How exactly does elasticity relate to slope?

Question

My book (Goodwin's Microeconomics in Context, pg. 117) states the following about price-elasticity of demand:

Given two demand curves that go through a specific point on graphs with the same scale, the flatter demand curve will represent the relatively more elastic demand and the stepper one the relatively less elastic demand.

I have actually two questions about this:

1) Will the flatter demand curve be more elastic at any given point (for any given value of $p$) or just at the point that both curves pass through?

2) How can we show this mathematically using the definition of elasticity as  $$epsilon=frac{dQ}{dp}frac{p}{Q}$$?

Thanks very much in advance.

Pburg · Accepted Answer

1) Yes, the steeper curve is more inelastic at all prices, if they are linear.

2) For linear demand curves, we have $epsilon(P) = frac{1}{m}frac{P}{Q(P)}$ for a demand curve with slope $frac{Delta P}{Delta Q}=m$. Let the demand curve be represented $P=b+mQ$. This will reduce to $epsilon = frac{P}{P-b}$ where $b$ is the $P$-intercept.

By hypothesis, they both share a point $(Q,P)$, so the steeper slope corresponds to a greater value of $b$, and so the curve is more inelastic.

Giskard · Answer

The two demand functions $D_1(p),D_2(p)$ cross at the point $(Q,p)$. Their respective elasticities at price $p$ are
begin{align*}
epsilon_1(p) & = frac{text{d}D_1(p)}{text{d}p}frac{p}{D_1(p)}

epsilon_2(p) & = frac{text{d}D_2(p)}{text{d}p}frac{p}{D_2(p)}.
end{align*}
However since both function cross at the point $(Q,p)$ we know that
$$
D_1(p) = D_2(p) = Q.
$$
But then
$$
frac{p}{D_1(p)} = frac{p}{Q} = frac{p}{D_2(p)}.
$$
Meaning the only difference between their elasticities is $text{d}D_i(p)/text{d}p$, which is their slopes.

As for your 1. question, the conditions are not clear. Is the 'flatter' curve only 'flatter' locally, or for every price $p$? If you only mean locally, then no, the statement is only valid in the intersection point $(Q,p)$.

How exactly does elasticity relate to slope?

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