Migrating Football Players, Transfer Fees and Migration Controls

where T_i represents the exogenous part of talent availability and ei ≥ 1 indicates the
investment in home talent. An investment equal to e_i ≥ 1 involves costs equal to
c(e_i - 1). For simplicity we assume that training only determines the number of talents,
but their capability remains exogenously given.

Below, we first derive how the social welfare optimum looks like in a closed and an open
economy, respectively. Then we compare the market equilibrium, respectively, without
and with a transfer fee system, to the social-welfare optimum. Finally, we consider
whether a home-grown rule is better able to approach the social-welfare optimum than a
transfer fee system.

4.1 Optimal allocation

Given this set up, we first consider the case where the FFI is able to determine the
command optimum. It sets the optimal amount of training by the leagues, determines the
optimal allocation of players to the two leagues, and decides how the revenues from the
football product and the costs of the training facilities are shared between the leagues. So,
the FFI sets T_i^* and e_i^* under the restriction T₁^* + T₂^* = T₁ + T₂ . Moreover, product
revenue and training cost sharing is implicitly given shape in the form of a redistributive
scheme between the leagues. This scheme contains a transfer Γ , which can be positive or
negative, from the big league to the small league. The FFI maximizes the following
social welfare function:

W =N₁log(Y₁(1-αT₁)-c(e₁ -1)-Γ)+N₂ log(Y₂(1-αT₂) -c(e₂ -1)+Γ)(8)

The first-order conditions read:

∏1 =∏2

N1  N 2

e₁^* = e^*₂                                                                        (10)

Y₁((1-αT₁^*)logσ-1)≥Y₂((1-αT₂^*)logσ-1)                            (11)

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