C.3 Elasticity with Linear Relationships

We’ve established that the general formula for elasticity is \(\epsilon_{Y,X} = \frac{\Delta y}{\Delta x} \times \frac{x}{y}\) Let’s examine the case in which there is a constant slope: that is, where $\Delta y/\Delta x$ is constant. For example, if we have \(y = mx + b\) In this case $\Delta y/\Delta x = m$, so \(\begin{aligned} \epsilon_{Y,X} &= \frac{\Delta y}{\Delta x} \times \frac{x}{y}\\ &= m \times \frac{x}{mx + b}\\ &= \frac{mx}{mx + b}\end{aligned}\) Notice that we plug in $y = mx + b$ for $y$ in order to express elasticity as a function of the independent variable $X$.

Example

A common example of this in economics is a linear demand curve; for example, \(Q = 80 - P\) Applying the formula, we have that the demand elasticity is \(\epsilon_{Q,P} = -\frac{P}{80-P}\) You can try moving the price up and down to see the elasticity at different points along this demand curve:

(Graph: point_slope)

You can check the “show regions of elasticity” box in the diagram above to see which part of the demand curve is perfectly elastic, elastic, unit elastic, inelastic, and perfectly inelastic. You might notice that, regardless how you move the intercepts of the demand curve:

• the demand elasticity (in absolute value) increases as $P$ increases. That is, as price increases and quantity decreases, even though a $€1$ increase in price always leads to the same decrease in quantity, that same $€1$ increase in price represents a smaller and smaller percent of the price, while the quantity decrease represents a larger and larger percent decrease in quantity.

• in the extreme case when $Q = 0$ (along the vertical axis), demand is perfectly elastic ($\epsilon = -\infty$). This is because even the smallest decrease in price will result in an infinite percentage change in quantity (from zero to some positive number).

• in the other extreme case when $P = 0$ (along the horizontal axis), demand is perfectly inelastic ($\epsilon = 0$). This is because it would take an infinite percentage change in price (from zero to some positive number) to cause any change in quantity.

• demand is unit elastic at the midpoint of the demand curve. This is because, if demand is given by $Q = a - bP$, then the demand curve extends from $(a, 0)$ to $(0, \frac{a}{b})$, and the formula for elasticity is \(\epsilon_{Q,P} = -\frac{bP}{a-bP}\) This is equal to $-1$ when $bP = a - bP$, which occurs at the the point $(\frac{a}{2}, \frac{a}{2b})$ – i.e., the midpoint of the demand curve.

Putting this all together, we can see that the upper (i.e., left) half of the demand curve is elastic, and the lower (i.e. right) half of the demand curve is inelastic.