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The preferences are complete, transitive, continuous, strictly monotone and strictly convex

The textbook says it is because (1) Budget Set is compact so that a solution exists; (2) B is convex and the utility function representing this preference is strictly quasi-concave

What if I have no information on budget set, that is, I don't know what properties B might have. Can all arguements above still hold?

Can I use the strict increasing utility function to prove the solution is unique? Since a preference with all these assumptions ensures a continuous, strictly increasing and strictly quasi-concave utility function. Then by contradiction, there can not be two or more maximization point of utility function.

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In that case you would still have a problem showing the existence of a solution. Without existence, it makes little sense to think about uniqueness. The reason a continuous utility function (implied by some of those preference assumptions you gave) has a maximum at all is because a continuous function achieves a maximum somewhere on a compact set, which in this case is B.

Luckily, we don't really have to imagine scenarios in which we have no information about the budget set. This is not some abstract set, but rather something very concretely defined as:

$B(p,w)=\{x \in \Bbb R_{+}^{n}: px \leq w \}$,

where $p$ is the price vector, $x$ is the demand, $w$ is the budget and $n$ is the dimension of the vector.

Luckily, this is always compact.

So, no, you cannot show there is a unique solution, because in that case you can't show there is a solution in the first place (using standard tools).

However, you are correct in thinking that uniqueness itself does not hinge upon the compactness of the budget set.

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  • $\begingroup$ I am confused with your last sentence. Uniqueness of a solution does not even enter the picture without compactness, since a solution does not exist, so in what sense "it does not hinge upon compactness"? $\endgroup$ – Alecos Papadopoulos Oct 26 '16 at 1:09
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    $\begingroup$ I meant that given we have established existence the further proof of uniqueness does not require compactness. $\endgroup$ – BB King Oct 26 '16 at 8:29

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