American Economic Review 101 (August 2011): 1912–1932
http://www.aeaweb.org/articles.php?doi
=
10.1257/aer.101.5.1912
1912
In the absence of large income effects, a neoclassical model of labor supply pre-
dicts a positive wage elasticity of hours. However, Camerer et al. (1997) collected
data on the daily labor supply of New York City cab drivers who, unlike most work-
ers in modern economies, are free to choose their own hours, and found a strongly
negative elasticity of hours with respect to their closest analog of a wage, realized
earnings per hour. In Camerer et al.’s dataset, realized earnings per hour (which they
call the “wage”) is uncorrelated across days but positively serially correlated within
a day, so that high earnings early in a day signal higher earnings later that day, and
a neoclassical model predicts a positive elasticity even though realized earnings per
hour is not precisely a wage. If instead realized earnings per hour is serially uncor-
related within a day, as Farber (2005) shows is roughly true in his dataset (see, how-
ever, our analysis in Section IIC), then a driver with high early earnings experiences
a small change in income but no change in expected wage, and a neoclassical model
predicts an elasticity near zero. A neoclassical model could explain Camerer et al.’s
New York City Cab Drivers’ Labor Supply Revisited:
Reference-Dependent Preferences with Rational-
Expectations Targets for Hours and Income
†
By V P. C J M*
This paper proposes a model of cab drivers’ labor supply, building
on Henry S. Farber’s (2005, 2008) empirical analyses and Botond
Koszegi and Matthew Rabin’s (2006; henceforth “KR”) theory of
reference-dependent preferences. Following KR, our model has tar-
gets for hours as well as income, determined by proxied rational
expectations. Our model, estimated with Farber’s data, reconciles
his nding that stopping probabilities are signicantly related to
hours but not income with Colin Camerer et al.’s (1997) negative
“wage” elasticity of hours; and avoids Farber’s criticism that esti-
mates of drivers’ income targets are too unstable to yield a useful
model of labor supply. (JEL J22, J31, L92)
* Crawford: Department of Economics, University of Oxford, Manor Road, Oxford OX1 3UQ, United Kingdom,
and Department of Economics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0508
(e-mail: vincent.crawford@economics.ox.ac.uk); Meng: Guanghua School of Management, Peking University,
Beijing, 100871, China (e-mail: jumeng@gsm.pku.edu.cn). Meng’s work on this paper was completed while
she was at the Department of Economics, University of California, San Diego. We thank Henry Farber, Colin
Camerer, and Linda Babcock for sharing their data, and Nageeb Ali, Dan Benjamin, Camerer, David Card, Julie
Cullen, Gordon Dahl, Stefano Dellavigna, Kirk Doran, Farber, Uri Gneezy, Roger Gordon, Nagore Iriberri, Botond
Koszegi, Jaimie Lien, David Miller, Ulrike Malmendier, Ted O’Donoghue, Matthew Rabin, Adam Sanjurjo, Joel
Sobel, and the referees for their insightful comments and suggestions.
†
To view additional materials, visit the article page at
http://www.aeaweb.org/articles.php?doi=10.1257/aer.101.5.1912.
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strongly negative elasticities only via an implausibly large negative serial correla-
tion of realized earnings per hour.
To explain their results, Camerer et al. informally proposed a model in which driv-
ers have daily income targets and work until the target is reached, and so work less
on days when earnings per hour are high. Their explanation is in the spirit of Daniel
Kahneman and Amos Tversky’s (1979) and Tversky and Kahneman’s (1991) pros-
pect theory, in which a person’s preferences respond not only to income as usually
assumed, but also to a reference point; and there is “loss aversion” in that the person is
more sensitive to changes in income below the reference point (“losses”) than changes
above it (“gains”). If a driver’s reference point is a daily income target, then loss aver-
sion creates a kink that tends to make realized daily income bunch around the target,
so that hours have a negative elasticity with respect to realized earnings per hour.
Farber (2008, p. 1,069) suggests that a nding that labor supply is reference
dependent would have signicant policy implications:
Evaluation of much government policy regarding tax and transfer programs depends
on having reliable estimates of the sensitivity of labor supply to wage rates and income
levels. To the extent that individuals’ levels of labor supply are the result of optimiza-
tion with reference-dependent preferences, the usual estimates of wage and income
elasticities are likely to be misleading.
Although Camerer et al.’s analysis has inspired a number of empirical studies of
labor supply, the literature has not yet converged on the extent to which the evidence
supports reference dependence.
1
Much also depends on reference dependence’s
scope and structure: If it were limited to inexperienced workers or unanticipated
changes, its direct relevance to most policy questions would be small, though it might
still have indirect policy relevance via its inuence on the structure of labor relation-
ships. This paper seeks to shed additional light on these issues, building on two recent
developments: Farber’s (2005, 2008) empirical analyses of cab drivers’ labor supply
and KR’s (2006; see also 2007, 2009) theory of reference-dependent preferences.
Farber (2005) collected and analyzed data on the labor supply decisions of a new
set of New York City cab drivers. He found that, before controlling for driver xed
effects, the probability of stopping work on a day is signicantly related to realized
income that day, but that including driver xed effects and other relevant controls
renders this effect statistically insignicant.
Farber (2008) took his 2005 analysis a step further, introducing a structural
model based on daily income targeting that goes beyond the informal explanations
in previous empirical work. He then estimated a reduced form, treating drivers’
income targets as latent variables with driver-specic means and driver-indepen-
dent variance, both assumed constant across days of the week—thus allowing
the target to vary across days for a given driver, but only as a random effect.
1
KR (2006) and Farber (2008) survey some of the empirical literature. Gerald S. Oettinger’s (1999) eld study
found increased daily participation by stadium vendors on days on which the anticipated wage was higher, as sug-
gested by the neoclassical model, and in seeming contrast to Camerer et al.’s nding of a negative response of hours
to (partly unanticipated) increases in wage. Ernst Fehr and Lorenz Goette’s (2007) eld experiment found increased
participation by bicycle messengers, but reduced effort, in response to announced increases in their commission.
They argued that effort is a more accurate measure of labor supply and concluded that the supply of effort is refer-
ence dependent.
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Constancy across days of the week is violated in the sample, where Thursdays’
through Sundays’ incomes are systematically higher than those of other days, and
the hypothesis that income is constant across days of the week is strongly rejected
( p-value 0.0014, F-test with robust standard errors). Farber included day-of-the-
week dummies in his main specications for the stopping probability, but this
turns out to be an imperfect substitute for allowing the mean income target to vary
across days of the week.
Farber found that a sufciently rich parameterization of his targeting model ts
better than a neoclassical model, and that the probability of stopping increases sig-
nicantly and substantially when the target is reached; but that his model cannot
reconcile the increase in stopping probability at the target with the smooth aggre-
gate relationship between stopping probability and realized income. Further, the esti-
mated random effect in the target is large and signicantly different from zero, but
with a large standard error, which led him to conclude that the targets are too unstable
to yield a useful reference-dependent model of labor supply (Farber 2008, p. 1,078):
There is substantial inter-shift variation, however, around the mean reference income
level. … To the extent that this represents daily variation in the reference income level
for a particular driver, the predictive power of the reference income level for daily
labor supply would be quite limited.
KR’s (2006) theory of reference-dependent preferences is more general than
Farber’s (2008) model in most respects but takes a more specic position on how
targets are determined. In KR’s theory applied to cab drivers, a driver’s preferences
reect both the standard consumption utility of income and leisure and reference-
dependent “gain-loss” utility, with their relative importance tuned by a parameter.
As in Farber’s model, a driver is loss-averse; but he has a daily target for hours as
well as income, and working longer than the hours target is a loss, just as earning
less than the income target is. Finally, KR endogenize the targets by setting a driver’s
targets equal to his theoretical rational expectations of hours and income, reecting
the belief that drivers in steady state have learned to predict their distributions.
2
This paper uses Farber’s (2005, 2008) data to reconsider the reference depen-
dence of cab drivers’ labor supply, adapting his econometric strategies to estimate
models based on KR’s (2006) theory. Section I introduces the model. Following KR,
we allow for consumption as well as gain-loss utility and hours as well as income
targets; but when we implement the model we follow Farber (2008) in assuming that
drivers are risk-neutral in income.
To complete the specication, we must describe how a driver’s targets are deter-
mined and, for some of our analysis, how he forms his expectations about earnings
hour by hour during a day. In an important departure from Farber’s approach, we
follow KR in conceptualizing drivers’ targets and expected earnings as rational
2
In theory there can be multiple expectations that are consistent with the individual’s optimal behavior, given
the expectations. KR use a renement, “preferred personal equilibrium,” to focus on the self-conrming expecta-
tions that are best for the individual. Most previous analyses have identied targets with the status quo; but as KR
note, most of the available evidence does not distinguish the status quo from expectations, which are usually close
to the status quo. Even so, our analysis shows that KR’s rational-expectations view of the targets has substantive
implications for modeling cab drivers’ labor supply. KR’s view of the targets has also been tested and supported in
laboratory experiments by Johannes Abeler et al. (2011).
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expectations, but for simplicity we depart from KR in treating them as point
expectations rather than distributions. Because KR’s model has no errors, their
distributions are necessary for the existence of deviations from expectations, with-
out which their model reduces to a neoclassical model. Our model has errors and
so has deviations even with point expectations. We operationalize the targets and
expected earnings via natural sample proxies with limited endogeneity problems
as explained below, for expected earnings assuming for simplicity that earnings
per hour are serially uncorrelated within a day (as well as across days). This last
simplication is motivated by Farber’s (2005) nding, in a detailed econometric
analysis of his dataset, of only a weak and insignicant relationship, which led
him to argue that hourly earnings are so variable and unpredictable that “predict-
ing hours of work with a model that assumes a xed hourly wage rate during the
day does not seem appropriate.” We note however that Camerer et al. did nd
some within-day predictability of earnings in their dataset. It also seems plausible
that drivers on the ground may be able to predict their earnings better than even
the most careful econometrics. Given no serial correlation and risk-neutrality in
income, and ignoring option value in the stopping decision, a driver’s expected
hourly earnings are equivalent to a predetermined (though random) daily schedule
of time-varying wages.
3
If the weight of gain-loss utility is small, our model mimics a neoclassical labor-
supply model, so that the elasticity of hours with respect to earnings per hour is
normally positive. If the weight of gain-loss utility is large, perfectly anticipated
changes in earnings per hour still have this neoclassical implication because gain-
loss utility then drops out of a driver’s preferences. However, unanticipated changes
may then have nonneoclassical implications. In particular, when the income tar-
get has an important inuence on a driver’s stopping decision, a driver who values
income but is “rational” in the reference-dependent sense of prospect theory will
tend to have a negative elasticity of hours with respect to earnings per hour, just as
Camerer et al. found.
Section II reports our econometric estimates. In Section IIA we estimate probit
models of the probability of stopping with an index function that is linear in cumula-
tive shift hours and income as in Farber (2005), but splitting the sample according
to whether a driver’s earnings early in the day are higher or lower than his proxied
expectations. This “early earnings” criterion should be approximately uncorrelated
with errors in the stopping decision, limiting sample-selection bias. To avoid con-
founding due to our operationalization of the targets being partly determined by the
variables they are used to explain, we proxy drivers’ rational point expectations of
a day’s income and hours, driver/day-of-the-week by driver/day-of-the-week, by
their sample averages up to but not including the day in question.
3
Farber (2008) modeled a driver’s stopping decision by estimating a daily latent income target and continuation
value, assuming that a driver stops working when his continuation value falls below the cost of additional effort. He
dened continuation value to include option value; but if option value is truly important, his linear specication of
continuation value is unlikely to be appropriate. We simply assume that drivers’ decisions ignore option value, as
Thierry Post et al. (2008) did, and as seems behaviorally reasonable. Farber’s (2008) and our treatments of drivers’
decisions are both rst-order proxies for globally optimal stopping conditions that depend on unobservables, which
treatments both yield coherent results, despite their aws.
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In a neoclassical model, when earnings per hour is serially uncorrelated within a
day, it is approximately irrelevant whether early earnings are unexpectedly high or
low, because this affects a driver’s income but not his expected earnings later in the
day, and the income effect is negligible. But in a reference-dependent model, high
early earnings make a driver more likely to reach his income target before his hours
target, and this has important consequences for behavior. In our estimates drivers’
stopping probabilities happen to be more strongly inuenced by the second target
a driver reaches on a given day than by the rst. As a result, when early earnings
are high, hours (but not income) has a strong and signicant effect on the stopping
probability, either because the driver reaches his hours target or because his mar-
ginal utility of leisure increases enough to make additional work undesirable. When
early earnings are low, this pattern is reversed.
4
Such a reversal is inconsistent with a
neoclassical model, in which the targets are irrelevant; but it is gracefully explained
by a reference-dependent model. If preferences were homogeneous, as Farber’s and
our models assume, drivers’ stopping probabilities would either all tend to be more
strongly inuenced by the rst target reached on a given day, or all by the second.
Thus the pattern of signicance in our results is one of the two that are character-
istic of a reference-dependent model with homogeneous preferences, and as such
is powerful evidence for reference-dependence, even though with heterogeneous
preferences other patterns are possible.
5
Further, because the elasticity of hours with respect to earnings per hour is sub-
stantially negative when the income target is the dominant inuence on stopping
probability, but near zero when the hours target is dominant, and on a typical day
some drivers’ earnings are higher than expected and others’ lower, KR’s distinction
between anticipated and unanticipated wage changes can easily reconcile the pre-
sumably normally positive incentive to work of an anticipated increase in earnings
per hour, with a negative observed aggregate elasticity of hours. As KR put it (2006,
p. 1,136):
In line with the empirical results of the target-income literature, our model predicts
that when drivers experience unexpectedly high wages in the morning, for any given
afternoon wage they are less likely to continue work. Yet expected wage increases will
tend to increase both willingness to show up to work, and to drive in the afternoon
once there. Our model therefore replicates the key insight of the literature that exceed-
ing a target income might reduce effort. But in addition, it both provides a theory of
what these income targets will be, and—through the fundamental distinction between
unexpected and expected wages—avoids the unrealistic prediction that generically
higher wages will lower effort.
Finally, with a distribution of earnings the model can also reproduce Farber’s (2005)
ndings that aggregate stopping probabilities are signicantly related to hours but
not earnings, but nonetheless respond smoothly to earnings.
4
Our estimates reverse the patterns of signicance from the analogous results in Table 2 of the original version
of this paper, Crawford and Meng (2008), suggesting that those results were biased due to the endogeneity of the
sample-splitting criterion we used there: whether realized earnings were higher or lower than the full-sample aver-
age for a given driver and day-of-the-week.
5
Kirk Doran (2009), in an important study of yet another group of New York City cab drivers, with enough
data to estimate individual-level effects, nds considerable heterogeneity in drivers’ behavior, with some reference
dependent and others neoclassical.