As we know that elasticity varies from infinity to zero as we move along a linear demand curve, then is it correct to label any demand curve elastic or inelastic as a whole?
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1$\begingroup$ Often you are just interested in the elasticity in a relevant range. While we define elasticiy at a point on the demand curve, we are also able to say whether the price elasticity is >1 or <1 in a range of prices. $\endgroup$– BayesianCommented Nov 4, 2020 at 13:53
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1$\begingroup$ could you also clarify if you are asking solely for linear demand functions? I gave you bit more general answer below, if you are solely interested in a linear demand I will edit it. $\endgroup$– 1muflon1 ♦Commented Nov 4, 2020 at 13:55
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$\begingroup$ Yes, I was asking for linear demand functions. Thanks in advance. $\endgroup$– Peeyush ShivamCommented Nov 4, 2020 at 16:54
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From your Q I am not sure if you are asking solely for linear demand functions or demand functions in general.
When it comes to demand functions just in general actually, we do not a priori know that "elasticity varies from infinity to zero as we move along a linear demand curve" there is a class of constant elasticity demand functions for which price elasticity of demand is always the same. For example:
$$P = Q^{1/\epsilon}\implies Q= P^\epsilon$$
will have constant price elasticity. Since price elasticity is given as:
$$e_p= \frac{dQ}{dP} \frac{P}{Q}$$
In the above case the elasticity would be simply equal to $\epsilon$, as:
$$\frac{dQ}{dP} \frac{P}{Q} = \epsilon P^{\epsilon-1}\frac{P}{P^{\epsilon}}= \epsilon$$
The constant price elasticity functions are relatively popular in the literature and for these functions it clearly makes sense to label them as elastic or inelastic based on value of $\epsilon$.
However, even for a linear demand functions where price elasticity of demand changes along the function it would be reasonable to say that one demand curve is more inelastic relatively to some other demand curve, respectively being more or less elastic over some range. In such situations however the terminology might often be abused and demand function simply termed 'inelastic' - this happens especially in many undergraduate texts and I suppose you could call it an 'oversimplification'.
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$\begingroup$ A demand function with constant elasticity is not linear though. $\endgroup$– BayesianCommented Nov 4, 2020 at 13:48
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1$\begingroup$ @Bayesian that is true, but not all demand functions are linear, I thought user is using a linear function just as an example not saying that they are looking for answers only for linear functions. But maybe I misunderstood English is high context language - I will ask user for clarification $\endgroup$– 1muflon1 ♦Commented Nov 4, 2020 at 13:52