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«*This paper is a revised and shortened version of a paper with the same title, published in the Quarterly Journal of Economics, May 1997, 407-441. It ...»

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Regressions of (log) hours on (log) wages are shown in Table II for the three data sets. TRIP and TLC1 include multiple observations for each driver, so either the standard errors are corrected to account for the panel nature of the data, or driver fixed effects are included. A driver fixed effect is a dummy variable for each driver which adjusts for the possibility that each driver might systematically drive more or less hours, holding the wage constant, than other drivers. Several other variables controlling for weather conditions and shift dummy variables were also included; their effects were 2 In fact, we used three summary statistics of the distribution of hourly wages of other drivers that drove on the same day and shift (the 25th, 50th, and 75th percentiles) as instruments for a driver’s wage.

modest and are not shown in Table II.

The IV elasticities in Table II are negative and significantly different from zero, except in the TRIP sample when fixed effects are included. Indeed, the elasticities in the TLC samples are close to which is the number predicted by daily targeting theory. The results in Table II are quite robust with respect to various specifications we tried to control for outliers, such as median regression. The difference between the wage elasticities in the two TLC samples and the fixed-effects estimate in the TRIP sample can be explained by a difference in the composition of types of drivers across the three samples.3 How do Elasticities Vary with Experience?

Drivers may learn over time that driving more on high wage days and less on low wage days provides more income and more leisure. If so, the wage elasticities of experienced drivers should be more positive than for inexperienced drivers. There are good measures of driver experience in these data sets. In the TLC data sets, the TLC separated drivers into experience groups: for TLC1, those with greater or less than four years of experience and in TLC2, those with greater or less than three years of experience. These group measures are absent in the TRIP data. However, cab driver licenses are issued with six-digit numbers (called hack numbers), in chronological order, so that lower numbers correspond to drivers who obtained their licenses earlier. Using their license numbers, we use a median split to 3 TRIP consists entirely of fleet drivers (who pay daily) while the TLC samples also includes weekly and monthly lease-drivers, and owner-drivers. Lease-drivers and owner-drivers have more flexibility in the number of hours they drive (since fleet drivers are constrained to drive no more than 12 hours). Elasticities for the fleet drivers are substantially smaller in magnitude (less negative) than for lease- and owner-drivers (as we see below). The different results in the TRIP sample, which is all fleet drivers, reflects this compositional difference in driver types.

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

Table III presents the wage elasticities estimated separately for low- and high-experience drivers.

All regressions include fixed effects (except, of course for TLC2). In all three samples, the low-experience elasticity is significantly negative, and insignificantly different from -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), and insignificantly smaller in the TLC1 sample.

How do Elasticities Vary with Payment Structure?

The way drivers pay for their cabs might affect their responsiveness 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 IV presents elasticity estimates in the three payment categories from the TLC1 sample.

All regressions are estimated using instrumental variables and include driver-fixed effects.

All wage elasticities in Table IV are negative. The elasticity which is smallest in magnitude, for fleet drivers, is not significantly different from zero. The lease and owner-driver wage elasticities are approximately -.9 and are significantly different from zero. Part of the explanation for the lower elasticity for fleet drivers is a technical one. Since they are constrained to drive no more than 12 hours, the dependent variable is truncated, biasing the slope coefficient towards zero.

Could Drivers Earn More by Driving Differently?

One can simulate how income would change if drivers changed their driving behavior. Using the TLC1 data, we take the 234 drivers who had two or more days of data in our sample. For a specific driver i, call the hours and hourly wages on a specific day t, hit and Wit. respectively, and call driver i's mean hours over all the days in the sample hi. By construction, the driver's actual total wages earned in our sample is Σt hitWit.

One comparison is to ask how much money that driver would have earned if he had driven hi hours every day rather than varying the number of hours. Call this answer "fixed-hours earnings" (FHE), Σt hiWit.

Is FHE greater than actual earnings? We know that, on average, hit and wit are negatively correlated so that the difference between FHE and actual earnings will be positive in general. In fact, drivers would increase their net earnings by 5.0 percent on average (std.error =.4 percent) if they drove the same number of hours (hi) every day, rather than varying their hours every day. If we exclude drivers who would earn less by driving fixed hours (because their wage elasticity is positive), the improvement in earnings would average 7.8 percent. And note that if leisure utility is concave, fixed-hours driving will improve overall leisure utility too.

These increases in income arise from following the simplest possible advice -- drive a constant number of hours each day. Suppose instead that we hold each driver's average hours fixed, but reallocated hours across days as if the wage elasticity was +1. Then the average increase in net income across all drivers is 10 percent. Across drivers who gain, the average increase is 15.6 percent.

–  –  –

Wage elasticities estimated with instrumental variables are significantly negative in two out of three samples. Elasticities are also significantly higher for experienced drivers in two of three samples, and significantly more negative for lease- and owner-drivers than for fleet drivers. These two empirical regularities, along with other patterns in the data, and information gleaned from our telephone survey of fleet managers, allow us to evaluate four alternative explanations for the observed negative elasticities.

Ruling out these alternatives is important (see Camerer et al, 1997 for details), because it leaves daily targeting as the most plausible explanation for anomalous negative elasticities.

One hypothesis is that drivers are “liquidity-constrained”-- they don’t have much cash to pay everyday expenses (and cannot borrow), so they cannot quit early on low-wage days. But drivers who own their cab medallions are presumably not liquidity-constrained (because medallions are worth $130,000), and their elasticities are negative too.

A second possibility is that drivers finish late on low-wage days, but take lots of unrecorded breaks on those days, so they actually work fewer hours. But we excluded long breaks from the TRIP sample and found no difference in the results.

A third possibility is that drivers quit early on high-wage days because carrying a lot of passengers is especially tiring. But the fleet managers we surveyed said the opposite; most of them thought that fruitlessly searching for fares on a low-wage days was more tiring than carrying passengers.

A fourth alternative is more subtle: We only have observations of work hours on the days that drivers chose to to work at all (or “participate”, in labor economics jargon). Omitting non-working days can bias the measured elasticity negatively if the tendency for a driver to work unexpectedly on a certain day is correlated with the tendency to work unusually long hours (Heckman, 1979). But drivers usually participate on a fixed schedule of shifts each week (and often must pay their lease fee, or some penalty, if they do not show up for scheduled work), so there is little unexpected participation and probably very little bias.

A fifth alternative is that drivers like happy endings: They drive until they earn a lot in a final unit of time (such as their final trip, or final hour). Ross and Simonson (1996) report evidence that people like "happy endings" and will end event sequences happily when they can. Drivers who create happy endings will drive longer on slow days (if the earnings that constitute a happy ending are not too responsive to earnings earlier in the day) than drivers on good days. We tested this hypothesis by comparing earnings in the final hour with earlier earnings, but found no evidence of a happy-ending effect.

Daily Income Targeting As explained in the introduction, the prediction we sought to test in our study is based on two assumptions: Cab drivers take a one-day horizon, and set a target (or target range) and quit when the target is reached.

Taking a one-day horizon is an example of narrow "bracketing” (Read and Loewenstein,

1996) simplifying decisions by isolating them from the stream of decisions they are embedded in.

For example, people are risk averse to single plays of small gambles, even though they typically face many uncorrelated small risks over time which diversify away the risk of a single play. Bettors at horse tracks seem to record the betting activity for each day in a separate "mental account" (Thaler, this volume). Since the track takes a percentage of each bet, most bettors are behind by the end of the day. Studies show that they tend to shift bets toward longshots in the last race in an attempt to `break even' on that day (McGlothlin, 1956). Read and Loewenstein [1995] observed an unusual kind of bracketing among trick-or-treaters on Halloween. Children told to take any two pieces of candy at a single house always chose two different candies. Those who chose one candy at each of two adjacent houses (from the same set of options) typically chose the same candy at each house.

Normatively, the children should diversify the portfolio of candy in their bag, but in fact they only diversify the candy from a single house. Isolation of decisions has also been observed in strategic situations: Camerer et al (1993) found that subjects in a three-stage `shrinking-pie’ bargaining experiment often did not bother to look ahead and find out how much the `pie’ they bargained over would shrink if their first-stage offers were rejected.

The notion that drivers are averse to falling below a target income is consistent with other evidence that judgments and decisions depend on a comparison of potential outcomes against some aspiration level or reference point [Helson, 1949; Kahneman and Tversky, 1979; Tversky and Kahneman, 1991], and people are dispropoportionally sensitive to losing, or falling short of a reference point.4 Both narrow bracketing and loss-aversion are analytically necessary to explain negative wage elasticities. A one-day horizon is necessary because drivers who take a longer horizon, even two days, can intertemporally substitute between the two days and will have positive wage elasticities. Therefore, if their elasticities are negative they must be taking a one-day horizon.

Aversion to falling short of the target is a necessary ingredient because if drivers do take a one-day 4Other applications of loss-aversion include Kahneman, Knetsch and Thaler (1990) on “endowment effects” in consumer choice and contingent valuation of nonmarket goods, Samuelson and Zeckhauser (1988) on “status quo biases”, and Bowman et al (1997) and Shea (1995) on anomalies in savings-consumption patterns.

horizon, elasticities will only be highly negative if the marginal utility of daily income drops sharply around the level of average daily income, which is just a labor-supply way of saying they really dislike falling short of a daily average (compared to how much they like exceeding it).

Furthermore, the daily targeting hypothesis rang true to many of the fleet managers we surveyed. They were asked to choose which one of three sentences "best describes how many hours cab drivers drive each day?". Six fleet managers chose "Drive until they make a certain amount of money". Five chose the response "Fixed hours". Only one chose the intertemporal substitution response "drive a lot when doing well; quit early on a bad day".

Several other studies with field data have used the same ingredients-- narrow bracketing and loss-aversion-- to explain anomalies in stock market behavior and consumer purchases. For example, the “equity premium puzzle” is the tendency for stocks (or “equity”) to offer much higher rates of returns than bonds over almost any moderately long time interval, which cannot be reconciled with standard models of rational asset pricing. Benartzi and Thaler (1995) argue that the large premium in equity returns compensates stockholders for the risk of suffering a loss over a short horizon. They show that if investors evaluate the returns on their portfolios once a year (taking a narrow horizon), and have a piecewise-linear utility function which is twice as steep for losses as for gains, then investors will be roughly indifferent between stocks and bonds, which justifies the large difference in expected returns. If investors took a longer horizon, or cared less about losses, they would demand a smaller equity premium. Two experimental papers have demonstrated the same effect (Thaler, Tversky, Kahneman and Schwartz, 1997; Gneezy and Potters, 1997).

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