F13-final exam

Advanced Microeconomics I-Final Examination(2hours,Closed book)Version M:2013/12/9,9-11amI.Short questions.Limit your answer to two lines.(a)Definition of Arrow-Pratt measure of absolute risk averse R a(w,u)?give an example ofCARA(constant absolute risk averse)?[5points](b)Let a preference relation on G satisfy axioms G1to G6defined in the lecture.then itmust also satisfy the following axiom:Axiom G4’:Let g,g and h be three lotteries with g g and let t∈(0,1],then(t◦g,(1−t)◦h) (t◦g ,(1−t)◦h).[5points](c)let f(x)be a production function,and c(w,y)be the associated cost function.Supposeand c(w,10)=4,what is c(w,5)?[5points]f is homogeneous of degree12(d)What is the definition of the following statement:”A preference relation on theconsumption set X⊂R n+is convex.”[5points](e)What is the optimal price for a monopolist facing a market with demand function q(p)=p−2,assuming the cost function c(q)=12q.[5points](f)Consider a two-consumer,two-good exchange economy.Utility function and endowmentsareU1(x1,x2)=(x1x2)2013and e1=(18,4),U2(x1,x2)=x501x100and e2=(3,6),2Find a Walrasian equilibrium and compute the WEA.[10points]II.Longer questions:Show how you got your answers in detail.1.Consider a matching model with4men and4women.Their preferences are:P(m1)=w1,w2,w3,w4P(w1)=m2,m3,m1,m4P(m2)=w1,w2,w3,w4P(w2)=m2,m3,m1,m4P(m3)=w1,w2,w3,w4P(w3)=m2,m3,m1,m4P(m4)=w1,w2,w3,w4P(w4)=m2,m3,m1,m4Using Gale&Shapley’s Deferred Acceptance Algorithm tofind the stable matching with women proposing.[10points]12.Suppose that there are n goods,and a consumer’s preference relation is represented by someutility function u(·).Suppose that the indirect utility function isv(p,y)=ln y−ni=1a i ln(p i),where a1,...,a n>0are parameters with ni=1a i=1.Derive the Marshallian demandfunctions x i(p,y),expenditure function e(p,u),and Hicksian demand functions x h i(p,u).[15points]3.Samsung produces liquid crystal display(LCD)panels,while LG produce plasma panels fortelevisions sets.As a result,the two manufacturers produce imperfect substitutes.Suppose Samsung can produce a LCD panel for a marginal cost of$50per unit and LG can producea plasma panel for a marginal cost of$80per unit.Samsung’s demand function is q S=50000−100p S+100p L and LG’s demand function is q L=50000−100p L+100p S.(a)If Samsung and LG have to set prices simultaneously,what is the Bertrand-Nash equi-librium in this market?In other words,what price does eachfirm charge and what quantity does eachfirm produce?[10points](b)Suppose instead that Samsung chooses its pricefirst,followed by LG.What are theequilibrium prices and quantities for eachfirm in the sequential Stackelberg game?[5 points]4.Suppose an individual initially has w0dollars of wealth.There are twofinancial assets thathe can allocate his initial wealth to invest over a year.Thefirst one is a risk-free asset with rate of return7%,i.e.for each dollar of investment it yields1.07dollars with certainty aftera year.The second one is a risky asset:if the economic environment of the year is good thenits rate of return is21%;if the economic environment of the year is bad then its rate of returnis−19%.Suppose that the probability of having a good year is23;the probability of having abad year is13.Suppose that the individual has a VNM utility function given by u(w)=w r,where w∈R+denotes the individual’s wealth after a year,where r∈(0,1).(a)Is this individual risk averse,risk neutral,or risk loving?what’s the relative risk aversemeasure for this individual?[5points](b)suppose that the individual can invest his initial wealth in both of the twofinancialassets.Denote the amount of investment in the risky asset as x(and the amount of investment in the risk-free asset as w0−x).Write down the maximization program and the corresponding F.O.C.(you don’t need to solve for the optimal x.)[5points](c)Let x∗(w o)denote the optimal amount,i.e the solution to the F.O.C you found in part(b).Prove that the ratio x∗(w0)w0is independent of w0,i.e.,∂∂w0x∗(w0)w0=0.Briefly explainwhy.[5points]5.Show the market clearance theorem:In an exchange economy E=i,e ii∈I,if p 0andall markets but one are cleared,then all markets are cleared.(note:if you use Walras’Law in your proof,you should prove itfirst.Assume the preferences are strictly monotonic.)[10points] Total:100points2。