Athlete Pay and Competitive Balance in College Athletics

Abstract

In this paper we analyze the argument—which has been used by both the National Labor Relations Board and the National Collegiate Athletic Association—that unionization and/or player pay will hurt competitive balance in college sports. We present a theoretical analysis of universities that recruit athletes and examine the assumptions that are needed for player compensation to decrease competitive balance. We also empirically illustrate the differences in balance between professional and college sports. Given the theoretical and empirical analysis, we argue that unionization and/or player pay is unlikely to hurt competitive balance.

Appendix

If we allow for a non-zero restriction—$\bar{\rho }$—that does restrict both teams, then Eqs. (9) and (10) become

$$ \left. {\frac{{\partial \pi_{L} }}{{\partial \varphi_{L} }}} \right|_{{\rho = \bar{\rho }}} = \frac{{\sigma c\left( {\alpha_{S} + b\bar{\rho } + c \varphi_{S} } \right)}}{{\left( {\alpha_{L} + b\bar{\rho } + c \varphi_{L} + \alpha_{S} + b\bar{\rho } + c \varphi_{S} } \right)^{2} }} - 1 . $$

and

$$ \left. {\frac{{\partial \pi_{S} }}{{\partial \varphi_{S} }}} \right|_{{\rho = \bar{\rho }}} = \frac{{c\left( {\alpha_{L} + b\bar{\rho } + c \varphi_{L} } \right)}}{{\left( {\alpha_{L} + b\bar{\rho } + c \varphi_{L} + \alpha_{S} + b\bar{\rho } + c \varphi_{S} } \right)^{2} }} - 1 . $$

Therefore, the optimal level of indirect investment for the small-market team is given by

$$ \varphi_{S} = \frac{\sigma }{{\left( {1 + \sigma } \right)^{2} }} - \frac{{\alpha_{S} + b\bar{\rho }}}{c} . $$

Since investment must be non-negative, indirect investment will stop when

$$ \bar{\rho } = \frac{c\sigma }{{b\left( {1 + \sigma } \right)^{2} }} - \frac{{\alpha_{S} }}{b} . $$

As the restriction on direct investment increases, the next qualitative change in investments is either that the large-market ceases its indirect investment, or the restriction on direct investment will not be restricting for the small-market team. We will first analyze the case where large-market indirect investment stops first. In this case, $\varphi_{S} = 0,$ and $\rho_{S} = \bar{\rho },$ so the impact of indirect investment by the large-market team is given by

$$ \left. {\frac{{\partial \pi_{L} }}{{\partial \varphi_{L} }}} \right|_{{\rho = \bar{\rho }}} = \frac{{\sigma c\left( {\alpha_{S} + b\bar{\rho }} \right)}}{{\left( {\alpha_{L} + b\bar{\rho } + c \varphi_{L} + \alpha_{S} + b\bar{\rho }} \right)^{2} }} - 1 . $$

Therefore

$$ \varphi_{L} = \frac{{\sqrt {\sigma c\left( {\alpha_{S} + b\bar{\rho }} \right)} - \alpha_{L} - \alpha_{S} - 2b\bar{\rho }}}{c} . $$

And this will equal zero when

$$ \bar{\rho } = \frac{{\sigma c - 4\left( {\alpha_{S} + \alpha_{L} } \right) \pm \sqrt {\sigma^{2} c^{2} + 8\sigma c\left( {\alpha_{S} - \alpha_{L} } \right)} }}{8b} . $$

If it is the case that $a_{S} = a_{L}$, this can be simplified to

$$ \bar{\rho } = \frac{\sigma c - 4\alpha }{4b} . $$

However, the small-market team might stop direct payments before the large-market team stops indirect payments. In this case, the marginal impact of direct payments to the small-market team is given by

$$ \left. {\frac{{\partial \pi_{L} }}{{\partial \varphi_{L} }}} \right|_{{\rho = \bar{\rho }}} = \frac{{\sigma c\left( {\alpha_{S} + b\rho_{S} } \right)}}{{\left( {\alpha_{L} + b\bar{\rho } + c \varphi_{L} + \alpha_{S} + b\rho_{S} } \right)^{2} }} - 1. $$

and

$$ \frac{{\partial \pi_{S} }}{{\partial \rho_{S} }} = \frac{{b\left( {\alpha_{L} + b\bar{\rho } + c \varphi_{L} } \right)}}{{\left( {\alpha_{L} + b\bar{\rho } + c \varphi_{L} + \alpha_{S} + \rho_{S} } \right)^{2} }} - 1, $$

and therefore

$$ \rho_{S} = \frac{b\sigma c}{{\left( {b + \sigma c} \right)^{2} }} - \frac{{\alpha_{S} }}{b}. $$

So, when $\bar{\rho }$ surpasses that point, the restriction is no longer restrictive to the small-market team.

Finally, we find the point of a superfluous restriction where it does not restrict either team. The first-order conditions are given by,

$$ \frac{{\partial \pi_{L} }}{{\partial \rho_{L} }} = \frac{{\sigma b\left( {\alpha_{S} + b\rho_{S} } \right)}}{{\left( {\alpha_{L} + b\rho_{L} + \alpha_{S} + b\rho_{S} } \right)^{2} }} - 1. $$

And

$$ \frac{{\partial \pi_{S} }}{{\partial \rho_{S} }} = \frac{{b\left( {\alpha_{L} + b\rho_{L} } \right)}}{{\left( {\alpha_{L} + b\rho_{L} + \alpha_{S} + \rho_{S} } \right)^{2} }} - 1, $$

which gives us

$$ \rho_{L} = \frac{{\sigma^{2} }}{{\left( {1 + \sigma } \right)^{2} }} - \frac{{\alpha_{L} }}{b}. $$

If we assume that $\sigma = 2,\alpha_{S} = \alpha_{L} = .1$, $b = 2$, and $c = 1$, then the small-market team will stop investing indirectly when $\bar{\rho } = 0.06111$; the small-market team will stop investing directly; and the large-market team will stop investing indirectly when $\bar{\rho } = 0.2$, and the large-market team will stop investing directly when $\bar{\rho } = 0.39444$.