Income Taxation when Markets are Incomplete



118


M. Tirelli

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

If HS - 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 (xh, θj , yj) » 0 for all j J2, θh = 0 for all j J1 and all
h) can be written as:

(1)

h,       Duh (xh) - λh = 0

NH

(2)

h,      - xh +e + W(q,y,t1 h = 0

NH

(3)

h,       λhW(q, y, t1) = 0

JH

(4)

j,       βj - UjDfj(yj) = 0, βj =V θhλhU(11 + t1 )

h

NJ2

(5)

j,        fj(yj) =0

J2

(6)

j J1, Σ θJ= 0
h

J1

j J2, V θJ- 1 = 0
h

J2

(7)

v      Vf10 + 11fV θhRj + V θhvjʌʌ = 0

s,              ts + ts       θj Rs + θj ys    = 0,

hH           jJ1         jJ2

S

(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 × ΨRn', 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),
and
k = 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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