Question

5. Graph the reaction functions and find the Nash equilibria for the following games:

• Bertrand competition with two companies and constant and equal marginal costs.

• Consider that two companies are investigating to develop a new technology that will revolutionize the market. In this economy there are no patents, so if one company develops the technology, the other can use it at no cost. The cost of developing the new technology is c and each firm has a reserve price for the investigation of ri, such that ri<c for all i . Therefore, the companies know that they have to do joint research if they want to develop the technology, but they do not know how to divide the costs. The representatives of each company will meet to announce their offer b; to the other company. If the sum of the offers is greater than c, each person will pay their offer and the investigation will be carried out. The surplus will be donated to charity. If the sum of the bids is less than c, no one will pay anything and the investigation will not be done either.

Answer 1

Bertrand Model-

Firms can compete on several variables, and levels, for example, they can compete based on their choices of prices, quantity, and quality. The most basic and fundamental competition pertains to pricing choices. The Bertrand Model is examines the interdependence between rivals’ decisions in terms of pricing decisions. The assumptions of the model are: 1. 2 firms in the market, i ∈ {1, 2}. 2. Goods produced are homogenous, ⇒ products are perfect substitutes. 3. Firms set prices simultaneously. 4. Each firm has the same constant marginal cost of c. What is the equilibrium, or best strategy of each firm? The answer is that both firms will set the same prices, p1 = p2 = p, and that it will be equal to the marginal.

This is a very powerful model in that it says that price competition is so intense that all you need is two firms to achieve the perfect competitive outcome. We will show this through logical arguments and contradictions, as well as through the use of a diagram. Using logical arguments: 1. Firm’s will never price above the monopoly’s price: Suppose not. And suppose firm 1 believes that firm 2 would choose a price p2 above the monopoly’s price, then the best response of firm 1 is to price at the monopoly’s price since at that point, its profit is maximized. And firm 2 would be driven out of the market. Therefore no firm would ever price above the monopoly’s price. 2. In equilibrium, all firm’s prices are the same: Suppose firm 2 chooses to price at the monopoly’s price, what is the best response of firm 1? Firm 1 would realize that by pricing at a slightly lower price, it would be able to capture the entire market since the goods are perfectly substitutable, that is p1 = pM + , where pM is the monopoly’s price , and > 0. Then only one firm is left. Therefore the equilibrium where firms charges a different prices cannot be an equilibrium, p1 = p2 = p. 3. In equilibrium, prices must be at the marginal cost: Suppose not, than p1 = p2 = p > c. However, either firm would always find it is in their best interest or their best response to under cut its competition and obtain the entire market for itself, by reducing its prices a little bit more, say > 0. By induction, it is in fact not possible then to have an equilibrium above the marginal cost, since it is only at the marginal cost that firms have no incentives to deviate from the equilibrium prices. ∴ in equilibrium, p1 = p2 = p = c. Notice that in making the arguments we have always stated the firm’s choice as a function of the other firm’s choice, p ∗ i (pj ), where i 6= j, and i, j ∈ {1, 2}. This is known as a reaction function. Depicting our argument on a diagram with prices on both the axes. It is obvious that equilibrium is achieved only at the point where the reaction functions meet, since it is only at the intersection that each firms best response corresponds with the other’s. Any other point cannot be an equilibrium since the actions that one believes the other would do would never be realized. Only at c does their expectations match, and the equilibrium is sound since both firms are the same, symmetric.

P1 45° p (1) PM PP (P2) 1 с Bertrand-Nash Equilibrium 1 1 с PM P2