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Promotions as work incentives.


Why do professional partnerships like law firms, accounting practices, and management consulting groups to name a few, rely almost exclusively on promotions for incentives and do not typically use incentive pay to motivate their associates? We offer a possible explanation for this observation that is based upon the fact that these firms face similar capital constraints. That is, when capital is required in professional partnerships the funds are usually obtained internally from the firm's partners since partnerships are legally prohibited from incorporating and selling shares to the public. It is this common capital restriction which may explain the widespread use of promotion-based contests in this class of firms.(1)

We argue that principals cannot effectively set compensation rules without at least some knowledge of the production environment. While such knowledge is costly to acquire, it is jointly produced while performing supervisory services. Supervision may also provide information on the states of the world, knowledge which is important for the design of compensation mechanisms. It appears, therefore, that principals are likely to be active participants in supervision rather than passive bystanders who delegate all production responsibilities. This improves the efficiency of owner-managed firms relative to that of firms owned by passive shareholders.

Most papers in the literature, such as Lazear and Rosen |1981~, Nalebuff and Stiglitz |1983~ and Holmstrom |1982~, concentrate on the moral hazard arising from the inability of the passive principal to observe workers' effort. We allow for a similar moral hazard on the part of the supervising principal. Generally, within the type of organization considered here it is at least as difficult for workers to observe supervisory effort as it is for the supervisor to observe worker's effort. Hence, both workers and their supervisor are subject to moral hazard which must be resolved by the appropriate choice of compensation scheme.

Lazear and Rosen |1981~, Green and Stokey |1983~ and Nalebuff and Stiglitz |1983~ have shown that where the agents and principal are risk neutral, both piece rates, which are related to an individual's measured output, and tournaments, which pay a monetary prize to the highest producers, yield first-best results. In contrast, when workers are risk averse, neither mechanism is efficient. The relative efficiency of each compensation mechanism has been shown to depend upon the relative magnitude of the common errors (in the measurement of total group output) and the individual errors (associated with the measurement of the individual's effort), on the one hand, and the degree of workers' risk aversion on the other hand.

We believe such comparisons between economic contests and piece rates are unduly restrictive and may obscure some important elements in the theory of compensation. Both tournaments and individualistic piece rates are imperfect instruments which omit different pieces of relevant information. Tournaments ignore information on quantitative differences among participants, while individualistic piece rates ignore information about the group.

It is quite straightforward to show that the existence of common error and risk aversion cannot explain the prevalence of tournaments in these firms. Relative piece rates, for example, will reduce the variance in compensation due to errors in the measurement of effort associated with group output without the additional variance associated with the discrete nature of the tournament reward. Risk averse employees would, therefore, prefer relative piece rate schemes to tournaments. What we see, however, is the widespread use of contests as compensation mechanisms in the type of firms examined in this paper. To explain this phenomenon, it is necessary to focus on the often ignored fact that most contests in these firms take the form of promotions to partnership which in turn entails supervisory responsibilities. Indeed, while Lazear and Rosen |1981~ motivate their analysis by the observation that the difference in pay of chief executives and their immediate subordinates seems to be greater than the difference in their abilities, or marginal products, they (and everyone else) proceed to analyze the tournament as if prizes are purely monetary, rather than promotions.

This tying of prizes to supervisory positions raises an immediate question. It is clear that, in general, such a link may reduce the efficiency of the system by reducing the degrees of freedom available to the firm. In particular, if the firm is free to determine separately the level of supervision and the number and size of prizes, it cannot do worse than it could by tying supervision to prizes. Why then do firms choose to tie promotions and prizes together?

We argue that supervisory tournaments may arise in owner-managed firms as a result of limits on the availability of capital to potential supervisors. These limits create the possibility of bankruptcy and, hence, induce a potentially serious moral hazard problem, particularly for the supervisor. Limitations on worker's capital reinforce the results, but in the interests of simplicity we exclude them. The moral hazard arises directly as a result of the possibility of bankruptcy. Most papers in the literature, in contrast, implicitly assume that the principal is sufficiently endowed, or has sufficient access to capital, to overcome the problem of bankruptcy.(2)

Recall that in the partnerships examined here, an outside capitalist cannot supply the necessary bond that would eliminate this moral hazard problem. In any case, the introduction of outside shareholders reduces the size of the residual claimed by managers and introduces incentive problems. See, for example, Jensen and Meckling |1976~.

Once the possibility of bankruptcy is introduced, each of the schemes examined is shown to be inefficient under conditions of risk neutrality. We then show that both relative-input-based piece rates and a monetary prize tournament yield the same results for any level of capital endowment of the supervisor. However, since the promoted employee is a partner in the firm, the liability of the firm (to pay her salary) is significantly reduced by tying the prize to the supervisory position.

Of course, a promotion contest may also be an efficient selection mechanism when workers have different abilities. However, in this paper we concentrate on the tournament as a device to motivate effort. Consequently, we choose to illustrate our point with equal ability workers.

The conversion of a cash prize to a residual claim reduces the bankruptcy risk of the firm and, in turn, the capital requirement of the supervisor. This suggests that supervisory contests are superior to piece rate schemes and monetary prize tournaments when firms face capital constraints and may explain their popularity in professional partnerships. Finally, we demonstrate that the implementation of supervisory tournaments requires the existence of firms to insure continuity over time and to internalize intergenerational conflicts.

Section II describes the basic model and compares output and input based schemes. Section III analyzes the capital requirements for the alternative mechanisms and discusses the nature of the supervisory tournament. Finally, section IV summarizes the results and suggests further extensions.


We consider an economic environment in which production takes place in teams composed of a supervisor and n workers. Examples include an audit team within an accounting firm, the information systems group within a management consulting company, or the litigation group within a law firm. To simplify the analysis, n is assumed to be exogenously determined. There are no substantial changes to our results if we allow the number of workers to be endogenously derived within the model. Following the literature, all workers are assumed to be identical with utility functions separable in income (y) and effort (e).(3) In order to focus on the unobscured comparison of the alternative compensation mechanisms, we assume risk neutrality with respect to income:

(1) U = y - U(e), |U.sub.e~ |is greater than~ 0, | |is greater than~ 0.

Workers are assumed to work for two periods, after which time they retire. Supervisors must be experienced workers, and as such they have a residual claim on the output of the period in which they supervise.

It is convenient to express the production function for each group as

|Mathematical Expression Omitted~

where Q is net group output (after payments to other factors). Note that Q |is less than~ 0 is possible since Q is defined as net group output, after payments to other factors of production. The variable q is output per unit of effort extended by each worker, a quasi-concave function dependent upon the supervisor's effort devoted to coordination (c) and the number of workers in the group (n); and |Epsilon~ is a random element,(4) reflecting production and market conditions with E|Epsilon~ = 0 and ||Sigma~.sub.|Epsilon~~ = |Sigma~ for all

|Mathematical Expression Omitted~

The random element |Epsilon~ is not directly observable by anyone either prior to or after production, although the supervisor may be able to guess at it using the private information obtained in the course of supervision. Any individual worker's level of effort is observed only imperfectly by other team members. Although the supervisor's effort level cannot be observed by workers, the supervisor may be able to observe workers' effort subject to a random error |v.sub.i~ which is equal to the difference between each worker's measured effort (|e.sub.i~|prime~) and actual effort (|e.sub.i~) with |Mathematical Expression Omitted~, Ev = 0, ||Sigma~.sub.v~ = |Sigma~(n) and |Delta~||Sigma~.sub.v~/|Delta~n |is greater than or equal to~ 0.(6)

Equation (2) captures the group production aspects which are critical to the analysis of team production. In contrast to other team production models, such as Holmstrom |1982~, which assume complementarity among team members, we assume that team complementarity evolves through the efforts of the supervisor, i.e., through coordination. The basic argument is that communication among members of a team is costly in terms of the time and effort required to reach a consensus, where differences in information or opinions may exist among members. One of the principal jobs of the supervisor is to channel information in an efficient way and resolve possible disputes by assuming responsibility for decisions about the mode of production and allocation of tasks--hence, the complementarity between supervisory effort and number of workers.(7)

As any experienced worker can become a supervisor, competition among them must insure that, in the absence of rationing, they receive the same utility as they would have received as workers. In setting up groups, supervisors must compete with each other to attract workers by offering the best deal possible to the workers in their group. Within accounting firms, for example, the partner in charge of auditing competes with the tax partner as well as other accounting firms for new staff. We assume, moreover, that employees require a positive subsistence wage level. Thus, the problem facing supervisors is to choose a method of compensation for workers, and themselves, which will maximize workers' welfare subject to the production function and the constraint that supervisor's welfare must at least equal that of workers.

The approach taken is to first compare an input-based relative-piece-rates scheme with a tournament offering a monetary prize. These particular compensation mechanisms are chosen as "strawmen" because of their popularity in the literature and among firms in the real world. We show that these two schemes are both socially optimal and yield the same capital requirements on the part of the supervisor in order to overcome the moral hazard problem. When these schemes are then compared to a tournament in which the prize is a promotion, we show that the promotion tournament has a smaller capital requirement than either relative piece rates or a monetary prize tournament. Consider the following two methods of payment to workers.(8)

(a) A linear piece rate scheme where individual j's income (|y.sub.j~) is based upon measured individual effort less the average of the measured effort levels of the other workers in the group, i.e.,

|Mathematical Expression Omitted~

where a represents the fixed component of the wage payment; b is the piece rate per unit of effort; (e.sub.j~|prime~ is the measured effort of worker j and |Mathematical Expression Omitted~ is the average measured effort of the other workers in the group. A relative piece rates mechanism is chosen as the benchmark for comparison because the supervisor has an incentive to systematically underestimate all workers' effort levels as long as her own effort level and that of other workers is not observable by any individual worker. Including the average measured effort of the other workers in the remuneration scheme removes the incentive for the supervisor to bias downward her estimates of workers' effort.(9)

Inclusion of the average group effort also eliminates the common error for the group, removing the statistical inefficiency of individually based piece rates. Note as well that the relative piece rate scheme satisfies the sufficient statistic requirement for efficiency discussed by Holmstrom |1982~.(10) It is, therefore, a particularly useful benchmark for comparison.

(b) A tournament in which the worker who is credited with exerting the most effort is rewarded by a monetary prize of M.(11) That is,

(4) |y.sub.j~ = f + |P.sub.j~M

where f is a fixed payment (or entry fee if negative) and |P.sub.j~ is the probability of worker j winning the tournament with

|Mathematical Expression Omitted~

where g = g(v) is the density function of the measurement error, v, and G is the probability that |e.sub.i~ - |e.sub.j~ + |v.sub.i~ |is less than or equal to~ |v.sub.j~ for any given |v.sub.i~. At the symmetric equilibrium, |P.sub.j~ = 1/n(12)

|Mathematical Expression Omitted~

Maximizing the expected utility of workers, evaluated at the symmetric equilibrium, where |e.sub.i~ = |e.sub.j~ = e, yields the first-order condition for workers' effort supply under tournaments, i.e.,

(6) sM = |U.sub.e~;


(7) |Mathematical Expression Omitted~

is the marginal increase in the probability of winning the prize by increasing own effort above that of all other workers. This probability depends on the variance of the effort measurement error |Mathematical Expression Omitted~ and, hence, on the number of workers (n).

When the supervisor is a residual collecting principal and the workers are the agents, the solutions to these problems are straightforward and well known.(13) Compensation schemes based on inputs, i.e., on individually measured effort, whether cardinal (linear piece rates) or ordinal (monetary tournaments), yield the socially optimal results. With each scheme, all individuals receive their socially optimal levels of compensation with the marginal productivities of workers' and supervisor's effort equal to their respective dis-utilities.


The results summarized above ignore the capital requirements of the supervisors and, by doing so, the possibility of bankruptcy. These capital requirements constitute bonds which must be available in the event that negative cash flows accrue to the supervisor (recall that net group output can be negative). As Holmstrom |1982~ has shown, these bonds are essentially a way of breaking the budget constraint, so that the sum of the marginal payments to effort by all workers and the supervisor may exceed their collective marginal output. This allows the workers and their supervisor to be compensated in a way which reflects the externalities of their effort, without exceeding their expected output.

The necessary capital requirements may be very substantial and if the supervisor does not possess the required capital, bankruptcy is possible in some states of the world. The risk of bankruptcy by any member of the group creates a serious moral hazard problem, which in turn prevents attainment of the socially efficient outcome.(14) In the appendix we derive the first-order conditions for the piece rates and monetary prize reward schemes where the capital resources of supervisors are insufficient to avoid bankruptcy in some states of the world.

The capital resources available to the supervisor include the value of the group's output which is subject to a random error, |Epsilon~, and the supervisor's own initial capital endowment. This sum must be sufficient to cover the aggregate wage bill in order to avoid bankruptcy. In bad states of the world (i.e., a large negative value of |Epsilon~), these resources will be insufficient, and bankruptcy will occur. Given a capital endowment of K for the supervisor, we define |Epsilon~* for input piece rates and |Epsilon~|prime~ for the tournament such that for all |Epsilon~ |is less than or equal to~ |Epsilon~*(|Epsilon~|prime~) supervisors are bankrupt. The error terms |Epsilon~* and |Epsilon~|prime~ are defined respectively by the following budget constraints for the supervisor:

|Mathematical Expression Omitted~


(9) qne + |Epsilon~|prime~ + K - nf - M = 0.

When bankruptcy does occur, the supervisor loses K and workers share equally the amount qne + |Epsilon~ + K.(15)

The optimal effort levels for the supervisor and workers are determined by the solution to the following conditions for the piece rate system (with b replaced by M in equation (11) for the monetary prize tournament) together with the respective first-order condition for optimal workers' effort:

(10) |q.sub.c~neF + |U.sub.c~


(11) n(q - |U.sub.e~)|e.sub.b~ + (|q.sub.c~ne - |U.sub.c~)|c.sub.b~ = 0


|Mathematical Expression Omitted~

is the probability of no bankruptcy.

It follows immediately that, for any bankruptcy probability (F), the results are identical for the two systems. Furthermore, when there is a positive probability of bankruptcy, coordination effort falls short of its socially optimal level as does workers' effort.(16)

The reason for this inefficiency is the moral hazard of both the supervisor and workers which is induced by the possibility of bankruptcy. This is a result of the fact that in bankruptcy states, supervisor's effort is not rewarded at all and workers receive only partial compensation for their additional effort. The problem can be resolved for workers by increasing the marginal piece rate, or prize, in order to restore the expected marginal return to effort to the optimal level, while reducing the fixed component of wages. However, limits on the financial resources of the supervisor prevent the imposition of an appropriate penalty on the supervisor that would prevent shirking. Any penalty of this nature must be imposed in states of the world in which output is low, because only in such states can shirking by the supervisor be detected. Since these states already induce bankruptcy, any additional penalty would be meaningless.

The moral hazard problem cannot be overcome through borrowing by the supervisor

since the original conditions for bankruptcy remain unchanged, only the identity of the creditor changes. Likewise, outside shareholders reduce the residual claimed by the supervisor and, therefore, her incentives since she is no longer entitled to 100 percent of the residual earnings. In either case, the supervisor's moral hazard problem is exacerbated, not eliminated.(17)

It follows that social efficiency can be maintained only if capital constraints on the supervisor are not binding. The differences in capital requirements to avoid bankruptcy between the two reward schemes are, therefore, likely to be important determinants of the optimality and relative desirability of these schemes. Capital requirements are calculated so that, in the worst case, income plus capital requirements are non-negative. Denoting the supervisor's capital requirements using input based piece rates as |C.sub.s~ and using a monetary prize tournament as |K.sub.s~, we have (refer to the appendix)

|Mathematical Expression Omitted~

Notice that the supervisor's capital requirements are identical for input based piece rates and tournaments with monetary prizes, because both schemes entirely eliminate the effects of group errors on payments to workers.(18)

By tying the prize to the supervisory position, the contest offering only a monetary reward can be significantly improved upon in terms of the supervisor's capital requirements. It is straightforward to show, moreover, that given any capital endowment, a promotion contest yields superior results to both an ordinary monetary prize tournament and an input-based piece rate scheme, because the effective capital constraint is less binding.(19) Intuitively, this follows because while both the pure monetary and promotion contests entail rents to motivate increased worker effort, the promotion prize rent is retained within the firm and, therefore, reduces the probability of bankruptcy.


Up to this point we have not considered the intergenerational incentives associated with promotion tournaments. An important feature of this reward mechanism is that it ensures the continuity of the firm through time. In order to address this issue, let us consider a new firm composed of a group of inexperienced workers and an experienced (as a worker) supervisor recruited from among experienced workers of an existing firm. Assume all workers are homogeneous with respect to their ability and wealth, and each one lacks sufficient capital to avoid bankruptcy if made a supervisor. The group then forms a contest in which a promotion to supervisor together with a prize is given to one member of the group at the end of the period, i.e., the group promises to make one of themselves a supervisor during the next period with a rent of M.(20)

Therefore, in the next, or second, period, the supervisor has a claim on additional capital of M which is held in trust by the firm. It is important that the prize won at the end of the first period not be withdrawn by the supervisor until after she has completed her supervisory tenure. If M is sufficiently large, this eliminates the possibility of her bankruptcy and, hence, eliminates any moral hazard on her part.

During the next period, groups composed of experienced workers who did not win the contest and a supervisor who won the contest in the previous period are, consequently, free from the risk of the supervisor's bankruptcy and can choose any method of compensation. Thus, the promotion tournament among inexperienced workers serves the dual purpose of motivating effort in the first period and insuring efficiency of production in the second period by removing the bankruptcy constraint in that period.

However, the promise of promotion does not solve the bankruptcy problem of the supervisor of the inexperienced workers in the first period. Unless these workers can find a wealthy supervisor, they must either operate inefficiently or bribe the supervisor to supply optimal effort. In either case, their utility falls short of that attainable with a wealthy supervisor.

A wealthy supervisor is therefore valuable. Young workers can create a wealthy supervisor costlessly by promoting two workers rather one, with prizes adjusted so as leave work incentives and expected lifetime income unchanged. At the end of the first period, workers have two wealthy supervisors--one available for themselves and the other to supervise a team of new first-generation workers.

Assuming a constant or increasing population of workers, which seems quite reasonable when talking about lawyers or accountants, the next generation of new employees may join an existing firm with a newly promoted supervisor, or they may form a group with a supervisor chosen from among the experienced workers who have not been promoted. The utility attainable is higher with the promoted supervisor than with the supervisor chosen from the ranks of workers not promoted. In subsequent periods, young workers would, therefore, be required to pay an entry fee into the established firm. This fee is used to compensate the older workers who "own" the promoted supervisor.

Essentially, any prizes in a given generation (in addition to the internal promotion prize to supervise the same group in the next period) creates a wealthy experienced worker, who can become an efficient supervisor in future generations. Thus, such a prize constitutes an externality from one generation to the next.

We can think of the firm as an institution which spans generations of workers and thus internalizes these externalities. It forms the necessary intergenerational link whereby each generation of new workers benefits from the promotional prize system instituted by its predecessors. The firm guarantees that the supervisors of the current group of new workers are the winners of the competition held by the previous generation of workers.

The resulting wage profile yields low wages for young workers, higher wages for older workers and additional prizes (rents) to supervisors. This picture corresponds closely to the observations made by Lazear |1981~, that older workers receive wages in excess of their marginal productivity while younger workers receive wages below the value of their marginal product. It should be noted that payment of the prize is part of the compensation of the supervisor in the second period and, consequently, it is held internally by the firm as a bond to prevent shirking by the supervisor. This is frequently the case in law and accounting firms where compensation to new junior partners takes the form of partnership shares, the value of which depends on the future performance of the firm.

What are the necessary conditions for the existence of such a firm? The answer to this question depends upon the relationship between the size of the optimal prize in a two-prize promotion contest (|M.sup.*~) relative to the capital requirement of supervisors (|C.sub.s~ = |K.sub.s~). We distinguish between two cases: |M.sup.*~ |is greater than or equal to~ |C.sub.s~ and |M.sup.*~ |is less than~ |C.sub.s~.

The first case, in which the optimal prizes exceed the supervisors' capital requirements, calls for a pure promotional tournament structure for the firm. In this situation the monetary reward associated with the promotion is sufficient to eliminate the possibility of supervisor bankruptcy. In the second period either a monetary prize or piece rates may be used since the experienced workers exit the labor force after that period.

The second situation is one in which the optimal prizes are too small to remove the supervisors' bankruptcy risk; the optimal solution is unattainable. If M* |is less than~ |C.sub.s~, prizes cannot be increased to eliminate bankruptcy without eliciting too much work. The solution must involve a compromise between the removal of moral hazard on the part of the supervisor and the disutility of excessive effort by workers. The precise nature of the second-best contracts is beyond the scope of this paper and must be left for future research.


Economic contests for job promotions are a prevalent means of eliciting effort from workers and supervisors in new firms and professional partnerships like law firms and accounting practices. In spite of the restrictive nature of these mechanisms and the fact that they do not utilize all available information, promotion tournaments are widespread among these types of firms. In attempting to explain this phenomenon, we have compared various compensation schemes and have derived the conditions under which a promotion tournament is superior to the alternative compensation mechanisms examined. The types of firms analyzed are characterized by group production, risk neutrality, potential moral hazard by both workers and the supervisor and possible wealth constraints on the part of workers and the supervisor.

We find that under risk neutrality and in the absence of binding wealth constraints, input based piece rates and monetary prize contests, not surprisingly, yield socially efficient outcomes. Tournaments and relative piece rates, moreover, are found to be indistinguishable in terms of their capital requirements for the supervisor. When the possibility of supervisor bankruptcy is allowed, monetary prize tournaments and piece rates are also shown to be equally inefficient. By tying prizes to supervisory positions, the capital requirements of the supervisor are significantly reduced. The conversion of a cash prize to a residual claim reduces the bankruptcy risk of the firm and, therefore, the capital requirement of the supervisor. This improvement makes supervisory tournaments generally superior to the other schemes.

The difficulty in implementing promotion tournaments, however, requires the existence of firms (capitalists) to insure continuity over time and to internalize potential intergenerational conflicts. This leads to a possibly important rationale for the existence of firms.

Of course, the reason we suggest may not be the only explanation for the use of promotion contests in owner-managed firms. Throughout our discussion we have assumed that all workers are alike, including their supervisory capabilities. If this is not the case, tournaments may simply be an efficient selection mechanism. Supervisory tournaments may then serve a dual purpose--incentive and selection. The more homogeneous are the young lawyers hired by a law firm, for example, the greater is the incentive motive for employing promotion tournaments in the firm. Although the factor which is most responsible for the widespread use of these contests is purely an empirical consideration, it can be argued that both large law firms and accounting firms use sophisticated screening mechanisms that result in very homogeneous classes of new hirees, lending some support for the incentive structure of the promotion contests used in these organizations.


a. Input based piece rates

|Mathematical Expression Omitted~

such that

i. bF + (1-F)q/n = |U.sub.e~;

ii. |q.sub.c~neF = |U.sub.c~

iii. |Mathematical Expression Omitted~

We recall that e|prime~ = e + v and define F as the probability of not going bankrupt, i.e.

|Mathematical Expression Omitted~


|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

Constraint i represents the workers' condition for optimal effort; ii is the supervisor's optimal effort condition; and iii is the supervisor's budget constraint with Z |is less than~ 0 implying bankruptcy. It follows:

(A.1) |Delta~L/|Delta~a = 1 - |Lambda~ - |Lambda~n = 0;

as |Mathematical Expression Omitted~. Since |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~, it follows that

(A.2) |Delta~L/|Delta~b - n(q - |U.sub.e~)|e.sub.b~ + (|q.sub.c~ne-|U.sub.c~)|c.sub.b.~ = 0.

From |Delta~L/|Delta~|Lambda~,

|Mathematical Expression Omitted~.

Capital requirements (defined where F = 1) for the supervisor are then:

|Mathematical Expression Omitted~

b. Monetary prize tournament

|Mathematical Expression Omitted~

such that

i. sMF = |U.sub.e~;

ii. |q.sub.c~neF = |U.sub.c~

iii. Z = qne + |Epsilon~|prime~ + K - nf - M = 0

(A.3) |Delta~L/|Delta~f = 1-|Lambda~-|Lambda~n = 0;

as |Mathematical Expression Omitted~. Since |Mathematical Expression Omitted~, |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~,

(A.4) |Delta~L/|Delta~M = n(q - |U.sub.e~)|e.sub.M~ + (|q.sub.c~ne - |U.sub.c~|c.sub.M~ = 0.

From |Delta~L/|Delta~|Lambda~, f = a - M/n where a is defined in part a. It follows that the supervisor's capital requirement (defined where F = 1) is

|Mathematical Expression Omitted~

c. Restricted Capital

Given |Epsilon~* = |Epsilon~|prime~, we have, from the definition of bankruptcy under piece rates and tournaments, that K* - K|prime~ = na - nf - M.

From the constraint defining equal utilities for the supervisor and workers,

(n+1)Fnf = qne|(n+1)F-1~ -(n+1)(1-F)K|prime~ + |Epsilon~(n+1)+n|U(e)-U(c)~

and (n+1)an = qne|(n+1)F-1~ -(n+1)(1-F)|K.sup.*~ + |Epsilon~(n+1)+n|U(e)-U(c)~.

As equations A.1-A.2, defining the optimal parameters of the piece rate scheme, are identical to equations A.3-A.4, defining the corresponding optimal parameters for a tournament, it follows that

K* - K|prime~~ = (1-F)(K|prime~ - K*)/(n+1)F = 0.

Therefore, given K* = K|prime~, there exists a solution where all relevant variables are equal for both systems. Assuming concavity of the Hamiltonian, these solutions are unique and, therefore, for any capital endowment (K* = K|prime~), the corresponding effort levels of supervisors and workers are identical.

1. This paper does not explain the case in which a worker is promoted, assumes more responsibility and is paid a higher salary as compensation for the additional responsibilities undertaken. Moreover, it does not explain the common up-or-out promotion systems within professional partnerships since we focus here only on the incentive properties of the contest and not the selection/screening properties that would be required to explain this behavior within our model.

2. One exception is Eswaran and Kotwal |1989~ who use the limited liability of the entrepreneur to explain why the owners of capital actively participate in the operations of the firm. Another paper that does not assume away the bankruptcy issue is Sappington |1983~ who introduces limits on the ex post liability of the agent and examines the properties of the contracts that emerge between the principal and risk neutral agent in this situation.

3. The assumption of identical workers simplifies the analysis without compromising the results. If several ability groups are present, self-selection into the appropriate tournament can be obtained through a judicious choice of prizes and monitoring. See, for example, O'Keeffe et al. |1984~.

4. We assume additive errors for simplicity. The qualitative results remain unaltered if the errors are multiplicative.

5. The logic of our argument is unchanged in the absence of a lower bound on |Epsilon~, but the presentation would be much more complicated without any substantial change in results.

6. The supervisor's coordination effort is assumed to yield a basic minimum level of information on workers' effort as a side product, so that ||Sigma~.sub.v~ is finite. For simplicity, we ignore monitoring effort by the supervisor. It is straightforward to show that, in the absence of bankruptcy, no monitoring is optimal since this activity has no social value within the framework adopted in this paper. Proofs are available from the authors.

7. In order to simplify the exposition, we assume constant returns to individual effort. This does not affect the results. It is implicitly assumed throughout the analysis as well that the workers' participation constraint is not binding, i.e., workers expect to receive at least their reservation wage by working for their firm.

8. We do not consider piece rates based upon group output since it is straightforward to show that this scheme is not efficient. Holmstrom |1982~ has shown that in the absence of a binding budget constraint and the presence of an independent external supplier of capital group-output-based incentives are efficient. In our model, we have ruled out an external supplier of funds which results in the group output scheme being inefficient even without capital restrictions on workers and the supervisor.

9. Payments to workers are

|Mathematical Expression Omitted~

Therefore payments are unaffected by the measurement error, v. For a discussion of this problem, see Carmichael |1983~, Malcomson |1984~, Bhattacharya |1987~, and Mookherjee |1984~. Mookherjee also points out that relative payment schedules are vulnerable to agent collusion. This problem can be resolved by making payments conditional on group output, as well as relative input. For example, in the piece rates case above, the payment scheme |Mathematical Expression Omitted~ will eliminate a collusive outcome.

10. That is, basing compensation on any additional variables only adds random noise to the compensation mechanism without improving its efficiency.

11. Although we specifically consider only a single prize among the general set of n-tuple wage tournaments, since relative piece rates can be approximated by a tournament with many different prizes, our results are quite robust.

12. The group size (n) must be sufficiently small to satisfy the "no complete shirking" assumption, i.e. f - U(|e.sub.j~ = 0) |is less than~ f + M/n - U(|e.sub.j~ = e*).

13. The interested reader may obtain a copy of the proofs from the authors.

14. An interesting variation of the piece rate scheme examined here is one in which each worker is paid her effort times total net output divided by total observed effort of the group. While this scheme balances the budget constraint, worker's wages can still be negative since net output can be negative. Hence, bankruptcy is still possible since workers require non-negative wages. The likelihood of bankruptcy is, however, reduced as compared to the piece rate scheme considered here. We thank the anonymous referee for pointing this out.

15. We assume

|Mathematical Expression Omitted~

so there are no other unsatisfied creditors. This simplifies the analysis without affecting the results.

16. From (10) it is clear that F |is less than~ 1 implies too little coordination effort being extended relative to the socially optimal level when F = 1. Substituting (10) into (11), n(q - |U.sub.e~)|e.sub.b~ = -(1-F)|q.sub.c~ne|c.sub.b~. Since |c.sub.b~ |is less than~ 0 and |e.sub.b~ |is greater than~ 0, it follows that q - |U.sub.e~ |is greater than~ 0, implying insufficient effort extended by workers.

17. This suggests that wealthy workers have an advantage in becoming supervisors, because they are able to post the required bond to prevent insolvency. However, we assume there is an insufficient number of rich workers to form all firms. Limits on managers' capital are clearly important in the formation and success of new firms. Unless entrepreneurs have some internal start-up capital, it is extremely difficult for them to generate external financing.

18. When the supervisor's capital endowment is insufficient to prevent bankruptcy, the two systems yield equal bankruptcy probabilities (|Epsilon~* = |Epsilon~|prime~) and identical effort levels for both supervisors and workers (see appendix, part c, for proof).

19. Adding the capital constraint to the Lagrangean expression in the appendix, section b, yields the additional term |Mu~(|K.sup.*~-K) where |K.sup.*~ is the supervisor's initial capital endowment and |Mu~ is the shadow price. Differentiating with respect to K then yields |Delta~L/|Delta~K = -|Mu~ |is less than or equal to~ 0, for a maximum, after substitution from A.3. This yields a positive shadow price for capital when the constraint is binding and, hence, indicates an increase in workers' welfare associated with an increase in capital.

20. As shown by Ferrall |1991~, rents which more than compensate for higher ability appear to be common in law firms.


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Author:Berkowitz, M.K.; Kotowitz, Y.
Publication:Economic Inquiry
Date:Jul 1, 1993
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