**Unit 7** The firm and its customers

## 7.10 Markets with few firms: Strategic price setting

- Nash equilibrium
- A Nash equilibrium is an economic outcome where none of the individuals involved could bring about an outcome they prefer by unilaterally changing their own action. More formally, in game theory it is defined as a set of strategies, one for each player in the game, such that each player’s strategy is a best response to the strategies chosen by everyone else.
*See also Game theory.*

### Before you start

To understand the model in this section, you will need to know about game theory and the concept of **Nash equilibrium**, introduced in Unit 4. If not, you can either skip this section, or read Sections 4.2, 4.3, and Section 4.13 before beginning work on it.

So far we have assumed that a firm selling a differentiated product sets its price taking its demand curve as given. But the elasticity of demand is determined by how easily consumers can find satisfactory substitute products, and therefore it depends on the decisions of the firm’s competitors. So, ‘taking the demand curve as given’ means that the firm assumes that its own pricing decision will have little or no effect on what its competitors do.

This is a plausible assumption when consumers have a choice between many substitutes, with each reaching a small proportion of the consumers in the market as a whole. For example, a city will have many high street fashion shops, with each selling different selections and styles of clothes. The demand curve for an individual shop is determined by the actions of many competitors, and—because it is a very small part of the market—whether it sets a high price or a low price will have little effect on what other shops choose to do.

But this is not always the case. Where there are only a few competing firms, a change in price by an individual firm will change the demand available to the others. If there are three fast food providers in a shopping mall, each knows that their own prices will affect the others. In this situation, pricing decisions are mutually interdependent. In choosing its own price, the firm will need to think strategically: not only about what its competitors will do, but also how its strategies will be affected by its own decisions.

Imagine a beach where tourists can hire a windsurfing board or a kitesurfing board for the day. The owner of the windsurfers (Wanda) and the owner of the kiteboards (Kit) both face the same costs, of *c* = €10 per board per day. For a simple model of price-setting firms interacting strategically, we will assume that they can choose either a high or low price: H = €36 or L = €20 per day.

Suppose that there are 60 potential customers each day. Half prefer windsurfing and half prefer kitesurfing, but some care more about the price, while others are driven more by their preferences between the activities:

**22 customers are loyal:**They choose their preferred activity whatever the prices (11 for each activity).**18 customers are sensitive to relative prices:**They will pay H for their preferred activity, but will switch if the price of the other is lower.**20 customers are mainly price-driven:**They will pay L but not H for either activity, choosing their preferred one if it is available.

The left panel of Figure 7.24 summarizes the demand for each firm, depending on the prices they choose. When both prices are high, the 20 price-driven consumers don’t buy, and the others choose their preferred board. If both are low, everyone buys their preferred board. But if one is high and the other low, only the 11 customers loyal to the high-priced activity will choose it; everyone else will choose on price.

How should Wanda set her price? Her decision will depend on the price set by her rival, Kit. Suppose she expects his price to be H. Using the information about demand, she can calculate her profit at each price, L and H, and then choose the price that gives higher profit.

The right panel shows how the profits each firm achieves depend on the prices they choose. For example, if Kit chooses H ($36) and Wanda chooses L ($20), Wanda will hire out 49 boards, with a profit of (L – *c*) = €10 per board. Her total profit will be €490, shown in the bottom left of the corresponding square.

- Nash equilibrium
- A Nash equilibrium is an economic outcome where none of the individuals involved could bring about an outcome they prefer by unilaterally changing their own action. More formally, in game theory it is defined as a set of strategies, one for each player in the game, such that each player’s strategy is a best response to the strategies chosen by everyone else.
*See also Game theory.*

What will the market outcome be? The profits in the right-hand panel are the pay-offs in the strategic price-setting game between Wanda and Kit. This is a coordination game like the one in Section 4.13. If Kit chooses H, Wanda obtains a higher profit by choosing H too, but if he chooses L, then she gets higher profit from L. Her best strategy is to match his price. Kit faces exactly the same situation. The game has two **Nash equilibria**: one in which both choose the high price, and one in which both price low.

Either of these is a possible outcome of the game. Both players would prefer the (H, H) equilibrium, since profits are considerably higher. On the other hand, choosing H is a more risky strategy: it gives them high profits if the other does the same, but the fall in profits if the other chooses L, after all, is greater.

In choosing their prices, Wanda and Kit face a trade-off. A high price is feasible because their products are differentiated: there are 22 loyal customers who care more about the particular activity they offer than the price. But a low price attracts other customers away from their rival. The trade-off depends on how many loyal customers there are.

Figure 7.25 shows how the pay-offs change when there are more loyal customers (26, in the left panel), or fewer (14, on the right). When both players choose the same price, the pay-offs are as before. But they are different when one player sets a higher price than the other.

With 26 loyal customers, Kit and Wanda both have a dominant strategy: to choose H. So there is only one Nash equilibrium, in which both firms price high. In this case, product differentiation gives the firms a lot of market power; demand is not very responsive to price differences and they can profit from setting high prices.

When there are only 14 loyal customers and customers are more sensitive to prices, competition between the two firms is more intense. Demand is more elastic: Wanda has only seven loyal customers, so if she increases her price from L to H her demand falls more: from 30 to seven if Kit’s price is L, and from 53 to 20 when his price is H.

This game is a prisoners’ dilemma: although Wanda and Kit would still make the highest profits if both chose H, L is a dominant strategy. The only Nash equilibrium is for both to set low prices (to the benefit of their customers).

This model illustrates how firms producing differentiated products behave strategically when the number of competing firms is small. In previous sections, firms have taken their demand curve as given. Here Wanda’s demand shifts, depending on the price set by Kit, and she knows that her price also affects his decision.

But the main conclusions are as before. Product differentiation gives them market power: when consumers care strongly about the characteristics of the product (the case of more loyal consumers) competition is less intense, the firms’ demand curves are less elastic, and they set high prices. When consumers are more price-sensitive (fewer loyal consumers; product differentiation is less important) demand is more elastic and prices are lower.

**Question 7.14** Choose the correct answer(s)

Based on the information given in Figure 7.25, read the following statements and choose the correct option(s).

- When demand is more elastic, L is a dominant strategy for both Wanda and Kit. Both make 300, which is less than if they had both chosen H.
- The unique Nash equilibrium is (H, H) when demand is less elastic, and (L, L) when demand is more elastic.
- The fact that Wanda and Kit’s profits depend on what the other person chooses is due to there being only two sellers in the market, which is conceptually different from a coordination game (which was shown in Figure 7.24).
- When consumers are less price-sensitive (sellers have more market power), the Nash equilibrium is for both sellers to charge high prices. When consumers are more price-sensitive (sellers have less market power), the Nash equilibrium is for both sellers to charge low prices.