Spillovers, complementarities, and sorting in labor markets with an application to professional sports.

• Author: Kendall, Todd D.

1. Introduction

A moral of Aesop reads, "The strong and the weak cannot keep company." Casual empiricism indicates that law firms, hospitals, academic university departments, and many other industries seem to consist of companies with relatively homogeneous quality workers and that top firms are generally averse to hiring low-quality workers, even at significantly lower wages. One explanation for this behavior is the presence of labor complementarities caused by spillovers across workers, as when the presence of better coworkers improves a worker's own productivity.

Spillovers by themselves do not necessarily affect the sorting of workers into firms. However, spillovers can affect worker-firm sorting if the size of the spillover depends not only on the characteristics of the individual worker but also on those of his coworkers. In this case, the wage a worker can demand for his labor may differ across firms because the composition of workers in those firms differs. In particular, if higher-quality workers receive greater spillover benefits from high-quality colleagues, it is optimal for homogeneous quality workers to be matched together in firms as complementary labor inputs.

This paper provides theory to this effect and considers empirically the specific case of basketball teams. To illustrate the previous discussion, when a player with high offensive talent is added to the team, he tends to draw double- and triple-teaming defenses to himself, opening up clearer paths to the basket for his teammates and improving their likelihood of scoring. This constitutes a spillover. If the improvement in the likelihood of scoring the teammates receive is higher for high-quality teammates than for low-quality teammates, this constitutes a complementarity between players and suggests that the optimal matching of players to teams will tend toward homogeneous teams, with the best players matched together, since this is where their marginal physical product is highest.

I find that there are significant spillovers to offensive production in professional basketball: A 10% increase in teammates' productivity leads to a 4.5% increase in own productivity. However, there is little evidence for offensive complementarities: Teams are not significantly homogeneous in offensive talent.

This paper makes two primary contributions to literature. First, it provides a general equilibrium structural model linking productivity spillovers to worker sorting. Sorting models like the one developed here are part of the lengthy literature on "assignment" problems, in which a number of heterogeneous units are to be assigned into groups in such a way as to optimize the aggregate output. Von Neumann (1953) and Koopmans and Beckman (1957) present mathematical derivations of basic assignment problems. Becker (1973) provides a canonical model of coordination and assignment in marriages. Applying this model to the labor market, Kremer (1993) and Saint-Paul (2001) consider assignment of workers to firms when there are complementarities in production. The present paper extends this literature to specifically account for assignment when spillovers to productivity exist and to provide empirically estimable specifications.

The paper also adds to the literature on the estimation of production processes in sports. Scully (1974) presented estimates of baseball players' marginal revenue products. (1) Scott, Long, and Somppi (1985) and Berri (1999) use similar methods in basketball The accuracy of these estimates, however, depends on the absence of complementarities between players since productivity depends not only on the player but also on the teammates he played with, as Scully (1974) notes. Idson and Kahane (2000) find some evidence of labor complementarities in the National Hockey League, and Carmichael and Thomas (1995) provide evidence on complementarities between labor and other inputs in rugby. Chapman and Southwick (1991) find that baseball managers have different productivity levels when matched with different teams. This paper takes a different approach to the problem than previous studies; it seeks not to measure complementarities directly but instead to measure spillovers to offensive productivity between professional basketball players and then to seek appropriate evidence of matching to check for the existence of complementarities. Unlike previous studies, I utilize a panel data set to distinguish the existence of spillovers from any sorting effects they may generate. Moreover, I utilize the fact that offensive production in basketball is sometimes performed in teams (floor shooting) and sometimes individually (free throw shooting) to distinguish spillover effects from other exogenous effects on labor productivity. (2)

Section 2 develops a model of spillovers in a generalized workplace and shows the conditions under which these effects generate the complementarities between workers that lead to homogeneous matching of workers into firms. Section 3 explores professional basketball as a case study. Section 4 concludes.

2. The Model

   The workforce consists of a unitary measure of workers. Workers can be classified as one of N > 1 types, indexed in the set Q = {1, 2, ..., N}. A worker of type j [member of] Q is of quality [[phi].sub.j], which might measure an exogenous endowment of human capital, for instance. There is a finite measure [m.sub.j] of type j workers, where [[summation of].sub.j[member of]Q] [m.sub.j] = 1.

   Firms consist of coalitions of workers, and a particular firm k consists of an (endogenous) measure [n.sub.jk] of type j workers, so that the total measure of workers at firm k is [n.sub.k] = [[summation of].sub.j[member of]Q] [n.sub.jk].

   All workers provide an inelastic quantity of labor; however, since the workers differ in quality, the input of quality-adjusted labor hours differs by worker type. Let the quality-adjusted labor input ("productivity") of a worker of type j at firm k be [a.sub.jk].

   To quantify the spillover effects across coworkers, let [a.sub.jk] be a function of j's own quality, [[phi].sub.j], the total productivity of the other workers at firm k, and the size of the measure of other workers at firm k:

   (1) [a.sub.jk] = g([[phi].sub.j], [A.sub.k], [n.sub.k]) where [A.sub.k] = [summation over (j[member of]Q)][n.sub.jk][a.sub.jk].

   If [a.sub.jk] is increasing (decreasing) in the second argument, then there are positive (negative) spillovers. If larger teams are more difficult to coordinate than smaller teams, then g should be decreasing in [n.sub.k]. This means that if an additional worker is not supplying enough productivity to the coalition through his quality, he may be a net drag on team output because there are costs to including him in the production process. An example of this might be a case where a group of inventors are working on a new drug in a pharmaceutical company. Adding additional inventors increases the probability that someone will come up with a patentable idea, but in the process of developing such an idea, the inventors may argue among themselves about the best way to proceed in research or may find it difficult to come to a unified decision regarding production in a reasonable amount of time.

   The output of the firm is [A.sub.k]:

   (2) [A.sub.k] = [summation over (j[member of]Q)][n.sub.jk][a.sub.jk].

   Equations 1 and 2 jointly describe the production process of firms in the economy. Together, these equations imply a production function for the firm:

   (3) [A.sub.k] = [summation over (j[member of]Q)][n.sub.jk]g([[phi].sub.j], [A.sub.k], [n.sub.k]).

   Define an "assignment" for a firm like that described in Equation 3 to be a set of numbers [n.sub.jk] for every j and every k such that

   (i) [n.sub.jk] [greater than or equal to] 0 for all j and all k and

   (ii) [[summation of].sub.k] [n.sub.jk] [less than or equal to] [m.sub.j].

   An assignment both defines the number of firms in the economy and indicates which workers work at which firms. The first condition indicates that nonnegative assignments must be made of each type to each firm. The second condition indicates that the total number of type j workers assigned must not be more than the number of type j workers who exist in the labor force.

   Now define an "equilibrium" in this economy as

   (i) a wage function [w.sub.jk],

   (ii) a positive number K indicating the number of firms, and

   (iii) a set of numbers {[n.sub.jk]} for all j [member of] Q and for all k [less than or equal to] K, satisfying the following conditions:

   (4) {[n.sub.jk]} is an assignment

   (5) [summation over (j[member of]Q)][n.sub.jk][w.sub.jk] [less than or equal to] [A.sub.k] [for all] k [less than or equal to] K

   (6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

   (7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

   Equation 5 indicates every firm can cover its labor costs. Equation 6 indicates that the assignment {[n.sub.jk]} is individually rational in the sense that no subcoalition of workers at any firm or any coalition of workers...