Microeconomic Theory: Basic Principles and Extensions , 12th Edition Solution Manual

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Microeconomic Theory:Basic Principles and Extensions12thEditionSolutions ManualWalter Nicholson & Christopher Snyder

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The problems in this chapter are primarily mathematical.They are intended to give studentssome practice with the concepts introduced in Chapter 2, but the problems in themselves offerfeweconomic insights.Consequently, no commentary is provided. Results from some of theanalytical problems are used in later chapters, however, and in those cases the student will bedirectedbacktothis chapter.Solutions2.122( ,)43.f x yxya.8 ,xfx6 .yfyb.Constraining( ,)16fx ycreates an implicit function between the variables.Theslope of this function is given by86xyfdyxdxfy for combinations ofxandythat satisfy the constraint.c.Since(1, 2)16f,we know that at this point8 126 23dydx  .d.The( ,)16fx ycontour line is an ellipse centered at the origin.The slope of theline at any point is given by86.dy dxxy Notice that this slope becomesmore negative asxincreases andydecreases.2.2a.Profits are given by2240100.RCqq  The maximumvalue isfoundby setting the derivative equal to 0:4400d=q +dq,implies*10qand*100. b.Since240,2d=dqthis is a global maximum.c.702 .MR= dR dq =q230.MC = dC dq =qSo,*10qobeys50.MRMCCHAPTER 2:Mathematics for Microeconomics

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Chapter 2: Mathematics for Microeconomics22.3First, use the substitutionmethod.Substituting1yxyields2( )( ,1)(1).fxfxxxxxxTaking the first-order condition,( )120,fx =xand solving yields*0.5,x =*0.5y =,and***()(,)0.25.fxfxy=Since*()20,fx thisis a local and global maximum.Next, use the Lagrangemethod.The Lagrangian is(1).xyxyLThefirst-order conditions are0,0,10.xy= y== x==xy =LLLSolving simultaneously,.xyUsing the constraint gives**0.5,xy0.5,=and**0.25.x y2.4Setting up the Lagrangian,(0.25).xyxyLThe first-order conditionsare1,1,0.250.xyyxxyLLLSo.xyUsing the constraint2(0.25)xyxgives**0.5xyand2.=Note thatthe solution is the same here as in Problem 2.3, but here the value for the Lagrangianmultiplier is the reciprocal of the value in Problem 2.3.2.5a.The height of theball is given by2( )0.540 .f tgtt The value oftforwhichheight is maximized is found byusingthe first-order condition:400,dfdt =gtimplying*40.tgb.Substituting for*,t2*4040800()0.540.f tgggg Hence,*2()800 .df tdgg c.Differentiation of the original function at its optimalvalue yields**2()0.5() .df ttdg Because the optimal value oftdepends on,g

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Chapter 2: Mathematics for Microeconomics32**2()408000.5()0.5,2df t=tdggg as wasalsoshown inpart(c).d.If32,g*5 4.tMaximum height is800 3225.If32.1,gmaximumheight is800 32.124.92,a reduction of0.08.This could have been predictedfrom the envelope theorem, since*280025()(0.01)0.08.3232df tdg 2.6a.This is the volume of a rectangular solid made from a piece ofmetal, which isxby3xwith the defined corner squares removed.b.Thefirst-order condition for maximum volume is given by22316120.VxxtttApplying the quadratic formula to this expression yields22162561441610.60.225 .2424xxxxxtxThe second value given by the quadratic(1.11 )xis obviously extraneous.c.If0.225 ,tx33330.670.040.050.68.VxxxxSovolume increases without limit.d.This would require a solution using the Lagrangian method.The optimal solutionrequires solving three nonlinear simultaneous equations,a tasknotundertakenhere.Butit seems clear that the solution would involve adifferent relationshipbetweentandxthan inparts(a–c).2.7a.Set uptheLagrangian:12125ln().xxkxxLThefirst-order conditions are212125100,0.,xx=x=kxxLLLHence,215.x With10,ktheoptimal solution is**125.xxb.With4,ksolving thefirst-order conditions yields*11x and*25.xc.If all variables must be nonnegative,it is clear that any positive valuefor1xreduces

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Chapter 2: Mathematics for Microeconomics4d.If20,koptimal solution is*115,x*25,xand*155ln5.yBecause2xprovides a diminishing marginal increment toyas its value increases,whereas1xdoes not, all optimal solutions require thatonce2xreaches 5, anyextra amounts be devoted entirely to1.xIn consumer theory,this function can beused to illustrate how diminishing marginal usefulness canbe modeled in a verysimple setting.2.8a.BecauseMCis thederivativeof,TCTCis an antiderivative of.MCBythefundamental theorem of calculus,0( )( )(0),qMC x dxTC qTCwhere(0)TCis the fixed cost, which we will denote(0)TCKfor short.Rearranging,00202( )( )(1)2.2qqxqxTC qMC x dxKxdxKxxKqqKb.Forprofit maximization,( )1,pMC qqimplying1.qpBut15pimplies14.qProfitare2( )1415 1414298.TRTCpqTC qKKIf the firm is just breaking even, profitequals 0, implying fixed cost is98.Kc.When20pand19,qfollow the same steps as inpart(b), substituting fixedcost98.KProfitare

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Chapter 2: Mathematics for Microeconomics52( )1920 19192180.59882.5.TRTCpqTC qKd.Assuming profit maximization, we have22()( )(1)(1)(1)982(1)98.2ppqTC qpp pppe.i.Using the above equation,(20)(15)82.5082.5.ppii.The envelope theorem states that*().ddpqpThat is, thederivative ofthe profitfunction yields this firm’s supply function.Integrating overpshowsthe change in profits by the fundamentaltheorem of calculus:2015201520215(20)(15)(1)218097.582.5.ppddpdppdpppAnalytical Problems2.9Concave andquasi-concavefunctionsThe proof is most easily accomplished through the use of the matrix algebra of quadratic forms.See, for example, Mas Colellet al.,1995,pp. 937–939.Intuitively, because concave functions liebelow any tangent plane, their level curves must also be convex.But the converse is not true.Quasi-concave functions may exhibit “increasing returns toscale”;even though their levelcurves are convex, they may rise above the tangent plane when all variables are increasedtogether.

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Chapter 2: Mathematics for Microeconomics6A counter example would be the Cobb–Douglas function,which is always quasi-concave, but convex when1.2.10The Cobb–Douglasfunctiona.121221111222121112211112120,(1)0,(1)0,0.0,fx xfxxfxxxffxxfx  Clearly, all the terms in Equation 2.114 are negative.b.A contour line is found by setting the function equal to a constant:12,ycx ximplying121.xcx Hence,210.dxdxFurther,22210,d xdximplying the countour line is convex.c.Using Equation 2.98,2222211221212(1),fffxxwhich isnegative for1.2.11Thepower functiona.Since0y and0,y the function is concave.b.Because1122,0ffand12210,ffEquation 2.98 is satisfied,and the functionis concave.Because12,0,ffEquation 2.114 is also satisfied, so the function isquasi-concave.c.yis quasi-concave as is.yHowever,yis not concave for1. This can beshown most easily by1212(2, 2)2(,).fxxfxx2.12Proof ofenvelope theorem

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Chapter 2: Mathematics for Microeconomics7a.The Lagrangian for this problem is121212(,, )(,, )(,, ).xxaf xxag xxaLThefirst-order conditions are1112220,0,0.fgfggLLLb.,c.Multiplication of eachfirst-order condition by the appropriate deriviative yields221112120.dxdxdxdxffggdadadada d.The optimal value offis given by12( ),( ),.fxaxaaDifferentiation of thiswith respect toashows how this optimal value changes witha:*1212.adadxdxddffffdaae.Differentiation of the constraint12( ),( ),0gxaxaayields11220.adadxdxdggggdadaf.Multiplying the results frompart(e)byand using parts(b)and(c)yields*.aaadffgdaLThisproves the envelope theorem.g.In Example 2.8,we showed that8.P This shows how much an extra unit ofperimeter would raise the enclosed area.Direct differentiation of theoriginalLagrangian shows also that*.PdAdPLThisshows that the Lagrange multiplier does indeed show thisincremental gain inthis problem.2.13Taylorapproximationsa.Afunction in one variable is concave if( )0.fxUsing the quadratic Taylorformulato approximatethis function at pointa:2( )( )( )()0.5( )()( )( )().fxf afaxafaxaf afaxaTheinequality holds because( )0.faBut theright-hand side of this equation is

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Chapter 2: Mathematics for Microeconomics8the equation for the tangent to the function atpoint.aSo we have shown that anyconcave function must lie on or below the tangent to the function at that point.b.Afunction in twovariables is concave if21122120.fffHence,the quadratic form22111222)(2f dxf dx dyfdywill also benegative.But this says that the final portion of the Taylor expansion will benegative (by settingdxxaanddyyb),and hence the function will bebelow its tangent plane.2.14More onexpected valuea.The tangent to( )g xat the point( )E xwill have the form( )cdxg xfor allvalues ofxand( )(( )).cdE xg E xBut,because the linecdxis above thefunction( )g x,we know( ( ))()( )(( )).E g xE cdxcdE xg E xThis proves Jensen’sinequality.b.Usethe same procedure asin part(a), but reverse the inequalities.c.Let1( ),uF x( ),dufx ,xvand.dxdv0001( )(1( ))( )0( )( ).xxF xdxF xxfxxdxE xE xd.Usethe hint to break up the integral defining expected value:1011( )( )( )( )( )( )().tttttE xtxfx dxxfx dxttxfx dxttfx dxfx dxP xte.1.Show that this function integrates to 1:

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Chapter 2: Mathematics for Microeconomics93211( )21.xxfx dxxdxx 2.Calculate the cumulative distribution function:32211( )21.xtxtF xtdttx 3.Using the result frompart(c):21111( )1( )1.xxE xF xdxxdxx 4.To show Markov’s inequality use21( )()1( ).E xP xtF ttttf.1.Show that the PDF integrates to 1:222311811.3999xxxxdx2.Calculate theexpected value:223411155( ).312124xxxxE xdx3.Calculate( 10Px ):0023111 .399xxxxdx4.All we must do is adjust the PDF so that it now sums to 1 over the new,smaller interval.Since()8 9,P A2( )3(|)defined on 02.8 98fxxfxAx5.The expected value is againfound through integration:223400333(|).8322xxxxE xAdx6.Eliminating the lowest values ofxincreases the expectedvalue of theremaining values.2.15More onvariancesa.This is justan application of the definition of variance:

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Chapter 2: Mathematics for Microeconomics1022222222Var( )( )2( )[( )]().2[( )][( )]()[( )]xE xE xExxE xE xE xE xE xE xE xb.Here,we letxyxand apply Markov’s inequality toyand remember thatxcan only take on positive values.222222()()().xE yP ykP ykkkc.Let,ix1,,inbenindependent random variables each with expected valueand variance2.1.niiExn 2221Var.niixn Now,let1.niixxn().nE xn222Var().nxnnd.Let12(1)Xkxk xand()(1).E Xkk222222Var()(1)(221).Xkkkk2Var()(42)0.dXkdkHence,varianceisminimized for0.5.kIn this case,2Var()0..5XIf0.7,k2Var()0.58X(not much of an increase).e.Suppose that21Var()xand22Var().xrNow222222Var()(1)(1)2.Xkkrr kkrr2Var()2(1)20.dXr krdk.1rkrFor example, if2,rthen2 3k,and optimal diversification requires that thelower risk asset constitute two-thirds of the portfolio.Note, however, that it is still

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Chapter 2: Mathematics for Microeconomics11optimal to have some of the higher risk asset because asset returns areindependent.2.16More oncovariancesa.This is a direct result of the definition ofcovariance:Cov( ,)(( ))(())[()( )( )()]()( )()( )( )( )()()( )().x yExE xyE yE xyxE yyE xE x E yE xyE x E yE y E xE x E yE xyE x E yb.222222222222Var()[() ][()]()2()()[( )]2E( )E()[E()]Var( )Var()2Cov( ,).axbyEaxbyE axbyaxabE xybyaE xabxybyaxbyaExEbyThe final line is a result ofProblems 2.15a and 2.16a.c.The presence of the covariance term in the result of Problem 2.16b suggeststhatthe results would differ.In the two-variable case, however, this isnot necessarilythe situation.For example, suppose thatxandyare identically distributed and that2Cov( ,).x yrUsing the prior notation,22222Var()(1)2.(1)Xkkkk rThefirst-order condition for a minimum is2(4224)0,krrkimplying*220.5.44rkrRegardless of the value of.rWith more than tworandom variables, however,covariances may indeed affect optimal weightings.d.If12,xkxthe correlation coefficient will be either1(ifkispositive) or1(ifkisnegative),sincekwill factor out of the definition leaving only the ratio ofthe common variance of the two variables.With less than a perfect linearrelationship0.5| Cov( ,) |Var( )Var(.)x yxye.If,yxCov( ,)(( ))(())[(( ))(( ))]Var( ).x yExE xyE yExE xxE xx

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Chapter 2: Mathematics for Microeconomics12Hence,Cov( ,) .Var( )x yx 

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1313These problems provide some practice in examining utility functions by looking atindifference curve maps and at a few functional forms.The primaryfocus is onillustrating the notion of quasi-concavity (a diminishingMRS) in various contexts.Theconcepts of the budget constraint and utility maximization are not used until the nextchapter.Comments on Problems3.1This problem requiresstudents to graph indifference curves for a variety offunctions, some of which are not quasi-concave.3.2This problem introducesthe formal definition of quasi-concavity (from Chapter2) to be applied to the functions studied graphically in Problem 3.1.3.3This problem shows that diminishing marginal utility is not required to obtain adiminishingMRS. All of the functions are monotonic transformations of oneanother, so this problem illustrates that diminishingMRSis preserved bymonotonic transformations but diminishing marginal utility is not.3.4This problem focuses on whether some simple utility functions exhibit convexindifference curves.3.5This problem is an exploration of the fixed-proportions utility function.Theproblem also shows how the goods in such problems can be treated as acomposite commodity.3.6This problem asks students to use their imaginations to explain how advertisingslogans might be captured in the form of a utility function.3.7This problem shows how utility functions can be inferred fromMRSsegments. Itis a very simple example of “integrability.”3.8This problem offers some practice in deriving utility functions from indifferencecurve specifications.CHAPTER 3:Preferences and Utility

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Chapter 3: Preferences and Utility14Analytical Problems3.9Initial endowments.Thisproblem shows how initial endowments can be treatedin simple indifference curve analysis.3.10Cobb–Douglas utility.This problem providessome exercises with the Cobb–Douglas function,including how to integrate subsistence levels of consumptioninto the functional form.3.11Independent marginal utilities.This problem showshow analysis can besimplified if the cross-partials of the utility function are zero.3.12CES utility.This problem showshow distributional weights can be incorporatedinto the CES form introduced in the chapter without changing the basicconclusions about the function.3.13The quasi-linear function.Thisproblem provides a brief introduction to thequasi-linear form,which (in later chapters) will be used to illustrate a number ofinteresting outcomes.3.14Preferencerelations.This problem provides a very brief introduction to howpreferences can be treated formally with set-theoretic concepts.3.15Thebenefit function.This problem introducesLuenberger’snotion of reducingpreferences to a cardinal number of replications of a basic bundle of goods.Solutions3.1Here we calculate theMRSfor each of these functions:a.3.1xyUMRSUMRSis constant.b.0.50.50.5().0.5()xyUy xyMRSUy xxConvex;MRSis diminishing.c.0.50.5.1xyUxMRSUMRSis diminishing.d.220.5220.520.5()2.0.5()xyUxxMRSxyyUxyyMRSisincreasing.

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