**
where Π1 = Y1(1 -αT1 ) - c(e1 - 1) - Γ is defined as total profit for league 1, andΠ2 is
defined accordingly. Eq. (9) reproduces the well-known fiscal-federalism result that
transfers should be such that the profit per capita is equalized. According to eq. (10) both
leagues should invest the same amount in training facilities; in other words the total
number of talents should be trained such that total training costs are minimized. Eq. (11),
finally, governs the optimal allocation of talents. Talents should be located firstly in the
league with the large potential. Dependent on the parameters of the model, the marginal
loss of moving a player trained in the small league can in equilibrium be smaller than the
gain of this move for the big league, in which case the >-sign holds in eq. (11). In this
corner solution all talents will be located in the big league while part of their training
takes place in the small league.
Note that it is optimal that the two leagues have training facilities even if talents will
never play in the small league. As we shall see below, this is typically not one of the
properties of the equilibrium generated by the market.
4.1.1 The optimum in a closed league
Before proceeding it is of interest to consider how the optimum looks like in a closed
league. Welfare maximization then obviously coincides with profit maximization in the
closed league. The first-order condition for optimal training reads,
αδY (logσ(1 -αT) - 1)/e = c
(12)
Training talents will be extended until the marginal profits generated by the training of
talents equal the marginal training costs.
By totally differentiating first-order condition (12) it follows that the relationship between
investment in players and the parameters of the model is nonlinear. Table 3, that gives the
optimum investment in players, and the corresponding number of players, demonstrates
this. The market sizes considered are N=100 and N=50, respectively.
The optimal investment in training players first increases with α and then decreases. It is
clear that market size is an important determinant of the investment in talents. Large
markets have more incentives to train players than small markets as, team size being
identical in the two leagues, the training costs per capita are lower in the big than in the
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