Question

# A firm faces a demand function D(p), for which the revenue- maximizing price is \$10. The...

A firm faces a demand function D(p), for which the revenue- maximizing price is \$10. The demand
function is altered to 2D(p). What is the new revenue maximizing price? wih explanation

Revenue(R) = p*D where p = price and D = quantity demand and D is a function of p

Max : R = p*D

First order condition:

dR/dp = 0

=> p(dD/dp) + D = 0 ----------------(1)

It is given that revenue maximizing price = 10

Hence, Solving (1) we get p = 10

Now Demand = D' = 2D

Now Revenue(R') = pD' = 2p*D

Max : R' = 2p*D

First order condition :

dR'/dp = 0 => 2(pdD/dp + D) = 0 => (pdD/dp + D) = 0 -------------(2)

Hence we have to find p that satisfies (2)

If we look at (1) and (2) then we can see that both are the same and it is given that solution of (1) is p = 10 and thus solution of (2) will also be p = 10.

Hence, The new revenue maximizing price = 10.

Note:

If demand doubles then price and quantity that maximizes Revenue will be same for both but revenue when demand doubles is twice of the Revenue under initial demand.

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