Price Elasticity Calculation

Question

I have a question concerning price elasticity calculation. Let us say that I have 26 points in time where for each one I have recorderd price $P_t$ and demand $D_t$ for a product.

One way to calculate elasticities is to take two consecutive points in time and use the elasticity formula

$$ \varepsilon_t := \frac{\Delta D_t}{\Delta P_t} \frac{P_t}{D_t}, $$

for each time point $t=2,\dotsc,26$. Another way is to run a regression using all 26 points, i.e.

$$ \log D_t = \beta_0 + \beta_1 \log P_t + u_t, $$

and take $\beta_1$ as the elasticity.

What are the pros and cons of these two methods?
Which one is considered to be more accurate?

Now let us say that I am using the regression method. If new points in time are added with more data, how should I calcualte the new price elasticity? E.g. if three points of time with relevant data are added, should I run a regression with 29 points (26+3) or should I use data on a rolling horizon basis?

Answer 1

Several comments:

1. Depending on what you precisely mean by $D_t$, you may have a serious identification problem. Anytime the price and quantity change, you don't know if the supply curve is shifting (and you're moving along the demand curve) or the demand curve is shifting (and you're moving along the supply curve). You need supply curve shifts to estimate the price elasticity of demand and demand curve shifts to estimate the price elasticity of supply.

   • If this is some survey data or data where only the supply curve is shifting, you're ok. A shift in the supply curve allows you to estimate the price elasticity of demand.
   • If it's not, you typically need some instrument for shifts in supply to estimate a demand curve.

2. Assuming your data is from supply curve shifts, two problems you can now worry about are:

   • Noise (eg. the variance of $u_t$)
   • Specification error (eg. the elasticity isn't constant)

   If the elasticity is constantly changing, running linear regression under the assumption that the elasticity is constant may not work terribly well, and using the two most recent points hypothetically may work better. On the other hand, if noise is high, using only a few observations will lead to horribly imprecise estimates.

3. Assuming your data is from supply curve shifts, what you probably want to do is estimate the elasticity from your regression equation. If the fit is horrible or doesn't make sense, you may want to think more deeply about what might be going on.