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PUBLIC EDUCATION EXPENDITURES AND MOBILITY

57

Partial derivation with respect to τ gives
∂a ∗
φ(1 + r )
γ
= −E αBt τ −1
E [(α + γ )(1 − τ ) − τ ]
∂τ
(1 − τ )D At

− C[α(1 − τ ) − τ ] < 0
γ

γ

where E At = τ γ (1 − eB )γ Y t−1 ; E αBt = (τ eB Y t−1 )α C = 2 + r ; and
γ
D = E αBt (E At − C). Clearly, this derivative is always negative since we
have γ (1 − τ ) > 0.
The wealth threshold level given in Equation (8) can be written as
follows:
γ
(1 + i) φ − (1 − τ )a(τ e B Yt−1 )α [τ γ (1 − e B )γ Yt−1 − (2 + r )]
xt∗ (a) =
(1 + r )(i − r )
The derivation of this expression with respect to τ yields

γ
−a E αBt
∂ xt∗ (a)
=
E γ (1 − τ )(1 − α)
∂τ
τ (1 + r )(i − r ) At

− (2 + r )[(1 − τ ) α − τ ] < 0
This derivative is always negative since we have γ (1 − τ ) > 0.


C 2010 The Authors. Journal compilation
C 2010 Blackwell Publishing Ltd and the Board of Trustees of
the Bulletin of Economic Research.