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

Summary

Function of \(p\)

\[R'\left( p \right)=q\left( p \right)\left( 1-\left| ε \right| \right)\]

Function of \(q\)

\[R'\left( q \right)=p\left( q \right)\left( 1- \frac{1}{\left| ε \right|} \right)\]

Example

\(q=20-2p\) , \(0≤p≤10\) . \(p=5-q/2\) , \(0≤q≤20\)
\(\left| ε \right|>1\) (elastic) when \(p>5\) ( \(q<10\) )
\(R\left( p \right)=20p-2p^2\) . \(R\left( q \right)=10q- \frac{q^2}{2}\)
\(R'\left( p \right)=20-4p\) . \(R'\left( p \right)<0\) if \(p>5\) (elastic)
\(R'\left( q \right)=10-q\) . \(R'\left( q \right)>0\) if \(q<10\) (elastic)