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Tournaments and Liquidity Constraints for the Agents

Kosmas Marinakis

Theofanis Tsoulouhas

Department of Economics

North Carolina State University

Raleigh, NC 27695-8110

September 4, 2008

Abstract. A celebrated result in the theory of tournaments is that relative performance evaluation (tour-

naments) is a superior compensation method to absolute performance evaluation (piece rate contracts) when

the agents are risk-averse, the principal is risk-neutral or less risk-averse than the agents and production is

subject to common shocks that are large relative to the idiosyncratic shocks. This is because tournaments

get closer to the …rst best by …ltering common uncertainty. This paper shows that, surprisingly, tourna-

ments are superior even when agents are liquidity constrained so that transfers to them cannot fall short

of a predetermined level. The rationale is that, by providing insurance against common shocks through a

tournament, payments to the agents in unfavorable states increase and payments in favorable states decrease

which enables the principal to satisfy tight liquidity constraints for the agents without paying any ex ante

rents to them, while simultaneously providing higher-power incentives than under piece rates.

Keywords: Piece rates, tournaments, liquidity constraints.

JEL Codes: D82, D21.

Acknowledgments: We are grateful to Metin Balikcioglu for comments on an earlier draft, and to

Elias Dinopoulos, Duncan Holthausen, Charles Knoeber, Wally Thurman and the participants of the 2008

conference on Tournaments, Contests and Relative Performance Evaluation at North Carolina State Uni-

versity, the 2008 c onfe rence on Research in Economic Theory and Econometrics in Naxos, Greece, and the

2008 Econometric Society European Meetings in Milan, Italy, for useful suggestions.

1. Introduction

Even though linear contracts are only a proxy of the theoretically optimal non-linear con-

tracts, they are popular in several occupations or industries (e.g., sales, physician contracts

with HMOs, contracts between processors and farmers, and faculty raises), partly because

they are simple to design and easy to implement and enforce.

1

The most common linear

contracts are the piece rate contract and the cardinal tournament. Under the piece rate

contract each agent is evaluated according to his absolute performance or according to his

performance against a predetermined standard, while under the tournament each agent is

evaluated relative to the performance of his peers. In particular, under both schemes each

agent receives a base payment and a bonus payment, but the bonus payment is determined

by absolute performance in piece rates and by relative performance in tournaments.

2

Fol-

lowing the footsteps of Lazear and Rosen (1981), Green and Stokey (1983), Nalebu¤ and

Stiglitz (1983) and others, the comparison of these two alternative evaluation methods has

been the subject of current literature (Tsoulouhas (1999), Wu and Ro e (2005 and 2006),

Marinakis and Tsoulouhas (2007) and Tsoulouhas and Marinakis (2007)). This comparison

is important because it allows us to contrast the e¢ ciency of absolute performance evaluation

against relative performance evaluation.

Absent liquidity considerations, when agents are risk averse and production is subject to

su¢ ciently large common shocks, the tournament is a superior incentive scheme to the piece

rate. This is because the tournament uses the information generated by the performance of

the group of participating agents as a whole, while the piece rate does not. Speci…cally, if

the disturbance in the output of each agent is correlated with the disturbances in the output

of the other agents, the information contained in the average production can be very useful

to the principal in creating a contract which is a step closer to the First Best. Moreover,

under the tournament, if the principal is risk-neutral or is less risk-averse than the agent,

an insurer–insured relationship can be developed between principal and agent allowing for

a Pareto improvement of the contract. That is, the principal will o¤er insurance to the

risk averse agent by …ltering away the common shock from his responsibility. Insurance will

make the agent more tolerant to a higher-power incentive scheme and, therefore, the agent

is expected to increase his e¤ort level.

1

To some extend, the non-linearity of the theoretically optimal contract is due to the fact that contracts

accomodate all possible events. Holmström and Milgrom (1987), however, have argued that schemes that

adjust compensation to account for rare events may not provide correct incentives in ordinary high probability

circumstances.

2

The base payment ensu res agent participation and the bonus provides incentives to perform. Under

tournament an agent receives a bonus if his performance is above that of his peers, and a penalty otherwise.

1

One might conjecture that the superiority of tournaments over piece rates may not

survive under liquidity constraints. Marinakis and Tsoulouhas (2007) have shown that the

optimality of tournaments over piece rates breaks down when the risk-neutral principal is

subject to a limited liability (bankruptcy) constraint, which limits the payments a principal

can make, provided that the liquidation value of the principal’s enterprise is su¢ ciently

small. This is so because tournaments increase payments in unfavorable states, but these

are the states in which the limited liability constraint comes into play. The intuition is

that contracts with risk neutrality and limited liability for the principal look very much like

those that would have been obtained with risk aversion. In other words, if the principal is

concerned about the allocation of pro…t across states, he will no longer o¤er insurance against

common shocks via tournaments and will resort to piece rate contracts or …xed performance

standards. This paper investigates the optimality of tournaments over piece rates when the

agent, instead, is subject to a liquidity constraint which introduces ex post limitations on

the minimum payment the agent can accept or the maximum penalty that can be imposed

on him (Innes 1990, 1993a and 1993b). The liquidity constraint prevents the principal from

compensating the agent by an amount smaller than a predetermined level in all states of

nature.

The models used by Lazear and Rosen (1981), Green and Stokey (1983), Meyer and

Vickers (1997) and others, allow the payments to the agents to be negative. In particular,

under both the piece rate and the tournament payment schemes, if the agents produced a

su¢ ciently low output they would usually have to pay the principal. Thus, according to

the standard literature, if the production of an agent is su¢ ciently low the principal will

penalize the agent by imposing a negative compensation and acquire whatever output the

agent produced. This is certainly inconsistent with what we observe in reality.

The liquidity constraint is partly an institutional constraint on contracts. It is imposed

by law for several industries in numerous countries. Such legislation aims at removing the

burden of excessive penalties imposed on agents for negative outcomes beyond their control,

rather than at maximizing social welfare. However, a liquidity constraint for the agent

may alter the choice the principal makes between tournaments and absolute performance

contracts. This can be due to a number of reasons. Some of these reasons are in favor of

tournaments and some are in favor of piece rates. First, by increasing payments to the agents

in unfavorable states, tournaments are more likely to satisfy tight liquidity constraints for the

agents. Second, by providing insurance, tournaments may satisfy the liquidity constraints

for the agents without paying rents to them. This is so because tournaments increase the

compensation to the agents in unfavorable states but they reduce the payments in favorable

states. By contrast piece rates may pay the agents ex ante rents when the liquidity constraints

2

are tight (i.e., when the minimum required payment to the agents is high), which reduces

the principal’s pro…t. If piece rates pay ex ante rents to the agents, they could be dominant

over tournaments from the principal’s perspective only if implemented e¤ort under piece

rates were higher. But, in general, tournaments allow the principal to implement higher-

power incentives than piece rates, which enhances the dominance of tournaments. Third,

agents may be unable to pay for insurance especially in low states of nature if the liquidity

constraints are tight, which works against tournaments. Fourth, the attitude of the principal

and the agents toward risk may change. Liquidity constraints may make the agents more

tolerant to risk, in the sense that if the agents know that their liability is limited, they may

become indi¤erent among the range of states over which the liquidity constraint is binding.

This is certainly in accord with La¤ont and Martimort (2002) who state (see p.121):

“A limited liability constraint on transfers implies higher-powered incentives for

the agent. It is almost the same as what we would obtain by assuming that

the agent is a risk lover. The limited liability constraint on transfers somewhat

convexi…es the agent’s utility function.”

On the other hand, the liquidity constraints for the agents are expected to make the principal

care about the allocation of payments and, hence, pro…t across states to satisfy the liquidity

constraints and ensure agent participation. When the principal b ecomes less tolerant to risk,

while agents simultaneously become more tolerant to risk and, therefore, they are not willing

to pay enough for insurance, the principal may …nd it suboptimal to o¤er insurance to the

agent through a tournament and may resort to piece rates again. Thus, in all, it is not a

priori clear if tournaments, which are normally superior over piece rates when production

is subject to common shocks, maintain their superiority under liquidity constraints for the

agents.

Our analysis shows that, surprisingly, in the presence of su¢ cient common uncertainty

a principal contracting with risk averse agents will prefer to o¤er a tournament even when

agents are liquidity constrained. This …nding is diametrically opposite to the result for the

case when the principal, instead, is subject to limited liability. The rationale for this result

follows directly from the discussion above. It turns out that by providing insurance against

common shocks through a tournament, so that payments to the agents in unfavorable states

increase and payments in favorable states decrease, the principal can satisfy tight liquidity

constraints for the agents without paying any ex ante rents to them while simultaneously

providing them with higher-power incentives than under piece rates. The individual ratio-

nality constraints for the agents are always binding under tournaments, whereas under piece

rates they are non-binding (that is, the agents receive ex ante rents) when the liquidity

3

constraints for the agents are really tight (that is, when the minimum payment required to

satisfy the liquidity constraints is high). This …nding establishes our claim that the principal

can satisfy tight liquidity constraints for the agents without paying any ex ante rents to

them under tournament. Our second claim, that the principal can implement higher-power

incentives under tournament, follows from the fact that the piece rate contract cannot be

de…ned for a piece rate larger than one (in the sense that the principal would not make an

o¤er such that marginal cost exceeded marginal revenue) whereas the tournament is de…ned

for a larger bonus factor. The larger the minimum payment satisfying an agent’s liquidity

constraint, the higher the power of incentives the principal provides. In other words, the

principal counterbalances the increase in the base payment, which is required to satisfy the

liquidity constraint, with higher-power incentives in order to curb agent rents and in order

to reduce the likelihood that output is low. Tournaments provide the principal with added

‡exibility in the determination of this power when the liquidity constraints are really tight.

On the other hand, regardless of whether the principal o¤ers a piece rate or a tournament,

the liquidity constraints for the agents are non-binding (that is, in some sense, agents receive

ex post rents) when the minimum payment required to satisfy the liquidity constraints is

low. In that case, the analysis is similar to the benchmark case in Lazear and Rosen (1981),

Green and Stokey (1983) and Nalebu¤ and Stiglitz (1983), and tournaments are optimal

under su¢ cient common uncertainty.

The empirical application that stems from our analysis is that …rms should adopt relative

performance evaluation via tournaments over absolute performance evaluation via piece rates

regardless of whether the agents are liquidity (wealth) constrained or not. This …nding

enhances the generality of the results obtained in Lazear and Rosen (1981), Green and Stokey

(1983) and Nalebu¤ and Stiglitz (1983). For instance, in the case of processor companies

contracting with farmers who most often are liquidity constrained, processors need not fear

that the farmers’liquidity issues detract from the superiority of tournaments.

3

Even though the issue we analyze has been largely overlooked by the current tour-

nament literature, the introduction of liquidity constraints on the agent side is not novel.

Bhattacharya and Guasch (1988) examine the e¢ ciency of tournaments with heterogeneous

agents. They argue that tournaments that are based on comparisons across ability levels

are more e¢ cient than tournaments that are based on comparisons within cohorts of similar

ability agents. However, this result is reversed when agents are subject to limited liability

(liquidity) constraints, because tournaments with comparisons across cohorts are more likely

to lead to negative payments. Kim (1997) analyzes a setting with a risk neutral principal and

a risk neutral agent when the agent’s liability is limited. He shows that the optimal contract

3

Wealth constraints can certainly be a c once rn in contracts for salesmen as well.

4

is a bonus contract in which the principal and the agent share the output, and the agent

receives an additional …xed bonus only when output is greater than some predetermined

level. Demougin and Garvie (1991) examine two forms of constraints for risk neutral agents:

non-negativity constraints for the transfers to the agents and ex post individual rationality

constraints for the agents. They show that the principal cannot implement the First Best

and agents earn informational rents. Courty and Marschke (2002) analyze a framework with

liquidity constraints and budget balancing. They show that when the di¤erence in agent

budgets is large enough, the liquidity and budget balancing constraints bind, thereby re-

ducing the e¤ectiveness of incentives. Demougin and Fluet (2003) focus on examining the

cost of providing incentives through rank-order tournaments when agents care about the

fairness of their payo¤s relative to that of others, and agents are subject to limited liabil-

ity which makes rents possible. They show that the presence of more envious contestants

reduces the principal’s cost of providing incentives, when rents must be paid, because the

agents will motivate themselves to perform even with lower rents from the principal. Kräkel

(2007) analyzes a model with risk-neutral agents who are subject to limited liability, but

face no common uncertainty in‡icted on their productive activities, to make the point that

the Lazear and Rosen (1981) …nding of equal incentive e¢ ciency for piece rates and for

rank-order tournaments does not necessarily carry over when limited liability is introduced.

In particular, piece rates dominate tournaments if idiosyncratic risk is high. This is an in-

tuitive result because, even absent limited liability, a tournament would be suboptimal by

introducing idiosyncratic noise from the activity of other agents onto the payment to any

given agent. Therefore, the introduction of limited liability should not change that, but

it should change the speci…cation of the piece rate. Namely, given risk-neutrality, limited

liability should entail a move from the "selling the enterprise to the agent" solution (i.e., a

piece rate of 1) to a piece rate of less than one, because the liquidity constraint prevents

the sale of the enterprise to the agent. We di¤er from Kräkel in a numb er of important

respects. We assume the existence of su¢ cient common uncertainty which provides scope

for tournaments. We also assume that agents are risk-averse to incorporate the insurance

aspect of tournaments, and we show that tournaments in our setting are dominant over piece

rates with or without limited liability. Last but not least, note that similar to Lazear and

Rosen (1981), Green and Stokey (1983), Nalebu¤ and Stiglitz (1983) and Malcomson (1984)

we are not looking for the optimal contract, instead, we contrast the e¢ ciency properties of

absolute to relative performance evaluation.

Section 2 presents our model, section 3 presents the benchmark case without liquidity

constraint and section 4 presents our results when the agents are liquidity constraint. Section

5 determines the dominant compensation scheme and section 6 concludes.

5

2. Model

A principal signs a contract with n homogeneous agents.

4

Each agent i produces output

according to the production function x

i

= a + e

i

+ + "

i

, where a is the agent’s known

ability, e

i

is his e¤ort, is a common shock and "

i

is an idiosyncratic shock. The idiosyncratic

shocks, "

i

; and the common shock follow independent distributions. Each agent’s e¤ort and

the subsequent realizations of the shocks are private information to him, but the output

obtained is publicly observed. The principal compensates agents for their e¤ort based on their

outputs by using a piece rate contract or a tournament. Agent preferences are represented

by a CARA utility function u(w

i

; e

i

) = exp

w

i

+

1

2a

e

2

i

;where the agent’s coe¢ cient

of absolute risk aversion is set equal to 1 for simplicity. The cost of e¤ort is measured in

monetary units. Each agent has a reservation utility exp(u).

3. Piece Rates and Tournaments without Liquidity Constraints

We start by deriving the optimal contractual variables for the piece rate and the tournament

without liquidity constraints for the agents. We assume that the total production distur-

bance, "

i

+ ; follows a normal distribution with zero mean and variance equal to c=

p

2;

and the idiosyncratic shock, "

i

; follows a normal distribution with zero mean and variance

equal to d=

p

2.

5

The piece rate contract (R) is the payment scheme in which the compensation to the

i

th

agent is w

i

= b

R

+

R

x

i

, where (b

R

;

R

) are the contractual variables to be determined

by the principal. The principal determines these parameters by backward induction. Thus,

the principal calculates each agent’s expected utility

EU

R

= exp

b

R

R

(a + e

i

) +

e

2

i

2a

+

2

R

c

2

p

2

: (1)

To ensure the compatibility of the contract with agent incentives to perform, the principal

calculates the e¤ort level that maximizes (1). First order conditions yield

6

e

i

= a

R

: (2)

4

Agent heterogenity has been examined in a number of recent papers. Konrad and Kovenock (2006)

examine discriminating contests with stochastic contestant abilities. Ganuza and Hauk (2006) analyze c om-

petition in tournaments with cost di¤erentiation among the contestants. Tsoulouhas et al (2007) consider

CEO contests that are open to heterogeneous outsider contestants. Kolmar and Sisak (2007) analyze discrim-

inating contests among heterogeneous contestants. Bhattacharya and Guasch (1988) allow for agents who

are heterogeneous ex ante. Instead, Tsoulouhas and Marinakis (2007) analyze ex post agent heterogeneity

to make the point that agent heterogeneity compromises the insurance function of tournaments. Münster

(2007) examines sabotage in a model with heterogeneous contestants.

5

As will become obvious in the remaining analysis, this assumption on the variance simpli…es the expo-

sition.

6

Note that the concavity of the utility function implies that …rst order conditions are su¢ cient.

6

To ensure the compatibility of the contract with agent incentives to participate, the

principal selects the value of the base payment, b

R

, that satis…es the agent’s individual

rationality constraint with equality so that the agent receives no rents but still accepts the

contract. The agents individual rationality constraint satis…es EU

R

= exp(u), where

EU

R

is determined by (1) and (2). Solving for b

R

implies

b

R

= u +

c

p

2

a

2

2

R

a

R

: (3)

Thus, by choosing the piece rate

R

, the principal can precisely determine the agent’s e¤ort

because the agent will optimally set his e¤ort according to (2). In addition, by setting b

R

in

accordance with (3) the principal can induce agent participation at least cost. That is, agent

incentives to perform are only determined by the piece rate

R

, whereas agent incentives to

participate are determined by the base payment b

R

.

Given conditions (2) and (3) the principal maximizes his expected total pro…t

ET

R

=

P

n

i=1

[Ex

i

Ew

i

] = n

a + a

R

c

p

2

+ a

2

2

R

u

: (4)

The solution to this problem satis…es

R

=

a

a +

c

p

2

: (5)

Condition (3) then implies

b

R

= u

a

2

2

c

p

2

+ 3a

h

c

p

2

+ a

i

2

: (6)

Given conditions (5) and (4) expected pro…t per agent is

E

R

= a +

1

2

a

2

a +

c

p

2

u: (7)

The tournament (T) is the payment scheme in which the compensation to each agent is

determined by a relative performance evaluation. Speci…cally, w

i

= b

T

+

T

(x

i

x); where

x is the average output obtained by all agents and (b

T

;

T

) are the contractual variables to

be determined by the principal. Under a tournament the agent’s expected utility is

EU

T

= exp

b

T

T

n 1

n

(a + e

i

) +

T

1

n

j6=i

(a + e

j

) +

e

2

i

2a

+

1

2

n 1

n

2

T

d

p

2

: (8)

7

The e¤ort level that maximizes (8) satis…es

e

i

=

n 1

n

a

T

: (9)

Further, the individual rationality constraint EU

T

= exp(u) implies

b

T

= u +

1

2

n 1

n

n 1

n

a +

d

p

2

2

T

: (10)

Then, given conditions (9) and (10), the principal maximizes expected total pro…t

ET

T

= n

a +

n 1

n

a

T

1

2

n 1

n

n 1

n

a +

d

p

2

2

T

u

: (11)

The solution to the principal’s maximization problem satis…es

T

=

a

n1

n

a +

d

p

2

; (12)

therefore,

b

T

= u +

1

2

a

2

a +

n

n1

d

p

2

: (13)

Given (12) and (11) expected pro…t per agent is

E

T

= a +

1

2

a

2

a +

n

n1

d

p

2

u: (14)

By comparing (4) to (14) it can easily be shown that

E

T

> E

R

,

n

n 1

d < c; (15)

that is, tournaments are superior when total uncertainty is large relative to the idiosyncratic

uncertainty (equivalently, when common uncertainty is relatively large) and when the number

of agents is large. This so because tournaments eliminate common uncertainty but they add

the average individual noise of others. It is also straightforward to show that

T

>

R

(16)

and

b

T

> b

R

: (17)

8

The rationale behind (17) is that the expected bonus payment under tournament is zero,

whereas that under piece rate is positive. Therefore, agents expect to be compensated for

e¤ort through the base payment in a tournament. The intuition behind (16) is that the

principal implements higher-power incentives when common uncertainty is removed from

the responsibility of the agent under tournament.

4. Piece Rates and Tournaments with Liquidity Constraints

Next we turn to the case with liquidity constraints for the agents. The liquidity constraint

is

w

i

w; (18)

where w is the minimum p ermissible payment. The liquidity constraints for the agents

necessitate a support for the production shocks which is bounded below and above. The

support must be bounded below so that in the worst possible output state the liquidity

constraints are still satis…ed (obviously they cannot be satis…ed with an output space which

is unbounded below). For a similar reason, the support must be bounded above to eliminate

the case when the payment under tournament is below the minimum required to satisfy

the liquidity constraint when average output is unbounded above.

7

With b ounded support

for the production shocks one might expect that the First Best is always implementable by

punishing the agent severely for outcomes outside the support (see p. 140 in Bolton and

Dewatripont (2004)). Note, however, that the liquidity constraints of the agents prevent

severe punishment of them.

The requirement of bounded support eliminates unbounded distributions such as the

normal we used in section 3 (which is typically used in the literature for the setting without

liquidity constraints). The normal distribution is necessary to obtain a closed form solution

for the case without liquidity constraints. Further, a truncated normal distribution pro-

vides neither a closed form solution nor a numerical one. However, we were able to obtain

signi…cant insight through a numerical analysis by assuming that the idiosyncratic and the

common shocks follow independent uniform distributions, in which case the sum of these

shocks follows a triangular distribution. Speci…cally, the idiosyncratic shocks, "

i

; follow in-

dependent uniform distributions with support [d; d] and, therefore, the total production

shock, v

i

"

i

+ ; follows a triangular distribution with density f(), the support of which

is assumed to be [c; c] with zero mean. The following lemmata apply to piece rates and

tournaments with liquidity constraints.

7

An alternative approach would be to consider a modi…cation of the payment schemes such that the

agent still receives the minimum payment required to satisfy his liquidity constraint, w; speci…cally, consider

maxfw; w

i

g whe re w

i

is determined by the scheme. However, the analysis in this case is intractable.

9

Lemma 1 Under piece rates, when the agents are subject to liquidity constraints in addition

to individual rationality constraints, at least one of the individual rationality and the liquidity

constraints for each agent binds depending on the values of parameters w, u, a and c.

Proof. The proof is straightforward by noting that if both constraints were non-binding,

then, the principal would reduce the payments to the agent until one of the two constraints

became binding (that is, until the agent received no rents in an ex ante or in an ex post

sense). As shown in section 3, solving without the liquidity constraint for each agent (in which

case the individual rationality constraint is obviously binding) implies that the contractual

variables (b

R

;

R

) satisfy conditions (6) and (5) and therefore the payment w

i

may or may not

satisfy the liquidity constraint in all states depending on the values of parameters w, u , a and

c. Therefore, when the individual rationality constraint is binding, the liquidity constraint is

binding or non-binding (the latter when w is relatively low). Solving without the individual

rationality constraint (in which case the liquidity constraint is obviously binding in the lowest

possible state) implies that the payments to the agent may or may not satisfy the individual

rationality constraint depending on the values of parameters w, u, a and c again. Therefore,

when the liquidity constraint is binding the individual rationality constraint is binding or

non binding (the latter when w is relatively large).

Lemma 2 Under tournaments, when the agents are subject to liquidity constraints in ad-

dition to individual rationality constraints, and assuming that the regularity condition (n

1)a > nd holds, the individual rationality constraint for each agent is always binding and

the liquidity constraint for each agent is binding or non-binding depending on the values of

parameters w, u, a and d.

Proof. First, similar to Lemma 1, the two constraints cannot simultaneously be non-binding.

Solving without the individual rationality constraint (in which case the liquidity constraint

is obviously binding in the lowest possible state) implies that b

T

= w +

T

d. This is so

because w

i

= b

T

+

T

(x

i

x) = w and, given that "

i

2 [d; d], if the number of agents is

su¢ ciently large x

i

x

D

! uniform[d; d]: Then, since the principal’s pro…t per agent is

i

= x

i

b

T

= a w +

n1

n

a d

T

+ + "

i

, it follows that expected pro…t per agent

is E

T

= a w +

n1

n

a d

T

. To maximize this expected pro…t the principal chooses

the maximum

T

that satis…es the individual rationality constraint with equality so that the

agent accepts the contract. Therefore, the individual rationality constraint is always binding.

As shown in section 3, solving without the liquidity constraint (in which case the individual

rationality constraint is obviously binding) implies that the contractual variables (b

T;

T

)

satisfy (13) and (12) and therefore the payment w

i

may or may not satisfy the liquidity

10

constraint in all states depending on the values of parameters w, u, a and d. Therefore,

when the individual rationality constraint is binding, the liquidity constraint is binding or

non binding (the latter when w is relatively low).

Note that the regularity condition (n 1)a > nd requires that agents are of su¢ ciently high

ability. The proof of Lemma 2, then, shows that the principal’s pro…t is increasing in the

bonus factor

T .

The rationale why the individual rationality constraint is always binding

for the tournament case but not for the piece rate case is that pro…t is decreasing in the piece

rate

R

. Therefore, unlike the tournament case in which the principal bene…ts by increasing

the bonus factor

T

until it yields no rents to the agent, in the piece rate case the principal

may prefer to provide the agent with rents in order to increase his pro…t. Thus, there is

a fundamental di¤erence between tournaments and piece rates in this respect, which drives

the results in our paper.

We start by analyzing the piece rate case. The piece rate scheme can be written as

w

i

= b

R

+

R

(a + e

i

+ v

i

) : As Lemma 1 indicates, the individual rationality constraint

can b e binding or not. Because of this, the procedure for determining the contractual

variables is somewhat di¤erent than the one we followed above for the case without liquidity

constraints (without liquidity constraints the individual rationality constraints are always

binding). With liquidity constraints, we determine the base payment b

R

through these

constraints, and the piece rate

R

from the pro…t maximizing condition. Then we check

whether this solution satis…es the individual rationality constraints.

Clearly, if the payment satis…es the liquidity constraint (18) in the lowest possible state,

then, it satis…es the constraint in all states because the payment scheme is increasing in the

state. Therefore, if the constraint is binding in the lowest state, then it is non-binding in all

states. From the agent’s perspective, given that the principal controls incentives through the

payment scheme, the worst state is the one in which the principal provides him no incentives

to perform and the production state turns out to be the worst, that is, e

i

= 0 and v

i

= c.

In the remaining analysis we focus on the case when the liquidity constraint is binding in

the lowest possible state.

8

Therefore, the principal will set

b

R

= w

R

(a c) : (19)

8

Recall that the liquidity constraint is an institutional constraint which prohibits penalizing the agent

for obtaining a low output, and it should hold regardless of whether the contract is optimal or not. The

agent’s optimal response under the contract should not be included in the calculation of the required base

wage, because we cannot assume the optimal contract in setting up the constraint. Instead, the cons traint

determines th e agent’s optimal response under contract and the optimal contract.

11

The expected utility for the agent is

EU

i

=

Z

c

c

exp (

R

v

i

) f(v

i

)dv

i

exp

w

R

c

R

e

i

+

e

2

i

2a

: (20)

To provide correct incentives to the agent, the principal calculates the e¤ort level e

i

that

maximizes (20). First order conditions yield

Z

c

c

exp (

R

v

i

) f(v

i

)dv

i

exp

w

R

c

R

e

i

+

e

2

i

2a

R

+

e

i

a

= 0 (21)

and, because

Z

c

c

exp (

R

v

i

) f(v

i

)dv

i

and exp

w

R

c

R

e

i

+

e

2

i

2a

cannot be equal to

zero, it follows that

e

i

= a

R

: (22)

The principal’s pro…t per agent is

i

= (1

R

) x

i

b

R

= a + a

R

a

2

R

w

R

c +

(1

R

) v

i

. Then the expected pro…t per agent is

E

R

= a + a

R

a

2

R

w

R

c: (23)

Maximizing the expected pro…t with respect to

R

yields

R

=

a c

2a

: (24)

Hence, given the contractual variables and the optimal e¤ort level for the agent, the expected

pro…t per agent is

E

R

=

5

4

a +

1

4

c

2

a

1

2

c w: (25)

Note that condition (25) indicates that the principal will make an o¤er only if w is relatively

low, otherwise production is unpro…table.

Given conditions (19) and (24), the individual rationality constraint requires

Z

c

c

exp

a c

2a

v

i

f(v

i

)dv

i

exp

w

3

4

c +

5

8

c

2

a

+

1

8

a

exp(u): (26)

Clearly, (26) may or may not hold, depending on the values of parameters w, u, a and c. If

it holds, then the contractual variables to be o¤ered by the principal satisfy (19) and (24).

If (26) do es not hold, that is, if

R

in (24) violates the individual rationality constraint, then

the individual rationality constraint is binding. In this case,

R

must be determined through

the individual rationality constraint with equality. Given (20), (22) and the density function

12

for v

i

, the individual rationality constraint is written as

Z

c

c

exp (

R

v

i

)

c jv

i

j

c

2

dv

i

exp

1

2

a

2

R

R

c

exp(w) = exp(u): (27)

Given that c > 0; (27) is equivalent to

[1 + exp (2

R

c) 2 exp (

R

c)] exp (

R

c)

2

R

c

2

exp

1

2

a

2

R

R

c

exp(u w) 1 = 0: (28)

A closed form solution for

R

is impossible to obtain from (28). As a result we have to rely on

computational methods in order to determine the piece rate values

R

which are individually

rational. Our computations proceed as follows: We derive the contractual variables from

equations (19) and (24) assuming that the liquidity constraint is binding in the lowest state

and ignoring the individual rationality constraint. Then we check if the individual rationality

constraint (26) is satis…ed by the solution (in which case it is non-binding) or if it is violated

(in which case it is binding). If (26) is found to be binding, then the piece rate

R

is derived

by the solution of (28) using a Newton algorithm and b

R

is still determined by (19). In this

case, when we have multiple solutions for

R

, we keep the one maximizing the principal’s

pro…t. If (26) is found to be non-binding we keep the solutions from equations (19) and (24).

Next, we turn to the tournament case. Recall that under the tournament the compensa-

tion to each agent is w

i

= b

T

+

T

(x

i:

x) ; which can be written as w

i

= b

T

+

T

(e

i

e) +

T

#

i

; where #

i

"

i

"; with e denoting the average e¤ort and " denoting the average

idiosyncratic shock. Given that the agents are homogeneous, the contract is uniform for all

agents and the optimal e¤ort level is equal in equilibrium for all agents. Thus, the compen-

sation to each agent can be expressed as w

i

= b

T

+

T

#

i

: As shown in the proof of Lemma

2,

b

T

= w +

T

d: (29)

Similar to piece rates, if the liquidity constraint is binding in the lowest state, then it is

non-binding in all states, because the payment under tournament is also increasing in the

state. The agent’s expected utility is

EU

i

=

Z

d

d

exp (

T

#

i

) f(#

i

)d#

i

exp

w

T

d

T

n 1

n

e

i

+

T

1

n

n

j=1

j6=i

e

j

+

e

2

i

2a

:

(30)

The e¤ort level that maximizes the agent’s expected utility satis…es

Z

d

d

exp (

T

#

i

) f(#

i

)d#

i

exp

w

T

d

T

n 1

n

e

i

+

T

1

n

n

j=1

j6=i

e

j

+

e

2

i

2a

13

T

n 1

n

+

e

i

a

= 0: (31)

Because the product of the …rst two terms in the equation cannot be equal to zero, it follows

that

e =

n 1

n

a

T

: (32)

Given Lemma 2, which states that the individual rationality constraint is always binding

under tournaments with liquidity constraints, the principal chooses the value of the piece

rate

R

that satis…es the agent’s individual rationality constraint with equality. Thus, (29)

and (32) imply

EU

i

=

Z

d

d

exp (

T

#

i

) f(#

i

)d#

i

exp

w

T

d +

1

2

n 1

n

2

a

2

T

!

= exp(u):

(33)

Note that in equilibrium x

i

x = #

i

D

! uniform[d; d], when the number of agents is

su¢ ciently large. Hence,

Z

d

d

exp (

T

#

i

) f(#

i

)d#

i

converges to

Z

d

d

exp (

T

#

i

)

1

2d

d#

i

=

exp (

T

d) exp (

T

d)

2

T

d

(34)

Then, (33) becomes

exp (

T

d) exp (

T

d)

2

T

d

exp

w

T

d +

1

2

n 1

n

2

a

2

T

!

= exp(u) ,

, exp

1

2

n 1

n

2

a

2

T

T

d

!

exp(u w) =

2

T

d

exp (

T

d) exp (

T

d)

: (35)

Clearly, similar to the piece rate case, equation (35) has no closed form solution. A solution

can only be obtained by computational methods (recall that we use a Newton algorithm).

The principal’s pro…t per agent is

i

= x

i

b

T

= aw +

n1

n

a d

T

+"

i

+: Hence, given

the optimal base payment and the optimal e¤ort level for the agent, the expected pro…t per

agent is

E

T

= a w +

n 1

n

a d

T

: (36)

where

T

can only be determined numerically by solving (35).

5. The Dominant Contract Under Liquidity Constraints

The principal’s decision about which compensation scheme to o¤er depends entirely on ex-

14

pected pro…ts. Clearly, under both schemes, expected pro…ts decline when a liquidity con-

straint is introduced in addition to the other constraints. Our analysis indicates that these

pro…ts decline faster under piece rates as the liquidity constraint becomes tighter. The in-

tuition behind our result is that the liquidity constraint distorts the agent’s incentives to

perform because it reduces the penalty the principal can impose for unfavorable outcomes.

Therefore, the principal needs to provide higher-power incentives. By …ltering common

shocks from the responsibility of the agent, tournaments make the agent more tolerant to

higher-power incentives, hence, it is easier for the principal to implement higher-power incen-

tives under tournament than under piece rates. Moreover, the piece rate

R

cannot exceed 1

(i.e., because marginal cost cannot exceed marginal revenue). By contrast, the bonus factor

T

can exceed 1 which enables the implementation of higher-power incentives.

Figure 1 illustrates that tournaments are dominant over piece rates when liquidity con-

straints are introduced. In particular, panel (a) shows that expected pro…t is always strictly

larger under tournament regardless of the value of w, that is, regardless of how tight the

liquidity constraint is. Note that in our numerical analysis we assume that condition (15)

holds, that is, we assume that common uncertainty is su¢ ciently large relative to the idiosyn-

cratic uncertainty. For the case without liquidity constraints expected pro…ts per agent are

calculated by using conditions (7) and (14). For the case with binding liquidity constraints

expected pro…ts per agent are calculated by using condition (23) where

R

is determined

either by (24) or by the numerical solution of (28), and condition (36) where, again,

T

is

determined numerically by (35). Obviously, for the range over which the liquidity constraint

is non-binding, expected pro…t is ‡at and independent of w. We con…rmed this result for all

possible values of common uncertainty that satisfy condition (15). A su¢ cient increase in

the minimum permissible wage w decreases the expected pro…t under both schemes, but it

does so much faster under piece rates. In fact, piece rates cannot be de…ned at all after a

critical value of w is passed (see point B in panel (a)), because the principal needs to o¤er

a piece rate larger than 1 to provide correct incentives to the agent. However, given that

R

cannot exceed 1, piece rates cannot be de…ned.

9

In interpreting the results depicted in

Figure 1, note that for w in the range up to A the individual rationality constraint under

piece rates is binding and the liquidity constraint is non-binding. For w in the AB range the

individual rationality constraint under piece rates is binding or non-binding and the liquid-

ity constraint is binding. Under tournaments, the individual rationality constraint is always

binding (see Lemma 2). Lastly, for w in the range up to C the limited liability constraint

under tournament is non-binding.

Panel (b) indicates that the base payment is always larger under tournament, but it

9

In other words, the principal does not …nd it pro…table to make an o¤er that the agent will accept.

15

Figure 1: The expected pro…t p er agent and the contractual variables for the piece rate

contract and the tournament.

16

increases with w, that is, when the minimum acceptable payment increases the base payment

must also increase to provide correct incentives to the agent to participate. Further, panel

(c) indicates that both the piece rate

R

and the bonus factor

T

increase when w increases.

There are two reasons for this: First, because the base payment increases when w increases,

the principal must provide the agents with higher-power incentives in order to exert more

e¤ort and make up in lost pro…t due to the increase in the base payment. Second, when w

increases, the principal provides the agents with higher-power incentives in order to minimize

the likelihood that output is low and the principal is forced by the liquidity constraint to pay

the minimum acceptable wage to the agent when, absent the constraint, it would have been

optimal to pay less or impose a penalty. Again, note that piece rates are not de…ned for a piece

rate above 1, whereas under tournaments the principal can continue to provide incentives to

the agents through a bonus factor larger than 1, which explains the increased dominance of

tournaments over piece rates for large values of the minimum acceptable payment w.

6. Conclusion

A familiar result in the principal-agent literature is that when agents are risk averse and

production is subject to relatively large common shocks the tournament is a superior com-

pensation scheme to the piece rate. The superiority of tournaments over piece rates may not

survive under liquidity constraints. Prior research (for instance, Marinakis and Tsoulouhas

(2007) for limited liability on the principal) would lead someone to expect the same result

even when limited liability is imposed on the agent instead of the principal. In addition,

one might also expect that limited liability would make the agents more tolerant to risk (in

the sense that liquidity constraints convexify the agent’s utility function) and the principal

less tolerant to risk (in the sense that the principal cares about the allocation of payments

across states in order to satisfy the liquidity constraints). The reduced interest of agents in

getting insurance, as well as the reduced ability of the principal to provide it, might also

diminish the scope for tournaments. However, there is a fundamental di¤erence between

limited liability on the principal side and limited liability on the agent side. Under limited

liability for the principal, the agents cannot be suckered by the prospect of payments the

principal cannot make, therefore, the principal introduces a constraint to provide correct

incentives to the agents. The constraint puts a maximum on the payment to the agents in

low states and, hence, the solution looks like that if the principal were risk averse. Under

liquidity constraints for the agents, instead, the agents will not sign a contract with the

principal unless it satis…es these constraints and the principal incorporates the constraints

to make sure that the agents participate. The constraints put a minimum on the payments

to the agents in low states. Our analysis builds on this fact to show that in the presence

of common uncertainty a principal contracting with risk averse agents will prefer to o¤er a

17

tournament even when agents are liquidity constrained.

The rationale for our result is that by providing insurance against common shocks

through a tournament, so that payments to the agents in unfavorable states increase and

payments in favorable states decrease, the principal can satisfy tight liquidity constraints for

the agents without paying any ex ante rents to them while simultaneously providing them

with higher-power incentives than under piece rates. The larger the minimum payment

satisfying an agent’s liquidity constraint, the higher the power of incentives the principal

provides. In other words, the principal counterbalances the increase in the base payment,

which is required to satisfy the liquidity constraint, with higher-power incentives in order

to curb agent rents and in order to reduce the likelihoo d that output is low. Tournaments

provide the principal with added ‡exibility in the determination of this power.

18

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20