i. labour supply - university of british columbia

Fortin – Econ 560 Lecture 1D

I. Labour Supply 4. Challenges to the Neo-Classical Theory

Plan 1. Issues and Alternatives to Neo-classical Theory

2. Evidence from Workers with Flexible Hours

a. Bicycle Messengers b. Taxi Drivers


1. Issues and Alternatives to Neo-classical Theory

In principle, the lifecycle labour supply model offers an explanation for four main aspects of individual hours choices:

1) mean hours over the lifecycle 2) the age profile of hours 3) aggregate movements in hours 4) individual-specific variation in hours around the lifecycle profile. o All of these components being tied together by a combination of intertemporal

substitution effects and wealth effect.

David Card’s (1991) argues that the model has been successful only towards 3), despite much effort directed at 4).

His negative assessment of the explanatory power of the intertemporal labour supply model is partly based on its inability to account for some simple differences in the profile of hours and wages over the life-cycle. o In particular, the fact that over the life-cycle hours rise much more quickly than

wages.


More generally, one potential problem is that individual hours are influenced directly

by employer-specific demand conditions.

There is a substantial literature demonstrating that workers are not to be free to set their hours of work.

In the hours distribution, we saw earlier a substantial mass of male workers at 40 hours a week. o Ham (1982) finds the labor supply functions of constrained and unconstrained

workers differ substantially. o Altonji and Paxson (1988) using longitudinal data on hours of work demonstrate that

the temporal variation in workers’ annual hours is larger for workers who change jobs than for those who remain on the same job.

o Dickens and Lundberg (1993) find that a model of labor supply in which workers choose among a finite set of alternative jobs with fixed wage-hours combinations fits the observed hours distribution quite well.

Much recent research has thus focused on evaluating the intertemporal elasticity of labour supply in case where workers are free to choose their hours of work.


There is an interesting literature on labour supply in settings where workers are free to set their labour supply.

Oettinger (1999) finds that stadium vendors at baseball games are more likely to go to work on days when their wage can be expected to be higher.

Fehr-Goette (2007) show bicycle messengers sign up for more shifts when their

commission is experimentally increased. Camerer et al. (1997) , Farber (2005, 2008), Crawford-Meng (2009): collecting data on

the daily labor supply decisions of New York City cabdrivers, who found a strongly negative elasticity of hours with respect to realized earnings, especially for inexperienced drivers, implying large daily income effects. o Debate on whether cab drivers are rational or have a daily income target

Farber (2008) further provides further test of the theory of reference-dependent utility

and find that, the data show more smoothness in the relationship between income and the continuation and stopping probabilities than an important role for reference-dependent preferences would suggest. o Most shifts end before reaching the reference income level, which varies

substantially day by day for each driver.


The theory of reference-dependent preferences has its roots in the concept of loss aversion. o Tversky and Kahneman (1979, 1991) present an analysis of choice in a riskless

framework where the concavity of utility as a function of income changes at some reference or status quo value of income RY .

o This core idea has been credited as the explanation for the “endowment effect” (Richard Thaler,1980) (e.g. mug experiment, WTA (willingness to accept) >WTP (willingness to pay)

The idea is that the interpersonal/intertemporal comparisons set up a reference point that

introduces an asymmetry between gains and losses relative to this reference point.

29

h=T/w

w

h Fig. 1bThe Labor Supply Curve -Daily Income Targeting

y

v

(t,v(t))

Fig. 1aIncome Utility Under Target Income Theory

Source: Chou (2002)


2. Evidence from Flexible Hours Workers a. Couriers (Field Experiment)

The life-cyle model predicts that people should increase

their working hours if they experience a positive transitory wage shock.

Fehr and Goette (2007) analyze the labour supply behaviour of bicycle messengers in Zurich with a randomized field experiment.

They randomly allocate an exogenous and transitory increase of 25 percent of the

commission rate which lasted only for 4 weeks (very small effect on lifetime wealth). They study 2 bicycle messenger firms in Zurich, Veloblitz and Flash, with 50-60

bicycle messengers.

The messengers receive no fixed-pay wage component and are solely paid on commission. o Commission rate for male treatment group increased from 0.39 to 0.49 and for

females from 0.44 to 0.54.


The messengers can freely choose how many shifts (one shift is 5 hours) they want to

work per day and how much effort to exert (no constraints to adjust working hours). o Observe response in both number of shifts and effort (longer deliveries) during

shifts. o Total labour supply is revenues (shifts*effort)

They have fixed shifts where they agree to work, say every Wednesday morning, and

flexible shifts where everybody can sign up at will. They split employees from one firm (Veloblitz) willing to participate in the

experiment into two groups A and B. The authors then increased the commission rate by 25 percent as follows:

1. Group A: higher pay in September 2000. 2. Group B: higher pay in November 2000.

During all other times they received their normal pay.

Their experimental design also allows them to control for the small positive wealth effect because both groups eventually received the additional income and the additional wages were paid at the same time in December 2000.

III. Results

This section reports the results from our fieldexperiment. Our analysis is based on the fourweeks prior to the first experimental period andthe two subsequent experimental periods inwhich first group A and then group B receiveda wage increase. The data contain the day ofeach delivery, the messenger’s identificationnumber, and the price for each delivery. Thus,we have, in principle, two measures of laborsupply: the amount of revenue generated andthe number of deliveries completed. Sincelonger deliveries command a higher price andrequire more effort, the revenue is our preferredmeasure of labor supply. Our estimates of thetreatment effect, however, are almost identicalfor either choice of the labor supply measure.

A. The Impact of the Wage Increase on TotalRevenue per Messenger

The first important question is whether thereis a treatment effect on total revenue per mes-senger during the first and second experimentalperiods. Tables 1 and 2 present the relevantdata. The tables show the revenue data forgroups A and B, and the messengers at Flashand Veloblitz who did not participate in theexperiment. Table 1 shows the “raw” revenue

per messenger—uncontrolled for individualfixed effects. Table 2 controls for individualfixed effects by showing how, on average, themessengers’ revenues deviate from their per-son-specific mean revenues. Thus, a positivenumber here indicates a positive deviation fromthe person-specific mean; a negative numberindicates a negative deviation.

Tables 1 and 2 show that group A and groupB generate very similar revenues per messengerduring the four weeks prior to the experiment. Ifwe control for individual fixed effects, we findthat the revenues per messenger are almostidentical across groups and close to zero. Forexample, the difference in revenues betweengroup A and group B is only CHF 71.03 if wecontrol for person-specific effects with a stan-dard error of CHF 475.37 (see Table 2). Thisdifference is negligible compared to the averagerevenue of roughly CHF 3,400 that was gener-ated by a messenger during the preexperimentalperiod. Thus, in the absence of an experimentaltreatment, the messengers in group A and groupB behave in the same way.

During the first experimental period (hence-forth, “treatment period 1”), however, in whichgroup A received the higher wage, the totalrevenue generated by group A is much largerthan the revenue of group B, indicating a largetreatment effect. On average during this period,

TABLE 1—DESCRIPTIVE STATISTICS

Participating messengers Differencegroups

A and B

Nonparticipatingmessengers,

VeloblitzMessengers,

FlashGroup A Group B

Four-week periodprior toexperiment

Mean revenues 3,500.67 3,269.94 241.67 1461.70 1637.49(2,703.25) (2,330.41) [563.19] (1,231.95) (1,838.61)

Mean shifts 12.14 10.95 1.20 5.19 6.76(8.06) (7.58) [1.75] (4.45) (6.11)

N 21 19 21 59Treatment period 1 Mean revenues 4,131.33 3,005.75 1,125.59 844.21 1,408.23

(2,669.21) (2,054.20) [519.72] (1,189.53) (1,664.39)Mean shifts 14.00 9.85 4.15 3.14 6.32

(7.25) (6.76) [1.53] (4.63) (6.21)N 22 20 21 65

Treatment period 2 Mean revenues 2,734.03 3,675.57 �941.53 851.23 921.58(2,571.58) (2,109.19) [513.2] (1,150.31) (1,076.47)

Mean shifts 8.73 12.55 �3.82 3.29 4.46(7.61) (7.49) [1.65] (4.15) (4.74)

N 22 20 24 72

Notes: Standard deviations in parentheses, standard error of differences in brackets. Group A received the high commissionrate in experimental period 1, group B in experimental period 2.Source: Own calculations.

307VOL. 97 NO. 1 FEHR AND GOETTE: DO WORKERS WORK MORE IF WAGES ARE HIGH?

Nicole

Rectangle


Messengers increase labour supply in higher wage periods (treatment effects very similar for both groups). o Including additional messengers (who did not participate in the experiment) in

the control group does not affect the results.

The implied intertemporal labour supply elasticity is between (1,000/3,568)/0.25 =1.12 and (1,000/3,205)/0.25 = 1.25.

These results provide strong evidence in favour of the life-cycle model: people increase labour supply when wages are high.

But it is interesting to see whether the messengers increase both the number of shifts

and the effort per shift. To control for individual effects, the authors estimate

푟 = 훼 + 훿푇 + 푑 + 휀 where 푟 measures the revenue generated by messenger i during a four-week period t, 훼 is a fixed effect for messenger i, 푇 is a dummy variable that is equal to 1 if the messenger is on the increased commission rate, 푑 is a time dummy estimated for treatment period 1 and for treatment period 2, and 휀 is the error term.

What do the authors implicitly assume?

experiment. Therefore, this dummy measureswhether the nontreated group at Veloblitz be-haved differently relative to the messengers atFlash, and the treatment dummy measureswhether the treated group at Veloblitz behaveddifferently relative to the messengers at Flash.In this regression, the coefficient of the treat-ment dummy indicates a treatment effect ofroughly CHF 1,000. In addition, the dummy forthe whole nontreated group at Veloblitz is smalland insignificant, indicating that the nontreatedgroup was not affected by the wage increase forthe treated group. This result suggests that thewage increase for the treated group did notconstrain the opportunities for working for thenontreated group at Veloblitz. The result is alsoconsistent with the permanent existence of un-filled shifts and with survey evidence. The over-whelming majority of the messengers statedthat they could work the number of shifts theywanted to work.14

In summary, the results above indicate a largeand highly significant effect of a temporarywage increase on the total effort of the treatedgroup. In contrast to many previous studies, ourresults imply a large intertemporal elasticity ofsubstitution. We have seen that the treatmenteffect is roughly CHF 1,000. The average rev-enue across group A and group B is CHF 3,568in treatment period 1; in treatment period 2 it is3,205. Thus, the intertemporal elasticity of sub-stitution is between (1,000/3,568)/0.25 � 1.12and (1,000/3,205)/0.25 � 1.25, which is sub-stantially larger compared to what previousstudies have found (see, e.g., Oettinger 1999).15

14 It is also noteworthy that we find a negative effect oftime on revenues per messenger in all three regressions. Thetime effect is never significant for the first treatment period,but it is higher for the second treatment period and reachessignificance at the 5-percent level in some of the regres-

sions. These time effects suggest that a comparison of therevenues of the same group over time is problematic be-cause revenue is likely to be “polluted” by monthly varia-tions in demand. It is thus not possible to identify thetreatment effect by comparing how a group behaved intreatment period 1 relative to treatment period 2.

15 It is even possible that our measure of the elasticity oflabor supply with regard to a temporary wage increaseunderestimates the true elasticity because we use revenuesper messenger as a proxy for labor supply per messenger. Ifwages w affect effort e and effort affects revenue r, theelasticity of e with respect to w, which we denote by �ew, isgiven by �rw/�re, where �rw is the elasticity of r with respect

TABLE 3—MAIN EXPERIMENTAL RESULTS

(OLS regressions)

Dependent variable:Revenues per four-week period

Dependent variable:Shifts per four-week period

(1) (2) (3) (4) (5) (6)

Observations arerestricted to

Messengersparticipating in

experiment

Allmessengers at

Veloblitz

Allmessengers at

Flash andVeloblitz

Messengersparticipating in

experiment

Allmessengers at

Veloblitz

Allmessengers at

Flash andVeloblitz

Treatment dummy 1,033.6*** 1,094.5*** 1,035.8** 3.99*** 4.08*** 3.44**(326.9) (297.8) (444.7) (1.030) (0.942) (1.610)

Dummy for nontreatedat Veloblitz

�54.4 �0.772(407.4) (1.520)

Treatment period 1 �211 �370.6 �264.8 �1.28 �1.57 �0.74(497.3) (334.1) (239.9) (1.720) (1.210) (0.996)

Treatment period 2 �574.7 �656.2 �650.5** �2.56 �2.63** �2.19**(545.7) (357.9) (284.9) (1.860) (1.260) (1.090)

Individual fixed effects Yes Yes Yes Yes Yes YesR squared 0.74 0.786 0.753 0.694 0.74 0.695N 124 190 386 124 190 386

Note: Robust standard errors, adjusted for clustering on messengers, are in parentheses.*** Indicates significance at the 1-percent level.** Indicates significance at the 5-percent level.* Indicates significance at the 10-percent level.

Source: Own calculations.

309VOL. 97 NO. 1 FEHR AND GOETTE: DO WORKERS WORK MORE IF WAGES ARE HIGH?


Table 3 shows that the average number of shifts worked during the two treatment

periods is 11.925 and 10.64, respectively, the wage elasticity of shifts is between (4/11.925)/0.25=1.34 and (4/10.64)/0.25=1.50. o Thus, the shift choices are even more responsive to the wage increase than total

revenue per messenger. o A first indication that the elasticity of effort per shift is negative.

There are two alternative explanations to why there is a negative effect on effort

Denote labour supply between 푡 and 푡 + ∆ by 푒 , and let 푒 = 0, if the individual is not

working during period 푡 . o Neoclassical model o If utility is time-separable across periods, intertemporal maximization implies that the

utility from working in period t can be represented as 푉 (푒 ) = 휆 ∙ (푤 푒 + 푧 ) − 푔(푒 ; 휆푝 )

where 푤 is the discounted wage per unit of effort e in period t, 휆 is the marginal utility of lifetime wealth, 푧 is income unrelated to effort in period t, and 푝 is the discounted price of consumption. The function 푔(푒 ; 휆푝 ) is a money-metric disutility of effort, which is increasing and convex in e .


Because any additional income is used to smooth out consumption over the rest of life, to a first order approximation, small changes in income have no wealth effects.

o Without time-separability, we could have 푔(푒 (1 + 훼푒 )), last period’s effort influence on this period’s effort level.

o If the messenger worked more shift yesterday, this may affect the effort level today. o Messengers who work more shifts when the wage is high may rationally decide to

reduce the effort per shift.

o Loss aversion (reference-dependent utility): disutility of earnings below a particular target level greater than utility above this level.

o Individuals tend to evaluate outcomes as gains and losses relative to a reference point or target, and thus may especially dislike a low daily income, because it feels like a loss.

A parsimonious way to model behaviorally income targets is to assume that individuals

maximize 푉 (푒 ) = 푣(푤 푒 + 푧 − 푟)− 푔(푒 ; 휆푝 )


where 푣(∙) = 휆 ∙ (푤 푒 + 푧 )− 푔(푒 ; 휆푝 ) if 푤 푒 + 푧 − 푟 ≥ 0, and 훾휆 ∙ (푤 푒 + 푧 )−푔(푒 ; 휆푝 ) otherwise, where 훾 > 1 measures the degree of loss aversion, and r is the daily income goal that serves as the reference point.

o The KT value function used has a discrete drop in the slope of 푣(∙) if 푤 푒 + 푧 − 푟

becomes positive.

o Since higher commission implies that they reach the income level faster this may explain lower effort.

o Effort may change during the day as they move close to their target: an increase in the wage, or a windfall gain in the morning, increases the probability of quitting early, because surpassing the income target reduces the marginal valuation of working another hour.

Eight months after the experimental wage increase, Fehr and Goette measure the

messengers’ loss aversion by observing choices under uncertainty in the following experiment.

They use the following lotteries

o Lottery A: Get CHF 8 with probability ½ and pay CHF 5 with probability ½. o Lottery B: Six independent repetitions of lottery A.

compared to the control period. This patternsuggests that the negative effect of wages oneffort per shift may be driven solely by theloss-averse messengers.

To examine this possibility in more depth, weran the regressions in Table 6. In these regres-sions, log daily revenue of messenger i at day tis again the dependent variable and we controlfor messenger fixed effects in all regressions, asloss-averse messengers may differ in more thanone dimension from other messengers. In thefirst regression, we split the treatment groupaccording to behavior in lottery A. If a messen-ger rejects lottery A, the messenger is more lossaverse than if lottery A is accepted. In regres-sion (1), we estimate the treatment effect sepa-rately for loss-averse messengers (who rejectedlottery A) and messengers who did not displayloss aversion (who accepted lottery A). Theresults show that loss-averse messengers gener-ated roughly 10-percent lower revenue per shiftwhen they received the high wage. In contrast,the treatment effect is much lower and insignif-icant for the messengers without loss aversion.

Regression (2) of Table 6 provides a robust-ness check for this result because we use a finerscale of messengers’ loss aversion which yieldstreatment effects for three separate groups: mes-sengers accepting both lotteries (labeled “notloss averse”), messengers rejecting one of thetwo lotteries, and messengers rejecting both lot-

teries. The theory predicts that the strongesttreatment effect should occur for the group thatrejects both lotteries, followed by the group thatrejects only one lottery. We do find evidence ofthis, although the differences between thosewho reject both and those who reject only onelottery are small. Regression (2) also shows thatthe wage increase triggers no significantlynegative impact on messengers who exhibitno loss aversion in the lotteries, while theother two groups exhibit clear reductions inrevenues during the wage increase. These re-sults suggest that the negative impact of thewage increase on revenue per shift is associ-ated with the messengers’ degree of loss aver-sion, lending support to the target incomemodel discussed in Section IIC.

V. Summary

This paper reports the results of a randomizedfield experiment examining how workers, whocan freely choose their working time, and theireffort during working time, respond to a fullyanticipated temporary wage increase. We find astrong positive impact of the wage increase on

TABLE 6—DOES LOSS AVERSION EXPLAIN

THE TREATMENT EFFECT?(Dependent variable: log (revenues per shift)

during fixed shifts, OLS regressions)

(1) (2)

Treatment effect notloss averse

�0.0273 �0.027(0.033) (0.032)

Treatment effect rejectslottery A

�0.105**

(0.046)Treatment effect rejects

one lottery�0.0853*

(0.062)Treatment effect rejects

both lotteries�0.12**(0.053)

Log(tenure) 0.00152 0.0074(0.061) (0.060)

Day fixed effects Yes YesIndividual fixed effects Yes YesR-Squared 0.243 0.26N 1137 1137

Note: Robust standard errors, adjusted for clustering onmessengers, are in parentheses.

*** Indicates significance at the 1-percent level.** Indicates significance at the 5-percent level.* Indicates significance at the 10-percent level.

Source: Own calculations.

Deviation of log(daily revenues) from individual means of log(daily revenues)

-0.1

-0.05

0

0.05

0.1

Control period Treatment period

naem nosrep - )seunever yliad(gol) se unever yliad(gol fo

Not loss averse

Loss averse

FIGURE 2. THE BEHAVIOR OF LOSS-AVERSE AND NOT-LOSS-AVERSE SUBJECTS DURING CONTROL AND TREATMENT

PERIOD IN FIXED SHIFTS

Note: Error bars are standard errors of means.

314 THE AMERICAN ECONOMIC REVIEW MARCH 2007


Result: Among the 42 messengers who belong to either group A or group B, 19 messengers rejected both A and B; 8 accepted A and rejected B; 1 accepted B but rejected A; 14 accepted both A and B.

They show that only the messengers who displayed loss aversion in the lottery choices, exhibit a lower effort per shift in the treatment period


b. Taxi Drivers (Observational Data)

Camerer et al. (1997) investigate the daily hours of work of New York City taxi drivers. o These drivers lease their cabs for a pre-specified period (day, week, or month) for a

fixed fee, are responsible for fuel and some maintenance, o They keep 100 percent of their fare income after paying fixed costs. o They are free to drive as much or as little as they want during the lease period. This

leasing arrangement is close to the incentive theorist’s first-best solution to the firm-worker principal-agent problem of selling the firm to the worker.

Camerer et al. (1997) regress the logarithm of daily hours on the logarithm of the average hourly earnings rate and find a significant and substantial negative elasticity of labor supply.

The core of their analysis consists of computing a daily wage rate as the ratio of daily income to daily hours. o This leads to a classic case of “division bias” in which, if there is any

misspecification or measurement error, there will be a negative bias on the coefficient of the wage.

o The authors instrument for that problem using the average daily wage of other workers on the same calendar date.

LABOR SUPPLY OF NYC CABDRIVERS 415

TABLE I SUMMDARY STATISTICS

Mean Median Std. dev

TRIP (n = 70) Hours worked 9.16 9.38 1.39 Average wage 16.91 16.20 3.21 Total revenue 152.70 154.00 24.99 # Trips listed on sheet 30.17 30.00 5.48 # Trips counted by meter 30.70 30.00 5.72 High temperature for day 75.90 76.00 8.21 Correlation log wage and log hours = -.503. The standard deviation of log hours is .159, log wage is .183, and log total revenue is .172. The within-driver standard deviation of log revenue is .155 and across drivers standard deviation is .017. TLC1 (n = 1044) Hours worked 9.62 9.67 2.88 Average wage 16.64 16.31 4.36 Total revenue 154.58 154.00 45.83 # Trips counted by meter 27.88 29.00 9.15 High temperature for day 65.16 64.00 8.59 Correlation log wage and log hours = -.391. The standard deviation of log hours is .263, log wage is .351, and log total revenue is .347. The within-driver standard deviation of log revenue is .189 and across drivers standard deviation is .158. TLC2 (n = 712) Hours worked 9.38 9.25 2.96 Average wage 14.70 14.71 3.20 Total revenue 133.38 137.23 40.74 # Trips counted by meter 28.62 29.00 9.41 High temperature for day 49.29 49.00 2.01 Correlation log wage and log hours = -.269. The standard deviation of log hours is .382, log wage is .259, and log total revenue is .400.

are presented in Appendix 2. In the TRIP data the average trip duration was 9.5 minutes, and the average fare was $5.13.

One feature of the data is that the variation in hours worked and number of trips in the TRIP sample is substantially lower- about half as large as in the TLC1 and TLC2 samples. Recall that a key difference is that TRIP consists of only fleet drivers who rent their cabs daily, while TLC1 consists of fleet, lease, and owner-drivers and the TLC2 consists of lease and owner-drivers. Figure II below is a distribution of hours broken up by driver- type for the TLC1 data. It is clear from the histograms that the differences in variation in the key variables across data sets (see Appendix 2) are driven by the differences in driver-types across the data sets.

Source: Camerer et al. (1997)

418 QUARTERLY JOURNAL OF ECONOMICS

TLC1 TLC2

3 3P0 00 000

, o o-" 0 0o

1 04 3 4 0 0 0 0

00 0

0 0OD

0 ~~~~~~~00

2- 0 2- g0 I

0 0~0

2 23 3

Hours-Wag Relaionhip

00 0 00

L ~~~~~~~0 0

g ~~~~~000 og 0 0

warmer day has a higher opportunity cost (perhaps because for- gone leisure is more pleasurable). Also included is a dummy variable for the shift driven and a dummy variable for a weekday versus weekend day (although all shifts are during the week in the TLC2 data).9

9. Shifts are described in detail in Appendix 1. Briefly, in the TRIP and TLC2 samples, the dummy indicates night shift (versus day or afternoon) and in the TLC1 sample there are two shift dummy variables (night and day, versus "other") reflecting the greater heterogeneity of driving arrangements in this sample. The estimates are changed very little if no shift designations are used. No additional



TABLE II OLS LOG HOURS WORKED EQUATIONS

Sample TRIP TLC1 TLC2

Log hourly wage -.411 -.186 -.501 -.618 -.355 (.169) (.129) (.063) (.051) (.051)

High temperature .000 -.000 .001 .002 -.021 (.002) (.002) (.002) (.002) (.007)

Shift during week -.057 -.047 -.004 .030 -

(.019) (.033) (.035) (.042) Rain .002 .015 - -.150

(.035) (.035) (.062) Night shift dummy .048 -.049 -.127 -.294 -.253

(.053) (.049) (.034) (.047) (.038) Day shift dummy - - .000 .053

(.028) (.045) Fixed effects No Yes No Yes No Adjusted R2 .243 .484 .175 .318 .146 Sample size 70 65 1044 794 712 Number of drivers 13 8 484 234 712

Dependent variable is the log of hours worked. Standard errors are in parentheses and are corrected for the nonfixed effects estimates in coulmns 1 and 3 to account for the panel structure of the data. Explanatory variables are described in Appendix 1.

In TRIP the wage elasticities depend substantially on whether or not driver fixed effects are included in the model. In the first column (no driver fixed effects), the estimated wage elasticity is -.411 and is significantly different from zero. Including driver fixed effects, which are jointly significant, lowers the estimated elasticity to -.186, which is no longer significantly different from zero.10

improvement in fit is obtained if day of the week dummy variables are included rather than a weekday versus weekend dummy variable.

10. One way to make use of the large amount of screened-out data in TRIP is to impute missing hours for the incomplete trip sheets, by multiplying the driver-listed hours by the ratio of meter-recorded trips to the number of driver- listed trips. For example, if a driver listed only 16 trips in 5 hours of driving, but the meter recorded 24 trips, this method would impute 7.5 total hours of driving. This method yields OLS estimates of -.549 (se = .156, n = 162) and -.276 (se =

.071, n = 158) for the TRIP sample without and with fixed effects. These estimates are slightly more negative and more precisely estimated than those for the screened sample, reported in Table II. Another method of imputation assumes that drivers stopped filling out their trip sheets when they got busy (so that the average wage during the missing hours is higher than during the listed hours). This method scales up the number of hours by a factor that is less than the ratio of meter-recorded trips to driver-recorded trips (since it assumes the hours-per- trip is smaller for the missing trips) and actually makes the estimates even more negative.



TABLE IV IV LOG HOURS WORKED EQUATIONS BY DRIVER EXPERIENCE LEVEL

Sample TRIP TLC1 TLC2

Experience level Low High Low High Low High Log hourly wage -.841 .613 -.559 -1.243 -1.308 2.220

(.290) (.357) (.406) (.333) (.738) (1.942) Fixed effects Yes Yes Yes Yes No No Sample size 26 39 319 458 320 375 P-value for difference .030 .666 .058 in wage elasticity

Dependent variable is the log of hours worked. Standard errors are in parentheses. Regressions also include weather and shift characteristics (dummy variable for rain, high temperature during the day, dummy variable for shift on a weekday, and time of shift dummy variables) as explanatory variables. Instruments for the log hourly wage include the summary statistics of the distribution of hourly (log) wages of other drivers on the same day and shift (the 25th, 50th, and 75th percentiles).

split to divide drivers into low- and high-experience subsamples for the TRIP data.14

Table IV presents the wage elasticities estimated separately for low- and high-experience drivers. All regressions use instru- mental variables, and all include fixed effects (except, of course for TLC2). In all three samples, the low-experience elasticity is strongly negative, generally close to -1. The wage elasticity of the high-experience group is significantly larger in magnitude, for the TRIP and TLC2 samples (p = .030 and .058, respectively).'5

How Do Elasticities Vary with Payment Structure?

The way drivers pay for their cabs might affect their respon- siveness of hours to wages if, for example, the payment structure affects the horizon over which they plan. Alternatively, it might affect the degree to which they can significantly vary hours across days. The TLC1 sample contains data from three types of payment schemes: daily rental (fleet cabs), weekly or monthly rental (lease cabs), or owned. Table V presents elasticity estimates in

14. The number of observations in the low- and high-experienced samples for the TRIP data are not equal because the median split is done on drivers, not trip sheets, and there are different sample sizes for each driver.

15. An alternative approach is to use the median wage directly as a regressor, skipping the first-stage regression. This lowers the adjusted R2 substantially (as is expected) but does not alter the sign or magnitude of the estimates reported in Table III systematically (TRIP and TLC2 estimates become more negative and TLC1 estimates become less negative). The large estimate and standard error on the high-experience TLC2 elasticity reported in Table IV do become smaller (-.135 and .968, respectively), but that does not change the conclusion that experience makes elasticities less negative.



o But if there are calendar date effects on the daily wage that are also correlated with labour supply conditional on the wage, this instrument will be ineffective in purging the estimated labour supply elasticity of bias.

The authors’ explanation for the negatively slopped labour supply function is income targeting.

A driver’s reference point is a daily income target and loss aversion creates a kink that tends to make realized income bunch around the target, so realized hours have little or none of the positive wage elasticity predicted by the neoclassical model.

There are objections from the traditional model of labour supply to target earning o It implies that on days in which fares are scarce, drivers work longer hours to meet

the target. o But on days in which it is easy to make money, the drivers quit early. o If workers can substitute labor for leisure intertemporally across days, then they

should work more on days with higher wage rates relative to other days. o This implies very strong daily income effects on daily labor supply, so strong as to

overwhelm any substitution effect.


Farber (2005) revisits the Camerer et al. (1997) using their TRIP data and also analyzed new data on the labour supply of New York City taxi drivers using a different econometric framework. o He estimate a model of the decision to stop work or continue driving at the

conclusion of each trip and show that the primary determinant of the decision to stop work is cumulative hours worked on that day.

o In that study, he found no evidence of target earnings behavior.

Daily target earnings behavior implies that income effects dominate substitution effects so that the elasticity of hours with respect to changes in the marginal wage rate is minus one.

Farber (2005) estimates the same the labour supply model used by Camerer et al. (1997)

and find similar results that he dismiss as reliable estimates of labour supply elasticities because of the division bias induced by the computation of the wage rate.

He then use a probit model of the probability that a driver stops after a given trip as a function of hours worked to that point, income earned to that point, current location, weather, and fixed effects for driver, calendar date, hour of the day, and day of the week.

70 journal of political economy

TABLE 5Hazard of Stopping after Trip: Normalized Probit Estimates

Variable X* (1) (2) (3) (4) (5)

Total hours 8.0 .013(.009)

.037(.012)

.011(.005)

.010(.005)

.010(.005)

Waiting hours 2.5 .010(.010)

�.005(.012)

.001(.006)

.004(.006)

.004(.005)

Break hours .5 .006(.008)

�.015(.011)

�.003(.005)

�.001(.005)

�.002(.005)

Shift income�100 1.5 .053(.022)

.036(.030)

.014(.015)

.016(.016)

.011(.015)

Minimum tempera-ture ! 30

.0 … … … .002(.007)

.002(.006)

Maximum tempera-ture ≥ 80

.0 … … … �.016(.007)

�.016(.007)

Hourly rain .0 … … … .005(.124)

.015(.113)

Daily snow .0 … … … .007(.006)

.007(.005)

Downtown .0 … … … … .000(.006)

Uptown .0 … … … … .000(.005)

Bronx .0 … … … … .075(.057)

Queens .0 … … … … .044(.038)

Brooklyn .0 … … … … .078(.036)

Kennedy Airport .0 … … … … .056(.031)

LaGuardia Airport .0 … … … … .061(.029)

Other .0 … … … … .134(.071)

Driver (21) no yes yes yes yesDay of week (7) no no yes yes yesHour of day (19) 2:00 p.m. no no yes yes yesLog likelihood �2,039.2 �1,965.0 �1,789.5 �1,784.7 �1,767.6

Note.—The sample includes 13,461 trips in 584 shifts for 21 drivers. Probit estimates are normalized to reflect themarginal effect at of X on the probability of stopping. The normalized probit estimate is , where f(7) isˆX* b 7 f(X*b)the standard normal density. The values of chosen for the fixed effects are equally weighted for each day of theX*week and for each driver. The hours from 5:00 a.m. to 10:00 a.m. have a common fixed effect. The evaluation pointis after 5.5 driving hours, 2.5 waiting hours, and 0.5 break hour in a dry hour on a day with moderate temperaturesin midtown Manhattan at 2:00 p.m. Robust standard errors accounting for clustering by shift are reported in parentheses.

and income. Total hours is defined as the sum of hours spent on trips,hours “waiting” between trips, and hours on break. The probability ofstopping is significantly positively related to income and not significantlyrelated to time worked. The magnitude of the income effect is that a$10 increase in income implies a 0.53-percentage-point increase in theprobability of stopping work. The estimates in column 2 include driverfixed effects, and, on the basis of the improvement in the log likelihood,there are significant differences across drivers in their probability of

Source: Farber (2005)

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Nicole

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The results from the probit model show consistently that the probability of stopping daily work after a particular trip is strongly related to hours worked to that point and not significantly related to cumulative income earned.

Farber argues that this pattern is consistent with a conventional neoclassical intertemporal labour supply model with weak or nonexistent income effects due to transitory changes to the marginal daily wage.

Farber (2008) gives another chance to a model based on reference-dependent preferences.

He develops an empirical model of daily labour supply that incorporates reference-dependent preferences but does not require that the reference level of income be observed or defined in advance.

He assumes that a driver’s utility (as a function of accumulated income and hours) after trip t during a shift that incorporates a reference point T

1

1)])([1(),( tttt hTYTYIhYU

where ][I is the indicator function and th hours worked on shift at end of trip t, tY income earned on shift at end of trip t, T is the reference income, is a parameter determining sharpness of “kink” in utility function ( 0 ), parameter determining


disutility of work, and is the elasticity parameter (wage elasticity of labor supply/1 , defining WhY ).

This utility function will yield an expected marginal utility of the next trip that will

incorporate the kink, but rather than providing a complete solution to the optimal stopping problem, Farber uses a reduced-form approach.

He estimates the model using his previous data on the daily labor supply of New York

City taxi-drivers, allowing taxi drivers to have a reference level of daily income.

The econometric framework considers the decision of whether to continue driving or to stop, at the end of each passenger trip, by reference to person-shift specific level of income.

Letting driver i on shift j continue driving after trip t if the latent variable 0ijtC and stop if 0ijtC where

ijtijijtijt TYC ][βX ijt The probability of continuing conditional on the reference level ijT of driver i on shift j

is ])[()0Pr( ijijtijijt TYTC βX ijt


The reference level is not known and estimated as ijiijT

where i is an individual mean reference income level and ij is a random component distributed normally with mean 0 and variance 2

.

What would represent convincing evidence that reference-dependent preferences are important?

1) A substantial positive value for would imply a significant increase in the probability of stopping when the reference income level is reached.

2) An estimated value of the dispersion of the random component that is small relative to the values estimated for the mean reference income levels (the mean of the distribution of i )

3) A driver’s mean reference income level ( i ) strongly predictive of daily income.

4) Attainment of the reference income level on most shifts in the sample. The estimates suggest that, while there may be a reference level of income on a given

day such that there is a discrete increase in the probability of stopping when that income level is reached, the reference level varies substantially day to day for a particular driver.

junE 20081078 THE AMERICAn ECOnOMIC REVIEW

V. Estimation of the Labor Supply Model

In this section I present maximum likelihood estimates of the parameters 1b, ui, s2m, and d 2

of the reference-dependent labor supply model based on the likelihood function described in Section III and on the data described in the previous section.

The first column of Table 2 contains estimates of a restricted model that constrains all driv-ers to have the same mean reference income level and the X vector to contain only a constant. The evidence is mixed with regard to the role of reference income levels in determining labor supply. The estimated mean reference income level is reasonable at $159.02. The estimate of d, which indexes (inversely) the change in the probability of continuing to drive once the reference income level for the day is reached, is substantial at 3.40 and statistically significantly different from zero. This would seem to be strong evidence of the importance of reference-dependent preferences in this context.

There is substantial inter-shift variation, however, around the mean reference income level. The estimated variance of 3,199.4 implies a standard deviation of $56.60. 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.

The second column of Table 2 contains estimates of the model that include a set of variables in the continuation function (equation (6)) that are meant to capture earnings opportunities and other factors that would affect a driver’s continuation probability. These are:

• Indicators for eight categories of hours worked at trip end; • Six indicators for the day of week; • Indicators for 18 clock hours at trip end; • A day-shift indicator;

Table 2—Maximum Likelihood Estimates of Reference-Dependent Labor Supply Model

Parameter (1) (2) (3) (4)

b (contprob) (constant) 20.691 — — —(0.243)

u (mean ref inc) 159.02 206.71 250.86 —(4.99) (7.98) (16.47)

d (cont increment) 3.40 5.35 4.85 5.38 (0.279) (0.573) (0.711) (0.545)

s 2 (ref inc var) 3,199.4 10,440.0 15,944.3 8,236.2 (294.0) (1,660.7) (3,652.1) (1,222.2)

Driver ui (15) No No No Yes Vars in cont prob Driver FE’s (14) No No Yes No Accum hours (7) No Yes Yes Yes Weather (4) No Yes Yes Yes Day shift and end (2) No Yes Yes Yes Location (1) No Yes Yes Yes Day-of-Week (6) No Yes Yes Yes Hour-of-Day (18) No Yes Yes Yes Log(L) 21,867.8 21,631.6 21,572.8 21,606.0Number of parameters 4 43 57 57

note: The sample includes 12,187 trips in 538 shifts for 15 drivers. Models 2–4 also include a constant in the X vec-tor for the continuation probability. The hours from 5 am to 10 am have a common fixed effect. Standard errors are reported in parentheses.

Source: Farber (2008)

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o But most shifts end before the reference income level is reached. o Further the random effect is large, although imprecisely estimated

Farber (2008) finds that a sufficiently rich parameterization of his income-targeting

model has a better fit than a standard neoclassical specification.

But he argues that his model cannot reconcile the strong increase in stopping probability at the target with the aggregate smoothness of the relationship between stopping probability and realized income.

He concludes that drivers’ income targets are too unstable and imprecisely estimated to

yield a useful reference-dependent model of labor supply.

Crawford-Meng (2009) use Farber’s data to estimate a model based on Koszegi and Rabin’s (2006) theory of reference-dependent preferences.

Estimating linear and nonlinear probit models of the probability of stopping as in Farber’s (2005) analysis, but with the sample split according to whether realized income is higher or lower than our proxy for a driver’s expected income on a given day, they find very clear evidence of reference-dependence, with the probability of stopping work strongly influenced by realized income (but not hours) when the realized wage is higher than expected, and by hours (but not income) when the wage is lower than expected.

i. labour supply - university of british columbia

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