multiplicative-calculus.md

Multiplicative Calculus

In search of better formalisms for the reason described in my economics math write up, I found my way multiplicative integral wikipedia page.

When I was first taking economics masters classes at John Jay, that Wikipedia article was in a lot worse shape, and didn't give me the answers I was looking for. But now it is much better, and in particular it now cites doi:10.1016/j.jmaa.2007.03.081 which is fantastic! You should just go read the paper yourself.

I don't like doing the "unoriginal recap as blogpost" thing, but there are a few things I want to highlight, and also a few more basic precaluculus notions (that the paper did not cover) that I want to place alongside them.

Sequence pre-calculus

$\mathbb{R} \to \mathbb{R}$ function calculus

Multiplicative derivative

The multiplicative derivative is defined as follows:

$$f^* := a \mapsto \lim_{x \to a} \left( \frac{f(x)}{f(a)} \right)^\frac{1}{x -a}$$

The crucial things to note are that:

• Compared to the usual "additive" derivative, each arithmetic operation of "output" values is "promoted": addition to multiplication, multiplication to exponentiation.

• There is no addition (or subtraction) of "output" values.

This indicates we are now working on the level of multiplication, and not cheating. Indeed, the output type of $f$ could be something for which addition is not even defined!¹

After some manipulations (which you should definitely read work through!) the paper shows this definition equivalent to

$$f^* := x \mapsto e^{(\ln \circ f)'(x)}$$

Rewritten in point-free style, where $D_+$ is the regular (additive) differential operator and $D_*$ is our new one:

$$\begin{aligned} D_* &= (\exp \circ \_) \circ D_+ \circ (\ln \circ \_) \\\ &= (\exp \circ \_) \circ D_+ \circ (\exp^{-1} \circ \_) \\\ &= (\exp \circ \_) \circ D_+ \circ (\exp \circ \_)^{-1} \end{aligned}$$

Note: the inner $\circ$ is for function composition for real functions, $\mathbb{R} \to \mathbb{R}$, whereas the outer $\circ$ is for function composition for real-to-real functions, $(\mathbb{R} \to \mathbb{R}) \to (\mathbb{R} \to \mathbb{R})$.

We can now see the very nice way our new form of differentiation looks something like a group conjugation: tweak (the function), differentiate, and then untweak.

Multiplicative integral

Defining integrals formally is pain, so let's do something short of that.

$${\huge \mathscr{P}}_a^b f(x)^{dx} := \lim_{\Delta x \to 0} \prod f(x)^{\Delta x}$$

The multiplicative Riemann integral is "defined" as the limit of the product of increasingly many samples of $f(x)$ taken to the $\Delta x$ power.

The geometric intuitions off this are not as clear as the additive Riemann integral because are multiplicands of this are not "little rectangles" the way the addends of that are. Indeed, as $x$ and $f(x)$ must be dimensionless, any geometric/visual intuition is inherently suspect!

This definition can be reworked into a regular integral:

$$\begin{align} {\huge \mathscr{P}}_a^b f(x)^{dx} & = \lim_{\Delta x \to 0} \prod f(x_i)^{\Delta x} \\\ & = \lim_{\Delta x \to 0} \prod \exp\left((\ln f(x_i)) \cdot {\Delta x}\right) \\\ & = \lim_{\Delta x \to 0} \exp\left(\sum (\ln f(x_i)) \cdot {\Delta x}\right) \\\ & = \exp\left(\lim_{\Delta x \to 0} \sum (\ln f(x_i)) \cdot {\Delta x}\right) \\\ & = \exp\left(\int_a^b \ln f(x) \cdot {\Delta x}\right) \end{align}$$

This saves us from actually needing to rigorously define the multiplicative integral from scratch.

Footnotes

1. However, needing exponentiation means we still need some ring-like structure. The other flavours of product integrals on the Wikipedia page get into this more, including changing up the definition to dodge the exponentiation requirement. I am a bit conflicted on this, because on one hand it is useful to make it work with less structured / more general codomains of the integrand, but on the other hand, those changes, when applied to scalars, go against what I am arguing in this piece! ↩