Econ 301Intermediate MicroeconomicsProf. Marek WeretkaFinal (Group A)Problem 1.Monica consumes bana nas x1and kiwis x2: The prices of bot h goods ar e p1= p2= 1 and Mo nica’sincome is m = 100. Her utility function isU (x1; x2) = (x1)37(x2)37a) Find analyti cally Mo nica’s MRS as a function of (x1; x2) (give a function) and …nd its value for theconsumption bundle (x1; x2) = (80; 20) :Give its economic and geometric interpretation (one sentence and…nd MRS on the graph)b) Give two Monica’s secrets of happiness that determine her o ptimal choice of fru its (give two eq uation).Explain why violation of any of them implies that the bundle is not optimal (one senten ce for each condition).c) Show geometrically the optimum bundle of Monica –do not calculate it.Problem 2.Georgina loves two types of ‡owers: Cu ban lilies x1and calla lilies x2: Her utility from having a bouquet(x1; x2) isU (x1; x2) = 2x1+2x2a) Propose a utility functi on that gives a h igher level of utility for any (x1; x2), but represents the samepreferences (give utility function).b) Suppose the price s of bo th types of lilies are p1= 2 and p2= 1 and t he Georgina ’s total incomem = $10. Plot her budget set. Find the optimal bouq uet (x1; x2) and mark it in your graph (give twonumber s)c) Are the ‡owers Gi¤en goods (yes or no and one sentence explaining why)?d) Sup pose in the ‡ower shop currently there are only six calla lilies x2in stock (hence x2 6). Plota budget set with the extra con straint and …nd (geometrically) an optimal l evel of consump tion given theconstraint.Problem 3. (Equilibrium)Tomor row i t may rain or shine and the chances are 50%  50%. Today, there are two co mmoditiestraded on the market: umbrellas x1and swimming suits x2. Jeremy has ten umbrellas and no swimm ingsuits (!J= (10; 0) ) :Bill has twenty swimming suits and no umbrellas (!J= (0; 20)). Jeremy and Bill haveidentical utility functions given byUi(x1; x2) =12ln (x1) +12ln (x2)a) Plot an Edgewor th box and mark the point corresponding to endowments of Jeremy and Bill.b) Show in a gra ph a set of all Pareto e¢ cient allocations (do not calculate it).c) Find prices and an alloca tion of umbrellas and swimming suits in a competitive equilibrium and markit in your graph.d) Is the outcome of market interactions Pareto e¢ cient (yes or no, give an argument involving twonumber s)?Problem 4.(Short questions)a) You are going to pay taxes of $300 every year, forever. Find the Present Va lue of your taxes if theyearly interest rate is r = 1%.b) Consider a lottery that pays 0 with probability14and 4 with probability34and a Bernoulli utilityfunction is u (x) = x2. What is a von Neuman-Morgenstern utility function? Find the certainty equivalentof the lottery. Is it bigger or smaller than the expected value of the lottery? Why? (give a utility function,two numbers and one sentence.)1c) Give an exa mple of a Co bb-Douglass production fun ction that is associated with increasing returns toscale, increasing MPK and decreasing MPL (give a function). Without any calculations, sketch the averagetotal cos t function (AT C) associated with your produc tion function.d) Suppose the cost function is such that AT CM ES= 2 and yM ES= 1 and the demand is D (p) = 7  p:Determine a number of …rms in the indust ry given the f ree entry (and price taki ng). Is the industrymonopolistic, duopoli stic, oligopolistic or perfectl y competitive.f) In a mar ket for second-hand vehicles two types of cars can be traded: lemon s (bad quality cars) andplums (good quality ones). T he value of a car depends on its type and is given byLemon PlumSeller 0 5Buyer 2 6W ill we observe plums traded on the market if the probability of a lemon is equal to12? Is the outcomePareto e¢ cient?Problem 5.(Market Power)Consider an industry with the inverse demand equal to p (y) = 100  y; and suppose that the total costfunction is T C = 0.a) What are the total gains to trade in this industry? ( give one number)b) Find the level of production a nd the price if there is only one …rm in the industry (i.e., we have amonopoly) charging a u niform pric e (give two numbers). Illustrate the choice using a graph. Mark a DWL.c) Find the pro…t of the mon opoly and a DWL given that monopoly uses the …rst degree price discrimi-nation.d) Find the individual and aggregate prod uction th e p rice in a Cournot-Nash equilibrium given thatthere are two …rms (give three numbers). Show DWL in the graph.e) In which of the three cases the outcome is Pareto e¢ cient? (monopoly with un iform price, …rst degreeprice discrimination or a duopoly)Problem 6.(Externality)A bee keepe r chooses th e number of hives h. Each hive produces ten pounds of honey which sells at theprice of $1 per pound. The cost of holding h hives is T C (x) =12x2: Consequ ently the pro…t of bee keeperis equal toh(h) = 10h12h2The hives are located next to an apple tree orchard. The bees pollinate the trees and hence each tree in theorchard produces h apples (the more bees the higher the producti on). Apples sell for $1 and the cost of ttrees is T Ct(t) =12t2: Therefore the pro…t of an orchard grower ist(t) = t  h 12t2a) Find t he level of hives h that maximizes the pro…t of a be ekeeper and the number of trees thatmaximiz es the pro…t of an orchard owner. (two numbers)b) Find the Pareto e¢ cient level of h and t: Are the two values high er or smaller then the one s in a) ?W hy? (two numbers + on e sentenc