You sell software downloaded from via a website, which costs $1000 per month to maintain. Marginal cost is essentially $0 per download.
There are some competitors for your product, but none of them is exactly like your product. You have estimated that your demand follows the following pattern:
Qd= 100 - 4P + 100A
where A is an ad you can purchase for $500 per ad (you have to purchase a whole ad).
a) If you don't spend any money on advertising and charge $15, calculate the (own) price elasticity of demand.
b) Again, if A = 0, what is the optimal price to charge? (note, you are maximizing revenue since MC=0)
c) At this optimal price, should you buy any advertising (yes or no, and support your answer).
Answer
(a)
Demand is given by: Qd= 100 - 4P + 100A => dQd/dP = -4
Also when A = 0 and P = 15 then Qd = 100 - 4*15 = 40
Own Price Elasticity of demand (E) = (dQd/dP)(P/Qd)
= -4(15/40)
= -1.5
=> Own Price Elasticity of demand (E) = -1.5
(b)
In order to maximize profit a firm should produce that quantity at which MR = MC
Qd = 100 - 4P + 100A when A = 0 the Qd = 100 - 4P => P = (100 - Qd)/4
where MR = Marginal Revenue = d(TR)/dQd = d(PQd)/dQd = d(((100 - Qd)/4)Qd)/dQd = (1/4)(100 - 2Qd) and MC = 0
MR = MC => (1/4)(100 - 2Qd) = 0 => Qd = 50
Hence P = (100 - Qd)/4 = (100 - 50)/4 = 12.5
Hence, the optimal price to charge = 12.5
(c)
Lets find the total cost function
TC = Fixed cost + Variable cost + Advertisement cost
Advertisement cost for A ads = 500A , Fixed cost = 1000 and as Marginal cost = 0 for all Q => Variable cost = 0
=> TC = 1000 + 500A
If A = 0 then Profit = TR - TC = PQd - TC = 12.5*50 - (1000 + 500*0) = -375
If it buy A unit of ad then his profit = PQd - TC = 12.5(100 - 4*12.5 + 100A) - (1000 + 500A)
= 12.5(50 + 100A) - 1000 - 500A
= 625 + 1250A - 1000 - 500A
= -375 + 750A where A > 0
He should buy ad If profit after buying an ad is greater than profit when ad(A) = 0
profit after buying A unit of ad = -375 + 750A and Profit when (A = 0) = -375
As A > 0 => -375 + 750A > -375
So, profit after buying an ad is greater than profit when ad(A) = 0
Hence He should buy an advertisement.
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