Showing UMP and EMP do not exhibit duality if the assumption of local non-satiation is absent

Question

I have been trying to use the contradiction method to prove this, but it does not seem to be working.

Suppose $x^*$ is optimal in both EMP and UMP. Then $u(x^*) \geq u(x')$ for all $x'$ in $B_pw$.

And $p.x^*$ < $p.x'$ $\leq w$

Also $x^*$ is optimal in UMP. So $u(x^*)$ > $u(x')$ for all $x'$.

I can't find a way to reach a contradiction here. Is it that the demand correspondence lies on the budget plane so there is no way LNS can hold because it would lie outside Bp,w?

This is basically MWG proposition 3.E.1 without the LNS assumption.

Show that if $x^*$ is optimal in the utility maximization problem when $w > 0$, then $x^*$ is optimal in the expenditure minimization problem when the constraint is to attain a utility level $u(x^*)$.

Answer 1

A counterexample would work as well.

Consider the utility function $U(x) = 0$, and the budget constraint $1 \cdot x \leq 1$. The solution $x^* = 1$ is feasible and maximizes utility, but it does not minimize expenditure for this utility level (0) for this utility function, $x' = 0$ would achieve a lower expenditure with the same utility.