2025-05-02
Imagine you own an ice-cream parlor in a hypothetical town with exactly $\infty$ residents, not including yourself. This town is isolated from all external contact, and uses a local currency called beta ($\beta$), that has no smallest denomination and may be transacted in infinitely granular quantities.
You, being the owner of the parlor, decide to run a simple experiment. Each morning at 8:00 AM (which is the opening time of your parlor), you change the price of your ice-cream cone to a random real number greater than $1\beta$, which is your production cost per ice-cream cone. Your selection method ensures a uniform distribution and prevents the same price from being chosen twice.
As the day progresses, you keep a running tally of how many residents enter the parlor, and how many of those that enter purchase an ice-cream cone before 6:00 PM, the closing time of your parlor.
Nothing about the ice-cream parlor, the residents, or the entire town changes from day to day, except for the price of the ice-cream cone. Additionally, on each day, each individual resident has no knowledge of the new price of the ice-cream cone until they enter the parlor.
You continue this experiment for an infinite number of days, until you have collected data for every possible price.
With the resultant data, you would be able to construct a perfect approximation of a function $Z(p, x, c, s, t)$, which returns the probability that a given customer from population $c$ will purchase a product $x$ at price $p$ within some timeframe $t$, given that they've satisfied some prerequisite state $s$.
$$Z(p, x, c, s, t) : \mathbb{R} \times X \times C \times S \times T \rightarrow [0, 1]$$$$\text{such that } Z = P(c_{i} \in_{R} c \text{ purchases } x \text{ at price } p \mid s \cap t)$$Z of p, x, c, s, t is a function that maps from a real number and four complex objects of type X, C, S, and T to the interval from 0 to 1, such that Z equals the probability that a given customer from population c will purchase product x at price p given s and t.
In this scenario, $x$ was our ice-cream cone, $c$ was the $\infty$ residents, $s$ was the action of entering the parlor, and $t$ was defined as the hours between 8:00 AM to 6:00 PM, all of which were held constant. The only variable argument was $p$, which is the price of the ice-cream cone.
For notational simplicity, I will encode all of $x, c, s, t$ into a tuple $k$:
$$k = (x, c, s, t) : K = X \times C \times S \times T$$Tuple k of x, c, s, and t exists in complex configuration space K.
Z of p and k is equivalent to Z of p, x, c, s, and t.
Now, given that we have function $Z(p, k)$, the process of finding optimal price is quite simple. I define optimal price $\psi$ as the particular case of $p$ that maximizes a quantity $R$, called return, where $R$ is proportional to $n \times p$ (number of sales times price).
Note that:
$$p \times Z(p, k) \propto R$$Z of p and k times p is proportional to R.
Therefore, psi equals the price that maximizes p times Z of p and k.
Or, equivalently:
$$\left.\frac{\partial Z}{\partial p} \right|_{p=\psi} \left( p \times Z(p, k) \right) = 0$$The rate of change of p times Z of p and k with respect to p will be zero when psi is plugged in.
Therefore, psi equals the maximum of all prices times their corresponding outputs where the price makes the rate of change of p times Z of p and k with respect to p equal to zero.
In words, after constructing $Z(p, k)$ while keeping configuration $k$ constant, we simply derive the function and find the value of $p$ that maximizes the output.
Now that we've shown that finding the optimal price is trivial if we have $Z(p, k)$, we can turn our attention to the problem of actually approximating such a function.
It goes without saying that the ice-cream parlor was a thought experiment that is impossible to execute in practice. However, luckily for us, the accuracy of our approximation with respect to the quantity of data we collect has diminishing returns, meaning that even (relatively) small samples can yield a more-than-adequate approximation.
The first step is to define our configuration tuple $k$. It is imperative that our configuration remain constant throughout the data collection process, otherwise our derivation of $Z(p, k)$ will not be solely with respect to $p$. Furthermore, the optimal price yielded by $Z(p, k)$ will only remain valid for as long as our configuration remains roughly the same.
Object
Potential, Inexhaustible Causes of Variance
$x : X$
Any modifications to the product itself; a change in how the product is marketed or advertised; a material change in the product's relevance or utility relative to other substitutes; anything that may impact the perceived "value proposition" of the product, so to speak.
$c : C$
Changes in what individuals or how many individuals the population consists of; material changes in the population's circumstances; a lasting change in the population's preferences; anything that could effect the behavior of the population en masse.
$s : S$
Any modification to the arbitrarily defined prerequisite state that must be satisfied for a transaction to be considered a success (sale) or failure (no sale). Ideally, you want this condition to be as far from the point-of-sale as possible without compromising on the rigorosity of the data.
$t : T$
Any modification to the arbitrarily defined timeframe within which a transaction must occur to be considered a success (sale).
Once $k$ is clearly defined, we can begin the data collection process. We start by designating a sample size $n$, which is the number of individuals we will sample from population $c$. This sample should meet standard conditions for statistical inference, such as randomness and independence.
Then, we must define a price range $[a, b]$, which is the domain of $p$ across which we wish to know the underlying distribution of $Z(p, k)$. Ideally, you want this range to be as small as possible while still containing the brunt of the elasticity.
At this point, we are met with a classic optimization problem: what distribution of samples across the domain $[a, b]$ will yield the closest approximation of $Z(p, k)$? For example, if $n = 100$ and our price range is $[5, 25]$, we could take $20$ samples at prices $5, 10, 15, 20, 25$ each, or we could take $10$ samples at prices $5, 7.5, 10, \cdots, 25$ each, or any other combination thereof. The former trades accuracy for precision, while the latter trades precision for accuracy.
There are several statistical methods to resolve this problem, both parametric and non-parametric, and which one you should choose is a question that is not only heavily context-dependent, but also far beyond the scope of this "thought" (which has already become long-winded).
Once a distribution is chosen, you simply collect your samples accordingly, ensuring that state $s$ is being satisfied as strictly as possible, as if this were not an experiment (you may even consider allowing the customer to actually purchase the product at the sample price). Opinion surveys will not suffice -- you have to replicate the real thing, and sampled individuals must be unbeknownst to the experiment.
After data collection, you conduct some form of regression analysis, typically informed by your optimization method, to construct an approximation of $Z(p, k)$.
And once you have that -- well... you know what to do;)