How exactly does elasticity relate to slope? 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.
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) &amp; = \frac{\text{d}D_1(p)}{\text{d}p}\frac{p}{D_1(p)} \\
\\
\epsilon_2(p) &amp; = \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 $(Q,p)$.