The distribution of earnings and the rules of the game.

AuthorPorter, Philip K.
  1. Introduction

    Extensive research links the distribution of earnings to a variety of exogenous and endogenous factors: innate ability, motivation, acquired human capital, skill, occupation, race, sex, unionization, mobility, work-leisure preference, risk preference, luck, and so on. Overlooked in the analysis of income inequality is the institutional technology, or constitutional setting, or rules of the game in which income is earned. One of the authors [11] has examined the effect of differences in the degree of economic, legal (civil rights), and political freedom on income inequality across nations. He found that a substantial portion of the variance in income inequality was due to differences in the rights of various groups to compete for income streams. Much of constitutional history's evolution can be interpreted as investing previously unequal groups (peasants, blacks, women, the disabled, and so on) with equal rights to compete for income streams.

    In general, individuals are endowed with, or acquire through investment and training, talent that can be utilized in productive activities. The realization of an individual's talent in an activity is his or her performance. The dimensions of talent are too numerous, and perhaps too mysterious, to be precisely quantified. But, it is convenient to think of talent as being composed of broad dimensions like intelligence, strength and coordination, and of further refinements of these dimensions (e.g., language, math, or reasoning ability). In this way, an individual's "talent" for a particular activity can be thought of as a weighted total of the dimensional attributes he or she possesses, where the weight in each dimension is determined by the demands of the activity.

    Performance is the realization of talent plus a random component. Its expectation is an exact proxy for talent. Any discrepancy between talent and expected performance is the result of omitting relevant attributes of talent. For example, ignoring the intangible attributes of talent (determination, intensity, hustle) that contribute to one's success leads to the apparent observations of individuals that fail to perform at levels commensurate with their talent and individuals whose performance appears to surpass the measure of their talent. It must be said of the former that they are talented but lack something, say, concentration and of the latter that they make up for a lack of talent with something else, for example, effort. When the abilities to concentrate and to put forth effort are included as attributes of talent this discrepancy disappears and except for random occurrences (luck) performance is synonymous with talent.

    In production activities the weights (rewards per unit) placed on the dimensions of talent are determined by derived demand, through production technology, the supplies of the attributes of talent, and the organization of the talent market (rules that compensate the various dimensions of talent). Where the rules of the talent market are the same for all, the distribution of income is a function of the distribution of performance weighted by labor effort, where labor effort refers to the number of performances an individual undertakes and is not considered an attribute of talent. This is most transparent in piecework activities, where performance can be measured as output per period and labor effort can be measured as the number of periods committed to piecework activities. Then, if one controls for labor effort, income is a scaler multiple (determined by the value of output) of performance, and the distribution of income and the distribution of performance are identical.

    Within different talent markets, unique characteristics make for predictable differences in the relation between the distribution of talent and income. For example, when the payments for performance are rank ordered (that is, where the margin of performance between individuals determines their rank in the payment hierarchy but not the margin of compensation), income will be only rank correlated with performance and the distribution of income will (depending on the payment schedule) be more or less equal than the distribution of talent. In team production settings, where the assessment of individual performance is more difficult, or in instances where the consequence of the employment of labor effort and talent is uncertain, the distribution of income will tend to be more equal than in piecework settings. Finally, when there are rents (defined as returns not associated with the application of productive talent) to be distributed, income and its distribution may bear little resemblance to the underlying distribution of talent.

    In this paper we examine the effects of the rules of the game on the distribution of player earnings in the sports economy. The analysis builds on Lazear and Rosen [6], by extending the rank order tournament model to team sports and by investigating the implications of the rules of the game (and changes in the rules) on the distribution of earnings in individual and team sports. We focus on the earnings of athletes for several reasons. First, the rules of the game in sports are known and in some sports have undergone substantial change. Second, the dimensions of athletic performance and the technology of production, although complex, lends itself to fairly unambiguous modelling. Third, evidence presented below suggests that the distribution of talent within a sport is similar across time and across samples so that observed differences in the distribution of earnings can be ascribed to differences in the rules of the game. Comparisons across sports is somewhat more problematic as comparisons of talent are not possible. George Stigler noted that in each academic discipline there were at any time only a few superstars. Rosen [7] makes the same observation about entertainment fields, including sports. Implicit in these observations is the notion that the distribution of talent, at least the upper tail, is similar across these occupations. For our purposes it is sufficient that differences in the distribution of talent across sports, if such differences exist, are not positively correlated with differences in the distribution of income that our analysis predict. Finally, data exist to test our propositions about the effect of the rules of the game on the distribution of earnings within and across sports. Several propositions concerning the effects of different compensation formulas, of team versus individual production, of uncertainty in the labor and product markets, of employer monopsony power, and of changes in technology are developed and tested. Many of the implications of these findings, mutatis mutandis, can be generalized to any production economy.

  2. Individual Production

    In productive activities, talent is a function of physical, intellectual and emotional attributes valued by the requisite technology. For our purpose we think of each attribute as quantifiable, with some, like height, exogenous (endowed), and others, like strength, endogenous (determined by investment). Let there be K attributes of talent denoted k. Each activity determines a set of weights that, when applied to the attributes of individual i, defines his or her talent as [t.sub.i] = [[Sigma].sub.k][b.sub.k][a.sub.ik], where the measures t, b and a are, respectively, talent, weight and attribute.

    The realization of talent is an individual's average performance. Some factors beyond talent will influence performance (e.g., conditions of health, the environment, time of day, even how much sleep one got the night before). Thus, at any time the observation of individual i's performance, [P.sub.i], is a random variable with expectation, [t.sub.i], and a random component, [e.sub.i].(1)

    [P.sub.i] = [t.sub.i] + [e.sub.i]. (1)

    The [e.sub.i] are assumed to be independently and identically distributed (iid), with E(e) = 0.

    We think of individual performance as a serially repeated and rewarded event. Among the individuals competing in professional sports the distribution of earnings is determined by the distribution of awards for performance, the distribution of talent (expected performance), and by work effort (the number of competitions undertaken). Even at the same wage, work effort is not equal among all, because people differ in their willingness to trade leisure for income. In what follows, expected earnings should be understood to mean expected earnings per event, a labor-effort-constant measure. Such a measure is appealing for comparisons of income distributions, because it compares potential earnings before the choice to purchase leisure. Observed earnings that measure potential earnings after the purchase of leisure treat the leisure choice negatively.

    In a rank-order tournament the reward schedule is pre-determined and a competitor achieves the next higher reward by advancing one position in the rank of competitors. Consider random pairings of individuals in one-on-one, rank-order tournaments. Prior to the revelation of the pairings, each individual confronts the field of competitors. To compare the expected earnings of two individuals engaged in repeated rank-order contests, define [p.sub.12] as the probability that individual 1 will out-perform individual 2 in any given contest. That is,

    [p.sub.12] [equivalent to] prob([P.sub.1] [greater than] [P.sub.2]) = prob([e.sub.2] - [e.sub.1] [less than] [t.sub.1] - [t.sub.2]) [equivalent to] G ([t.sub.1] - [t.sub.2]), (2)

    where E([e.sub.2] - [e.sub.1]) = 0 and G is the cumulative density function (cdf) of [e.sub.2] - [e.sub.1]. Given a reward structure that pays [W.sub.1] to the winner and [W.sub.2] (less than [W.sub.1]) to the loser, the expected incomes are

    E[Y.sub.1] = [P.sub.12][W.sub.1] + (1 - [P.sub.12])[W.sub.2] and E[Y.sub.2] = (1 - [p.sub.12])[W.sub.1] + [p.sub.12][W.sub.2]. (3)

    Since some attributes of talent are given (endowed), the cost of...

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