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I'm reading a paper from Pindyck (1988) where $$P_t=X_t-\gamma Q_t.$$ $P_t$ is the price, $X_t$ is the demand state variable, $Q_t$ production and $\gamma\geq0$ is apparently the ''(constant) demand elasticity''. I'm not quite sure how to interpret this number (percentage change of what wrt what?)

According to wikipedia, I could consider \begin{align*} \text{Elasticity}&=\frac{\partial X_t}{\partial P_t}\frac{P_t}{X_t}=\frac{X_t-\gamma Q_t}{X_t}=1-\gamma\frac{Q_t}{X_t}\\ \text{Elasticity}&=\frac{\partial P_t}{\partial X_t}\frac{X_t}{P_t}=\frac{X_t}{X_t-\gamma Q_t}\\ \text{Elasticity}&=\frac{\partial Q_t}{\partial P_t}\frac{P_t}{Q_t}=-\frac{1}{\gamma}\frac{X_t-\gamma Q_t}{Q_t}=-\frac{1}{\gamma}\frac{X_t}{Q_t}+1 \\ \text{Elasticity}&=\frac{\partial P_t}{\partial Q_t}\frac{Q_t}{P_t}=\frac{-\gamma Q_t}{X_t-\gamma Q_t}=\frac{X_t}{\gamma Q_t-X_t}+1 \end{align*} None of these expressions is really constant

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  • $\begingroup$ Could you please provide reference to the paper? $\endgroup$
    – 1muflon1 ♦
    Dec 19 '20 at 23:58
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    $\begingroup$ @1muflon1 I linked Pindyck's paper. It's from 1988, AER. The equation is the first one of section II. (He uses $\theta_t$ instead of $X_t$) $\endgroup$
    – Alex
    Dec 20 '20 at 0:00
  • $\begingroup$ Thanks, also where do they mention elasticity of demand should be constant? I tried to search the paper for that but could not get any match for word elasticity $\endgroup$
    – 1muflon1 ♦
    Dec 20 '20 at 0:05
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    $\begingroup$ I think that was a mistake. There is no indication that $\gamma$ is elasticity here. $\endgroup$
    – 1muflon1 ♦
    Dec 20 '20 at 0:11
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    $\begingroup$ Yes that sort of function would give you constant elasticity of demand although it would not be the only sort of function that can give you that. But linear functions generally wont have constant elasticity of demand $\endgroup$
    – 1muflon1 ♦
    Dec 20 '20 at 0:24
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When people talk about 'elasticity of demand' without further qualification they normally mean 'price elasticity of demand':

$$\text{E}_p=\frac{\partial Q_t}{\partial P_t}\frac{P_t}{Q_t}$$

However, note your calculations are not entirely correct as in this case the elasticity should be:

$$\text{E}_p=-\frac{1}{\gamma}\frac{P_t}{(X_t-P_t)/ \gamma} = - \frac{P_t}{X_t-P_t}$$

However, the paper never claims $\gamma$ is elasticity and that the demand has constant elasticity in this case. Based on discussion in the comments which mentions that the claim was actually not made by the paper but by some student I assume it was a mistake (confusing slope of a demand with elasticity is actually surprisingly common among students).

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