Question

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:

Q_d= 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 1

(a)

Demand is given by: Q_d= 100 - 4P + 100A => dQ_d/dP = -4

Also when A = 0 and P = 15 then Q_d = 100 - 4*15 = 40

Own Price Elasticity of demand (E) = (dQ_d/dP)(P/Q_d)

= -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

Q_d = 100 - 4P + 100A when A = 0 the Q_d = 100 - 4P => P = (100 - Q_d)/4

where MR = Marginal Revenue = d(TR)/dQd = d(PQd)/dQd = d(((100 - Q_d)/4)Q_d)/dQ_d = (1/4)(100 - 2Q_d) and MC = 0

MR = MC => (1/4)(100 - 2Q_d) = 0 => Q_d = 50

Hence P = (100 - Q_d)/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 = PQ_d - TC = 12.5*50 - (1000 + 500*0) = -375

If it buy A unit of ad then his profit = PQ_d - 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.