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M. Tirelli
Matching the equations in system (7) with the unknowns it is immediate
that if H ≤ S the system has equations outnumbering the unknowns (r).27
Therefore, we are left to show that system (7) has independent equations.
The proof can be divided into the following two cases (a) and (b).
Case (a): rh1 = 0 for all h∖
The following perturbations can be used: ((dr.2,s)sJ=1,dr1), respectively,
for Equations (II), (III); drh3 for h = H, for (VI); dr7 for (VIII); dAh for
(I)h (given rh1 = 0) for all h. Finally, use ((drH2,s)s>J,drH3 ) for (VII).
Case (b): rh1 = 0 for some~h∣
Since, as we are going to show, rh1 = 0, for some h, implies that rh1 = 0
for all h, we cannot use quadratic perturbations of utilities in the present
case. Yet, our result continues to hold when r1 = 0, and the uh are fixed in
U.
Consider system (7). Now rh1 = 0, say for consumer h = H, implies that
rH2 - rH8 DuH = 0. Then, consumer H’s foc’s and (II) imply r6 = 0. Since
r6 is independent of h, rh2 ∈ {W)⊥ holds for all h such that rh = 0. By (III),
r1 = WrhT, i.e., either r1 ∈ {W) or rh = 0 (and r1 = 0) for all h = H.
Then, post-multiplying (I)h by rh1, and using the latter result, together with
strict quasi-concavity of uh, we have rh1 = 0. This holds for all h.
Also r1 = 0 implies r3 = 0 by (III). Then, consumers’foc’s ( λh ∙ Rj = qj )
imply that we can rewrite (VII) as Yh rhλhs Y1j Rs ij = Yh rhλhszhs for all
s ∈ S, or, using the notation introduced in Theorem 2, as
( r 18, ...,rH ) AZ = 0,
where AZ in (3) reduces to
λ λ1 z1 ∙∙∙ λk!
AZ =
(8)
..
..
..
λH H λH H
∖λ 1 z 1 ∙ ∙ ∙ λSzS
Since we are considering economies in Ψ *, under the assumption that
H ≤ S — J, Theorem 2 implies that r8 = (rf, ... , rH ) = 0. This im-
mediately implies r = 0. System (7) has no solution.
Part 2 (production GEI) J2 = 0.
We prove the result for the case of a pure production economy (with
J1 =0, J2 =0), since then extending the argument to mixed cases is
27 This has a natural interpretation: the number of policy objectives (the utility levels)
should not exceed the number of tax instruments (contingent taxes/transfers). The latter
condition is an incomplete market analogue of the one due to Tinbergen (1952). Thus, for
example, this bound could be dropped by introducing date 0, lump-sum transfers, τ ∈ RH .