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I have a question I have been asked to solve:

Given that $(a_1, a_2,...,a_n)\in R_{++}^n$ and $(x_1, x_2,...,x_n)\in R_{++}^n$, and $A>0, \mu >0, p \neq 0$, if there is a function $f(x)=A(a_1x_1^p+a_2x_2^p+...+a_nx_n^p)^\frac{\mu}{p} \space \space \forall x \in R_{++}^n$, show that function $f$ is a quasiconcave function over $R_{++}^n$ when $p \in (0,1]$.

I understand that to show a function is quasiconcave, it needs to satisfy this definition:

$f:𝑅_+^𝑁→𝑅$ is quasiconcave on $𝑅_+^𝑁$ if and only if $βˆ€π’™,π’šβˆˆπ‘…_+^𝑁$ and for all $𝑑\in(0,1)$

$f(𝑑𝒙+(1βˆ’π‘‘)π’š)β‰₯π‘šπ‘–π‘›\{f(𝒙),f(π’š)\}$

I am wondering how I would use this definition to show that a generalised function $f$ is quasiconcave, or if I should be approaching this from another concept/definition.

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    $\begingroup$ What do you mean by "generalized" function? This is an ordinary function. $\endgroup$ – Michael Greinecker May 8 at 13:33
  • $\begingroup$ Sorry, I omitted CES when typing it out. It should read generalised CES function $f$ $\endgroup$ – DoubleRainbowZ May 8 at 13:39
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Note first that the function $r\mapsto A(r)^{\mu/p}$ is strictly increasing on $\mathbb{R}_+$. When you look at the minimum in the definition of quasi-concavity, you can therefore ignore this part and it suffices to show that the function $$(x_1,x_2,\ldots,x_n)\mapsto a_1x_1^p+a_2x_2^p+...+a_nx_n^p$$ is quasi-concave. Actually, the function is even concave. The sum of concave functions is concave, so it suffices to show that the function $$(x_1,x_2,\ldots,x_n)\mapsto a_ix_i^p$$ is concacave for each $i=1,\ldots,n$.

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    $\begingroup$ Correct me if I'm wrong but your argument seems to be valid only for $p \in (0,1]$? For example if $p < 0$ then $A(r)^{\mu/p}$ is not increasing. Also if $p > 1$ then $x^p$ is convex? $\endgroup$ – tdm May 8 at 16:55
  • $\begingroup$ Yes, but that assumption is made in the question. $\endgroup$ – Michael Greinecker May 9 at 17:19
  • $\begingroup$ Ah ok, I saw the $p \ne 0$ qualification but looked over the $p \in (0,1]$ condition. Sorry. $\endgroup$ – tdm May 11 at 6:05

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