Income Taxation when Markets are Incomplete

Income taxation when markets are incomplete

Next, restricting attention to economies in Ψ *,we turn to proving Theo-
rem 1. We do so by showing that G = FF, u¹, ... , uH) (ξ', tI, ω) behaves
as a submersion, when it is evaluated at a no-tax equilibrium of ψ ∈ Ψ *.
As we have already argued, this is equivalent to showing that the following
system (locally) has no solution:

To this end, it is sufficient to show that there exists a local parameterization
of the economy such that (4) has the desired (local) properties. For reasons
that will soon become clear, the type of parameterization that does the job
is one that allows us to perturb the Hessian of u and f without altering
the values of the functions and their gradients. The latter is convenient,
since it implies that perturbations do not modify the equilibrium first-order
conditions. One such class of functions is the one that admits quadratic
perturbations. Denoting by clN the closure of the set N , let

u^{h (}x^h; A^h) = u^h (x^h) + 1 (x^h - x^h)TA^hγ^h (x^h) (x^h - x^h) if x^h ∈ N(x^h)
= u^{h (}x^h) if x^h / clN_e(j^^h),

f^{j (} y^j ; B^{j )} = f^{j (} y^{j )} + 1 (y^j - y ^j)TB^jδ^{j (} y^{j )} (y^j - У^j) if y^j ∈ N (y^j)
= f^j (y^j) if y^j / clNAy^j).

Here A^h is a square symmetric N -dimensional matrix of parameters, which
are taken to be sufficiently small to preserve strict quasi-concavity of the
utility function. Define by N, and N_e, neighborhoods of x^h, possibly empty,
such that clN ⊂ N_e ⊂ clN_e ⊂ R+₊, e > 0. Further, γ^h are smooth bump
functions (see Hirsch (1976), p. 41), taking value 1 if x^h / N,and0if
x^h / clN_e. The same construction has been used for f^j.

Without loss of generality, the Jacobian of G with respect to (ξ', t¹ ),
evaluated at t^I = t ¹ = 0, is represented in the following table with the first
column referring to the equation numbering in (2) above, and the first row
reporting the variables with respect to which derivatives are computed.^25
25 Observe that delating the last row block from the above Jacobian provides the represen-
tation of the Jacobian of the equilibrium first-order conditions of an E - t^I (i.e., the Jacobian
of F), evaluated at t^I = t ¹ = 0.

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