R revenue, P price, Q quantity sold
Revenue R=PQ
increase P by an amount \Delta P ( \Delta is the Greek letter delta, uppercase)
Q will change by \Delta Q and R by \Delta R
R=PQ (1)
R+\Delta R = (P+\Delta P)(Q+\Delta Q)\approx PQ+P\Delta Q+Q\Delta P (2) (expand and neglect the term \Delta P\Delta Q because it is smaller than the other terms)
subtract (1) from (2)
\Delta R = P\Delta Q+Q\Delta P
marginal revenue
MR=\frac{\Delta R}{\Delta Q}=P+Q\frac{\Delta P}{\Delta Q}=P\left(1+\frac{Q}{P}\frac{\Delta P}{\Delta Q}\right) (3)
everything is straightforward up to here.
The `trick’ now is to realize that price elasticity of demand
E_{p}=-\frac{\Delta Q}{Q}/\frac{\Delta P}{P} so that we can write (3) as
MR=P(1-\frac{1}{E_{p}})
If you prefer derivatives to deltas, then
R=PQ
MR=\frac{dR}{dQ}=P+Q\frac{dP}{dQ}=P\left(1+\frac{Q}{P}\frac{dP}{dQ}\right)
and again you can write this in the desired form using price elasticity of demand
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thank you so much!
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