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Appendix C / Elasticity

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:

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.

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