Federal Tax-Transfer Policy and Intergovernmental Pre-Commitment

For case (iii) and (iv) τ_ti < 0 and sⁱ_ti ≥ 0. Thus, Lemma 1 predicts t^d > t^N . In case (iv),
τ_ti < 0 and sⁱ_ti = 0 and consequently t^d > t^N . In case (i), (ii), and (v), the transfer and labor
tax response are opposite in sign. Lemma 2 provides a condition which allows us to sign the
net-effect in these three cases.

Lemma 2: In case (i), (ii), and (v), condition (27) is equivalent to the condition

₁    <>

β(t(tikβ + ki - t^jk^ji)+ αJ = >0 ≠- t^dl = \t^N.                  (28)

²                      I >I <

Proof: Substituting Eqs. (16) and (18) into condition (27) yields

i b⁰⁰ ( д^г ) b⁰⁰ ( g^j ) t^j kji + b⁰⁰ ( д^г ) b⁰⁰ ( g^j )( tⁱ kβ + ki )    < i β ( tⁱkti_i + ki ) + '>

^-l----------------И---------------- S  ^{b (g} )------И------.

Multiplying by -|A| > 0 gives

lⁱ (b⁰⁰(gⁱ)b⁰⁰(g^j)t^jkji + b⁰⁰(gⁱ)b⁰⁰(g^j)(tikβ + ki)´    S    b⁰(gⁱ) ∖β(tⁱk*i + ki) + α∖.      (29)

Using the fact that in equilibrium b⁰(gi) = ^+1^ (see Eq. (11)) and rearranging condition (29)
leads to

2 β ( tikii + ki + t^j kji ) S ∖β ( tikⁱ_ti + ki ) + α∖

with β > 0 as given by Eq. (20). Since in case (i), (ii), and (v), β(tⁱk_tⁱi + kⁱ) + α > 0, inequality
(30) can be rewritten to

0 S   2 β ( tiktii + ki — t^j k^ji ) + α.

Note, the inequality is equivalent to

-lⁱτti     S     ^∖b⁰(gⁱ)sⁱti^∖

28

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