Question

Let’s return to Tuftsville (Chapter 10) where everyone lives along Main Street, which is 10 miles long. There are 1,000 people uniformly spread up and down Main Street, and every day they each buy fruit smoothie from one of the two stores located at either end of the street. Customers ride their motor scooters to and from the store, using $0.50 worth of gas per mile. Customers buy their smoothies from the store offering the lowest price, which is the store’s price plus the customer’s travel expenses getting to and from the store. Ben owns the store at the west end of Main Street and Will owns the store at the east end of Main Street. The marginal cost of a smoothie is constant and equal to $1 for both Ben and Will. In addition, each of them pays Tuftsville $250 per day for the right to sell smoothies.

a. Ben sets his price p1 first and then Will sets his price p2. After the prices are posted consumers get on their scooters and buy from the store with the lowest price including travel expenses. What prices will Ben and Will set?

b. How many customers does each store serve and what are their profits?

Answer 1

Firm 1 is again the leader and sets its price first and firm 2 is the follower setting its price second. Otherwise, the model is exactly the same as before. Each firm produces an identical good at the same, constant marginal cost, c, will purchase the good at the lower priced firm. If they set the same prices then each firm will serve half the market.

In setting its price, firm 1 must of course anticipate firm 2’s best response. Clearly firm 2 will have an incentive to price slightly below firm 1’s price whenever firm 1 sets a price greater than unit cost c and less than or equal to the monopoly price. In that case, by undercutting, firm 2 will serve the entire market and earn all the potential profits. On the other hand, if firm 1 sets a price less than unit cost c, then firm 2 will not match or undercut firm 1’s price because firm 2 has no interest in making any sales when each unit sold loses money. Finally, if firm 1 sets a price equal to unit cost c firm 2’s best response is to match it. The anticipated behavior of firm 2 in stage 2 puts firm 1 in a tight bind. Any price greater than unit cost c results in zero sales and there is no sense in setting a price less than c. The best firm 1 can do then is set a price equal to unit cost c. Firm 2’s best response in the next stage is to match firm 1’s price.

Matters are very different, however, if the two firms are not selling identical products. In this case, not all consumers buy from the lower priced firm. Product differentiation changes the outcome of price competition quite a bit. To illustrate the nature of price competitio with differentiated products, recall the spatial model of product differentiation that we developed previously. The setup is the following. There is a product spectrum of unit length along which consumers are uniformly distributed. Two firms supply this market. One firm has the address or product design x = 0, on the line whereas the other has location, x = 1. Each of the firms has the same constant unit cost of production c.

A consumer’s location in this market is that consumer’s most preferred product, or style. “Consumer x” is located distance x from the left-hand end of the market. Consumers differ regarding which variant or location of the good they consider to be the best, or their ideal product, but are identical in their reservation price V for their most preferred product and we assume that the reservation price V is substantially greater than the unit cost of production c. Each consumer buys at most one unit of the product. If consumer x purchases a good that is not her ideal product she incurs a utility loss of tx if she consumes good 1 (located at x = 0), and t(1 − x) if she consumes good 2 (located at x = 1). The two firms compete for customers by setting prices, p1 and p2, respectively. However, unlike the simple Bertrand model, firm 1 sets its price p1 first, and then firm 2 follows by setting p2. In order to find the demand facing the firms at prices p1, and p2 we proceed as in the previous chapter by identifying the marginal consumer x m, who is indifferent between buying from either firm 1 or firm 2. Indifference means that the consumer x m gets the same consumer surplus from either product and so satisfies the condition:

In the simultaneous price game the two firms set the same prices p1 * = p2 * = c + t, whereas in the sequential game firm 1 sets a price in stage 1 that is greater than c + t, and firm 2 responds by setting a slightly lower price, but still higher than c + t. A second difference is that the two firms in the sequential price game have different market shares and earn different profits. In the simultaneous price-setting game, each firm served one half the market and earned the same profit equal to Nt/2. In the sequential game, on the other hand, firm 1 serves 3/8 of the market, and earns a profit equal to 18Nt/32, whereas firm 2 serves 5/8 of the market and earns a profit equal to 25Nt/32. Finally, note that unlike the Stackelberg output game, the sequential price game just described presents a clear second mover advantage. Firm 2 enjoys a larger market share and higher profit than firm 1. Both are better off than in the simultaneous game but firm 2, the second mover, is even better off than firm 1. However, this advantage diminishes as consumer preference for differentiation, as measured in our example by the parameter t, decreases. When the goods are perfect substitutes there is no second mover advantage.