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Price elasticity of demand $\mathrm{e_{D,P} = \dfrac{dD}{dP}. \dfrac{P}{Q^*}}$ clearly depends on the levels of price and quantity.

Then why is everywhere (research papers, textbooks etc.) a constant estimate provided?

Also, doesn't the non-constancy of the price elasticity render the comparative statics $e_{P^*,\alpha} = \dfrac{e_{D,\alpha}}{e_{S,P}- e_{D,P}}$ invalid?

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Premise of your question is simply false. You state (emphasis mine):

Price elasticity of demand $\mathrm{e_{D,P} = \dfrac{dD}{dP}. \dfrac{P}{Q^*}}$ clearly depends on the levels of price and quantity.

Incorrect. Consider trivial counter example. A perfectly reasonable demand can be given by:

$$ Q = A p^{-\epsilon} $$

where $Q$ is quantity, $p$ price and $A$ and $\epsilon$ are constant parameters.

Hence the price elasticity of demand $e$ will be given by:

$$e = \frac{dQ}{dp} \cdot \frac{p}{Q} = -\epsilon A p^{-\epsilon-1} \frac{p}{ A p^{-\epsilon}} = -\epsilon $$

Consequently, clearly elasticity does not always depend on the levels of price and quantity and can be completely constant.

Then why is everywhere (research papers, textbooks etc.) a constant estimate provided?

Again false premise, in several textbooks and papers you will find elasticity being expressed as a function of price and quantity. Since you made statement about everywhere it is enough to show one example to the contrary which would be Varian Microeconomic Analysis, where there are several examples where elasticity in a model won't be constant.

In empirical papers elasticity will be most often estimated as a single number, but this is due to the way how empirical models are set up. It is too broad to discuss all empirical models in a single question, but typically coefficients estimated by model are values that hold on average. For example, in a regression $y_i = \beta_0 + \beta_1 E_i + e_i$ where $y$ would be income and $E$ years of education, the $\beta_1$ would tell us how much on average $y$ increases with education, it would not tell us what happen to every individual kids. That is if $\beta_1=10$ that does not mean that if your kid stays in school one more year they are guaranteed that their income will increase by 10. That is just a value we would expect for average kid. Similarly, even if researcher assumes elasticity is not constant because they are using linear demand, they might be forced to estimate elasticity at a point rather than as a whole function and convention is to estimate point elasticity at average quantity and price (see discussion in Espey Espey & Shaw (1997).

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