In my job as a data scientist, I am required to model the relationship between the price of a product and the sales or number of unit sold. I am trying to build a simplistic model, the assumptions of which are given below. I am not sure if all the assumptions will hold true simultaneously, or if there is a missing assumption or if some assumptions are wrong. Can any expert have a look and comment/give suggestions?
Assumptions:
Different customers have different income so the product priced at $P$ may be affordable to some and not affordable to others. But from a practical business point of view, we cannot build a model of each individual. So we assume that average income of the customers remains fairly constant in the short term and so we assume that we are trying to model the behavior of a single customer whose income is constant and equal the average income of all the customers.
- All other conditions external conditions such as macro economics, competitive products, government policies and business rules remain constant
- The product has demand, it is mature and stable and so at present the no. of units of the product sold depends only on its price.
- As the price changes, the probability that th customer will purchase the product changes. Let $a_1$ be the probability that the customer purchases the product at price $P_1$ and $a_2$ be the probability that the customer curchases the product at price $a_2$. Then $a_2$ depends only on the initial price, initial probability and the new price i.e. $a_2 = f(a_1,P_1,P_2)$ where $f$ is some probability function whose behavior we are trying to find out assuming that such a function exists in the first place.
- If price remains constant, then the purchase probability remains unchanged i.e. $f(a_1,P_1,P_1) = a_1.$
- Hypothetically, if the product is critical for survival (e.g. purchasing your daily oxygen in a Mars colony) the customer will buy it no matter what the price is i.e. $f(1,P_1,x) = 1$ for all $x$.
- If the product is not critical for survival then it will be out of demand at infinite price i.e. $ Lim_{x \to \infty} f(a_1,P_1,x) = 0.$
- If the product is important and very cheap then everybody will but it i.e. $ Lim_{x \to 0} f(a_1,P_1x) = 1$.
- As price increases, sales decreases i.e. if $P_1 < P_2 < P_ 3$ then $0 \le f(a_1,P_1,P_3) < f(a_1,P_1,P_2) \le 1.$
- Since $f$ is a probability density function, its value must be between 0 and 1 i.e. $0 \le f(a,x,y) \le 1$ for all $x$ and $y$.
- Mathematically, since $f$ is a probability density function, and price cannot be negative the total area under the probability density curve must be 1 i.e. $$\int_{-\infty}^{\infty} f(a_1,P_1, x) dx = \int_{0}^{\infty} f(a_1,P_1,x) dx = 1.$$
Now there will be infinitely many functions $f$ satisfying all the above conditions. For example
$$ a_1^{P_2/P_1}, \frac{a_1\log(1+P_2/P_1)}{\log 2} $$
We eleminate the ones that do not fit the observed data. But even then we can still be left with multiple functions that satisfies all the assumptions of the framework as well as fits the data within an acceptable error range as defined by the business.
Question 1: What other criteria/conditions/assumptions can we use to choose one mathematical model over another?
One way is to use Occam's Razor and go with the simplest model defined as the one which uses the least number of parameters in which case $$ f(a_1,P_1,P_2) = a_1^{P_2/P_1} $$
Question 2: Lets ignore my entire framework. What are the other models in economics that can be used to estimate the purchase probability given $a_1, P_1$ and $P_2$.
Question 3: Which assumption is simple impossible and should be removed? What additional assumptions are required if any?