<p><span class="math-container">$u:\mathbb R^n\to\mathbb R$</span> is a quasi-concave utility function so the indifference curves are convex.</p>
<p><span class="math-container">$a,b\in\mathbb R^n$</span> are two points. Our budget set is the (one-dimensional) segment <span class="math-container">$[a,b]$</span> that connects <span class="math-container">$a$</span> and <span class="math-container">$b$</span>.</p>
<p>Given: <span class="math-container">$$x^*=\arg\max_{x\in[a,b]}u(x)$$</span></p>
<p>Let <span class="math-container">$b'$</span> be a point in the segment <span class="math-container">$[a,x^*]$</span>. That is: <span class="math-container">$b'=\lambda a+(1-\lambda)x^*$</span> for any <span class="math-container">$\lambda\in[0,1]$</span>.</p>
<p>Prove that:</p>
<p><span class="math-container">$$b'=\arg\max_{x\in[a,b']}u(x)$$</span></p>
<p>Graphically this result is very straight forward but I don't know how to mathematically prove it.</p>
<p>I think we could start of proving that <span class="math-container">$u(\lambda a+(1-\lambda) x^*)$</span> is monotonically decreasing with <span class="math-container">$\lambda$</span>.</p>
<p>Are there named theory related?</p>

CC BY-SA 4.0
2020-08-07T07:05:12.970Z
By $\mathbb R^n$ did you simply mean the real line $\mathbb R$? The notation $[a,b]$ implies $b>a$, but if we take $n=2$ and the budget set be the line segment connecting $(0,1)$ and $(1,0)$, then it's not obvious which point should be $a\in\mathbb R^2$ and which be $b\in\mathbb R^2$.
Herr K.
CC BY-SA 4.0
2020-08-07T09:39:03.180Z
@HerrK. Either way, the argument holds geometrically, I think. The notation is a bit unconventional I suppose but it is pretty clear that $a,b$ are any $\mathbb R^n$ points.
High GPA
CC BY-SA 4.0
2020-08-07T16:25:43.080Z
I know the result is unaffected. I'm just saying that maybe you should be a little more careful with notations.
Herr K.
<utility><consumer-theory><optimization><demand>
When the global optimal is outside of the constraint set, what will be the demand?
High GPA
# When the global optimal is outside of the constraint set, what will be the demand?

(Asked by: username_0 on 2020-08-06)

$u:\mathbb R^n\to\mathbb R$ is a quasi-concave utility function so the indifference curves are convex.

$a,b\in\mathbb R^n$ are two points. Our budget set is the (one-dimensional) segment $[a,b]$ that connects $a$ and $b$.

Given: $$x^\*=\arg\max\_{x\in[a,b]}u(x)$$

Let $b'$ be a point in the segment $[a,x^\*]$. That is: $b'=\lambda a+(1-\lambda)x^\*$ for any $\lambda\in[0,1]$.

Prove that:

$$b'=\arg\max\_{x\in[a,b']}u(x)$$

Graphically this result is very straight forward but I don't know how to mathematically prove it.

I think we could start of proving that $u(\lambda a+(1-\lambda) x^\*)$ is monotonically decreasing with $\lambda$.

Are there named theory related?

**Tags:** utility, consumer-theory, optimization, demand

> **Comments:**
> **username_1**: By $\mathbb R^n$ did you simply mean the real line $\mathbb R$? The notation $[a,b]$ implies $b>a$, but if we take $n=2$ and the budget set be the line segment connecting $(0,1)$ and $(1,0)$, then it's not obvious which point should be $a\in\mathbb R^2$ and which be $b\in\mathbb R^2$.  
> **username_0**: @HerrK. Either way, the argument holds geometrically, I think. The notation is a bit unconventional I suppose but it is pretty clear that $a,b$ are any $\mathbb R^n$ points.  
> **username_1**: I know the result is unaffected. I'm just saying that maybe you should be a little more careful with notations.  

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## 0 answers

