<p>Let <span class="math-container">$f_1:\mathbb{R}\to\mathbb{R}$</span> be given by <span class="math-container">$f_1(x)=0$</span> for <span class="math-container">$x&lt;0$</span> and <span class="math-container">$f_1(x)=x^2$</span> for <span class="math-container">$x\geq 0$</span>. Also, let <span class="math-container">$f_2:\mathbb{R}\to\mathbb{R}$</span> be given by <span class="math-container">$f_2(x)=x^2$</span> for <span class="math-container">$x&lt;0$</span> and <span class="math-container">$f_2(x)=0$</span> for <span class="math-container">$x\geq 0$</span>. Both <span class="math-container">$f_1$</span> and <span class="math-container">$f_2$</span> are quasi-concave but their sum <span class="math-container">$f_1+f_2$</span> given by <span class="math-container">$(f_1+f_2)(x)=x^2$</span> for all <span class="math-container">$x\in\mathbb{R}$</span> is not quasi-concave.</p>
<p>However, the result does hold true for concave functions. Show that if <span class="math-container">$p\geq 0$</span> and <span class="math-container">$f$</span> is concave, then <span class="math-container">$pf$</span> is concave too. Then show that if <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are concave, then <span class="math-container">$f+g$</span> is concave too. The rest follows by induction.</p>

CC BY-SA 4.0
2022-10-15T23:26:13.040Z
2022-10-15T23:31:49.220Z
2022-10-15T23:31:49.220Z
Michael Greinecker
<p>Let <span class="math-container">$f_1,…, f_m$</span> be quasiconcave functions on a convex subset X on <span class="math-container">$IR^n$</span>. Let <span class="math-container">$p_1,…, p_m$</span> be nonnegative real numbers.</p>
<p>Then, how can I show that <span class="math-container">$\sum_{i=1}^mp_if_i $</span> is also quasi concave function?</p>
<p>Similarly, Let <span class="math-container">$g_1,…, g_m$</span> be concave functions on a convex subset X on <span class="math-container">$IR^n$</span>. Let <span class="math-container">$p_1,…, p_m$</span> be nonnegative real numbers.</p>
<p>Then, how can I show that <span class="math-container">$\sum_{i=1}^mp_ig_i $</span> is also concave function?</p>
<p>—</p>
<p>If <span class="math-container">$f_i$</span> is quasiconcave then, <span class="math-container">$\{u,v\}\in$</span> domain of f for <span class="math-container">$0&lt;a&lt;1$</span>, and for <span class="math-container">$f_i(u)&gt; f_i(v)$</span> <span class="math-container">$f_i(au+(1-a)v)\ge f_i(v)$</span></p>
<p>Similarly, for concavitiy, <span class="math-container">$g_i(au+(1-a)v)\ge ag_i(u)+(1-a)g_i(v)$</span></p>
<p>And then, how can I proceed the proof?</p>
<p>Thank you. All helps will be appreciated.</p>
<p>*duplicated question</p>

CC BY-SA 4.0
2022-10-15T23:27:28.083Z
What does the "*duplicated question" refer to? Duplicated from where?
Michael Greinecker
<microeconomics><mathematical-economics><self-study>
Proof on quasi-concavity and concavity
studentp
# Proof on quasi-concavity and concavity

(Asked by: username_0 on 2022-10-15)

Let $f\_1,…, f\_m$ be quasiconcave functions on a convex subset X on $IR^n$. Let $p\_1,…, p\_m$ be nonnegative real numbers.

Then, how can I show that $\sum\_{i=1}^mp\_if\_i $ is also quasi concave function?

Similarly, Let $g\_1,…, g\_m$ be concave functions on a convex subset X on $IR^n$. Let $p\_1,…, p\_m$ be nonnegative real numbers.

Then, how can I show that $\sum\_{i=1}^mp\_ig\_i $ is also concave function?

—

If $f\_i$ is quasiconcave then, $\{u,v\}\in$ domain of f for $0<a<1$, and for $f\_i(u)> f\_i(v)$ $f\_i(au+(1-a)v)\ge f\_i(v)$

Similarly, for concavitiy, $g\_i(au+(1-a)v)\ge ag\_i(u)+(1-a)g\_i(v)$

And then, how can I proceed the proof?

Thank you. All helps will be appreciated.

\*duplicated question

**Tags:** microeconomics, mathematical-economics, self-study

> **Comments:**
> **username_1**: What does the "*duplicated question" refer to? Duplicated from where?  

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## 1 answers

### Answer by username_1 on 2022-10-15. Score: 3

Let $f\_1:\mathbb{R}\to\mathbb{R}$ be given by $f\_1(x)=0$ for $x<0$ and $f\_1(x)=x^2$ for $x\geq 0$. Also, let $f\_2:\mathbb{R}\to\mathbb{R}$ be given by $f\_2(x)=x^2$ for $x<0$ and $f\_2(x)=0$ for $x\geq 0$. Both $f\_1$ and $f\_2$ are quasi-concave but their sum $f\_1+f\_2$ given by $(f\_1+f\_2)(x)=x^2$ for all $x\in\mathbb{R}$ is not quasi-concave.

However, the result does hold true for concave functions. Show that if $p\geq 0$ and $f$ is concave, then $pf$ is concave too. Then show that if $f$ and $g$ are concave, then $f+g$ is concave too. The rest follows by induction.


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