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next32/b
To move from QE to CE we require that expenditure minimization implies utility maximization, that if $x_n^*$ minimizes expenditure at price $p^*$ on the set $P(x^*_n)$, then $x^*$ is preference optimal on the set $\{z:p^*z\leq p^*x^*_n\}$.
Cheaper Point Lemma. Suppose at a price $p$, $x'_n$ minimizes expenditure on $R(x'_n)$. Suppose that $P(x'_n)$ is open and that there is an $x^0_n\in X_n$ such that $px^0_n\lt px'_n$. Then $x'_n$ is preference-maximal on the set $\{x''_n\in X_n:px''_n\leq px'_n\}$.Proof. If $x'_n$ is expenditure minimizing on $R(x'_n)$, then $x''_n\succ_nx'_n$ implies $px''_n\geq px'_n$. We must show that the inequality is strict. Suppose to the contrary that $p''x_n=px'_n$. Since $px^0_n\lt px'_n$, $x''_n\succ _nx'_n\succ_n x^0_n$. For all $0\lt t\lt 1$, $p(tx''_n+(1-t)x^0_n)\lt px'_n$, and for $t$ near enough to 1, $(tx''_n+(1-t)x^0_n)\succ_nx'_n$, contradicting expenditure minimization. ∎