Question

XYZ Candy makes specialty chocolate bars to distribute online, shipping across North America. XYZ has established a great deal of monopoly power in pricing given its specialty status. Let’s assume XYZ’s customers are identical with individual (inverse) demand as P = 2.50 – 0.1Q, where Q is number of bars the customer orders per month. Marginal cost to supply another bar is constant (equal to average cost) at $0.50. If XWZ acts like a single-price non-discriminating monopoly, its profit-maximizing price is $1.50 and quantity is 10 (you can check this!)

a. Suppose XYZ offers a quantity discount where it sells customers the first 10 units at $1.50 each but discounts the price for each bar ordered over 10 units (i.e., a lower price for the 11th bar and all beyond that). What will be the discounted price that maximizes profit? (Hint: The relevant portion of the demand curve for the discount pricing are quantities beyond price $1.50 and 10 units; it may help to think of this portion of the demand curve as a new demand curve beginning with the 11th unit, or a shift in the demand of 10 units for price $1.50 or below.)

b. How many bars does XYZ sell at the discounted price?

c. Compare the profit in the discounted pricing scheme to the monopoly single price of $1.50.

d. Suppose XYZ adds a three-tier pricing policy. Explain what would happen to profits.

e. Explain what happens to consumer surplus as XYZ adds more discount price points.

Answer 1

a) Relevant portion of demand curve is given by

P= 2.50 - 0.1(Q - 10 ) or P = 1.50 - 0.1Q ; P, Q $\geq$ 0

Profit at discounted price = P*Q - Average cost* Q = (1.50 - 0.1Q)*Q - 0.5*Q = Q- 0.1Q^​​​2

first order Condition of profit maximisation d(Profit)/dQ = 0 or (1-0.2Q) = 0 or Q = 5 bars

P = 1.50 - 0.1Q = 1.50 - 0.1*5 = $1

b) 5 bars are sold at discounted price of $1

And 10 are sold at $1.5 . In total 15 bars are sold

c) Profit from single price = Revenue - cost = P*Q - c*Q = 10*1.5 - 0.5*10= $ 10

Profit from remaining portion of demand curve when sold at discounted price = Revenue - cost = 5*1 - 0.5*5 = $2.5

When there is two layer pricing, monopolist earns an additional $2.5 profit . In total he earns $12.5

d) Adding three tier pricing

The relevant portion of demand curve would be P = 2.50 - 0.1(Q- 10-5) or P= 1- 0.1Q

Profit from remaining portion of demand curve = P*Q - 0.5*Q = (1-0.1Q)Q - 0.5Q = 0.5Q - 0.1Q^​​​2

first order Condition of profit maximisation gives

d(profit)/dQ = 0 or 0.5 - 0.2Q = 0 or Q = 2.5 units

P= 1-0.1Q = $0.75

Profit = 0.5Q - 0.1Q^​​​2 = 0.5*2.5 - 0.1*(2.5)² =$ 0.625

Hence we find that three tier pricing leads to an additional profit of $0.625 .

e) As more and more discount price is added , it enables consumer to buy more and and unit of commodity which otherwise in monopoly was not possible.

But with each discount price, the monopolist is able to capture more and more part of consumer surplus. As shown in the figure below, square abcd is the relevant area which represents capturing of consumer surplus by discriminatory monopolist. If price had been $1.50 area abcd would have been consumer surplus.

00 Pri ($) P=105 a a To 10 15 quantity