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How do you show that the price elasticity of demand is a constant if the demand function is log-linear? To show this, how do you differentiate the log-linear demand curve to determine dQ/dp, and substitute that expression into the definition of the elasticity of demand?

There is an explanation in the textbook but I don't get it at all for the log-linear demand curve, although I get it for the exponential demand curve. How do you differentiate the log-linear demand curve? Can you please explain each step in detail? Because the steps are already in textbook but I don't understand it even as I reread it multiple times.

Thank you!!

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  • $\begingroup$ What are the steps that your textbook takes? You should include the steps in your questions and say exactly which step(s) you don't understand. $\endgroup$
    – Herr K.
    Aug 27 '20 at 16:15
  • $\begingroup$ Log-linear? Do you mean what is sometimes termed double-log, ie $ln(Q)=a-b (ln(P))$? $\endgroup$ Aug 27 '20 at 18:48
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If you mean: $Q=a-b \ln(p)$ then

Elasticity $E = \frac{dQ}{dp} \frac{p}{Q}=-\frac{b}{p}\frac{p}{Q}$.

Or, equivalently, $E = \frac{d \ln(Q)}{d \ln(p)} = -b/Q$

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