Econ 301Intermediate MicroeconomicsProf. Marek WeretkaFinal ExamYou have 2h to complete the exam. The …nal consists of 6 questions. T he last question (about theequilibrium ) is harder than the other ones.Problem 1. (Consumer Choice)Jeremy reads books, x1, while drinking co¤ee, x2. His utility function is g iven byU (x1; x2) = (x)48(x2)24a) Plot Jeremy’s indi¤erence curve map (graph), …nd his MRS analytically (give a formula). Depict hisMRS in the graph at the co nsumption bundle (3; 3) :b) Using the "magic formula," …nd the optimal l evel of consumption x1and x2if p1= p2= 2 andm = 30 and show it in the graph. Plot carefully the budget line and indi¤erence curve passing through theoptimal point (a grap h + two numbers).c) Suppose Jeremy’s prefer ences change and now they are given by U (x1; x2) = min (x1;x2) : In aseparate gra ph, plot his i ndi¤erence curves, and …nd his M RS for consumption bundles (2; 1) and (1; 2) (agraph +two numbers).d) Find Je remy’s optimal choice given p1= 2 = 2 and m = 30 and new preferences (two numbers).Problem 2. (Technology)Suppose a producer has acces s to the technology given by the Cobb -Douglass prod uction function y =K14L14:a) What can you say about the returns to scale (chose: IRS, CRS or DRS) and MPK (chose: increas ing ,constant, or decreasing).b) Find a (variable) cost function C (y) given the prices of inputs wK= 2; wL= 2 (give a function):c) Suppose that in order to have access to the technology, the producer …rst has to pay the …xed costF = 4; and hen ce the total cost is given by T C = 4 + C (y) : Find the supply function of the individual…rm and plot it in the graph (give t he formu la, in the graph mark the prices for whi ch the market will notopen).d) Assume that produc ers are competitive and there is free entry. Determine the number of …rmsoperating in the industry with demand D (p) = 10 p (a numbe r).e) What is the microstructure of the industry (choose: competitive, ol igopolistic, duopolistic or mon-polisti c).Problem 3. (Short Questions)a) You are leasing a car for which you are going to pay $8000 in each of the following three years. Findthe pr esent value of your payment if interest is equal to r = 100% (number).b) A Bernoulli utility function is u (x) =px. and two states of the world are equally likely. Whi ch ofthe two options will be preferred: lottery ($0; $16) or $8 for sure (choose one)? Explain why (one se ntence).Find the certainty equivalent of lottery (0; 16) (a number).c) Signalli ng: Suppose there are two types of managers: talented with productivity 1 and not talentedwi th productivity 0: The types are unobservable to employers and wages are competitive - they correspondto the expected productivity. Is an MBA program that takes e = 2 to complete a su¢ cient signal to separatethe two types if the cost of e¤ort of a not talented manager isc (e) = 0:25e(yes/no + one sentece explaining why).d) Find the minimal e that is su¢ cient for separation (one number).Problem 4. (Market Power)1Consider an industry with t he inverse demand equal to p (y) = 10  y; and suppos e that the tota l costfunction i s T C = 2y.a) Find the l evel of produc tion in the industry and the price if there is only one …rm (monopoly) charging auniform price (give two numbers). Illustrate the cho ice using a graph, depicting Consumer, Producer Surplusand DWL (give three numbers and m ark them on the graph).b) Is the outcome Pareto e¢ cient (yes/no + one se ntence explaining why or why not). If not, whichstrategy would you suggest to increase both pro…t and restore e¢ ciency?c) Find the elasticity of the demand at the equilibrium level of production (number). Is a monopolyoperating on elastic or inelastic part of the supply? (chose elastic or inelastic)d) Find the indi vid ual and aggregate production a nd the price in a Cournot-Nash equilibrium given thatthere are three …rms operating in the industry (give three numbers). Show DWL in the graph. ( Hint: usethe sy mmetry of the …rms. )Problem 5. (Provision of Public Good)There are two railroad transportation companies A and B whose pro…t depend s on how the systemof high-spreed railroad system is developed. tAdenotes the miles of railroad s built by A and tBare therailroads constructe d by …rm B: Suppose that the railroads are public good: once constructed none of the…rms can prevent the other …rm from using the whole railroad system, t = tA+ tB:The pro…t of …rm A is given byAtA; tB= 100tA+ tB12tA2;where 100  t is the revenue from selling tickets and12tA2is the cost of constructing the railroad track.Similarly, the pro…t of …rm B is given byBtA; tB= 100tA+ tB12tB2:a) Find the number of miles o f railroads in the country i f each …rms individually chooses the level ofconstruction to maximize pro…t (two numbers).b) Find Pareto e ¢ cient level of tAand tB: Are the two values higher or smaller then the ones in a) ?W hy? (two numbers + one sentence)c) Should the railroad track system in our example be determ ined by free mar ket or a goverment (onesentece)?Problem 6. (Di¢ cult: Equilibrium With Interteporal Choice)Consider an interter mporal choice problem in which Jey is a manager who earns 0 today, and 100;tomorrow, !J= (0; 100), while Kate is an athlete with an income of 100 and 0 tomo rrow. !K= (100; 0)The utility function of Jey and Kate is the sa me and given by?Uixi1;xi2=xi1+xi2where i = J; Ka) Plot the Edgeworth box and mark the point corresponding to endowments of Jey and Kate (graph).b) Give a general de…nition of Pareto e¢ ciency (one sentence), give the condi tion in terms of MRS (onesentence + a formula). Is the endowment Pareto e¢ cient?c) In the Edgeworth box, …nd all the alloca tions that are Pareto E¢ cient (contract curve).d) Find (one) competitive equilibriu m. Calculate borrowing and savings for both agents in your equilib-rium. (Hint: in the intertemporal cho ice 1 + r =p1p2).e) Is a c ompetitive e quilibrium (allocation) unique (yes or no answer)? If not characterize the set of al lallocations observed in competitive