# Real world price elasticity when making pricing decisions

This is my first time trying to calculate price elasticity and put it to work in a real world situation. Let me state my problem first:

I work at a company (let’s call it a store) that was selling a particular product for £6.00. We raised the price to £7.00 (on a hunch) and it didn’t seem to have an impact on units sold. Our marketing team now wants to raise the price again, but before doing so, I wanted to see whether I could prove the impact of a price rise.

One issue I’ve identified is that the location of our store saw a significant increase in footfall in the same period where we raised our price, leading to an overall increase in units sold.

Therefore, this is what I have worked out so far:

Units sold increased by 115, while total visitors increased from 1,480 to 1,850. This meant that our “capture rate” (for want of a better term) decreased by (0.27ppts).

Therefore, had our total visitors remained at 1,480 we would have actually lost 4 units linked to the increase in price from £6.00 to £7.00.

When isolating the impact of the increase footfall results in increased Gross Revenue of +£452 of which -£24 is linked to the decrease in units sold and +£476 is linked to the increase in price. Now for price elasticity.

If the price increases by +£1, while the units decrease by (4) units, then we have a 17% increase in price with a (1%) decrease in volume. If I am not mistaken, this means the price is inelastic with the price elasticity at -0.05. If we plot this on a graph, we get the following chart. Meaning that our units sold theoretically changes by 4 units for every £1 change in price, with an intercept of 504. Here’s where I get stuck. And please forgive my extreme newbieness with this vocabulary (if anyone wants to teach me the correct way to talk about this, please feel free).

• Does this suggest that an increase in price to £8.00 would have a marginal impact on units sold?

My intuition would be that the price is inelastic due to demand being solid enough. If this is the case, we could expect additional revenue of £444, of which £472 was linked to the increased price and (£28) to the decrease in units sold. Obviously, if you have any other suggestions/guidance, that would be very much appreciated.

1. before going to the answer to your question let me point out what you do is futile.

Sales (demand) and price are parts of endogenous supply-demand system. If you just collect data on Sales (demand) and price, you will get a lot of nonsense (in statistics we call it spurious) results.

If you want to correctly calculate elasticity you need to use some model that can account for endogeneity (e.g. TSLS). Also you need more than 2 data points. For example, the simplest possible way to do it is to run simple IV such as:

$$s = \beta_0 + \beta_1 \hat{p} +e \tag{II stage}$$ $$p = \alpha_0 + \alpha_1 c + \epsilon \tag{I stage}$$

where s is sales, p hat predicted price from first stage and c would be production cost which is usually good instrument for price.

If you don't have enough data and do not correct for endogeneity, you are just wasting your time. You can do the calculations you do for fun, but I would advise against using them in any decision making process, especially if money is on line.

1. Does this suggest that an increase in price to £8.00 would have a marginal impact on units sold?

Yes, your graph (that you should not take at face value because of 1) suggests that at price £8.00 the quantity sold would be 472, so changing price from £7.00 to £8.00 would have only small impact.

My intuition would be that the price is inelastic due to demand being solid enough.

No, for a linear demand function you have:

$$y=4x+504$$

elasticity will be different for different price, the same linear demand function is elastic for some prices and inelastic for different prices.

The elasticity of demand is:

$$el=\frac{dq}{dp} \frac{p}{q}$$

In your case the elasticity is:

$$el= \frac{-4x}{-4x+504}$$

Depending on x (price) the elasticity might be elastic, unit elastic or inelastic. At price 8 we get that the elasticity of 0.067 so it is inelastic at that point but not inelastic in general.

Obviously, if you have any other suggestions/guidance, that would be very much appreciated.

My guidance is to either collect more data and make some structural model (something like suggested at the beginning or even more fancy) or just do not bother with trying to calculate the elasticity and do decisions with some other method that does not require a lot of data/modelling.

• Is the estimate of $-0.067$ is so unreliable that the real value is quite possible on the other side of $-1$? Can you please include some stuff about the size of the bias in case of a biased (non-TSLS) estimation? Aug 24, 2022 at 6:30
• @Giskard well but the size of the bias is unknown as it depends on cov(p,e)/var(p) from naive regression (and also parameters values of first stage). The error term can’t be even directly observed (we can only observe residuals), so the bias can lead true elasticity to be anything from $-(\infty,0]$ (or even positive if we allow for Giffen goods). The OP estimate -0.067 is as good as completely randomly generated number
– 1muflon1
Aug 24, 2022 at 7:48
• What is a "completely randomly generated number" though. $x \sim N(0,0.000001)$ is completely randomly generated but quite reliably within $[-0.1,0.1]$. Aug 24, 2022 at 8:25
• @Giskard in this case anything drawn from (almost) uniform distribution between $\pm \infty$ (unless you impose structural restrictions for elasticity to be zero or less then it’s from almost uniform distribution between $-\infty$ and 0
– 1muflon1
Aug 24, 2022 at 8:54
• While I understand that the distortion could conceivably take any value form $-\infty$ to $\infty$, I am struggling to see why it would do so with an (almost) uniform distribution. Can you please explain? Should I post this as a new question? (If yes, please help me get the wording right.) Aug 24, 2022 at 9:30