Micro Qualification
aws代写 | project代写 – 这是一个aws面向对象设计的practice, 考察aws的理解, 涵盖了aws等方面, 这个项目是project代写的代写题目
Micro Qualification
- This has 3 questions/4 pages.
- If you must make assumptions beyond what is given in a problem to reach a solution, state them clearly.
- Good luck.
- Determine whether each of the following statements is true or false. If true, provide a proof. If false, provide a counter example.No credit is given unless a proof or a counterexample is provided. (a) True/false: Suppose a decision maker has a strictly concave and twice differentiable Bernoulli utility functionu(x)and wealthW > 0 .Assume that a risk free asset exists that can be purchased at price (normalized to) 1 and pays 1 for sure. Also assume that a risky asset exists that can be purchased at price (normalized to) 1 that paysRH> 1 with probabilitypandRL< 1 with probability 1 p, wherepRH+ (1p)RL> 1. Both assets can be purchased in perfectly divisible quantities. The decision maker puts all wealth in the safe asset. (b) True/false: In a signaling game, every sequential equilibrium passes the test of equi- librium domination. (c) True/False. In a signaling game in which the sender has two types and two actions, every weak perfect Bayesian equilibrium is a sequential equilibrium. (d) True/False. Every weak perfect Bayesian equilibrium is subgame perfect. (e) True/False. Suppose that there is no pure strategysithat strictly dominatessi,then there is no mixed strategyithat strictly dominatessi. (f) True/False. For any normal form game, ifsis a Nash equilibrium, thenssurvives iterated elimination of strictly dominated strategies. (g) True/False. Consider an auction setup where agentihas utilityitiif being allocated the (indivisible) good andtiotherwise, wheretiis the transfer from the agent to the auctioneer (who is not modeled as a player). Let(p,)be a direct mechanism withp : ni=1i n+1being the allocation rule that determines the probability distribution over who gets the object as a function of announced type profiles and : ni=1iRnbeing the transfers. Let(i) =
iipi()fi(i)dibe the
perceived probability for agentito get the object as a function of own type only. Then,
must be weakly increasing iniif(p,)is incentive compatible.
(h) True/False. Consider an infinite repetition of
s w
S 1 , 1 7 , 1
W 1 , 7 2 , 2
Assume that =^45 .Nash reversion be used to sustain an equilibrium in which both
players earn an average payoff that is strictly higher than 2.
- Consider a potential seller (playerS) holding an asset with returns being distributed in ac- cordance with probability densityfGif the asset is good andfBif the asset is bad. There is also a potential buyer (playerB). Both the buyer and the seller arerisk neutral, so they only care about the expected return, x
xfK(x)dx
given state of the worldK {G,B}. Trading proceeds as follows. First the buyer and the
seller dr aws (independent) noisy signalsJ{g,b}forJ=S,Bwhere the joint probability
distribution is
G B
g qJ 1 qJ
b 1 qJ qJ
,
whereqJ>^12 forJ=S,B.The buyer and the seller privately observe their signals. Then,
the buyer and seller simultaneously announce ask pricea(announced by seller) and bid price
b(announced by buyer). Ifabtrade occurs, but the buyer needs to pay trading cost > 0
to a broker (not a player in the game) if there is trade.
(a) What is the expected return conditional onJ=gand the expected return conditional
onJ=b.How do they compare?
(b) Is it possible to construct an equilibrium in which the asset is sold whenS =band
B=g.If yes, construct such an equilibrium. If no, prove your claim.
(c) Suppose that the buyer gets utilityx+tfrom the asset witht > .What is the efficient
allocation of the asset as a function of the state?
(d) In the model with buyer utilityx+tis there always an equilibrium that implements the
efficient allocation? If yes, construct the equilibrium. If no, construct an example with
an inefficiency.
- Consider a principal-agent model with limited liability. An entrepreneur has a project which needs $1 investment. If successful, the project yields$Rand if unsuccessful, it yields zero. The probability of project success depends on entrepreneurs action. Specifically, if the entrepreneur exerts effort, the probability of project success isp(0,1), and it costs him g(0,1). If he shirks, the project fails for sure, and the cost is zero. Unlike in the model you saw in the class, the entrepreneur has no cash, but he has some collateral-eligible asset (e.g. real estates),W, that could be used for securing loan. Liquidat- ing any part ofWto self-finance the project is extremely costly. So the entrepreneur has to enter credit market to finance the project. The credit market is competitive with risk neutral lenders, each of whom can either deposit their wealth on a risk-free asset at a gross rate of returnr(1,pR)or lend some wealth to the entrepreneur. Assume thatpRg > r. Due to the moral hazard problem, the entrepreneur proposes a collateral-based lending con- tract to a lender. Specifically, the contract specifies the payment[0,R]if the project success, and a fraction of his collateral assetk[0,W]. In case the project fails, the project does not generate any revenue, but the lender receives the collateral assetk. We assume that collateral is costly in that the lenders valuation of the collateral is a fraction(0,1)of the entrepreneur s valuation. That is, if the entrepreneur posts collateralkand the project fails (so he has to default), the lenders net payoff isk. The difference,(1)k, is the repossession cost of collateral because transferring control of these assets typically involves legal and other administrative costs.
(a) Derive the participation constraint for lenders that the lending contract(,k)must
satisfy.
(b) Consider the first best case where the entrepreneurs effort is contractible. Derive the
optimal contract. Specifically, whats the value ofandk?
(c) Consider the second best case where the entrepreneurs effort is not contractible. Derive
the incentive-compatibility constraint under which entrepreneur finds it optimal to exert
effort. Is the first best contract incentive compatible?
(d) Derive the second best optimal contract. Compare the value ofkin the first best con-
tract and the second best contract to discuss the social benefit and cost of collateral
assets.