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Problem 5. Bertrand Competition in discrete increments (64 points) (Note: This problem resembles one on last year's exam, but it's not the same, read carefully) Consider a variant of the Bertrand model of competition with two firms that we covered in class. The difference from the model in class is that prices are not a continuous variable, but rather a discrete variable. Prices vary in multiples of 1 cent. Firms can charge prices of 0, .01, .02, .03.... etc. The profits of firm i are (pi - c) D (p:) if pi Pj- Demand is extremely inelastic, that is, D (p) = @ for all p. Both firms have the same marginal cost o = c2 = c, and the marginal cost c is a multiple of 1 cent. (The firm can charge c-.01, c, c+ .01, c+ .02, etc.) Consider first the case in which the two firms move simultaneously (as we did in class), and apply the Nash Equilibrium concept. 1. Write down the definition of Nash Equilibrium as it applies to this game, that is, with p; as the strategy of player i and ;(pi, p; ) as the function that player 1 maximizes. Provide both the formal definition and the intuition. Do not substitute in the expression for a;- (8 points) 2. Show that pi = p; = c (that is, marginal cost pricing) is a first Nash Equilibrium of this game. (8 points) 3. Show that pi = p; = c + .01 is a second Nash Equilibrium of this game. (8 points) 4. (Harder) Can you find another Nash Equilibrium (you need to prove that it is a Nash Equilibrium) [Hint: The peculiar feature of this setup is that the firm can only charge prices that are multiples of 1 cent] Why does it matter that demand is inelastic? (10 points) 5. Now, we change the setup in just one way. The game is now played sequentially, that is, firm 1 moves first, and firm 2 follows after observing the price choice of firm 1. We apply Subgame Perfection to solve this game, and therefore start from the last period, from the choice of player 2. Player 2's strategy will be a function of Player I's price p1- Find the best response for player 2 as a function of pi, that is, find p; (pi) . (10 points) 6. Now let's continue with the backward induction and go back to player 1. Player 1 anticipates the best response of player 2 and chooses the price p, that will yiled the highest profit. What is this price pi? To simplify the solution, assume that player 2 responds to a price of c + .02 by also setting price c + .02 (8 points) 7. Write down the subgame perfect equilibrium. How does it differ from the set of Nash Equilibria of the symultaneous game? (8 points) 8. Can you conjecture how the solution of the dynamic Bertrand game will differ if firms can set price continuously? (4 points)2. The supply side of a perfectly competitive market is represented by n identical producers each one endowed with a technology that uses two inputs, a, and T2, and produces one output y described by the following cost function: c(w1, W2. y) = (Bwiwz) y where y denotes the output, w, and w2 the input prices and we assume B > 0. (i) [6 marks] What is the value of $ compatible with the properties of cost functions? Explain your answer. (ii) [4 marks] Identify the supply correspondence of each producer in this economy. (iii) [4 marks] Identify the aggregate supply correspondence of commodity y in this market. The demand side of this market is represented by n identical consumers. They consume only the output commodity y and their preferences are represented by the strictly monotonic utility function u(y). Each consumer has a trust fund that pays her an exogenous income equal to 2p where p denotes the price of commodity y. (iv) [4 marks] Identify the Marshallian demand of each individual consumer. (v) [3 marks] Identify the consumers' aggregate demand in this market. (vi) [6 marks] What is the equilibrium price and quantity in this market? Explain your answer. (vii) [3 marks] Can you identify the quantity supplied by each individual producer in this market? Explain your answer. (viii) [3 marks] What is the profit each individual producer obtains at the market equilibrium price? Explain your answer.3. Consider a two-consumer (Ms. A and Mr. B), two-commodity (z, and x2) pure exchange economy. Consumer A's preferences are represented by the utility func- tion: UA(ri, 17 ) = ari+12 Consumer B's preferences are represented by the utility function: UB(x, 14) = 15 + 315 where ; denotes the quantity of commodity j