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