Price elasticity when relationship between sales, price and other factors is not linear

Question

For commercial deployment, price elasticity is calculated through linear regression which assumes that there is a linear relationship between price and sales. I have a)price and b)social media ratings for my product. I know that social media ratings influence consumer decision to purchase or not purchase something. As per assumption of linear regression if social media rating is low we need to reduce price to improve sales. However, beyond a point price cuts will not help in increasing sales. This is a violation of linearity.

So my question is what if

1) Relationship is not linear even after log transformation of the data?

2) How can i know about the point at which i need to stop from cutting price further.

To give a background, i have already read this Econometrics: Is elasticity meaningful in my, or any, regression? However, it didn't seem to answer my second question, which is will my sales continue to increase if i keep dropping price although i have a low social media rating?

My background: I am a statistician and my economics knowledge is all i read in college. I am working a revenue optimization problem

Answer 1

There is a confusion between a "linear relationship between two variables" and an "econometric equation that is linear in the unknown parameters to be estimated".

The first has to do with what happens in reality, and it implies that the marginal relation is constant. The second may be obtained even if the actual relation is not linear, but non-linear in specific ways that permit to obtain it by a suitable transformation of the data.

To illustrate this, for the OP's case, the actual (deterministic part of the) relation may be

$$X_d = AR^aP^{-b} \tag{1}$$

Demand $X_d$ for the product is a positive non-linear function of social ratings $R$ and a negative non-linear function of its own price $P$. The marginal effect of price for this relation is not-constant

$$ \frac {\partial X_d}{\partial P} = -\frac{b}{P}X_d <0$$

By assuming $(1)$ we have already made a series of assumptions about the interplay of the variables involved. This specification permits us to obtain "an econometric equation that is linear in the unknown parameters to be estimated" since by taking logs we have

$$\ln X_d =\ln A +a\ln R +(-b)\ln P \tag{2}$$

So while the marginal effect of price on demand is not linear and not constant, the elasticity of demand with respect to price is constant, and equal to $-b$ (the sign indicating direction of influence).

But is $(1)$ an adequate way to represent the actual relation?

So the proper way to proceed here is to

1) To the best of our knowledge, using evidence and logical arguments, we determine the qualitative interrelations between the variables involved: is the effect positive/negative? Is the relation of their levels linear/non-linear? Is it monotonic or, say "inverted-U", etc.

2) We construct a mathematical form that reflects qualitatively the conclusions/assumptions arrived at in step 1. For example, if we believe that an "inverted-U" relationship exists between levels of $Y$ and $Z$, this could be modeled by $Y = a + bZ + cZ^2$ with $c<0$

3) If the mathematical expression we obtain in step 2 is not linear in the unknown parameters of interest, we check whether it can be transformed into one that it is. Of course, there are estimation methods for non-linear relationships, non-linear least-squares being the easy example. But experience has taught us that our estimation techniques are better when they estimate equations linear in the unknown parameters, this is why we always try to arrive at such a specification, even if we may accept in the process certain approximating steps to what we have obtained in step 2 (and not just exact transformations).