A Theoretical and Empirical Comparison of Free Agent and Arbitration-Eligible Salaries Negotiated in Major League Baseball.

Phillip A. Miller [*]

This paper presents a theoretical and empirical comparison of determination of negotiated salaries in baseball's free agent market to that in its final-offer arbitration (FOA) system. The theoretical bargaining model of each system is based on Nash (1950). It is argued that Farber's (1980) model of FOA is not fully applicable in explaining baseball's FOA process. The free agent market and the arbitration system each determine negotiated salaries that are dependent on the different disagreement outcomes of the negotiators in the respective systems. It is thus concluded theoretically that the two systems will determine salaries differently. The theoretical analysis also suggests that there may be selection bias present when one empirically analyzes only negotiated settlements. Using a straightforward regression model that controls for productivity, playing experience, and the potential selection bias, the empirical analysis substantiates the theory's results. A method is then developed to estimate the effect th at free agent salaries have on salaries for arbitration-eligible players. It is found that there is a significant positive relationship between them, but the systems do not determine equal salaries for comparable players.

1. Introduction

Final-offer arbitration (FOA) is a dispute resolution system in which disputants give their final offers to a third party who then chooses one of the offers as the binding settlement. The most public example of FOA is that found in Major League Baseball.

The labor market for major league baseball players is unique in that it is a three-tiered market. All players with fewer than six years of experience are subject to the reserve clause. However, 17% of two-year players and all players with at least three years of playing experience in the major leagues are eligible for salary arbitration. Players with six or more years of experience are eligible for free agency.

As Vrooman (1996) has argued and others (e.g., Zimbalist 1992) have shown empirically, the free agent market, as structured, has given rise to player monopoly power. Zimbalist argues that this monopoly power may arise because each free agent may have unique abilities that the team that signs him needs (p. 93). Zimbalist argues that this, coupled with competitive pressures, may lead to additional risk taking by teams that results in free agents being paid above their marginal revenue product (p. 94).

Conversely, the reserve clause has caused the labor market for players who are not arbitration eligible to be monopsonistic and, as Vrooman argued, allows teams to recoup their player development expenses. However, during the 1994-1995 players' strike, team owners proposed to allow players to move between teams after four years rather than six years in return for the removal of player rights to binding arbitration after three years. This proposal indicates that team owners believe that arbitration may raise player salaries such that teams in general are unable to recoup all their player development costs. This is most likely through the mechanism by which arbitrators make their decisions. In deciding a particular case, arbitrators are allowed to base their decisions on salaries of other players with comparable productivity to the particular player involved. This raises two interesting questions: (i) Is there a difference between the structures of negotiated salaries of free agents and arbitration eligibles? ( ii) What is the effect of the former on the latter?

In economics, the literature on FOA is vast, as is that on Major League Baseball. This paper contributes to and complements both bodies of literature by providing a theoretical and empirical comparison of negotiated salaries determined in baseball's free agent system to those determined in its FOA system. To my knowledge, there is no research developing a formal model describing the bargaining behind free agency that also compares it to a formal bargaining model describing salaries negotiated in baseball's FOA system. The present paper does this.

Moreover, it is argued that Farber's (1980) model of FOA is not fully applicable to explain baseball's arbitration system, and thus a more adequate model is described. In addition, the theoretical model of arbitration is used to explain how free agent salaries affect salaries negotiated within the arbitration system. Then, using a straightforward empirical model and readily available productivity and salary measures, the paper presents regression estimates of the relationship between player characteristics and the salaries of free agents and arbitration-eligible players and examines the effect the salaries of free agents have on those negotiated by arbitration eligibles.

The paper is organized as follows. Section 2 presents the theoretical models. Section 3 describes the empirical models and methodology. Section 4 presents the variables used as regressors, the data, and the empirical results. Section 5 discusses and concludes.

2. The Theory

This section presents theoretical models of bargaining that result in negotiated salaries in baseball's FOA system and in its free agent system. Note that I do not analyze the situations in which a particular case was decided by an arbitrator. We will return to this point later.

Baseball's FOA system consists of two stages: a noncooperative initial stage of bargaining in which negotiators set final offers to be presented to an arbitrator in case a dispute ensues, and a cooperative second round of bargaining in which the negotiators bargain over the salary to be paid to the player (see Faurot and McAllister 1992; Miller 1998). If the negotiators do not arrive at a voluntary settlement in the second round, the parties go to arbitration. Even if they arrive at a negotiated salary, it is influenced by their final offers since these in part determine the disagreement outcome if a dispute occurs.

There is no direct analogue to this initial noncooperative round in the free agent system. The bargaining in this system is essentially a one-stage cooperative process in which the player and the team bargain over the salary to be paid to the player. However, when a disagreement occurs in free agency between a particular player and team, the player may sign with another team or may leave the league, perhaps playing in another league (such as the minor leagues) or leaving baseball altogether. Since the disagreement outcome will affect negotiated salaries and because the disagreement outcomes are different, it is likely that the two systems will determine negotiated salaries differently.

However, negotiated salaries in the two systems are interconnected. In deciding a particular case, the arbitrator can consider salaries of players with similar abilities and productivity to the player in question, including those of comparable free agents. Consequently, the disagreement outcome in FOA and, thus, salaries negotiated in the second stage of this system will be influenced by salaries negotiated for free agents. It is therefore important to formally analyze and compare negotiated salaries determined in the free agent system to those determined in the arbitration system.

Bargaining and Salaries Negotiated in the Free Agent System

As noted previously, the bargaining that occurs between teams and free agents is essentially cooperative. The bargaining model employed here is thus based on that of Nash (1950).

For simplicity, consider a two-team league with teams 1 and 2. Let the free agent earn the team gross benefits of B. If the player receives a wage of W, the team receives a surplus of B -- w. Assume that each negotiator is risk averse [1] and has increasing preferences over the share of the benefits he receives. Let the preferences of the player and the two teams over shares of the pie be represented by, respectively, [U.sub.p] and [[U.sup.i].sub.t] (i = 1, 2), continuously differentiable, strictly increasing and concave von Neumann and Morgenstern utility functions. Note that for either team, [[U.sup.i].sub.t] is decreasing in w. Assume that the player and the team are utility maximizers and each negotiates in his own behalf. Assume that the player is negotiating with team 1.

The Nash solution for free agents is characterized by

[[w.sup.*].sub.FA] = [arg.sub.[w.sub.FA]] max [U.sub.p][[w.sub.FA] - [d.sub.p])([[U.sup.1].sub.t][B - [w.sub.FA]] - [[d.sup.1].sub.t]) (1)

where [[w.sup.*].sub.FA] is the negotiated salary received by the free agent; [d.sub.p] and [[d.sup.1].sub.t] are the disagreement outcomes for the player and team 1 and the outcomes should the player not sign with this particular team, respectively.

In free agency, the disagreement outcome for the player is the utility from not being signed by the team in question. This will include the utility of outside opportunities such as playing for another team or of leaving the league (perhaps signing a minor-league contract or playing in a foreign league). For the team, the disagreement outcome is the utility of not signing the player in question: the value of not having the player on the team. Essentially, it is the value to the team of signing another player to take the place of the unsigned player. We assume that the team's utility is not a function of having the other team signing the player in question.

First consider the player's disagreement outcome, [d.sub.p]. If the player and team 1 do not reach an agreement, then the player can possibly sign with team 2 for [w.sub.2]. If he is unable to sign with team 2, then assume he will work outside the league for a salary of [[w.sup.p].sub.r]. The term [[w.sup.p].sub.r] can be interpreted as a reservation wage arising from playing in the minor leagues or in another independent league or by working outside of baseball. Consequently, [w.sub.2] [greater than] [[w.sup.p].sub.r].

When the player is negotiating with team 1, he is not precluded from informally negotiating with team 2. These latter negotiations are important in that they can be used to gather information about the reservation price of team 2 for the player. Assume that the player uses the negotiations for this purpose. Let this information be summarized by the continuously differentiable probability distribution G(.) with density g(.). Let [[w.sup.2].sub.r] denote team 2's reservation price for the player, a random variable from the player's viewpoint. Now, we have

pr(not sign with 2) = pr(work for [[w.sup.p].sub.r]) = pr([[w.sup.2].sub.r] [less than] [w.sub.2]) = G([w.sub.2]).

Therefore, the player's disagreement outcome is given by [2]

[d.sub.p] = [1 - G([w.sub.2])][U.sub.p]([w.sub.2]) + G([w.sub.2])[U.sub.p]([[w.sup.p].sub.r]). (2)

It is assumed that [w.sub.2], B, and [[w.sup.p].sub.r] are exogenously determined and are known to both the player and team 1.

Now consider team l's disagreement outcome [[d.sup.1].sub.t]. If the team does not sign the player, then it must substitute another player for him. Assume that this substitute player generates a surplus of s for team 1. Consequently, the disagreement outcome of team 1 is given by [[d.sup.1].sub.t] = [[U.sup.1].sub.t](s). Note that for the parties, negotiating a settlement must give no less utility than [d.sub.p] and [[d.sup.1].sub.t], respectively. Also note that information held by the player and team 1 is perfect, symmetric, and complete. It is also uncertain when G(.) [neq] 1.

Substituting [[d.sup.1].sub.t] from (2) into (1) gives

[[w.sup.*].sub.FA] = [arg.sub.[w.sub.FA]] max ([U.sub.p]([w.sub.FA]) - {[1 - G([w.sub.2])][U.sub.p]([w.sub.2]) + G([w.sub.2])[U.sub.p]([[w.sup.p].sub.r])})([[U.sup.1].sub.t][B - [w.sub.FA]] - [[U.sup.1].sub.t][s]). (3)

Maximizing the right-hand side over [w.sub.FA] yields the first-order condition

[U'.sub.p]([w.sub.FA])[[[U.sup.1].sub.t](B - [w.sub.FA]) - [[U.sup.1].sub.t](s)]

+ [[U.sup.1]'.sub.t](B - [w.sub.FA])([U.sub.p][[w.sub.FA]] - {[1 - G([w.sub.2])][U.sub.p][[w.sub.2]] + G([w.sub.2])[U.sub.p][[[w.sup.p].sub.r]]}) = 0. (4)

Note that the second-order condition for a maximum is satisfied with the given assumptions. Therefore, Equation 4 implicitly defines the function

[[w.sup.*].sub.FA] = [w.sub.FA](B, [w.sub.2], [[w.sup.p].sub.r], s). (5)

Standard comparative static analysis reveals that [[w.sup.*].sub.FA] is increasing in B and [[w.sup.p].sub.r] and decreasing in s. However, this analysis does not reveal how [[w.sup.*].sub.FA] changes as [w.sub.2] changes (see Appendix 1 for a proof). This result occurs because a change in the level of [w.sub.2] has two offsetting effects on the player's disagreement outcome: It will increase the utility of the player if he indeed signs with team 2 but decreases the probability that he signs with that team. The counterexample derived in Appendix 1 suggests that the probability effect outweighs the utility effect when the spread between [w.sub.2] and [[w.sup.p].sub.r] is large and vice versa.

Bargaining and Negotiated Salaries in FOA

As indicated previously, the arbitration-eligible bargaining process is essentially a two-stage process. In the first stage, the negotiators choose final offers noncooperatively. In the second stage, the negotiators bargain cooperatively to reach a negotiated settlement. If they are unable to reach a settlement in the second stage, then the case goes to an arbitrator who chooses one of the final offers as the binding settlement.

In this paper, we are concerned primarily with the second stage of the arbitration process. Consequently, we will model the second stage with only brief comments regarding the first stage. For further analyses of this bargaining model, the interested reader is directed to Crawford (1982) and Miller (1998).

Unless specified otherwise, all the previously made assumptions hold. However, additional assumptions and new notations are needed. Let [w.sub.t] and [w.sub.p] be the final offers set by the team and the player, respectively. Once the final offers are chosen, they may not be altered.

Assume that the negotiators have common information about the arbitrator's preferences. Let the arbitrator's preferences be represented by his preferred settlement, [w.sub.a], which is assumed to be determined exogenously. [3]

Let the negotiators' information about [w.sub.a] be represented by the continuously differentiable cumulative distribution denoted by F(.) with corresponding density f(.). As in Farber (1980), it is assumed that the arbitrator chooses that offer closest to [w.sub.a]. Thus, since optimally [w.sub.t] [leq] [w.sub.p], the probability the arbitrator chooses [w.sub.t] is given by F[([w.sub.t] + [w.sub.p])/2].

The Cooperative Bargaining Process in Arbitration

In the second stage, the negotiators have already set their final offers, which both observe. Information in this stage is symmetric, complete, and perfect. If they have not voluntarily settled when the stage ends, they give the final offers to an arbitrator who renders a decision as described previously. If they do voluntarily settle, the payoffs in the second stage are just the utilities evaluated at the negotiated wage. Note that if the negotiators reach a voluntary settlement, there is no uncertainty in the second stage. Thus, in this framework, the Nash bargaining solution that characterizes negotiated settlements in the second stage is given by

[[w.sup.*].sub.AE] = [arg.sub.[w.sub.AE]] max ([U.sub.p]([w.sub.AE]) - [d.sub.p])([U.sub.t](B - [w.sub.AE]) - [d.sub.t]), (6)

where [[w.sup.*].sub.AE] is the negotiated salary that the arbitration-eligible player receives; [d.sub.p] and [d.sub.t] are the disagreement outcomes of the player and team, respectively, and their respective expected utilities from proceeding to arbitration. These are given by

[d.sub.p] = F[lgroup][frac{[w.sub.t] + [w.sub.p]}{2}][rgroup] [U.sub.p]([w.sub.t]) + [1 - F[lgroup][frac{[w.sub.t] + [w.sub.p]}{2}][rgroup]][U.sub.p]([w.sub.p])

[d.sub.t] = F[lgroup][frac{[w.sub.t] + [w.sup.p]}{2}][rgroup] [U.sub.t](B - [w.sub.t]) + [1 - F[lgroup][frac{[w.sub.t] + [w.sub.p]}{2}][rgroup]][U.sub.t](B - [w.sub.p]). (7)

Note that in Farber's (1980) model of FOA, the negotiators choose their final offers to maximize their respective expressions in Equation 7. In the present model, negotiators choose their final offers to maximize the expression on the right-hand side of Equation 6. Consequently, [partial][d.sub.p]/[partial][[w.sup.*].sub.p] [neq] 0, [partial][d.sub.t]/[partial][[w.sup.*].sub.t] [neq] 0, and at least one of the negotiators sets a different equilibrium final offer than in the Farber model. [4] Consequently, the Farber model is not an appropriate model to fully explain baseball's FOA process.

Maximizing the right-hand side of Equation 6 over [w.sub.AE] yields the solution to the second stage, with the first-order condition given by

[U'.sub.p]([w.sub.AE])[[U.sub.t](B - [w.sub.AE]) - [d.sub.t]] - [U'.sub.t](B - [w.sub.AE])[[U.sub.p]([w.sub.AE]) - [d.sub.p]] = 0. (8)

The second-order condition is given by

[U".sub.t](B - [w.sub.AE])[[U.sub.p]([w.sub.AE]) - [d.sub.p]] - [2U'.sub.t](B - [w.sub.AE])[U'.sub.p]([w.sub.AE]) + [U".sub.p]([w.sub.AE])[[U.sub.t](B - [w.sub.AE]) - [d.sub.t]] [less than] 0.

Note that

[U.sub.p]([w.sub.AE]) - [d.sub.p] [geq] 0 and [U.sub.t](1 - [w.sub.AE]) - [d.sub.t] [geq] 0

since the utility from reaching a negotiated settlement cannot be less than the expected utility of disagreeing for either negotiator. Consequently, the second-order condition is satisfied at any nondisagreement solution.

Therefore, Equation 8 implicitly defines the function

[[w.sup.*].sub.AE] = [w.sub.AE][[d.sub.t]([w.sub.t], [w.sub.p]),[d.sub.p]([w.sub.t], [w.sub.p])]. (9)

Standard comparative statics show that [[w.sup.*].sub.AE] is increasing in [d.sub.p] and decreasing in [d.sub.t].

Before proceeding, it is necessary to consider the possibility that, in either model, the negotiators may choose to disagree. For example, in the model of FOA, in the face of high negotiating costs, the negotiators may find it optimal to forgo bargaining and instead opt for an arbitrated solution. The Nash bargaining models described previously have disagreement outcomes below the utility possibility frontier because of the parties' risk aversion and do not incorporate negotiating costs. Thus, the models predict that there will always be a negotiated solution and, consequently, do not describe the processes that cause disagreements. Myerson (1984) overcomes this problem by introducing incomplete information about rivals' preferences. However, this is not the primary concern here. Since arbitrated outcomes sometimes occur in practice, I note that disagreements in the context of the present models can occur because of (unmodeled) negotiating costs.

A Comparison of Bargaining Outcomes

One can now compare the Nash solution given by Equation 5 to that derived in the previous section for the FOA system (Eqn. 9). By comparing Equations 5 and 9, under the given assumptions, salaries negotiated in the two systems will differ because the disagreement outcomes are determined differently. The disagreement outcome for free agents is a function of [[w.sup.p].sub.r], [w.sub.2], and s. That of arbitration eligibles is a function of the arbitrator's preferences and the negotiators' expectations regarding them. Since these preferences are determined by the criteria that the arbitrator is allowed to consider (described in section 3), the disagreement outcome is determined differently from that of free agents. This is not to say that salaries negotiated in the two systems are determined by different sets of variables but rather that they put different weights on the variables.

The Theoretical Relationship between Free Agent Salaries and Arbitration-Eligible Salaries

It has been argued that increases in free agent salaries will cause the salaries of arbitration eligibles to increase. For example, Andy MacPhail, president of the Chicago Cubs, was concerned that the contract given to Gary Sheffield by the Florida Marlins prior to the 1997 season would increase player salaries. It was shown previously that the model for free agents is consistent with this observation when the players are substitutes for one another. We now show that a similar result holds for the arbitration-eligible model.

We assume that the player and the team have common information on comparable salaries, information that is also complete, perfect, and certain. This is not an unrealistic assumption to make since all player salaries are made public each year. We also assume that [partial][Ew.sub.a]/[partial][w.sub.c] [greater than] 0, where [Ew.sub.a] is the expected value of [w.sub.a] and [w.sub.c] is a representative comparable salary. We also assume that the player and team are equally risk averse. This implies that F[([w.sub.t] + [w.sub.p])/2] = 1/2, an implication consistent with observed practice (Faurot and McAllister 1992).

Initially assume that [w.sub.c] increases before the setting of final offers. Now, since both negotiators have common information on [w.sub.a], an increase in [w.sub.c] will cause them to update their expectations on [w.sub.a]. Since they are equally risk averse, to keep the probability that either offer is chosen at 1/2, they will each set their final offers higher than they would if [Ew.sub.a] had not increased. This implies that [partial][w.sub.e]/[partial][Ew.sub.a] [greater than] 0 and [partial][w.sub.f]/[partial][Ew.sub.a] [greater than] 0, increasing [d.sub.p] and decreasing [d.sup.t]. That is [[w.sup.*].sub.AE] is increasing in [d.sub.p] and decreasing in [d.sub.t] implies that [partial][[w.sup.*].sub.AE]/[partial][w.sub.c] [greater than] 0.

Note that if the change in [Ew.sub.a] occurs after the setting of final offers, [partial][[w.sup.*].sub.AE]/[partial][w.sub.c] [greater than] 0. In this case, the probability that the arbitrator will choose [w.sub.p] increases. This in turn increases [d.sub.p] and decreases [d.sub.t], giving the same result. Note that this latter result does not depend on a comparison of the levels of negotiator risk aversion.

3. The Empirical Model

Faurot and McAllister (1992) described baseball's FOA system, so it will not be completely described here. The interested reader is directed to that paper.

In baseball, arbitrators are allowed to consider six general factors in determining salaries: (i) the quality of the player's prior season's performance, (ii) the length and consistency of the player's career, (iii) the player's past compensation, (iv) the recent performance of the player's team, (v) comparable baseball salaries, and (vi) any physical or mental disabilities the player might have. In the empirical analysis, it is implicitly assumed that these general factors that determine arbitration-eligible salaries are also the same general factors that determine salaries in the free agent market.

Two regressions will be run, one for arbitration eligibles and one for free agents. In the regressions, the structures of negotiated salaries will be estimated. A Chow test will be performed to test for equality of the salary structures in the two systems. However, even if the two salary models are shown not to be equal in terms of structures, then it could still be the case that the two result in similar salary levels for comparable players. Consequently, we will investigate further whether the arbitration system and the free agent system determine significantly different salary levels. The method used to examine this is described next.

4. The Data and Results

This paper uses proxies for the five general criteria listed previously to estimate the salary structure for arbitration eligibles and free agents. These proxies are described here.

Players included in the data set were those who filed for arbitration or free agency immediately prior to each of the baseball seasons from 1991 through 1994. Salary data, arbitration filing data, and some free agent filing data were obtained from various issues of USA Today and The Sporting News. Additional free agent filing data were obtained from various issues of Baseball America's Almanac (1993, 1994). Pro-rata signing bonuses are included in the salary figures. Incentive bonuses are not included because they were not available from the sources from which the data were obtained. Since salaries for a given player in a given year are determined before the season begins, these offers are a function of the player's expected performance for his team, expectations formed on the basis of performance in previous years. Consequently, this paper uses player salary data for the years 1991-1994 and player productivity and team data for 1990-1993. Individual productivity data were obtained from The Sports Encycloped ia: Baseball (Neft and Cohen 1994) and from Total Baseball (Thorn, Palmer, and Gershman 1995). Since the structures of salaries negotiated for pitchers and position players are determined by very different factors, the empirical analysis is separated into two general sections: one for position players and one for pitchers.

Position Players' Data

Position players are all nonpitchers on a team roster, including designated hitters. A position player's productivity and consequently his worth to his team is measured by the number of runs he produces offensively and saves defensively (Scully 1989).

The basic equation estimated for position players is

In SAL = [eta] + [[vartheta].sub.1]ln PREVSAL + [[vartheta].sub.2]RC + [[vartheta].sub.3]FR + [[vartheta].sub.4]MID + [[vartheta].sub.5]WPCT + [[vartheta].sub.6]CGINV + [[vartheta].sub.7]NL + [[vartheta].sub.8]1991 + [[vartheta].sub.9]1992 + [[vartheta].sub.10]1993 + [[vartheta].sub.11]LAMBDA + [psi],

where ln SAL is the logarithm of a player's current year's salary and ln PREVSAL is the logarithm of the previous year's salary and captures the effects of offensive and defensive performance of all previous years. For example, ln PREVSAL for a player who had his 1993 salary determined by free agency or by the arbitration system captures his performance for the years up to and including 1991. In addition, ln PREVSAL can also serve as an additional proxy for the consistency of a player's career: The more consistent his career contribution, the higher his salary.

The term RC is runs created, a statistic developed by Bill James (Thorn, Palmer, and Gershman 1995) to measure the number of runs a player offensively contributed to his team. There are several versions of this statistic, and the use of one version or another depends on the availability of various statistics to the researcher. The version used in this study is

RC = A.B/C 

where A = hits + walks - caught stealing, B = total bases + 0.55.stolen bases, and C = atbats + walks.

FR is a statistic that measures the number of runs a player saved defensively above league average at that position. A negative value indicates the player contributed below average defensively. It helps capture a player's ability to save runs and therefore his ability to defensively help his team win games.

MID is a dummy variable equal to one for catchers, shortstops, and second basemen and is equal to zero for all other position players (including those used primarily as designated hitters). A player was given a value of MID = 1 if he played more than 50% of the games in which he appeared at the catching, shortstop, or second-base position or combination thereof. These players are often those playing primarily for defensive purposes, and their values to their teams (and therefore their salaries) may not be adequately captured by their offensive statistics.

CGINV is the inverse of a player's career games played up to the prior season. It serves as a proxy for years of experience as well as the length and consistency of the player's career. It also controls for unobservable characteristics acquired with experience such as name recognition by fans. [5] A quadratic form of career games played was tried but was not found to be significant.

WPCT is the prior season's winning percentage for a player's team and serves as a measure of the team's performance. In the cases where a player played for more than one team, the WPCT in the sample observation is that of the team for which the player had the most at-bats.

Player salaries increase each year, and, consequently, there is an upward salary trend that is observed between subsequent years. To control for the trend and to control for other year-specific effects on salaries, three dummy variables are used in the regressions: 1991 = 1 if the year for which the salary was determined was 1991, 0 if otherwise; 1992 = 1 if 1992, 0 if otherwise; and 1993 = 1 if 1993, 0 if otherwise.

As described previously, we will compare the structures of negotiated salaries between free agents and arbitration eligibles. However, if the negotiators cannot negotiate an agreement, we do not observe a negotiated settlement. For these negotiators the potential settlement lies below the disagreement outcome. Consequently, by restricting our attention to negotiated settlements, there is a potential selection bias that arises. Heckman (1979) argues that not controlling for nonrandom selection introduces an omitted-variable bias into the regression. He argues that including the inverse Mills ratio in the regression equation controls for this bias. Therefore, an inverse Mills ratio, designated LAMBDA, is calculated for each player.

As noted previously, in baseball, arbitrators are allowed to consider the salaries of comparable baseball players. This makes an empirical comparison of free agent salaries and those determined through the arbitration systems difficult: What salaries does an arbitrator consider, and how do we control for them empirically? However, if the comparable salaries are functions of the same variables as the arbitration-eligible salaries, it is as if the arbitrator does not consider the comparable salary but considers only these variables. Consequently, it is not necessary to control for comparable salaries directly.

Position Player Results

Comparison of the Salary Structures. First we make a comparison of the structures of negotiated salaries. The means of the variables for the position players are given in Table 1. In the first-stage probit models used to calculate LAMBDA for free agent and arbitration-eligible position players, the variables RC, FR, MID, and CG were used. Regressions were then performed on two data sets, one for arbitration eligibles and one for free agents. Initially, a Chow test (Chow 1960) was performed to determine whether the two data sets should be analyzed separately. The null hypothesis that the two systems have the same salary structures was rejected (F = 4.6455, p [less than] 0.01). It is thus concluded there is a difference in salary structure for arbitration-eligible and for free agent position players. The two regressions were then performed, the results of which are given in Table 2.

In both regressions, ln PREVSAL, RC, CGINV, 1991, and 1992 are significant with p values no greater than 0.1. Moreover, ln PREVSAL and RC are both positive and highly significant, which suggests that previous compensation and prior season offensive performance weigh highly in both systems. In the arbitration-eligible regression, MID is significant. In the free agent regression, 1993 is significant. In addition, in the free agent regression, a decrease in CGINV (an increase in CG) decreases ln SAL, whereas the opposite effect is present in the arbitration-eligible regression. As a player appears in more games, he gains additional experience. However, the more games in which a player appears, the more wear and tear occurs on the player's body. This, over time, could cause his productivity to fall. The data suggest that the experience factor outweighs the wear-and-tear factor for arbitration eligibles, but the opposite holds for free agents.

In both regressions, NL, FR, WPCT, and LAMBDA are all insignificant. That FR is insignificant indicates that position players are paid primarily for their offensive contribution to their teams and not their defensive contribution. That WPCT is insignificant indicates that a position player is not rewarded (penalized) as his team's winning percentage improves (worsens). That LAMBDA is insignificant indicates that, for position players, there is no selection bias present in the data.

Comparable Baseball Salaries. In 1976, the players' union was contemplating in what year a player should be allowed to become a free agent. Marvin Miller, the head of the union, wanted to choose the year in order to maximize the salaries of players in general, realizing that salaries negotiated for free agents would affect those of all reserved players, including arbitration eligibles (Helyar 1994). Although it was argued previously that it is not necessary to control for salaries of comparable players when estimating the salary structure of arbitration eligibles, it is interesting to analyze the correlation between free agent salaries and salaries negotiated within the arbitration system. In the theory, it was predicted that there would be a positive relationship between comparable salaries and salaries within the arbitration system.

Faurot and McAllister (1992) did not estimate the effect that this criteria had on arbitrated salaries because of the salary escalation over the years covered in their study. Burgess and Marburger (1993) used average baseball salaries in their study as a proxy for comparable baseball salaries.

Based on the estimated salary structures in the current analysis, a free agent salary can be estimated for each player in the arbitration-eligible sample. This is a prediction of the salary a player would have negotiated had he been eligible for free agency, all else equal. By using the prediction as a regressor, one can estimate the effect that free agent salaries have on those negotiated for arbitration eligibles. Moreover, since this method does not rely on an overall average, it controls for differences between players that using average salaries does not. To my knowledge, no such estimation has been performed.

To estimate the logarithm of the salary that a position player would have earned as a free agent, which we denote FASAL, it is assumed that the free agent model given in Table 2 is applicable. The regression result is given at the bottom of Table 2. One can see that the slope estimate is positive and statistically significant, indicating a strong relationship between the two salaries as predicted by the theory. To test for the equality of the two salaries, an F-test was performed on the null hypothesis that the intercept is zero and the slope is one. The null was rejected (F = 27.6824, p [less than] 0.01).

Suppose that a particular arbitration-eligible position player has a predicted free agent salary equal to the mean of the overall sample ($1.4035 million, from Table 1). Using this value, the predicted salary that he would negotiate in the arbitration system is $1.2557 million, approximately 10% lower than his predicted free agent salary. While this difference will vary with the predicted free agent salary, it will be positive for predicted free agent salaries over $0.6543 million.

We now summarize the results of this section. Although there seems to be a strong relationship between market and arbitrated salaries, the two systems determine different salary levels. Note that we have controlled for playing experience. Moreover, the Chow test showed that there is a structural change that also occurs when a position player becomes a free agent. In general, it is concluded that there is a statistical difference in salaries determined in FOA compared to those determined through free agency for position players. In addition, the arbitration system will result in lower negotiated salaries relative to the free agent system for comparable players.

Pitchers' Data

Pitchers are used primarily for defensive purposes: Their main objective is to prevent their opponent from scoring runs. Therefore, a pitcher's value to his team will be greater as the opponents he faces score fewer runs. The general regression equation for pitchers is

ln SAL = [[phi].sub.1] + [[Delta].sub.1] ln PREVSAL + [[Delta].sub.2]L + [[Delta].sub.3]WPCT + [[Delta].sub.4]PD + [[Delta].sub.5]ln CG + [[Delta].sub.6]ln CIP + [[Delta].sub.7]RAT + [[Delta].sub.8]SV + [[Delta].sub.9]1991 + [[Delta].sub.10]1992 + [[Delta].sub.11]1993 + [[Delta].sub.12]LAMBDA + [epsilon].

ln SAL, ln PREVSAL, WPCT, 1991, 1992, 1993, and LAMBDA are as described previously in the position players' model.

L is a dummy variable equal to one for each left-handed pitcher and zero for right-handed pitchers. This controls for the possibility that left-handed pitchers may be more valuable to clubs than right-handed pitchers.

PD is pitcher's defense, a statistic similar to fielding runs described previously for position players and is defined as runs prevented after the ball has been pitched (Thorn, Palmer, and Gershman 1995).

RAT is the average number of batters that reach base via a hit, a walk, or being hit by a pitch for each nine innings pitched. The fewer base runners allowed by the pitcher, the lower the RAT and the higher the effectiveness of the pitcher.

SV is the number of saves a pitcher earns and is another measure of pitcher productivity.

ln CG is the logarithm of career games played and partly controls for experience.

Pitchers also gain experience from working innings. To further control for experience and the length and consistency of a pitcher's career, ln CIP, the logarithm of innings pitched in a career, was used as a measure of length and consistency of a pitcher's career contribution. As in the position player's case, the logarithmic form is used to control for the decreasing value of each additional inning pitched.

The means of the pitcher variables are given in Table 3.

Pitcher Results [6]

Comparison of the Salary Structures. In the probit model used to calculate LAMBDA for each pitcher, the variables PD, RAT, and SV; the number of games pitched in the previous season; and the number of innings pitched in the previous season were used. A Chow test indicated that the structures of negotiated salaries are significantly different between the arbitration and free agent systems (F = 3.0567, p [less than] 0.01). It is thus concluded that, as in the position players' model, the systems have different salary structures. The two regressions were then performed, and the results are given in Table 4.

In both regressions, the intercept, ln PREVSAL, L, ln CG, WPCT, and LAMBDA are all significant with p-values no greater than 0.08. All other parameters are insignificant. Note that in both regressions, ln PREVSAL is positive as in the position player regression.

In both regressions, the coefficient on ln CG is negative. Over a career, pitching additional innings gives more experience, which should increase a pitcher's effectiveness. However, it is plausible to expect that the marginal effect on productivity of each career inning pitched falls. In addition, each additional career inning pitched may cause wear and tear on the pitcher's body, making a pitcher more susceptible to injury, which, in turn, lowers his effectiveness. The data suggest that the wear-and-tear factor dominates.

Unlike the position player regression, WPCT has a significant effect on pitcher salaries. Also note that, in the arbitration-eligible regression, its coefficient is positive, while it is negative in the free agent regressions. This suggests that arbitration-eligible pitchers are rewarded by an improvement in team winning percentage. However, the data suggest that free agent pitchers are penalized if they played for teams with high winning percentages. The reason for the negative coefficient on winning percentage in the free agent model can be explained by the theoretical model.

Recall that, in the theory, as the team's disagreement outcome (the surplus the team receives if it signs the alternative player) increases, the negotiated salary decreases. It is plausible that winning teams have better alternatives than do losing teams, which causes winning teams' disagreement outcomes to be larger than those of losing teams. Suppose that the free agent pitcher signs with the team for which he played the previous year and that team has a high winning percentage. Since the high winning percentage causes the team to have a higher disagreement outcome than if that team had a lower winning percentage, this causes the pitcher's salary that he negotiated to be lower.

However suppose that the pitcher did not sign with his former team but that team was his next-best alternative. Since the salary he could have negotiated with his former team is decreasing in the team's disagreement outcome, he would receive a lower salary if his former team had a high winning percentage than if its winning percentage were lower.

Now consider the salary that the player actually negotiated. This is increasing in the player's disagreement outcome. This disagreement outcome is in turn decreasing in that particular team's disagreement outcome. Thus, when the alternative team's winning percentage is high, that increases that particular team's disagreement outcome, which in turn decreases the players disagreement outcome in the negotiated salary that he actually receives.

That the coefficient on LAMBDA is significant indicates that the data suggest that there is selection bias present in both the free agent and the arbitration-eligible pitchers' models.

Comparable Salaries. Comparable free agent salaries for arbitration-eligible pitchers were calculated in the same manner as those for position players. The predicted comparable salary was regressed against the actual arbitration system salary to analyze the correlation between the two. The regression results are given at the bottom of Table 4.

An F-test was performed to test for equality between the two salaries (intercept = 0 and slope = 1). Equality was rejected (F = 8.6073, p [less than] 0.01). As predicted by the theory, it is concluded that the free agent and arbitrated salaries for relievers and starters are positively related but significantly different.

Now suppose that a pitcher has a predicted free agent salary equal to the mean of the overall sample ($l.5496 million). We predict that he would have negotiated a salary of $l.1993 million had he gone through the arbitration system, a decrement of $0.3503 million (over 22%). Note that the difference between predicted free agent salary and predicted arbitration-eligible salary will be positive for every predicted free agent salary over $0.02 million.

5. Summary and Discussion

Baseball's FOA system is one of the most public examples of an arbitration system. In this system, most cases in which final offers are submitted never reach the arbitrator. Consequently, it is important to understand the process underlying negotiated salaries.

In this paper, I developed a bargaining model that describes negotiated salary determination in baseball's arbitration system. In this system, bargaining continues after final offers are chosen. It was argued that negotiators in such a system do not choose final offers under the same incentive structure as that described in Farber's (1980) model. In the present model, the final offers are chosen to maximize the utility received from negotiating a settlement prior to going to arbitration, whereas in Farber's model the negotiators choose their final offers to maximize their expected utility from proceeding to arbitration. In the present model, the final offers in a particular case in part determine the negotiated salary because they help determine the outcome should the case ultimately go to arbitration.

Salaries negotiated under the threat of arbitration are dependent on salaries negotiated under free agency. It is therefore also important to understand the latter bargaining environment so that we can examine the relationship between salaries negotiated in the two systems. A formal model of the bargaining in baseball's free agent system based on Nash's (1950) cooperative bargaining model is thus developed. In free agency, the disagreement outcome for the player is the expected utility of not signing with the team. For the team, it is the utility from signing a substitute player.

In theoretically comparing negotiated salaries determined in both systems, it was argued that negotiated salaries will be determined differently since the disagreement outcomes are different in the two systems. The structures of salaries negotiated within baseball's FOA system and within its free agent system were then empirically analyzed. Using straightforward regression models, the empirical analysis shows that the structures of negotiated salaries of arbitration eligibles are significantly different from those of free agents as predicted by the theory.

Another contribution of the paper is that it provides an understanding of how salaries determined outside of an arbitration system affect negotiated salaries within that system. Many wage arbitration systems allow arbitrators to base their decisions in part on wages and salaries of workers that are comparable to those within the scope of the dispute. It is therefore important to understand how the arbitrators take this information into account when making their decisions.

In baseball, arbitrators can consider the compensation of players with comparable productivity and experience, including free agents, to a player in a particular case. The data from baseball thus allow one to examine the relationship between comparable salaries and actual negotiated salaries for arbitration eligible. Predicted free agent salaries were calculated for each arbitration-eligible player and regressed on the actual negotiated salary. It was found that although there is a statistical difference between the salary structures of the two systems and that although the two systems determine different salary levels, there is a significant positive relationship between salaries determined in the two systems for comparable players.

This result shows the effect that free agent salaries have on those negotiated within baseball's arbitration system. Vrooman (1996) argued that the reserve clause helps teams recoup their player development costs. On the other hand, players become eligible for free agency after six years of service, a threshold set in part to maximize the salaries of all players (Helyar 1994). That teams wanted to abolish the arbitration system and replace it with four-year free agency indicates that teams believe this threshold does not allow them to fully recapture their player development expenses (although it was argued that most arbitration-eligible players will negotiate lower salaries in the arbitration system than in the free agent system). This is a phenomenon that requires future research.

Another important contribution of the paper is that it further examines the data offered by baseball. Because of its availability to the researcher, these data are often used to examine various economic phenomena, and it is therefore necessary to understand these data. Since we do not observe negotiated salaries in cases in which a dispute occurred and since the disagreement outcomes of the negotiators lie above the negotiated outcome in these cases, it is argued that there is possible selection bias present in baseball salary data. The data suggest that there is selection bias present in the salaries of pitchers but not in the salaries of position players. The empirical evidence on this question is thus mixed. Consequently, more empirical research is needed to further examine this question.

Finally, it has been asserted that, in baseball's arbitration system, in order to appear fair and remain employed as arbitrators, arbitrators will roughly split their decisions over time. How does such behavior affect negotiated settlements? Suppose that a player wins an arbitration case and files for arbitration the following year. Given the final offers, if the player and the team expect the arbitrator to choose the team's final offer with a higher probability than if the team had won the previous arbitration case, then the team's disagreement outcome increases, while that of the player decreases. These effects will in turn move the negotiated settlement in the favor of the team.

(*.) Department, of Economics and Center for Economic Policy Analysis, 10 Professional Building, University of Missouri-Columbia, Columbia, MO 65211, USA; E-mail [e-mail].

I thank Dave Mandy, Peter Mueser, Michael Podgursky, Ken Troske, Xinghe Wang, Robert Weagley, and two anonymous referees for valuable comments. Any remaining errors are my own.

Received January 1998; accepted September 1999.

(1.) The assumption of risk aversion is common in the economic literature on FOA and allows the model of FOA described in this paper to be compared to that of Farber's model. For an analysis of FOA when one negotiator is risk averse while the other is risk neutral, the reader is directed to Faurot and McAllister (1992).

(2.) Note that having the player negotiate with team 1 does not mean that he has not already negotiated with team 2. If he has and did not sign, then this is a special case of the model where G(.) = 1. If the player already signed with the other team, G(.) = 0, and the model becomes trivial.

(3.) For a discussion of endogenously formed arbitrator preferences, see Gibbons (1988).

(4.) A proof of this result can be found in Miller (1998).

(5.) Fizel (1996) argues that career games played also serves as a proxy for physical and mental defects: Players with mental or physical disabilities would play in fewer games. However, this would be an inadequate proxy for player disabilities unless one controlled for years played. There were insufficient data to calculate years played for each observation in the data set.

(6.) It is plausible that salary structures for relievers and starters may be different within the two systems. Pitching is an endurance activity: The longer a pitcher pitches in a game, the more fatigued he becomes, and the less effective he is. Most starters do not pitch entire games, usually pitching five to six innings per start. Starters generally go four or five days without pitching in a game, while relievers may pitch as many as three to five times a week. When a pitcher, particularly a starter, is able so pitch more innings in a given appearance, he gives the other pitchers on the team a rest, increasing, or at least prolonging, their endurance. Consequently, there may be a difference in salary structures between pitchers who are primarily starters and those who are primarily relievers.

It is assumed that, in general, forecasts of a pitcher's specialty are made based on his specialty from the previous season. A statistic, REL, was calculated to determine the proportion of games each pitcher relieved in the previous season. REL = 0 indicates a full-time starter; REL = 1 indicates a full-time reliever, Of the 377 pitchers in the two data sets, only 22 had REL-values between 0.69 and 0.3. Consequently, each data set was divided into two subsets, one for relievers (REL [geq] 0.50) and one for starters (REL [less than] 0.50). Chow tests were performed to test for the equality of salary structures between starters and relievers within each system. The null was rejected for arbitration eligibles (F = 2.495, p = 0.01) but not for free agents (F = 1.6046, p [greater than] 0.50). Although arbitration-eligible pitcher salaries are determined by their productivity as specialists, to compare the two systems for pitchers, one regression for relievers will be estimated within each system.

References

Baseball America's almanac. 1993. Baseball America Inc., Durham, NC.

Baseball America's almanac. 1994. Baseball America Inc., Durham, NC.

Burgess, Paul L., and Daniel Marburger. 1993. Do negotiated and arbitrated salaries differ under final-offer arbitration? Industrial and Labor Relations Review 46:548-59.

Chow, Gregory C. 1960. Tests of equality between sets of coefficients in two linear regressions. Econometrica 28:591-605.

Crawford, Vincent P. 1982. Compulsory arbitration, arbitral risk and negotiated settlements: A case study in bargaining under imperfect information. Review of Economic Studies 69:69-82.

Farber, Henry S. 1980. An analysis of final offer arbitration. Journal of Conflict Resolution 24:683-705.

Faurot, David J., and Stephen McAllister. 1992. Salary arbitration and pre-arbitration negotiation in major league baseball. Industrial and Labor Relations Review 45:697-710.

Fizel, John. 1996. Bias in salary arbitration: The case of major league baseball. Applied Economics 28:255-65.

Gibbons, Robert. 1988. Learning in equilibrium models of arbitration. American Economic Review 78:896-912.

Heckman, James J. 1979. Sample selection bias as a specification error. Econometrica 47:153-61.

Helyar, John. 1994. Lards of the realm. New York Ballantine Books.

Miller, Phillip A. 1998. A two-stage bargaining game defined by final-offer arbitration. Ph.D. dissertation, University of Missouri-Columbia. Columbia, MO.

Myerson, Roger B. 1984. Two-person bargaining problems with incomplete information. Econometrica 52:461-87.

Nash, John. 1950. The bargaining problem. Econometrica 18:155-62.

Neft, David S., and Richard Cohen. 1994. The sports encyclopedia: Baseball. New York: St. Martin's Press.

Scully, Gerald W. 1989. The business of major league baseball. Chicago: University of Chicago Press.

Thorn, John, Pete Palmer, and Michael Gershman. 1995. Total baseball, edited by David Pietrusza. New York: Viking Press, pp. 665-1407.

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Zimbalist, Andrew. 1992. Baseball and billions. New York: Basic Books.

Appendix: Free Agent Bargaining Model

In this appendix, I show that the effect of is [w.sub.2], a salary for which a player might sign if he signs with team 2, on [[w.sup.*].sub.sfa], the free agent salary the player negotiates with team 1, is ambiguous.

From totally differentiating Equation 4, we have

[frac{[partial][[w.sup.*].sub.sfa]}{[partial][w.sub.2]}] + [frac{-[U'.sub.f](1 - [w'.sub.sfa]){[1 - G([w.sub.2])][U'.sub.e]([w.sub.2]) - g([w.sub.2])[[U.sub.e]([w.sub.2]) - [U.sub.e]([[w.sup.p].sub.r])]}}{A}], (A.1)

where

A = [[U.sup.n].sub.e]([[w.sup.*].sub.sfa])[[U.sub.f](1 - [[w.sup.*].sub.sfa]) - [U.sub.f](1 - [w.sub.k])] - [U'.sub.e]([[w.sup.*].sub.sfa])[U'.sub.f](1 - [[w.sup.*].sub.sfa]) + [U".sub.f](1 - [[w.sup.*].sub.sfa])[[U.sub.e]([[w.sup.*].sub.sfa]) - {[1 - G([w.sub.2])][U.sub.e]([w.sub.2]) + G([w.sub.2])[U.sub.e]([[w.sup.p].sub.r])}].

Since the denominator is negative and [U'.sub.f](1 - [[w.sup.*].sub.sfa]) [greater than] 0,

sign[frac{[partial][[w.sup.*].sub.sfa]}{[partial][w.sub.2]}] = sign{[1 - G([w.sub.2])][U'.sub.e]([w.sub.2]) - g([w.sub.2])[[U.sub.e]([w.sub.2]) - [U.sub.e]([[w.sup.p].sub.r])]}.

Now, let [U.sub.e](w) = [w.sup.1/2] and let g([w.sub.2]) be a uniform density with denominator parameters 1 and 0 (i.e., g([w.sub.2]) = 1). Since [w.sub.2] and [[w.sup.p].sub.r] are assumed to be exogenous, all that is required is to pick values for these variables with the restriction that 1 [greater than] [w.sub.2] [greater than] [[w.sup.p].sub.r] [greater than] 0. It is straightforward to show that if [w.sub.2] = 0.9 and [[w.sup.p].sub.r] = 0.1, then [partial][[w.sup.*].sub.sfa]/[partial][w.sub.2] [greater than] 0. However, if [w.sub.2] = 1/3 and [[w.sup.p].sub.r] = 0.1, then [partial][[w.sup.*].sub.sfa]/[partial][w.sub.2] [greater than] 0. Thus, with the restrictions given in the model, the general sign of [partial][[w.sup.*].sub.sfa]/[partial][w.sub.2] is ambiguous.