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

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 $$u(x^*)$$.

• I think that contrary to your title. you don't want a contradictory example (which is easy to give), but a proof by contradiction. Please consider editing the title. Nov 27, 2023 at 13:53
• A counterexample would work as well. I was not able to come up with a counterexample, so just thought that maybe proving by contradiction could be the easier approach (as I could tie it with MWG's explanation), but apparently that's not the case either. Nov 27, 2023 at 14:55
• I just realized if a person has the greatest integer function as his utility function, it could work as a valid counterexample. However, I would appreciate it someone could provide an answer with a more formal approach to my question. Nov 27, 2023 at 15:00

## 1 Answer

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.