The Welfare Effects of Incentive Schemes

The Welfare E(cid:11)ects of Incentive Schemes (cid:3) y z Adam Copeland Cyril Monnet December, 2002 Abstract This paper computes the change in welfare associated with the introduction of incentives. Speci(cid:12)cally, we calculate by how much the welfare gains of increased output due to incentives outweigh workers’ disutility from increased e(cid:11)ort. We accomplish this by studying the use of incentives by a (cid:12)rm in the check-clearing industry. Usingthis(cid:12)rm’sproductionrecords,we modelandestimate theworker’s dynamic e(cid:11)ort decision problem. We (cid:12)nd that the (cid:12)rm’s incentive scheme has a large e(cid:11)ect on productivity, raising it by 12% over the sample period. Using our parameter estimates, we show that the cost of increased e(cid:11)ort due to incentives is equal to the dollar value of a 7% rise in productivity. Welfare is measured as the output produced minus the cost of e(cid:11)ort, hence the net increase in welfare due to the introduction of the (cid:12)rm’s bonus plan is 5%. Under a (cid:12)rst-best scheme, we (cid:12)nd that the net increase in welfare is 6%. (cid:3)WewouldliketothankTomHolmesforhisadviceandencouragement. WealsothankZviEckstein, Gautam Gowrisankaran,Matt Mitchell, and Andrea Moro for their helpful comments. In addition, this paper has bene(cid:12)ted from participants at numerous seminars. Finally, we thank Jon Stam and Deb Rahn for their assistance in gathering the data and answering our questions about check processing. This paper is based on the (cid:12)rst part of Adam Copeland’s University of Minnesota Ph.D. disseration and much of it was written while both authors were Research Associates at the Federal Reserve Bank of Minneapolis. The views expressed herein are those of the authors and not necessarily those of the European Central Bank, the Eurosystem, the Federal Reserve Bank of Minneapolis, or the Federal Reserve Board. yFederal Reserve Board (Adam.M.Copeland@frb.gov) zEuropean Central Bank (Cyril.Monnet@ecb.int) 1

1 Introduction Incentives are often used by (cid:12)rms to encourage their employees to work hard. The bonus plans used vary widely, from complicated stock option o(cid:11)erings to simple employeeof-the-month awards. But how well do incentive plans work? Within the theoretical framework of the principle-agent problem, we can gauge the success of incentives by answering three questions: Do incentives matter? How signi(cid:12)cant are they? What are the welfare gains to the worker and the (cid:12)rm from using incentives? This last question is particularly important as it captures both the bene(cid:12)ts and costs to using incentives. While a particular bonus scheme may increase worker productivity by a large amount, this gain is costly as it is the result of workers exerting a higher level of e(cid:11)ort. Only by measuring workers’ disutility from e(cid:11)ort and comparing it to the gains in productivity, will we have an accurate account of how well incentives work. Our paper answers these three questions by studying the use of incentives by the Check Department of the Federal Reserve Bank of Minneapolis. Given this (cid:12)rm’s production records, we develop, solve and estimate a dynamic model of worker behavior. Using these estimates, we determine that incentives matter and have signi(cid:12)cant e(cid:11)ects on worker behavior. We then compute the welfare gains from using incentives, by determining how much output and the disutility from e(cid:11)ort increase in response to di(cid:11)erent bonus plans. This case study is particularly interesting due to special properties of the (cid:12)rm’s incentive pay scheme that allow us to identify e(cid:11)ort’s e(cid:11)ect on output and the cost of e(cid:11)ort. The structure of the bonus system creates a dynamic e(cid:11)ect within the worker’s problem. This scheme is designed so that employees are only eligible for incentive pay if their daily productivity is above a threshold level. Conditional on being eligible, bonus pay is an increasing function of the distance between the worker’s productivity and the threshold level. For any level of productivity below the threshold, workers simply earn zero bonus pay. This kink in the bonus pro(cid:12)le creates the perverse incentive for a worker to quit working hard in the later part of the day if the worker’s measure of productivity is low (due, for example, to a bad shock) in the early part of the day. Consequently, the worker’s history within the day, and expectation over the probability of the daily bonus pay, a(cid:11)ect the worker’s e(cid:11)ort decision. This dynamic aspect of the worker’s problem is crucial in our model as it is the main source of identi(cid:12)cation. Theory provides us with the intuitive result that a worker with a small probability of earning incentive pay chooses a lower level of e(cid:11)ort compared 2

to the case where the worker has a high probability. Because of the structure of the bonus scheme, we know that after experiencing a particularly unproductive morning, a worker’s chances of beating the (cid:12)rm’s daily productivity threshold, and so being eligible for incentive pay, are small. Consequently, the worker will exert a low level of e(cid:11)ort. In contrast, if this worker had a productive morning, the chances of earning a bonus are large enough that a high level of e(cid:11)ort is exerted. Knowing that a worker’s e(cid:11)ort level di(cid:11)ers under these two circumstances enables us to identify both e(cid:11)ort’s e(cid:11)ect on output and the cost of e(cid:11)ort to the worker. We can identify that e(cid:11)ort is costly, as workers with a low probability of earning incentive pay will have an unexplained drop in productivity compared to those workers with a high probability. In addition, we can identify e(cid:11)ort’s e(cid:11)ect on output by comparing the di(cid:11)erence in productivity between workers with low and high probabilities of earning a bonus. We are able to exploit this method of identi(cid:12)cation as the (cid:12)rm has provided us with a detailed data set on worker productivity. We have the (cid:12)rm’s production records for 15 full-time, experienced workers over a 15 month time period. These records are at such a (cid:12)ne level of detail that we can track a worker’s productivity within the day. The data also contain information on a large number of characteristics of the sorted checks. In addition, the (cid:12)rm provided information on its incentive plan, allowing us to measure how well a worker is performing relative to the (cid:12)rm’s benchmark productivity level. As such, we can compute a worker’s chances of earning a bonus throughout the day and so take advantage of the avenue of identi(cid:12)cation described above. We model the worker’s problem as having to make a number of e(cid:11)ort decisions within a day, where at the end of the day the incentive pay formula computes the worker’s bonus. We assume e(cid:11)ort is a binary variable, and so allow workers to choose a low, costless e(cid:11)ort level or a high, costly one. When making an e(cid:11)ort decision, the worker knows the past history of events as well as the number of checks left to sort in the day. This information allows the worker to determine how well the worker has performed relative to the incentive pay scheme, and to compute the probability the worker will be eligible for incentive pay at the end of the day. As the bonuses are calculated on a daily basis, a worker starts each day anew. Withthismodel, wegeneratetwo kindsofresults. First, weusereduced formanalysis to check that the data supports the basic implications of the model. Second, we employ a simulated maximum likelihood approach to estimate the model’s structural parameters. Using reduced form techniques, we test if a worker’s e(cid:11)ort level is increasing in the 3

probability ofearninga bonus. To seeif thisrelationshipholdsinthedata, welookatthe set of workers’ jobs at the end of the day, where the e(cid:11)ects of incentives are strongest. We then check whether the unexplained portion of worker productivity is correlated with the worker’s probability of earning a bonus. In support of our model, we (cid:12)nd that this correlation is signi(cid:12)cant and has the predicted sign. Using the set of all jobs, we then obtain an upper bound on the e(cid:11)ect of e(cid:11)ort on productivity. We accomplish this by comparing the set of productivity residuals where workers have an extremely high probability of earning a bonus to the set of residuals where this probability is extremely low. The di(cid:11)erence between these two groups of productivity residuals is large and statistically signi(cid:12)cant. With these positive results in hand, we turn to estimating the structural parameters of the model. Our estimation method is the simulated maximum likelihood approach, common in the discrete choice estimation literature. This involves solving the worker’s problem for a given set of parameters to obtain the worker’s policy function. Next, using these policy rules and the data, the model generates a distribution of the time taken to sort checks, which we use to compute the model’s likelihood. Our parameter estimates show that workers readily respond to incentives, and that e(cid:11)ort signi(cid:12)cantly a(cid:11)ects worker output. Over the sample period, the (cid:12)rm’s bonus plan increased worker productivity by 12%. Using our estimate of the cost of e(cid:11)ort, we also compute the dollar value of the disutility of workers due to the additional e(cid:11)ort they exert under the incentive scheme. We (cid:12)nd that this utility cost to workers is equal to a 7% gain in productivity, which is two-thirds of the gain in output due to incentives. As welfare is measured as the amount of output produced minus the cost of e(cid:11)ort, our results show the net increase in welfare from the (cid:12)rm’s bonus plan is 5%. We also show that under a (cid:12)rst-best scheme, the net increase in welfare is 6%. Most of the existing empirical work on incentives has focused on answering the (cid:12)rst two of three questions listed at the beginning of the introduction: Do incentives matter and, if so, are they signi(cid:12)cant? The typical approach has been to use reduced form analysis to quantify how much output is a(cid:11)ected by a change in incentives. The results from this avenue of research vary widely, attributing an increase in productivity from zero to over twenty percent due to the introduction of incentives.1 A limitation of the reduced formapproach however, is that this methodology cannot answer the third question previ- 1Lazear (2000) is a recent paper that quanti(cid:12)es the relationship between output and incentives. Blinder (1990) and Prendergast (1999) are good surveys on this literature. 4

ously posed: What are the welfare gains from using incentives? Answering this question requires estimating the worker’s disutility from e(cid:11)ort, which can only be done using a structural approach. There are a small number of papers within the personnel literature that use structural estimation. These papers, however, focus on answering other important questions within the literature, rather than attempt to measure the welfare gains from incentives. Paarsch and Shearer (2000), for example, study the optimal contract choice of a (cid:12)rm while Margiotta and Miller (2000) estimate the cost of moral hazard at the senior managerial level. Within this group of papers, our work is closest to Ferrall and Shearer (1999). They estimate the structural parameters of a principal-agent model using a (cid:12)rm’s payroll records from the 1920’s. With a static model of e(cid:11)ort, they analyze the classic tension between risk averse workers and the (cid:12)rm’s use of incentives. We, on the other hand, develop a dynamic model of e(cid:11)ort and abstract away from risk aversion. Assuming risk neutrality simpli(cid:12)es the computation of the worker’s policy function. In addition, while Ferrall and Shearer study a biweekly bonus scheme, the incentive scheme we analyze is daily and only in e(cid:11)ect for part of a worker’s shift. As such, given that the gambles workers take are relatively small, using risk neutrality provides us with a plausible approximation of worker behavior. Paarsch and Shearer (1999) is another paper close to ours. They use a structural approach to estimate worker e(cid:11)ort under piece rates in the tree-planting industry. Our papers di(cid:11)er in our methods of identi(cid:12)cation and in our modeling of the worker’s problem. Two other papers related to our work are Asch (1990) and Oyer (1998). These two papers describe data on the behavior of Navy recruiters and salespeople, respectively, under incentive schemes. They both show that employees signi(cid:12)cantly alter their behavior during a pay period, conditional on their history of events and future expectations. Hence, they (cid:12)nd that dynamic models of worker behavior, such as the one we use, are signi(cid:12)cant improvements over their static counterparts. Our analysis also points to the importance of dynamic models. Finally, our work is also related to the general literature on testing contract theory. Papers such as Chiappori, Heckman, and Pinquet (2000) and Chevalier and Ellison (1999) use data that captures agents’ dynamic behavior to measure the e(cid:11)ect of moral hazard in, respectively, automobile insurance and mutual fund markets. We also use this ‘dynamic data’ approach to identify the moral hazard e(cid:11)ect of incentives. Chiappori and Salani(cid:19)e (2000) provides a survey of recent work done in this general research area. The rest of our paper is organized as follows. Section 2 describes the data. Section 3 5

lays out the model and derives the worker’s problem. Section 4 describes the estimation procedure and reports results. Section 5 concludes. 2 Data Description In this section, we describe our data set. We (cid:12)rst explain the nature of the check-sorting job and in which set of workers we are interested. We then describe the (cid:12)rm’s incentive pay scheme and summarize workers’ performance under it. Our data comes from the production and human resource records of the Check Department of the Federal Reserve Bank of Minneapolis. This ‘(cid:12)rm’ provided us with information over a 15 month period (3/01/99 - 5/27/00) on its employees that sort checks in the Low Speed Check Processing Department.2 While these workers have several responsibilities, their main task is to sort checks by running them through a sorting machine (see Figure 1). Ideally, this sorting machine would process checks without any worker input. However, two events occur that require worker interaction: checks get jammed in the machine, and (cid:12)elds on the check cannot be electronically read by the machine.3 In the (cid:12)rst instance, workers clear the jammed checks and reset the sorting machine. In the second, workers type in the (cid:12)eld which the machine failed to read. Checks are processed in batches. We de(cid:12)ne a job as a batch of checks that needs to be sorted. The production data we received from the (cid:12)rm is at the job level and includes information on which worker ran a job, what time it was run, how long it took, and its characteristics (e.g. the number of jams that occurred). The human resource component of the data set provides us with information about the tenure of theworker, as well asthe worker’s wage-gradelevel. Using thisinformation, we selected those workers who were full-time and were employed in the check-sorting department for at least 6 months before the beginning of our sample period. Excluding new workers allows us to ignore the e(cid:11)ects of learning-by-doing, simplifying our analysis. Of the original 52 workers, only 15 met these requirements. However, this group of workers completed 34,077 jobs in the data set, which account for roughly half of all checks sorted in the 15 month sample period. 2This Department processes all checks that are ‘rejected’ from the High Speed Check Processing Department. Checks are (cid:12)rst processed by the High Speed Department. But if the High Speed sorting machines have any di(cid:14)culty processing a check, that check is immediately diverted to the Low Speed Department for further processing. 3An example of a (cid:12)eld is the account number. 6

Figure 1: Reader/Sorter Machine Looking at this subset of the data, we (cid:12)nd there is a wide range in the number of jobs a worker completed within a day (1-90) as well as the number of checks sorted by job (1-9000). Typically however, workers run 9 jobs a day, where each one averages 660 checks in length and takes 18 minutes to complete. In addition, workers on average clear 9.3 jams and type in 128 (cid:12)elds per job. As the large number of jams and (cid:12)eld corrections indicate, worker input is a large determinant of how fast a job is completed. The (cid:12)rmuses anincentive pay arrangement that rewards workers based ontheir daily performance above and beyond their hourly wage. This mechanism works by using a formula that provides a benchmark time for each worker, given the characteristics of the jobs the worker ran that day. Some characteristics that the (cid:12)rm uses are the number of checks sorted, the number of jams cleared, and the number of (cid:12)elds manually typed in. Letting z(cid:22) be a vector of C characteristics of a job, we denote this formula as (cid:11)(z(cid:22)). This 7

formula is linear in job characteristics, having the following form: (cid:11)(z(cid:22)) = (cid:26) +(cid:26) z1 +:::+(cid:26) zC: (1) 0 1 C If a worker completes N jobs in a day, and z(cid:22) denotes the characteristics of job n, then P n N (cid:11)(z(cid:22) ) determines the overall time that a worker needs to beat in order to earn any n=1 n bonus pay. Conditional on achieving this, the amount of bonus pay a worker receives is an increasing function of the di(cid:11)erence between the worker’s actual and benchmark time. P Let (cid:28) be the actual time a worker spends on a job and de(cid:12)ne s = N [(cid:11)(z(cid:22) ) − (cid:28) ]. n n=1 n n The variable s is the amount of time a worker is behind or ahead the benchmark time at the end of the day. Note that s is a function of all the jobs a worker completed in a day. We can then write the (cid:12)rm’s bonus payment scheme as 8 < 0 if s (cid:20) 0 ~b(s) = (2) : K (cid:1)s otherwise where K > 0 is some constant. So given s > 0, the bonus amount that a worker earns is a function of the total time s, in hours, by which the worker beat the benchmark formula, multiplied by a wage K. The (cid:12)rm has provided us with the constant K as well as its incentive pay formula, (cid:11), which enables us to reproduce the daily cuto(cid:11) times each worker faced. An important aspect of this data set is a change in the incentive pay formula in January of 2000, roughly two-thirds of the way through the sample period. Before the switch, K = $7:17 for all employees. After the switch, K = $9:75 for workers with a grade of 4 or 5 and K = $12:76 for those with a grade of 6 or 7.4 In addition, the parameters of the formula (cid:11) and the set of job characteristics z(cid:22), were changed in order to raise the threshold level of productivity. So the switch in the incentive scheme involved two opposing e(cid:11)ects: on one hand, it became harder to earn incentive pay, while on the other hand, bonuses were potentially bigger as K was increased. Table 1 contains the average bonus payments for workers under the two regimes, as well as the fraction of total jobs where a positive incentive amount was earned. This table demonstrates that after the switch, workers earned smaller bonus payments less often. Notice that the average daily bonus of all workers decreased from $10.06 to $6.64. 4Lower grade employees earn lower wages and are typically newer employees relative to their higher grade counterparts 8

Table 1: Bonus Payments First IP Regime Second IP Regime Worker Grade Mean (Std) Frac Mean (Std) Frac All n/a $10.06 (8.54) 0.95 $6.64 (7.62) 0.81 High grade 6 & 7 $10.94 (9.19) 0.96 $8.63 (8.50) 0.85 Low grade 4 & 5 $8.46 (6.94) 0.92 $3.32 (4.07) 0.73 1 7 $14.60 (12.43) 1.00 $11.19 (11.30) 0.96 2 7 $8.52 (7.96) 0.95 $7.28 (6.33) 0.93 3 7 $9.74 (11.33) 0.97 $10.50 (8.80) 0.92 4 6 $14.29 (10.03) 0.98 $14.04 (9.32) 0.99 5 6 $18.22 (6.48) 1.00 $16.13 (5.33) 1.00 6 6 $15.23 (8.94) 0.98 $12.59 (8.88) 1.00 7 6 $9.14 (4.78) 1.00 $2.75 (2.80) 0.72 8 6 $6.10 (6.33) 0.93 $5.44 (5.78) 0.82 9 6 $5.45 (3.98) 0.84 $0.22 (0.74) 0.12 10 6 $6.95 (6.38) 0.95 $4.59 (4.58) 0.85 11 5 $11.05 (8.14) 0.98 $4.67 (4.63) 0.92 12 5 $5.35 (6.02) 0.80 $4.64 (3.31) 0.89 13 5 $14.71 (6.24) 1.00 $5.30 (5.37) 0.85 14 4 $5.27 (3.39) 0.90 $0.54 (1.07) 0.33 15 4 $5.40 (3.42) 0.95 $1.86 (2.32) 0.76 Frac: Fraction of days with positive bonus pay In addition, the fraction of days where a worker earned a bonus fell from 95% to 81%. Table 1 also shows the large amount of heterogeneity in worker productivity. Worker 5 is one of the most productive workers in the sample, earning an average bonus of over $16 under both bonus regimes. Also, this worker always earned a bonus. At the other extreme, Worker 9 received a mean bonus of $5.45 under the (cid:12)rst regime and $0.22 under the second. Under the second regime, this worker earned a bonus only 12% of the time. Workers also di(cid:11)er in how they reacted to the change in incentives. Workers 3 and 7 both earned an average bonus payment of roughly $9 a day under the (cid:12)rst incentive pay regime. Under the second regime however, Worker 3’s average bonus increased to $10.50 while Worker 7’s dropped to $2.75. How important, though, are these bonus payments to workers? To check this, we computed the mean of the ratio of bonus pay to total wages. As these workers performed tasks throughout the day not under the purview of the incentive scheme, the focus of this 9

ratio is restricted to time spent on sorting machines. Total wages are thus computed as thetimespentonasortermultipliedbytheworker’shourlywageplusthebonusamount.5 The mean value over all workers for this ratio was 0.25 under the (cid:12)rst incentive scheme and 0.18 for the second. The mean value by worker ranged from 0.10 to 0.34 and 0.003 to 0.33 for the (cid:12)rst and second incentive schemes respectively. These results suggest that when looking at the time spent sorting checks, incentives are indeed signi(cid:12)cant to workers. While incentives are bene(cid:12)cial in that they increase worker productivity, they also have the disadvantage of adversely a(cid:11)ecting the quality of output. In this check-sorting environment, workers, in an attempt to increase productivity, could decrease quality by entering incorrect numbers for the (cid:12)elds that cannot be electronically scanned. However, there is another department within the (cid:12)rm that is able to detect when incorrect (cid:12)eld numbers have been entered. When a mistake is found, an analyst goes through the records to (cid:12)x the error and (cid:12)nd the responsible worker. Due to the time this errorchecking process takes, the (cid:12)rm views such mistakes seriously. A worker who makes such an error is docked pay, and repeat o(cid:11)enders are (cid:12)red. By reviewing the payroll records for workers over the sample period, we found that workers rarely made these quality errors, and no worker continually made them over time. As such, in this paper we do not model a quality trade-o(cid:11). 3 The Model In this section we layout the model and describe some theoretical predictions. We (cid:12)rst de(cid:12)ne the environment of the worker and derive the worker’s e(cid:11)ort decision problem. We then prove that a worker’s e(cid:11)ort level is increasing in the probability of earning a bonus. Finally, we test whether this implication holds in the data. 3.1 The Environment Each day the (cid:12)rm needs to hire a worker to sort N checks.6 As in the standard moral hazard model, the (cid:12)rm cannot determine the e(cid:11)ort level exerted by the worker.7 We 5Hourly wages for employees range from $8 to $14 an hour 6Each day workers are shown a detailed schedule of their day. This information, along with their experience on the job, enables workers to forecast the amount of checks they will process in their shift. 7See Grossman and Hart (1983) and Holmstrom and Milgrom (1987). 10

accomplish this by modeling the time it takes a worker to process a check as a function of three variables: the worker’s e(cid:11)ort level and two random shocks. We model e(cid:11)ort as a binary choice, e 2 f0;1g. The (cid:12)rst random shock is a vector z(cid:22) of the characteristics of the check that are observed both by the (cid:12)rm and the econometrician. An example of this characteristic would be whether or not a check jams the machine. We assume that z(cid:22) is independent over checks and let F denote its cdf. The second shock " is a characteristic of a check that is unobservable to the (cid:12)rm and the econometrician. An example of this shock would be a particularly tricky jam or ripped check. This shock is also independent over checks, and we assume it is distributed normally, with mean (cid:22) " and variance (cid:27)2. Using these three variables, we can write the time it takes to complete " a job as (cid:28)(e;z(cid:22);"). Under this formulation, even though the (cid:12)rm knows (cid:28) and observes z(cid:22), it cannot determine the worker’s e(cid:11)ort level because of the unobserved e(cid:11)ects of ". A strong assumption on (cid:28) is that both shocks are i.i.d. throughout the day. Focusing (cid:12)rst on z(cid:22), it might be the case that the probability of a check jamming increases if the previously sorted check jammed. Using the data however, we were able to reject the hypothesis that check characteristics are correlated within a day.8 Turning next to ", there are two stories why this variable may be correlated within a day. The (cid:12)rst story is that shipments of checks may have some common unobserved component. Checks from Bank A, for example, may be packaged and shipped in such a way that they rip more often than checks from Bank B. As we have data on the orginating location of checks, we were able to explore and reject this hypothesis. The second story for why " may be correlated within a day focuses on the worker, rather than the checks. Workers might have good and bad days, which would a(cid:11)ect the time it takes them to sort checks. Rather than capture this e(cid:11)ect through (cid:28), however, we include a daily cost shock in the worker’s cost of e(cid:11)ort function, c. Hence, when workers feel particularly good or bad, the resulting e(cid:11)ect on their ability to sort checks will come through their disutility of exerting e(cid:11)ort.9 Turning to the cost of e(cid:11)ort, we specify it as a function of both the level of e(cid:11)ort exerted, and a daily cost shock, (cid:14) (cid:24) N((cid:22) ;(cid:27) ). As mentioned previously, this daily shock (cid:14) (cid:14) captures the fact that workers may have good and bad days. We assume that e(cid:11)ort is costly, c(e = 1;(cid:14)) (cid:21) c(e = 0;(cid:14)) 8(cid:14) and that dc(0;(cid:14)) > 0 and dc(1;(cid:14)) > 0. d(cid:14) d(cid:14) 8In the data we are also able to identify which machine a worker used to sort checks. As workers mentioned in interviews, we found that all the machines performed the same in sorting checks. 9Weestimatedaversionofthe modelwheredailyshocksa(cid:11)ectedthe(cid:28) functiondirectly. Therewere no substantial di(cid:11)erences in the results. 11

The timing of events plays an important part in the model. At the beginning of the day, workers draw their daily cost shock. We then assume that workers choose an e(cid:11)ort level e before they process each check. While the check is being sorted, the two random shocks, (z(cid:22);"), are realized. This implies that when the worker makes an e(cid:11)ort decision before sorting a check, allchecks lookidentical. It is only after the check has been sorted, after the realization of (z(cid:22);"), that checks are distinguishable. As this paper focuses on worker behavior, we take the contracts o(cid:11)ered to workers as exogenous. Hence, we do not model the (cid:12)rm’s problem. As discussed in section 2, this contract includes a (cid:12)xed wage w(cid:22) and a variable incentive component, the function ~b (see equation 2). Note that the (cid:12)rm’s incentive scheme is how this model di(cid:11)ers from the standard moral hazard problem. Unlike in the standard model, workers in this (cid:12)rm do not receive compensation after every e(cid:11)ort decision, but rather at the end of the day after sorting N checks and making N e(cid:11)ort decisions. As shown in the following section, this change makes theworker’s problem dynamic, which is adeparture fromthe standard model. Workers’ utility depends upontheir wageandhowmuch e(cid:11)ort theyexert. We assume that workers are risk-neutral and that there is no discounting. Utility is then separable in wage and e(cid:11)ort, and can be speci(cid:12)ed as (cid:16)XN (cid:17) XN w(cid:22) + ~b (cid:11)(z )−(cid:28)(e ;z(cid:22) ;" ) − c(e ;(cid:14)); (3) n n n n n n=1 n=1 where the summation is over the N checks a worker processed in a day. The (cid:12)rst term in the utility function is the workers’ base wage, the second term is their daily bonus, and the last term the cost of e(cid:11)ort over the entire day. The assumption that a worker is risk neutral, as opposed to risk averse, simpli(cid:12)es the worker’s policy function and allows us to directly compare the dollar gains from output with the cost of e(cid:11)ort in our welfare analysis. This assumption however, is not crucial for our results. As the variation in a worker’s income due to bonuses is small and as the bonuses are paid out at a high frequency, worker’s behavior under risk neutrality is a good (cid:12)rst order approximation of the worker’s behavior under risk aversion.10 10To provide a stronger basis for this claim, we computed the certainty equivalent of an agent with a standard level of risk aversion when faced with the varying income stream of a typical worker in the data. We found that the percentage di(cid:11)erence between the certainty equivalence and the mean level of income to be less than one-half of a percent. 12

3.2 The Worker’s Problem The worker’s problem is to decide, for every check, whether e(cid:11)ort should be exerted. To form expectations on the amount of bonus pay a worker will receive at the end of the day, a worker needs to know the history of events in the day as well as the number of checks left in the day. The history of events a worker observes is the triplet (e;z(cid:22);") for all checks already sorted. This information allows the worker to determine how well the worker is doing with respect to the (cid:12)rm’s formula (cid:11). A su(cid:14)cient statistic for this history is the variable s, which is the sum of the di(cid:11)erence between (cid:11)(z(cid:22)) and (cid:28)(e;z(cid:22);") for all checks a worker has already processed in a day. Hence, s = 0 for the (cid:12)rst job in the day. Letting (e;z(cid:22);") be the choice of e(cid:11)ort and realizations of the two random shocks that occurred when sorting the latest check, we can de(cid:12)ne the law of motion of s as s0 = s+(cid:11)(z(cid:22))−(cid:28)(e;z(cid:22);"). The number of checks a worker has left in the day to sort is simply denoted as n, where n 2 f0;1;2::: ;Ng. Using the two variables (s;n) and the daily cost shock (cid:14), we can write the worker’s problem recursively as an N period stochastic dynamic problem. In this environment, the worker only gets paid at the end of the day, n = 0, but incurs the cost of e(cid:11)ort, c(e;(cid:14)), each period. The worker’s value function is 8 n o <max −c(e;(cid:14))+E[V(n−1;s~(e;z(cid:22);";s);(cid:14)] if n = f1;2;::: ;Ng; V(n;s;(cid:14)) = e2f0;1g (4) : w(cid:22) + ~b(s) if n = 0; where the expectation is taken over (z(cid:22);") and s~(e;z(cid:22);";s) = s+(cid:11)(z(cid:22))−(cid:28)(e;z(cid:22);"): To solve for the worker’s policy function e~(n;s;(cid:14)), we use backward induction and determine for which values of s a worker will exert e(cid:11)ort, in every period n and for every (cid:14). In deciding whether or not to choose e = 1, the worker computes whether the expected value of the bonus at the end of the day is larger than the cost of e(cid:11)ort this period and the expected cost of e(cid:11)ort in future periods. 13

3.3 Theoretical Results To better understand the worker’s problem, we analyze the comparative statics of the worker’s policy function e~(n;s;(cid:14)). We (cid:12)rst examine how e~(n;s;(cid:14)) changes with respect to n, holding (s;(cid:14)) (cid:12)xed. Increasing n, or increasing the number of checks a worker has left to sort, provides the worker with more opportunities to a(cid:11)ect s. For a highly skilled worker, this increase in opportunities is bene(cid:12)cial as the worker has more chances to increase s and so o(cid:11)set any bad draws of (z(cid:22);"). Conversely, for an unskilled worker who struggles to sort checks faster than the (cid:12)rm’s estimated time, increasing n is not bene(cid:12)cial. More opportunities mean that this worker is less likely to have a positive s at the end of the day, and so be eligible for incentive pay. Hence, without knowing the skill level of a worker, we are unable to predict whether e~(n;s;(cid:14)) is increasing or decreasing in n.11 Next, we turn to examining how e~(n;s;(cid:14)) changes with respect to s, for a (cid:12)xed (n;(cid:14)). Due to the structure of the bonus scheme, the probability and size of the worker’s bonus is increasing in s. As such, as s increases, so does the worker’s e(cid:11)ort level. Below, we formally prove this result, assuming that e(cid:11)ort is a continuous variable and that c(e;(cid:14)) is increasing and convex. We assume continuity as it simpli(cid:12)es the proof of the theorem and highlights the relevant forces at work. Theorem 1. Given e(cid:11)ort is a continuous variable, e 2 [0;1], for all (n;(cid:14)), the policy function e~(n;s;(cid:14)) is increasing in s. Proof. We (cid:12)rst need to show that V(n;s;(cid:14)) is increasing and convex in s for all (n;(cid:14)). This can be shown by induction, using the necessary (cid:12)rst order condition, the envelope theorem, and the fact that V(s;0;(cid:14)) is increasing and convex in s, 8 (cid:14). Then, assuming an interior solution, the (cid:12)rst order condition from the worker’s e(cid:11)ort decision problem is Z (cid:16) (cid:17) dc(e;(cid:14)) dV(n−1;s0;(cid:14)) d(cid:28)(e;z(cid:22);") − + − = 0; de ds0 de where s0 = s+(cid:11)(z(cid:22))−(cid:28)(e;z(cid:22);") and the integral is over all possible pairs of (z(cid:22);"). Given that c, (cid:28), and V are convex functions, we use the implicit function theorem to show that de~(n;s;(cid:14)) (cid:21) 0. Whenever the solution is not interior for a given s, the optimal e(cid:11)ort level ds de~(n;s;(cid:14)) is con(cid:12)ned to either 0 or 1, and we have = 0. ds 11Using a probit model, we infer that the probability of earning a bonus is increasing in n for all workers. 14

Figure 2: A Plot of G(n;s;(cid:14)) 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −200 −150 −100 −50 0 50 100 150 200 S, Minutes Ahead or Behind troffE morf niaG ralloD ,)0,S,2(G For a given set of parameters, we can calculate the gain from e(cid:11)ort, G(n;s;(cid:14)), or the di(cid:11)erence in utility from high versus low e(cid:11)ort. Let V (n;s;(cid:14)) = c(e;(cid:14))+E[V(n−1;s;(cid:14))]; (5) e then G(n;s;(cid:14)) = V (n;s;(cid:14)) − V (n;s;(cid:14)). Theorem 1 implies that G is increasing in s 1 0 for a given (n;(cid:14)). Using our results from the estimation section of our paper, we plot G over s given n = 2 and (cid:14) = 0 . As demonstrated in (cid:12)gure 2, G is non-linear and, as predicted, increasing in s. 3.4 Empirical Tests Theorem 1provides away forustotest ourmodelusing reduced formanalysis. It implies that for (cid:12)xed (n;(cid:14)), a worker’s e(cid:11)ort level is increasing in s. To test this implication 15

and to estimate an upper bound on the e(cid:11)ect of a high level of e(cid:11)ort on the time taken to sort checks, we perform two experiments. First, we examine whether worker’s e(cid:11)ort levels vary with s as predicted by theory. We accomplish this by looking at the last job workers processed in a day. Looking at this subset of the data allows us to examine worker behavior for a relatively small range of n while still observing a wide range of s. In addition, theory predicts that within a day, workers are more likely to alter their e(cid:11)ort levels at the end of the day. As such, analyzing this set of jobs will most clearly highlight the e(cid:11)ect of incentives on worker behavior. The second experiment we run estimates the upper bound on the di(cid:11)erence between high and low e(cid:11)ort on the time it takes to sort checks. We do this by measuring the di(cid:11)erence in workers’ productivity residuals for the case when workers have extremely high and extremely low s. In e(cid:11)ect, we are measuring the di(cid:11)erence in worker productivity when a worker will earn a bonus with probability one versus the case when a worker will earn a bonus with probability zero. Using the data, we can compute two of the worker’s state variables, (n;s), for every job (i.e. for every observation). E(cid:11)ort, naturally, is unobserved. However, we can measure the e(cid:11)ect of e(cid:11)ort on the time taken to process a job by looking at the residuals of a productivity regression. The regression we run has the time taken to complete a job as the dependent variable. The independent variables are the job’s observable characteristics, day e(cid:11)ects, worker dummy variables, and the worker’s state variable n. Recall that (cid:28) denotes the time taken to sort a job j, z(cid:22) are a job’s characteristics, and j j n is the number of checks in the day that the worker processing job j has left to sort, j including those checks in job j. Let d be the day in which the job was processed, and j i be the worker who processed the job. Note there are 366 days and 15 workers in the j sample. The regression we ran is (cid:16) (cid:17) X366 X15 ln((cid:28) ) = ln (cid:12)0z(cid:22) + (cid:24) (cid:1)1 + (cid:17) (cid:1)1 +(cid:23) (cid:1)n +" ; (6) j j l l=d k k=i j j j j l=1 k=1 where(cid:12) isavector ofcoe(cid:14)cients and1 isadummy variableequalto1when x = y. As x=y e(cid:11)ort is unobserved, its e(cid:11)ect on time is captured by the residual term of this regression. While the above regression captures day and operator e(cid:11)ects, it does not control for operator-dayshocks. Assuch, our results fromusing theresiduals ofthe above regression provide us with an upper-bound on the e(cid:11)ect of e(cid:11)ort. In the next section, we use a structural approach to separate out the e(cid:11)ects of daily shocks and e(cid:11)ort on the time 16

Table 2: Conditional Means of Residuals Obs Mean Std Error High Prob 13941 -0.0037 0.0023 Low Prob 319 0.0791 0.0205 Note: ‘High Prob’ is de(cid:12)ned as s being greater than half an hour ‘Low Prob’ is de(cid:12)ned as s being less than half an hour taken to sort checks. After computing these residual terms, we then look at the subset of the data containing the last job of the day that a worker completed, and measure the relationship between these residuals and s. Theorem 1 predicts a negative correlation between s and the residuals, as a higher s induces higher e(cid:11)ort, reducing the time taken to sort checks. Con(cid:12)rming the theoretical prediction, we (cid:12)nd that the Pearson correlation coe(cid:14)cient is -0.06 and is strongly signi(cid:12)cant. To compute an upper bound on the e(cid:11)ect of e(cid:11)ort on the time taken to sort checks, we look at the subsets of the data where theory predicts workers will exert high e(cid:11)ort and where they will exert low e(cid:11)ort. As workers spend, on average, 2 to 3 hours a day sorting checks, a worker with an s greater than half an hour has a high probability of earning a bonus and so will exert a high level of e(cid:11)ort. In contrast, a worker with an s less than negative thirty (i.e. they are half an hour behind) has a low probability of earning a bonus and will exert a low level of e(cid:11)ort. To analyze the gain in time due to a high level of e(cid:11)ort, we create two subsets of the data based on s being larger than thirty, and less than negative thirty. We then compute and compare the mean value of the residual terms in both subsets. As mentioned above, the residual term captures the e(cid:11)ect of e(cid:11)ort on the time taken to process a job. Table 2 contains the results. The di(cid:11)erence between the conditional means signi(cid:12)es that a high level of e(cid:11)ort reduces the time taken to sort checks by at most 8%, compared to the case where low e(cid:11)ort is exerted. Hence, despite the automated nature of the check sorting process, workers do play a large role in determining how fast checks are sorted. Another interesting aspect of Table 2, is that unlike the conditional mean in the low probability case, the conditional mean of the residuals in the high probability case is not signi(cid:12)cantly di(cid:11)erent from zero. This is a function of the observations in each category. In the high probability case, there are almost 14,000 observations, while the low probability case only contains 319. This reflects thefact that workers earned a bonus 95%ofthe time under the(cid:12)rst incentive pay 17

regime and 81% under the second. As the entire data set has 34,076 observations, it is not surprising that the conditional mean in the high probability case is not signi(cid:12)cantly di(cid:11)erent from zero. To check the sensitivity of our results to the number of observations, were-computedourresultsusing di(cid:11)erent cuto(cid:11)sfors. There werenosigni(cid:12)cant changes in our results. These results suggest that e(cid:11)ort plays an important role in the time taken to process checks and that workers do respond to incentives as predicted by theory. To better understand how worker’s react to incentives and to compute the cost of e(cid:11)ort to the worker, we estimate the structural parameters of the model. 4 Estimation In this section, we describe how we estimate the structural parameters of this model. We begin by stating and justifying our functional form speci(cid:12)cations. Then we show how the model is identi(cid:12)ed and summarize our estimation technique. Finally, we report our parameter estimates and discuss their implications. 4.1 Speci(cid:12)cation In order to estimate this model, we need to know the functional form of (cid:28) and c, and the distributions of z(cid:22);", and (cid:14). In specifying the functional form of (cid:28), there are three main issues that we consider. First, workers are heterogeneous in their productivity, as demonstrated by the wide range in mean bonus payments listed in Table 1. To capture this, we include (cid:12)xed e(cid:11)ects, (cid:17) , in (cid:28). i The second issue we consider is which characteristics of a job are important in determining how long a job takes to (cid:12)nish. We have previously mentioned three observed characteristics of jobs, the number of jams cleared, (cid:12)elds corrected, and checks sorted. There are, however, a number of other characteristics in the data, such as the area of the country from which the checks originated and which sorting machine was used, that may help determine (cid:28). To (cid:12)nd the set of characteristics that are important predictors of time spent processing a job, we regressed all observable characteristics in the data on time. The end result is that three characteristics - the number of jams, the number of (cid:12)elds corrected, and the number of checks sorted - explain 90% of the variation in time. Other characteristics either do not add any explanatory power or only have a marginal 18

e(cid:11)ect. For instance, we checked and rejected the hypothesis that reader-sorter machines processed checks at di(cid:11)erent speeds. In our speci(cid:12)cation of (cid:28), we only include the number of jams and (cid:12)elds corrected as a job’s observable characteristics. We do this because of a problem computing the worker’s policy functions. We found that solving the worker’s problem for a large number of checks, N, took a prohibitively long time. As such, we decided to approximate the model by assuming that workers made an e(cid:11)ort decision every 1000 checks. This, however, raised a problem with the structure of our data. As discussed in the Data Description (Section 2), anobservationinour dataset isa job, where a jobranges from1 toover 8000 checks. To bring the model speci(cid:12)cation and data into line, we re-arranged the data to construct observations of 1000 checks. We accomplish this by (cid:12)rst chronologically lining up the jobs for every worker in a day. Then, starting at the end of the day, we cut and spliced jobs together to make new jobs of uniform length. Depending on their number, the residual checks left at the beginning of the day were either discarded or expanded into a 1000 check job. Under this modi(cid:12)cation, the number of checks per job in the data is constant and so can be captured by (cid:28)’s intercept term. Consequently, we construct the random variable z(cid:22)of a job’s characteristics, as a 2(cid:2)1 vector, where z1 is the number of jams that occurred and z2 is the number of (cid:12)elds corrected. Reducing the vector of characteristics z(cid:22) to two dimensions is advantageous as it decreases the computation burden of solving the worker’s problem. However, it also complicates the problem of computing a worker’s expectations over (cid:11)(z(cid:22)). The (cid:12)rm’s actual incentive pay formula uses a number of characteristics other than jams and (cid:12)elds corrected. As such, when we compute the worker’s expectations over the state variable s in the next period, we use an approximation of the (cid:12)rm’s actual incentive pay formula. Like the (cid:12)rm’s actual formula, the approximation we use is a linear function of jams and (cid:12)elds corrected.12 Finally, the last issue we consider with respect to (cid:28)’s speci(cid:12)cation is how e;z(cid:22); and " interact with one another. The result mentioned above, that a linear regression of the job’s observable characteristics on time has an R-squared of 90%, is strong evidence for a linear speci(cid:12)cation of (cid:28) in z(cid:22). To check the magnitude of a non-linear relationship between the observable characteristics and time, we re-ran the above regression adding squared and cubed terms of the observable characteristics to the set of regressors. These 12We estimated the coe(cid:14)cients and intercept term of this function using ordinary least squares. The R-squared for this regression under the (cid:12)rst regime is 0.70 while under the second regime it is 0.98 . 19

additional non-linear terms did not add any explanatory power, adding to the credibility of (cid:28) being linear in z(cid:22). Lastly, with regard to e(cid:11)ort, our observations in the workplace and conversations with workers lead us to believe that e(cid:11)ort has a direct e(cid:11)ect on how quickly jams are cleared and (cid:12)elds are entered. Consequently, the e(cid:11)ort term in (cid:28) needs to interact with both elements of z(cid:22). With these issue in mind, we specify (cid:28) as h (cid:16) (cid:17) (cid:16) (cid:17) i (cid:28)i (e;z(cid:22);") = (cid:12) + (cid:12) −(cid:12) (cid:1)e (cid:1)z1 + (cid:12) −(cid:12) (cid:1)e (cid:1)z2 (cid:1)exp((cid:17) )(cid:1)exp("); (7) 0 1 3 2 4 i where (cid:12) ;(cid:12) (cid:21) 0, " (cid:24) N(0;(cid:27)2), and i = 1;::: ;15. Taking logs, we get the form of the 3 4 " equation we actually estimate, h (cid:16) (cid:17) (cid:16) (cid:17) i ln((cid:28)i (e;z(cid:22);")) = ln (cid:12) + (cid:12) −(cid:12) (cid:1)e (cid:1)z1 + (cid:12) −(cid:12) (cid:1)e (cid:1)z2 +(cid:17) +": (8) 0 1 3 2 4 i Inthis speci(cid:12)cation, theheterogeneity inworkers iscapturedby adjusting thecoe(cid:14)cients (cid:12) through (cid:12) by a fraction(cid:17) . We constrain the coe(cid:14)cients (cid:12) and (cid:12) to be non-negative 0 4 i 3 4 as our prior is that e(cid:11)ort lowers the time it takes to complete a job. This also reduces the parameter space and so speeds up the estimation algorithm. Having speci(cid:12)ed (cid:28) and the distribution of ", we are left with F, the cdf of z(cid:22). As we have data on realizations of z(cid:22), we use these observations to construct an empirical distribution of F. Turning to the cost of e(cid:11)ort, we specify c as c(e;(cid:14)) = e(cid:1)γ (cid:1)exp((cid:14)); (9) where γ (cid:21) 0 and (cid:14) (cid:24) N(0;(cid:27)2). Hence, the cost of low e(cid:11)ort is normalized to zero, while (cid:14) the cost of high e(cid:11)ort is γ (cid:1)exp((cid:14)). Note that as γ is the same for all workers, they are only heterogeneous in (cid:17) . i With these functional forms, we can compute the likelihood of the model. We have a panel data set of I individuals, where we observe each individual for D days. A day i is composed of J jobs. Individuals are heterogenous in that they have di(cid:11)erent (cid:12)xed i;d e(cid:11)ects, (cid:17) . In this data set, we have information on the job’s observable characteristics, i z^ , the number of checks left to sort in the day, n^ , the time taken to sort jobs, (cid:28)^ , i;d;j i;d;j i;d;j 20

and the benchmark times computed by the (cid:12)rm, (cid:11)^ . The likelihood is then i;d;j L((cid:12) ;(cid:12) ;(cid:12) ;(cid:12) ;(cid:12) ;γ;f(cid:17) g;(cid:27) ;(cid:27) jfz^ ;n^ ;(cid:11)^ ;(cid:28)^ g) = 0 1 2 3 4 i " (cid:14) i;d;j i;d;j i;d;j i;d;j YI YD i Z 1 YJ i;d (cid:16) (cid:17) P ln((cid:28)^ )jfz^ ;n^ ;(cid:11)^ g;(cid:12) ;::: ;(cid:12) ;γ;(cid:17) ;(cid:27) ;(cid:14) (cid:30) (d(cid:14)); (10) i;d;j i;d;j i;d;j i;d;j 0 4 i " (cid:14) −1 i=1d=1 j=1 where (cid:30) x = N(0;(cid:27) x 2). From equation 8, we know that the likelihood (cid:16)of the observatio(cid:17)n ln((cid:28)^ ), conditional on the data and parameters speci(cid:12)ed above, is (cid:30) ln((cid:28)^ )−(cid:18) , i;d;j " i;d;j i;d;j where h (cid:16) (cid:17) (cid:16) (cid:17) i (cid:18) = ln (cid:12) + (cid:12) −(cid:12) (cid:1)e (cid:1)z^ 1 + (cid:12) −(cid:12) (cid:1)e (cid:1)z^ 2 +(cid:17) : (11) i;d;j 0 1 3 i;d;j i;d;j 2 4 i;d;j i;d;j i Thevariablee isthee(cid:11)ortlevelexertedbyworkeri, whilesortingajobj ondayd. We i;d;j compute which e(cid:11)ort level a worker chooses by solving the worker’s dynamic problem given the parameter values and obtaining the worker’s policy function. This policy function depends upon the worker’s state variables (n;s;(cid:14)). We observe n directly in the data, n^ , and can compute s from the two sequences f(cid:11)^ ;(cid:28)^ g. We integrate over i;d;j i;d;j i;d;j the distribution of the daily shock (cid:14). 4.2 Identi(cid:12)cation The general identi(cid:12)cation issue in this check-sorting environment is determining if e(cid:11)ort signi(cid:12)cantly a(cid:11)ects the time taken to sort checks. The main source of identi(cid:12)cation comes from the incentive e(cid:11)ects of the (cid:12)rm’s kinked bonus system. Looking back at the incentive pay program de(cid:12)ned by equation 2, note that this formula generates a bonus wage pro(cid:12)le that is flat at $0 dollars for all negative values of the state variable s. Then, for positive values of s, the wage pro(cid:12)le is linearly increasing in s. This kink at 0 creates a perverse incentive for workers to quit working hard once they have fallen too far behind in terms of s. In Theorem 1, we prove this intuitive result, showing that a worker’s e(cid:11)ort level is increasing in the probability of earning a daily bonus. So for large, positive values of s, a worker will choose high e(cid:11)ort while for values of s that are negative and large in absolute value, workers will choose low e(cid:11)ort. Theory, then, predicts that workers with low, negative values of s will have low productivity (high (cid:28)), while those with large positive values of s will have high productivity (low (cid:28)). Conversely, in an environment where incentives did not matter, there is no correlation between s and (cid:28). In the data, 21

we observe the values of s and (cid:28) for a worker for every job. As the distribution of s in the data ranges from under a low -180 minutes to over a high 200 minutes, we have a straightforward way of determining if e(cid:11)ort matters in this environment. Having daily shocks in our model does not a(cid:11)ect this avenue of identi(cid:12)cation. The e(cid:11)ect of daily shocks is captured by the variation in worker’s productivity across days, while e(cid:11)ort’s e(cid:11)ect on productivity is captured by within day variation in productivity. For instance, when a worker’s productivity is low throughout the day, the model ascribes this behavior to a bad daily cost shock, (cid:14), and constant low e(cid:11)ort. Similarly, if a worker’s productivity is high throughout the day, the model interprets this sequence of events as a good daily shock and constant high e(cid:11)ort. In contrast, when a worker begins the day with high productivity but ends the day with low productivity, the model ascribes this behavior to a bad job productivity shock, ", and the worker’s subsequent declining e(cid:11)ort level. The identi(cid:12)cation of e(cid:11)ort, then, comes from the dynamics of the data within the day. Estimates of the signi(cid:12)cance of the daily shock are derived from variation of productivity across days. Forour particularmodelandfunctionalformspeci(cid:12)cations, thisgeneralidenti(cid:12)cation problem reduces to showing that we can measure e(cid:11)ort’s e(cid:11)ect on time, captured by ((cid:12) ;(cid:12) ), and the cost of e(cid:11)ort, γ. As described above, theory predicts when e(cid:11)ort will 3 4 and will not be exerted in certain cases. Using this information, we are able to precisely measure (cid:12) and (cid:12) . Further, the model imposes a speci(cid:12)c structure on the relationship 3 4 among γ;(cid:12) ; and (cid:12) through the utility-maximizing behavior of workers. These cross- 3 4 equation restrictions allow us to estimate γ. In addition to this general source of identi(cid:12)cation, we can also identify e(cid:11)ort through another feature of the (cid:12)rm’s bonus scheme. The (cid:12)rm uses an adhoc formula (equation 1) to determine how well a worker is performing. Hence, it is possible for a worker to be doing well relative to a true measure of productivity, but to be performing badly relative to the (cid:12)rm’s incentive scheme. In such a case, despite the fact that the worker is actually performing well, the worker will give up and exert low e(cid:11)ort. This di(cid:11)erence between true productivity and the (cid:12)rm’s formula provides us with another way to separate out the e(cid:11)ect of random daily shocks and the e(cid:11)ect of bonus-maximizing e(cid:11)ort decisions by workers. A third source of identi(cid:12)cation comes from the (cid:12)rm’s change of parameters in the (cid:12)rm’s incentive scheme roughly two-thirds of the way through the sample period. As described in the data section of the paper, the (cid:12)rm altered the productivity thresholds 22

workers need to beat to earn bonus pay, as well as the formula used to compute bonuses conditional on eligibility. A consequence of this switch in the bonus scheme is that workers systematically changed when they would exert e(cid:11)ort, influencing their productivity. In an environment where e(cid:11)ort did not matter, however, a change in the incentive schemes would have no e(cid:11)ect on worker productivity. These di(cid:11)erent predictions on how productivity should change with the switch in incentives provides us with a third avenue of identi(cid:12)cation. 4.3 Estimation Algorithm As mentioned in the introduction, we use a simulated maximum likelihood approach to estimate the structural parameters of the worker’s problem. To compute the model’s likelihood, we use a simple three step algorithm that is common to the discrete-choice structural estimation literature.13 The (cid:12)rst step is to specify the functional forms of the cost of time function, (cid:28), and the cost of e(cid:11)ort function, c. Note that we have already assumed that workers’ utility is additive in wages and e(cid:11)ort. In addition, we need to choose values for all the parameters in the model. The second step involves using the parameter values and newly speci(cid:12)ed functions to solve the worker’s problem. For the functional forms we consider in this paper, the worker’s policy rules are cuto(cid:11) rules. Hence, for every period n, we (cid:12)nd a threshold value s(cid:22) where a worker will only choose n e = 1 if s > s(cid:22) . These policy rules are computed using backward induction. Finally, n the last step is to use the policy rules to infer the worker’s e(cid:11)ort decisions. Using this information along with the data, we then calculate the likelihood. Our technique is simulated maximum likelihood, as computing the model’s likelihood entails integrating over the distribution of the daily cost shock. To (cid:12)nd the set of parameters that maximize the likelihood, we use a two step approach. We (cid:12)rst search over the parameter space using a simulated annealing program with a large tolerance setting.14 This algorithm, while slow, performs well at searching over a large parameter space. In addition, the likelihood we are maximizing is a step function along certain dimensions, which the simulated annealing algorithm is adept at handling. We then take the result from the simulated annealing algorithm, and plug it into a standard simplex based algorithm with a small tolerance setting. This algorithm is faster than the simulated annealing one, and searches well within a local area. 13See, for example, Rust (1987) and Pakes (1986). 14A good source on how a simulated annealing algorithm works is Go(cid:11)e, Ferrier, and Rogers (1994). 23

Table 3: Parameter Estimates Parameter Estimate Standard Error (seconds) Intercept (cid:12) 357.357 1.5e-4 0 Jams (cid:12) 31.465 1.4e-5 1 Fields (cid:12) 3.831 1.2e-6 2 E(cid:11)ort on Jams (cid:12) 6.560 1.5e-5 3 E(cid:11)ort on Fields (cid:12) 0.839 1.1e-6 4 (dollars) Cost of e(cid:11)ort γ 0.455 6.2e-6 (no units) Standard deviation of period shock (cid:27) 0.179 1.3e-6 " Standard deviation of daily shock (cid:27) 1.747 3.6e-3 (cid:14) We repeat this second step several times, until the maximum likelihood results from consecutive searches are within 0.1 of each other. 4.4 Parameter Estimates Using these functional speci(cid:12)cations, we obtained the parameter estimates listed in Table 3. All parameter estimates in Table 3 are highly signi(cid:12)cant as are the (cid:12)xed e(cid:11)ect estimates listed in Table 4. These estimates imply that when a worker does not exert e(cid:11)ort, entering a (cid:12)eld takes 3.8 seconds. With e(cid:11)ort, a (cid:12)eld is entered 0.8 seconds faster, a 21% reduction in time. Clearing a jam when not exerting e(cid:11)ort typically takes a worker 31 seconds. With e(cid:11)ort, a worker takes 24 seconds, a 23% reduction in time. The gain from e(cid:11)ort with regard to clearing jams is easy to see. Motivated workers will simply perform the necessary steps required to clear a jam faster than an unmotivated one. E(cid:11)ort improves the speed with which a worker enters a (cid:12)eld (e.g. the account number on a check) in a di(cid:11)erent manner. Entering numbers on a reader/sorter machine is a task much like touch-typing, where typing speed depends upon the worker’s concentration. For this task, then, e(cid:11)ort improvestheworker’sfocusontheparticularjobathand. Togainabetterunderstanding of how important e(cid:11)ort is in reducing time, consider that a typical batch of 1000 checks requires a worker to clear 14 jams and type in 193 (cid:12)elds. Our parameter estimates imply that with e(cid:11)ort, the time a worker spends processing checks decreases by 17%. 24

Table 4: Worker Grade and Fixed E(cid:11)ect High Grade Worker 1 2 3 4 5 6 7 8 9 10 Grade 7 7 7 6 6 6 6 6 6 6 Fixed E(cid:11)ect -0.031 0.085 0.096 0.028 0.004 0.014 0.315 0.175 0.429 0.135 Low Grade Worker 11 12 13 14 15 Grade 5 5 5 4 4 Fixed E(cid:11)ect 0.104 0.240 0.104 0.385 0.301 Turning to the cost of e(cid:11)ort, notice that γ is 0.45. As we assumed that utility is additively separable in e(cid:11)ort and wage, this value can be interpreted as the dollar cost to a worker for choosing e = 1. Thus, the disutility from working hard while sorting 1000 checks is $0:45. We included (cid:12)xed e(cid:11)ects in order to capture worker heterogeneity. As shown in Table 4, workers widely di(cid:11)er in skill levels. The di(cid:11)erence between the best (worker 1) and the worst (worker 9), is considerable. Our results imply that worker 9 takes over 40% longer to sort checks than worker 1. Interestingly, both these workers have high grades, suggesting that skill alone does determine a worker’s grade. It is true, however, that most of the high grade workers sort checks faster than the low grade workers. 4.5 Analysis of E(cid:11)ort Decisions Using these parameter estimates, we can analyze workers’ e(cid:11)ort decisions. The model infers that workers exerted e(cid:11)ort 75% and 66% of the time respectively, under the (cid:12)rm’s two incentive schemes. Hence, the changes the (cid:12)rm made to the bonus scheme decreased the number of times workers exerted e(cid:11)ort by 9%. Recall that the (cid:12)rm’s bonus scheme was changed in two ways. First, the (cid:12)rm made it harder for workers to be eligible for a bonus by altering the parameters in the formula time (cid:11). Second, the (cid:12)rm increased the marginal return of e(cid:11)ort to workers, conditional on being eligible for a bonus, by increasing the bonus wage B. Note that these two changes have opposing e(cid:11)ects on the worker’s e(cid:11)ort decision problem. The result that workers exert less e(cid:11)ort after the change indicates that overall, the (cid:12)rm’s decision to increase the di(cid:14)culty of earning a bonushadamuchlargere(cid:11)ectonworkers’e(cid:11)ortdecisionsthanincreasingthebonuswage B. Table 5 shows the e(cid:11)ects of the change in the incentive scheme at the worker level 25

Table 5: Percentage of High E(cid:11)ort Decisions Worker Grade First IP Regime Second IP Regime Di(cid:11)erence All n/a 75% 66% -9% 1 7 78% 75% -3% 2 7 75% 75% 0% 3 7 77% 74% -3% 4 6 76% 75% -1% 5 6 76% 75% -1% 6 6 76% 75% -1% 7 6 75% 69% -6% 8 6 75% 73% -2% 9 6 69% 16% -53% 10 6 75% 74% -1% 11 5 76% 74% -2% 12 5 73% 73% 0% 13 5 75% 74% -1% 14 4 73% 27% -46% 15 4 74% 69% -5% While none of the workers increased the percentage of times they exerted e(cid:11)ort under the second bonus regime, there is a lot of variation in how much workers decreased how often they exerted e(cid:11)ort. Workers 2 and 12 continue to exert the same amount of e(cid:11)ort before and after the incentive change. In contrast, workers 9 and 14 drastically lower the amount of e(cid:11)ort they exert. This dramatic change, however, is not surprising given the data on their daily bonuses. As shown in Table 1, both these workers went from earning daily bonuses of over $5, to bonuses of less than a dollar. By looking at the worker’s policy rules, we were able to determine why these two workers behaved so di(cid:11)erently from the group. Under the (cid:12)rst regime, all workers start the day exerting e(cid:11)ort, as long as they do not receive a bad daily shock. This is also true under the second regime, except for workers 9 and 14. They perceive that their chances of earning a bonus are so low under the new bonus scheme, that they no longer exert e(cid:11)ort in the (cid:12)rst period unless they receive a good daily shock. So unlike their co-workers, these two workers only exerted e(cid:11)ort after receiving a particularly good daily or period shock. Knowing how often workers exert e(cid:11)ort allows us to determine the increase in productivity due to the (cid:12)rm’s bonus plan. Using our parameter estimates, we compute that exerting e(cid:11)ort typically decreases the time spent processing checks by 16.5%. This 26

implies that under the (cid:12)rst incentive pay plan, workers’ high e(cid:11)ort decisions decreased the time spent sorting checks by (0:75(cid:1)0:16) = 12:0%. The decrease in time under the second incentive pay scheme is slightly less, at 10.6%. Taking the average of these two numbers, weighted by the number of observations under each regime, we (cid:12)nd that over the sample period, the (cid:12)rm’s bonus plan decreased the time spent sorting checks by 11.9%. These results illustrate the large ine(cid:14)ciencies associated with the (cid:12)rm’s incentive scheme. The kinked nature of the bonus scheme discourages workers from exerting e(cid:11)ort after receiving a bad daily or period shock. Under the (cid:12)rst bonus scheme, this results in workers exerting e(cid:11)ort only 75% of the time, while under the second workers exert e(cid:11)ort 66% of the time. Consequently, the (cid:12)rm’s bonus scheme increases worker productivity by only 11.9%, which is signi(cid:12)cantly less than 16.5% increase in productivity associated with the (cid:12)rst best scheme, where workers exert e(cid:11)ort all the time. 4.6 Welfare Analysis Now that we understand how often workers choose to exert a high level of e(cid:11)ort, we analyze by how much the (cid:12)rm’s bonus plan increases welfare. We address this issue by (cid:12)rst considering the welfare in the economy under two extreme cases: a flat-wage scheme and a (cid:12)rst-best scheme. We choose these two schemes as we are interested in comparing the case without any incentives to the one where incentives are the most e(cid:11)ective. By looking at the worker’s problem, it is clear that if workers are only paid an hourly rate, they will never exert a high level of e(cid:11)ort. To solve for the (cid:12)rst-best scheme, we take the environment we speci(cid:12)ed earlier and make the additional assumptions that the (cid:12)rm wants to induce the worker to always exert e(cid:11)ort and to minimize the cost (i.e. wages).15 In this environment there are several schemes the (cid:12)rm can use to achieve the (cid:12)rst-best. A simple one is for the (cid:12)rm to pay workers a bonus after every e(cid:11)ort decision, conditional on whether or not the (cid:12)rm observes the worker exerting e(cid:11)ort. With these compensation schemes in mind, we turn to measuring welfare. Our strategy is to compare the welfare associated with processing a typical day’s worth of checks, 15By (cid:12)rst-best scheme, we mean the optimal scheme for the (cid:12)rm in an environment where the (cid:12)rm can costlessly observethe worker’sactions. To solve for this scheme, we use the fact that the (cid:12)rm signs contracts with its customers to sort checks within a short period of time. In this deadline oriented environment,processingchecksasfastaspossibleisvaluabletothe (cid:12)rm. Hence,weassumethatevenif the worker receives a large daily cost, it is still worthwhile for the (cid:12)rm to motivate the worker to exert e(cid:11)ort. 27

Table 6: Welfare Analysis per Worker per Day Flat Wage First Best Di(cid:11)erence in Welfare (e=0) (e=1) Minutes Spent Sorting 128 107 21 Cost of E(cid:11)ort (γ) $0 5 (cid:1) $0.45 -$2.25 about 5000 checks, under the two schemes. From the data, we know that while sorting 5000 checks, a worker will typically clear 70 jams and type in 966 (cid:12)elds. Using these numbers, we (cid:12)nd that in the flat-wage case (where no e(cid:11)ort is exerted), a worker will typically take 128 minutes to sort 5000 checks. Naturally, as a worker never exerts e(cid:11)ort under this scheme, the disutility from e(cid:11)ort is $0. Moving to the (cid:12)rst-best, we (cid:12)nd that a worker will typically sort 5000 checks in 107 minutes, 21 minutes faster than without e(cid:11)ort. The expected disutility the worker experiences from exerting e(cid:11)ort is 5(cid:1)$0:45 = $2:25, as e = 1 is chosen 5 times (once for every 1000 checks). To compute the net change in welfare when moving from a flat-wage to the (cid:12)rst best scheme, we need to compare the gain of 21 minutes to the disutility from exerting e(cid:11)ort. The (cid:12)rm gains in two ways from the decrease in time taken to sort checks. First, the worker now has an extra 21 minutes to perform other tasks for the (cid:12)rm. To compute the value of this extra time to the (cid:12)rm, we use the mean wage of workers, $10.55. Second, the sorting machine is free for 21 minutes. The value of this extra time to the (cid:12)rm is harder to quantify, though it is clearly positive. As such, we only consider the welfare e(cid:11)ects of freeing up the worker’s time, and interpret our estimates of the welfare gain as a lower bound. The welfare e(cid:11)ects of freeing up the worker’s time equals $10:55(cid:1) 21 = $3:69. The total 60 expected gain in welfare from using the (cid:12)rst-best scheme is then $3:69−$2:25 = $1:44, which is a 6.3% increase in welfare. Table 6 summarizes these results. Using this method of analysis, we can easily compute the welfare gain from using the (cid:12)rm’s incentive schemes. As previously mentioned, our estimates imply that workers exerted a high level of e(cid:11)ort 75% of time under the (cid:12)rst bonus scheme and 66% of the time under the second. Hence, under the (cid:12)rst incentive pay scheme, welfare was increased by 0:75(cid:1)6:3% = 4:7%. We similarly calculate the welfare increase under the second incentive pay regime and then take the average of the two percentages, weighted by number of observations under each regime. We (cid:12)nd that over the sample period, welfare increased by 4.5% due to the (cid:12)rm’s bonus plan. 28

4.7 Goodness of Fit To determine the goodness-of-(cid:12)t of our estimated parameters, we perform two tests. First, we perform a likelihood ratio test. Second, we use the (cid:12)rm’s policy shift in the bonus scheme to perform an out-of-sample prediction which we can then check against the data. The null hypothesis in the likelihood ratio test is that both (cid:12) and (cid:12) are equal to 3 4 zero. The alternative hypothesis, that these parameters are greater than zero, is the model we originally estimated. Finding the maximum likelihood of the model under the null hypothesis is straightforward, as workers will always choose low e(cid:11)ort. Taking the ratio of the likelihood for each model yields a test statistic which allows us to strongly reject the null hypothesis. This implies that accounting for workers’ e(cid:11)ort decisions is important when analyzing worker productivity in this environment. The second goodness-of-(cid:12)t test is an out-of-sample prediction. As discussed in Section2, two-thirds of the way throughour data sample the(cid:12)rm changed the bonus scheme in two ways. First, the formula time (cid:11) was altered to give less time to workers to complete each job. Second, the bonus wage B was raised, increasing the marginal return to e(cid:11)ort for workers, if they are eligible for a bonus. As the (cid:12)rst e(cid:11)ect creates less incentive for workers to work hard, while the second e(cid:11)ect creates more incentive, it is unclear what will happen to worker productivity after the policy change. This policy shift provides an excellent opportunity to test how well the model is (cid:12)tting the data. As we have data before and after the policy change, we can estimate the model’s parameters on one part of the data and use it to predict worker productivity on the second part. Then, we can compare the model’s predictions to the actual data and judge how well the model performs. To measure the change in worker productivity in the data from the policy change, we run the following regression: X15 ln((cid:28) ) = ln((cid:12)0z(cid:22) )+ (cid:17) (cid:1)1 +’(cid:1)1 (cid:1)+" ; (12) i;t i;t j j=i IP i;t j=1 where (cid:28) is the time taken to complete job t by worker i. The dummy variable 1 i;t j=i is equal to 1 when j = i, while 1 is equal to 1 after the switch in the bonus scheme. IP As shown in Table 7, the estimate of ’ from the data is 0.04, which implies that the 29

Table 7: Actual and Predicted Productivity Change Estimate of ’ Data 0.041 (0.0030) Simulation 0.019 (0.0005) Note: Standard errors are in parenthesis time taken to sort checks increased by 4% after the change in the bonus scheme.16 We then estimated the model only using the data from after the policy change. We use this subset of the data as there are more instances of workers not earning a daily bonus in this time frame. These observations are crucial for identifying the e(cid:11)ect of e(cid:11)ort on the time taken to sort checks. Using these parameter estimates, we simulated the model to get predicted times on the jobs completed. Using these simulated times, we re-ran the above regression and estimated ’ to be 0.019. Hence, the model correctly predicts the sign of the productivity change, but is o(cid:11) on the magnitude. 5 Summary and Conclusion Most empirical work on incentive pay has only focused on measuring by how much incentives increase output. As such, it is unclear at what cost this extra output is obtained. Our paper adds to this literature by examining both the increase in output andthecorrespondingriseindisutilityfromhighere(cid:11)ortduetoincentives. Thisallowsus to measure by how much the welfare of the (cid:12)rm and workers rises due to the introduction of incentives. We accomplish this by studying a check-clearing (cid:12)rm’s use of incentives. Using the (cid:12)rm’s production records, we develop and estimate a dynamic model of worker behavior. This allows us to determine by how much the welfare gains of increased output due to incentives outweigh the disutility from increased e(cid:11)ort. We (cid:12)nd that compared to an environment without incentives, the (cid:12)rm’s bonus scheme lowers the time taken to sort checks by 11.9%. Roughly two-thirds of this gain, however, is needed to compensate workers for their higher e(cid:11)ort levels. By comparing these two welfare changes, we compute that the introduction of the (cid:12)rm’s incentive scheme increases the welfare of the (cid:12)rm and workers by 4.5%. 16Note that we use the transformed data, where each job has 1000 checks. 30

Although this paper only looks at one (cid:12)rm, we believe our results have broad applicability. The ‘continuous flow’ production technology used by the (cid:12)rm has general characteristics common to a large portion of the manufacturing sector of the economy. Speci(cid:12)cally, the automation of the check-sorting process and the worker’s role in maintaining the operation of a machine, are production characteristics found throughout a variety of manufacturing industries. In these industries, then, we believe that the introduction of incentives would increase the welfare of (cid:12)rms and workers by a similar amount. Another lesson we draw from our results concerns the dynamic e(cid:11)ects of contracts. In the case of the check-clearing (cid:12)rm, the combination of compensating workers after they have made multiple e(cid:11)ort decisions, with a minimum productivity requirement signi(cid:12)cantly reduces the bonus system’s e(cid:11)ectiveness. The general lesson, then, is that when designing incentive schemes, close attention should be paid to how the bonus scheme a(cid:11)ects the worker’s e(cid:11)ort decision over time. The results from this paper suggest several areas of future research. One extension would be to investigate the e(cid:11)ect of incentives on new workers. In particular, do incentives encourage workers to learn faster? More generally, fully modelling the (cid:12)rm’s problem andunderstanding the constraints it faces is of interest. With such anapproach, the e(cid:11)ect of adverse selection can be measured. In addition, within this general framework other relationships, such as the connection between worker turnover and incentives, could be explored. 31

References Asch, B.(1990): \DoIncentivesMatter?,"Industry Labor Relations Review,43,89{107. Blinder, A. (ed.) (1990): Paying for Productivity: A Look at the Evidence. Brookings Institution, Washington, DC. Chevalier, J., and G. Ellison (1999): \Risk Taking by Mutual Fuds as a Response to Incentives," Journal of Political Economy, 105, 1167{1200. Chiappori, P., J. Heckman, and J. Pinquet (2000): \Testing for Moral Hazard on Dynamic Insurance Data," University of Chicago mimeo. Chiappori, P., and B. Salani(cid:19)e (2000): \Testing Contact Theory: A Survey of Some Recent Work," Invited Lecture at the World Congress of the Econometric Society. Ferrall, C., and B. Shearer (1999): \Incentives and Transactions Costs Within the Firm: Estimating an Agency Model Using Payroll Records," Review of Economic Studies, 66, 309{338. Goffe, W. L., G. D. Ferrier, and J. Rogers (1994): \Global Optimization of Statistical Functions with Simulated Annealing," Journal of Econometrics, 60, 65{99. Grossman, S., and O. Hart (1983): \An Analysis of the Principle-Agent Problem," Econometrica, 51, 7{46. Holmstrom, B., and P. Milgrom (1987): \Aggregation and Linearity in the Provision of Intertemporal Incentives," Econometrica, 55, 303{328. Lazear, E. P. (2000): \Performance Pay and Productivity," American Economic Review, 90, 1346{1361. Margiotta, M., and R. Miller (2000): \Managerial Compensation and the Cost of Moral Hazard," International Economic Review, 41, 669{719. Oyer, P. (1998): \Fiscal Year Ends and Nonlinear Incentive Contracts: The E(cid:11)ect on Business Seasonality," Quarterly Journal of Economics, 113, 149{185. Paarsch, H., and B. Shearer (1999): \The Response of Worker E(cid:11)ort to Piece Rates," Journal of Human Resources, pp. 643{667. 32

(2000): \Fixed Wages, Piece Rates, and Incentive E(cid:11)ects: Statistical Evidence from Payroll Records," International Economic Review, 41:1, 59{92. Pakes, A. (1986): \Patents as Options: Some Estimates of the Value of Holding European Patent Stocks," Econometrica, 54, 755{784. Prendergast, C. (1999): \TheProvision ofIncentives inFirms," Journal of Economic Literature, 37, 7{63. Rust, J. (1987): \Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher," Econometrica, 55, 999{1033. 33