# 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 \(0≤p≤a/b\)
- \(p= \frac{a}{b}- \frac{q}{b}\)
- \(R\left( p \right)=ap-bp^2\)
- \(R\left( q \right)= \frac{aq}{b}- \frac{q^2}{b}\)