Assignment:
Suppose a firm offers two products, a word processor and a spreadsheet, and produces each at a marginal cost of zero. There are two consumers: Justin values a word processor at $120 and a spreadsheet at $100, while Ubaldo values a word processor at $100 and a spreadsheet at $120. It is easy to show that the optimal pricing strategy is to offer both products as a bundle at a price of $220.
But in the real world, we often see "mixed bundling" of products - that is, the individual products are offered at separate prices and they are also offered as a bundle (usually at a discount from the sum of the two separate prices). The example above does not give us mixed bundling because only the bundle is offered, and not the separate components. Let's look at an example where the firm might like to offer mixed bundling.
(a) Draw a graph, similar to the one in class, with valuations of the word processor and the spreadsheet on the horizontal and vertical axes. Put a J and U on the graph to represent the preferences of Justin and Ubaldo, respectively.
(b) Suppose we add a third customer, Michael, who values a word processor at $150 and a spreadsheet at $10 (perhaps Michael is a professional writer, or perhaps he is the best defensive center fielder in baseball). Add an M to your graph. What are the profits if we only offer a bundle, at $220, as before?
(c) If we still use only pure bundling, what is the optimal bundle price and what are profits?
(d) If we now use mixed bundling, what are the optimal bundle and component prices? What are profits?
(e) Suppose now we add a fourth customer, Chris, who values a word processor at $20 and a spreadsheet at $160 (perhaps Chris is an accountant, or perhaps he's a late-inning reliever with emotional problems). Add a C to your graph. Now what are the optimal mixed bundling prices and profits?
(f) Graph the prices from mixed bundling. Break this graph up into four regions: those that consume only word processors, those consuming only spreadsheets, those consuming both, and those consuming nothing.
2. Suppose the demand for a new pharmaceutical drug, on which the manufacturer has a patent monopoly, is given by:
Q(P,A) = (100 - P) · A0.5
where Q is output per period, P is the price, and A is the current period promotional expenditure.
Total production costs are given by C(Q) = 60Q.
(a) Calculate the profit-maximizing price, advertising expenditure, and profits for the firm.
(b) At the firm's optimal choices, what are the: (i) price elasticity of demand; (ii) elasticity of demand with respect to advertising expenditure; and (iii) advertising / sales ratio for this product? What does all this have to do with the Dorfman-Steiner condition?
3. PRN Problem 2, p. 634 (f denotes the fraction of consumers who purchase the service).
Assume that consumers contemplating buying a network service have reservation prices uniformly distributed on the interval [0, 50] (measured in dollars). Demand by a consumer with reservation price wi for this service is
?????? = {0 ???? ?????? < ??
{1 ???? ?????? ≥ ??
(a) Calculate the demand function for this service.
(b) What is the critical mass if the price is set at $5?
(c) What is the profit maximizing price for the service?