The sign of the derivative of the revenue function when demand is elastic and inelastic

Summary

Function of pp

R(p)=q(p)(1|ε|)R(p)=q(p)(1|ε|)

  • R(p)>0R(p)>0 Demand is inelastic ( |ε|<1|ε|<1 )
  • R(p)=0R(p)=0 Demand is unit inelastic ( |ε|=1|ε|=1 )
  • R(p)<0R(p)<0 Demand is elastic ( |ε|>1|ε|>1 )

Function of qq

R(q)=p(q)(11|ε|)R(q)=p(q)(11|ε|)

  • R(q)>0R(q)>0 Demand is elastic ( |ε|>1|ε|>1 )
  • R(q)=0R(q)=0 Demand is unit inelastic ( |ε|=1|ε|=1 )
  • R(q)<0R(q)<0 Demand is inelastic ( |ε|<1|ε|<1 )

Example

    • q=202pq=202p , 0p100p10 . p=5q/2p=5q/2 , 0q200q20
    • |ε|>1|ε|>1 (elastic) when p>5p>5 ( q<10q<10 )
    • R(p)=20p2p2R(p)=20p2p2 . R(q)=10qq22R(q)=10qq22
    • R(p)=204pR(p)=204p . R(p)<0 if p>5 (elastic)
    • R(q)=10q . R(q)>0 if q<10 (elastic)