<p>In the case of linear demand <span class="math-container">$d_i=a_i-x_iP$</span> (assuming <span class="math-container">$d_i$</span> is quantity demanded by individual <span class="math-container">$i$</span>), the price elasticity of demand at point <span class="math-container">$(d_i,P)$</span> is
<span class="math-container">\begin{equation}
\epsilon_i(d_i,P)=x_i\cdot \frac{P}{d_i}.
\end{equation}</span>
As @the_rainbox noted in their answer, price elasticity of demand varies along a linear demand curve. So in order to compare elasticities between different demand curves based only on the slope coefficients (the <span class="math-container">$x_i$</span>'s), you need to fix <span class="math-container">$P$</span> and <span class="math-container">$d_i$</span>; that is, assume that the demand curves of individuals <span class="math-container">$1$</span> and <span class="math-container">$2$</span> cross at some point <span class="math-container">$(Q_0,P_0)$</span>.
Then, you can say things like
<span class="math-container">\begin{equation}
\epsilon_1(Q_0,P_0)\ge \epsilon_2(Q_0,P_0) \quad\Leftrightarrow\quad x_1\ge x_2.
\end{equation}</span>
Or in words: <span class="math-container">$1$</span>'s demand is more elastic than <span class="math-container">$2$</span>'s at <span class="math-container">$(Q_0,P_0)$</span> if and only if <span class="math-container">$1$</span>'s demand curve is flatter than <span class="math-container">$2$</span>'s. [Note that since by convention demand curves are plotted in the <span class="math-container">$(Q,P)$</span>-plane, a flat demand curve actually corresponds to a high <span class="math-container">$x_i$</span>.]</p>
<p>Beware, however, that in contexts where a high degree of mathematical rigor is not required, it is sometimes taken as a rule-of-thumb that flat demand curves are &quot;generally&quot; more elastic than steep ones. Hence slope becomes a proxy for elasticity in those less rigorous discourses.</p>

CC BY-SA 4.0
2020-11-01T22:57:16.707Z
2020-11-01T22:57:16.707Z
Herr K.
<p>I have some question about the elasticity of the demand functions of two different traders/consumers. Suppose that <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> are the elasticities of the demand functions of these agents. Furthermore, they have linear demands, that are <span class="math-container">$d_1=a_1-x_1P$</span> and <span class="math-container">$d_2=a_2-x_2P$</span>, where <span class="math-container">$a_1,a_2$</span> are the intercept points of the demand functions and <span class="math-container">$P$</span> is the price of the asset/commodity. What does it mean that the elasticity of trader <span class="math-container">$1$</span> is higher than that of trader <span class="math-container">$2$</span> i.e. <span class="math-container">$x_1&gt;x_2$</span> or the opposite, <span class="math-container">$x_1&lt;x_2$</span>, in case they buy the asset/commodity or they sell it.</p>

<microeconomics><mathematical-economics><finance><financial-economics>
Elasticity of demand functions
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# Elasticity of demand functions

(Asked by: username_0 on 2020-10-31)

I have some question about the elasticity of the demand functions of two different traders/consumers. Suppose that $x\_1$ and $x\_2$ are the elasticities of the demand functions of these agents. Furthermore, they have linear demands, that are $d\_1=a\_1-x\_1P$ and $d\_2=a\_2-x\_2P$, where $a\_1,a\_2$ are the intercept points of the demand functions and $P$ is the price of the asset/commodity. What does it mean that the elasticity of trader $1$ is higher than that of trader $2$ i.e. $x\_1>x\_2$ or the opposite, $x\_1, in case they buy the asset/commodity or they sell it.

**Tags:** microeconomics, mathematical-economics, finance, financial-economics

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## 1 answers

### Answer by username_1 on 2020-11-01. Score: 5 | Accepted answer

In the case of linear demand $d\_i=a\_i-x\_iP$ (assuming $d\_i$ is quantity demanded by individual $i$), the price elasticity of demand at point $(d\_i,P)$ is
\begin{equation}
\epsilon\_i(d\_i,P)=x\_i\cdot \frac{P}{d\_i}.
\end{equation}
As @the\_rainbox noted in their answer, price elasticity of demand varies along a linear demand curve. So in order to compare elasticities between different demand curves based only on the slope coefficients (the $x\_i$'s), you need to fix $P$ and $d\_i$; that is, assume that the demand curves of individuals $1$ and $2$ cross at some point $(Q\_0,P\_0)$.
Then, you can say things like
\begin{equation}
\epsilon\_1(Q\_0,P\_0)\ge \epsilon\_2(Q\_0,P\_0) \quad\Leftrightarrow\quad x\_1\ge x\_2.
\end{equation}
Or in words: $1$'s demand is more elastic than $2$'s at $(Q\_0,P\_0)$ if and only if $1$'s demand curve is flatter than $2$'s. [Note that since by convention demand curves are plotted in the $(Q,P)$-plane, a flat demand curve actually corresponds to a high $x\_i$.]

Beware, however, that in contexts where a high degree of mathematical rigor is not required, it is sometimes taken as a rule-of-thumb that flat demand curves are "generally" more elastic than steep ones. Hence slope becomes a proxy for elasticity in those less rigorous discourses.


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