Federal Tax-Transfer Policy and Intergovernmental Pre-Commitment

The sign of |A¹| and |A²| depends on the sign and magnitude of α and β and is thus ambiguous.

Using Eqs. (25) and (26) the slopes of the best-response functions (24) are:

∂sⁱ _ β ( tⁱkii + kⁱ ) + α

∂tⁱ =            ∣A∣         ’

∂s^j      βt^j k^ji - α

∂tf  = Й ’

∂τ         b⁰⁰(gⁱ)b⁰⁰(g^j)t^jk_t^ji + b⁰⁰(gⁱ)b⁰⁰(g^j)(tⁱk_tⁱi + kⁱ)

∂)tⁱ =                   Й

A.2 Proof of Proposition 3

Proposition 3 is proved by establishing two lemmata. Lemma 1 states conditions for t^d to be
greater, equal, or lower than t^N . Lemma 2 presents a refined condition pertaining to cases (i),
(ii), and (v) listed in table 1.

Lemma 1: If sⁱ_ti ≥ 0 the capital tax rate under decentralized leadership exceeds the capital
tax rate under both-sided Nash behavior, i.e. t^d > t^N . If sⁱ_ti < 0, the effect on the tax rate tⁱ is

<

>

Proof: The optimal state behavior is characterized by Eq. (21). Thus, starting from the
Nash outcome t^N , as characterized by Eq. (9), state i’s incentive to deviate from t^N depends on
the terms V_gⁱsⁱ_ti and V_τⁱτ_ti . By Eq. (5) V_τⁱτ_ti = -lⁱτ_ti which following Eq. (18) is positive. Note,
V_gⁱsⁱ_ti = b⁰(gⁱ)sⁱ_ti . If b⁰(gⁱ)sⁱ_ti is non-negative (case (iii) and (iv)), state i can increase after-tax
labor income as well as local public good provision by setting t^d > t^N . To prove the second
assertion, consider the opposite case. If sⁱ_ti < 0 and -lⁱτ_ti > (<) ^Ib⁰(gⁱ)sⁱ_ti ^I, state i can improve
utility of the representative household by choosing t^d > (<) t^N . For the special case sⁱ_ti < 0 and
-lⁱτ_ti = I b⁰(gⁱ)sⁱ_ti I the opposite effects nullify each other and t^d = t^N. □

27

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