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EC 326 Problem set No 3

Question No. 1 (30%)

A profit maximizing firm produces three products X, Y and Z. The firm has no costs. There are three customers 1, 2 and 3. Each customer is willing to purchase at most one unit of each of the three products. The firm cannot price discriminate between customers. The following table presents the willingness to pay of each of the three customers for each of the three products:

```Product
X                  Y                   Z
Customer

1                    10                 12                  5

2                    8                  14                  0

3                    4                  16                   7
```

So, for example, Customer 1 is willing to pay no more than \$10 for purchasing one unit of product X and Customer 3 is willing to pay no more than \$7 for purchasing one unit of product Z.

• What will be the price of each product if the firm decides to sell them separately?
• Suppose, instead, that the firm decides to sell the three products only as a bundle. What will be the price of the bundle in this case?
• Which of the two alternatives above is better for the firm?

Which of the two alternatives above is better for each of the three customers?

• Can you think of a pricing and bundling strategy that is more profitable for the firm than the two strategies discussed above?

Question No. 2 (30%)

You run a sport club that offers fitness guidance by the hour.

Your cost is \$40 per hour and you have no other costs.

You serve 2 types of customers in equal number: 100 youngsters and 100 seniors.

The weekly demand by each youngster is: where q is the number of hours per week and p is the price per hour.

Each senior customer is willing to pay no more than \$50 per hour, for the first 3 hours, and zero thereafter.

1. Since you can distinguish between the customers, you have decided to set different prices to different customers. What price will you charge the seniors? How many hours will they buy? What price will you charge the youngsters? How many hours will they buy? What will your profit be in this case?
2. Now suppose that you have decided to change your pricing policy and to sell each customer a weekly card that indicates the maximum number of hours he/she can spend in the club, during the week. Since you can distinguish between the youngsters and the seniors, you can sell different customers different cards with a different number of hours and different prices. What price and how many hours will you set for a young client? What price and how many hours will you set for a senior client? What will your profit be in this case?
3. Instead, you have decided to use TPT for each group (charging an entry fee plus a price per hour). What prices should you set for each group? For how many hours
4. How would your answers to part c above change, if you could still use a TPT strategy but you could not discriminate between the customers?

Question No. 3 (40%)

Consider a monopolist seller (with no production costs) and two buyers,such that buyer 1 makes her buying decision first. Each buyer wants to purchase exactly one unit of the supplier’s product. The buyers’ actual valuation of the product, , is either 6 or 2. The two realizations are thought to be equally likely by the buyers (and the seller) without any additional information. Neither the buyers nor the seller know about the buyers’ true values before any such information.

Consider the following stage game:

Stage 1: The seller chooses. (The seller’s choice becomes publicly observable.)

Stage 2: The buyers observe their private signal, and updates their assessment of valuation.

Stage 3: The seller sets a single uniform price for both the buyers.

Stage 4: Buyer 1 makes the purchase decision. If she buys, she gives a perfectly informative signal to buyer 2. If she does not, the only information available to buyer 2 is the advertisement.

Stage 5: Buyer 2 makes the purchase decision.

• For a fixed, what is the maximum price that the seller can charge to the buyer if they receive a signal? What is the maximum price that the seller can charge to each buyer if they receive a signal ? Call these prices and .
• Write down the ex-ante expected demand faced by the firm for a given choice of . Write down the profits of the firm as a function of the prices (and the accuracy level). (Hint: In writing the expected demand, think of the ranges of prices and the probabilities for which only one consumer buys or both the consumers buy the product.)
• Get the threshold level of accuracy of the advert such that the firm finds it optimal to charge the higher price.
• Plot the resulting profits as a function of . What is the optimum value of accuracy and price that the monopolist chooses? Does the firm find it profitable to sell the product to buyer 1, why/why not?
• Now think of the situation in which buyer 2 has a sufficiently large “high” valuation, i.e. her maximum valuation is large enough. Intuitively, do you think the equilibrium behavior of the monopolist (in terms of the choice of and price) should change?