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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.
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.
