Federal Tax-Transfer Policy and Intergovernmental Pre-Commitment

s¹ + s² = τ(l¹ + l²) are differentiated with respect to tⁱ which yields in matrix notation:

Applying Cramer’s Rule:

∂sⁱ ∣A¹1 ∂sj ∣A²1 ∂τ _ ∣A³∣
∂⅛ⁱ = 7^T^, ∂β = 7^T ^an ∂tⁱ = 7^T^,

where

^; = ^b'' ( gⁱ ) (-^b" ( gj ) Σ< ^ln+τln ) + ^— ₁+j ]+^b" ( gj ) _{dτ 1}+₇ <⁰.

∣A¹1 = -b"(g')(tⁱkii + kⁱ) ( -b"(gj) X(lⁿ + τl"_τ) + ^₊ ,j )

∖         n=1                       ^η /

r) 1                         . ■

⁺∂τ τ+η?^b''⁽ gj >^tj   .

.. . .. . . ,. ʌ                  .. ...      . ∂    1

O²1 = b''(g')b''(gj)tjkj £(lⁿ + τl';)+ b''(g`)(t<sup>i</sup>kii + k`)_dτ —

n =1                                           '

^-— τ+√^b''⁽gj)^tj^kji^{,   and}

∣A³1 = b''(gⁱ)b''(gj)tjkj + b''(gⁱ)b''(gj)(t`kβ + kⁱ) > 0.                                  (25)

Given by the second-order conditions of the federal optimization problem ∣A∣ < 0. The sign of
∣A³1 is strictly positive since tjkj_i > 0 and by assumption (A) tⁱk'_i + kⁱ > 0. Helpful in signing
∣A¹1 and ∣A²1 we rewrite both determinants as:

∣A¹1 = β(tⁱk'i + kⁱ) + α      and      ∣A²1 = β(tjkj_i) - α,                 (26)

with

.....     . ∂   1       ..     . ∂ ∂   1

α := -b '( g )( tⁱkii + k^l ) — —- + b' '( gj ) tj kj — —-    and

^t ∂τ 1 + ηj            ^t ∂τ 1 + ηⁱ

2

β := b' '( cf ) b' '( cf )£( lⁿ + τlⁿ_τ ).

n =1

26

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