Let us again indicate the number of home-grown talents in league i by T ( i = 1,2) and the
number of players that move from the small league to the big league by T21. Assume that
for each migrated player the big league pays a transfer fee equal to τtimes the wage paid
to the talent. Then for the two leagues total profit can be written as:
л Λ
π1 = ɪɪ(l - α(Tl + (1 + τ)T21)), π2 = Y2(1 - α(T2 - T21)) + ταY1T21 (3)
As before, all players that are in demand by the big league, will be migrating from the
small league. The first-order condition for the demand for players by the big league, i.e.
T21 , assuming an interior solution equals:
(4)
T21 = (l — (l + τ)/log σ - αTι)∕(α + ατ)
As expected the number of home grown talents and the size of the transfer fee have a
negative effect on the demand for foreign talents.
Given the market solution that is generated by eq. (4) and the definitions of total profit for
the two leagues in eq. (3), the FFI can calculate the tax rate that maximizes the social-
welfare function (1). Although this is a straightforward exercise we get a highly nonlinear
solution for the optimal transfer rate. Here, we only present the first-order derivative of
the social-welfare function, evaluated in τ = 0. Under the condition that Ti + Ti = l, we
get,
∂W T21
T21Y1 / Y2 5
(5)
l - a(T2 - T21)
---(τ = 0) = α(Nι - N2)T21 logσ + Nɪ + αN2-----7--+ αN2
dτ l — α( T2 — T21)
where T2'1 = ∂T21 /∂τ < 0 . Eq. (5) specifies the rationales for the implementation of a
transfer fee system. The first term indicates the loss of production value in league 1
relative to the gain in league 2. The second term represents the savings in wage payments
by league 1 that result from the lower demand for foreign talents. The third term
represents the additional wage payments by the small league due to the lower emigration
of talents, while the last term gives the gain in welfare for league 2 due to the transfer
payments that go along with the introduction of a transfer fee system.
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