Elasticity of demand functions

Question

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, $x_1<x_2$, in case they buy the asset/commodity or they sell it.

Answer 1

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.

Answer 2

Please note that the slope of the demand curve ($x_1, x_2$ in each case) is not the same as the price elasticity of demand. Especially in linear demand curves, we notice that price elasticity takes values in the range $(-\infty, 0)$.

It's easier to argue the differences in slope between two agents once that has been clarified.