This is the excess demand function of consumer t wrier the budget constrain* ;

t t with the tax rates t^. and rebate j\ Define s'Cp.rl = This is the aggregate excess demand function. Not« that if p is m equilibrium price t vector and r is the associated tax rebate for each consumer* liaen z (p,r) = 0.

We shall now show that, conversely, If a price vector p and the associate tax rebate r satisfies zt(p,r) * 0, then they constitute an equilibrium with t ^ t^xes. In fact, then, L^ip» r) ■ Moreover,

111 i 1

By the budget constraints, the first *erm 5s equal, to + 1-r. The second term is equal xo

Hence I^giy^P* P*w| * s Sr- P ^ equilibrium price vector t and r is the associated rebate for each consumer If and only If z [p,r) - 0.

T6j! prove the existence of an equilibrium along the Sine of Proposition t

1TC.1, note first that the aggregate excess demand function z (-) with taxes t t does not satisfy Walras1 law, (That H, p*S Ip.r) * Q.) So we modify 2 M'

to satisfy Walras' law:1 Define t t L

= z ip.r) - (p-z ip,r]/p*p)p « fit , then p-zw(p,r) = 0 for every (p,rJ. Just like the original excess demand f w function Z {-), this modified excess demand function 2 {-) is continuous (in both P and r). Moreover, for each fixed r £ 0, z (%r] satisfies properties

(iv) and (*) of Proposition 17.0,2. For this, since z (\r) satisfies these properties, it sufficient to prove that p-z'(p,r) is bounded above. But this follows froffl p-z^p.r) - p-(E|**(p> P-M. + r) - Zf^

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