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