The demand function for a well known economics textbook is:
P = 100 - .005Q
The publisher must pay $20 per book in printing and distribution costs and, in
addition, it must pay the author a $20 royalty for each book sold.
(a) Your job is to provide advice to the publisher. What price will maximise
the publisher’s profit? How much profit will the publisher earn? What will
be the total royalty payment earned by the author?
Ans: profit maximizes when MR=MC.
MR=50-10Q
since TC function is not given do i have to assume ATC is constant. If ATC is constant MC will be constant, but then how do I equal MC to MR (what would be the MC function) and how do I substitute that to demand function.As I am stuck here couldn't proceed.
(b) A consultant says that the publisher and the author have the wrong type of
agreement. He says that the author and the publisher should tear up their
original agreement, in which the author gets $20 per book sold, and enter
into a profit-sharing agreement. He recommends that the author gets 40%
of the profit and the publisher 60%. What price should the publisher set
with this profit-sharing agreement?
(c) Will both the author and the publisher prefer the profit-sharing agreement
to their original agreement? Which agreement will the students who buy
the textbook prefer?
(d) Given the demand and cost conditions indicated above suppose that the
royalty payment was such that the author received a payment which was
15% of sales revenue. Prove that there is an inherent conflict between the
author and the publisher in that the author has an interest in the book’s
price being lower than the price which maximises the publisher’s profit.
First, you missed on marginal revenue.
Total revenue is P*Q. = 100Q-.005Q2 . Marginal revenue is the first derivitive of total revenue, so MR=100-.01Q
Marginal cost is the cost, to the publisher, of producing 1 additional book. He pays $20 printing costs plus $20 royalities for each book, regardless of the number printed. Ergo, Marginal Cost = Average cost = $40.
Solve MR=MC = 100-.01Q=40 Q=6000.
For you, calculate optimal price, the net profit going to the publisher and the royalities going to the author.
b) Now things get a bit tricky. The publisher must still pay $20 for printing. However, under the new contract, he pays 40% of the net profit. So, Marginal cost becomes 20 + .4*(P-20) = 12+.4*P. Substitute the original demand equation for P. Thus MC=12+.4*(100-.001Q) = 52-.002Q.
Again, set MC=MR and solve for Q.
c) Again calculate the amounts going to the publisher and author. Compare these to the amounts you calculated for part a). Compare the prices, as well, to see which plan the students will prefer.
d) With a flat 15% of sales, the marginal cost for the publisher becomes 20+.15P Repeat the steps you did in b). The inherent conflict arises because the publisher must pay the printing costs, while the marginal cost for the author is zero.
To answer the questions:
(a) To maximize the publisher's profit, we need to set the marginal revenue (MR) equal to the marginal cost (MC).
1. The MR is the derivative of the total revenue with respect to quantity (Q). In this case, the total revenue (TR) is given by P * Q, which equals (100 - 0.005Q) * Q = 100Q - 0.005Q^2. Taking the derivative, we get MR = 100 - 0.01Q.
2. The MC is given by the printing and distribution costs ($20) plus the royalty payment to the author ($20), regardless of the quantity produced. Therefore, MC = $40.
Setting MR = MC, we have the equation 100 - 0.01Q = 40. Solving for Q, we get Q = 6000.
3. To find the price that maximizes profit, substitute Q = 6000 into the demand function: P = 100 - 0.005(6000) = $70.
So, the price that maximizes the publisher's profit is $70. To find the profit, we need to calculate total revenue and subtract total costs, including the printing costs and the total royalty payment to the author.
Total revenue (TR) at Q = 6000 is P * Q = $70 * 6000 = $420,000.
Total costs (TC) include the printing and distribution costs, which is $20 per book regardless of the quantity produced. At Q = 6000, TC = $20 * 6000 = $120,000.
In addition, the royalty payment earned by the author is $20 per book regardless of the quantity sold. So, at Q = 6000, the total royalty payment is $20 * 6000 = $120,000.
Net profit = TR - TC - total royalty payment = $420,000 - $120,000 - $120,000 = $180,000.
Therefore, the publisher will earn a profit of $180,000.
(b) Under the profit-sharing agreement, the publisher and the author will split the profit in a 60:40 ratio. The publisher gets 60% of the profit, and the author gets 40%.
1. We need to find the new MC considering the profit-sharing agreement. The publisher pays 40% of the net profit to the author. Therefore, MC becomes $20 (printing costs) + 40% * (P - $20).
2. Substituting the original demand function into the expression for MC, we have MC = $20 + 0.4 * (100 - 0.005Q) = $52 - 0.002Q.
3. Setting MR = MC, we have 100 - 0.01Q = $52 - 0.002Q. Solving for Q, we get Q = 5000.
4. To find the price under the profit-sharing agreement, substitute Q = 5000 into the demand function: P = 100 - 0.005(5000) = $75.
Therefore, under the profit-sharing agreement, the publisher should set the price at $75.
(c) Compare the amounts going to the publisher and the author from both agreements to determine their preferences:
1. Under the original agreement, the publisher earns a profit of $180,000 and pays the author a total royalty payment of $120,000.
2. Under the profit-sharing agreement, the publisher's profit is the total profit multiplied by 60%, which is $180,000 * 0.6 = $108,000. The author's payment is the total profit multiplied by 40%, which is $180,000 * 0.4 = $72,000.
3. Therefore, the publisher would prefer the original agreement since it results in a higher profit ($180,000 compared to $108,000). Likewise, the author would prefer the original agreement since the total royalty payment is higher ($120,000 compared to $72,000).
4. As for the students who buy the textbook, they would prefer the profit-sharing agreement since it leads to a lower price for the textbook ($70 under the original agreement compared to $75 under the profit-sharing agreement).
(d) When the royalty payment is fixed at 15% of sales revenue, there is a conflict of interest between the author and the publisher. This conflict arises because the author has an incentive for the book's price to be lower than the price that maximizes the publisher's profit.
Under this scenario, the publisher's MC would become $20 + 0.15P, where P is the price. Setting MR equal to this MC would yield a different equilibrium price than before, leading to a conflict between the interests of the author and the publisher.