<p>When people talk about 'elasticity of demand' without further qualification they normally mean 'price elasticity of demand':</p>
<p><span class="math-container">$$\text{E}_p=\frac{\partial Q_t}{\partial P_t}\frac{P_t}{Q_t}$$</span></p>
<p>However, note your calculations are not entirely correct as in this case the elasticity should be:</p>
<p><span class="math-container">$$\text{E}_p=-\frac{1}{\gamma}\frac{P_t}{(X_t-P_t)/ \gamma} = - \frac{P_t}{X_t-P_t}$$</span></p>
<p>However, the paper never claims <span class="math-container">$\gamma$</span> 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).</p>

CC BY-SA 4.0
2020-12-20T00:21:24.617Z
2020-12-20T00:21:24.617Z
1muflon1
<p>I'm reading  a paper from <a href="https://www.jstor.org/stable/1807160#metadata_info_tab_contents" rel="nofollow noreferrer">Pindyck (1988)</a> where <span class="math-container">$$P_t=X_t-\gamma Q_t.$$</span> <span class="math-container">$P_t$</span> is the price,  <span class="math-container">$X_t$</span> is the demand state variable, <span class="math-container">$Q_t$</span> production and <span class="math-container">$\gamma\geq0$</span> is apparently the ''<strong>(constant) demand elasticity</strong>''. I'm not quite sure how to interpret this number (percentage change of what wrt what?)</p>
<p>According to wikipedia, I could consider
<span class="math-container">\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*}</span>
None of these expressions is really constant</p>

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2020-12-19T23:58:38.213Z
Could you please provide reference to the paper?
1muflon1
CC BY-SA 4.0
2020-12-20T00:00:31.810Z
@1muflon1 I linked [Pindyck's paper](https://www.jstor.org/stable/1807160?seq=1#metadata_info_tab_contents). It's from 1988, AER. The equation is the first one of section II. (He uses $\theta_t$ instead of $X_t$)
Alex
CC BY-SA 4.0
2020-12-20T00:05:36.587Z
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
1muflon1
CC BY-SA 4.0
2020-12-20T00:10:58.907Z
@1muflon1 You're right, the word elasticity isn't mentioned in the paper. We had a fellow student summarizing the paper and he labelled $\gamma$ a constant elasticity. I tried to figure out what he could have meant.
Alex
CC BY-SA 4.0
2020-12-20T00:11:52.070Z
I think that was a mistake. There is no indication that $\gamma$ is elasticity here.
1muflon1
CC BY-SA 4.0
2020-12-20T00:14:10.320Z
@1muflon1 Thanks for the confirmation. Constant elasticity functions would be typically of the form $P_t= X_t Q_t^{\gamma}$, right? With such a price function, $P_t=X_t-\gamma Q_t$, what would be the corresponding price demand elasticity, $\frac{\partial Q_t}{\partial P_t}\frac{P_t}{Q_t}$?
Alex
CC BY-SA 4.0
2020-12-20T00:24:35.690Z
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
1muflon1
<microeconomics><supply-and-demand><elasticity><price>
Elasticity of demand
Alex
# Elasticity of demand

(Asked by: username_0 on 2020-12-19)

I'm reading a paper from [Pindyck (1988)](https://www.jstor.org/stable/1807160#metadata_info_tab_contents) 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

**Tags:** microeconomics, supply-and-demand, elasticity, price
> **Comments:**
> **username_1**: Could you please provide reference to the paper?  
> **username_0**: @1muflon1 I linked [Pindyck's paper](https://www.jstor.org/stable/1807160?seq=1#metadata_info_tab_contents). It's from 1988, AER. The equation is the first one of section II. (He uses $\theta_t$ instead of $X_t$)  
> **username_1**: 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  
> **username_0**: @1muflon1 You're right, the word elasticity isn't mentioned in the paper. We had a fellow student summarizing the paper and he labelled $\gamma$ a constant elasticity. I tried to figure out what he could have meant.  
> **username_1**: I think that was a mistake. There is no indication that $\gamma$ is elasticity here.  
> **username_0**: @1muflon1 Thanks for the confirmation. Constant elasticity functions would be typically of the form $P_t= X_t Q_t^{\gamma}$, right? With such a price function, $P_t=X_t-\gamma Q_t$, what would be the corresponding price demand elasticity, $\frac{\partial Q_t}{\partial P_t}\frac{P_t}{Q_t}$?  
> **username_1**: 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  

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## 1 answers

### Answer by username_1 on 2020-12-20. Score: 5 | Accepted answer

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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