Price elasticity of demand - logs, units and elastic vs inelastic demand

Summary

Price elasticity and logs

• An alternative (and equivalent) definition of price elasticity of demand

\[ε= \frac{dlog q}{dlog p}\]

• Example

\[q\left( p \right)=c/p\]

\[log q\left( p \right)=log c-log p\]

\[ε= \frac{dlog q}{dlog p}=-1\]

Price elasticity and units

• \(q\left( p \right)\) is a given demand function
• \(dq/dp \) depends on units
• Elasticitites are unit free
• Example: \(q\) measured in [ litres ] and \(p\) in [euro]
• \(dq/dp\) will be measured in [litres]/[euro] or litres per euro
• The price elasticity of demand \(ε\) will be unit free

\[ \frac{\left[ litres \right]}{[euro]}⋅ \frac{\left[ euro \right]}{\left[ litres \right]}=1\]

Elastic and inelastic demand

• \(ε\) is price elasticity of demand
• If \(\left| ε \right|>1\) : we say that demand is elastic . If \(p\) increases by 1%, \(q\) falls by more than 1% (demand is price sensitive)
• If \(\left| ε \right|<1\) : we say that demand is in e lastic . If \(p\) increases by 1%, \(q\) falls by less than 1% (demand is not price sensitive)

Constant price elasticity of demand

• If \(q\left( p \right)=cp^{-α}\) where \(c>0,α>0\) are arbitrary constants then

\[ε=-α\]

• for all \(p\) .
• For \(α=1\) , \(q\left( p \right)=c/p\) and \(ε=-1\) .