Semi-parametric estimation of the team wage frontier of European football technology

Abstract In this paper, the team wage function of the football teams in the 5 major European football leagues is estimated using information about their sports results and controlling for the national league they play, the international competitions (i.e., Champions League and Europe League) they participate each year and the fresh promotions. The dataset used covers seasons 2018/19 to 2024/25. The methodology used corresponds to stochastic semi-parametric envelopment of data with contextual variables (StoNEZD), which allows estimating the outputs shadow prices for each team and the marginal effects of the contextual variables. The inefficiency score of each team, as well as their scale efficiency and returns to scale in each season, have also been estimated. The results show significant wage differences between the different national leagues as well as higher wages for teams that participate in the international competitions (higher in the case of the Champions League than in the Europa League). Also, it was found that the wages of freshly promoted teams do not catch up immediately with respect to those of the more established clubs. We also tested the effect of the COVID-19 pandemic (seasons 2019/20 and 2020/21) but it was not found significant pandemic. The estimated average wage inefficiency is rather low for all 5 leagues, indicating that a substantial part of the wage expenses is not explained by the observed ex-post performance. Scale efficiencies are higher and have less dispersion, with a majority of teams (between 62 and 75%, depending on the league) exhibiting increasing returns to scale and another fraction (between 25 and 35%, depending on the league) exhibiting decreasing returns to scale. Interestingly, a wage floor seems to exist so that the wages become insensitive to sports results below certain thresholds.