# Elasticity of demand functions

Economics Asked on January 5, 2021

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.

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 $$begin{equation} epsilon_i(d_i,P)=x_icdot frac{P}{d_i}. end{equation}$$ 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 $$begin{equation} epsilon_1(Q_0,P_0)ge epsilon_2(Q_0,P_0) quadLeftrightarrowquad x_1ge x_2. end{equation}$$ 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.

Correct answer by Herr K. on January 5, 2021

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.

Answered by S. Iason Koutsoulis on January 5, 2021