Elasticity of demand 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}&amp;=\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}&amp;=\frac{\partial P_t}{\partial X_t}\frac{X_t}{P_t}=\frac{X_t}{X_t-\gamma Q_t}\\
\text{Elasticity}&amp;=\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}&amp;=\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
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).