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?
(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.
Take a shot.
Note that the publisher is a monopolist wrt to the textbook. Always maximize where marginal cost=marginal revenue. So, for each sub-problem, construct a marginal cost and marginal revenue equations.
(a) To find the price that will maximize the publisher's profit, we need to determine the quantity at which marginal cost (MC) equals marginal revenue (MR).
First, we need to calculate the publisher's total cost (TC) function. The publisher incurs a printing and distribution cost of $20 per book, so the TC can be expressed as:
TC = 20Q
Next, we find the publisher's revenue (TR) function. The price (P) is given by:
P = 100 - 0.005Q
The total revenue (TR) is the product of price and quantity:
TR = P * Q = (100 - 0.005Q) * Q = 100Q - 0.005Q^2
To determine the MC, we take the derivative of the TC function with respect to Q:
MC = d(TC) / dQ = d(20Q) / dQ = 20
To calculate the MR, we take the derivative of the TR function with respect to Q:
MR = d(TR) / dQ = d(100Q - 0.005Q^2) / dQ = 100 - 0.01Q
Setting MC equal to MR:
20 = 100 - 0.01Q
Rearranging the equation to solve for Q:
0.01Q = 100 - 20
0.01Q = 80
Q = 8,000
Substituting the value of Q back into the demand function to find the price (P):
P = 100 - 0.005Q = 100 - 0.005(8,000) = 100 - 40 = $60
Therefore, the price that will maximize the publisher's profit is $60.
To calculate the profit, we subtract the total cost from the total revenue:
Profit = TR - TC = (100Q - 0.005Q^2) - (20Q) = 100Q - 0.005Q^2 - 20Q
Substituting the value of Q:
Profit = (100 * 8,000) - 0.005 * (8,000)^2 - (20 * 8,000) = $400,000
The publisher will earn a profit of $400,000.
The total royalty payment earned by the author is equal to the royalty per book sold multiplied by the number of books sold. In this case, the royalty per book is $20, and the number of books sold is Q:
Total Royalty Payment = $20 * Q = $20 * 8,000 = $160,000
The total royalty payment earned by the author is $160,000.
(b) With a profit-sharing agreement, the author receives 40% of the profit, and the publisher receives 60%. We need to find the price that maximizes the total profit (TP) under this agreement.
Total Profit = 0.4 * Profit (Author's Share) + 0.6 * Profit (Publisher's Share)
Since we already know the profit from part (a), we substitute it into the equation:
Total Profit = 0.4 * $400,000 + 0.6 * $400,000
Total Profit = $160,000 + $240,000
Total Profit = $400,000
To find the price, we use the demand function:
P = 100 - 0.005Q
Substituting the value of Q from part (a):
P = 100 - 0.005 * 8,000 = 100 - 40 = $60
Therefore, with a profit-sharing agreement, the publisher should set a price of $60.
(c) Both the author and the publisher will prefer the profit-sharing agreement over their original agreement. Under the profit-sharing agreement, the author receives a larger share of the profit compared to the fixed royalty per book.
The students who buy the textbook may prefer either agreement depending on the price. If the price under the profit-sharing agreement is the same as the price under the original agreement, then the students would have no preference. However, if the price under the profit-sharing agreement is lower, they would prefer the profit-sharing agreement as it offers a lower price.
(d) To prove that there is an inherent conflict between the author and the publisher, we consider the situation where the author receives a payment that is 15% of sales revenue.
Let R represent the sales revenue. The royalty payment received by the author is 15% of R:
Royalty Payment = 0.15 * R
The publisher's profit can be calculated as:
Profit = R - TC = R - (printing and distribution cost) - (Royalty Payment)
Profit = R - 20Q - 0.15R
Simplifying the equation:
Profit = 0.85R - 20Q
To find the price that maximizes the publisher's profit, we need to set MC equal to MR:
MC = 20
MR = d(0.85R - 20Q) / dQ
MR = -20
Setting MC equal to MR:
20 = -20
This equation has no solution, indicating that there is no price at which the publisher's profit is maximized under this royalty payment condition.
Hence, there is an inherent conflict between the author and the publisher because the author's interest in receiving a higher royalty conflicts with the publisher's need to maximize profit.