pay spread and skewness,employee effort and firm productivity

7/27/2019 Pay Spread and Skewness,Employee Effort and Firm Productivity

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1 Introduction

Incomplete contracts and dysfunctional responses have been important elements in the

growing literature on the internal labour economics of the firm.1 Workers focus on

rewarded aspects of performance, in particular those that are measurable. Since

contracts incompletely specify desired worker behaviour, agents respond to objective

contracts for private benefit, which may be harmful to the employer. Multitask agency

models have extended this earlier literature to allow workers several instruments.Multi-dimensional effort gives agents greater scope to game a compensation system

to their private advantage. Rank-order tournaments are a setting where agents

compete against each other for fixed prizes. They provide a simple structure, which

can be extended to incorporate aspects of incomplete contracts and multitask agency.

Tournament theory provides many testable predictions, which articulate that in certain

contexts weak incentives may be more effective in eliciting desired performance than

high powered, but dysfunctional ones.

This paper tests several predictions from tournament theories of firm compensation

structures, and thus, adds to the rather small empirical literature on theories of

organizational hierarchies and pay structures.2 Previous studies have primarily been

concerned with special groups of employees, and managerial workers in particular.

Moreover, most previous studies have tested only a few theoretical predictions at one

time. As emphasized by Prendergast (1999), the literature suffers from empirical

identification problems, as many empirical outcomes are consistent with several

competing theories and distinguishing between them is difficult. Care is taken to

discuss alternative non-tournament behaviour also consistent with the data. The

burden of proof is somewhat greater than other studies in that firstly we examine

different parts of firms where tournaments as well as other theories should bedifferentially evident, and secondly we consider both individual and collective

effort/performance measures for which theories predict distinct outcomes.

There are three novelties distinguishing this from previous empirical tournament

studies as well as other analyses of the relationship between wage dispersion and

productivity. First, evidence is sought for tournaments in the whole pay distribution of

firms in terms of several outcome measures, which allow testing of common and

distinctive theoretical predictions.3 Second, predictions are examined from multi-task

1 See Gibbons and Waldman (1999) for a survey.

2 Prendergast (1999) provides a recent review.

3 To our knowledge the only other paper looking at tournaments and the whole pay structure of firms isWinter-Ebmer and Zweimller (1999). However, they do not use conventional corporate or individualperformance measures, but proxy them by wage level. This obviously leads to intractable simultaneityproblems in the relation between pay spread and pay level.


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The effort function is C = C(), with both first and second derivatives positive. The

probability of winning depends positively on the players own effort and negatively

on the competitors effort. Thus, the expected utility of player j is

(2) p(W1 - C(j)) + (1-p) (W2 - C(j))

where p is the probability of winning. The probability that j wins is then

(3) prob (qi > qj) = prob (k - j) < (j - k) = prob ((j - k) > ) =

H(j - k),

where = k- j ; h(), H is the cumulative distribution function of and E()= 0.

Each player maximizes (3) by choosing the effort level. Optimum conditions are:

(4) (W1 W2)(p/i) - C/i = 0

and

(5) (W1 W2)(2 p/i

2) - 2 C/i2 < 0

In Nash equilibrium, j = k, and the outcome of the game is random:

(6) (W1 W2) h(0) - C/i = 0

Given (6) and assuming that firms maximize profits per worker, the optimum pay

spread is:

(7) (W1 W2)/2 = C()

and

(8) W1 W2 = h(0)-1

According to equation (6), equilibrium effort is increasing in the prize spread. In the

case with several positions (players), it can be shown that prize increment increases

with higher prizes, or in other words, that the prize-position relationship is convex.

Furthermore, from (6) and (8) it can be seen that the convexity increases as

performance measurement becomes more noisy (i.e., h(0) decreases). Allowing for

several players, the probability of winning obviously becomes smaller. How effort is


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affected is examined by McLaughlin (1988), who shows that the prize spread is

increasing in tournament size, that is, pay rises associated with promotions increase as

the number of contestants grows. Another extension considered by Knoeber and

Thurman (1994) is mixed and biased tournaments, that is when players know their

own as well other players abilities. They demonstrate that mixed and biased

tournaments reduce performance for the best contestants as well as other players.4

Rosen (1986) extends the basic tournament model to account for dynamics by

examining sequential elimination contests.5 Why do firms run sequential contests?

One argument is that since rank is easier to observe than individual output, it is cost

efficient to have winners play winners. In Rosens model career trajectories are the

outcomes of competition among peers to attain higher rankings and better-paid jobs

over the life cycle. The reward structure influences competition at each stage of the

game. Since performance incentives at each stage include an option value on

competing in all successive higher stages, it follows that in order to maintain

incentives throughout the game, an extra reward is required for the overall winner

(typically, the CEO). If players are risk averse, the incentive maintaining prize

structure requires strictly increasing inter-rank pay spreads. Sequential eliminationcontests, therefore, give rise to skewed pay distributions. Consequently, in addition to

pay spread, we should expect pay skewness to affect performance in compensation

which is structured like a tournament.

The original model of Lazear and Rosen (1981) and their followers was formulated in

terms of one-dimensional effort. Lazear (1989) extends the rank-order tournament

model to allow that in addition to productive effort (from the firms perspective),

which furthers the agents, own success directly, the agent can also improve her own

chances of success by non-productive effort (sabotage or lobbying to makecompetitors look bad) to induce rivals failure. This can easily be modeled by

augmenting the players production and effort functions with a sabotage parameter, ,

showing the harm inflicted on the other player. The first-order conditions for the

employees maximization problem now become:

(9) (W1 W2)(p/i) = C/i

and

4 Baker, Gibbons and Murphy (1994) is an example of another one-dimensional effort paper where thesocial and private optimum effort diverge, but in a non-tournament framework.

5 In the working paper version, Lazear and Rosen (1979), of their 1982 article Lazear and Rosen had asection on sequential games. Rosen (1986) expands on this.


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(W1 W2)(p/) = C/

Without strategic behaviour the first-order conditions for the firms maximization

problem are:

(10) (1-(C/)) /W1 = 0

and

(1-(C/)) /W2 = 0,

whereas in presence of strategic behaviour they are:

(11) (1-(C/)) /W1 - (1 + (C/)) /W1 = 0

and

(12) (1-(C/)) /W2 - (1 + (C/)) /W2 = 0

Since (C/) > 0, equilibrium effort is lower when players behave strategically

against their competitors than when they do not. The optimum prize spread is smaller

with sabotage than in the case where one worker cannot affect co-workers

productivity. 6

The employer is assumed to only observe total individual worker effort and cannot

distinguish between effort, which is productive and unproductive. So individuals with

the lowest cost of effort will be promoted, regardless of effort type. If there is a

positive correlation between types of effort, i.e. productive individuals are also good

saboteurs, then the population of tournament winners will over-represent good

saboteurs (Lazear, 1989). Hence, saboteurs should be increasingly present higher up

in the hierarchy as competitive or aggressive players rise to the top. Relative payment

of higher-level workers is potentially quite damaging because of the greater degree of

potentially counterproductive behaviour among that group. Optimum prize spread will

be smaller at senior levels of the hierarchy since sabotage is of increasing relevance.

Alternatively, relative comparisons between competitive workers should only be

made where they cannot affect each others performance.7 Especially higher-level

6 Other two-dimensional effort papers, which do not assume a tournament compensation structure, areItoh (1992) and Milgrom and Roberts (1988).

7 See Carmichael and MacLeod (1993 and 2000) for empirical analyses of worker cooperation.


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workers, who can be particularly counterproductive, need to be kept apart.8 This can

be achieved for example by organising work by product rather than function. Firms,

which replicate organisational structures at several distinct sites, keep managers apart.

If there are only weak complementarities in production between managers then the

incentive benefits of the greater pay spread which will be allowed if they are located

separately may dominate. Thus, optimum pay spread is expected to be larger in multi-

plant compared to single-plant firms.

Allowing workers to affect the productivity of co-workers has implications for the

structure of relative compensation and for organizational form. When it is not possible

to keep competitive individuals apart, firms have to use less high-powered incentives

by compressing pay dispersion for workers to compete with each other less

aggressively in order to avoid counterproductive effort.9 It is simple to show that

allowing multitasking in a sequential tournament leads to a similar counter-

productivity result to that found in the repeated or one-shot tournament case, only

now it is too much skewness which induces the wrong type of effort.10

The important result of Lazear (1989) is that within relevant groups, some wagecompression is efficient. The efficiency argument does not need to appeal to notions

of fairness, whereby workers own utility includes a term in co-workers utility, which

is the key feature in fairness/reciprocity analyses. This literature typically builds on

psychological research using experiments in game settings or survey investigations to

postulate that firms and workers operate under a fairness constraint.11 Behavioral

game theory, motivated by apparent anomalies for standard game theory, which are

produced in experimental settings, has made progress analysing situations where

players care about social allocations, fairness and perceived intentions of other

players.12 If firms set wages for some workers below what is perceived as a fair level,their workers will respond by reducing effort.

Three other arguments for why it may be socially advantageous to compress the wage

structure have been suggested.13 The first is that reduced wage dispersion speeds up

8 Milgrom (1988) and Holmstrm and Milgrom (1991) are related analyses outside a tournamentframework with similar implications for organisational design.

9 Another multitask agency paper is Baker (1992) which is similar to Lazear (1989) in that contracting

on a single agents value contribution is ruled out.10 By extension to Lazear (1989).

11 Examples are Frank (1984), Akerlof and Yellen (1990) and Levine (1991a and 1991b).

12 Rabin (1993) is an early discussion of fairness equilibria. Camerer (1997) gives a brief introductionto behavioural game theory.

13 Agell (1999) provides an extended discussion.


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the rate of structural change by eliminating low-productivity sectors or firms and

subsidizing high-productivity performers. This is frequently referred to as a

motivation for centralised wage bargaining.14 A second argument is that smaller pay

differentials may provide incentives for skill formation as increased relative wages of

unskilled workers leads to a reduced demand for them, wage compression creates

incentives for schooling (Agell and Lommerud (1997) and Acemoglu and Pischke

(1999)).A third rationale for wage compression is that it may act as a social insurance

against changes in individuals relative position in the wage distribution (Agell and

Lommerud (1992)).

The applied gambling literature has long evidenced betting market inefficiency

through long shot bias. This is the phenomenon of positive rates of return for hot

favourites. Players are found to bet on long shots to a higher degree than justified by

rates of return. Recently this apparent negative risk aversion has been rationalised by

tastes for prize skewness.15

It is useful to summarize the essential predictions from the tournament papers we have

discussed, and contrast these with competing theories, so that they can be placed inthe context of our empirical work. In a one-shot tournament game between two

players, it is the prize spread, which creates incentives for effort. Extending this to

more players and more prizes, prize skewness is shown to be important. These results

are carried over to repeated tournaments. However, in a sequential elimination

tournament between two players, it is the option value of competing in future rounds

of the tournament which leads to prize skewness rather than spread driving incentives.

All of the prize spread and skewness results have to be moderated in a multi-task

setting where more than one type of effort is allowed, to the effect that after a pointspread and skewness become net counter-productive. The type of behaviour induced

depends, of course, on the incentive structure. However, in our empirical context we

do not know exactly what game the agents are playing. Furthermore, it is certainly not

explicitly set up as a rank-order tournament, especially for its own sake. But if some

pay structures exhibit important tournament features, do workers act according to

theory? Since we do not have access to narrowly defined job levels, we cannot

establish whether workers are on a career track or in a dead end job 16, whether they

14

See Hibbs and Locking (2000) for an aggregate time-series analysis on Swedish data of productivityeffects.

15 Golec and Tamarkin (1998) were the first to find tastes for skewness in betting markets, analysingthe apparent market inefficiency anomaly of long shot bias in horse race betting. Garret and Sobel(1999) find similar tastes for skewness among lotto players.

16 Baker, Gibbs and Holmstrm (1994) and a number of followers describe careers in largecorporations. The promotion perspectives differ widely between jobs, and an interesting researchquestion is whether incentives are found to differ accordingly.


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could be thought of as playing a sequential or repeated game, respectively. In practice

we would expect to find a mixture of the two, and it is an empirical question which

we shall be examining in this paper, whether incentives can explained accordingly.

What predictions are common to tournaments, tastes for skewness and fairness

theory? (1) Pay spread after a point reduces collective productivity for tournaments

and fairness. Pay skewness increases collective productivity both in tournament and

taste for skewness models. Although the recent applied gambling literature has not

tested whether utility decreases in skewness after a point, casual evidence that most

prize structures are not winner-takes-all would suggest that skewness too might also

reduce collective productivity if it becomes high enough. (2) The effects of pay

distribution in multi- versus single-plant firms may simply be a question of relevant

reference group for calculating spread. The part of the within-firm between-worker

pay spread which is between plants is irrelevant for fairness and makes sabotage more

difficult in tournaments. (3) Different parts of the hierarchy may have different effects

on productivity anyway, if there is a magnification effect of any behaviour at the top.

This suggests that tournaments, fairness and tastes for skewness may be

observationally equivalent.

What prediction does tournament theory deliver that tastes for skewness and fairness

theory do not? The Lazear (1989) model has essentially two different types of effort,

only one of which is collectively counter-productive. Own private effort should not

exhibit counter productive effects. Only tournament theory makes this distinction.

3 Previous empirical research

The empirical analysis of tournaments has followed two routes. One asks whether,given an explicit tournament reward structure, agents do respond as theory predicts.

These studies have used either experimental data (Bull, Schotter and Weigelt, 1987)

or data from sports (Ehrenberg and Bognano, 1990a and 1990b on golf). The only

study of a business setting is Knoeber and Thurman (1994) who examine how a

tournament compensation structure affects the behaviour of broiler chicken producers.

The other approach asks whether tournament-like settings are associated with

behaviour suggested by the theory and has primarily examined whether there is

evidence of tournaments in the structure of top managers compensation. The earlystudies focused on only a few predictions (examples are OReilly, Main and Crystal

(1989), Leonard (1990) and Lambert, Larcker and Weigelt (1993)), whereas two more

recent papers, Main, OReilly and Wade (1993) and Eriksson (1999), have attempted

to test a more comprehensive set of predictions. In the main, these studies have

produced evidence indicating tournaments are present in the compensation of

managerial employees. Thus, they as well as a number of other studies (for example


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Conyon and Peck (1999)) find that the pay-job level relationship is convex. There is

also support for the prediction that pay differentials increase with the number of

contestants and/or declining promotion probabilities. However, the evidence on the

effects of pay spread on firm performance is mixed. Leonard (1990) and Main,

OReilly and Wade (1993) do not find that greater pay dispersion is associated with

better company performance, whereas the results in Eriksson (1999) show a

significantly positive, albeit quantitatively relatively weak, relationship between

managerial pay spread and firm profitability.

Three studies have been concerned with multi-task settings. Cowherd and Levine

(1992) examine the determinants of customer-assessed quality in US and UK firms

and find that pay differentials between managers and blue collar workers as well as

within the management group reduce quality. Drago and Garvey (1998) use

Australian workplace survey data and find that increased promotional incentives give

rise to higher effort, as proxied by reduced absenteeism, and that a wider pay spread

leads to more individual effort but less time spent on helping co-workers.

In a recent study, Hibbs and Locking (2000) test for fairness amongst otherhypotheses on Swedish aggregate time-series data. They do not find that wage

compression within workplaces or industries have had productivity enhancing effects.

The authors do not discuss nor attempt to test predictions from tournament models.

Although there seems to be some empirical support for tournament predictions in

various settings, several limitations of the available evidence should be noticed. It is

not very surprising that explicit tournament settings (like sports) give rise to predicted

incentives. However, most labour contracts are considerably more complicated.

Hence, the results from explicit tournament contexts may not necessarily apply tomany businesses. As pointed out by Prendergast (1999), outside sports, the empirical

research on tournaments, and incentive schemes in general, has focused on agents,

such as executives, whose output is easily observed.17 Again, the strength of the

evidence of these studies seems limited, as it is obvious that most jobs are not like

this. Furthermore, the literature is to some extent plagued by an observational

equivalence problem in that many of the empirical outcomes studied are also

consistent with alternative theories and setting up tests which discriminate between

competing hypotheses is difficult. The burden of proof in this paper is somewhat

heavier than in other studies. Firstly, we examine different parts of firms wheretournaments as well as other theories should be differentially evident. Secondly, we

consider both individual and collective effort/performance measures for which

theories predict distinct outcomes.

17 An exception is Farrell (1996) on US law firm promotion tournaments.


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4 Data description and empirical strategy

Initial sample selection is the population of Danish private sector firms with 20 or

more full-time equivalent employees during any year in the period 1992-95. These

firms are kept for any year 1992-95 when they have 5 or more employees in each of

three occupation groups: management, white collar non-managerial and blue collar.

The resulting sample consists of 6,501 unique firms, each followed for on average 3.5

years, yielding 22,665 firm-year observations.

The important feature of the data is the link between firms and employees which is

consistent over time. Our data originates from two separate registers maintained by

Statistics Denmark: the integrated database for labour market research (IDA) and the

business statistics database (BSD).18 While the matched registers contain a rich set of

background variables, we have access to a subset for the current study, which are now

described.

Business statistics at the firm-year level include annual sales, wage bill, book value of

capital, industry and municipality. Worker characteristics at the person-year level

include gender, age, ongoing tenure, years of education, occupation, periods of

absence, home municipality, annual labour earnings at the current employer and

sufficient information to calculate income tax liability.

A primary goal of the analysis is to establish a link between moments of the pay

distribution within firms and firm-level outcomes of interest, and to evaluate whether

this relation is as predicted by tournament theory. In particular, the paper focuses on

the effect of pay spread and pay skewness on firm productivity and individual effort.

The economic model of Section 2 gives guidance on construction of measures of thepay distribution. Regardless of whether efficiency, fairness or tastes for skewness are

the causal links driving the relation between pay distribution moments and effort, it is

the part of compensation which is due to unobserved characteristics which drives

incentives, equality concerns or distributional tastes. Pay dispersion, all else equal, is

relevantly defined only within an appropriate reference group. Ideally, the part of

compensation which remains unexplained should be within precisely defined job

levels, controlling for human capital. In the absence of narrowly defined job levels,

we take three broad occupations within the same firm-year as relevant reference

groups and control for gender, age, education, and tenure.

Specifically, we run ordinary least squares (log) earnings equations on our sample of

persons once and for all, and calculate the residuals. Summary statistics and

18 It is the work of Sren Leth-Srensen and Claus W. Andersen in checking the consistency of thematch between IDA and BSD which makes this work possible.


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regression coefficients are presented in Table 1. Descriptive statistics show that the

sample of workers employed in medium-to-large establishments have age, education

and tenure which is slightly higher than the population of private sector employees as

a whole. Regression coefficients age and tenure are conventionally quadratic.

Education is negative quadratic due to the inclusion of occupation dummy variables,

and reverts to a conventional positive quadratic function if occupations are omitted.19

The residuals are grouped according to occupation-firm-year cell, and for each cell,

second and third moments of the residuals distribution are calculated.

Table 1. Worker Descriptives and Earnings Equation Estimates (OLS)

Variable Coefficient Std. Error Mean Std.dev.

Intercept 9.209 0.007

Male 0.287 0.001 0.689 0.462

Age 0.123 0.000 36.401 12.177

Age2 -0.001 0.000 1473.326 941.687

Education -0.027 0.001 11.325 2.489

Education2 0.003 0.000 134.446 56.520

Tenure 0.096 0.000 4.898 4.016

Tenure2 -0.005 0.000 40.111 55.591

Manager 0.211 0.001 0.101

Blue collar -0.110 0.001 0.583

1992 0.013 0.001 0.245

1993 -0.013 0.001 0.241

1995 0.016 0.001 0.260

Log(earnings) 12.179 0.709

Absence 1.648 2.295

R2 0.435

# observations 2280607

# persons 859574Notes: log (annual earnings) (2000 DKK) regression. Omited categories are white-collar non-

managerial and 1994.

We make use of two alternative performance measures: one measuring collective and

the other individual effort. Solow residual measures of total factor productivity are the

most widely used and accepted measures of firm productivity (Hulten, 2000). We

assume a Cobb-Douglas production function and compute Solow residual productivity

measures from that. Firm-wise descriptive statistics and production function estimates

are presented in Table 2. The population of Danish firms with 20 or more employeescomprises somewhat smaller firms than elsewhere in the Nordic countries, Europe

and the US. Business statistics on sales, wages and capital vary accordingly. Cobb-

Douglas production function estimates exhibit conventional signs.

19 Reference groups in the earnings regression are women, non-managerial white collar workers and theyear 1994.


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Table 2. Firm Descriptives and Production Function Estimates (OLS)

Variable Coefficient Std. Error Mean Std.dev.

Intercept 6.896 0.038

Log(capital) 0.375 0.003 8.462 1.707

Log(wagesum) -0.356 0.005 9.104 0.943

1992 -0.023 0.011 0.250

1993 -0.066 0.011 0.250

1995 0.022 0.011 0.250

Log(sales/worker) 6.818 0.798

Capital 15975 39327

Wagesum 16109 27872

sSles 84181 168159

siSe 75 241

R2 0.372

# observations 22665

# firms 6055

Notes: log (annual sales) (2000 DKK) regression. Cobb-Douglas production function. 1994 is the

omitted year

For our measure of individual effort we follow Drago and Garvey (1998) and proxy

effort by the inverse of the firms average rate of absenteeism. That is, we use the

log(1-annual proportion of full-time equivalent days lost though absence) as a

dependent variable. Tournament agency models require unobserved total effort.

Absenteeism is only part of productive effort. Employers cannot base payment on that

because of moral hazard problems of employees turning up for work when they are

sick. Even though absenteeism is a noisy proxy for individual productive effort,

employers not being able to contract on it, makes it a valid measure for testingtournament predictions.

Having defined the outcomes of interest: collective productivity and individual (or

collective productive) effort, and covariates of interest: moments of the pay residual

distribution; we next turn to the primary analysis. The discussion of tournament

theories in Section 2 focuses on the implications of moments of the relevant pay

distribution for effort incentives. Specifically, productive effort is increasing in both

pay spread and skewness up to a point, after which counterproductive behaviour

dominates. This hypothesis can be tested in a regression framework as follows:

(13) ft = 12ft + 22ft2 + 33ft + 43ft

2 + Xft + f+ ft

where ft is productivity of firmfin yeart,2 and 3 are second and third moments

of the pay residual distribution within the firm-year, s are associated vectors of


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coefficients, Xft is a matrix of exogenous characteristics, is an associated vector of

coefficients, fis a firm-specific error term and ft is an idiosyncratic error term. The

null hypothesis of the tournament models discussed is that the s are respectively

positive, negative, positive and negative. However, this finding alone is consistent

with non-tournament explanations of productivity, such as fairness concerns.

Employees in different parts of the firm may exhibit differential productivity effects

predicted by tournament theory. However, fairness arguments can be extended toallow magnified counter-productivity effects higher up the hierarchy, since managers

behaviour has greater impact than that of other workers. Similar productivity

magnification arguments can augment tastes for skewness predictions so that they also

have an occupational distributional effect similar to tournaments.

Moments are computed for three different levels in the occupational hierarchy --

managers, non-managerial white-collar workers and blue-collar workers -- within

each firm-year. Equation (13) is then estimated separately for each occupation group.

The productivity measure is always at the firm-year level since we are unable toallocate productivity between different occupations.

A major empirical problem facing all observational studies of the effect of pay on

firm performance is simultaneity. That is, firms which perform well may pass product

market rents on to top managers, or their employees in general. However, it may not

be the case that firms where top managers contracts have such a structure are

necessarily those which perform well. Failure to deal with the potential endogeneity

of the pay structure, could lead to biased estimates of the relationship of interest,

because of the spurious correlation. It is an important contribution of the current paperthat we are the first to address this issue.

The structure of the Danish income tax system provides institutional variation which

has the potential of being used in estimation for breaking this simultaneity. What is

needed is a source of variation which determines the between firm-year differences in

the within-firm pay distribution, but which at the same time does not also affect firm

performance. Specifically, we need instruments for moments of the within-firm pay

distribution in an equation explaining firm performance.

Income tax rates in Denmark vary according to workers municipality (of which there

are 275) of residence. Consequently, firms employing workers from different

municipalities, have workforces with differential marginal and average tax rates, over

and above that variation which is gross income related due to the piecewise linear (3

segment) nature of the income tax schedule. Assuming firm location is exogenous at

the margin for productivity, between municipality income tax rate variation provides


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candidate instruments for the within firm wage distribution. Alternatively stated:

exogenous regional differences in income tax progressivity identify the effect of

moments of the gross earnings distribution on productivity. The part of the variation

in moments of pay which is explained by progressivity is useful for determining the

effects of pay moments on productivity, because that part is purged of the spurious

correlation which would otherwise bias estimates. It is arguably sensible to assume

that firm location is exogenous at the margin, since it is the changes in progressivity

over time between regions (the difference in differences) which will identify measures

of interest.20 Having discursively motivated the identification strategy, it is of course

an empirical question whether our choice of instruments are statistically appropriate.

Diagnostic tests are presented in a later section.

Table 3. Analysis of Variance

Productivity Effort Std.dev. (Std.dev.)2 Skewness (Skewness)2

Firm 0.818 0.288 0.150 0.141 0.159 0.138

3 digit industry code 0.444 0.067 0.011 0.010 0.010 0.010

5 digit industry code 0.569 0.101 0.021 0.019 0.020 0.019

SIC3-year 0.477 0.079 0.057 0.059 0.055 0.059SIC5-year 0.640 0.160 0.058 0.060 0.056 0.059

Municipality 0.092 0.019 0.005 0.005 0.005 0.005

Municipality-year 0.105 0.051 0.056 0.059 0.055 0.006

Note: Each cell is an R2 measure from a separate dummy variable regression. Regressions are ofvariables defined in the column header on dummies defined in the row header. SIC3 and SIC5 arerespectively industry codes with 50 and 800 unique realisations.

The basic unit of the primary analysis is firm-year. Some important points can be

noticed from the variance of outcomes (productivity and effort) and the explanatory

variables (pay residual moments) of interest. Table 3 presents R2

measures ofgoodness-of-fit from regressions of variables defined in the column headers on

dummies defined in the row headers. Each cell represents a different regression. For

example, the first substantive cell is the R2 from a regression of Solow residuals on

firm dummies. Firm is the first level of dummy considered. Remaining variation must

be over years within-firm. To benchmark this consider 5- (800 realisations) and 3-

digit (50 realisations) industry codes, which respectively explain about a half and two

thirds as much variation in all dimensions as firms themselves. Sweeping across the

columns we can see that there is most heterogeneity between units in Solow residuals,

about a quarter as much in individual effort, and much less in pay moments.

21

Secondand third moments have very similar between-unit heterogeneity.

20 This builds on the assumption that the costs of re-location are higher than any gain from movingmunicipality in order to change tax progressivity. See Papke (1991) for a counter-argument.

21 Lazear (1999) finds that within firms, wages are compressed relative to output.


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It is important to appreciate that both the dependent variable and the regressors of

interest in the primary analysis are themselves generated, and as such have a variance

which should properly be accounted for in estimation. In order to avoid this problem,

it is often simpler to perform the analysis in a single step, but in our context this is not

feasible because of the different dimensionality of the first stage regressions: firm-

year and worker-year. We could appeal to the fact that we are analysing a population,

and leave it at that, but in the interests of generality of the results outside Denmark,

we have performed some bootstrap sensitivity checks on the standard errors. 22

5 Results and discussion

Selected productivity estimates are presented in Table 4. They are easiest divided into

9 blocks of 3 columns comprising different samples (all firms, single-plant firms and

multi-plant firms) and 3 rows comprising different occupation groups within the firm

(managers, non-managerial white collar and blue collar). Within each of the 9 blocks

are 2 columns of coefficients and associated t-statistics and 4 rows defining

coefficients on pay residual moments and moments squared. Consider for example

rows 1-4 in column 3, which refer to instrumental variables estimates of coefficients

on moments of managers pay in a productivity regression.23

Beginning with the estimates for all firms together, presented in columns 3-4, we may

see that for white collar workers (managers and non-managers), pay spread (second

moment) and skewness (third moment) have a quadratic relation with productivity.

That is, they increase productivity up to a point, after which they become counter

productive. The counter-productivity effect is greatest for managerial workers. The

latter is consistent with Lazears (1989) model according to which hawks rise to the

top and therefore optimum spread is lower in the upper parts of the hierarchy.However, it cannot be ruled out that the differences between employee-groups

observed, to a large extent may reflect magnification of effects at senior levels of the

hierarchy.

22 We drew 100 samples from the population of firms (of size 100% with replacement), and re-estimated the model; then we drew 100 samples of workers, and re-estimated. Re-sampling firms hadnegligible impact on standard errors. However, re-sampling persons increased standard errors often toinsignificance. This is mainly due to a large number of smaller firms falling below our size thresholdfor moment calculation. We base this conclusion on results from restricting the estimation sample toinclude only firms of size 50 and above, and re-sampling therefrom. Analytic and bootstrappedstandard errors coincide much more closely.

23 OLS estimates available on request.


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Table 4. Firm-Level Productivity Estimations

Productivity All firms Single-plant firms Multi-plant firms

coef. t-stat coef. t-stat coef. t-stat

Managers std.dev. 0.139 4.848 0.137 4.318 0.177 5.805

std.dev.2 -0.007 -3.371 -0.008 -3.379 -0.001 0.177

skew. 0.754 2.598 0.735 2.822 0.763 5.274

skew.2 -0.429 -2.632 -0.432 -3.484 -0.416 0.546

std.dev. 0.088 4.982 0.072 3.767 0.135 2.103White collar

workers std.dev.2 -0.005 -0.735 -0.004 -2.094 -0.008 -0.949

skew. 0.631 5.890 0.635 6.688 0.793 2.264

skew.2 -0.433 -2.512 -0.443 -2.434 -0.439 -0.311

std.dev. 0.042 1.102 0.037 0.333 0.053 3.486Blue collar

workers std.dev.2 -0.002 -0.121 -0.001 -1.723 -0.002 -0.248

skew. 0.417 2.014 0.440 2.834 0.361 2.524

skew.2 -0.346 -1.246 -0.355 -1.282 -0.308 0.110

Notes: Selected IV regression coefficients explaining Solow residual productivity. Each set of 4coefficients comes from a separate regression. The column header defines the firm sample andestimator, and the row header occupation. Other explanatories included but not presented areproportions of workers by age group, occupation, gender, education and tenure and firm size.

Blue-collar pay moments exhibit weak, if any, productivity effects. However, the

relative importance of skewness compared to spread increases as one moves down the

hierarchy. This finding is consistent with the idea that sequential tournaments are the

more important concept lower down the job hierarchy. In other words, promotion to

the next round of the tournament which allows competition for bigger prizes may be

the relevant incentive rather than just the single or repeated tournament at the current

job level.

Splitting the sample into single plant and multi-plant firms, we turn to compareestimates in columns 5-6 with those in columns 7-8. Direct comparison of regression

coefficients between the two samples is valid because the different means are washed

out of higher moments and units are consistently year 2000 Danish Kroner. The

incentive effects of pay moments appears to be greater in multi-plant firms.

Furthermore, counter productivity effects kick in somewhat later in multi-plant firms,

though these differences are hardly significant. Thus one conclusion to extract from

the estimates in Table 4 is that they provide moderate support for the organisational

form implications of industrial politics (see Lazear, 1989). To the extent that the

between-plant part of within-firm between-worker pay spread is irrelevant for what isconceived of as fair wage differences, these findings are not inconsistent with fairness

explanations.

So far we have been concerned with collective effort as measured by firm level

productivity. Individual effort is notoriously hard to measure, except in rather special

occupations. We use the inverse of the average firm rate of absenteeism as our proxy


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Table 6. Goodness-of-Fit R2

All firms Single-plant firms Multi-plant firms

Productivity managers 0.184 0.204 0.168

white collar workers 0.190 0.204 0.183

blue collar workers 0.214 0.236 0.180

Effort managers 0.049 0.047 0.050

white collar workers 0.050 0.048 0.049

blue collar workers 0.054 0.053 0.047

# obs. 22665 15466 7199

Note: Goodness-of-fit associated with Tables 4 and 5

Table 7. Instrument Diagnostics

All firms Single-plant firms Multi-plant firms

test-stat p-value test-stat p-value test-stat p-value

Managers F-test std.dev. 806 0.000 602 0.000 162 0.000

std.dev.2 1355 0.000 999 0.000 287 0.000

skew. 1034 0.000 794 0.000 181 0.000skew.2 1157 0.000 877 0.000 219 0.000

Hausman prod. 56.82 0.000 43.40 0.000 15.71 0.000

effort. 6.72 0.000 3.93 0.003 8.36 0.000

Overident. prod. 20.20 0.010 16.19 0.028 5.73 0.325

effort. 6.73 0.250 3.77 0.550 5.00 0.450

F-test std.dev. 713 0.000 710 0.000 62 0.000White collar

workers std.dev.2 1055 0.000 1054 0.000 78 0.000

skew. 656 0.000 669 0.000 39 0.000

skew.2 808 0.000 811 0.000 49 0.000

Hausman prod. 90.62 0.000 68.83 0.000 20.19 0.000

effort. 10.85 0.000 7.41 0.000 6.61 0.000

Overident. prod. 27.74 0.000 2.15 0.020 6.32 0.720

effort. 8.66 0.450 5.06 0.760 5.50 0.740

F-test std.dev. 836 0.000 947 0.000 72 0.000Blue collar

workers std.dev.2 1115 0.000 1269 0.000 91 0.000

skew. 521 0.000 576 0.000 46 0.000

skew.2 705 0.000 782 0.000 50 0.000

Hausman prod. 94.22 0.000 75.04 0.000 18.27 0.000

effort. 6.72 0.000 4.49 0.001 4.73 0.000

Overident. prod. 21.54 0.007 1.58 0.996 5.81 0.305

effort. 7.60 0.480 4.42 0.805 5.00 0.450Notes: F-tests are of significance of instruments explaining endogenous variables. Hausman tests are ofsignificance of first stage residuals in the second stage regression. Overidentification tests areuncentered R2 from regressing structural equation residuals on instruments.


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6 Summary and conclusions

Two types of theories predict that reduced pay spread may be productivity enhancing.

Tournament theories provide a simple structure, which articulates the point made by

theories of incomplete contracts and multi-task agency, that weak incentives can be

more effective in inducing desired worker performance than high-powered but

dysfunctional ones. Models based on notions of fairness and reciprocity show that

fairness concerns a may act as a constraint on firms wage setting behaviour, and thus,increases in pay dispersion may be counter-productive. Both tournament models and

tastes for prize skewness are consistent with skewness effects on productivity. A

number of predictions arise regarding individual and collective incentives, and this

paper tests these empirical implications together for the first time.

Our analysis differs from previous ones in that we consider the entire pay distribution

of firms. This is informative in that the burden of proof is greater and potentially

theory is more seriously evaluated if in the same general setting both presence and

absence of effects can be tested. In other words, tournament-like compensation

structures are expected to have certain effects for some groups and different effects

for others. The cuts in the data we have chosen which test a number of predictions

common to competing theories of tournaments, fairness and tastes for skewness are

between managers, non-managerial white collar workers and blue collar workers; and

single- and multi-plant firms. Those which distinguish tournaments are collective

productivity versus individual effort.

The application is to a longitudinal matched employer-employee dataset comprising

the population of Danish medium-to-large private sector firms. This together with the

nature of the Danish tax system, provides exogenous contract variation required toidentify the effect of moments of the pay distribution on productivity and effort. The

results of our empirical analysis can be briefly summarised as follows: In common to

all theories, for white collar workers, pay spread and skewness are found to increase

firm productivity up to a point, after which it becomes counter-productive. Among

white collars, counter-productive behaviour is more important higher up in hierarchy.

Only weak productivity effects are detected for blue-collar workers. More pay spread

and skewness is productive in multi-plant compared to single-plant firms. However,

importantly, there are no counter-productivity effects on individual effort. Individual

effort is increasing with pay spread and skewness, which is a distinctive prediction oftournament models.

The novelty of this paper has been to examine firm productivity and individual effort

effects of the whole pay distribution of firms. Differences in firm productivity effects

between occupational groups and types of firms give support to theories of fairness,

tournaments and tastes for skewness. Only individual effort effects, proxied by


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absenteeism, support tournament theory alone. A more convincing test would require

isolating predictions exclusive to fairness and skewness models not in common with

tournaments. Rejecting those predictions would base tournament theory on still firmer

empirical ground.


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