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)Problem 1: The Marginal Monopolist's Market Moves (29 Points) The graph above shows the marginal cost (MC), demand (D), and marginal revenue (MR) curves for a market with one firm and infinitely many consumers. P MC D MR Part A (2) Explain the shape of the marginal cost curve. Part B (2) Explain why the marginal revenue curve declines faster than the demand curve. Part C (2) At what quantity and price does this market operate? Is this efficient? Why or why not? Part D (3) On your answer paper, copy over the above graph (It does not need to be perfect! Just the same general shapes are fine). Add an average total cost curve such that the firm is productively efficient. Then, define productive efficiency. Part E (4) The firm produces widgets with a negative externality. Define negative externality of production and give an example. Explain one action the government could take to reduce or eliminate the externality. Part F (2) On your graph, add a marginal social cost curve that takes into account the negative externality. Part G (3) If the firm was forced to internalize the externality, but was still allowed to operate as a monopoly, where would it produce? Part H (2) What is the socially optimal level of production? 2 Part I (6) There are many ways to force markets to operate at efficient levels, but they are not equally applicable to every situation. For each of the policies below, explain in 2-4 sentences why it would or would not help the market achieve the optimal level of output: - An educational campaign on the dangers of producing widgets. - A tax for every widget produced. - A ban on the production of widgets. Part J (3) Of the policies suggested above, which is the most effective? Why?Part A (Exponential Discounting) (3) One of the things that we humans often prefer is to consume now rather than later. Ideally economists discount the future value of an object exponentially. For example, if I know I am going to receive a $1000 in 5 years, and my annual discount factor is 0.95, that means that the present value of $1000 in five years is (0.95)" * 1000, which is the minimum amount of money that I would like to receive right now to give up the $1000 in 5 years. We call d' the discount factor and d' *(future value) the present value of the future asset. Now, suppose my annual discount factor is 0.9. In the present, do I prefer receiving $1000 in 4 years or receiving $1100 in 5 years? Part B (Hyperbolic Discounting) (3) In reality, people might use a different discounting system, which skews upward payoffs in the more recent future. The formula is (discount factor) = (future value)/(1+t*k), where k is a parameter that represents my preference. Suppose k = 0.1, explain which of the following options is better today (in 2021): $1050 in 2026 or $1101 in 2031. Part C (Regretting) (2) In 2026, I offer you the same option as the above: $1050 right now or $1101 in 2031. If you use hyperbolic discounting, explain which option you prefer now. (Hint: it might be easier to put 2026's value in 2031 terms instead of doing division) Part D (Time Consistency) (4) Consider only two options: 1 in to years and vy in to years, where up and we are positive and fo and to are positive integers. Suppose without loss of generality that to