Question

KW has produced a new hit song, “Lovin’ KK.” Digital copies can be produced at marginal cost of zero, and he has no fixed costs. The market demand for KW’s song is given by Q_D = 8 – P, where quantity is measured in millions of copies.

• If KW seeks to maximize profits and sells the song directly, how many copies will he sell?
• What is his profit?  

Suppose KW cannot sell the song directly but must sell rights to copies of the song to the online retailer Hamazon, who in turn sells the copies to their customers at a price of $P per copy. Hamazon has no marginal or fixed costs of selling the song.

• Under double marginalization, the quantity of copies sold will be?
• Under double marginalization, the total profits of KW and Hamazon combined from sales of the song will be?

Answer 1

KW has produced a new hit song. The marginal cost of produce that song is zero. Now market demand curve for that song is given as Qd = 8 - P, where quantity is in millions of copies. There is no fixed cost to produce that song. Now we can rearrange the demand function as P = 8 - Q. Therefore TR = P*Q = 8Q - Q^2 , MR = 8 - 2Q. We know the profit maximising condition of a firm is MR = MC, as MC =0 we can write it as MR = MC =0, or 8 - 2Q = 0, or Q = 4. Therefore P = 8 - Q = 8 - 4 =4. So the profit maximising quantity is 4 millions of copies and profit maximising price is $4. The profit of KW is equal to TR, as there is no fixed cost and marginal cost is zero. So total cost is 0. Profit = TR - TC, as TC =0, so profit = TR. Therefore Profit = P*Q =$4*4millions = $16 millions.

If KW is not able to sell the copies directly and sell it to online retailer Hamazon. Hamazon sell the copy at $P. As Hamazon has no marginal cost and fixed cost of selling those copies we can say the only marginal cost of Hamazon will be the cost at which Hamazon is getting the copies from KW. Now KW sell those products to Hamazon at a price of $4. Because KW will sell those products to Hamazon at profit maximising price. Now the MC of Hamazon of selling those copies will be $4 +$0 = $4. As own selling marginal cost is 0, so overall marginal cost will be $4. Now we set profit maximising condition of MR = MC. We have already derived the MR function of market demand and MR = 8 - 2Q. Now equalising MR = MC we get 8 - 2Q = 4 , or Q = 2. Therefore P = 8 - 2 =6. Therefore under double marginalization the quantity sold will be 2 millions of copies. Total combined profit will be profit of KW plus profit of Hamazon. Profit of KW is P*Q = $4*2millions = $8millions. ( Q is 2 millions because Hamazon's profit maximising demand from KW is 2 millions of copies). Profit of Hamazon is $6*2millions - $4*2millions = $4 millions. So total combined profit = $8 millions + $4millions = $12 millions.