Consider two ice cream sellers competing at a beach that is 1000 metres long. Ice cream prices are fixed by the ice cream company, but companies can choose their locations simultaneously. Customers are located
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- Consider two ice cream sellers competing at a beach that is 1000 metres long. Ice cream prices are fixed by the ice cream company, but companies can choose their locations simultaneously. Customers are located uniformly (spread out evenly on the beach) and do not like walking. The cost of walking every metre is the same (i.e. linear cost).
a) Suppose there are three ice cream sellers that locate simultaneously. Find the Nash equilibrium is there is one. Else, explain why there is none. (Focus on pure strategy Nash equilibria)
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- Jane and Sara are competing orange juice salespersons in Amherst. Their stands are next to each other on a street and consumers regard them as identical. The marginal cost of an orange juice is $1. The demand for orange juice every hour is Q = 20 − P where P is the lowest price between the two salespersons. If their prices are equal they split demand equally. a. If they set prices simultaneously (prices can be any real number), what is the Nash equilibrium price? b. If, against what we have assumed in class, orange juice salespersons have to charge prices in whole dollars ($1, $2, $3, etc), what are the Nash equilibrium prices? c. Assuming whole dollar pricing, if Jane sets her price before Sara, what price would she charge? Answer all 3 partsConsider the following static game with two firms as the players. Each firm must decide either to upgrade (U) an existing good to a new version; or not upgrade it (N). The decisions are simultaneous. If a firm chooses to upgrade, they have to pay a fixed cost of 7. If they don’t upgrade, there is no fixed cost. The marginal cost is always equal to 3. The demand side of the market is as follows: If neither firm upgrades, each firm sells 2 units at price 4. If both firms upgrade, each firm sells 3 units at price 5. If only one firm upgrades, the one who upgrades sells 5 units at price 5, and the other firm does not sell anything.Two countries produce oil. The per unit production cost of Country 1 is C1 = $2 and of country 2 it is C2 = $4. The total demand for oil is Q = 40-p where p is the market price of a unit of oil. Each country can only produce either 5 units, 10 units or 15 units. The total production of the two countries in a Nash equilibrium is 10 15 20 25 30
- In the following two-user game, the payoffs of users Alice and Bob are exactly negative of each other in all the combinations of strategies (a,a), (a,b), (b,a), (b,b). This models an extreme case of competition, and is called a zero-sum game. Please point out every pure-strategy Nash equilibrium of this game, and explain why it is. a b a (1, -1) (3, -3) (2, -2) (5, -5)Two firms operating in the same market must decide between charging a high price or a low price. The Payoffs are as below. Firm A's profit is listed before the comma, B's profit after the comma. Firm B Firm A Low Price High Price Low Price 16, 17 7, 28 High Price 28, 7 22, 22 If each firm tries to choose a price that is optimal, regardless of the other firm's price, what is the Nash equilibrium? Does either firm have a dominant strategy?Exercise 6.7. Suppose two identical companies produce wood stoves and they are the only ones on the market. Its costs are given by: C1 (q1 )=200q1 and C2 (q2) = 200q2. And the inverse market demand curve is: P=2000-2Q, where Q =q1 + q2 Get the Cournot-Nash equilibrium. Calculate the profits of each company. Show graphically. Suppose that the two companies form a cartel to maximize joint profits. How many stoves will you produce? Calculate the profits of each company. Represent graphically. Managers now note that explicit agreements to collude are illegal. Each company must decide on its own whether to produce the amount of Cournot or that of the cartel.
- Consider a market organized along a 1 mile stretch of road (from D=0 to D=1). Consumers along the stretch of road are uniformly distributed. There are two firms, with Firm 1 located at mile marker .5 and Firm 2 located at mile marker 1.0 and they compete on prices. Customers choose which store to shop at according to P + cD, where c=2 and D is the distance to the store. What is the Nash Equilibrium?Consider the following game: PLAYER 2 Left Middle Right Up 0,0 2,5 5.4 PLAYER 1 Middle 1,2 7,3 4,3 Down 5,5 3, 1 2, 1 Which of the following statements is true? There are two Nash equilibria, and they are both Pareto efficient. There is a unique Nash equilibrium. There are two Nash equilibria, and they are both Pareto inefficient. There are two Nash equilibria, but only one is Pareto efficient. None of these.Two competing firms must choose their quantity of production simultaneously. Each firm can choose either a High quantity of 3 or a Low quantity of 2. The price for both firms is 9-Q, where Q is the sum of both their quantities. Costs are zero; the profit is simply price times quantity. For example, if firm 1 chooses High and firm 2 chooses Low, then price is 9-(3+2)=4; payoff for firm 1 is 12 while payoff for firm 2 is 8. What is the unique Nash equilibrium? (Firm 1's strategy will be written before firm 2's.)
- Economics Consider two firms that are choosing the price of competing products. The choices are contained in the payoff table. Each firm can raise price, lower price, or maintain their price. Suppose the game is played once each period forever. If both players play the strategy "always lower price" is this a Nash equilibrium? Let b = discount rate, 0 < b <1. Raise price Maintain price Lower price Firm B Raise price Maintain price Lower price 6. 4 8. 8 1.1 5, 5 4, 6 7.2 1.1 3.3 Firm A 2.7 No. because it is dominated by raising the price. No, because it is dominated by both firms raising the price. No, because it is dominated by maintaining the price. Yes.Consider two firms with a homogeneous product who face the market demand function p = 2 – q1 – 92, where q; and p are the quantities and price. Their constant marginal costs are given by c= 1. The firms compete in quantities in a simultaneous move game. Use this specific example (not a general case) to show that the Nash equilibrium is not Pareto efficient, and the cooperative solution is not an equilibrium (in the sense that both firms have an incentive to cheat). In your answer, use the fact that the firms are identical. Namely, they produce equal amounts (both in the simultaneous move game and in the cooperative case).The inverse market demand for fax paper is given by P=100-Q. There are two firms who produce fax paper. Firm 1 has a cost of production of C1= 15*Q1 and firm 2 has a cost of production of C2=20*Q2 a) Suppose that firm play a Stackelberg game. First firm 1 sets the quantity in t=1, then, knowing which quantityfirm 1 has set, firm 2 chooses the quantity in t=2. What are the Stackelberg quantities and prices? What arethe profits od firm 1 and 2? Compared to part a) which firm benefits and which firm loses?
