s permanent consumption increases

Grochulski and Zhang: Saving for Retirement

over time for as long as the agent does not lose his job. As wealth
approaches W , the agent’s human capital goes down to zero smoothly,
and his consumption increases smoothly to rW . If the agent loses his
job involuntarily at some date f before his …nancial wealth reaches
W , the agent’s human capital discontinuously jumps down to zero
and his permanent consumption jumps down to just the return on his
…nancial assets, rW f .

RETIREMENT SAVING AND THE
JOB LOSS RISK

In this section, we study the dependence of the optimal consumption
and saving plan on the job loss rate .
Proposition 1 At any Wt < W , the larger the job loss intensity
, the lower consumption ct , the higher the wealth accumulation rate
dWt =dt, and the shorter the time to planned retirement
t. If ! 1,
then ct ! rWt , dWt =dt ! y, and
t ! (W
Wt )=y.
Proof. In Appendix A.
This proposition shows that if we compare two agents identical in
all respects (same wealth, same income) except for the job loss rate ,
the agent with larger job loss risk will consume less and save more than
the other agent. Intuitively, the agent with higher holds less human
capital than the agent whose is lower. The labor income ‡ow rate y,
the same for both agents, therefore, is higher relative to total wealth
for the agent with higher , and so he will save a larger portion of y
than the other agent. In other words, labor income y is less permanent
for the agent with higher , so intertemporal consumption smoothing
implies he will save more. Figure 3 illustrates this point by plotting
optimal paths for consumption and wealth for several values of .
This comparative statics result can be interpreted as showing the
agent’s response to a completely unanticipated shock to the job loss
risk the agent faces in our model. Under the parametrization used in
Figure 3, if = 0:02, the agent will follow the highest of the three
accumulation paths plotted in that …gure. If at some point prior to
retirement the intensity parameter jumps to 0:1, the agent will switch
at that point to the lowest of the three paths. This means that his
saving rate will increase and consumption will decrease without any
change to his current income. This example illustrates a response of

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Figure 3 Optimal Accumulation Paths for Three Values of
the Job Loss Rate

Notes: Other parameters used in this plot: r = 0:04, y = 1,

such that W = 20.

optimal consumption to a change in the expectations the agent holds
about the future.9
If is very large, then, as Proposition 1 shows, the agent saves close
to 100 percent of his labor income y and consumes close to rWt . This
again is intuitive, as when is large, the agent’s human capital is close
to zero and …nancial wealth constitutes the bulk of his total wealth.
The level of permanent consumption he can a¤ord is thus close to the
level he could maintain if he had lost his job already, which with assets
Wt is exactly rWt .
9
The discussion in this paragraph assumes that the agent does not anticipate that
could jump, and that once it does jump, the agent …rmly expects it to never jump again.
Clearly, this is an oversimpli…cation. We can expect, however, that our conclusion here
continues to hold when the jumps in
are anticipated. That is, although the shape of
the accumulation paths in Figure 3 must be adjusted, we expect consumption to decline
when
increases in a model in which changes in
are anticipated by the agent.

Grochulski and Zhang: Saving for Retirement

Figure 4 Dependence of Time to Retirement on the Job Loss
Rate

Notes: Parameters used in this plot are the same as those in Figure 3.

Proposition 1 also shows that conditional on not losing the job
before the planned retirement date , the agent with larger will reach
his desired retirement wealth level W faster. Note that the theoretical
limit with ! 1 of the time to retirement, (W
Wt )=y, is consistent
with the agent saving 100 percent of his labor income and living only
o¤ his asset income already before retirement.
Figure 4 plots the planned time to retirement
t against wealth
Wt for several values of . In the example presented in that …gure, we
have r = 0:04, which makes one unit of time correspond roughly to one
year. Annual labor income y is normalized to 1, and W = 20, which
means that the agent wants to retire as soon as his stock of wealth
reaches the equivalent of 20 years of labor income. With = 0:02,
meaning the event of involuntary and permanent job loss on average
occurs once in 50 years, the agent who starts out with zero initial
wealth plans to retire after about 32 years. With = 0:04, i.e., when

Federal Reserve Bank of Richmond Economic Quarterly

involuntary retirement is a once-in-a-quarter-century event, the agent
plans to retire after roughly 28.5 years. With = 0:01, the permanent
job loss shock becomes a once-in-a-decade event in expectation. In this
case, the agent plans to retire after 25 years. These numbers illustrate
the fact that the agent can only partially insure himself against the
permanent job loss shock in our model. With = 0:02, the probability
that an agent with zero wealth reaches voluntary retirement is e 0:02 32 ,
which equals roughly 53 percent. For an agent with the same initial
wealth but with
= 0:1; this chance is only e 0:1 25 , i.e., about 8
percent.
Di¤erentiating with respect to the expressions for optimal consumption in (25) and (26), it is easy to check that the response of ct to
a given change in is stronger the longer the agent’s planned time to
retirement
t. In particular, the response of consumption to changes
in the job loss risk is the strongest in the case of = 0, where the agent
plans to never retire voluntarily. This result is very intuitive given that
fast planned retirement means human capital is a small portion of the
agent’s total wealth.

ADDITIONAL COMPARATIVE
STATICS RESULTS

With closed-form solution for the optimal path of saving and consumption, we can provide several additional comparative statics results.
We saw already in Figure 2 how the optimal path of consumption
and wealth accumulation depends on the parameter . In (14), higher
leisure utility implies a lower retirement threshold W . In Figure 2,
we see that lower W means faster retirement with a higher saving rate
along the optimal accumulation path.
We can also examine how consumption, saving, and the retirement
decision depend on the level of labor income y. We know from (14)
that the retirement threshold wealth level W is increasing in y. Using
(25), it is not hard to show that if two agents have the same …nancial
wealth Wt and face the same job loss rate , the agent with higher labor
income y will consume more and retire later. The numerical example
given in Figure 5 illustrates this point. In that …gure, paths leading to
lower retirement points are everywhere below those leading to higher
retirement wealth thresholds. Those higher paths correspond to higher
labor income y earned during employment.
Finally, we examine how the solution to the agent’s optimal consumption, saving, and retirement problem depends on the real interest
rate r. Dashed lines in Figure 6 show three accumulation paths, each
optimal at a di¤erent level of r. That the retirement wealth threshold

Grochulski and Zhang: Saving for Retirement

Figure 5 Consumption and Wealth Accumulation for Three
Di erent Values of Labor Income

Notes: Other parameters as in Figure 2.

W is lower at higher r can be seen from the terminal points of the
accumulation paths in Figure 6, or directly from the formula for W
given in (14). Since the marginal value of wealth in retirement u0 (rWt )
decreases in r, it is intuitive that when r is higher the agent chooses
to give up labor income y in return for utility earlier, i.e., at a lower
wealth threshold. Figure 6 shows that prior to retirement, at higher r
both the agent’s consumption ct and his interest income rWt are higher.
Interest income is represented in Figure 6 by the straight lines rW connecting the origin to the terminal points of the optimal accumulation
paths. How the wealth accumulation rate dWt =dt = rWt ct + y depends on r is determined by the magnitudes of ct and rWt . In fact,
the rate of wealth accumulation is increasing in r at high levels of
wealth Wt and decreasing in r at low levels of Wt . For example, at
Wt = 12; the vertical distance between (any two) dashed lines (representing ct ) is smaller than the vertical distance between the solid lines

Federal Reserve Bank of Richmond Economic Quarterly

Figure 6 Optimal Consumption and Wealth Accumulation
Paths at Three Levels of the Interest Rate

(representing rWt ). At Wt = 1, the opposite is true. The cumulative
e¤ect of these di¤erences on the agent’s wealth is positive. With some
algebra that we omit here, it can be shown that the agent retires faster
when r is higher. That is, for any given Wt the agent’s time to planned
retirement,
t, is shorter the higher the interest rate r.

CONCLUSION

This article studies optimal consumption and saving decisions in an
in…nite-horizon model that allows for endogenous retirement. Relative
to the standard model with no retirement, the optimal saving rate is
higher. An increase in the job loss risk decreases consumption, even
without the actual job loss occurring. Accounting for retirement subdues the magnitude of the response in consumption to changes in the
job loss risk. These results may be important for quantitative analyses
of observed consumption and saving decisions.

Grochulski and Zhang: Saving for Retirement

The strong assumption we make on the shape of the agent’s pro…le
of labor income lets us abstract in this article from borrowing constraints. Since his income can only decrease, the agent never wants
to borrow in our model, so the no-borrowing constraint is natural in
our analysis, and it never binds. Increasing and hump-shaped paths
of income are standard in life-cycle models. Incorporating such paths
into our model would require an extension of our analysis accounting
for the possibility of binding borrowing constraints.
Our analysis of optimal saving for and timing of retirement can be
extended to study other types of actions for which savings are important. For instance, due to down-payment requirements, the optimal
timing of a house purchase by a household will depend on the …nancial
wealth of the household. Our analysis in this article can be adapted
to study jointly the saving decisions and the optimal timing of this
purchase.

APPENDIX:

APPENDIX A

Proof of Lemma 1
Multiplying (1) by r and subtracting it from (23) we obtain a linear
di¤erential equation
d(cs rWs )
= (r + )(cs rWs ) ry;
ds
which, with the notation zs = cs rWs ; we can write more compactly
as
dzs
= (r + )zs ry:
(29)
ds
The solution to this equation is standard. Di¤erentiating zs e (r+ )s ;
we have
d zs e

(r+ )s

= dzs e

(r+ )s

(r + )zs e

(r+ )s

ds =

rye

(r+ )s

ds;

where the second equality uses (29). Integrating from t to and solving
for zt yields
ry
zt =
1 e (r+ )( t) + z e (r+ )( t) :
r+
Writing the boundary condition c = rW + y as z = y and using it in
the above general solution gives us (24). With the boundary condition
c = rW , we have z = 0, which gives us (25). For (26), we take a
limit of (25) with ! 1.

Federal Reserve Bank of Richmond Economic Quarterly

Proof of Proposition 1
Write (25) as
r
y 1 e (r+ )( t)
r+
and note that the right-hand side of this equality is decreasing in
and goes to zero as ! 1. This proves the proposition’s conclusions
about ct and, using (1), dWt =dt. Next, write (27) as
ct

Wt =

rWt =

(

t) y +

r
1
(r + )2

(r+ )(

and check that the right-hand side of this equality is strictly increasing
with respect to both and
t. Because W does not depend on ,
the left-hand side is constant. Thus, the time to retirement
t must
decrease when increases to keep the right-hand side constant.

APPENDIX:

APPENDIX B

Figure 7 provides the analog of Figure 2 for a nonquadratic utility function u. In particular, this …gure depicts numerically computed solution
paths to the system of di¤erential equations (1)–(22) for constant relative risk aversion (CRRA) preference represented by the utility function
u of the form
c1
u(c) =
.
1
Qualitatively, these graphs are similar to one another for all values of
> 0.
Our analysis determining the optimal accumulation path for a given
voluntary retirement wealth threshold W from Section 5 is unchanged.
The main di¤erence between CRRA preferences and quadratic preferences is that the permanent income hypothesis does not hold under
CRRA preferences. With CRRA preferences, agents have the so-called
precautionary motive for saving, which is absent under quadratic preferences. When the precautionary saving motive is present, the agent
will increase the amount he saves in response to an increase in the riskiness of his income process, holding his expected income constant. (See
Ljungqvist and Sargent [2004] for a general discussion of precautionary
savings.)
In Figure 7, precautionary savings are best seen by comparing the
solution path labelled C with the dotted line labelled P IH. The

Grochulski and Zhang: Saving for Retirement

Figure 7 Solution Paths for CRRA Preferences

solution path C in Figure 7 is analogous to the solution path C in
Figure 2. It is the single solution path that never leaves the middle
band of the graph bounded by the lines c = rW and c = rW + y. It is
the optimal solution path under CRRA preferences for an agent whose
= 0, i.e., an agent who never chooses to retire voluntarily. The line
labelled P IH in Figure 7 is the solution path that would be optimal
for that agent if he did not have a precautionary saving motive (i.e., it
is an exact replica of the solution path C from Figure 2). At any level
of wealth Wt , the vertical distance in Figure 7 between line P IH and
the solution path C measures precautionary saving of the agent with
CRRA preferences. As we see, precautionary saving is positive at all
wealth levels and its magnitude decreases in Wt . In fact, solution C
converges to line P IH as Wt ! 1.
As in Figure 2, each solution path crossing the line c = rW is
an optimal accumulation path for an agent whose value of the leisure
preference parameter
is strictly positive. In these cases, as well,
precautionary saving can be seen by comparing corresponding solution

Federal Reserve Bank of Richmond Economic Quarterly

paths in Figures 2 and 7. For any given voluntary retirement wealth
threshold W > 0, the solution path leading to the retirement point
(W; c) = (W ; rW ) will in Figure 2 be strictly above the path leading
to the same path in Figure 7. The vertical distance between these two
paths will represent precautionary saving. All solution paths in Figure
7 converge to zero consumption when wealth goes to zero, while in
Figure 2 they do not. Comparing solutions with voluntary retirement
in …nite time, as in the case of no voluntary retirement, we thus see
that the precautionary saving motive is the strongest at very low wealth
levels.

APPENDIX:

APPENDIX C

In this appendix, we discuss an extension of our model in which the
option value of delaying retirement is positive.
Let us add a positive labor income shock to our model. That is,
instead of assuming that at all times prior to retirement the agent’s
labor income is constant, suppose it can increase from y to y > y.
Suppose this upward jump arrives with Poisson intensity > 0. Also,
let’s assume the job loss shock is independent of the level of income
and, as before, it arrives with Poisson intensity > 0.
We will show that with this positive income shock, the agent with
income y will not choose to retire as soon as his wealth reaches the
threshold W but rather will prefer to keep working. The reason why
the agent prefers to keep working is that postponing retirement has
a positive option value when there is a chance that his labor income
increases in the future.
Let J(Wt ) be the maximal utility value the agent can obtain when
his income is already high, i.e., y. Because once it hits y income stays
constant until retirement; our previous analysis applies: The agent
whose income is y will want to retire exactly when his wealth hits the
threshold
1
W = u0
r

1
> u0
r

=W :

We will show, however, that with low income y the agent will not want
to retire as soon as his wealth reaches W . That is, the retirement
rule with wealth threshold level W that we obtained in Section 3 is
no longer optimal for the agent.

Grochulski and Zhang: Saving for Retirement

Consider the following strategy for an agent whose labor income
is low, y, and whose wealth is Wt . Suppose the agent works over a
small time interval [t; t + h). By time t + h, three things can happen.
The agent loses his job, gets a promotion, or neither. Suppose the
agent behaves optimally after a promotion thus obtaining in that event
the value J(Wt+h ). In the event of the job loss, he behaves optimally
in retirement and so he obtains V (Wt+h ). If neither promotion nor
job loss happen, suppose the agent retires voluntarily at t + h, thus
obtaining the value V (Wt+h ) in this event as well. Thus, the agent’s
strategy is to postpone retirement by h and see if he gets a promotion.
If he does not, he quits. Denote by V h (Wt ) the value that this strategy
gives the agent as of date t.
We proceed analogously to Section 4. We have
nR
h
h
V (Wt ) = max 0 e rs u(c)ds
c

+ 1

)J(Wt+h ) + e
o
V (Wt+h )

V (Wt+h )

with wealth following (1) between t and t + h, as the agent works
between t and t + h. Because h is small, we use the …rst-order approximation and express V h (Wt ) as
V h (Wt ) = max V (Wt ) + u(c) + J(Wt )
c

(r + )V (Wt )

t
h :
+ V 0 (Wt ) dW
dt

Next, we compare this value to the value of retiring immediately at t,
which we know to be V (Wt ). We have that the value of postponing
retirement by at least h is strictly preferred to retiring immediately,
i.e., V h (Wt ) > V (Wt ), if and only if
max V (Wt ) + u(c) + J(Wt )
c

+ V 0 (Wt )(rWt + y

c) h

> max V (Wt ) + u(c) +
c

(r + )V (Wt )

rV (Wt ) + V 0 (Wt )(rWt

c) h :

Dividing by h, simplifying terms, and removing the identical maximization problems with respect to c on both sides of this condition simpli…es
it to
J(Wt )

V (Wt ) + V 0 (Wt )y > :

Now we note that J(W ) V (W ) > 0 because with high labor income
y the agent only wants to retire with wealth W > W and not earlier.
By de…nition of W , we have V 0 (W )y = . Therefore,
J(W )

V (W ) + V 0 (W )y > :

Federal Reserve Bank of Richmond Economic Quarterly

This means that with low income y and wealth W ; the agent prefers to
postpone retirement.
The reason for this is that the term
J(W ) V (W ) is strictly positive. This term represents the option value of delaying retirement. For as long as the agent is not retired,
he has a chance to see his labor income increase, in which case he would
prefer to continue working until his wealth reaches W . Because retirement is permanent, by retiring with wealth W < W , the agent closes
this possibility to himself or, in other words, gives up this option. By
delaying retirement, he keeps this option open.
By continuity, the above condition holds in the neighborhood of
W , i.e., also for some wealth Wt > W . At that wealth level we have
V 0 (Wt )y < , i.e., in terms of his current payo¤ the agent would be
strictly better o¤ to retire immediately. He does not choose to do so,
however, because the option value
J(Wt ) V (Wt ) is larger than
the payo¤ from retiring
V 0 (Wt )y.

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