1) Consider the Akerlof's model of lemons. Assume the sellers have all the bargaining power: they can charge the buyers their expected value of buying the car. Suppose there are 3 possible car qualities: high, medium and low. Suppose also that the proportion of each type of car is 1/3. The values of buyers and sellers for different car qualities are tabulated below. Seller's value Buyer's value Low quality car Medium quality car High quality car 8 10 13 15 18 (a) Is there an equilibrium in which all types of qualities are traded? (b) Is there an equilibrium in which only low and medium qualities are traded? (c) Is there an equilibrium in which only low qualities are traded?
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- Suppose the equilibrium price for good quality used cars is $20,000. And the equilibrium price for poor quality used cars is $10,000. Assume a potential used car buyer has imperfect information as to the condition of any given used car. Assume this potential buyer believes the probability a given used car is good quality is .60 and the probability a given used car is low quality is .40. Assume the seller has perfect information on all cars in inventory. How does the informational imbalance result in adverse selection? a. The expectedprice offered by the buyer encourages the seller to sell a poor quality car. Hence only poor quality cars are sold, which harms sellers. b. The expected price offered by the buyer encourages the seller to sell a good quality car. Hence only good quality cars are sold, which harms buyers. c. The expected price offered by the buyer encourages the seller to sell a good quality car. Hence only good quality cars are sold, which harms sellers. d. The…For the adverse selection model by George Akerlof, for a given car, if the seller values the car at v, the buyer is willing to pay 1.6v. It is assumed that the value of a car for a seller lies uniformly between 0 to 2000 dollars. Demonstrate that a price of 1000 is not feasible.Consider two bidders – Alice and Bob who are bidding for a second-hand car. Each of them knows the private value she/he assigns to the car, but does not know the exact value of others. It is common knowledge that the value of other bidders is randomly drawn from a uniform distribution between 0 and $10000. Assume that Alice values the car at $8500 and Bob values the car at $4500. a) If Alice and Bob participated in the second-price sealed bid auction, what would they bid and what would be the result of the auction? Explain your answer. b) If they participate instead in a first-price sealed bid auction, what would they bid and what would be the result of the auction? Explain your answer. c) Calculate and compare the revenue of the seller in the above situations. Which type of auction should the seller use? Explain your answer
- Consider a first-price, sealed-bid auction, and suppose there are only three feasible bids: A bidder can bid 1, 2, or 3. The payoff to a losing bidder is zero. The payoff to a winning bidder equals his valuation minus the price paid (which, by the rules of the auction, is his bid). What is private information to a bidder is how much the item is worth to him; hence, a bidder’s type is his valuation. Assume that there are only two valuations, which we’ll denote L and H, where H > 3 > L > 2. Assume also that each bidder has probability .75 of having a high valuation, H. The Bayesian game is then structured as follows: First, Nature chooses the two bidders’ valuations. Second, each bidder learns his valuation, but does not learn the valuation of the other bidder. Third, the two bidders simultaneously submit bids. A strategy for a bidder is a pair of actions: what to bid when he has a high valuation and what to bid when he has a low valuation. a. Derive the conditions on H and L…Suppose two bidders compete for a single indivisible item (e.g., a used car, a piece of art, etc.). We assume that bidder 1 values the item at $v1, and bidder 2 values the item at $v2. We assume that v1 > v2. In this problem we study a second price auction, which proceeds as follows. Each player i = 1, 2 simultaneously chooses a bid bi ≥ 0. The higher of the two bidders wins, and pays the second highest bid (in this case, the other player’s bid). In case of a tie, suppose the item goes to bidder 1. If a bidder does not win, their payoff is zero; if the bidder wins, their payoff is their value minus the second highest bid. a) Now suppose that player 1 bids b1 = v2 and player 2 bids b2 = v1, i.e., they both bid the value of the other player. (Note that in this case, player 2 is bidding above their value!) Show that this is a pure NE of the second price auction. (Note that in this pure NE the player with the lower value wins, while in the weak dominant strategy equilibrium where both…Why can a pooling equilibrium be sustained in Spence’s signaling model but not in the Rothschild-Stiglitz screening model? Explain why in both models, the “good” type (high productivity in Spence, or low risk in Rothschild-Stiglitz) lose out in a separating equilibrium.
- Two bidders compete in a second price auction (i.e., the winning bidder pays the losing bidder’s bid, and the losing bidder does not pay anything). They submit sealed bids, and the one with the highest bid wins the contract and pays the other bidder’s bid. Each bidder i’s private valuation is vi and is distributed independently and uniformly between 0 and 50. 1. For any given bidder, prove that he has a dominant strategy bid and show what it is. 2. Assuming each bidder bids his dominant strategy noted above, if a bidder with vi = 40 wins, what price does he expect to pay?Consider three individuals with the following preferences: Suppose the following dynamic voting protocol is in effect: first, there is a vote between a and b; then the winner goes against c; and the winner of this contest will be implemented. Focus on SPE where voters do not use “weakly dominated” strategies at any stage. (a) Show that these preferences are single peaked, but sincere voting is not equilibrium behavior. (b) Characterize the SPE of this game under strategic voting by all players. (c) Consider a generalization in which the society H consists of H individuals and there are finite number of policies, R = {p1, p2,...,pM}. For simplicity, suppose that H is an odd number. Voting takes M − 1 stages. In the first stage, there is a vote between p1 and p2. In the second stage, there is a vote between the winner of the first stage and p3, until we have a final vote against pM. The winner of the final vote is the policy choice of the society. Prove that if…Suppose that optimal deterrence of a particular crime requires setting the expected punishment equal to $2000. Assume that the probability of apprehension is fixed at .2 Suppose an offender has a wealth of $5000 and incurs a cost of 1000 per month spent in prison. What combination of a fine and prison term achieves optimal deterrence at the lowest cost? What combination of a fine and prison term is optimal for an offender with a wealth of $7000 (assuming the same monthly cost of imprisonment)? Suppose considerations of fairness dictate that the prison term of offenders who commit the same crime must be the same. What fine and prison term must be imposed on the two offenders to maintain optimal deterrence?
- You are a bidder in an independent private values auction, and you value the object at $4,000. Each bidder perceives that valuations are uniformly distributed between $1,500 and $9,000. Determine your optimal bidding strategy in a first-price, sealed-bid auction when the total number of bidders (including you) is: a. 2. b. 10. c. 100Look again at Figure 21.12. It appears that the investor in panel b can't lose and the investor in panel a can' t win. Is that correct? Note: The solution should not be hand written.Consider a medieval Italian merchant who is a risk averse expected utility maximiser. Their wealth will beequal to y if their ship returns safely from Asia loaded with the finest silk. If the ship sinks, their incomewill be y − L. The chance of a safe return is 50%. Now suppose that there are two identical merchants, A and B, who are both risk averse expected utilitymaximisers with utility of income given by u(y) = ln y. The income of each merchant will be 8 if theirown ship returns and 2 if it sinks. As previously, the probability of a safe return is 50% for each ship.However, with probability p ≤ 1/2 both ships will return safely. With the same probability p both willsink. Finally, with the remaining probability, only one ship will return safely.(iv) Compute the increase in the utility of each merchant that they could achieve from pooling theirincomes (as a function of p). How does the benefit of pooling depend on the probability p? Explainintuitively why this is the case.