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 $ε=-1$ .