# Terminology, is elasticity used as “mean elasticity”?

Given a demand $$q$$ and a price $$p$$ sutch that $$q=q(p)$$, the elasticity of demand is given by,

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

which depends on the price.

But, when reading papers about estimation of elasticities (elasticity of oil or elasticity of gasoline just to mention some examples), I typically find that there is one single value reported as elasticity but not clear how it is computed, is this just the plain mean elasticity?

I would not call it average elasticity rather its elasticity at an average price. For example take the first paper about elasticity of oil you cite (that is Cooper, J. C. (2003). Price elasticity of demand for crude oil: estimates for 23 countries. OPEC review, 27(1), 1-8.).

In that paper the Cooper estimates elasticity using the following model:

$$\ln D_t = \ln \alpha + \beta \ln P_t + \gamma \ln Y_t + \delta \ln D_{t–1} + e_t$$

Where $$\beta$$ gives you the estimate of elasticity. However, $$\beta$$ is not necessary equal to $$\bar{\epsilon}$$ for any model specification (even though the above one would actually imply it -see the last paragraph), rather $$\beta$$ gives you point estimate elasticity at $$\bar{P}$$.

In fact generally OLS regression is constructed in such way that it intercepts point given by $$\bar{y}$$ and $$\bar{x}$$ - that is $$\bar{y}-\hat{\alpha} -\hat{\beta} \bar{x} = 0$$. Hence, more correct interpretation here would not be that $$\beta$$ gives you average elasticity (i.e $$\bar{\epsilon}$$) but rather that it gives you point estimate for elasticity at average price (i.e. $$\epsilon_{\bar{P}}= (\bar{P}/Q)/(dQ/d\bar{P})$$.

This being said note that using OLS model such as the one that is used in Cooper would actually imply that the elasticity is constant because its a linear model (linear in its parameters that is) where $$\beta$$ is assumed to be constant across all observations (even though you get perfect fit only at ($$\bar{P},\bar{D}$$). In a model with constant elasticity it would actually hold that $$\epsilon_\bar{P} = \bar{\epsilon}$$. Nonetheless, I would still caution against that interpretation. The reason for that is that in most cases it is commonly understood that the linear model is used as a simplification and not because people actually assume that elasticity of demand is constant. Mostly, people take this point estimates to be reasonable for small changes around mean price but not really claiming that they found a constant price elasticity of demand, and in most such models if you would look at errors they would get larger the further you get from mean estimate.

• Thanks a lot for such detailed explanation. I got a bit puzzled I must admit. With the linear fit that you show I would immediately think this assumes the Demand is a power law of the price with constant exponent $\beta$ (elasticity). If that was true then that is the end of the story, on that assumption $\beta$ does not depend on price. But then you say that actually that if the $D \sim P^\beta$ is not true, then the linear fit gives you the mean price. 1. I cannot really see that in the equations. 2. If that is the case then, for non constant elasticity the method is arbitrarily poor, right? – myradio Jun 30 '20 at 22:29
• @myradio well as explained in the +1 answer of Michael the authors do assume that elasticity is constant - even just using OLS in the form they use it implicitly assumes it. Rather the point I am trying to get across is that thus is assumption that is always assumed for convenience (ease of estimation) but rarely believed. I been to seminars when some younger scholars always try to take the implicit assumption to its logical conclusion and were scolded by older scholars on the grounds that this is really just simplification on multiple occasions. That’s why I also chose the words carefully not – 1muflon1 Jun 30 '20 at 22:34
• saying that it would be incorrect to say that it’s also mean elasticity (since elasticity is constant - which also implies that it does not depend on the price by extension) but rather that even though it’s technically true implied by the model it’s a undesirable byproduct of the estimation. It’s also a property of any OLS linear in parameters that it fits exactly mean Y and X, you can see the proofs of that in some textbooks such as statistics for business and economics by newbold et al. It’s also true that unless there is good reason to believe that elasticity is actually constant (so we – 1muflon1 Jun 30 '20 at 22:55
• have for example demand given by exponential function such as the one used by Martin) the model will be poor - in a sense that it gives us only correct information close to average P. However, without knowing more about the data authors had at their disposal I would not judge them too harshly. Also note that the results from such model are not useless if you interpret them in a way I suggested in my answer – 1muflon1 Jun 30 '20 at 22:55
• Yes, I understand it's a first approximation and it's true that if the result is useful is probably enough reason to use it. I think the link I was missing was also that relation but is now filled with newbold's ref. – myradio Jun 30 '20 at 23:04

The two papers you provide are explicit on how elasticities are computed. To take a simplified version of the specifications used in both papers, let $$\log D(p)=\beta\log p.$$ Now, $$D(p)=e^{\log D(p)}=e^{\beta\log p}=(e^{\log p})^\beta=p^\beta.$$

Therefore, $$\frac{p}{D(p)}\frac{d D(p)}{dp}=\frac{p}{p^\beta}(p^\beta)'=p^{1-\beta}\beta p^{\beta-1}=\beta.$$ So for the given functional form, the elasticity does not depend on the price.