Suppose your utility over money (x) is given by u(x)=x(1-1), where r-2/3. You are one of two bidders in a first price sealed bid auction. The other bidder places a bid randomly drawn from a uniform distribution between 0 and 10. 1. What is your optimal bid in this case? 2. Compare that number with what your optimal bid would be if r were equal to 0. What is the explanation for this difference?
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- There are three bidders participating in a first-price auction for a painting. Each bidder has a private, independent value vi for such a painting that is drawn uniformly from [0,1] Assume that each bidder i has a linear bidding function bi=avi, where a>0. What is the bidding strategy of bidder i , namely bi in the Bayesian equilibrium?which is the nash equilibrium? small pig does press large pig does press (20, 130), small pig doesn't press large pig does press (140, 10), small pig does press large pig doesn't press (-5, 150), small pig does not press large pig doesn't press (0,0).Assume you are one of the two bidders in a second-price sealed bid auction for a preserved grilled cheese sandwich which purportedly bore the portrait of the Virgin Mary: • This sandwich is worth $8000 to you. • You know that your rival bidder’s valuation is uniformly distributed between $0 and $12,000. a) What is the probability that your bid exceeds your rival’s bid given that you know your rival’s valuation is uniformly distributed between $0 and $12,000? (In other words, what is the probability that your rival’s bid is lower than your valuation?) Remember the equilibrium bidding strategies for a SPA in weakly dominant strategies ! b) What is your rival’s average bid, if you win the auction (considering that he must have bid below your bid)? c) What is your expected payoff from this auction, if you win the bidding ? (Hint: Combine b) and c), i.e. compute the payoff from winning giving the probability of winning)
- Jacob is considering buying hurricane insurance. Currently, without insurance, he has a wealth of $80,000. A hurricane ripping through his home will reduce his wealth by $60,000. The chance of this happening is 1%. An insurance company will offer to compensate Jacob for 80% of the damage that any tornado imposes, provided he pays a premium. Jacob’s utility function for wealth is given by U(w) = In (w). (A) What is the maximum amount Jacob is willing to pay for this insurance? Show work and explain.Consider the lottery that assigns a probability T of obtaining a level of consumption CH and a probability 1-T an individual facing such a lottery with utility function u(c) that has the properties that more is better (that is, a strictly positive marginal utility of consumption at all levels of c) and diminishing marginal utility of consumption, u"(c) CL. Consider du(c) for the first derivative of the utility function with respect to dc du(c) du' (c) consumption and u"(c) (which is also the derivative of the first derivative of the utility function). to be the second derivative of the utility function dc dc2 1. Provide a definition for the certainty equivalent level of consumption for the simple lottery described above.Use the expected value information to illustrate how having more bidders in an oral auction will likely result in a higher winning bid.
- In 'the dictator' game, one player (the dictator) chooses how to divide a pot of $10 between herself and another player (the recipient). The recipient does not have an opportunity to reject the proposed distribution. As such, if the dictator only cares about how much money she makes, she should keep all $10 for herself and give the recipient nothing. However, when economists conduct experiments with the dictator game, they find that dictators often offer strictly positive amounts to the recipients. Are dictators behaving irrationally in these experiments? Whether you think they are or not, your response should try to provide an explanation for the behavior.Suppose there are three bidders with values for the object that are independent, private and uniformly distributed over [0, 1]. a) Derive the seller's expected revenue in a second-price auction. Fully explain your answer. b) Derive the seller's expected revenue in a first-price auction. Fully explain your answer. c) Explain why the answers to parts (a) and (b) are the same. d) Show that the answers to parts (a) and (b) remain the same when there are N bidders. Fully explain your answer.Consider the following game, with a risk-neutral principal with preferences π = q - w hiring an agent 2, and the agent can = with preferences U = √w- - e.. The agent's reservation utility is given by U choose between an effort level of 0 or an effort level of 10. Output is either 0 or 400 and follows the following probability distribution, a function of effort level and some uncertain factor: Probability (q=0) Probability (q=400) 0.4 0.9 e=0 0.6 e-10 0.1 a) Illustrate this game of moral hazard using a fully labeled game tree with payouts.
- Consider the lottery that assigns a probability of obtaining a level of consumption Ch and a probability 1-7 of obtaining a low level of consumption c1 with cH > CL. Consider an individual facing such a lottery with utility function u(c) that has the properties that more is better (that is, a strictly positive marginal utility of consumption at all levels of c) and diminishing marginal utility of consumption, u"(c) < 0. As usual, we are using for the first derivative of the utility function with respect to du(c) dc du(c) dc2 the shorthand u'(c) du' (c) consumption and u" (c) (which is also the derivative of the first derivative of the utility function). to be the second derivative of the utility function dcWhen a famous painting becomes available for sale, it is often known which museum or collector will be the likely winner. Yet, the auctioneer actively woos representatives of other museums that have no chance of winning to attend anyway. Suppose a piece of art has recently become available for sale and will be auctioned off to the highest bidder, with the winner paying an amount equal to the second highest bid. Assume that most collectors know that Yakov places a value of $35,000 on the art piece and that he values this art piece more than any other collector. Suppose that if no one else shows up, Yakov simply bids $35,0002=$17,500 $35,000 2 = $17,500 and wins the piece of art. The expected price paid by Yakov, with no other bidders present, is. Suppose the owner of the artwork manages to recruit another bidder, Bob, to the auction. Bob is known to value the art piece at $28,000. The expected price paid by Yakov, given the presence of the second bidder Bob, is.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…