Income Taxation when Markets are Incomplete

122

M. Tirelli

	...x ^h...	...θ^h...	...λ^h...	...υ ^j...	...y ^j...	iiiq ^jiii	t¹	t⁰
1	[D²u^h]	---- 0	---- -[IN]	--- 0	---- 0	--- 0	----- 0	-- 0

2	-I_NH	[W]	0	0	...θ_j^hIN...	iiiiiiiiii θ^hT— θh 0S×J	iiiiii ^T 0,iii,0 t∑j ]	0 IS
			[W^T]		pï	i_λii₀_hii_Ii_Jii	iii, 0, iii iii, λ_s y_s , iii
3	0	0	[W^T]	0	pï	i_λii₀_hii_Ii_Jii	iii, 0, iii iii, λ_s y_s , iii	⁰ (5)
4	0	0	..iθjlN...	T [Df^jT]	-[D²f^j]	0	0, iii, 0 ∑ hθj [ λh ]	0
5	0	0	0	0	[Df^j]	0	0	0
6	0	...I_J ...	0	0	0	0	0	0
7	0	0	0	0	0	0	[Σ j ∈ J2 ys ]	HIS
8	[Du^h]	0	0	0	0	0	0	0
[c_s ] denotes a		diagonal matrix with typical diagonal element c_s					varying across s	>

Looking at Equation (2), it is immediate that, if DG has full row rank,
rDG = 0 has equations outnumbering the unknowns r = (r¹ ,... ,r⁸) ∈
Rⁿ⁺^H . Thus, it is sufficient to show that the Jacobian (5) has independent
rows. An explicit formulation of (4) is:

(I) (II)	∀h, r_h¹ D²u^h - r_h² + r_h⁸Du^h = 0 ∀ h,j,r^h Wj + rj = 0	NH JH	r¹ r³
(III)	∀ h, — r1 + r3 WT + ∑ j^θjrj = 0	NH	r²
(IV)	∀j ∈ J2,r_j⁴Df^jT = 0	J₂	r⁵
(V)	∀s^, ∀ j ∈ J2^, — ∑ h (^θ>h²s — ' )	NJ2	r⁴
	— r_j⁴Di²s f^j — r_j⁵Dsf^j = 0		(6)
(VI)	∀ j^, ∑ hrh,o (j—θh )—∑ hλ hrj=0	J	r⁶
(VII)	∑ ∑ ∑ ∑ h∩22- I г-3 h h τu^j∖ ∀s ∈ S, _j,h θ_j r_h,s Ws + r_h,j λ_s Ws	S	(r⁸)
	+ ∑j ∈ J2 (rj W^j + rl_sβ^j) = 0
(VIII)	∀s ∈ s^, ∑h rh s + Hrl = 0	S	r⁷
(IX)	Il r II — 1 = 0	1

where each block of equations is labelled in Roman numbers on the left-hand
side, and the last two columns on the right-hand side indicate the number of
equations in each block, with the unknown variables (r ) to which they can
be matched.

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