BA. How is the Pigou tax computed (write down carefully the way it is computed in a simple two-firm competitive environment)? What is its aim? 6B. The preferences of agents A and B are representable by expected utility functious such that uA(z) - 5z/3 + 30, and un(z) - kr following allocation of the expected returns of a risky joint investment of A and B as represented by lottery L= ((:1500), (:120)) is Pareto-efficient: (a) ZA = (500,100), zn (b) ZA - (100,100), rp (c) ZA (d) ZA - (750,60), n (e) NOPAC. 20. Then, the (1000,20) (1300,20) (80,80), n - (1420,40) (750,00)
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- 4) Consider investors with preferences represented by the utility function U = E(r) – Ao². (a) Draw the indifference curve representing a utility level of 10% for an in- vestor with a risk aversion parameter A = 3 in expected return-standard deviation space. (b) In the same graph, draw the indifference curve representing a utility level of 15% for an investor with a risk aversion parameter A = 3. (c) In the same graph, draw the indifference curve representing a utility level of 10% for an investor with a risk aversion parameter A = 5.Buying and selling prices for risky investments obviously are related to certain equivalents. This problem, however, shows that the prices depend on exactly what is owned in the first place. Suppose that your utility for wealth (A) can be represented by the utility function u(A) = In [(A)] You currently have R1000 in cash. A business deal of interest to you yields a reward of R100 with probability 0,5 and RO with probability 0,5. 2.1 If you own this business deal in addition to the R1000, what is the smallest amount for which you would sell the deal? 2.2 Suppose you do not own the deal. Formulate an appropriate equation and solve with algebra to find the largest amount you would be willing to pay for the deal. 2.3 Explain why the amounts in 2.1 and 2.2 are slightly different.You are a hotel manager and you are considering four projects that yield different payoffs, depending upon whether there is an economic boom or a recession. The potential payoffs and corresponding payoffs are summarized in the accompanying table. Recession (50%) -$ 10 $ 20 -$ 30 $ 50 Boom (50%) $ 20 Project A B -$ 10 $ 30 $ 50 If a manager adopted both projects A and B simultaneously, the varlance in returns assoclated with this joint project would be Multiple Choice 0. 10. 30. 50.
- A recently discovered painting by Picasso is on auction at Sotheby's. There are two main bidders Amy and Ben {1,2}. Bidding starts at £10M but the value of the painting is certainly not more than £20M. Each bidder's valuation v; is independently and uni- formly distributed on the interval [10M, 20M], and this is common knowledge among the players: A bidder knows their own valuation but not of their opponent. Consider an auction where an object is allocated to the highest bidder but the price paid by the bidder is determined randomly. With probability 3/4, the bidder pays their own bid, and with probability 1/4 the bidder pays the losing bid. The person bidding lowest pays nothing. If the bids are equal, each bidder gets the object with probability one-half, and in this case, pays their bid. Suppose that bidder 1 assumes that bidder 2 will bid a constant fraction, 7, of bidder 2's valuation (and similarly, bidder 2 assumes bidder 1 will bid the same constant propor- tional value of…You are considering a $500,000 investment in the fast-food industry and have narrowed your choice to either a McDonald's or a Penn Station East Coast Subs franchise. McDonald's indicates that, based on the location where you are proposing to open a new restaurant, there is a 25 percent probability that aggregate 10-year profits (net of the initial investment) will be $16 million, a 50 percent probability that profits will be $8 million, and a 25 percent probability that profits will be -$1.6 million. The aggregate 10-year profit projections (net of the initial investment) for a Penn Station East Coast Subs franchise is $48 million with a 2.5 percent probability, $8 million with a 95 percent probability, and -$48 million with a 2.5 percent probability. Considering both the risk and expected profitability of these two investment opportunities, which is the better investment? Explain carefully.4. The preferences of agents A and B are representable by expected utility functions such that uA(x) = 5x^1/3 +30, and uB(x)= 1/5x - 20. Then, the following allocation of the expected returns of a risky joint investment of A and B as represented by lottery L = ((2/3);1500), (1/3);120)) is Pareto efficient: (a) xA = (500,100), xB = (1000,20) (b) xA = (100,100), xB= (1300,20) (c) xA= (80,80), xB = (1420,40) (d) xA = (750,60), xB= (750,60) (e) NOPAC
- Suppose there is a joint project that generates an amount 10 for each of the individuals A and B, provided the sum total of A's and B's investments in the project is at least 7. Suppose A's investment is denoted by x and B's by y, where r and y can take any values in {0,1,2,3,4..8,9,10}. A player always has to incur the cost of the investment (equal to æ or y respectively) and gets the benefit if and only if the project goes ahead. Suppose each player is interested in his own net benefit (benefit-cost) and players choose their investment levels simultaneously. The following relate to pure strategies. (a) Is there a Nash equilibrium in which both players invest 0? (b) Is there a Nash equilibrium in which one player invests 0 and the other does not? (c) What is the lowest payoff a player can get in any Nash equilibrium?4) Increasing the fit of investment service to the long term objectives of each investor—moving from caveat emptor “product” sales to more durable, shared-understanding service relationships— would increase the duration or “loyalty” and thereby the economic value of client–manager relationships. (The Winners’ Game, by Charles D. Ellis, 2011) Select an alternative True FalseQuestion 1) Dave is an expected utility maximizer and his von Neumann-Morgenstern utility function is given by u(w) = = (w). He has an initial wealth of $27,000. 199 √99+7 (a) Determine the risk attitude of this economic agent. b) If John is facing a risk of losing $19,000 with probability 0.1, write down his prospective final wealth and the associated probabilities in the form of a lottery. (c) Facing the risk of losing $19,000 with probability 0.1, what would be the maximum that Dave is willing to pay for 19,000 units of a contingent claim that pays $1 if and only if the loss occurs.
- (Derive the coefficients of absolute and relative risk aversion of the following functions, and point out th conditions to make each function increase and concave: (а) и (х) —х- (b/2)x2 (b) µ (x) = (B/(B –- 1))x!-1/B (c) µ (x) = (1/ (B – 1)) [A +Bx]'¬1/BConsider the game below, which is infinitely repeated at t = 1,2, . Both players discount the future at rate d E (0, 1). The stage game is C D C 5,5 0,6 D 6,0 1,1 1. The following three questions are about "Grim Trigger" strategies. (a) Describe the "Grim Trigger" strategy profile of this game. (b) Draw the finite automata representation of this strategy profile. (c) Find the lowest value of d for this strategy profile to form a subgame perfect equilibrium. 2. Suppose that the players play (C,C) in period t = 1,3, 5, ... and plays (D,D) in period t = 2,4, 6, ... Compute the discounted payoff of each player. 3. Find a subgame perfect equilibrium such that, on the equilibrium path, (D,D) is played at periods t = 1,3, 5 .., and (C,C) is played at periods t = 2, 4, 6, ... Write down the finite automata representation for the strategy profile you proposed. Find the lowest value of ő for the profile to form a subgame perfect equilibrium.2. Consider a trader with initial fund given by To holding q shares of stock i is C(q) = 10 + q². The price (x;) at which this trader sells its position is stochastically distributed according to the following probability distribution: 15, and the transaction cost function of || S0.5, if a; = $8 |0.5, if x; = $2 P(x:) = Let a random variable îñ be the profit of trading at each time t, t = 1, 2, . ..,T, (b) Consider now that the trader's utility function is described by u(ñ) = µ(7). What is now the optimal level of position and the associated equilibrium profits?