How to Determine Concavity of the Product of Two Functions?

[MathewsMD]

Given two functions and their respective concavity, is there any method to determine the concavity of the product of these two functions? Also, can you determine concavity if these two functions are composite?

 

[mfb]

Given two functions and their respective concavity, is there any method to determine the concavity of the product of these two functions?

Not without additional information. If the functions are differentiable (twice), you can find a nice condition for that.

Also, can you determine concavity if these two functions are composite?

There is nothing special about composite functions, they just might need more work to analyze.

 

[economicsnerd]

Q: Is the product of two concave functions concave? i.e. If $f$ and $g$ are concave, and $h(x)=f(x)g(x)$, then is $h$ concave?
A: Not necessarily. Consider the case of $f(x)=-1$, the constant function.

Q: Is the composition of two concave functions concave? i.e. If $f$ and $g$ are concave, and $h(x)=f(g(x))$, then is $h$ concave?
A: Not necessarily. Consider the case of $f(y)=-y$.

In both settings, $f$ (weakly concave) was chosen so that $h=-g$. Then the only way $h$ can be concave is if $g$ was both concave and convex, i.e. a straight line.

A nice exercise: If $f,g$ are both (weakly) concave and f is (weakly) increasing, show that the composition $h(x)=f(g(x))$ is also concave.

 

[economicsnerd]

A nice exercise: If $f,g$ are both (weakly) concave and f is (weakly) increasing, show that the composition $h(x)=f(g(x))$ is also concave.

An example of this that shows up in probability sometimes: If $g$ is concave, then so is $\log(g)$. Some probabilistic theorems involve an assumption that some function has a concave logarithm, which is (according to the exercise I gave you, since the $\log$ function is increasing and concave) a less stringent requirement than the function itself being concave.

 

[MathewsMD]

Not without additional information. If the functions are differentiable (twice), you can find a nice condition for that.
There is nothing special about composite functions, they just might need more work to analyze.

Ok, so you say not without additional information. What is the minimal amount of information that must be provided to assess such a question? Also, what if the function could not have a degree of -1, 0 or 1, how would this change everything?

Thanks!

 

[mfb]

Ok, so you say not without additional information. What is the minimal amount of information that must be provided to assess such a question?

If you know f and g are both differentiable, just look at the second derivatives of f(x)*g(x) or f(g(x)).
If you don't know that, I don't think there is a general answer. There are some special cases, economicsnerd mentioned some of them.