Income Taxation when Markets are Incomplete

M. Tirelli

investment by a corresponding y₀^j θ_j^h, at date 0, and θ_j^h entitles h to receive
ys^j θ_j^h in state s (for all s in S), at date 1.

At time zero, consumers can also trade in bonds, which are assets in zero
net supply. The (after-tax) asset matrix is W = (W1, W2), where

^{W1= ...}  (1 +^qt^js¹)Rs^j j∈J₁

and M^j(y 1, 11 ) has typical s-element m^j = ( 1+t_s¹ )y^j .Lastly, agents face the
same financial structure W. Thus, they all have equal access opportunities
to the equity market. Bankruptcy or default is not allowed.

eh = I ehh + ∑q^jθ^,eh + t10
j∈J2

define the total after-tax endowment of a typical consumer h. The individual
budget set, for given financial opportunities W and total endowments e^h, is

B ^{w e}h ) = Ph ∈ R +:Xh ≥ eh= ∈Jφ

An individual optimum for h, at (W,eh), is a pair (xh, ^θ)h) such that x^ft
maximizes uh(xh) on B W ∙e^h), and ^θ^h satisfies x^h — eh = Wθh, and
θ_j^h ≥ 0forallj inJ2.

At an interior individual optimum, x^h' ^ 0, the present value vector (or
stateprice) of consumer h instate s isλh = λh(xh) = D_xsu(x^h)∕D_x0 0 u(xh).
Let ( W) denote the column span of W, a subspace of dimension (at most)
equal to J. Its orthogonal complement, ( W) ⊥, has (at least) dimension N — J.
If we assume, for the moment, that the no-short sale constraints are non-
binding, λh ∈ (W)⊥.

2.1.5. The firm’s problem If we keep on abstracting from the no-short sale
constraints, the fact that, at the individual optimum, present value vectors
satisfy λh ∈ (W)⊥, implies that consumers do not typically agree on the
evaluation of future income profiles. Since consumers are also firms’ share-
holders, the latter poses problems for the definition of the objective function
of the firm. Precisely, if the firm chooses its production plans in the best

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