Slope of linear demand curve and price elasticity regions along the curve

Question

My textbook mentions that the slope of a linear demand curve gives crucial information regarding price elasticity of demand; namely, that demand is inelastic for steeper curves and elastic for flatter ones. Later they also demarcate different regions of the curve, where the region to the northwest of the curve's midpoint shows elastic demand, and the region southeast of the midpoint inelastic demand.

I'm a little confused however on how to combine these two aspects. If you have for example a very steep linear demand curve, which is indicative of inelastic demand, but equilibrium price and quantity are in the elastic region, is demand then elastic or inelastic?

Answer 1

Elasticity is defined at a point.

$$ \frac{\partial Q}{\partial P} * \frac{P}{Q} $$

A more or less elastic curve just means that the elasticity is higher or lower in general. For example, if you have the two demand curves

$$ q = -\frac{1}{4}p + 8 $$ $$ q = -\frac{1}{2}p + 12 $$

If you plug in the same price points or quantity points, you will get higher elasticities for the second equation and lower elasticities for the first. So we talk about the second curve being more elastic and the first curve being more inelastic, but its because each is characterized as such by the elasticities at its points.