Revenue function

Summary

Revenue function

• Demand function: \(q=q\left( p \right)\)
• Inverse demand function: \(p=p\left( q \right)\)

\[R\left( p \right)=p⋅q\left( p \right)\]

• is called the revenue function (revenue as a function of price)

\[R\left( q \right)=q⋅p\left( q \right)\]

• is the revenue as a function of quantity

• Revenue function is also called the expenditure function
• Example: price elasticity of demand equal to -1
• \(q=cp^{-1}\) where \(c>0\) is an arbitrary constant
• \(p=cq^{-1}\)
• \(R\left( p \right)=R\left( q \right)=c\) . \(R\) does not depend on \(p\) or \(q\) in this example

Revenue function: linear demand

• \(q=20-2p\) , \(0≤p≤10\)
• \(p=10-q/2\) , \(0≤q≤20\)

\[R\left( p \right)=20p-2p^2\]

\[R\left( q \right)=10q- \frac{q^2}{2}\]

• Linear demand in general
• \(q=a-bp\) where \(a>0,b>0\) are constants and \(0≤p≤a/b\)
• \(p= \frac{a}{b}- \frac{q}{b}\)
• \(R\left( p \right)=ap-bp^2\)
• \(R\left( q \right)= \frac{aq}{b}- \frac{q^2}{b}\)