Final.pdfSolA.pdfSolB.pdfF.pdfFinal A.pdfFinal_A_sol.pdfFinal B.pdfFinal B sol.pdfEcon 301Intermediate MicroeconomicsProf. Marek WeretkaFinal Exam (A)You have 2h to complete the exam and the …nal consists of 6 questions (15+10+25+15+20+15=100).Problem 1. (Consumer Choice)Jeremy’s favorite ‡owers are tulips x1and da¤odils x2. Suppose p1= 2, p2= 4 and m = 40.a) Write down Jeremy’s budget constraint (a formula) and plot all Jeremy’s a¤ordable bundles in thegraph (his budget set). Find the slope of a budget line (number). Give an economic interpretation for theslope of th e budget line (one sentence).b) Jeremy’s utility function is given byU (x1; x2) =q(ln x1+ ln x2)2+ 7Propose a simpler utility funct ion that repr esents the same preferences (give a formula). Explain whyyour utility represents t he same preferences (one sentence).c)1Plot Jeremy’s indi¤erence curve map (graph), …nd MRS analytic ally (give a formula) and …nd itsvalue at bundle (2; 4) (one number). Give economic interpretation of this number (one sentence). Mark itsvalue in the graph.d) Write down two s ecrets of happiness (two equalities) that allow determining the opti ma l bundle. Pro-vide their geometric interpretation (one sentence for each). Find the optimal bundle (x1; x2) (two numbers).Is your solution interior? (a yes -no answer)e) Hard: F ind the optimal bundle given p1= 2, p2= 4 and m = 40 assuming U (x1; x2) = 2x1+ 3x2(two numbers). Is your solution interior? (a yes -no answer)Problem 2. (Producers)Consider producti on f unction given by F (K; L) = 3K14L14.a) Using the  argument demonstrate that production function exhibits decreasing returns to scale.b) Derive th e cost function given wK= wL= 9.c) Derive a sup ply function of a competitive …rm, assu ming the cost function from b) and …xed costF = 2 (give a formula for y(p)). Plot the supply function in a graph, marking the thresho ld price bel owwhich a …rm chooses inaction.Problem 3. (Competitive Equilibrium)Consider an economy with apples and oranges. Dustin’s endow ment of two commodities is given by!D= (8; 2) and Kate’s endowment is !K= (2; 8). T he utility functions of Dustin and Kate are the sameand given byUixi1; xi2= 5 ln xi1+ 5 ln xi2where i = D ; K.a) Plot the Edgeworth box and mark the point corresponding to the initial endowments.b) Give a general de…nition of Pareto e¢ cient allocation x (one sentence) and state its equ ivalent con-dition in terms of M RS (one sentence, you do not need to prove the equivalence).c) Using t he "MRS" condition verify that the initial endowments are not Pareto e¢ cient.d) Find a c ompetitive equilibrium (six numbers). Provide an e xampl e of a competitive equilibriu m wit hsome other prices (six numbers).e) Using MR S condition verify that the competitive allocation is Pareto e¢ cient.f) Har d: Find prices p1; p2in a competitive equilibri um for identical preferences of two agents U (x1; x2) =2x1+3x2(two numbers, no calculations). Explain why any two prices that give rise to a relative price higherthan p1=p2cannot be equilibrium prices (which condi tion of equilibrium fails?)1If you do not know the ans wer to b), to get partial credit in points c)-e) your can assume U (x1; x2) =x1x2.1Problem 4. (Short Questions)a) Uncertainty: Find the certainty equivalent of a lottery which, in two equally likely states, pays (0; 9).Bernoulli utility function is u(c) =pc (one number). Is the certainty equival ent smaller or bigger tha n theexpected value of a lo ttery 4:5. Why? (one sentence)b) Market for lemons: In a market for racing horses one can …nd two types of animals: champions(Plums) and ordinary recreational horses (Lemons). Buyers can distinguish between the two types only longafter they b uy a horse. The valu es of the two types of horse s for buyers a nd sellers are summarized in thetableLemon PlumSeller 1 4Buyer 2 5Are champions (Plums) going to be traded if probability of Lemons is12. (yes-no). Why? (a one sentenceargument that involves the expected value of a horse to a buyer)c) Signaling: The productivity of high ability workers (and hence the competitive wage rate) is 1000 whileproducti vity of low ability workers is only 400. To determine the type, e mployer can, …rst o¤e r a n internshipprogram with t he length of x months, during which a worker has to demonstrate her hig h productivity. Alow ability worker by putting extra e¤ort can mimic high ability performance, which costs him c(x) = 200x.Find the minimal length x for which the intern ship becomes a credib le signal of high ability. (one number)d) PV of Perpetuity: You can rent a n apartment paying 1000 per month (starting next month, till th e"end of the world") or you can bu y the apartment for 100:000. Which option are you going to chose ifmonthly interest rate is r = 2%? (…nd the PV of rent and compare two numbers)Problem 5. (Market Power)Consider a monopoly facing the invers e demand p (y) = 25  y, and with total cost T C(y) = 5y.a) Find the margin al revenue of a monopoly, M R (y) and depict it in a graph t ogether w ith the demand(formula +graph). Which is big ger: price or marginal revenue? Why? (one sentence)b) Find the optimal level of production and price (two number s). Illustrate the optimal choice in agraph, depictin g Consumer and Producer Surplus, and DWL (thr ee numbers +graph).c) Find equilibrium markup (one number).d) Fi rst Degree Price Discriminati on: Find Total Surplus, Consumer, Producer Surplus and DWL ifmonopoly can pe rfectly discriminate among buyers and quantities. (four numbers +graph)e) Hard : …nd th e individual level o f production and price in a C ournot-Nash equi librium with N identical…rms with cost T C(y) = 5y, both as a functi on of N (two formulas). Arg ue that the equi librium priceconverges to the marginal cost as N goes to in…nity.Problem 6. (Public good: Music downloads)Freddy and Mir iam share the same collection of songs downloaded from i-tunes (they have one PC ).Each song costs 1. If Freddy downloads xFand Miria m xM, their collection c ontains xF+ xMand utilityof Freddy (net of the cost) is given byuFxF= 200 ln(xF+ xM)  xF;while Miriam’s utility (net of the cost) isuMxM= 100 ln(xF+ xM)  xM;(Observe that Freddy is more into music than Miriam.)a) Find optimal number of downloads by Freddy xF(his best re sponse) for any choice of Miriam xM(formula xF= RF(xM)). Plot the best response in the coordin ate system (xF; xM).(Hint: You