<p><strong>Proof by contradiction</strong>. Suppose <span class="math-container">$(x_1^*, x_2^*)$</span> solves the problem and <span class="math-container">$x_1^* \neq x_2^*$</span> is true. By symmetry of the utility and equal prices, <span class="math-container">$(x_2^*, x_1^*)$</span> which is not the same bundle as <span class="math-container">$(x_1^*, x_2^*)$</span> also solves the given problem contradicting that the solution is unique.</p>
<p><strong>Direct Proof</strong>. Alternatively, you can also write the same argument directly. Given that <span class="math-container">$(x_1^*, x_2^*)$</span> solves the problem, by symmetry of the utility and equal prices, <span class="math-container">$(x_2^*, x_1^*)$</span> also solves the given problem. Since the solution is unique, <span class="math-container">$(x_1^*, x_2^*)=(x_2^*, x_1^*)$</span> must be true. Therefore, <span class="math-container">$x_1^*=x_2^*$</span>.</p>

CC BY-SA 4.0
2023-04-26T17:23:55.720Z
I feel stupid for not being able to see such a logical reason before. Thanks a lot.
mynameparv
CC BY-SA 4.0
2023-04-26T15:12:23.660Z
2023-04-26T15:28:58.460Z
2023-04-26T15:28:58.460Z
Amit
<p>Encountered the following problem during self study:
<a href="https://i.sstatic.net/Aj7I3.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Aj7I3.png" alt="enter image description here" /></a></p>
<p>My take on the problem is that if we can show that the equation of the income expansion path is <span class="math-container">$x_1=x_2$</span> for all such <span class="math-container">$U(x_1,x_2)$</span> then we have proved the claim. I can see the result holding in examples for different utility functions, for example, <span class="math-container">$U=x_1x_2$</span>, but I am unsure how to prove this more generally. Any help is appreciated</p>

<microeconomics><mathematical-economics><consumer-theory><optimization>
Existence and uniqueness of demand, and symmetry implies equal demands given equal prices
mynameparv
# Existence and uniqueness of demand, and symmetry implies equal demands given equal prices

(Asked by: username_0 on 2023-04-26)

Encountered the following problem during self study:
[![enter image description here](https://i.sstatic.net/Aj7I3.png)](https://i.sstatic.net/Aj7I3.png)

My take on the problem is that if we can show that the equation of the income expansion path is $x\_1=x\_2$ for all such $U(x\_1,x\_2)$ then we have proved the claim. I can see the result holding in examples for different utility functions, for example, $U=x\_1x\_2$, but I am unsure how to prove this more generally. Any help is appreciated

**Tags:** microeconomics, mathematical-economics, consumer-theory, optimization


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## 1 answers

### Answer by username_1 on 2023-04-26. Score: 5 | Accepted answer

**Proof by contradiction**. Suppose $(x\_1^\*, x\_2^\*)$ solves the problem and $x\_1^\* \neq x\_2^\*$ is true. By symmetry of the utility and equal prices, $(x\_2^\*, x\_1^\*)$ which is not the same bundle as $(x\_1^\*, x\_2^\*)$ also solves the given problem contradicting that the solution is unique.

**Direct Proof**. Alternatively, you can also write the same argument directly. Given that $(x\_1^\*, x\_2^\*)$ solves the problem, by symmetry of the utility and equal prices, $(x\_2^\*, x\_1^\*)$ also solves the given problem. Since the solution is unique, $(x\_1^\*, x\_2^\*)=(x\_2^\*, x\_1^\*)$ must be true. Therefore, $x\_1^\*=x\_2^\*$.

> **Comments:**
> **username_0**: I feel stupid for not being able to see such a logical reason before. Thanks a lot.  

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