Price elasticity of demand for linear demand functions

Summary

Price elasticity of demand: linear demand function

• \(q=a-bp\) where \(a>0,b>0\) are constants and \(0≤p≤a/b\)
• Inverse demand function: \(p=a/b-q/b\) for \(0≤q≤a\)
• Price elasticity of demand

\[ε= \frac{-bp}{a-bp}\]

• Middle of the demand curve: \(p=(a/b)/2\) , \(q=a/2\) we have \(ε=-1\)
• Top left: demand is elastic ( \(ε=-∞\) at \(q=0\) )
• Lower right: demand is inelastic ( \(ε=0\) at \(p=0\) )

• Example:
• \(q=20-2p\) for \(0≤p≤10\)
• \(p=10-q/2\) for \(0≤q≤20\)
• \(ε= \frac{-2p}{20-2p}=- \frac{p}{10-p}\)
• \(\left| ε \right|=1\) when \(p=5\) ( \(q=10)\)
• \(\left| ε \right|>1\) when \(p>5\) ( \(q<10\) )
• \(\left| ε \right|<1\) when \(p<5\) ( \(q<10\) )