The Response of Worker Effort to Piece Rates: Evidence from the British Columbia TreePlanting
Industry
Author(s): Harry J. Paarsch and Bruce S. Shearer
Source: The Journal of Human Resources, Vol. 34, No. 4 (Autumn, 1999), pp. 643-667
Published by: University of Wisconsin Press
Stable URL: http://www.jstor.org/stable/146411
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The Response of Worker Effort to
Piece Rates
Evidence from the British Columbia TreePlanting
Industry
Harry J. Paarsch
Bruce S. Shearer
ABSTRACT
We measure the elasticity of worker effort with respect to
piece rate using panel data collected from the payroll reco
Columbia tree-planting firm. Our data contain information
worker productivity and the piece rate received over a five-m
Using a structural model to control for the endogeneity of t
we estimate this elasticity to be approximately 2.14. We a
nonstructural lower bound to this elasticity equal to 0.77. S
mation also allows us to perform policy experiments and t
profits under alternative compensation systems. Our results
profits would increase by at least 17 percent were the firm t
the optimal static contract as predicted by principal-agent
crease in profits would be due to capturing worker rents aft
tion of private information over ability. Yet, only a negligib
of these rents could be captured while inducing workers to
truthfully, suggesting that dynamic considerations were imp
termining the firm's actual choice of contract.
I. Introduction and Motivation
The role of economic incentives in determining behavior is of major
interest to economists. Within the domain of labor economics, much theoretical attention
has been focused on the optimal form of contracts between the firm and its
Harry J. Paarsch is a professor of economics at the University of Iowa; Bruce Shearer is a professor
of economics at the Universite Laval, Quebec, Canada and is a member of CREFA and CIRANO. The
authors acknowledge research support from SSHRC, CIRANO, and FCAR (Shearer). Useful comments
and helpful suggestions were provided by participants at the American Compensation Association Academic
Research Conference in Islamorada, Fla., by seminar participants at the University of Iowa, Universite
Laval, and Universite de Quebec a Montreal, and by Jean Fares, Christopher J. Flinn, David
A. Green, Lawrence Katz, Joel L. Horowitz, John H. Pencavel, and an anonymous referee. Data used
in this article can be obtained from the authors beginning June 2000 through May 2003
[Submitted August 1997; accepted November 1998]
THE JOURNAL OF HUMAN RESOURCES * XXXIV * 4
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644 The Journal of Human Resources
workers (see, for example, Hart and Holmstrom 1987; Holmstr6m and Milg
1990; and Baker 1992). The ability of labor contracts to affect worker productiv
can also be applied to the analysis of personnel policies within the firm (as in
grom and Roberts 1992 and Lazear 1998). Some economists (for example, Blin
1990) have further argued that the increase in worker productivity in respon
the widespread adoption of performance-based pay would result in macroecon
benefits.1 For normative policy prescriptions to be valuable, however, they mus
based on empirical analyses of the benefits accruing to changes in compensat
systems. Empirically analyzing compensation policies and evaluating these ben
require measuring incentive effects; that is, how workers react to changes in th
economic incentives.
In the past, empirical work concerning incentive models has typically involved
cross-sectional or longitudinal comparisons of wages among workers who do and
do not receive incentive pay (see, for example, Pencavel 1977; Seiler 1984; Parent
1997; and Booth and Frank 1997). The strength of this approach is that it is based
on a wide sample of observations from different sectors of the economy and therefore
provides "general" results. Yet, whereas the results of these studies are usually
consistent with incentive models (thus supporting the existence of incentive effects),
problems exist with their interpretation. In particular, workers who do not receive
explicit incentive pay may be provided with incentives through other mechanisms,
such as the promise of future promotions (as in Lazear and Rosen 1981 or Goldin
1986) or termination contracts (as in Shapiro and Stiglitz 1984 or Macleod and Malcolmson
1989). This inability to document and to understand fully the personnel
policies implemented by different firms in a cross section of data makes it difficult
to identify incentive effects using these methods.
An alternative approach is to concentrate on industry- or firm-level data. Such an
approach combines elements of the traditional case-study methodology, once popular
in the industrial organization literature (see, for example, Wallace 1937), with econometric
estimation. Examples of this approach can be found in the recent work of
Ferrall and Shearer (forthcoming), Shearer (1996), Lazear (1996), Paarsch and
Shearer (forthcoming), and Treble (1996). Within this approach, the detailed study
of the personnel policies of the firm or firms in question yields knowledge of the
incentive system determining worker behavior. Measuring worker reaction to variation
in the compensation system then permits identification of incentive effects. Furthermore,
access to firm archives often yields direct measures of worker productivity,
so the presence or absence of incentive effects does not have to be inferred indirectly
through a comparison of wages.
One potential problem with both approaches is that the changes in the compensation
system may not be exogenous (Ehrenberg 1990; Brown 1990). To wit, the firm
may select a compensation system based on elements which are unobservable to the
econometrician, but which affect worker productivity. This suggests that regression
methods, which use the observed covariation between worker productivity and the
1. At least one country has taken these notions seriously. Booth and Frank (1997) report that the government
of the United Kingdom has introduced tax policy aimed at inducing firms to implement incentive
pay.
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Paarsch and Shearer 645
payment system to identify the incentive effect, may fail to provide a consisten
estimate of this effect.
In this paper, we measure incentive effects with particular emphasis on piece-rat
workers, those workers whose pay is proportional to their output. We use daily dat
on individual productivity and piece rates to measure how workers react to changes
in their compensation system. Knowledge of the elasticity of effort with respect t
changes in the piece rate has important implications for firms that are paying o
considering paying their workers piece rates. As Stiglitz (1975) has shown, the opti
mal piece rate which a firm should set is an increasing function of this elasticit
Intuitively, the higher is the elasticity of effort the more beneficial it is for the firm
to set a high piece rate. Although this elasticity may depend on the technology em
ployed in a particular industry or firm, the case-study approach can still be usef
as long as the characteristics of the firm are taken into account for policy proposa
Our data were collected from the payroll records of a tree-planting firm in th
province of British Columbia, Canada. This firm paid its planters piece rates excl
sively, and planters received no base wage. The tree-planting industry has man
advantages as a laboratory within which to estimate labor-market incentive models
Planter output is easily observable on a daily basis and compensation systems var
within firms. Moreover, the compensation systems are relatively simple, permittin
straightforward analyses of incentives. For example, in the firm that we study, no
team production existed and planters were not unionized. Because our data are pane
in nature, we observe the daily productivity of each planter as well as the piece rat
received by that planter over a period of approximately five months.
There are also practical reasons for studying the British Columbia tree-plantin
industry. British Columbia produces around 25 percent of the softwood lumber i
North America.2 The success this province has in managing its timber affects th
supply of timber to North America as well as to many other parts of the globe.
addition, the scope of reforestation in British Columbia is huge. At its peak, betwe
1981 and 1985, almost two billion seedlings were planted. This pace has slowed
somewhat but still remains important. Today, about 200 million seedlings are planted
per year. An average seedling costs about $0.50 to plant. Thus, a 10 percent improvement
can yield savings of about $10 million per year. Small improvements in personnel
policy can result in large savings because of the enormous scale involved.
Using our data, we highlight the endogeneity problems inherent in the empirical
analysis of compensation systems. In particular, employing regression methods,
which use the observed covariance between piece rates and productivity to identify
the incentive effect, we consistently estimate the elasticity of effort with respect to
changes in the piece rate to be negative. Though this result seems nonsensical from
the point of view of incentive theory, it obtains because piece rates are determined
endogenously by the firm in response to the relative difficulty of planting in different
areas. In particular, the firm chooses the observed piece rate to satisfy the laborsupply
constraint of the planter, the amount the firm must at least pay the planter
2. When statistics are reported for Canada, they are reported as "East of the Rockies" and British Columbia.
British Columbia is also broken up into three regions-the coast, the southern interior, and the northern
interior-each of which produces more timber than any province of Canada or state of the United States.
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646 The Journal of Human Resources
to induce him to accept the contract, implying that piece rates are negatively co
lated with average planting conditions. Because these planting conditions are un
servable to the econometrician, they enter the error term of the regression mod
Thus the piece rate is, in fact, a statistically endogenous variable, and the estim
of the elasticity of effort with respect to the piece rate is inconsistent.
Obtaining a consistent estimate of the effort elasticity requires controlling for
unobservable planting conditions during estimation. We accomplish this by mode
explicitly the firm's choice of the piece rate as a function of planting conditions
planter behavior as a function of the piece rate. In our model, we incorporate asy
metric information between the firm and the planter over planter effort and planti
conditions. We also allow for individual-specific heterogeneity across planters.
estimate the elasticity of effort with respect to the piece rate to be 2.14, which imp
that an increase in the piece rate of one cent from the sample mean of 25 ce
would increase average daily output by 67 trees, holding planting conditions c
stant.
Our structural estimates are sensitive to identifying assumptions, particularly with
respect to measures of a planter's alternative utility. Therefore, in addition to our
structural estimates, we provide a lower bound on the elasticity of effort. This lower
bound is calculated to be 0.77, suggesting that an increase in the piece rate of one
cent from the sample mean of 25 cents would increase average daily output by at
least 24 trees, holding planting conditions constant.
Estimating the model structurally has benefits beyond controlling for the endogeneity
of covariates. In particular, using estimates of the structural parameters, we
can investigate how the observed contract departs from the optimal contract as predicted
by theoretical incentive models. In particular, given risk-neutral planters, the
optimal contract involves the planters paying the firm a fixed fee to plant trees and
then receiving a piece rate equal to the price of output. The inclusion of the base
fee gives the firm two instruments in the contract: one to provide incentives (the
piece rate) and the other to extract rents from the planter (the base fee). In contrast,
the observed contract contains only one instrument and therefore allows planters to
earn rents. If the firm could charge the planters an up-front fee to plant trees, then
our results suggest that firm profits would increase, on average, by at least $31.77
per planter per day, an increase on the order of 17.25 percent. We argue that dynamic
considerations are important determinants of the firm's choice of contract. In particular,
by committing not to extract rents in a given period, the firm induces high-ability
planters to reveal their type and produce higher output.
Our structural estimates can further be used to calculate the proportion of rent
that the firm could capture while inducing workers to reveal their abilities. We estimate
this proportion to be negligible, suggesting that the observed piece-rate contract
was an effective means of inducing worker effort within a firm composed of heterogeneous
workers.
In the next section, we describe the tree-planting industry in British Columbia as
well as the compensation system with which we are concerned. In Section III we
describe the sample data and present some regression results which illustrate our
point concerning endogeneity. In Section IV we develop and estimate a simple theoretical
model of planter-effort choice for a given piece rate, and then the choice of
piece rate chosen by the firm in response to planter behavior. We use the estimated
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Paarsch and Shearer 647
parameters from the structural model of Section IV to investigate alternative co
tracts in Section V, and we conclude in Section VI.
II. Tree Planting in British Columbia
Timber is a renewable resource, but active reforestation can increase
the speed at which forests regenerate and also allows one to control for speci
composition, something that is difficult to do in the case of natural regeneratio
Reforestation is central to a steady supply of timber to the North American marke
In British Columbia, extensive reforestation is undertaken by both the Ministry of
Forests and the major timber-harvesting firms that hold Tree Farm Licenses.3
The mechanics of this reforestation are straightforward. Prior to the harvest o
any tract of coniferous timber, random samples of cones are taken from the tre
on the tract, and seedlings are grown from the seeds in these cones. This ensure
that the seedlings to be replanted are compatible with the local microclimates an
soil as well as representative of the historical species composition.
Tree planting is a simple but physically exhausting task. It involves digging a
hole with a special shovel, placing a seedling in this hole, and then covering its roo
with soil, ensuring that the tree is upright and that the roots are fully covered. Th
amount of effort required to perform the task depends on the terrain on which th
planting is done. The terrain can vary a great deal from site to site. In some case
after a tract has been harvested, the land is prepared for planting by burning whatever
slash timber remains and by "screefing" the forest floor. Screefing involves remov
ing the natural buildup of organic matter on the forest floor so that the soil is exposed
Screefing makes planting easier because seedlings must be planted directly in t
soil. Sites that are relatively flat or that have been prepared are much easier to pla
than sites that are very steep or have not been prepared. The typical minimum densi
of seedlings is about 1,800 stems per hectare, or an intertree spacing of about 2
metres, although this can vary substantially.4 An average planter can plant between
700 and 900 trees per day, about half a hectare, depending on conditions. An averag
harvested tract is around 250 hectares.
Typically, tree-planting firms are chosen to plant seedlings on harvested tracts
through a process of competitive bidding. Depending on the land-tenure arrangement,
either a timber-harvesting firm or the Ministry of Forests will call for sealedbid
tenders concerning the cost per tree planted, with the lowest bidder being selected
to perform the work. The price received by the firm per tree planted is called "the
3. In British Columbia, nearly 90 percent of all timber is on government-owned (Crown) land. Basically,
the Crown, through the Ministry of Forests, sells the right to harvest the timber on this land in two different
ways. The most common way is through administratively set prices to 34 firms that hold Tree Farm
Licenses. These licenses have been negotiated over the last 75 years and require that the licensee adopt
specific harvesting as well as reforestation plans. About 90 percent of all Crown timber is harvested by
firms holding Tree Farm Licenses. The second, and less common way, to sell timber is at public auction
through the Small Business Forest Enterprise Program. In this case, the Ministry of Forests assumes the
responsibility of reforestation.
4. One hectare is an area 100 metres square, or 10, 000 square metres. Thus, one hectare is approximately
2.4711 acres.
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648 The Journal of Human Resources
bid price." Bidding on contracts takes place in the late autumn of the year preced
the planting season, which runs from early spring to late summer. Before the bid
takes place, the principals of the tree-planting firms typically view the land t
planted and estimate the cost at which they can complete the contract. This estim
cost depends on the expected number of trees that a planter will be able to plant
a day, which, in turn, depends on the general conditions of the area to be plan
Planters are predominantly paid using piece-rate contracts, although fixed-w
contracts are sometimes used instead. Under piece-rate contracts, planters are
in proportion to their output. Generally, no explicit base wage or production stan
exists, although firms are governed by minimum-wage laws. Output is typically m
sured as the number of trees planted per day, although some area-based sche
are used as well. An area-based scheme is one under which planters are paid
proportion to the area of land they plant in a given day, assuming a particular st
density.
Our data were collected from a medium-sized tree-planting firm that employed a
total of 155 planters throughout the 1994 tree-planting season. This firm paid its
planters exclusively piece rates; daily earnings for a planter were determined by the
product of the piece rate and the number of trees the planter planted on that day.
Sites to be planted were divided into plots. For each plot, the firm decided on a piece
rate. This rate took into account the expected number of trees that a planter could
plant in a day and the expected wage the firm wanted to pay. Thus, the piece rate
should have been negatively correlated with good planting conditions. All planters
planting on the same plot received the same piece rate; no matching of planters to
planting conditions occurred, so even though planters may have been heterogeneous,
the piece rate received was independent of planter type. Planters were assigned to
plots as they disembarked from the ground transportation that took them to the planting
site. Thus, to a first approximation, planters were randomly assigned to plots.
III. Sample Data and Regression Results
Our data set contains information on the piece rate received by each
planter as well as that planter's daily productivity. We considered only those planters
who received the same piece rate for the whole day of planting. This eliminated the
problem of aggregating trees planted under different piece rates. The summary statistics
for the entire data set, which contains 4,578 observations on 155 planters planting
for 31 different contracts over a five-month period in the spring and summer of 1994,
are presented in Table 1. A contract was identified by a unique value for the piece
rate on a particular tract. The average piece rate received by planters was about 25
cents per tree and planters planted, on average, about 764 trees per day. The average
wage was about $178 per day.
Table 1 also suggests that outliers exist in the data. For example, the recorded
minimum number of trees planted in one day was 30 and the recorded minimum
daily wage was $9.30. There are two possible causes of outliers in our sample.5 First,
5. The existence of outliers also suggests that measurement error may be present in these data. We do
not incorporate this possibility into our estimation strategy since our structural model, developed in Section
IV, does not identify the elasticity of effort in the presence of measurement error; see footnote 9 below.
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Paarsch and Shearer 649
Table 1
Summary Statistics: Full Sample
Standard
Variable Observations Mean Deviation Minimum Maximum
Number of trees 4,578 764.26 319.30 30 2260
Piece rate 4,578 0.25 0.06 0.13 0.48
Daily earnings 4,578 178.32 62.09 9.30 530.00
.25 .2
-
0
oc)
.15 .1
-
.05 -
0 -
3 4 5 6
Logarithm of Daily
7 8
Figure 1
Histogram of the Logarithm of Daily Trees Planted, Full Sample;
4,578
Sample Size =
some planters may work less than the usual eight hours per day. This may be the
result of injury, sickness, or the planter's being asked to perform ancillary tasks such
as collecting or sorting trees. Second, some planters may be of very low ability. By
law, the firm is required to pay planters at least the minimum wage for an eighthour
day, or $48.00 per day in 1994; planters who are incapable of consistently
earning this amount through the piece-rate system are fired. In Figure 1, we present
a histogram of the logarithm of trees planted daily. The presence of outliers is clearly
evident from the long left-hand tail.
Our strategy for dealing with outliers is to eliminate them from the sample (see
Donald and Maddala, 1993); that is, we seek an equilibrium sample of planters possessing
satisfactory ability who work at least eight hours per day. Yet, identifying
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650 The Journal of Human Resources
.2.15
-
.1.05
-
4 5 6 7 8
Logarithm
Figure 2
Histogram o
Wage Daily
outliers simp
ductivity may
worked. Since
tions, we ide
eliminated fr
$48.00 per da
wage law for
to planters w
the sample t
logarithm of
and the sum
average num
as do averag
We first con
effort. We e
(1) logYit,
where Yi,t is
ual-specific) c
6. In Section IV
to individuals wh
sistency in this
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Paarsch and Shearer 651
Table 2
Summary Statistics: Daily Earnings above Minimum-Wage Daily Earnings
Standard
Variable Observations Mean Deviation Minimum Maximum
Number of trees 3,960 786.48 307.39 120 2,260
Piece rate 3,960 0.25 0.06 0.13 0.48
Daily earnings 3,960 183.97 57.38 48.00 530.00
Table 3
Simple Regression Results
Independent Variable (a) (b)
Constant 5.363 5.049
(0.031) (0.047)
Logarithm of piece rate -0.858 -0.893
(0.021) (0.019)
Maximum individual-specific effect 0.700
(0.055)
Minimum individual-specific effect -0.496
(0.071)
R2 0.292 0.556
Notes: Dependent vari
Sample size = 3,960.
and Uit is a zerozero
covariance with r.
We estimated Equation 1 in two different ways. First, we included as explanatory
variables only a constant and the piece rate. These results are presented in Column
a of Table 3. Note that the estimate of P, is negative, equal to -0.858, and has a
p-value which is virtually zero. Second, we included individual-specific intercepts
to control for heterogeneity across individuals. These results are presented in Column
b of Table 3. Again, the estimate of Pi is negative, but now equal to -0.893, and
has a p-value which is also virtually zero.
To provide visual confirmation of our regression results, we present in Figure 3
a scatterplot of the logarithm of the number of trees planted daily versus the logarithm
of the piece rate, along with the estimated regression line. Note the strikingly negative
relationship.
The negative coefficient estimate on the logarithm of the piece rate paid to planters
is troubling from the perspective of incentive theory. Taken literally, it suggests that
when the piece rate is high, planters work less intensively than when the piece rate
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652 The Journal of Human Resources
8-
75 1 -i
7 -,
I 5.5- . * - *
5- t ' i i
4.5 ,
: , I J l :
.15 .2 . 2 . 3 .35 .4 .45 .5
Piece Rate (Logarithic Sca
Figure 3
Scatterplot of Daily Trees Planted and Piece Rates, Sample Attaining MinimumWage
Daily Earnings; Sample Size - 3,960
is low; this seems counterintuitive. An alternative explanation is that the piece rate
is endogenous to the statistical model. In particular, if piece rates are correlated with
unobservable variables which also affect planter productivity, then the observed
piece rate will be correlated with the error term Ui, in Equation 1. This correlation
will result in biased estimates of the elasticity of effort with respect to piece rates
because one of the maintained assumptions of least-squares estimation has been vio-
lated.
Discussions with firm principals revealed that piece rates are chosen by the firm
after average planting conditions have been observed. The actual piece rate is chosen
to ensure that the planter's labor-supply constraint, the amount the firm must pay
the planter to induce him to accept the contract, is satisfied. A planter's productivity
is a function of how hard he works and the conditions under which he is planting:
it is easier to plant trees on flat terrain which is covered in loose soil than on steep
rocky hillsides. For planting conditions that are favorable to productivity, planter
output will be higher for any given level of effort. Since planters are paid in proportion
to the number of trees they plant and since effort is costly, these planters prefer
planting in favorable conditions-they can plant lots of trees for little effort. Therefore,
to induce planters to plant under unfavorable conditions the firm must increase
the piece rate.
The effect of this process on regression models is illustrated graphically in Figure
4. In the bottom panel, we represent the inverse relationship between the piece rate
r and average planting conditions gt caused by the labor-supply constraint. In the
upper panel of Figure 4, we illustrate the relationship between productivity Y and
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Paarsch and Shearer 653
y
Y(r;gh)
Y(r r;l)
r(gh) r(1l) r
r (9h) r(g/) r(g)
Figure 4
Example of Low- and High-Productivity Plots and the Piece Rate
the piece rate for two different levels (low [L and high gh) of planting conditions g.
The slope of the line Y(r; FL), the [productivity, piece-rate] locus, represents the incentive
effect that we seek to estimate. The fact that high piece rates r(gl) are associated
with poor planting conditions gt implies that productivity is lower for any value of
r, shifting down the [productivity, piece-rate] locus. Since average planting conditions
are unobservable to the econometrician, we have no way of controlling for the
fact that the locus has shifted down and a simple regression of productivity on the
piece rate connects the points on two separate loci producing the dotted line with a
negative slope.
To obtain a consistent estimate of PI requires controlling for planting conditions.
In general, three possible ways exist to do this. First, one could collect data on the
planting conditions which affect the firm's choice of a piece rate. Note, however,
that one would have to have all the information that the firm has when the firm
makes its choice. If the econometrician observed only a subset of the planting conditions,
then biased estimates of the incentive effect would still obtain because the set
of conditions that were unobserved could still be correlated with both a planter's
output and the piece rate. Gaining access to such data has proven impossible. A
second possibility would be to use an instrumental variable; that is, a variable correlated
with the piece rate but uncorrelated with the planting conditions. Though such
a variable is easy to define, finding such a variable is generally more problematic
in practice. One possibility might be to use the identity of the firm manager responsible
for setting the piece rate on a given contract. In setting piece rates, different
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654 The Journal of Human Resources
managers may exhibit indiosyncratic behavior that is independent of planting con
tions. However, such idiosyncracies are likely to be significant only over a lo
period of time. Given that we only have data from one planting season, during wh
each manager would have been responsible for at most one or two contracts,
chose not to pursue this strategy. A final approach, the one which we follow, i
model explicitly the firm's decision rule over r as a function of planting conditio
and to incorporate this decision rule directly into the estimation procedure.
IV. Deriving and Estimating a Structural Model
We model unobserved planting conditions as productivity shocks that
affect the output which obtains for a given level of effort on the part of the plan
We assume that productivity shocks S are draws from a particular random variab
having cumulative distribution function F(s) which depends on parameters g
a2. The firm decides on the piece rate r by considering the parameters g and (
contract is defined by the pair (g, a2) and a unique value of r. We model the firm
choice of r as satisfying the planter's labor-supply constraint, conditional on ave
planting conditions. Thus, the firm chooses r to induce the planter to participate
planting. Note that changes in average planting conditions lead to changes in r
We develop a simple model of planter-effort determination under piece rates w
a risk-neutral planter.7 We assume for planters the following utility function def
over earnings W and effort E:
U(W, E) = W- C(E)
where the earnings function is given by
W= rY
with Y being planter output and the function C(E) denoting the planter
effort, which is parameterized as
C(E) =- Enq > 1, K> 0.
Output is assumed to be determined by the following function:
Y = ES
where S is a random productivity shock drawn from the distribution F(
parameters g and &C2 and represents planting conditions beyond the planter
such as the slope of the terrain, hardness of the ground, and the amount o
cover. We assume that s, a realization of S, is observed by planters befo
choose their effort levels, but after they accept a contract. Note that the f
not observe s; it only observes the parameters of the distribution of S,
Thus, though a planter can observe average planting conditions before he
plant, he only learns the exact nature of the terrain to be planted once plantin
7. Interviews with planters suggest that variation in daily earnings is a relatively minor conce
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Paarsch and Shearer 655
The logarithm of the productivity shock is assumed to follow a normal distribution
with mean ,t and variance 62, so the probability density function of S takes the for
fs(s) Q (logs gs)
where q represents the standard normal probability density function.
The timing of events in our model is as follows:
1. For a particular contract to be planted, nature chooses the pair (g
parameters of the distribution of S.
2. The firm observes (,g, )2), and then chooses a piece rate r.
3. The planter observes (g, (2, r), and accepts or rejects the contrac
4. If the planter accepts the contract, then he is randomly assigned
particular plot of the contract.
5. For each plot, nature chooses s, a particular value of S.
6. The planter observes s, and chooses an effort level e producing ou
7. The firm observes y, and pays earnings ry.
To solve the model, we work backwards. First, we solve for the plan
effort level conditional on a given piece rate and productivity shock. Th
for the firm's choice of the piece rate, taking the reaction of the plan
Note that to induce the planter to accept the contract, the contract mu
planter's labor-supply constraint.
Conditional on s, a particular realization of S, planters choose effort t
their utility, so the optimal level of effort e is
e = -
K/
To simplify resulting expressions, let Y denote [1/(! - 1)]. Note that the secondorder
condition of the planter's problem is satisfied as long as y exceeds zero, qi
exceeds one. Making the appropriate substitutions, we write the expressions for optimal
effort and output on the part of the planter in response to a particular piece rate
r as
rs)
so
(K)y = s+l+1
Taking logarithms of both sides of the second equation above yields
logy = ylogr - ylogK + (y + l)log s
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656 The Journal of Human Resources
or, in terms of random variables,
(2) logY = ylogr - ylogKc + (y + l)log S
where
(y + l)log S - N[(y + 1)L, (y + 1)202].
Note that the parameter y gives a direct measure of the elasticity of planter effort
with respect to the piece rate.
Note too that Equation 2 has the same form as the regression model Equation 1
estimated above. From Equation 2, it is also clear why regression methods fail to
provide a consistent estimate of the incentive effect. To convert Equation 2 into an
equation with a mean-zero error term, we add and subtract (y + 1)t, which yields
(3) logY = ylog r - ylog K + (y + 1)gl + V
where V now equals (7 + 1)(log S - g), which is distributed normally with mean
zero and variance (y + 1)2&2. Comparing Equation 3 to Equation 1, one notes immediately
that Ui t, the error term in Equation 1, equals (y + 1)[ + Vi,t, but from Figure
4 we know that the covariance between [t and r does not equal zero.8 Thus, one of the
assumptions maintained in least-squares estimation (namely, the weak exogeneity of
the covariates) has been violated.
We assume that planters have an alternative utility given by u, so the labor-supply
constraint is
[W - C(e)] = u
where E is the expectation operator taken with respect to the random variab
Substituting optimal effort into the labour-supply constraint yields
?(S,+) = u.
(y + 1)KcY
Using the properties of the log-normal distribution, we obtain
(4) (y + 1)g = logu + log(y + 1) + ylogK - (y + l)logr - (y + 1)22
Substituting Equation 4 into Equation 3 yields an equation for the daily productivity
of individual i on contract t
(5) log Y, = log(y + 1) + log U - log r, - (y + 1)2 - + Vi
Incorporating the planter's expected-utility constraint eliminates the endogeneity
problem because V represents only unexpected deviations from average conditions.
8. Note that though g and (2 are fixed for a given contract, they vary across contracts, causing the correlation
between J and r.
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Paarsch and Shearer 657
Therefore, it is a mean-zero error term that is uncorrelated with r. Estimation bas
on Equation 5 can provide consistent estimates of y.9
It may be informative to consider our estimation approach within the context th
analysis-of-variance (ANOVA) framework often used to evaluate experimental dat
Consider the logarithm of the independent variable log Yijk, which measures the log
arithm of output by planter i planting on plot j for piece rate k. Ignoring highe
order effects, the logarithm of output can be decomposed into individual-specif
effects (cis), plot-specific effects (6js), piece-rate specific effects (rkS), and a rando
error Uijk in the following way using a three-way ANOVA:
log Yijk = + a + a j + nk + Uijk
We cannot use these methods to identify 6j and rk separately because there is no
independent variation between plots and the piece rate in our data; conditions o
any plot uniquely determine the piece rate. Furthermore, because the planting condi
tions on a given plot are negatively correlated with the piece rate, the estimate
piece-rate effect is inconsistent. By modeling behavior, we are able to rectify th
problem, but at the cost of making the estimation problem nonlinear.
A. Parameter Identification and Estimates
Our data set contains observations on 89 planters planting under 31 different co
tracts. Each contract t is specified by a pair (tL , 6o), which in turn determines th
piece rate rt through Equation 4. Therefore, the structural model consists of the p
rameter vector (y, K, g, , (. , . . ., 31, )T
Estimating Equation 5 requires a measure of alternative utility u.We used the dail
British Columbia welfare payments to a single individual with no dependents in
1994. This measure captures what an individual would receive were zero effort supplied.
In 1994, daily welfare payments were $27.05 per day. In principle, u defin
an indifference curve in (W, E) space and can therefore be measured at an infini
of different points. For each wage earned in the labor market, however, there i
corresponding positive and unknown labor-market level of effort that gives th
planter utility level equal to u. We know the level of effort for only one point o
the indifference curve: at zero labor-market effort. The planter's utility at this poi
is equal to his wage, which is the welfare payment he receives for not working.
Defining
Y*, - log Yi, + log r, - log u,
we can then rewrite Equation 5 as
-2
Yi = log(y + 1) - (y + 1)2 + V,t.
9. Equation 5 makes it clear why this model does not permit the identification of y in the presence of
measurement error. In particular, by introducing measurement error into daily productivity, the variance
of the error term in Equation 5 would no longer be equal to (y + 1)2 02, the coefficient on the contractspecific
dummy variable. Therefore, one of these contract-specific dummy variables would have to be
dropped and the intercept term, instead of measuring (y + 1), would measure (y + 1) - [(y + 1)2 2,/2]
where t' is the contract for which the dummy variable is suppressed.
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658 The Journal of Human Resources
Table 4
Parameter Estimates of Structural Models: with and without Individual-Specific
Heterogeneity
Parameter (a) (b)
y 5.876 2.135
(0.038) (0.183)
Maximum ( 0.081 0.141
Minimum o 0.014 0.006
Average ( 0.047 0.078
Maximum individual-specific effect 1.168
Minimum individual-specific effect 0.199
Average individual-specific effect 0.737
Logarithm of the likelihood function -1,257.790 -209.020
Note: Sample size = 3,960.
Estimating the above specification by the method of maximum likelihood is
to estimating a linear regression with the added complication that the contra
cific variance of the productivity shock a2 enters the conditional-mean funct
elasticity of effort with respect to the piece rate is identified by the constan
Note that K and the {Ij gt enter Equation 5 and Equation 4 additively. Thu
Equation 5 is estimated, we can recover an estimate of [(y + 1)gt - ylogK
substituting back into Equation 4. However, separately identifying K an
{t()} would require an additional identifying normalization, such as Rg's e
zero.
Results obtained by estimating Equation 5 are given in Column
estimate of y is 5.88, suggesting a very large elasticity of effort w
piece rate. Of the 31 contract-specific variances, we only report
minimum as well as the average of the 31 values.
B. Introducing Individual-Specific Heterogeneity
Estimates of y based on Equation 5 neglect the fact that planters m
neous with respect to their ability. To capture individual-specific h
admit planters who have different costs of effort. We then assu
chooses the piece rate to ensure that the least-able (highest-cost) pl
is indifferent between working and not working. Within this fram
planters earn rents.
Denoting the cost of effort for planter i by Ki and the cost of eff
able planter by kh, which is the max { K1, K2,..., Kn}, piece rates ar
by
ry+l
(6) (S7+l) = u
k(y + 1)
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Paarsch and Shearer 659
while the output for individual i is determined by
(7) Yi= r S (+)
Taking logarithms of both sides of Equation 7 yields
(8) log Yi = ylogr - ylogKi + (y + 1)g + V.
where V equals (y + 1)(log S - g), which is normally distributed with mean zero
and variance (y + 1)22. Taking logarithms of Equation 6 and substituting for the
term (y + 1)g in Equation 8 yields
(9) log Yi,, = log(y + 1) + logu - logr,
+ y(logkh - log Ki) - (Y + l)2 + Vit.
Comparing Equation 9 to Equation 5 suggests that ignoring individual-specific
heterogeneity will lead to an overestimation of y because planters with low KS will
produce more output, on average, than those planters with high Ks. In essence, by
allowing planters to be heterogeneous, the term that was estimated as log(y + 1) in
Equation 5 is now estimated as
log(y + 1) + y(log kh - logKi).
To estimate Equation 9, we simply add to Equation 5 individual-specific dummy
variables for each planter in the sample, except the planter corresponding to kh, for
whom the term (logkh - log Ki) equals zero. We take the planter corresponding to
kh to be the planter with the lowest average productivity in the firm. Results from
estimating Equation 9 are presented in Column b of Table 4. After controlling for
individual-specific effects, the estimate of y falls to 2.14. That the individual-specific
effects are jointly significant is evident from a comparison of the maximized values
of the logarithm of the likelihood functions, -1, 257.79 for the restricted model and
-209.02 for the unrestricted model.
These results suggest that, holding planting conditions fixed, a 1 percent increase
in the piece rate will increase productivity by 2.14 percent. In terms of measured
output, an increase of the piece rate by $0.01 from the sample mean of 25 cents
will increase output by 67 trees if the conditions are held constant.
C. Prediction
To evaluate the performance of the structural model, we consider its ability to predict
observed productivity for the different contracts. In Figures 5 and 6, we present the
average observed productivity per contract, denoted by the circles, and the 95-percent
and 99-percent confidence intervals for the average predicted productivity,
which are derived from the structural parameter estimates of the model. In total,
seven of the 31 95-percent confidence intervals and nine of the 31 99-percent confidence
intervals encompass the observed average productivity. Given the large sample
size, the structural parameters are estimated very precisely, leading to very narrow
confidence intervals. Nevertheless, Figures 5 and 6 are informative about the perforThis
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660 The Journal of Human Resources
7.5 :
7-
0
a 6.5-
a
1=
o
6 -
5.5 -
o
o =
o
L :ra
I '" I - I I i I
.15 .2 .25 .3 .35 .4 .45
Piece Rate (Contract)
Figure 5
Ninety-Five Percent Confidence
7.5;>
7-
4 6.5-
,.
ot
0
1- 6-
0r
6
- -r 0
0 1:=
I .0
j
i -It :
? ?t0 1111
o I
o
II
o
5.5 -I
I , , - , ,
.15 .2 .25 .3 .35 .4 .45
Piece Rate (Contract)
Figure 6
Ninety-Nine Percent Confidence In
? ?
-r = 0
1:Z 0
? 10 0 ~~~0
o
.5
I
.5
o
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Paarsch and Shearer 661
mance of the model. In particular, the model has trouble fitting contracts with ver
high and very low piece rates.
D. Sensitivity Analysis
It is clear from Equation 9 that identification of the model depends crucially on
measuring u. Furthermore, given the effort elasticity y is identified through the con
stant term in Equation 9, estimates of y are likely to be sensitive to the value chose
for u.
An alternative estimation strategy is to bound the value of y. Working at the mini
mum wage Wm implies a positive, unknown (to the econometrician) level of effort,
e,. Thus,
u = W, - C(em).
Letting
V = log[u + C(e)] - log(u) > 0
we can then write
log(W) = log(u) + vi.
Substituting into Equation 9 gives
(10) log Y,, = log(y + 1) + log(Wm) - logr, + y(log k, - log Ki) (Y
+ 1)2- + Vi,t.
The estimated constant term in Equation 10
log(y + 1) = log(y + 1) - < < log(y + 1)
provides a lower bound. Estimates of this Equation 10 are given in Table 5. Note
that the calculated lower bound to y is 0.77. This suggests that an increase in the
piece rate by $0.01 from the sample mean of 25 cents would increase output by at
least 24 trees.
V. Alternative Contracts and Rents
The contract used by the firm we have studied is restrictive in that this
firm only has one instrument (the piece rate) to accomplish two tasks: the provision of
incentives and the division of rents. With heterogeneous planters, some of the planters
will earn rents. The expected utility of planter i is
E(U) = r e(S+ ).
K1(y + 1)
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662 The Journal of Human Resources
Table 5
Parameter Estimates of Structural Model: Lower Bound with
Individual-Specific Heterogeneity
Parameter
y 0.767
(0.
Maximum a 0.250
Minimum ac 0.011
Average o 0.139
Maximum individual-specific effect 1.168
Minimum individual-specific effect 0.199
Average individual-specific effect 0.737
Logarithm of the likelihood function -209.020
Note: Sample size = 3,960.
Substituting for r from Equation 6 yields
( kh
?(U) - ul.
Therefore, the rent earned by planter i is
0(khi ]
Note that this rent depends only on the individual-specific eff
the measure of u, and is therefore independent of the estimate
An alternative contract, which nests the observed contract, p
form
W = B + rY
where B is a base "wage" (or fee) that is independent of planter productiv
advantage of introducing a base wage is that the firm can extract rents f
planter while still providing incentives. In particular, the optimal contract so
following problem:
max (P - r)Y - B subject to e(U) = u.
r,B
The solution to this problem is well known: with risk-neutral planters, the piece rate
is set equal to the price of output and the base wage is adjusted to ensure the planter
earns his alternative utility u.
It is impossible to calculate the difference in profits between the fully optimal
contract and the observed contract because the base wage under the optimal contract
would vary across individuals and would depend on each individual's cost of effort;
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Paarsch and Shearer 663
Table 6
Estimates of Expected Worker Rents, Per Day
Worker 1 2 3 4 5 6 7 8 9 10
Rent 5.96 7.20 8.15 8.23 10.39 12.02 12.36 13.57 13.85 15.48
Fee 0.81 0.85 0.87 0.87 0.96 1 1.01 . 1.04 1.04 1.09
Worker 11 12 13 14 15 16 17 18 19 20
Rent 16.40 16.79 17.01 17.14 17.37 17.95 18.06 19.92 19.96 20.89
Fee 1.10 1.11 1.11 1.11 1.11 1.11 1.11 1.17 1.17 1.18
Worker 21 22 23 24 25 26 27 28 29 30
Rent 21.61 22.58 23.64 23.89 25.64 25.84 26.02 26.09 26.86 27.07
Fee 1.19 1.20 1.22 1.22 1.26 1.26 1.26 1.26 1.27 1.27
Worker 31 32 33 34 35 36 37 38 39 40
Rent 27.16 27.49 27.56 27.69 27.94 28.41 28.81 29.34 29.67 29.68
Fee 1.27 1.27 1.27 1.27 1.28 1.28 1.28 1.28 1.29 1.29
Worker 41 42 43 44 45 46 47 48 49 50
Rent 30.37 30.53 30.61 31.26 31.56 31.95 32.43 32.51 32.65 32.67
Fee 1.29 1.29 1.29 1.30 1.30 1.30 1.30 1.30 1.30 1.30
Worker 51 52 53 54 55 56 57 58 59 60
Rent 32.70 32.75 32.80 33.28 33.45 33.75 34.70 35.96 36.16 36.77
Fee 1.30 1.30 1.30 1.31 1.31 1.31 1.32 1.34 1.34 1.34
Worker 61 62 63 64 65 66 67 68 69 70
Rent 37.25 37.45 37.61 38.43 38.90 39.15 39.37 40.45 40.83 40.98
Fee 1.34 1.35 1.35 1.35 1.36 1.36 1.36 1.37 1.37 1.37
Worker 71 72 73 74 75 76 77 78 79 80
Rent 41.01 42.25 42.49 43.07 43.58 43.92 44.85 45.10 46.01 46.13
Fee 1.37 1.39 1.39 1.39 1.39 1.40 1.40 1.41 1.41 1.41
Worker 81 82 83 84 85 86 87 88
Rent 46.16 46.42 46.86 47.39 47.96 48.73 49.41 59.91
Fee 1.41 1.41 1.42 1.42 1.42 1.43 1.43 2.39
these are not identified in our model. We d
cost of effort relative to kh, so we can estim
in our sample. This allows us to estimate a
that would accrue from the optimal compen
bound because we hold the piece rate fixed.
To estimate the rents that accrued to each
structural model in Equation 9. The estima
estimate that the planters are earning, on a
introducing a base wage could increase avera
per planter.
Firm profits under the observed system are given by (P - r)Y. Interviews with
the firm manager revealed that the bid price per tree received by the firm is typically
twice the piece rate paid to the planters. This suggests that average firm profits are
given by rY, which equals $178, implying that the optimal contract would increase
profits by at least [100 X (30.71/178)] or 17.25 percent. Yet, this contract is only
optimal within a static context. In particular, the base wage paid to each worker
depends on that worker's cost of effort (ability), ki. If the firm were subsequently
to use information on ability to extract rents from the workers, then workers would
not reveal their types, and high-ability workers would produce at the same levels
as low-ability workers; that is, the firm is faced with a ratchet-effect type problem
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664 The Journal of Human Resources
(Lazear 1986; Gibbons 1987; Kanemoto and Macleod 1992). To induce high-abi
workers to reveal their type and plant lots of trees, the firm must commit to n
exploiting information on worker type once it is revealed.
There exists a level of rent that the firm can extract while inducing the work
to reveal their types. By the revelation principle, the firm can charge the worke
base fee that will equate the worker's expected utility to the level of utility the work
would receive from misrepresenting his type.
Theorem 1.Without loss of generality, let workers be ordered on the basis o
their respective cost of effort so that
kh = K1 > K2 > . . . Kfn.
An upper bound to the base fee that the firm can charge worker i > 1 is given
by
Bi(k u - Ri_(i- 1) - (- Ki)
where Ri(j) denotes the utility an individual of type i receives from preten
to be of type j and
Ri(i) = R(i - 1) + 7u( khl - ( K_l
Ki- I/ Ki-1I
R1(1) = u.
The proof of this theorem is presented in the Appendix. Theorem 1 allows u
construct a series of base wages from the relative effort cost parameters identi
in Equation 9. These values are presented in the rows labelled "Fee" in Tabl
They suggest that the firm could capture only a very small part of the informa
rent that workers are presently earning without violating the incentive-compatib
constraints. This suggests that the observed piece-rate contract was an effe
means of inducing effort within a firm composed of heterogeneous workers.
VI. Discussion and Conclusions
In this paper, we have investigated the sensitivity of worker perfor
mance to changes in the compensation system with particular emphasis on ch
in the piece rate. Using data from the payroll records of a British Columbia
planting firm, we have highlighted the econometric problems inherent in evalu
changes in firm compensation policy when compensation systems are endoge
Modeling the decision rules of the worker explicitly with regard to effort, and
firm, with regard to the parameters of the compensation system, controls for
endogeneity. We estimate the elasticity of effort with respect to the piece ra
be 2.14.
This result adds to a small but growing body of direct evidence supporting the
existence of incentive effects. Lazear's (1996) study of auto-glass-worker productivity
under fixed wages and piece rates measured a 20 percent increase in productivity
associated with the switch from fixed wages to piece rates. Meanwhile, Paarsch and
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Paarsch and Shearer 665
Shearer (forthcoming) estimated the gain in productivity associated with paying tr
planters piece rates rather than fixed wages to be 21 percent. Furthermore, bot
these papers emphasize the misleading results that can be obtained through a straigh
comparison of productivity under different payment systems. Lazear's research high
lighted the importance of worker selection effects induced by the change in the com
pensation system, while the research of Paarsch and Shearer emphasized the importance
of the endogenous choice of the compensation system on the part of the firm
Structural analysis has benefits beyond being able to control for endogenous re
gressors. In particular, we are able to investigate the relative merits of alternativ
compensation systems. Our results suggest that dynamic considerations are importan
in determining the form of the observed contract. Although workers earn rents und
this contract, these rents are due to differences in worker ability. By committing n
to exploit information on ability once it is revealed, the firm induces high-abilit
workers to reveal their type and increase their productivity.
Appendix
Proof of Theorem 1
The proof follows Gibbons (1986). To satisfy the incentive compatibility (truthtelling)
constraint, the contract must satisfy
Ri(i) > Ri(),
Let individuals be ranked according to their cost of effort so that
kh = K :> K2 2 . > Kn.
Denote as Bi , the base fee that an individual declaring himself to be of type i
pays to the firm. Since the individual with the highest cost of effort earns zero rent,
it follows that RI =Rh= u and B = 0. For i > 1,
Kic
R^~(i)0 ~~-- -B,.
Ri(i - 1) = (-) U(7 + 1) (1 Y- ) - B.
K,i-l + 1 }Ki-_
Therefore truth-telling implies
kh ( K\
(Al) Ri(i) - R_(i - 1) ukh
Setting
Bi = - Ri_,(i- 1)- (khY t( - )
Ki K-i_ l Kiensures
that Equation Al hol
bound to the base wage that
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666 The Journal of Human Resources
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