OptimizationIntroduction.pptx

Faculty of EconomicsOptimizationLecture 1Marco HaanFebruary 14,20052IntroductionThis is a math course for AE students.The emphasis on applications.The goal is to provide you with the mathematical skills required for the more advanced courses,especially in micro and macro.Literature:Hoy,Livernois,McKenna,Rees,StengosMathematics for Economics,MIT-Press,2nd edition,2001.For those who took this course last year:this book is infinitely better than Lambert.For this course,practice is essential!Note:there is a Solutions Manual for this book,with solutions for all odd-numbered problems.3IntroductionThis is a math course for AE students.The emphasis on applications.The goal is to provide you with the mathematical skills required for the more advanced courses,especially in micro and macro.Literature:Hoy,Livernois,McKenna,Rees,StengosMathematics for Economics,MIT-Press,2nd edition,2001.For those who took this course last year:this book is infinitely better than Lambert.For this course,practice is essential!Note:there is a Solutions Manual for this book,with solutions for all odd-numbered problems.4Introduction(2)The Team:Dr.M.A.Haanm.a.haaneco.rug.nl(micro,semester 2a)Prof.dr.E.Sterkene.sterkeneco.rug.nl(macro,semester 2b)Drs.L.Daml.dameco.rug.nl(exercise hours,entire semester)Classes:selected Mondays,10:00-12:00,ZG 114selected Thursdays,15:00-17:00,ZG 1075Set-upLectures,exercise hours,take-home,exam.One lecture per week.One exercise meeting per week.Exercises to be discussed will be announced in advance.Do the exercises before the meeting!There will be 3 take-home problem sets.You have roughly two weeks for each.During these two weeks,there will be no(or fewer)lectures.Problem sets will be relatively tough.You can discuss with others,but you cannot cooperate.If you do,we will find out.With the sets,you can earn 2 bonus points on your final grade.The bonus points are likely to be crucial.All you need will be on Nestor.6Preliminary ScheduleFeb14Optimizationwithn-variables(H12)Feb17ExercisesH12Feb21ConstrainedOptimization(H13)Feb24ExercisesH13Feb28ComparativeStatics(H14)Mar3ExercisesH14Mar7ConcaveProgramming(H15)Mar10NoclassProblemSet1distributedMar14 NoclassMar17NoclassMar21 NoclassMar24NoclassProblemset1dueMar28 EASTERN(noclass)Mar31ExercisesH15Apr4Integration(H16)Apr7ExercisesH16ProblemSet2distributed7Preliminary Schedule(contd)May2DifferenceEquations(H18-20)May5ASCENSNDAY(noclass)Problemset2dueMay9DifferentialEquations(H21-23)May12 ExercisesH18-20May16 PENTECOST(noclass)May19 ExercisesH21-23May23 SystemsofDiffEquations(H24)May26 ExercisesH24May30 DynamicOptimization1(H25)June2ExercisesH251ProblemSet3distributedJune6NoclassJune9NoclassJune13 DynamicOptimization2H25)June16 ExercisesH252June20 DynamicOptimization3(add)June23 ExercisesadditionalProblemset3dueJuly14Exam8Today:Chapter 121.Concavity.2.Stationary values.3.Local optima.4.Second order conditions.5.Direct restrictions on variables.6.Shadow prices.Note:In these lectures I will focus on the main ideas and intuition.Definitions and theorems are in the book.9ConcavityThe function f is concave iffor all in 0,1.It is strictly concave if the strict inequality holds forall in 0,1.What this says is:1.Take any two points on the graph.2.Draw a line between those points.3.The entire line should be below the graph.If the second derivative is negative,then the function is strictly concave.1011ConvexityThe function f is convex iffor all in 0,1.It is strictly convex if the strict inequality holds forall in 0,1.What this says is:1.Take any two points on the graph.2.Draw a line between those points.3.The entire line should be above the graph.If the second derivative is positive,then the function is strictly convex.1213NoteConcAveConVex14How to find a local optimimum of a one-dimensional function1.Take the derivative.2.Set it equal to zero(first-order condition).3.If the function is concave at this point,we have a local maximum.4.If the function is convex at this point,we have a local minimum.The latter are the second-order conditions.15With more than one dimension.things get more complicated.A stationary value is a point where the derivative equals zero in every single dimension.Yet,a stationary value is not necessarily an extreme value.16OK!17OK!Problem19This is a saddlepoint.Hoy:“The function takes on a maximum with respect to changes in some of the variables and a minimum with respect to others”.Note that this definition is in terms of the existing variables.According to this definition,when a surface with a saddle point is rotated 45 degrees around the vertical axes,then the new surface does not necessarily have a saddle point as well.Opinions differ as to whether this is the proper definition.Nevertheless,well stick to it.20Second-order conditionsFor a local maximum of a function f(x)we need that,starting in a stationary point,the function is non-increasing in every direction.It is sufficient to have:or:theHessianmatrixHisnegativedefinite,with21Second-order conditionsFor a local minimum of a function f(x)we need that,starting in a stationary point,the function is nondecreasing in every direction.It is sufficient to have:or:theHessianmatrixHispositivedefinite,with22Second-order conditionsFor a saddlepoint of a two-dimensional function f(x)we need that:theHessianmatrixHisindefinite,andeither23A global optimum may also involve a corner solution.24252627ThusThe global maximum x*of a function g(x)on some interval xmin,xmax has either one of the following properties:g(x*)=0 x*=xmin x*=xmaxThis implies that one or both of the following must hold:g(x*)0 and(x*xmin)g(x*)=0 g(x*)0 and(xmax x)g(x*)=028g(x)=0(x xmin)g(x*)=0(x xmax)g(x*)=0g(x)0(xmax x)g(x*)=0g(x)=0(x xmin)g(x*)=0(xmax x)g(x*)=0These conditions are necessary,not sufficient!Condition satisfiedg(x)0(x xmin)g(x*)=0(xmax x)g(x*)029NoteIf a function is increasing in xmin,then xmin cannot be a global maximum.Similar necessary conditions can be derived for a minimum.With more dimensions,conditions have to be satisfied in each dimension.30g(x)0(xmax x)g(x*)=0g(y)=0(ymax y)g(x*)=031A monopolist faces two submarkets,with demandsand costsProfits then equalExample(pg.553)32First-order conditions:hence we have a maximum.33Cournot duopoly.Suppose that a monopolist sells two products.He will solve:Now suppose we have two firms that each sell one of these products.They will solve:34Example(pg.574)A monopolist supplies its product from two plant,with cost functionsand demandProfits then equalFirst-order conditions:Infeasible!35We also have to take non-negativity constraints into account.Three possibilities:We already know that(a)is not possible.Recall that we need(qi*)0 and(qi*qmin)(qi*)=0Case(c)would then implyNot feasible either.36So we must have:This requires:Feasible.3738This weeks exercisespg.558:1a,1bpg.568:1a,1b.pg.580:1a1d.pg.581:1a1d.pg.582:3.