<p>What you call &quot;<span class="math-container">$g(1)-g(0)$</span>&quot; is sufficient because it is actually a condition on <span class="math-container">$g_{\mathbf{x}, \mathbf{y}}(1) - g_{\mathbf{x}, \mathbf{y}}(0)$</span>, which is required to hold <strong>for all <span class="math-container">$x$</span> and <span class="math-container">$y$</span></strong>. In other words, your suggestion to require <span class="math-container">$g(p) - g(q) \leq g'(q)(p - q)\,\,\forall p,q\in[0,1]$</span> is already contained in the proof's requirement, since for given <span class="math-container">$x$</span> and <span class="math-container">$y$</span> any choice of <span class="math-container">$p$</span> and <span class="math-container">$q$</span> just results in another (shorter) line segment between some point <span class="math-container">$a$</span> and some point <span class="math-container">$b$</span> in <span class="math-container">$U$</span> which can also be represented by setting <span class="math-container">$x=a$</span> and <span class="math-container">$y=b$</span>, scaling again by <span class="math-container">$t\in[0,1]$</span> between these two endpoints.</p>

CC BY-SA 4.0
2023-10-13T09:29:11.220Z
2023-10-13T09:29:11.220Z
VARulle
<h2><strong>Background Information</strong></h2>
<p>I am studying concave and convex functions using Mathematics for Economists by Simon and Blume. I have difficulties understanding their proof of the following theorem:</p>
<blockquote>
<p><strong>Theorem 21.3</strong><span class="math-container">$\quad$</span> Let <span class="math-container">$f$</span> be a <span class="math-container">$C^1$</span> function on a convex subset <span class="math-container">$U$</span> of <span class="math-container">$\mathbb{R}^{\mathbf{n}}$</span>. Then, <span class="math-container">$f$</span> is concave on <span class="math-container">$U$</span> if and only if for all <span class="math-container">$\mathbf{x}$</span>, <span class="math-container">$\mathbf{y}$</span> in <span class="math-container">$U$</span>:
<span class="math-container">\begin{align*}
f(\mathbf{y}) - f(\mathbf{x}) \leq Df(\mathbf{x})(\mathbf{y} - \mathbf{x});
\end{align*}</span>
that is,
<span class="math-container">\begin{align*}
f(\mathbf{y}) - f(\mathbf{x}) \leq \frac{\partial f}{\partial x_1}(\mathbf{x})(y_1 - x_1) + \dots + \frac{\partial f}{\partial x_n}(\mathbf{x})(y_n - x_n).
\end{align*}</span></p>
</blockquote>
<p>Here is the proof in textbook:</p>
<blockquote>
<p><strong>Proof</strong><span class="math-container">$\quad$</span> Let <span class="math-container">$\mathbf{x}$</span> and <span class="math-container">$\mathbf{y}$</span> be arbitrary points in <span class="math-container">$U$</span>. Let
<span class="math-container">\begin{align*}
g_{\mathbf{x}, \mathbf{y}}(t) &amp; \equiv f(t\mathbf{y} + (1 - t)\mathbf{x}) \\
&amp; = f(x_1 + t(y_1 - x_1), \dots, x_n + t(y_n - x_n)).
\end{align*}</span>
Then, by the Chain Rule,
<span class="math-container">\begin{align*}
g'_{\mathbf{x}, \mathbf{y}}(t) = \sum_{i = 1}^{n} \frac{\partial f}{\partial x_i}(\mathbf{x} + t(\mathbf{y} - \mathbf{x}))(y_i - x_i)
\end{align*}</span>
and
<span class="math-container">\begin{align*}
g'_{\mathbf{x}, \mathbf{y}}(0) = \sum_{i = 1}^{n} \frac{\partial f}{\partial x_i}(\mathbf{x})(y_i - x_i) = Df(\mathbf{x})(\mathbf{y} - \mathbf{x}).
\end{align*}</span>
By Theorem 21.1 and 21.2, <span class="math-container">$f$</span> is concave if and only if every such <span class="math-container">$g_{\mathbf{x}, \mathbf{y}}$</span> is concave if and only if for every <span class="math-container">$\mathbf{x}$</span>, <span class="math-container">$\mathbf{y} \in U$</span>,
<span class="math-container">\begin{align*}
g_{\mathbf{x}, \mathbf{y}}(1) - g_{\mathbf{x}, \mathbf{y}}(0) \leq g'_{\mathbf{x}, \mathbf{y}}(0)(1 - 0) = g'_{\mathbf{x}, \mathbf{y}}(0)
\end{align*}</span>
if and only if for every <span class="math-container">$\mathbf{x}$</span>, <span class="math-container">$\mathbf{y} \in U$</span>,
<span class="math-container">\begin{align*}
f(\mathbf{y}) - f(\mathbf{x}) \leq Df(\mathbf{x})(\mathbf{y} - \mathbf{x}).
\end{align*}</span></p>
</blockquote>
<p>The proof uses the following two theorems, Theorems 21.1 and 21.2:</p>
<blockquote>
<p><strong>Theorem 21.1</strong><span class="math-container">$\quad$</span> Let <span class="math-container">$f$</span> be a function defined on a convex subset <span class="math-container">$U$</span> of <span class="math-container">$\mathbb{R}^{\mathbf{n}}$</span>. Then, <span class="math-container">$f$</span> is concave if and only if its restriction to every line segment in <span class="math-container">$U$</span> is a concave function of one variable.</p>
</blockquote>
<blockquote>
<p><strong>Theorem 21.2</strong><span class="math-container">$\quad$</span> Let <span class="math-container">$f$</span> be a <span class="math-container">$C^1$</span> function on an interval <span class="math-container">$I$</span> in <span class="math-container">$\mathbb{R}$</span>. Then, <span class="math-container">$f$</span> is concave on <span class="math-container">$I$</span> if and only if
<span class="math-container">\begin{align*}
f(y) - f(x) \leq f'(x)(y - x) \quad \forall x, y \in I.
\end{align*}</span></p>
</blockquote>
<h2><strong>My Question</strong></h2>
<p>I have problem with the part &quot;By Theorem 21.1 and 21.2, <span class="math-container">$f$</span> is concave if and only if <span class="math-container">$g$</span> is concave if and only if <span class="math-container">$g(1) - g(0) \leq g'(0)(1 - 0) = g'(0)$</span>.&quot; Apparently, &quot;<span class="math-container">$f$</span> is concave if and only if <span class="math-container">$g$</span> is concave&quot; is by Theorem 21.1, because <span class="math-container">$g$</span> is a restriction of <span class="math-container">$f$</span> to any line segment in <span class="math-container">$U$</span>.</p>
<p>But, why is it that &quot;<span class="math-container">$g$</span> is concave if and only if <span class="math-container">$g(1) - g(0) \leq g'(0)(1 - 0) = g'(0)$</span>&quot;? I guess this is supposed to be based on Theorem 21.2, because <span class="math-container">$g$</span> is a <span class="math-container">$C^1$</span> function on an interval <span class="math-container">$I = [0,1]$</span> in <span class="math-container">$\mathbb{R}$</span>. However, if this is the case, then shouldn't it have written: <span class="math-container">$g$</span> is concave (on <span class="math-container">$[0,1]$</span>) if and only if
<span class="math-container">\begin{align*}
g(p) - g(q) \leq g'(q)(p - q)\quad \forall p, q \in [0,1]?
\end{align*}</span>
Why is <span class="math-container">$g(1) - g(0)$</span> sufficient?</p>
<p>Did I miss anything or is this proof wrong? Could someone please help me? Thanks a lot in advance!</p>

<microeconomics><mathematical-economics>
About Proof of Theorem 21.3 in Mathematics for Economists by Simon and Blume
Beerus
# About Proof of Theorem 21.3 in Mathematics for Economists by Simon and Blume

(Asked by: username_0 on 2023-10-12)

## **Background Information**

I am studying concave and convex functions using Mathematics for Economists by Simon and Blume. I have difficulties understanding their proof of the following theorem:

> **Theorem 21.3**$\quad$ Let $f$ be a $C^1$ function on a convex subset $U$ of $\mathbb{R}^{\mathbf{n}}$. Then, $f$ is concave on $U$ if and only if for all $\mathbf{x}$, $\mathbf{y}$ in $U$:
> \begin{align\*}
> f(\mathbf{y}) - f(\mathbf{x}) \leq Df(\mathbf{x})(\mathbf{y} - \mathbf{x});
> \end{align\*}
> that is,
> \begin{align\*}
> f(\mathbf{y}) - f(\mathbf{x}) \leq \frac{\partial f}{\partial x\_1}(\mathbf{x})(y\_1 - x\_1) + \dots + \frac{\partial f}{\partial x\_n}(\mathbf{x})(y\_n - x\_n).
> \end{align\*}

Here is the proof in textbook:

> **Proof**$\quad$ Let $\mathbf{x}$ and $\mathbf{y}$ be arbitrary points in $U$. Let
> \begin{align\*}
> g\_{\mathbf{x}, \mathbf{y}}(t) & \equiv f(t\mathbf{y} + (1 - t)\mathbf{x}) \\
> & = f(x\_1 + t(y\_1 - x\_1), \dots, x\_n + t(y\_n - x\_n)).
> \end{align\*}
> Then, by the Chain Rule,
> \begin{align\*}
> g'\_{\mathbf{x}, \mathbf{y}}(t) = \sum\_{i = 1}^{n} \frac{\partial f}{\partial x\_i}(\mathbf{x} + t(\mathbf{y} - \mathbf{x}))(y\_i - x\_i)
> \end{align\*}
> and
> \begin{align\*}
> g'\_{\mathbf{x}, \mathbf{y}}(0) = \sum\_{i = 1}^{n} \frac{\partial f}{\partial x\_i}(\mathbf{x})(y\_i - x\_i) = Df(\mathbf{x})(\mathbf{y} - \mathbf{x}).
> \end{align\*}
> By Theorem 21.1 and 21.2, $f$ is concave if and only if every such $g\_{\mathbf{x}, \mathbf{y}}$ is concave if and only if for every $\mathbf{x}$, $\mathbf{y} \in U$,
> \begin{align\*}
> g\_{\mathbf{x}, \mathbf{y}}(1) - g\_{\mathbf{x}, \mathbf{y}}(0) \leq g'\_{\mathbf{x}, \mathbf{y}}(0)(1 - 0) = g'\_{\mathbf{x}, \mathbf{y}}(0)
> \end{align\*}
> if and only if for every $\mathbf{x}$, $\mathbf{y} \in U$,
> \begin{align\*}
> f(\mathbf{y}) - f(\mathbf{x}) \leq Df(\mathbf{x})(\mathbf{y} - \mathbf{x}).
> \end{align\*}

The proof uses the following two theorems, Theorems 21.1 and 21.2:

> **Theorem 21.1**$\quad$ Let $f$ be a function defined on a convex subset $U$ of $\mathbb{R}^{\mathbf{n}}$. Then, $f$ is concave if and only if its restriction to every line segment in $U$ is a concave function of one variable.

> **Theorem 21.2**$\quad$ Let $f$ be a $C^1$ function on an interval $I$ in $\mathbb{R}$. Then, $f$ is concave on $I$ if and only if
> \begin{align\*}
> f(y) - f(x) \leq f'(x)(y - x) \quad \forall x, y \in I.
> \end{align\*}

## **My Question**

I have problem with the part "By Theorem 21.1 and 21.2, $f$ is concave if and only if $g$ is concave if and only if $g(1) - g(0) \leq g'(0)(1 - 0) = g'(0)$." Apparently, "$f$ is concave if and only if $g$ is concave" is by Theorem 21.1, because $g$ is a restriction of $f$ to any line segment in $U$.

But, why is it that "$g$ is concave if and only if $g(1) - g(0) \leq g'(0)(1 - 0) = g'(0)$"? I guess this is supposed to be based on Theorem 21.2, because $g$ is a $C^1$ function on an interval $I = [0,1]$ in $\mathbb{R}$. However, if this is the case, then shouldn't it have written: $g$ is concave (on $[0,1]$) if and only if
\begin{align\*}
g(p) - g(q) \leq g'(q)(p - q)\quad \forall p, q \in [0,1]?
\end{align\*}
Why is $g(1) - g(0)$ sufficient?

Did I miss anything or is this proof wrong? Could someone please help me? Thanks a lot in advance!

**Tags:** microeconomics, mathematical-economics

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## 1 answers

### Answer by username_1 on 2023-10-13. Score: 2 | Accepted answer

What you call "$g(1)-g(0)$" is sufficient because it is actually a condition on $g\_{\mathbf{x}, \mathbf{y}}(1) - g\_{\mathbf{x}, \mathbf{y}}(0)$, which is required to hold **for all $x$ and $y$**. In other words, your suggestion to require $g(p) - g(q) \leq g'(q)(p - q)\,\,\forall p,q\in[0,1]$ is already contained in the proof's requirement, since for given $x$ and $y$ any choice of $p$ and $q$ just results in another (shorter) line segment between some point $a$ and some point $b$ in $U$ which can also be represented by setting $x=a$ and $y=b$, scaling again by $t\in[0,1]$ between these two endpoints.


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