Federal Tax-Transfer Policy and Intergovernmental Pre-Commitment

The state’s marginal cost of public funds (r.h.s. of Eq. (9); henceforth SMCPF) is greater than
unity, as states perceive an outflow of capital if they increase their capital tax rate for a given
tax rate of the other state. At an optimum the benefits of taxation given by the marginal utility
of local public goods are equated to the marginal costs of taxation represented by the term ɪɪ.

The state first-order condition (9), together with the state budget constraint (6), determines
the reaction functions {tⁱ

Federal Government
optimization problem is:

max
^{s ,s ,τ}           i=1,2                                      i=1,2

subject to {kⁱ = kⁱ(t^τ,t^j)}_τ==ι,2 _τ=j and {l^τ = lⁱ(τ)}_i=ι,2. μ denotes the Lagrangian multiplier
associated with the federal budget constraint. The first-order conditions are

si :  Vg + μ = 0 and τ : X VT + μ(-lⁱ — τlT_τ) = 0.               (10)

i=1,2

Making use of (5) and rearranging

1τ
b⁰(g ) =  -----г > 1 with η := l-_r.                            (11)

1 + η^{i                          τ} lⁱ

The federal government sets transfers and the tax rate so as to equate the marginal rate of
substitution to the federal marginal cost of public funds (r.h.s. of Eq. (11); henceforth FM-
CPF). As labor taxes are distortionary (l_τ^τ < 0) the FMCPF exceed the social marginal rate of
transformation (= 1) which yields b⁰ (g^τ) > 1. Note, federal transfer policy equalizes the SMCPF
and FMCPF; thereby aligning the tax base elasiticities e^l and η^τ.

The federal first-order conditions (11) and the federal budget constraint (6) define the reac-
tion functions {s^τ = s^τ(t^τ, t^j), τ = τ(t^τ, t^j)}_,j_=1,2 6₌j.

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