<p>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.</p>

<p>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:</p>

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

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

<p>Luckily, this is always compact.</p>

<p>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).</p>

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

CC BY-SA 3.0
2016-10-26T01:09:37.607Z
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"?
Alecos Papadopoulos
CC BY-SA 3.0
2016-10-26T08:29:17.550Z
I meant that given we have established existence the further proof of uniqueness does not require compactness.
BB King
CC BY-SA 3.0
2016-10-24T20:03:46.317Z
2017-01-31T21:20:53.277Z
2017-01-31T21:20:53.277Z
BB King
<p>The preferences are complete, transitive, continuous, strictly monotone and strictly convex</p>

<p>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</p>

<p>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?</p>

<p>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.</p>

<consumer-theory>
Suppose I have a preference satisfy all basic assumptions, how to show consumer problem solution is unique
Eric Chen
# Suppose I have a preference satisfy all basic assumptions, how to show consumer problem solution is unique

(Asked by: username_0 on 2016-10-22)

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.

**Tags:** consumer-theory


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## 1 answers

### Answer by username_1 on 2016-10-24. Score: 3 | Accepted answer

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.

> **Comments:**
> **username_2**: 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"?  
> **username_1**: I meant that given we have established existence the further proof of uniqueness does not require compactness.  

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