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

Function of qq

R′(q)=p(q)(1−1|ε|)R′(q)=p(q)(1−1|ε|)

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

Example

    • q=20−2pq=20−2p , 0≤p≤100≤p≤10 . p=5−q/2p=5−q/2 , 0≤q≤200≤q≤20
    • |ε|>1|ε|>1 (elastic) when p>5p>5 ( q<10q<10 )
    • R(p)=20p−2p2R(p)=20p−2p2 . R(q)=10q−q22R(q)=10q−q22
    • R′(p)=20−4pR′(p)=20−4p . R′(p)<0R′(p)<0 if p>5p>5 (elastic)
    • R′(q)=10−qR′(q)=10−q . R′(q)>0R′(q)>0 if q<10q<10 (elastic)