Concavification

In mathematics, concavification is the process of converting a non-concave function to a concave function. A related concept is convexification – converting a non-convex function to a convex function. It is especially important in economics and mathematical optimization.^[1]

Concavification of a quasiconcave function by monotone transformation[]

An important special case of concavification is where the original function is a quasiconcave function. It is known that:

• Every concave function is quasiconcave, but the opposite is not true.
• Every monotone transformation of a quasiconcave function is also quasiconcave. For example, if ${\displaystyle f(x)}$ is quasiconcave and ${\displaystyle g(\cdot )}$ is a monotonically-increasing function, then ${\displaystyle g(f(x))}$ is also quasiconcave.

Therefore, a natural question is: given a quasiconcave function ${\displaystyle f(x)}$, does there exist a monotonically increasing ${\displaystyle g(\cdot )}$ such that ${\displaystyle g(f(x))}$ is concave?

Positive and negative examples[]

As a positive example, consider the function ${\displaystyle f(x)=x^{2}}$ in the domain ${\displaystyle x\geq 0}$. This function is quasiconcave, but it is not concave (in fact, it is strictly convex). It can be concavified, for example, using the monotone transformation ${\displaystyle g(t)=t^{1/4}}$, since ${\displaystyle g(f(x))={\sqrt {x}}}$ which is concave.

A negative example was shown by Fenchel.^[2] His example is: ${\displaystyle f(x,y):=y+{\sqrt {x+y^{2}}}}$. He proved that this function is quasiconcave, but there is no monotone transformation ${\displaystyle g(\cdot )}$ such that ${\displaystyle g(f(x,y))}$ is concave.^[3]: 7–9

Based on these examples, we define a function to be concavifiable if there exists a monotone transformation that makes it concave. The question now becomes: what quasiconcave functions are concavifiable?

Concavifiability[]

Yakar Kannai treats the question in depth in the context of utility functions, giving sufficient conditions under which continuous convex preferences can be represented by concave utility functions.^[4]

His results were later generalized by Connell and Rasmussen,^[3] who give necessary and sufficient conditions for concavifiability. They show an example of a function that violates their conditions and thus is not concavifiable. It is ${\displaystyle f(x,y)=e^{e^{x}}\cdot y}$. They prove that this function is strictly quasiconcave and its gradient is non-vanishing, but it is not concavifiable.

References[]