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1. Consumer 1 has expenditure function ,(p. p2, wi) = wjypipz and consumer 2 has utility function us($1, ry) = (a) What are Marshallian (market) demand functions for each of the goods by each of the consumers? Denote the income of consumer 1 by my and the income of consumer 2 by my. (b) For what value(s) of the parameter a will there exists an aggregate demand functions that is independent of the distribution of income? 2. Suppose that utility maximization problems and expenditure minimization problems are well defined and utility and expenditure functions satisfy all necessary "nice" properties. (a) Show (prove) that utility maximization implies expenditure minimization and vice versa. (b) List all relevant identities that are result of a. (c) Derive Roy's identity. (d) Derive Slutsky equation. 3. An economy has two kinds of consumers and two goods. Type A and type B consumers have the following utility functions UN(11, 12) = 41 - -+ 1. U"(1,1) = 2n - -+1 Consumers can only consume nounegative quantities. The price of good 2 is 1 and all consumers have incomes of 100. There are A type A consumers and NV type B consumers. (a) Suppose that a monopolist can produce good 1 at a constant unit cost of c per unit and cannot engage in any kind of price discrimination. Find its optimal choice of price and quantity. For what values of c will it be true that it chooses to sell to both types of consumers? (b) Suppose that the monopolist uses a "two-part tariff" where a consumer must pay a lump sum & in order to be able to buy as much as he likes at a price p per unit purchased. Consumers are not able to resell good 1. For p *(p) 2 #-(p). (b) If for all p, a(p) = "(p) = * (p), what you can say about YO, Y, and Y/? Provide formal arguments. (c) For (p'. y' ) = ([1, 1]. [-3, 4]), and (p3, y') = ([2, 1], [-1, 2]) construct YI and YO (graphically). What can you say about returns to scale in the technology these observations are coming from? Hint: think y = (-c,y). 2. Given the production function f(21, 12, 13) = 17 min (x2, 23)", (a) Calculate profit maximizing supply and demand functions, and the profit function. What re- striction you have to impose on a? (b) Fix y. Calculate conditional demands and the cost function c(w1, w2. (). (c) Solve the problem py - c(un, w2, y) - maxy, do you obtain the same solution as in 20? Explain your findings. 3. Given the production function f(21, 12) = 21 + 29, where b > 0, calculate the cost function c(1, 1, y). How would costs respond to the changes in wi, wa, and y? How would factor demands respond? 4. Consider a firm with conditional factor demand functions of the form (output has been set equal to 1 for convenience): 12 = 1+ dufw5. What are the values of the parameters a, b, c, and d and why? 5. The cost function is c(w1, w2, y) = wjugg. (a) What do we know about a and b? (b) What are the conditional factor demands? What is the production function? (c) What can you tell about returns to scale? 6. Let e(w1, u2. 7) = 2(g) be the isocost and y = f(21, 12) = 7 be the isoquant corresponding to a fixed output level 7 = 1. (a) What are the slopes of these lines? (b) Draw the isocost and isoquant for Cobb-Douglas technology ? = 27 2; corresponding to J = 1. (c) Suppose c(w1, w2, 1) = aw + buy. Draw the isocost and the corresponding isoquant (use the slopes to obtain the shape of the isoquant). (d) Repeat for c(w1, w2, 1) = minfawn, buy}. (e) Draw conditional demand z, as a function of " for 7c and 7d. Hint: If you have trouble in 7c - 7e, think what technology these costs came from.1. Consider the simultaneous move game below: Left Center Right Top -1,3 3,-1 5,0 Middle 3,-1 -1,3 5.0 Bottom 0.5 0,5 1000,4 (a) Show that Player 2s action of "right" is dominated by a mixed strategy of Left and Center. (b) Show that there is no nash equilibrium in dominant strategies. (c) Determine the NE in mixed strategies as formally as possible. (Hint: remember that in order to mix the expected value of both choices must be equal. Start with the row player - determine the y that will make choosing Top and Middle equal as a function of the mixing probability of the column player. Do the same thing for the column player. The point where both of these equations are satisfied will be the mixed strategy nash equilibrium). 2. Will and Davy are playing the following game of "Liar's poker" With the following form: . A deck of cards consisting of 4 kings and 4 aces is shuffled and put in front of Will. Will looks at the top card and makes a (potentially untruthful) announceent of "King" or "Ace". - If he announces "King" the game ends with no exchange of money. - If he announces "Ace", Davy gets to take an action: . Davy has the option of "Folding" or "Calling"- - If he folds, Davy pays Will 8.50. - If he Calls and Will is holding the Ace, Davy pays Will $1.00. - If he Calls and Will is holding the King, Will pays Davy $1.00. (a) Draw the extensive form and normal form of this game. (b) Show that there is no pure strategy Nash Equilibram. (c) Given that Will has a King, how often should he bluff? (d) Given that Will announces a King, how often should Davy call him? 3. Consider an industry with 3 firms, each having marginal costs equal to 0. The inverse demand curve facing this industry is: P(91. 92.93) = 60 - (q1 + 92 + 91) (a) If each firm behaves as a Cournot competitor (quantity competition), what is firm 1's best response function? (b) Calculate the Cournot equilibrium. (c) Firms 2 and 3 decide to merge and form a single firm (MC still 0). Calculate the new industry equilibrium. Is firm 1 better or worse off as a result? Are the combined profits from firm 2 and 3 greater or less than before? Would it be a profitable idea for all three firms to organize into a cartel? (d) Suppose firm I can commit to a certain level of output in advance. If the choice of firm 1 is qu,what would be the optimal choices of firms 2 and 3? 4. Consider the following game. An incumbent makes a decision to advertise at a cost K or not. This action is observed by a challenger who has the option of entering the market at cost F or staying out. If the challenger stays out of the market, the incumbent firm is a monopolist. If the challenger enters the market, the two parties compete a la Cournot. (a) Draw the extensive form of this game. (5 points) (b) Advertisement increases demand at any given price. The inverse demand curves when incumbent advertises and when not are (correspondingly): Pad(91: 42) = 60 - Q PNeAd(91+ 92) = 48 - Q Assume that all parties have zero marginal costs. 1. Calculate the profits of each firm when the challenger enters depending on whether the in- cumbent advertises or not. (9 points) 2. Suppose that F = 350. Should the incumbent advertise? (3 points) 3. Suppose that F = 100. For what levels of K should the incumbent advertise? (3 points) 5. *Consider the following game. There are N players, each with an initial endowment of k. Each agent has the opportunity to invest a' ( [0, ] in the spraying of mosquitos, effects of which are enjoyed evenly by all individuals. The remaining amount y' = k-r is privately consumed. If X=ritz' t...+2" is invested in bug spraying, the total benefits from the public good that is generated is G = aX, with each individual receiving g' = 4. Assume that all agents have a linear utility function U(g', y' ) = g' + y'. (a) Solve for NE of this game as a function of a. (b) What is socially optimal level of investment as a function of a. (c) Suppose that the local, peoples' government can observe individual contributions and distribute the outcome of investment as it wishes, not necessarily equally. Can you think of a scheme that would obtain the socially optimal level of investment? (d) "What if the government can observe the contributions, but does not know how much original resources each individual has? Can you think of a scheme that would obtain the socially optimal level of investment?3. Consider the following infinitely repeated game. . There is an Incumbent and a (potential) Entrant. . Each date t has two stages: - At the first stage, the Entrant decides whether to Enter or Stay Out. - If the Entrant decides to Enter, it bears an entry cost of k = 0.1. The two players then play a standard "Cournot" game: * Each of the two firms simultaneously decide on a non-negative output qe and qr- * The demand curve is given by p = 1-Q, where Q = qe + qr, so the price of output is given by p = 1 -Q. (Notice that we allow negative prices here; but you should be able to ignore that for the sake of this problem.) * The marginal cost of production is zero, so the profits are given by: qIP = qEP - k. - If the Entrant Stays Out, it gets a payoff of 0, and the Incumbent decides on a production level or to produce, facing the same demand curve. . The game is infinitely repeated, with a discount factor 6 e (0, 1). . Notice that the entrant bears the cost & in each round in which she enters. (a) (10 points) Find the subgame perfect equilibrium of the non-repeated version of this game (the game that takes place at any t). (b) (10 points) Prove that the following strategy profile is a subgame perfect equilib rium for sufficiently high o, and find the minimal o for which it is: . The Entrant Enters every round. . Every producer produces q = 1, so long as no producer has ever produced a quantity other than 3- . If any producer has ever produced a quantity other than }, "trigger" to play- ing the subgame perfect equilibrium from part (a) in every period. (c) (10 points) For o = .9, is there a subgame perfect equilibrium in which entry is always deterred (i.e., in which the Entrant never Enters)? If so, find one. If not, explain why not. (d) (5 points) Describe strategies that can implement the same outcome as part (b) for some 6