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I get that PED varies along linear (strictly speaking, affine) demand curves in a way that for a demand function $Q(P)=\alpha - \beta P$:

$$|\epsilon_D|=1 \iff \frac{\alpha}{2\beta}=P \land |\epsilon_D| \lessgtr 1 \iff \frac{\alpha}{2\beta} \gtrless P$$.

However, in many textbooks, it says a demand curve is elastic/inelastic (e.g. Mankiw):

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How does that make sense if the PED varies along affine demand curves? Why do they speak of "inelastic" vs "elastic" demand while referring to a whole demand curve? What sort of elasticity are they referring to?

(I know there are non-linear/non-affine demand curves such as those resulting from Cobb-Douglas-preferences, or, more generally, that have the following form: $Q(p)=Ap^{\epsilon}\ \forall\ A \gt 0 \land \epsilon \lt 0$, where PED is constant or iso-elastic but I'm referring to affine/linear demand curves only).

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You are right that this is a bit of an abuse of a terminology textbooks do but it is actually not without any sense at all.

For example consider two linear demand function functions:

$$ Q = 10 - 2p \tag{1}$$

$$ Q = 10 -1/2p \tag{2}$$

In a intro textbook such as in the Mankiws principles or any other undergraduate textbook the demand given by 1 would be called elastic while the demand given by 2 would be called inelastic.

Now, taken literally this does not make sense as both 1 and 2 are elastic at some prices but inelastic at others. However, note if you compare elasticity between these two functions point for point:

       Elasticity 
Price    1        2
  4     -4      -0.25
  3    -1.5     -0.176
  2    -0.667   -0.111
  1    -0.25    -0.052

As you can see, in a point for point comparison function 1 is more elastic than function 2. This is what the textbook means when it says one demand is more elastic than another demand.

Granted this is a sort of misnomer, a more correct way of stating this would be to say that at relevant price range demand 1 is relatively more elastic to demand 2, but that is quite a mouthful, it is easier just to say demand 1 is more elastic to demand 2. However, you are right this is not very rigorous terminology but undergraduate textbooks are less rigorous in general, in graduate texts this sort of terminology is not used anymore widely and terms are more precise.

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