Rank-Order Tournaments as Optimum Labor Contracts
Author(s): Edward P. Lazear and Sherwin Rosen
Source: The Journal of Political Economy, Vol. 89, No. 5 (Oct., 1981), pp. 841-864
Published by: The University of Chicago Press
Stable URL: http://www.jstor.org/stable/1830810
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Rank-OrderTournamentsas Optimum Labor
Contracts
EdwardP. Lazear and SherwinRosen
University
of Chicagoand NationalBureau of EconomicResearch
This paper analyzes compensation schemes which pay according to
an individual's ordinal rank in an organization ratherthan his output level. When workersare riskneutral,itis shown thatwages based
upon rank induce the same efficientallocation of resources as an
incentivereward scheme based on individual output levels. Under
some circumstances,risk-averseworkersactually prefer to be paid
on the basis of rank. In addition, if workersare heterogeneous in
ability, low-quality workers attempt to contaminate high-quality
firms,resultingin adverse selection. However, if abilityis known in
advance, a competitivehandicapping structureexistswhichallows all
in the same organization.
workersto compete efficiently
I.
Introduction
It is a familiar proposition that under competitive conditions workers
are paid the value of their marginal products. In this paper we show
that competitive lotteries are often efficient and sometimes superior
to more familiar compensation schemes. For example, the large
salaries of executives may provide incentives for all individuals in the
firm who, with hard labor, may win one of the coveted top positions.
This paper addresses the relation between compensation and incentives in the presence of costly monitoring of workers' efforts and
We are indebtedto JerryGreen,MertonMiller,JamesMirrlees,GeorgeStigler,
and EarlThompsonforhelpfulcomments.
JosephStiglitz,
The researchwassupported
in part by the National Science Foundation. The research reported here is part of the
NBER's research program in Labor Studies. Any opinions expressed are those of the
authors and not those of the National Bureau of Economic Research.
Journalof PoliticalEconomy,1981, vol. 89, no. 5]
1981 by The Universityof Chicago. 0022-3808/81/8905-0006$01.50
?
841
842
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ECONOMY
output. A wide varietyof incentive payment schemes are used in
practice. Simple piece rates, which have been extensivelyanalyzed
(see, e.g., Cheung 1969; Stiglitz1975; Mirrlees 1976), gear payment
to output. We consider a rank-order payment scheme which has
not been analyzed but which seems to be prevalent in many labor
contracts.This scheme pays prizes to the winnersand losers of labor
market contests.The main differencebetween prizes and other incentiveschemesis thatin a contestearningsdepend on the rank order
of contestantsand not on "distance." That is, salaries are not contingent upon the output level of a particular game, because prizes are
fixed in advance. Performanceincentivesare set by attemptsto win
the contest.We argue that in many circumstancesit is optimal to set
up executivecompensation along these lines and thatcertainpuzzling
features of that market are easily explained in these terms.
Central to this discussion are the conditions under which mechanismsexist for monitoringproductivity(Alchian and Demsetz 1972).
If inexpensive and reliable monitorsof effortare available, then the
best compensation scheme is a periodic wage based on input. However, when monitoringis difficult,so that workers can alter their
input with less than perfect detection, input-wage schemes invite
shirking.The situation often can be improved if compensation is
related to a more easily measured output level. In general, inputbased pay is preferablebecause it changes the riskborne by workers
in a favorableway. But when monitoringcosts are so high that moral
hazard is a serious problem,the gain in efficiencyfromusing outputbased pay may outweigh the risk-sharinglosses. Paying workers on
the basis of rank order alters costs of measurement as well as the
nature of the risk borne by workers.It is for these reasons that it is
sometimesa superior way to bring about an efficientincentivestructure.
In the development below we start with the simplestcase of risk
neutralityto illustratethe basic issues. Then the more general case of
risk aversion is treated in Section III. Section IV considers issues of
sortingand self-selectionwhen workersare heterogeneous.
II.
Piece Rates and Tournaments with Risk Neutrality
To keep thingssimpleand to avoid sequential and dynamicaspects of
the problem,we confineattentionto a single period in all thatfollows.
Therefore, the reader should thinkof the incentiveproblem in terms
of career development and lifetime productivityof workers. The
worker's(lifetime)output is a random variable whose distributionis
controlledby the workerhimself.In particular,the workeris allowed
to control the mean of the distributionby investingin costly skills
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prior to enteringthe market. However, a given productivityrealization also depends on a random factor which is beyond anyone's
control. Employers may observe output but cannot ascertain the extentto whichit is due to investmentexpenditure or to good fortune
or to both,thoughworkersknowtheirinputas well as output. Worker
j produces lifetimeoutput qj according to
qj =/ tj
+ Ej,
(1)
where pj is the level of investment,a measure of skill or average
output,chosen by the workerwhen youngand priorto a realizationof
the random or luck component, Ej. Average skill, ,j, is produced at
cost C(,), withC', C" > 0. The random variable E, is drawn out of a
knowndistributionwithzero mean and varianceC2.1 Here E is lifetime
luck such as life-persistent
person-effectsor an abilityfactor,whichis
revealed veryslowlyover the worker'slifetime.The crucial assumption is thatproductivityriskis nondiversifiableby the workerhimself.
That is another reason forchoosing a long period forthe analysis.For
example, if the period were very short and the random factorwas
independentlydistributedacross periods, the workercould diversify
per period risk by repetitionand a savings account to balance off
good and bad years. Evidentlya persistentperson or abilityeffect
cannot be so diversifiedwhen it is undiscoverablequickly,as appears
true of managerial talent,for example. It is assumed, however,thatE
is i.i.d. across individuals, so that owners of firmscan diversifyrisk
either by pooling workers together in one firm or by holding a
portfolio.
To concentrate on incentive aspects of various contractual arrangements,we adopt the simplesttechnologyfor firms.Production
requires only labor and is additively separable across workers. By
virtue of the independence assumptions, managers act as expected
value maximizersor as if they were risk neutral. Free entryand a
competitiveoutput marketset the value of the product at V per unit.
Again, these assumptionsare adopted to illustratebasic issues in the
simplestway. The analysis also applies when there are complementaritiesamong workersin production,whichis more realisticbut more
difficultto exposit.
Piece Rates
The piece rate is verysimple to analyze when workersare riskneutral.
It involves paying the worker the value of his product. Let r be the
1 In thispaper the workerhas no choice over o-.This does not affectthe risk-neutral
solution but does have an effectif workers are risk averse, since they tend to favor
overlycautious strategies.Also, virtuallyall the resultsof this paper hold true if the
error structureis multiplicativerather than additive.
844
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piece rate. Ignoringdiscounting,the worker'snet income is rq - C(pu).
Risk-neutralworkerschoose u to maximize expected net return
E[rq
-
C(g)] = r,
-
C(u).
The necessaryconditionis r = C'(tk) or the familiarrequirementthat
investmentequates marginalcost and return.On the other hand, the
expected profitof a firmis
E(Vq
-
rq) = (V -r),
so free entryand competitionfor workersimplyr = V. Consequently
V
= C'(p.).
The marginalcost of investmentequals its social return,yieldingthe
standard result that piece rates are efficient.
Rank-OrderTournaments
We shall consider two-playertournamentsin which the rules of the
game specifya fixedprize W1to the winnerand a fixedprize W2to the
loser. All essential aspects of the problem readily generalize to any
number of contestants.A worker's production follows (1), and the
winner of the contestis determinedby the largestdrawing of q. The
contestis rank order because the margin of winningdoes not affect
earnings. Contestants precommit their investmentsearly in life,
knowingthe prizes and the rules of the game, but do not communicate witheach otheror collude. Notice thateven though thereare two
playersin a given matchthe marketis competitiveand not oligopolistic,because investmentis precommittedand a given player does not
know who his opponent will be at the time all decisions are made.
Each person plays against the "field."
We seek to determinethe competitiveprize structure(W1,W2).The
method proceeds in two steps. First,the prizes W1 and W2 are fixed
arbitrarilyand workers' investmentstrategies are analyzed. Given
these strategies,we then find the pair (W1,W2) that maximizes a
worker'sexpected utility,subject to a zero-profitconstraintby firms.
It will be seen that a worker's incentivesto invest increase with the
spread between winningand losing prizes, W1 - W2. Each wants to
improve the probabilityof winning because the return to winning
varies with the spread. The firmwould always like to increase the
spread, ceterisparibus, to induce greater investment and higher
productivity,because its output and revenue are increased. But as
contestantsinvest more, their costs also rise. That is what limitsthe
spread in equilibrium:Firmsofferingtoo large a spread induce excessive investment.A competing firmcan attractall of these workersby
decreasing the spread because investmentcosts fall by more than
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expected product, raising expected net earnings. Increasing marginal cost of skill implies a unique equilibrium spread between the
prizes that maximizes expected utility.
More precisely,consider the contestant'sproblem, assuming that
both have the same costs of investmentC (,), so thattheirbehavior is
identical. A contestant'sexpected utility(wealth) is
(P)IW1
- C(y)]
= PW1 + (1 - P)W2
- C(Z)]
+ (1 - P)[W2
-
(2)
where P is the probabilityof winning.The probabilitythatj wins is
P = prob (qj >
qk)
= prob (Uj
-
Ek -
>
Ilk
Ej)
(3)
= prob (g j - t4k > G) G (g j - t),
- g(e), G(-) is the cdf of A,E(f) = O, andE(f2) =
where
Ek
Ej, 6
2cr2(because Ej and Ek are i.i.d.). Each player chooses pi to maximize
(2). Assuming interiorsolutions, this implies
(W1-W2)
-
a
OtFi)
= ?
and
(W1 - W2)
=j,k.
P2 - C"()<Oi
(4)
We adopt the Nash-Cournot assumptionsthat each player optimizes
against the optimum investmentof his opponent, since he plays
against the marketover whichhe has no influence.Therefore,j takes
and converselyfork. It then
/k as givenin determininghis investment
followsfrom (3) that, for playerj
aP/luj
= aG(gj
-
=
/Lk)/tL
g(/Lj
-
which upon substitutioninto (4) yieldsj's reaction function
(W1 - W2)gQ(
-
.k)
-
C'
)
=
0.
(5)
Player k's reaction functionis symmetricalwith (5).
Symmetryimpliesthatwhen the Nash solutionexists,Pj = /k and P
G (0) = 1/2,
so the outcome is purelyrandom in equilibrium.Ex ante,
each player affectshis probabilityof winning by investing.2
-
2
However, it is not necessarilytrue that there is a solution because with arbitrary
densityfunctionsthe objectivefunctionmay not be concave in the relevantrange. It is
large:
possible to show thata pure strategysolutionexistsprovided that.j2 is sufficiently
Contestsare feasible only when chance is a significantfactor.This resultaccords with
intuitionand is in the spiritof the old sayingthata (sufficient)differenceof opinion is
necessaryfora horse race. Stated otherwise,sinceOP/0,uj= g(lij - AOk)and g( ) is a pdf,
=
=
of second-orderconditionsin (4)
g'; (jIlk) may be positive,and fulfillment
implies sharp breaks in the reaction function.If U2 is small enough the breaks occur at
verylow levels of investment,and a Nash equilibrium in pure strategieswill not exist.
Existence of an equilibrium is assumed in all that follows.
846
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,j
=
,tk
OF POLITICAL
ECONOMY
at the Nash equilibrium,equation (5) reduces
i =j,k,
C'(p~i) = (W1 - W2)g(0),
(6)
verifyingthe point above that players' investmentsdepend on the
spread between winningand losing prizes. Levels of the prizes only
influencethe decision to enter the game, which requires nonnegativityof expected wealth.
The risk-neutralfirm'srealized gross receiptsare (qj + qk) V, and
its costs are the totalprize money offered,W1 + W2.Competitionfor
labor bids up the purse to the point where expected total receipts
equal costs W1 + W2
=
(9,i
+
9.k)
-
V. But since
equilibrium,the zero-profitcondition reduces to
pj
=
9k
=
g in
(7)
V11= (W1 + W2)/2.
The expected value of product equals the expected prize in equilibrium. Substitute(7) into the worker'sutilityfunction(2). Noting that
P = 1/2in equilibrium,the worker'sexpected utilityat the optimum
investmentstrategyis
Vpu- C(g).
(8)
The equilibriumprize structureselectsW1and W2to maximize(8), or
[V - C'(,W)](au/(Wi)
=
0
i
=
1, 2.
(9)
The marginalcost of investmentequals itsmarginalsocial return,V =
C'(g), in the tournamentas well as the piece rate. Therefore,competitive tournaments,like piece rates, are efficientand both result in
exactlythe same allocation of resources.
Some furthermanipulationof the equilibriumconditionsyieldsan
interestinginterpretationin terms of the theoryof agency (see Ross
1973; Becker and Stigler 1974; Harris and Raviv 1978; and Lazear
1979):
W, = VpL + C'(,u)/2g(O)
=
Vt + V/2g(O)
W2 = Vg - C'U(j)2g(0)
=
- VI2g(O).
(10)
The second equality followsfrom V = C'(/i). Now thinkof the term
C'(g)/2g(O) = V/2g(0) in (10) as an entrancefee or bond thatis posted
by each player. The winningand losing prizes pay off the expected
marginal value product plus or minus the entrance fee. That is, the
players receive theirexpected product combined witha fair winnertake-allgamble over the totalentrancefeesor bonds. The appropriate
social investmentincentivesare given by each contestant'sattemptto
win the gamble. This contrastswiththe main agency result,where the
bond is returnedto each workeraftera satisfactoryperformancehas
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847
been observed. There the incentive mechanism works through the
employee's attemptsto work hard enough to recoup his own bond.
Here it works through the attemptsto win the gamble.
Comparative staticsfor this problem all follow from (9) and (10)
once a distributionis specified. For example, if E is normal with
variance _2, theng(0) = 2uV'cr.It followsfrom(10) thatthe optimal
spread varies directlywith V and oC2. While several other interesting
observationscan be made of this sort,we note a somewhatdifferent
but important practical implication of this general scheme. Even
though the optimal prize structuredetermines expected marginal
product throughits effecton workerchoice of g and the zero-profit
condition (7) implies that expected prizes equal expected productivity,neverthelessactual realized earnings definitelydo not equal
productivityin either an ex ante or ex post sense. Consider ex ante
first.Since gj = 9k= a, expected productsare equal. Since W1> W2is
required to induce any investment,the paymentthat receives never
equals the payment that k receives. It is impossible that the prize is
equal to ex ante product,because ex ante productsare equal. Nor do
wages equal ex post products. Actual product is Vq rather than Vtk.
But q is a random variable,the value of whichis not knownuntilafter
the game is played, while W1and W2are fixedin advance. Only under
the rarest coincidence would W1 = Vqj and W2 = Vqk.
Consider the salary structurefor executives. It appears as though
the salary of, say, the vice-presidentof a particular corporation is
substantiallybelow thatof the presidentof the same corporation. Yet
presidentsare oftenchosen fromthe ranksof vice-presidents.On the
day that a given individual is promoted fromvice-presidentto president, his salary may triple. It is difficultto argue that his skillshave
tripled in that 1-day period, presenting difficultiesfor standard
theorywhere supply factorsshould keep wages in those two occupations approximatelyequal. It is not a puzzle, however, when interpreted in the context of a prize. The president of a corporation is
viewed as the winner of a contest in which he receives the higher
prize, W1. His wage is settledon not necessarilybecause it reflectshis
currentproductivityas president,but rather because it induces that
individual and all other individuals to performappropriatelywhen
they are in more junior positions. This interpretationsuggests that
presidentsof large corporations do not necessarilyearn high wages
because they are more productive as presidents but because this
particular type of payment structuremakes them more productive
over theirentireworkinglives. A contestprovides the proper incentives for skill acquisition prior to coming into the position.3
3 If E is a fixed effect,there is additional informationfromknowing the identityof
winnersand losers. The expected productivityof a winner is g + E(Ej I qj > qk), while
that of a loser is g + E(Ej I qj < qk). In a one-period contest there is no possibilityof
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Comparisons
Though tournamentsand piece rates are substantiallydifferentinstitutionsfor creatingincentives,we have demonstratedthe surprising resultthatboth achieve the Pareto optimal allocation of resources
when workersare riskneutral. In factother schemes also achieve this
allocation. For example, instead of playing against an opponent, a
workermightbe compared witha fixedstandard q, withone payment
awarded if output falls anywhere below _qand another, higher, paymentawarded ifoutput fallsanywhereabove standard. Attemptingto
beat the standard has the same incentiveeffectsas attemptingto beat
another player. Using the same methodsas above, it is not difficultto
show thatthere are spread-standardcombinationsthatinduce Pareto
optimum investments.Since all these schemes involve the same investmentpolicy,and since average payout by the firmequals average
product forall of them,theyall yieldthe same expected rewardsand,
therefore,the same expected utilityto workers.4
In spite of the apparent equality of these schemes in termsof the
preferences of risk-neutralworkers, considerations of differential
costsof informationand measurementmayserve to break thesetiesin
practical situations.The essential point follows from the theory of
measurement (Stevens 1968) that a cardinal scale is based on an
underlyingordering of objects or an ordinal scale. In that sense, an
ordinal scale is "weaker" and has fewerrequirementsthan a cardinal
scale. If it is less costlyto observe rank than an individual's level of
output, then tournamentsdominate piece rates and standards. On
the other hand, occupations for which output is easily observed save
resources by using the piece rate or standard, or some combination,
and avoid the necessityof making directcomparisons withothers as
the tournamentrequires. Salesmen, whose output level is easily observed,typicallyare paid by piece rates,whereas corporate executives,
whose output is more difficultto observe, engage in contests.
In a modern, complex business organization, a person's productivityas chief executive officer is measured by his effect on the
profitability
of the whole enterprise.Yet the costsof measurementfor
taking advantage of this information.However, in a sequential contest with no firmspecificcapital, the informationwould be valuable and would constrain subsequent
wage paymentsin successive rounds through competitionfrom other firms.It is not
difficultto show that this does not affect the general nature of the bond-gamble
elements or firmsadopt
solution. Alternatively,if the investmenthas firm-specific
policies thatbind workersto it (as in Lazear 1979), these restrictionsdo not necessarily
apply.
4The level of the standard is indeterminate,since for anyq a correspondingspread
can be chosen to achieve the optimal investment.This is also true of contestsamong
more than two players.WithN contestants,the prizes of N - 2 of them are indeterminate. When risk neutralityis dropped, the indeterminacyvanishes in both cases.
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849
each conceivable candidate are prohibitivelyexpensive. Instead, it
mightbe said thatthose in the runningare "tested"by assessmentsof
performanceat lower positions.Realizations fromsuch testsare sample statisticsin these assessments,in much the same way that grades
are assigned in a college classroom and IQ scores are determined.
The point is that such tests are inherentlyordinal in nature, even
though the profitability
of the enterpriseis meteredby a well-defined,
cardinal ratio scale. It is in situationssuch as this that the conditions
seem ripe for tournaments to be the dominant incentive contract
institution.
Notice in this connection that the basic prize and piece-rate structuressurvivea broad class of revenue functionsotherthan summable
ones. Even if the production functionof the firmincludes complicated interactionsinvolvingcomplementarityor substitutionamong
individual outputs, there exists the possibilityof paying workers
either on the basis of individual performanceor by rank order. The
revenue functionitselfcan even involve rank-orderconsiderations,
and both possibilitiesstillexist.For example, spectatorsat a horse race
generally are interestedin the speed of the winning horse and the
closeness of the contest. Then the firm's(track) revenue function
depends on the firstfew order statistics;yetthe horses could be paid
on the basis of theirspeed ratherthan on the basis of win,place, and
show positions. Both methods would induce them to run fast.5
There has been verylittletreatmentof the problem of tournament
prize structureand incentives in the literature.Little else but the
well-knownpaper by Friedman (1953) based on Friedman-Savage
preferencesforlotteriesexistsin economics. In the statisticsliterature
there is an early paper by Galton (1902) that is worthyof brief
discussion. Galton inquired into the ratio of first-and second-place
prize money in a race of n contestants,assuming the prizes were
divided in the followingratio:
W11W2= (Q1 - Q3)/(Q2 - Q3).
Here Qi is the expected value of the first-(fastest)order statistic,etc.
While a moment'sreflectionsuggeststhis criterionto be roughlyrelated to marginal productivity,Galton proposed it on strictlya priori
grounds.He wenton to show the remarkableresultthatthe ratioabove
5The reader is reminded that throughout this section and the next workers are
identicala prioriand differonlyex post throughthe realizationof E. In the real world,
where there is population heterogeneity,marketparticipantsare sorted into different
contests.There players (and horses, for that matter)who are known to be of higher
quality ex ante may play in games with higher stakes. If it can be accomplished, the
sorting is by anticipated marginal products. In that sense, pay differencesamong
contestantsof known quality resemble the effectof a "piece rate." These issues are
more thoroughlydiscussed below.
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is approximately3 when the parent distributionof speed is normal.
Hence, thiscriterionresultsin a highlyskewed prize structure.From
what we know today about the characteristicskew of extreme value
distributions,a skewed reward structurebased on order statisticsis
less surprising for virtuallyany parent distribution.In the more
modern statisticalliterature,the method of paired comparisons has
tournament-likefeatures. Samples from differentpopulations are
compared pairwise, and the object is to choose the one with the
largest mean. Comparing all samples to each other is like a roundrobin tournament. An alternativedesign is a knockout tournament
withsingle or double elimination.The latterrequires fewersamples
and is thereforecheaper, but does not generate as much information
as the round robin (David 1963; Gibbons, Olkin, and Sobel 1977).
Galton's original work and the more modern developments it has
given rise to are not helpful to us; theydeal withsamples fromfixed
populations, so the reward structureis irrelevantfor resource allocation.The problemwe have treatedhere is thatof choosing the reward
structureto provide the proper incentiveand elicitthe sociallyproper
distributions.
III.
Optimal Compensation with Risk Aversion
All compensation systemscan be viewed as schemes which transform
the distributionof productivityto a distributionof earnings. A piece
rate is a linear transformationof output,so the distributionof income
is the same apart froma change in locationand scale. A tournamentis
a highly nonlinear transformation:It converts the continuous distributionof productivityinto a discrete,binomial distributionof income. When workers are risk neutral, both schemes yield identical
investmentsand expected utilitybecause their firstmomentsare the
same. In thissection,it is shown thatwithriskaversionone method or
the other usually yields higher expected utility,because the interaction between insurance and action implies substantiallydifferentfirst
and second moments of the income distributionin the two cases.6
We have been unable to completely characterize the conditions
under which piece rates dominate rank-ordertournamentsand vice
versa,but we show some examples here. Truncation offeredby prizes
implies more control of extreme values than piece rates but less
control of the middle of the distribution.Differentutilityfunctions
6
One might think that risks could be pooled among groups of workers through
sharing agreements,but that is false because of moral hazard. A worker would never
agree to share prizes since doing so would resultin ,u = 0, and consequentlyE(qj + qk)
= 0 and bankruptcyforthe firm.As a result,firmsofferingtournamentsor piece rates
in the pure sense yield higher expected utilitythan the sharing arrangement.
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weight one aspect more than the other so that tournaments can
actually dominate piece rates.
Linear Piece Rate7
Optimum
The piece-ratescheme analyzed pays workersa guarantee,I, plus an
incentive,rq,wherer is the piece rate per unit of output. The problem
for the firmis to pick an r, I combination that maximizes workers'
expected utility
max [E(U) = maxf U(y)6(y)dy],
(11)
where
y =I + rq - C()
=
I + rp.+
rE
(12)
-
C(k)
and 0(y) is the pdf of y.
The worker'sproblem is to choose tkto maximize expected utility
given I and r. If E f(E), the worker's problem is
maxE(U) = f U[I + rp. +
rE
-C
(g)]f (E)dE.
The first-ordercondition is
9E(U) = f [U'(y)][r- C'(It)]f(E)dE
~49
=
0,
which convenientlyfactorsso that
r = C'(Q).
(13)
Condition (13) is identical to the risk-neutralcase, because E is independent of investmenteffort,g.
Assuming risk-neutralemployers,Vgtis expected revenue from a
workerand I + rg is expected wage payments.Therefore, the zeroprofitmarketconstraintis
( 14)
V,(A= I + rgu.
Solving (14) for I and substitutinginto (12), the optimum contract
maximizes
f U{Vg(r) + rE- C[g(r)]}f(E)dE
withrespectto r, where ,t = ,u(r)satisfies(13). Aftersimplificationthe
7 The followingis similarto a problemanalyzed byStiglitz(1975). A linear piece-rate
structureis a simplification.A more general structurewould allow for nonlinear piece
rates (see Mirrlees 1976).
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marginal condition is
[V
-
d_ EU'
C'()]
dr
+ EEU'=
0.
(15)
Since riskaversion impliesEEU' < 0, (15) shows thatV > C'(pu) in the
optimum contractfor risk-averseworkers.This underinvestmentis
the moral hazard resultingfrominsuranceI > 0 and r < V implied by
(15).
Using familiarTaylor series approximationsto the utilityfunction
and a normal densityfor E, the optimum is approximated by
c'-1( 1 + sC"o-2)
(16)
and
0_2
202
(17)
(
(1 + sC"o-2)2
-U"/U' evaluated at mean income is the measure of absowhere s
lute riskaversion. Investmentincreases (see [16]) in V and decreases
in s, C", and 0_2, because all these changes implysimilarchanges in the
marginal piece rate r which influencesinvestmentthroughcondition
(13). The same changes in V, s, and C" have correspondingeffectson
the variance of income (see [17]), but an increase in 0-2 actually
reduces variance,if 0-2 is large, because it reduces r and increasesJ*8
Prize Structure
Optimum
The worker'sexpected utilityin a two-playergame is
E(U) = P{U[W1
-
C(tk*)]} + (1
-
P){U[W2
-
C(tk*)]},
(18)
where * denotes the outcome of the contestratherthan the piece-rate
scheme. The optimum prize structureis the solution to
max (E(U*)
W1 sW2
= max{P
*
U[W1
-
C(Q*)]
+ (1 -P)
U[W2 -C(W)]})
(19)
subject to the zero-profitconstraint
V11* = PW1 + (1 - P)W2.
(20)
The workerselectsg* to satisfyoE(U)/h,* = 0. Since cost functions
are the same and Ej and Ek are i.i.d., the Nash solutionimpliespi = gk
8 Furthermore,r-V/(1 + sC"o-2)and I-sV2o-2/(1 + sC"o2)2,so thatr = V andI = 0 in
the case of riskneutrality(s = 0). All these approximationsuse first-orderexpansions
for termsin U'( ) and second-orderexpansions fortermsin U( ). The same is true of
the approximations below for the tournament.
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and P =
plifiesto
as before. Then the worker's investmentbehavior sim-
'/2
2[U(l) - U(2)]g(0)
U'( 1) + U'(2)
where U(T) UIWT - C(p*)] and U'(r)- U'[WT C(,*)]forT
Equation (21) implies
C' (u*
/I=
(21)
(1
= 1, 2.
(22)
1*(W,,W2),
and the optimum contract(W1,W2)maximizes
E(U*) =
12 U[W1 - C(,*)]
+
1/2U[W2 -C(9*)]
(23)
subject to (20), withP = '/2,and (22). Increasing marginal cost of
investmentand risk aversion guarantees a unique maximum to (23)
when a Nash solution exists. Again, assuming a normal densityforE,
second-order approximations yield
- Cr-l (1
V
)
(24)
and
V
7rVOfr;i
(1 + ITCrsC
o-2)2'
(25)
(5
2
where
ye == W, -- C (/i*) ifq j
W2 C (,*) ifq j
>
<
qk
qk
and E j- N(Oo2),Ek- N(0,o-2), and cov (Ej,Ek) = 0. The comparative
staticsof (24) and (25) are similarto the piece rate (16) and (17) and
need not be repeated.
Comparisons
Equations (16) and (24) indicate that investmentand expected incomel0are lower forthe contestthan forthe piece rate at givenvalues
of s. Moreover, for values of o.2 in excess of 1/sC"\/, the variance of
income in the tournament is smaller than for the piece rate. This
would seem to suggest that contests provide a crude formof insurance when the variance of chance is large enough, but the problem is
significantly
more complicated than that because there is no separation between tastesand opportunitiesin this problem: The optimum
9 Futhermore,C'(p.*) - g(O)(W1 - W2), so the spread is stillcrucial for investment
incentives,as in the risk-neutralcase.
10Since y = Vu CQ(g),and since ,u is below the wealth-maximizinglevel of u when
workersare riskaverse, lower,uimplieslowerybecause revenue fallsbymore than cost.
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JOURNAL OF POLITICAL ECONOMY
mean and variance themselvesdepend on utility-function
parameters.
Thus, for example, for the constant, absolute risk-aversionutility
functionU = -e-slYs, the insurance provided by the contest is insufficient
to compensate foritssmallermean: It can be shown thatthe
expected indirectutilityof the optimal piece rate exceeds that of the
optimal tournamentfor all values of o.2, at least withnormal distributions and quadratic investment-costfunctions.However, when there
is declining absolute risk aversion, we have examples where the contest dominates the piece rate.
Illustrativecalculations are shown in table 1 using the utilityfunction U = aya, which exhibitsconstantrelativebut declining absolute
riskaversion,s(y) = (1 - a)Iy. Again quadratic costsand normal errors
are assumed. However, this utilityfunction is defined for positive
incomes only,so an amount of nonlabor income yo is assigned to the
worker to avoid a major approximation error of the normal, which
admits negative incomes (i.e., the possibilityof losses).
Table 1 shows that when yo = 100 so thats = .005, the contestis
preferred until o.2 ? 3. However, if yo = 25 so that s = .020, the
contestis only preferredfor a2 < .2. The intuitionis that piece rates
concentratethe mass of the income distributionnear the mean, while
contestsplace 50 percentof the weightat one value significantly
below
the mean and the other value significantly
above. Stronglyrisk-averse
workersseem to dislike the binomial nature of thisdistributionwhen
0J2 is high because it concentratestoo much of the mass at low levels of
utility.However, when o-2 is small,the contestwhichtruncatesthe tails
TABLE 1
CONSTANT
'2
RELATIVE
RISK AVERSION
yA*
E (U)
E(U*)
5.012155
5.012150
5.012100
5.011940
5.011800
5.011420
5.012465
5.012445
5.012295
5.011925
5.011415
5.010515
2.524665
2.524616
2.524237
2.519930
2.524725
2.524575
2.523437
2.514282
Yo = 100; s(y0)= .005
.1
.5
1
3
6
12
.9995
.9975
.9950
.9852
.9710
.9436
.9984
.9922
.9846
.9552
.9142
.8420
Yo = 25; s(y0)= .020
.1
.2
1
12
-
No]
p/2:
.9980
.9960
.9807
.8094
F.-U
a2.
=
ay';y
ye
.9938
.9878
.9419
.5741
+I +rq -C(y)forpiecerate;y
=yo + W -C(yA) forcontest(i = 1, 2);a = .5, V = 1,C (?)
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855
of the income distributionassociated with a linear piece rate has
higher value.
IncomeDistributions
While it is not possible to make a general argument based on an
example, table 1 suggests that persons with more endowed income
and smaller absolute risk aversion are more likelyto prefercontests,
and those withlow levels of endowed wealth and larger absolute risk
aversion are more likelyto preferpiece rates. Consider a situationin
which all persons have the same utilityfunction,such as the one in
table 1, and face the same costs and luck distribution,the only difference being the fact that some workers have larger endowed incomes than others.If thisdifferenceis large enough, it can be optimal
to pay piece ratesto those withsmall values of endowed income and to
pay prizes to those with large values. Individuals will self-selectthe
paymentscheme in accordance withtheirwealth. The distributionof
earnings among those selecting the piece-ratejobs is normal with
mean Vg and variance r2o-2.It is binomial with mean Vp)* and variance (AW)2/4for those who enter tournaments.Note that , and A*
depend upon s(y), which is smaller for workerswho select contests,
and it can turn out as it does in table 1 thatexpected income is larger
in the contestthan in the piece rate; for example, if o-2 = 1 then the
rich prefercontests(5.012295 > 5.012100) and the poor preferpiece
rates (2.524237 > 2.523437), but /* = .9846 exceeds p. .9807. This
situationis shown in figure 1.
The overall distributionis the sum of a binomial and a normal with
lower mean, weighted by the number of individuals in each occupation (see fig. 1). It is positivelyskewed because Vp.* > Vp.. Note also
that the distributionof wage income will be less skewed than that of
total income. The reason is thatyo and mean-wage income are positivelycorrelatedbecause the likelihoodof choosing a contestincreases
withyo. These implicationsconform to the standard findingson the
distributionof income in an economy.
This example is interestingbecause it is verycloselyrelated to some
early resultsof Friedman (1953), who studied how alternativesocial
arrangementscan produce income distributionsthatcater to workers'
risk preferences. He showed that the Friedman-Savage utilityfunction leads to a two-classdistribution.Persons in the risk-averseregion
are assigned to occupations in which income follows productivity,
while persons in the risk-preferring
region buy lotteryticketsin very
riskyoccupations in which few win very large prizes. The overall
distributionis the sum of these two and exhibitscharacteristicskew.
The Friedman-Savage utilityfunction implies that a person's risk
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f(y)
.5.
Y,+ Vp
WZ+Y(
yO+VtL*
W1+yO
Income
FIG. 1
preferences depend on the part of his wealth that is not at risk.
Therefore, Friedman's assignment of people to jobs really follows
endowed wealth(yo),justas in our example. However, our framework
offerstwo improvements.First,the problem of incentivesis directly
incorporated into the formulationof the optimum policy. Second,
workersin this model are risk averse for all values of incomes, but
even so gambles can be the optimal policy.
Error Structure
Relative costs of measurementare stillimportantin choosing among
incentiveschemes,but the error structureplays additional roles when
workersare riskaverse. Suppose the output estimatorforworkeri in
activityT is qi=
qir + pT + PiT, where viT is random error and p, is an
errorthatis specificto activityi- but common to all workerswithinthat
activity.In the piece rate the common error p adds noise which
risk-averseworkersdislike, while the common noise drops out of a
rank-ordercomparison because it affectsboth contestantssimilarly.
That is, the relevantvariance for the contestis 2o-2,while thatfor the
piece rate is o-2 + o-2. It is evident thatthiscan tip the balance in favor
of tournamentsif o-2 is large enough and/or workersare sufficiently
risk averse.
The common error p bears two interestinginterpretations.One is
activity-specific
measurement error. For example, j and k may have
the same supervisorwhose biased assessmentsaffectall workerssimilarly.This is similarto monitoringall workersby a mechanical count-
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857
ing device that mightrun too fastor too slow in any given trial.The
other interpretationof p is true random variation that affectsthe
enterpriseas a whole. For example, suppose all firmsproduce with
the same technology,but that in a given period some firmsdo better
or worse than others. Then risk-averseworkers prefer not to have
their incomes vary with conditions facing the firmas a whole, and
wages based on a contesteliminatethis kind of variation.Withoutits
eliminationthere would be excessive losses due to moral hazard.
It must be pointed out that,in the absence of measurementerror,
using a contestagainst a fixed standard q discussed above has lower
variance than playing against an opponent. As shown in Section II,
the relevant variance in a contest is that of ( = Ek - Ej, which has
variance 20-2 against an opponent and only o-2 against a standard
(since the standard is invariant, Ek
0). Consequently, we might
expect risk-averseworkers to prefer absolute standards."1Again,
however,the crucial issue is the costs of measurementand the error
structure.For the complex attributesrequired for managerial positions,it is difficultto observe output and thereforedifficultto compare to an absolute standard. Insofar as samples and testsare necessary,it bears repeatingthatthese are inherentlyordinal in nature. But
thisleads us back to the problem of common error,where it is often
impossible to know whethera person's output is satisfactorywithout
comparisons to other persons. Further, when there are changing
productioncircumstancesin the firmas a whole, it is difficultto know
whetherthe person failed to meet the standard because of insufficient
investmentor because the firmwas generallyexperiencingbad times,
a problem of measuring "value added." Risk-averseworkersincrease
utilityby competingagainst an opponent and eliminatingthiskind of
firmeffect.
IV.
Heterogeneous Contestants
Workersare not sprinkledrandomlyamong firmsbut ratherseem to
be sorted by abilitylevels. One explanation for this has to do with
complementaritiesin production. But even in the absence of complementarities,sorting may be an integral part of optimal laborcontract arrangements. Informational considerations imply that
11 Playingagainst a standard is like Mirrlees's(1976) notion of an "instruction."It is
clear thatusingstandardsas well as piece ratesmustbe superiorto using one alone. That
schemewould allow workersto be paid I ifq < q and lo + rq forq -q. This is important
because it truncatesthe possibilitieswhen Vq < 0. Given the technology,it is possible
thatverylarge negativevalues of output can occur, and since it is impossibleto always
tax workersthe fullextentof thisloss, some formof truncationis desirable. A contestis
an alternativeway to control the tails of this distribution.
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compensation methods may affectthe allocation of worker types to
firms.Therefore, thissectionreturnsto the case of riskneutralityand
analyzes tournamentstructureswhen investmentcosts differamong
persons. Two typesof persons are assumed, a's and b's, withmarginal
costsof thea's being smallerthan those of the b's: CQ(pu)< CQ4) forall
pt.The distributionof disturbancesf(E) is assumed to be the same for
both groups. Many of the following results continue to hold, with
usually obvious modificationof the arguments,if the a's and b's draw
from differentdistributions.The following section addresses the
question of self-selectionwhen workersknow theiridentitiesbut firms
do not. The next section discusses handicapping schemes when all
cost-functiondifferencescan be observed by all parties.
AdverseSelection
Suppose that each person knows to which class he belongs but that
thisinformationis not available to anyone else. The principalresultis
that the a's and b's do not self-sortinto theirown "leagues." Instead,
all workersprefer to work in firmswith the best workers(the major
leagues). Furthermore,there is no pure price-rationingmechanism
that induces Pareto optimal self-selection. But mixed play is
inefficientbecause it cannot sustain the proper investmentstrategies.
Therefore, tournamentstructuresnaturallyrequire credentials and
other nonprice signals to differentiatepeople and assign them to the
appropriate contest. Firms select their employees based on such informationas past performances,and some are not permittedto compete.
The proof of adverse selection consistsof two parts. Firstwe show
that players do not self-sortinto a leagues and b leagues. Second, we
show that the resultingmixed leagues are inefficient.
1. Playersdo notself-sort.
-Assume leagues are separated and consider the expected revenue Ri generated by playingin league i = a, b
with an arbitraryinvestmentlevel g. Then
(26)
Ri(pu)= Wi + (WI,- W)Pi, i = a, b,
where (Wi ,W12)
is the prize money,and Pi is the probabilityof winning
in league i. Recall thatPi depends on the individual's level of investment and thatof his rivals.Therefore,pa = G (IL -E*)
and pb = G (a
,a*), where ua* is the existingplayers' investmentsin the a league,
where V = C'(/u4*),
and similarlyforpu. Recalling from(6) and (9) that
= V/g(O)and from (10) that W' = VI* - V12g(0),equation
I(26) becomes
Ri(,) = Vg* -
V0)[1/2-G (
-
g*)].
(27)
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Note thatRi(p) = VAluwhen , =
and thatdRi/dpu R2'(Q) = Vg(p.
=
V and C'(,u*) = V, thentz* < a* so that
- u*)/g(O) > 0. Since CQ(*)
= R'(pi). Therefore,
Rj(1_0)> Ri(Ipc). Furthermore,Rb[1, -(qa* -(l)]
V fork = p* and
Rb(pu) is a pure displacementofRa4L). Since R=*)
Ri'(t) < V elsewhere,and sinceRi(,) is increasing,the revenue functions never cross. So Rb(,U) lies to the southwestof Ra(pu)(see fig. 2).
Therefore,independent of cost curves,it is alwaysbetterto play in the
a league than the b league: Workers will not self-select.
are inefficient.
2. Mixedcontests
-Suppose the proportionsof a's and
b's in the population are a and (1 - a), respectively.If pairingsamong
a's and b's are random, then expected utilityof a player of typei is
W2 + [aPr + (1 - a)P'](W1 - W2)
-
Qbi),
where (W1,W2) is the prize money in mixed play and Pj7is the probabilitythata player of typei defeats a player of typeJ.The first-order
condition for investmentof type i in this game is
a
L
+
a+(l-a)
a
a)
i
i
] *- ( WI
-
-
W2)
=
Ci'
1i)
A development similar to Section II implies equilibrium reaction
functions
[ag(0)
+ (1
-
a)g(ia
-,b)](WI
W2) Ca(ja)
Rj(p)
Vfl
R
ib
2
FIG. 2
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for a's and
[ag (jib -')
+ (1 - a)g(O)](Wi
-
W2)
for b's. If the solution is efficient,then Cb'(Qb) = V
implies
ag(O)
+ (1 - a)g(ji;a
-/b)
ag(~i2b -a)
Cb'Vb)
=
Ca(,ua), which
+ (1 -a)g(O).
Since g is symmetricand nonuniform,thisconditioncan hold onlyifa
= 1/2.Therefore, except in thatveryspecial case, mixed contestsyield
inefficientinvestment:One type of player overinvestsand the other
underinvestsdepending upon whether or not a ] l2.
We conclude that a pure price systemcannot sustain an efficient
competitiveequilibrium in the presence of population heterogeneity
withasymmetricinformation.Marketscan be separated, but only at a
cost. Consider, for example, the case where a's want to prevent b's
from contaminatingtheir league. By making the spread, W7 - V2,
sufficiently
large, Ra(4) becomes steeper than Rb(4) in figure2 and
crosses it so that the envelope covers Rb(Q) at low values of p and
Ra(IL) at high values. Then, for some high levels of [L, it is more
profitableto play in the a league and, forlow levels of p., the b league
is preferable.Individuals may self-sort,but the cost is thata's overinvest. The result is akin to that of Akerlof (1976) and to those of
Spence (1973), Riley (1975), Rothschildand Stiglitz(1976), and Wilson (1977). As they show, a separating equilibrium need not exist,
but, even if it does, thatequilibriummay be inferiorto a nonseparating equilibrium.
The obvious practical resolution of these difficultiesis the use of
nonprice rationing and certificationto sort people into the appropriate leagues based on past performance.Similarly,firmsuse nonprice factorsto allocatejobs among applicants. The rules for allocating thosejobs may be importantfor at least two reasons thatwe can
only brieflydescribe here.
First,sortingworkersof differentskilllevels into appropriate positionswithina hierarchymaybe beneficial.In thispaper, productionis
additive, so it does not matterwho works withwhom. To the extent
that the production technologyis somewhat more complicated, sorting may well be crucial. A series of pairwise,sequential contestsmay
efficientlyperform that function. Suppose that qit = pi + 8i + qit,
where 6i is an unobserved abilitycomponent forplayeri and q is white
noise. Suppose it is efficientforthe individual withthe highest8 to be
the chief executive. There will be a tendency to have winners play
winners because
E(8j
I qjl
>
qkl)
> E(5k
qjil >
qkl)
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in the firstround. A sequential elimination tournament may be a
way to select the best person.
cost-efficient
Second, workersmay not know preciselytheirown abilitiesor cost
functions.A worker who is ignorant about his cost functionvalues
information before selecting a level of investment expenditure.
Therefore, firmsmay offer "tryouts"to provide informationabout
optimal investmentstrategies.In fact,one can imagine the existence
of firms which specialize in running contests among young
workers-the minor leagues-which provide informationto be used
when and if the workersopt to increase the stakesand enter a bigger
league.
These issues point up an importantdifferencebetween piece rates
and contests. In the pure heterogeneous case, where informationis
asymmetricand workersare riskneutral,a piece rate alwaysyieldsan
However, once slotefficientsolution, namely,V =
CA( = G(b).
tingof workersis importantbecause of complementaritiesin production, or if it is desirable for workersto gain informationabout their
type,it is no longer obvious that a series of sequential contestsdoes
not result in a superior allocation of resources.
Handicap Systems
This sectionmoves to the opposite extremeof the previous discussion
and assumes that the identitiesof each type of player are known to
everyone. Competitivehandicaps yield efficientmixed contests.
Consider again two types a and b now known to everyone. Prize
structuresin a-a and b-b tournaments satisfying(11) and (12) are
efficient,but those conditions are not optimal in mixed a-b play.
Denote the sociallyoptimal levels of investmentby /ua*and /4*,their
differenceby AA, and the prizes in a mixed league by W1and W2.Let
h be the handicap awarded to the inferiorplayer b. Then the Nash
solution in the a-b tournamentsatisfies
-~a
-ab
h)AW = QA.)
(28)
and
g
(-a
lib-
h)i\W=
CG(b)-
(The second condition in [28] followsfrom symmetryof g[~I) Since
the efficientinvestmentcriterionis V = C'(,u*) = C'(,u*), independent
of pairings, the optimum spread in a mixed match must be
A JW= V/g(A/,u - h).
(29)
From (28), condition (29) insures the proper investmentsby both
contestants.The spread is larger in mixed than pure contestsunless a
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gives b the full handicap h = a* - ib*. Otherwise, the appropriate
spread is a decreasing functionof h. Prizes W, and W2mustalso satisfy
the zero-profitconstraintW, + W2 = V * (qu* + g4) independent of h
since the spread is alwaysadjusted to induce investmentsgua*and /1t*.
The gain to an a from playing a b with handicap h, rather than
another a with no handicap, is the differencein expected prizes:
ya(h)= PW1 + (1
-P)W2
-
= pW1 + (1 -P)1/V2
C(La*)
-2[(Wy + Wa)/2 - Ct(a*)]
(30)
- (Wal + Wa2)/2,
whereYa(h) is the gain to a and P = G (A - h) is the probabilitythata
wins the mixed match. The corresponding expression for b is
Yb(h) =
(1 - P)W1 + PW2 - (Wbl+ Wb2)/2.
(31)
The zero-profitconstraintsin a-a, a-b, and b-b require that ya(h) +
yb(h)= 0 forall admissibleh. The gain of playingmixed matchesto a
is completelyoffsetby the loss to b and vice versa.
If Ca(11) is not greatlydifferentfromCb(U), then LA.t = /4*- lit is
- h). This approximation and
small and P
l/2 + [g(4g - h)](A
the zero-profitconstraintreduce (30) to
Ya(h)
V .
/
-) h
(32)
The expression foryb(h)is the same, except its sign is reversed,so the
gain to a decreases in h, and the gain to b increases in h. Therefore,h*
= gI/2is the competitivehandicap, since it impliesya(h*) = yb(h*)=
0. If the actual handicap is less than h*, then Ya is positive and a's
preferto play in mixed contestsratherthan withtheirown type,while
b's prefer to play withb's only. The opposite is true if h > h*.
A two-playergame is said to be fair when the players are handicapped to equalize the medians. The competitivehandicap does not
result in a fair game, since h* = LAtI2 < Ag. The a's are given a
competitiveedge in equilibrium,because theycontributemore to total
output in mixed matches than the b's do. This same result holds if Ea
has a differentvariance than Eb, but it may be sensitiveto the assumption of statisticalindependence and output additivity.
h can be constrainedto be zero. In thiscase, different
Alternatively,
wage schedules would clear the market. Since Ya(O) = -Yb(O)
3,
paying W1 - 3 and W2 - 3 to a's, while paying W1 + /3,W2 + /3to b's,
leaves the spread and, therefore,the investmentsunaltered. It is easy
to verifythata's and b's are stillindifferentbetween mixed and pure
contests,because expected returnsare equal between segregated and
integratedcontests for each type of player. With no handicaps, the
market-clearingprizes available to a's in the mixed contestare lower
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863
than those faced byb's. Still,expected wages are higherfora's than b's
in the mixed contest,because their probabilityof winningis larger.
The b's are given a superior schedule in the mixed contest as an
equalizing differencefor having to compete against superior opponents. This yields the surprisingconclusion that reverse discrimination, where the less able are given a head start or rewarded more
lucrativelyif they happen to accomplish the unlikely and win the
contest, can be consistentwith efficientincentive mechanisms and
mightbe observed in a competitivelabor market.
V.
Summary and Conclusions
This paper analyzes an alternativeto compensationbased on the level
of individual output. Under certain conditions,a scheme which rewards rank yields an allocation of resources identical to that generated by the efficientpiece rate. Compensatingworkerson the basis of
their relative position in the firmcan produce the same incentive
structurefor risk-neutralworkers as does the optimal piece rate. It
might be less costly,however, to observe relative position than to
measure the level of each worker's output directly.This results in
paying salaries which resemble prizes: wages which differ from
realized marginal products.
When riskaversionis introduced,the prize salaryscheme no longer
duplicates the allocation of resources induced by the optimal piece
rate. Depending on the utilityfunctionand on the amount of luck
involved, one scheme is preferred to the other. An advantage of a
contestis thatit eliminatesincome variationwhichis caused by factors
common to workersof a given firm.
Finally,we allow workers to be heterogeneous. This complication
adds an importantresult: Competitivecontestsdo not automatically
sort workers in ways that yield an efficientallocation of resources
when informationis asymmetric.In particular,low-qualityworkers
attemptto contaminatefirmscomposed of high-qualityworkers,even
if there are no complementaritiesin production. Contaminationresults in a general breakdown of the efficientsolution if low-quality
workers are not prevented from entering. However, when player
types are known to all, there exists a competitive handicapping
schemewhichallows all typesto workefficiently
withinthe same firm.
References
Akerlof,George. "The Economics of Caste and of the Rat Race and Other
Woeful Tales." Q.J.E. 90 (November 1976): 599-617.
Alchian, Armen A., and Demsetz, Harold. "Production, InformationCosts,
and Economic Organization." A.E.R. 62 (December 1972): 777-95.
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OF POLITICAL
ECONOMY
Becker, Gary S., and Stigler,George J. "Law Enforcement,Malfeasance, and
Compensation of Enforcers."J. Legal Studies3, no. 1 (1974): 1-18.
Cheung, Steven N. S. The Theoryof Share Tenancy:WithSpecial Applicationto
Asian Agriculture
and theFirstPhase of Taiwan Land Reform.Chicago: Univ.
Chicago Press, 1969.
David, Herbert A. TheMethodofPaired Comparisons.
London: Charles Griffin,
1963.
Friedman, Milton. "Choice, Chance, and the Personal Distributionof Income."J.P.E. 61, no. 4 (August 1953): 277-90.
Galton, Francis. "The Most Suitable Proportion between the Values of First
and Second Prizes." Biometrika1 (1901-2): 385-90.
Gibbons, Jean D.; Olkin, Ingram; and Sobel, Milton. Selectingand Ordering
Populations:A New StatisticalMethodology.
New York: Wiley, 1977.
Harris, Milton,and Raviv,Artur."Some Resultson IncentiveContractswith
Applications to Education and Employment,Health Insurance, and Law
Enforcement."A.E.R. 68 (March 1978): 20-30.
Lazear, Edward P. "Why Is There Mandatory Retirement?"J.P.E. 87, no. 6
(December 1979): 1261-84.
Mirrlees,James A. "The Optimal Structure of Incentives and Authority
withinan Organization." BellJ. Econ. 7 (Spring 1976): 105-31.
Riley,John G. "Competitive Signalling."J. Econ. Theory10 (April 1975):
174-86.
Ross, Stephen A. "The Economic Theory of Agency: The Principal's Problem." A.E.R. Papers and Proc. 63 (May 1973): 134-39.
Rothschild, Michael, and Stiglitz,Joseph E. "Equilibrium in Competitive
Insurance Markets:An Essay on the Economics of ImperfectInformation."
Q.J.E. 90 (November 1976): 630-49.
Spence, A. Michael. "Job Market Signaling." Q.J.E. 87 (August 1973):
355-74.
Stevens, S. S. "Measurement, Statistics,and the Schemapiric View." Science
161 (August 30, 1968): 849-56.
Stiglitz,Joseph E. "Incentives,Riskand Information:Notes towardsa Theory
of Hierarchy."Bell J.Econ. and ManagementSci. 6 (Autumn 1975): 552-79.
Wilson, Charles A. "A Model of Insurance Markets with Incomplete Information."J. Econ. Theory16 (December 1977): 167-207.