Econ 301 Intermediate Microeconomics Prof. Marek Weretka Final Exam (Group A) You have 2h to complete the exam. The exam consists of 6 questions (20,15, 15, 15, 15 and 20 points) Problem 1. (Consumer Choice) Lionel watches movies x while drinking beer, x . His utility function from consuming the two types of goods is 1 2 given by U(x ,x )=2ln(x )+ln(x ) 1 2 1 2 a) Plot Lionel’s indifference curve map (graph). Find his MRS analytically (give formula). Find the value of MRS at the consumption bundle (2,4) and depict it in the graph. b) Using “magic formulas,” find the optimal level of consumption of x and x if p =p =4 and m=15. Plot 1 2 1 2 carefully the optimal point along with the budget line and the indifference curve passing through the optimal point (a graph + two numbers). c)Arguethatthecommoditiesareneithergrosscomplementsnorgrosssubstitutes(argueusingthemagicformula, one sentence). For the rest of the problem suppose that Lionel’s preferences change and now they are given by U(x ,x ) = 1 2 3x +3x . 1 2 d) In a separate graph plot his indifference curves, and find MRS for consumption bundle (2,1). (a graph +one number). e)FindLionel’soptimalchoicegivenpricesp =p =4andm=15(twonumbers). Istheoptimalchoiceunique? 1 2 (yes/no answer) Problem 2. (Equilibrium with Intertemporal Choice) Consider an intertemporal choice problem with Ambrosia and Fergus. Ambrosia is an athlete with income of 4 when young and 1 when old, i.e, ωA =(4,1). Fergus is a manager who earns 1 when young and 4 when old, i.e., his endowment is ωF =(1,4). Ambrosia and Fergus have the same utility function given by 1 1 Ui(xi,xi)= ln(xi)+ ln(xi) 1 2 2 1 4 2 where i=A,F. a) Plot the Edgeworth box and mark the point corresponding to endowments of Ambrosia and Fergus (graph). b) Give a general economic definition of Pareto efficiency (one sentence). Write the equivalent condition for ParetoefficiencyintermsofMRS(equation). UseyourequationtoverifywhethertheendowmentisParetoefficient. (argue formally using your condition) c)FindalltheallocationsthatareParetoefficient(deriveformulaforthecontractcurve). Plotthecontractcurve in the Edgeworth box. d)Findthecompetitiveequilibrium. Calculateborrowingandsavingsforbothagentsinyourequilibrium. (Hint: use that for the intertemporal choice 1+r= p1 ). p2 1 Problem 3. (Technology) Suppose a producer has access to the technology given by the Cobb-Douglass production function y=2K41L14. a)Whatcanyousayaboutthereturnstoscale(chose: IRS,CRSorDRS)andMPK(chose: increasing,constant, or decreasing). b) Find the (variable) cost function c(y) given that the prices of inputs are w =w =2 (give a function). K L For the rest of the problem suppose that in order to have access to the technology, the producer first needs to pay fixed cost F =9 and hence the total cost is given by TC =9+c(y). c) Find the supply function of the individual competitive firm and plot it in the graph (give the formula, in the graph mark the prices for which the market will not open). d) Assume that the producers are competitive and there is free entry. Determine the number of firms operating in the industry in the long run if the demand is D(p)=24−p (one number). Problem 4. (Short Questions) a) A Bernoulli utility function is u(x)=x2 and there are two states of the world which are equally likely. Find the certainty equivalent and the expected value of lottery (1,7) (two numbers). Which of the two is bigger (choose one)? Explain why (one sentence). b) Consider an industry in which the market share of the dominant firm is 30%, while the market shares of the tenotherfirmsis10%each. FindHHIindexforthisindustry. Istheindustrycompetitive,moderatelyconcentrated, or concentrated? c) Suppose there are two types of managers: talented with productivity 9 and not talented with productivity 3. The types are unobservable to employers and the competitive wage is given by the expected productivity of the manager. Is an MBA diploma from a program that takes one year to complete e=1 a credible signal if the cost of effort for the not talented agent is c(e)=3e (yes/no + one sentence explaining why). Find minimal e for which the MBA diploma is a credible signal. d) Externality: Give two methods through which a government can achieve market efficiency in the presence of a negative externality. (two sentences for each method). Problem 5. (Market Power) Consider an industry with an inverse demand p(y)=6−y and the total cost TC =0. a)Findthelevelofproductionandthepricechosenbyamonopolywhoisnotallowedtopricediscriminate(give two numbers). Illustrate the choice using a graph. Find the consumer surplus (CS), the producer surplus (PS) and the deadweight loss (DWL) (give three numbers and mark them on the graph). b) Which pricing strategy of a monopoly gives rise to the Pareto efficient outcome (one sentence)? Find the consumer surplus (CS), the producer surplus (PS) and the deadweight loss (DWL) under your proposed strategy (give three numbers and mark them on the graph). c) Find the individual level of production, the price and the profit of each firm in the Cournot-Nash equilibrium if there are two identical firms in the industry with cost fucntions TC =0 (give three numbers). d) Find the joint profit of the two firms from part c) if they form a cartel. Explain the mechanism that prevents the formation of a (short-run) cartel in a Cournot-Nash equilibrium. 2 Problem 6. (Provision of a Public Good) There are two countries, the USA and a country that represents “the rest of the world” (denoted by R). The nationalproductsofbothcountriesareincreasingintheworld’sspendingonresearch,x=xUS+xR. Thusresearch is a public good. The “profit” of the USA, net scientific expenses is given by πUS =12ln(xUS+xR)−xUS. The net profit of country R is less sensitive to the scientific advancements, and is given by, πR =3ln(xUS+xR)−xR. a)FindanalyticallythebestresponseoftheUStoanylevelofspendingxR (deriveafunction)andplotitinthe coordinate system xUS,xR. (Make sure you show optimal choice xUS for xR >12). b) Find analytically the best response function of country R and add it to the graph in part a). c) Find the Nash equilibrium. What is the world’s spending on science, x? Is the predicted outcome associated with free riding? If so by which country? d) Find the Pareto efficient level of spending on reserch? Is it greater, smaller or equal to the one observed in markets (part c)? Explain intuitively why is it so? 3 Econ 301 Intermediate Microeconomics Prof. Marek Weretka Final Exam (Group B) You have 2h to complete the exam. The exam consists of 6 questions (20,15, 15, 15, 15 and 20 points) Problem 1. (Consumer Choice) Lionel watches movies x while drinking beer, x . His utility function from consuming the two types of goods is 1 2 given by U(x ,x )=5ln(x )+10ln(x ) 1 2 1 2 a) Plot Lionel’s indifference curve map (graph). Find his MRS analytically (give formula). Find the value of MRS at the consumption bundle (1,1) and depict it in the graph. b) Using “magic formulas,” find the optimal level of consumption of x and x if p =p =5 and m=45. Plot 1 2 1 2 carefully the optimal point along with the budget line and the indifference curve passing through the optimal point (a graph + two numbers). c)Arguethatthecommoditiesareneithergrosscomplementsnorgrosssubstitutes(argueusingthemagicformula, one sentence). For the rest of the problem suppose that Lionel’s preferences change and now they are given by U(x ,x ) = 1 2 1x + 1x . 3 1 3 2 d) In a separate graph plot his indifference curves, and find MRS for consumption bundle (2,1). (a graph +one number). e)FindLionel’soptimalchoicegivenpricesp =p =5andm=45(twonumbers). Istheoptimalchoiceunique? 1 2 (yes/no answer) Problem 2. (Equilibrium with Intertemporal Choice) ConsideranintertemporalchoiceproblemwithAmbrosiaandFergus. Ambrosiaisanathletewithincomeof800 when young and 200 when old, i.e, ωA =(800,200). Fergus is a manager who earns 200 when young and 800 when old, i.e., his endowment is ωF =(200,800). Ambrosia and Fergus have the same utility function given by 1 Ui(xi,xi)= ln(xi)+ln(xi) 1 2 2 1 2 where i=A,F. a) Plot the Edgeworth box and mark the point corresponding to endowments of Ambrosia and Fergus (graph). b) Give a general economic definition of Pareto efficiency (one sentence). Write the equivalent condition for ParetoefficiencyintermsofMRS(equation). UseyourequationtoverifywhethertheendowmentisParetoefficient. (argue formally using your condition) c)FindalltheallocationsthatareParetoefficient(deriveformulaforthecontractcurve). Plotthecontractcurve in the Edgeworth box. d)Findthecompetitiveequilibrium. Calculateborrowingandsavingsforbothagentsinyourequilibrium. (Hint: use that for the intertemporal choice 1+r= p1 ). p2 4 Problem 3. (Technology) Suppose a producer has access to the technology given by the Cobb-Douglass production function y=3K41L14. a)Whatcanyousayaboutthereturnstoscale(chose: IRS,CRSorDRS)andMPK(chose: increasing,constant, or decreasing). b) Find the (variable) cost function c(y) given that the prices of inputs are w =w =4.5 (give a function). K L For the rest of the problem suppose that in order to have access to the technology, the producer first needs to pay fixed cost F =16 and hence the total cost is given by TC =16+c(y). c) Find the supply function of the individual competitive firm and plot it in the graph (give the formula, in the graph mark the prices for which the market will not open). d) Assume that the producers are competitive and there is free entry. Determine the number of firms operating in the industry in the long run if the demand is D(p)=20−p (one number). Problem 4. (Short Questions) a) A Bernoulli utility function is u(x)=x2 and there are two states of the world which are equally likely. Find the certainty equivalent and the expected value of lottery (0,2) (two numbers). Which of the two is bigger (choose one)? Explain why (one sentence). b) Consider an industry in which the market share of the dominant firm is 30%, while the market shares of the fiveotherfirmsis10%each. FindHHIindexforthisindustry. Istheindustrycompetitive,moderatelyconcentrated, or concentrated? c) Suppose there are two types of managers: talented with productivity 5 and not talented with productivity 3. The types are unobservable to employers and the competitive wage is given by the expected productivity of the manager. Is an MBA diploma from a program that takes three years to complete e=3 a credible signal if the cost of effort for the not talented agent is c(e)=e (yes/no + one sentence explaining why). Find minimal e for which the MBA diploma is a credible signal. d) Externality: Give two methods through which a government can achieve market efficiency in the presence of a negative externality. (two sentences for each method). Problem 5. (Market Power) Consider an industry with an inverse demand p(y)=8−y and the total cost TC =0. a)Findthelevelofproductionandthepricechosenbyamonopolywhoisnotallowedtopricediscriminate(give two numbers). Illustrate the choice using a graph. Find the consumer surplus (CS), the producer surplus (PS) and the deadweight loss (DWL) (give three numbers and mark them on the graph). b) Which pricing strategy of a monopoly gives rise to the Pareto efficient outcome (one sentence)? Find the consumer surplus (CS), the producer surplus (PS) and the deadweight loss (DWL) under your proposed strategy (give three numbers and mark them on the graph). c) Find the individual level of production, the price and the profit of each firm in the Cournot-Nash equilibrium if there are two identical firms in the industry with cost functions TC =0 (give three numbers). d) Find the joint profit of the two firms from part c) if they form a cartel. Explain the mechanism that prevents the formation of a (short-run) cartel in a Cournot-Nash equilibrium. 5 Problem 6. (Provision of a Public Good) There are two countries, the USA and a country that represents “the rest of the world” (denoted by R). The nationalproductsofbothcountriesareincreasingintheworld’sspendingonresearch,x=xUS+xR. Thusresearch is a public good. The “profit” of the USA, net scientific expenses is given by πUS =7ln(xUS+xR)−xUS. The net profit of country R is less sensitive to the scientific advancements, and is given by, πR =ln(xUS+xR)−xR. a)FindanalyticallythebestresponseoftheUStoanylevelofspendingxR (deriveafunction)andplotitinthe coordinate system xUS,xR. (Make sure you show optimal choice xUS for xR >7). b) Find analytically the best response function of country R and add it to the graph in part a). c) Find the Nash equilibrium. What is the world’s spending on science, x? Is the predicted outcome associated with free riding? If so by which country? d) Find the Pareto efficient level of spending on research? Is it greater, smaller or equal to the one observed in markets (part c)? Explain intuitively why is it so? 6 Econ301 IntermediateMicroeconomics Prof. MarekWeretka FinalExam(GroupA) Youhave2htocompletetheexam. Theexamconsistsof6questions(20,15,15,15,15and20points) Problem1. (ConsumerChoice) a)PlotLionel’sindifferencecurvemap(graph). FindhisMRSanalytically(giveformula). Find thevalueofMRSattheconsumptionbundle(2,4)anddepictitinthegraph. (4pt) MU 2/x 2x MRS=− 1 =− 1 =− 2 MU 1/x x 2 2 1 2∗4 MRS(2,4)=− =−4 2 TheMRSat(2,4)istheslopeoftheindifferencecurveatthatpoint: b) Using “magic formulas,” find the optimal level of consumption of x and x if p = p = 4 1 2 1 2 andm=15. Plotcarefullytheoptimalpointalongwiththebudgetlineandtheindifferencecurve passingthroughtheoptimalpoint(agraph+twonumbers). (5pt) x∗ = a .m = 2.15 = 5 1 a+b p 3 4 2 1 x∗ = b .m = 1.15 = 5 2 a+b p 3 4 4 2 Thegraphlookslikethis: 1 c)Arguethatthecommoditiesareneithergrosscomplementsnorgrosssubstitutes(argueusing themagicformula,onesentence). (2pt) From the magic formula, x∗ = a .m, we can see that the optimal consumption of x does not 1 a+b p1 1 depend on the price of the other good, p . This implies that if p goes up or down, the optimal 2 2 consumption of x wouldn’t change. So, x is neither a gross complement, nor a gross substitute 1 1 for x . The reverse argument is identical. You can also show this by taking the derivative of the 2 optimal x with respect to p (and the derivative of the optimal x w.r.t. p ) and show that these 1 2 2 1 derivativesequal0. d)Inaseparategraphplothisindifferencecurves,andfindMRSforconsumptionbundle(2,1). (agraph+onenumber). (4pt) MRS=−MU1 =−3 =−1atanybundle. MU2 3 Thegraphlookslikethis: e) Find Lionel’s optimal choice given prices p = p = 4 and m = 15 (two numbers). Is the 1 2 optimalchoiceunique? (yes/noanswer)(5pt) |MRS|=1= p1. Then,anybundle(x ,x )thatsatisfiesthebudgetconstraint,i.e.,4x +4x =15 p2 1 2 1 2 isanoptimalbundleandsotheoptimalbundleisnotunique. Problem2. (EquilibriumwithIntertemporalChoice) a)PlottheEdgeworthboxandmarkthepointcorrespondingtoendowmentsofAmbrosiaand Fergus(graph). (2pt) Thegraphlookslikethis,wheretheblackdotistheendowmentallocation: b)GiveageneraleconomicdefinitionofParetoefficiency(onesentence). Writetheequivalent conditionforParetoefficiencyintermsofMRS(equation). Useyourequationtoverifywhetherthe endowmentisParetoefficient. (argueformallyusingyourcondition)(4pt) 2 IfanallocationisParetoefficient,neitherofthetwoagentscanbemadebetteroffwithadifferent allocation,withoutmakingtheotheragentworseoff. Formally,weneedMRSA =MRSF Fromtheutilityfunction,wecanfindtheMRS:MRS=−2x2 x1 Attheendowment,MRSA =MRS(4,1)=−1 andMRSF =MRS(1,4)=−8,sothemarginalrates 2 of substitution are not equal for the two agents and thus the endowment allocation is not Pareto efficient. c)FindalltheallocationsthatareParetoefficient(deriveformulaforthecontractcurve). Plot thecontractcurveintheEdgeworthbox. (4pt) ThecontractcurvecontainsallallocationsforwhichtheMRSofthetwoagentsareequaland whichexactlyexhaustthetotalendowmentofthetwogoods. Formally: xA xF 2 = 2 (1) xA xF 1 1 xA+xF =5 =⇒ xF =5−xA (2) 1 1 1 1 xA+xF =5 =⇒ xF =5−xA (3) 2 2 2 2 Pluggingin(2)and(3)into(1),weget: xA 5−xA 2 = 2 xA 5−xA 1 1 Cross-multiplyingandsolvinggives: xA =xAandxF =xF. Thecontractcurveisthediagonalofthe 1 2 1 2 Edgeworthbox: d)Findthecompetitiveequilibrium. Calculateborrowingandsavingsforbothagentsinyour equilibrium. (Hint: usethatfortheintertemporalchoice1+r= p1 )5pt First,calculatethe”incomes”ofthetwoagentsbasedontheirpe2ndowments,normalizingp =1: 2 mA =4p +p =4p +1 1 2 1 mF =p +4p =p +4 1 2 1 Using”magicformulas”,expresstheoptimalconsumptionofx forbothagents: 1 xA = a .mA = 2.4p1+1 (4) 1 a+b p 3 p 1 1 xF = a .mF = 2.p1+4 (5) 1 a+b p 3 p 1 1 3 Thetwoagentstogethermustexhaustthetotalendowmentofx ,so: 1 2.4p1+1 + 2.p1+4 =5 =⇒ p =2 3 p 3 p 1 1 1 Substituting p = 2 into (4) and (5) and similarly for the optimal consumption of good 2, we get 1 xA =xA =3,xF =xF =2. 1 2 1 2 SinceAmbrosiaisendowedwith4whenyoungandsheonlyconsumesxA =3,Ambrosiasaves 1 1dollarwhenyoung. SinceFergusisendowedwith1whenyoungandheconsumesxF =2,Fergus 1 borrows1dollarwhenyoung. Problem3. (Technology) a) What can you say about the returns to scale (chose: IRS, CRS or DRS) and MPK (chose: increasing,constant,ordecreasing). (2pt) Thetechnologyexhibitsdecreasingreturnstoscale(DRS)andtheMPKisdecreasing. b)Findthe(variable)costfunctionc(y)giventhatthepricesofinputsarew = w = 2(givea K L function). (5pt) First,tofindtheoptimallabor-to-capitalratio,set MPK = wK MPL wL MPK = L43L41 = L MPL K34K14 K w 2 K = =1 w 2 L L =1 =⇒ L=K K Then,expressthetotalcostasafunctionofoneoftheinputsbyusingL=K C=w K+w L=2K+2L=2K+2K=4K (6) K L Now, express output y as a function of that same input and use equation (6) to derive the final answer: y2 y=2K14L41 =2K14K14 =2K12 =⇒ K= =⇒ C(y)=4K= y2 4 c)Findthesupplyfunctionoftheindividualcompetitivefirmandplotitinthegraph(givethe formula,inthegraphmarkthepricesforwhichthemarketwillnotopen). (4pt) Tofinditsoptimaloutput,thefirmmaximizesitsprofitfunctionwithrespecttoy: π(y)=TR−TC=py−9−y2 ∂π p =p−2y=0 =⇒ y= ∂y 2 Tofindthepricesforwhichthefirmwouldnotproducebecauseitmakesnegativeprofits,weneed tofindyMES: MC=ATC 9 2y= +y y =⇒ yMES =3 =⇒ ATCMES =6 Then,thesupplyfunctionisgivenby:  y(p)=02p iiffpp≥<66 Thesupplyfunctionisgiveninred: 4