<p>I think you have the inequality sign wrong. The inequality as you write it cannot hold. To see this, consider the numerical case where <span class="math-container">$y=4 &gt; y'=3, d=1 &lt; d'=2$</span> and <span class="math-container">$u= \sqrt(x)$</span>. I let you do the calculation yourself. </p>

<p>I suppose the correct inequality is: 
<span class="math-container">\begin{equation} u(y−d)−u(y−d') \leq u(y'−d)−u(y′−d′) \end{equation}</span></p>

<p>This equation holds true due to the curvature of a concave function. If you're on the upward sloping part of the function, the closer you are to the summit, the flatter the slope. So the vertical distance between the images of the two points in the domain on the left side of the equation is smaller than the same on the right side if <span class="math-container">$y&gt;y'$</span> and <span class="math-container">$d \leq d'$</span>, and equal if <span class="math-container">$y=y'$</span>. On the downward sloping part it's the other way around. The set of inequality conditions imply each other. Changing one inequality requires changing the others. You can check this for yourself by picking points on a concave curve that satisfy the inequalities.</p>

<p>It is really just an elaborate version of <span class="math-container">$|x| \leq |y|$</span>, where if <span class="math-container">$x \leq 0, y \leq x$</span> and if <span class="math-container">$x \geq 0, y \geq x$</span>. 
I hope this answers your question. </p>

CC BY-SA 4.0
2019-01-06T21:21:04.200Z
Thanks. I realized that I wrote the inequality of $y$ wrong. I edited it.
shk910
CC BY-SA 4.0
2019-01-06T10:09:53.293Z
2019-01-06T11:13:26.577Z
2019-01-06T11:13:26.577Z
hrrrrrr5602
<blockquote>
<p>Suppose that we have the following inequality:</p>
<p><span class="math-container">$u( y- d) - u(y-d') \ge u(y' - d) - u( y' -d')$</span>.</p>
<p>The concavity of <span class="math-container">$u$</span> together with <span class="math-container">$y\le y'$</span> then implies that <span class="math-container">$d \le d'$</span>.</p>
</blockquote>
<p>I sometimes come across this problem in economic papers, but I don't understand the logic behind this. Can anyone explain this?</p>

CC BY-SA 4.0
2019-01-06T09:54:34.147Z
Please consider a more concise title, e.g. Implication of the concavity of $u$
emeryville
<microeconomics><theory>
Implication of the concavity of u
shk910
# Implication of the concavity of u

(Asked by: username_0 on 2019-01-06)

> Suppose that we have the following inequality:
>
> $u( y- d) - u(y-d') \ge u(y' - d) - u( y' -d')$.
>
> The concavity of $u$ together with $y\le y'$ then implies that $d \le d'$.

I sometimes come across this problem in economic papers, but I don't understand the logic behind this. Can anyone explain this?

**Tags:** microeconomics, theory
> **Comments:**
> **username_1**: Please consider a more concise title, e.g. Implication of the concavity of $u$  

---

## 1 answers

### Answer by username_2 on 2019-01-06. Score: 0 | Accepted answer

I think you have the inequality sign wrong. The inequality as you write it cannot hold. To see this, consider the numerical case where $y=4 > y'=3, d=1 < d'=2$ and $u= \sqrt(x)$. I let you do the calculation yourself.

I suppose the correct inequality is:
\begin{equation} u(y−d)−u(y−d') \leq u(y'−d)−u(y′−d′) \end{equation}

This equation holds true due to the curvature of a concave function. If you're on the upward sloping part of the function, the closer you are to the summit, the flatter the slope. So the vertical distance between the images of the two points in the domain on the left side of the equation is smaller than the same on the right side if $y>y'$ and $d \leq d'$, and equal if $y=y'$. On the downward sloping part it's the other way around. The set of inequality conditions imply each other. Changing one inequality requires changing the others. You can check this for yourself by picking points on a concave curve that satisfy the inequalities.

It is really just an elaborate version of $|x| \leq |y|$, where if $x \leq 0, y \leq x$ and if $x \geq 0, y \geq x$.
I hope this answers your question.

> **Comments:**
> **username_0**: Thanks. I realized that I wrote the inequality of $y$ wrong. I edited it.  

---

