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I am working on this question:

Let f and g be two strictly concave functions and let function $h$ be defined by $h = af + bg$, where $ a > 0$ and $ b > 0$ are constants. Using the definition of strict concavity (i.e., without the use of derivatives) show that $h$ is strictly concave.

How can I show that $h$ is strictly concave? Should I show that because both $af$ and $bg$ are increasing then their sum must also be increasing?

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  • $\begingroup$ Excuse me, but where are stuck in establishing the strict concavity of $af+bg$ with the onlyuse of the definition? $\endgroup$
    – agt
    Nov 23, 2011 at 14:46
  • $\begingroup$ Apply the definition of strict concavity directly should give you the result. Which step are you stuck on? $\endgroup$
    – Willie Wong
    Nov 23, 2011 at 14:46

1 Answer 1

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I don't think thinking about $f$ and $g$ as being increasing/decreasing is the right approach, since a concave function like $-x^2$ can be increasing in some places and decreasing others. Try working directly from the definition. A function is strictly concave if, for every two distinct real numbers $x$ and $y$,

$$f\left(\frac{x+y}{2}\right) < \frac{f(x)+f(y)}{2}.$$

Knowing that this inequality holds for $f$ and $g$, can you prove that it also holds for $af+bg$?

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