What to do about the third derivative of a twice differentiable function?

Question

In my economics dissertation, I have a general production function: $F:\mathbb{R^n}\rightarrow \mathbb{R}$ that maps input onto unit output. $F$ is unspecified (besides that $F'>0$ and $F''>c$, for some $c<0$). The assumption (widespread in the literature) is that $F$ is twice differentiable. However, when performing analysis, I derive equations that involve the third derivative of $F$. Including $F'''$ without assigning it a particular value would probably result in indeterminate results. I could assume $F'''=0$ but this strikes me as somewhat contrived. Are there any other options?

Answer 1

Some things to check:

• Can you get by with a weaker condition, e.g., Lipschitz or Hölder continuity of $F^{(\text{ii})}$?
• Do you need $F^{(\text{iii})}$, or just integrals of functions multiplied by $F^{(\text{iii})}$? If the latter holds, you only need weak derivatives.
• Much of microeconomic theory can be expressed in the context of convex optimization—functions don't need to be assumed differentiable for monotonicity and strong convexity to make sense. Are you absolutely sure that you need the third derivative of a function?
• One differentiability requirement that has become common in the convex optimization literature is self-concordance: $$F^{(\text{iii})}(t)^2\leq 4F^{(\text{ii})}(t)^3\text{.}$$ Maybe this property is what you actually need?