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.
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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