Income Taxation when Markets are Incomplete

M. Tirelli

for a policy reform dζ (e.g., see Equation (71) in Magill and Shafer (1991),
and Section 3 in Citanna et al. (2001)).

If H ≤ S - J, our main result shows that there exists a generic set of
economies in which equilibria possess sufficient utility variation (see (iv)
in Theorem 2 below). This upper bound condition is only sufficient, as we
have shown in Examples 1, 2 above.

4. Proofs

First-order (sufficient) conditions (Foc’s) for the existence of an interior
E - tI (with (x^h, θj , y^j) » 0 for all j ∈ J2, θ^h = 0 for all j ∈ J1 and all
h) can be written as:

(2)

where λ, υ are Lagrange multipliers.²² Each subsystem of equations is in-
dexed by Arabic numbers on the left-hand side, and its dimension is indicated
on the right-hand side.

Let F : Ξ' × TI × Ψ → Rⁿ', where F(ξ', t¹, ψ) = 0 represents the
left-hand side of the system of equilibrium first-order conditions, excluding
the government budget constraint, where Ξ' ⊂ Rⁿ has typical element
ξ' = (x, θ, λ, y, υ, q, t⁰), with n' = 2HN + HJ + JN + 2J + S, and
Ψ ⊂ R + denotes the parameter space, with typical element ψ = (e, η, θ),
andk = N (H + J)+HJ. Here, without loss of generality, we let the policy

²² In particular, the first two multipliers are attached to the constraints of the consumer
problem (ρ refers to the no-short sale constraints). Here υ is the vector of multipliers related
to the firms’ optimization problem.

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