Homework #5

Due 19 February 03

Plan to turn this assignment in on paper during class time, instead of emailing it. Instead of writing, this week we will solve a few mathematical problems on price discrimination.

You are encouraged to check answers with your classmates while solving these problems, but you must write up your solutions independently. Copying someone else's work will be a serious violation of academic integrity, and will be punished appropriately.

  1. Suppose you are a monopoly firm producing an information good (zero marginal cost). Suppose there are three segments of customers: 100 Power User customers willing to pay up to $10 for the good, 200 Regular customers willing to pay up to $5 for the good, and 300 Occasional customers willing to pay up to $3 for the good.
    1. (5 points) Suppose you cannot identify a consumer's group, so you must charge a uniform price to all customers. Which price would be optimal? What profits would you earn at that price. Explain your reasoning.
    2. (5 points) Suppose you could identify each of the groups, and prevent resale among them. What would then be your optimal set of prices? What total profits would you earn? Explain your reasoning.
    3. (5 points) Explain why you are able to achieve perfect price discrimination in (b). Give me a simple modification to the problem that would still allow you to engage in group pricing, but would prevent you from achieving perfect price discrimination.
  2. Let's reconsider problem 1, but change the number of Power User customers to 300.
    1. (5 points) Now what is your optimal uniform price, and what profits would you achieve?
    2. (5 points) When you can identify each of the groups, and prevent resale among them, what is your optimal price schedule? What are your profits?
    3. (5 points) Consider both consumer surplus and producer surplus. Which situation has more economic efficiency (total surplus), (a) or (b)? Which situation has more consumer surplus, (a) or (b)? Explain your reasoning.
    4. (5 points) Go back to the situations described in problem 1. Which situation has more economic efficiency, (a) or (b)? Which situation has more consumer surplus, (a) or (b)? Explain your reasoning.
    5. (5 points) Why do you get qualitatively different answers in part (c) than you do in part (d)? Explain.
  3. Now let's look at an example of versioning - that is, the firm makes a menu of versions for the customers to choose from. You have created a Premium version of your product, but you know you could disable some of the features in order to create a Normal version of your product. Both products have zero marginal cost.

    Suppose you have two groups of customers. Thirty (30) Major Users are willing to pay up to $20 for the Premium version of your product and $0 for the Normal version of your product. Seventy (70) Minor Users are willing to pay up to $7 for the Premium verison of your product and $5 for the Normal version of your product. Each consumer will choose the product that gives her the highest surplus.
    1. (5 points) If you offer only the Premium version of your product, what price would you charge in order to maximize your profits, and what profits would you make?
    2. (5 points) If you offer both versions of the product, what prices would you charge in order to maximize your profits, and what profits would you make?
    3. (5 points) Now suppose that instead of valuing the Normal version at $0, the Major Users value the Normal version at $12. Keep all other values the same as before. Show that at the prices you chose in (b), Major Users would actually prefer to buy the Normal good instead of the Premium good. Then show that if this happened, you would be better off choosing the single-product strategy in (a).
    4. (10 points) Consider keeping the Normal version at the same price as in (b), but lowering the Premium price until the Major Users will be willing to buy the Premium good instead of the Normal good. What pair of prices will you now be charging? What profits will you make? Is this better or worse for the firm than the single-product strategy described in (a)?
    5. (5 points) Using the work you have done above, show that firms can sometimes can use versioning to extract some but not all of the consumer surplus generated.
  4. Now let's look at bundling, a special case of versioning. Shapiro and Varian (page 75) use an example where a firm has two zero-marginal-cost products (word processor and spreadsheet), and two potential customers. Mark values a word processor at $120 and a spreadsheet at $100, while Noah values a word processor at $100 and a spreadsheet at $120. They show that the optimal pricing strategy is to offer both products as a bundle at a price of $220.

    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). This example 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.
    1. (5 points) Suppose we add a third customer, Oscar, who values a word processor at $150 and a spreadsheet at $10 (perhaps he's a professional writer). What are the profits if we offer only a bundle at $220, as before?
    2. (5 points) Remember that you may wish to offer individual products at their own prices, and the bundle at a separate price. Having added Oscar, what are the optimal product prices? What are the profits?
    3. (5 points) Suppose we now add a fourth customer, Peter, who values a word processor at $20 and a spreadsheet at $160 (perhaps he's an accountant). Now what are the optimal product prices and profits?
  5. On page 78, Shapiro and Varian discuss pricing of "mass customized" products. Suppose that all four of our customers from question 4 have the following values for two different music recordings (zero marginal cost) offered by a new music service:

      Widespread Panic Stephen Sondheim
    Mark $1.20 $1.00
    Noah $1.00 $1.20
    Oscar $1.50 $0.10
    Peter $0.20 $1.60

    1. (5 points) Suppose the music service offers a uniform price per song. What will that price be, and what will be the resulting profits?
    2. (5 points) Now suppose the service can offer a nonlinear pricing scheme. If it charges one price for the first recording and another price for the second recording purchased by each individual, what will be the two optimal prices? What will profits be?
    3. (5 points) Nonlinear pricing is a form of versioning, and hence a form of price discrimination. In part (b), which customers are discriminated against, and how?