Monthly price elasticity and possibility of using daily values

Question

I am calculating the price elasticity as a starting point to find a theoretical optimal price that would maximize our revenue.
I am looking at 2 years data and to use the price elasticity formula, I am considering monthly values separately (this also because given the strong seasonality we have, i would expect different elasticity per month and I would like to suggest different prices every month).

To get the monthly Price Elasticity, I calculate for each month Prices and Quantities of the respective year:

$$ \frac{(Q^{2018} - Q^{2017})/Q^{2017}}{(P^{2018} - P^{2017})/P^{2017}} $$

Question 1: Is this correct? Using aggregated monthly data is enough? What if I had also 2016 data? How could I put this extra information in the formula?

Question 2: Could I use daily data to calculate the price elasticity? Could I plot 2 years data and use the linear trend line for the elasticity calculation?

Thanks in advance, any help is highly appreciated!

Answer 1

There are a couple of things that you are going to want to keep in mind as you do this type of analysis. You want to be careful not to base your conclusions on too little data. It sounds like you would be basing your monthly elasticity estimates on two data points (the month in 2018 and the month in 2017). Unless your sales data is remarkably stable, this is likely to be highly influenced by noise in the data.

Additionally, the time period that you use in estimating elasticity should be determined by the frequency of price changes and the frequency of sales. For example, if you only observe price changes once per year, this type of analysis will likely be ineffective. I would recommend plotting your observed price and sales points at the weekly or monthly level. This will give you a sense of how noisy your data is. If the plots that you create look relatively stable, then, as you discussed, you can think about fitting a demand curve to the data and calculating elasticity using the equation of the fitted curve.

Answer 2

What you're trying to approximate is the quantity

$$ \gamma(t) = \frac{{\rm d}\ln Q(t)}{{\rm d}\ln P(t)} $$

$\gamma = \gamma(t)$ is a function of time, the better sampling you have of $P$ and $Q$ the better estimation of $\gamma$ you are going to get. Imagine you have one sample every $h$ units of time. From what I can read $h = 1 ~{\rm month}$, what you can do to measure $\gamma$ at time $t_k$ is calculate

$$ \frac{{\rm d}\ln P(t_k)}{{\rm d}t} \approx \frac{\ln P(t_k) - \ln P(t_k - h)}{h} \tag{1} $$

with a similar expression for $Q$. But this is really bad in general, here's a better approach

$$ \frac{{\rm d}\ln P(t_k)}{{\rm d}t} \approx \frac{-2\ln P(t_k + 2h) + 8\ln P(t_k + h) - 8\ln P(t_k - h) + \ln P(t_k - 2h)}{12h} \tag{2} $$

So to give you an idea, if you are measuring the elasticity on ($t_k$ = November 2018) with data measured each month, you will need to know the prices and quantities for ($t_k - 2h = $ September 2018), ($t_k - h = $ October 2018), ($t_k + h = $ December 2018) and ($t_k + 2h = $ January 2019).