数理经济学lecture12

数理经济学教学大纲期末复习《数理经济学》教学大纲一、课程基本信息课程代码 03323990 课程类别专业发展课程—选修课中文名称数理经济学英文名称Mathematical Economics适用专业经济学专业开课单位经济系总学时 54 (理论：48 其他：6 ) 学分 3先修课程经济数学基础，西方经济学后续课程计量经济学，数学建模，毕业设计二、课程性质、地位和任务数理经济学是经济学专业的选修课，强调运用数学方法，主要是微积分、线性代数、数理统计方法来解决经济学中的一些原理问题。

从性质上说，数理经济学是利用数学方法对经济问题进行分析演绎，因而是进行经济理论研究必备的重要工具。

因此，本课程既不能等同于纯粹的中高级西方经济学，也不能等同于一般的经济数学。

本课程通过对数学方法与经济模型结合，主要目的在使学生熟习如何应用数学模型来表述经济学，旨在提高经济学分析中的数理思维能力，培养学生运用数学工具进行经济学研究的能力，为将来的深入学习和研究打下必要的方法论基础。

三、课程基本要求本课程通过讲授、练习和阅读文献等形式，使学生掌握主要的数理经济学研究方法，了解其主要工具和概念，掌握微分、矩阵、最优化的基本原理和方法，并能运用所学数理工具分析一些理论与现实经济问题，从而对现实经济系统进行预测与决策。

四、课程内容及学时分配序号内容学时第一章导论 3第二章单变量微分法及其经济学应用 6第三章单变量最优化问题 12第四章线性模型与矩阵代数 9第五章多变量微分法及其经济学应用 12第六章多变量最优化问题 12合计 54第一章导论【教学内容】数理经济学的概念、数理经济学的起源和发展、数理经济学的研究对象、研究方法。

【教学基本要求】通过本章的学习，使学生了解数理经济学的产生和发展、数理经济学的研究方法，理解经济模型的构成及其作用。

【教学安排】课堂讲授3学时【重难点】数理经济学的研究方法、经济模型第一节数理经济学的产生和发展第二节数理经济学的研究方法第三节经济模型第二章单变量微分法及其经济学应用【教学内容】连续函数，导数和微分，经济学中的“边际”概念，反函数，经济学中的增长率【教学基本要求】复习单变量微分法基本数学原理；掌握经济学中各种函数连续性及其可微性；深刻认识边际、弹性与增长率的概念，理解它们与微分之间的关系。

Mathematical Preliminaries1.1.Sets•A set is a well-defined collection of elements・Example: A = {a, b, c} is a set with three elements・a G A means that the element a is in the set Ad g A means that the element d is not in the setA.•0 denotes the null set or empty set, i.e., the set with no element.•set A is a subset of set B 讦a w B for all a w A. This is denoted by A u B, or B A・•union of sets: A u B = {x: x G A or X G B}.•intersection of sets: A n B = {x: x G A and x G B}.•A and B are disjoint sets if A n B = 0.•the complement of set A in set B is the set B\A = {x: x G B and x G A}・•Cartesian product: Let x G X and y e Y, and let (x, y) be an ordered pair. Then (x, y) 6 XxY, where XxY is the Cartesian product of X and Y・1.2.LogicConsider two propositions P and Q・If P implies Q、then P is a sufficient coixlition for Q, and Q is a necessary condition for P. This is denoted by P => Q.If P implies Q and Q implies P, then “P holds if and only 讦Q holds，： P and Q are equivalent, and P is a necessary and sufficient condition for Q. This is denoted by P <=> Q.Let {not P} denote the statement that P is not true.CorHrapositive: If P implies Q, then {not Q} implies {not P}.Example 1: Let x be a real number, P = {x > 0) and Q = {x2 > 0). Then, P => Q and {not Q}=> {not P} hold, but {Q => P} is false.Example 2: Let P = {x > 0) and Q = {x3 > 0}・ Then P o Q, and {not P) <=> {not Q) hold.1.3.Numbers•natural numbers: N = {1, 2, 3,…}・integers: Z = {-3, -2, -1,(), L 2, 3,...}•rational numbers: Q = {a/b: a G Z, be Z, bHO}・irrational numbers: 2 ： 3 …•real numbers: R = {x: x is rational or irrational}.complex numbers: C = {a + b i: a e R, b G R, i = (-1)12), where a is the real part, and b is the imaginary part・the n-fold Cartesian product of R: R n = Rx...xR = {(x h x2, •••, x n): Xj G R, i = 1,n), where Xj is the i-th coordinate of x = (x b x2,x n)1.4.FunctionsLet X and Y be sets.Definition: f: X t Y is a mapping associating each element of X with an element of Y.X is the domain of ff(x) is the image of x under ff(X) = {f(x): x G X} is the image of X under f•a function: if only one point in Y is associated with each point in X・ a coirespondence: if more than one point in Y can be associated with each point in X •inverse function: x = f *(y) if and only if y = f(x).a function f: X T Y is onto 讦f(X) = Y.It means that the equation f(x) = y has at least one solution for each y.•if f(x) and f ly) are both single・valued, then f is on—to-one.It means that the equation f(x) = y has at most one solution for each y.•composite function: h = g(f(x)) = g • f is the composition of f with g satisfying f: A t B u C, g:C t D, g • f: A t D.L5e BoundsLet S c R.•S is bounded from above (from below)讦there exists a G R (b G R) such that x < a (x > b) for all x G S. Then, a is an upper bound of S, and b is a lower bound of S・•The least upper bound (lub) or suDremum (sup) of S is the upperbound of S such that there does not exist a smaller upper bound. It is denoted by sup(S).•The supremum of S is called a maximum (max) of S if sup(S) G S. It is denoted by max(S). •The greatest lower bound (gib) or infimum (inf) of S is the lower bound of S such that there does not exist a larger lower bound. 1( is denoted by inf(S).•The infimum of S is called a minimum (min) of S if inf(S) e S. It is denoted by min(S).• Property:If S c R and S has an upperbound, then S has a supremum.If S u R and S has a lowerbound, then S has an infimum.16 Vector SpaceConsider a set V.LI- associative law: x + (y + z) = (x + y) + z, for all x, y, z, w VL2- identity: there exists O G V such that x + 0 = x for all x G VL3- inverse: there exists (-x) e V such that x + (-x) = 0 for all x e VL4- commutative law: x + y = y + x for all x, y G VL5- associative law: a・([3・x) = (a-p) x for all a, p e R, and for all x G VL6- identity: there exists 1 G V such that l x = x for all x e VL7- distributive law: a・(x + y) = a x + a y for all ae R, and for all x, y w VL8- distributive law: (a + p)-x = a x + p x for all a,卩G R, and for all x G V L9- closure: x e V and y G V implies that (x + y) G V LIO- closure: x G V and ae R implies that (a・x) e V. Definition: A set V is vector space (or linear space) if it satisfies Ll-Ll0. Then x G V is called a vector. Examples: R l\ or C n is each a vector space.1.7. Norms and distancesConsider a function d(x, y) satisfying:M1: d(x, y) = 0 if and only if x = y M2: d(x, y) + d(y, z) > d(z, x) M3: d(x, y) > 0 for all x, y M4: d(x, y) = d(y, x).Definition: For a given set X,讦a function d: XxX —> R satisfies M1-M4, then: X is a metric space, denoted by (X, d)d is a metricd(x, y) is the distance between points x and y.Examples:di(x，y) = [Zi (xj - yi)2!12 = Euclidian distance，denoted by ||x - y||d2(x, y) = maxi 风-y s|d3(x, y) = Zi lx, - y.lNote: Topology consists in studying the properties of sets that are independent of the distance measure chosen.Definition: Let V be a vector space・ A real value function N: V t R is called a norm on V if: N(x) > 0 for all x G V N(x) = 0 if and only if x = 0N(r x) = |r| N(x) for all re R and x e V, andN(x + y) < N(x) + N(y) for all x, y G V.Example: N(x) = d|(x, 0) = [Xi (xj)2]12 = ||x|| is the Euclidian noim of x in R.R n, with Euclidian norm and Euclidian metric, is a normed vector space.Every normed vector space is a metric space with respect to the induced metric defined by d](x, y) = llx - yll.Convex SetsLet X be a vector space (e.g・，X = R n).Definition: A set S c: X is convex if any x, y G S implies that(0x + (1-0) y) e S, for all 0 e R, 0 < 0 < 1.Note: (0 x + (1-0) y) is called a linear combination of x and y.Properties:・Any intersection of convex sets is convex.•Let Si, i = 1,m, be convex sets in vector space X. Then:•(Ejei (Xj Si) = {x: x = Z i=i■•…m oti Xi, XjG Sj, otf R, i = 1,…，Hi} is a convexset.•(SixS2x...xS m) = x i=1■•…m (Si) is a convex set.L9. Compact SetsLet S u R n.Definition: An open ball about x0 e R n with radius r e R, r > 0, is defined as:B r(x0) = {x: x G S, d(x, x0) < r}, where d(x, x0) is the Euclidian distance betweenpoints x and x0.Definition: An open set S u R n is a set S such that, for each x e S, there exists an open ball B r(x) completely contained in S.• The union of open sets is open..A finite intersection of open sets is open.Definition: The inierior of a set S, denoted by int(S), is the union of all open sets contained in S..A set S is open 讦and only 讦S 二int(S).Definition: A set S c R n is closed 讦the set (R n\S) is open.・The intersection of closed sets is closed・.A finite union of closed sets is closed・Definition: The closure of a set S, denoted by cl(S), is the intersection of all closed sets containing S..A set S is closed if and only if S = cl(S).Definition: The boundary of a set S u R" is the set cl(S)ncl(R n/S).Definition: A set S is bounded if there exists an open ball with a finite radius which contains S・Definition: A collection of open sets (S a)a€A in a metric space X is said to be an open cover of a given set S u R n if S u 低入 S a.The open cover (S a)ae A of S is said to admit a finite subcover if there exists a finitesubcollection (SQ^F such that S u S®Definition 1: A set S c R n is compact if and only if it is closed and bounded・Definition 2: A subset S of a metric space X is compact if and only if every open cover of S has a finite subcover.Note: The definition 2 of compactness applies to sets in any metric space, while definition 1 applies only to sets in R n.LIO. SequencesLet (X, d) be a metric space (e.g., X = R n), and let S c X.Definition: A sequence {xj： j = 1,g} in S converges to y if, for any e > 0, there exists a positive integer j" such that j hj，implies d(y, xj) < e.This is denoted by y = limj* {xj, where y is the limit of {Xj}.Note: It does not follow that y = lirOjT® {xj} G S・Definition: A sequenee {Xj： j = 1，…，g} in S is a Cauchy sequence if for any e > 0, there exists a positive integer such that, for any ij > j\ d(x i9 Xj) < £.Definition: If every Cauchy sequence in a metric space is also a convergent sequence, then the* A sequence {xp j = 1,…严} in R" is a Cauchy sequence if and only if it is a convergent sequence, i.e. if and only if there is y G R n such that linij》{Xj} —> y.metric space is said to be complete.By the above definition, this implies that R n is complete (although not all metric spaces are complete)・Definition: Let m(j) be an increasing function: m: {12 3,…} T {1, 2, 3,such that m(k+l) > m(k).Given a sequence {xj： j = 1, 2,…，g}, {m = 1,…，g} is a subsequence of {xj： j =1,2,…，oo}..A set S c R n is closed if and only if every convergent sequenee of points in S converges to a point in S・・ A set S e R n is compact if and only if every sequence in S has a convergent subsequence whoselimit is in S.・ A sequence {xj： j = 1, 2,…，g} in R n converges to y if and only if every subsequence of {xj： j =1,…，g} converges to y.・Every bounded sequence contains a convergent subsequence・Definition: A sequence {xj： j = 1,2,…，oo} is (strictly) increasing if, for all m > n, x m > (>) x n for all n.A sequence {xj： j = 1,2,…，<»} is (strictly) decreasing if, for all m > n, x m < (<) x n forall n・Let X u R, X H 0. If X is bounded from above (below), there exists an increasing (decreasing) sequence in X converging to sup(X) (inf(X)).Definition: Assume that ±oo are allowed as limits of a sequence.The lim sup of the sequence {xj： j = 1,in R is defined as limj^oo {可:j =1,2 •••}, where aj = sup{xj, Xj+b Xj+2. ...}・ It is denoted by linij^oo supk>j x k, or simply by lim SUpjTco Xj・The lim inf of the sequence {Xj： j = 1,in R is defined as limj^ {bj： j = 1,2 …}, where bj = inf{x jt Xj+i, x i+2, ...}. It is denoted by inf k>j x k, or simply by lim infj†Xj・† A sequence xj in R converges to a limit y G R if and only if y = lim supj* Xj = lim in与* Xj.。

《数理经济学》课程参考文献（著作、教材类）注：以下所列文献难度各异，如需阅读建议请联系任课老师。

[1]Avriel, M., Nonlinear Programming: Analysis and Methods, EnglewoodCliffs: Prentice-Hall, 1976.[2]Barro, R. J. and X. Sala-I-Martin, Economic Growth, New YorK:McGraw-Hill, 1995.[3]Bazaraa, M. S., H. D. Sherali, and C. M. Shetty. Nonlinear Programming:Theory and Algorithms. New York: John Wiley & Sons, 1979.[4]Blanchard, M. and S. Fischer, Lecture on Macroeconomics, Cambridge:MIT Press, 1989.[5]Carlson, D. A. A. B. Haurie, A. Leizarowitz, Infinite Horizon OptimalContral, Berlin: Springer-Verlag, 1991.[6]Dixit, A. K. Optimization in Economic Theory, Oxford: OxfordUniversity Press, 1990.[7]Girsanov, I. V., Lecture on Mathematical Theory of Extremum Problems,Berlin: Springer-Verlag, 1972.[8]Hestenes, M. R., Calculus of Variations and Optimal Control Theory, NewYork: John Wiley & Sons, 1966.[9]Intriligator, M. D. Mathematical Optimization and Economic Theory,Englewood Cliffs: Prentice-Hall, 1971.[10]Ioffe, A. D. and V. M. Tihomirov, Theory of Extremal Problems,Amsterdam: North-Holland, 1979.[11]Jehle, G. A. and P. J. Reny, Advanced Microeconomic Theory, 2nd ed., 上海财经大学出版社，影印本。

数理经济学
数理经济学是一门研究数量和经济行为的综合学科，它对数学、统计学和经济学的应用相结合。

它的出现开拓了经济学的发展范围，深入剖析经济存在的问题，提供有效的解决解决方案，并实施经济政策。

数理经济学主要通过定量分析及模型去研究社会经济现象和政策，比如微观经济，宏观经济，货币市场，国际经济等等。

数理经济学运用了数学、统计、技术分析和实验方法来建模经济各类问题和政策，推导出有效的经济分析结果以及经济政策可行性分析。

数理经济学还有助于更好地理解复杂的经济系统，比如，金融市场中各类金融资产价格的变化，这些价格变化受多种因素共同影响，既有宏观因素也有微观因素，数理经济学使分析师们能够深入分析相关问题，并利用概率模型来研究当前的经济形势和走势。

总而言之，数理经济学运用了数学、统计、技术分析和经济学的原理，以及实验和模型等，来研究经济现象。

它为经济研究和经济政策制定提供了有效的方法，这极大地推动了经济发展和改善了现实经济环境。

Syllabus of『Fundamental Methods of Mathematical Economics』(数理经济学基础)Course DetailsSemester: 2005-2006学年,第2学期Lecturer 张明恒PhD., Associate ProfessorOffice: (教学行政大楼) Teachers Official Buildings 512RoomOffice Time: P.M.6:30 – 8:30, TuesdayTelephone: +86 21 6590 2567E-Mail: mingheng@Coordinator:Date: 3/4,Wed;1/2,FridayRoom: 1405Rm, No.1 Teaching Buildings(1405,国定路一教) Website: /~mingheng/MathEcon.htmlStudent Background(i)the pre-session mathematics course(ii)the background mathematical material(iii)the fundamental of EconomicsGeneral AdviceThe key to success in a course like mathematical economics is to work carefully and consistently on the course material as the semester progresses. Falling behind with the course material is a recipe for disaster. Therefore students are strongly urged to attempt to complete the exercise sheets prior to attending the examples class and to read the appropriate chapters in the course text in conjunction with their lecture notes.The learning resources(i)the office hours of the lecturer when he will be available to see students;or make appointment by email or phone to meet the lecturer at anothertime(ii)the textbooks (incl. the materials on the classroom)(iii)the materials on the course web page(iv)the discussion with (the advice from) colleagues and friends on the course AimsThe aims of this course are to(i)Provide an introduction to mathematical methods for economics that willassist students in understanding mathematical research in their field;(ii)Enable students to apply these mathematical methods in their own research.ObjectivesAt the end of the course students should be able to(i) Demonstrate their understanding of the mathematical methods foreconomics;(ii)Understand and study Equilibrium Analysis, Comparative Statics Analysis, Optimization (multivariate), Dynamic Economic, Simultaneous Equations andLinear Programming etc.;(iii)Ready to advanced microeconomics, macroeconomics and econometrics.List of Readings1)Chuang,A.C., Fundamental Methods of Mathematical Economics,McGraw-Hill Publishing Company, 1984, 3rd Ed. (刘学译, 数理经济学的基本方法, 商务印书馆,2003年)(Text Book).2)Silbergerg, E., The Structure of Economics – A Mathematical Analysis,McGraw-Hill Education - Europe, 1990, 2nd ed, paperback.3)Klein,M.W., Mathematical methods for Economics, Addison-Wesley, 1998.4)Dowling,E.,T., Theory and Problems of Introduction to Mathematical Economics, 3nd5)Handbook of Mathematical Economics Vol.1-6,6)International Journal on Mathematical Economics such asi.Journal of Mathematical Economicsii.Journal of Economics Theoryiii.Journal of Economics and Businessiv.Journal of Mathematical Financev.Journal of Econometricsvi.Mathematical Social Sciencesvii.Economics Modellingviii.Economics Lettersix.Economics Theoryx.Game TheoryLecturesChapter 1 Nature of Mathematical Economics/数理经济学的实质Chapter 2 Economic Models/经济模型Chapter 3 Equilibrium Analysis in Economics/静态均衡分析，市场均衡模型，国民收入均衡模型Chapter 4 Linear Models and Matrix Algebra/线性模型与矩阵代数Chapter 5 Linear Models and Matrix Algebra/线性模型与矩阵代数，市场模型，国民收入模型，Leontief投入产出模型Chapter 6 Comparative Statics and the Derivative/比较静态分析，变化率/导数/曲率，极限定理，函数的连续性与可微性Chapter 7 Rules of Differentiation and Comparative Statics，微分法则，偏微分，Jacobi行列式Chapter 8 Comparative Statics of General Function Models，全微分/全导数，隐函数存在定理，一般函数模型的比较静态分析Chapter 9 Optimization (univariate)，最优值与级值，导数检验，Maclaurin/Taylor 级数Chapter 10 Exponential and Logarithmic Functions/指数函数与对数函数，自然指数与增长问题，最优时间安排Chapter 11 Optimization (multivariate)/多变量的最优化，最优化条件的微分形式，二次型，目标函数，Hessian矩阵检验，函数的凹/凸性与极值问题，多产品厂商问题，价格歧视，投入决策Chapter 12 Optimization with Equality Constraints/具有约束方程的最优化，Lagrange Multiplier, Bordered Hessian Matrix检验，二阶条件效用最大化与消费需求，齐次函数/生产函数，投入最小成本组合Chapter 13 Economic Dynamics and Integral Calculus/动态经济学与积分学，动态与积分，定积分/不定积分/广义积分，Domar增长模型Chapter 14 First Order Differential Equations/连续时间的一阶微分方程，市场价格的动态过程，恰当微分方程，非线性微分方程Solow增长模型Chapter 15 Higher Order Differential Equations/高阶微分方程，复数，三角函数，复数根，具有价格预期的市场模型、通货膨胀与失业模型Chapter 16 First Order Difference Equations/离散时间的一阶差分方程，均衡的动态性，蛛网模型，具有存货的市场模型，非线性差分方程Chapter 17 Higher Order Difference Equations/高阶差分方程，Samuelson数乘加速模型，通货膨胀与失业模型Chapter 18 Simultaneous Differential and Difference Equations/联立微分和差分方程动态投入产出模型Chapter 19 Linear Programming/线性规划，单纯形SimplexChapter 20 Linear programming - cont./线性规划（续），对偶定理，活动的微观和宏观水平Chapter 21 Non-Linear Programming/非线性规划，Kuhn-Tucker定理，约束规范，Arrow-Enthoven定理Chapter 22 Game Theory / 搏弈论Chapter 23 1st Seminarspetitive Equilibrium Hyperinflation under Rational Expectations, Also see epge.fgv.br/portal/arquivo/1792.pdf2.Multiple Equilibria with Externalities，Also see/depts/econ/dp/0409.pdfChapter 23 2nd Seminars1.Dynamic Analysis of a Solow-Romer model of endgenous Economical GrowthAlso see .au/policy/ftp/workpapr/ip-68.pdf2.Seignorage, Productive Government Spending and Growth in a Lucasian General Equilibrium Model Also see /economics/staff/hmk/paper/HMKAW96.PDFSupplemental Materials for a ChoiceChapter 51)Leontief, Wassily W. “Input-Output Economics.” Scientific American, October1951, pp.15-21.2)Iris Jensen ,The Leontief Open Production Model or Input-Output Analysis3)Internet Resources for the Leontief Model/mathews/n2003/leontiefmodel/LeontiefModelBib/Links/LeontiefMo delBib_lnk_1.htmlChapter 71)The Financial Returns to Education2)Why Does’t Capital Flow to Poor Counters?, See Robert E. Lucas,Jr., American EconomicView, V ol.80,No.2,pp.92 – 96, 1990(May)Chapter 81)Growth Accounting,Cobb-Douglas Production Function2)The Division of national IncomeChapter 111)Properties of a Valid Cost Function2)Government Revenue3)Isabel Correia and Pedro Teles,The Optimal Inflation Tax4)Jens Suedekum, Profit maximization and firm supply under perfect competition,University of Konstanz,Chapter 121)Dan Segal, A Multi-Product Cost Study of the U.S. Life Insurance Industry, RotmanSchool of Management,University of Toronto2)Martin Feldstein, “College Scholarship Rules and Private Savings”, American EconomicReview, Vol.85, No. 3,pp. 552 –566(June, 1995)Chapter 131)Economics Growth modelChapter 141)Charles I. Jones, A Note on the Closed-Form Solution of the Solow Model2)Fadhel Kaboub, Long-run Keynesian Growth Theory: Harrod and Domar vs.Solow Chapter 151)Tor Jacobson, Johan Lyhagen†, Rolf Larsson and MarianneNessén,Inflation, Exchange Rates and PPP in a Multivariate Panel Cointegration Model, Sveriges Riksbank Working Paper Series No. 145,December 20022)Pierre L. Siklos, INFLATION AND HYPERINFLATION, Department of EconomicsWilfrid Laurier UniversityChapter 161)Money and Prices Model2)Philip Cagan “The monetary Dynamics of Hyperinflation”, Studies in the Quanlity Theory ofMoney, ed., Milton Friedman (Univ. of Chicago Press, 1956)Chapter 171)Nicholas Rau,Introduction to Growth Theory, University College London2)Economic Problems of Developing AreasChapter 181)The Dynamic Variable Input-Output Model: An Advancement From The Leontief DynamicInput-Output Model,Annals of Regional Science, Vol. 34, No. 4, January 20012)Obst,N.M., “Stabilization Policy with an Inflation Adjustment Mechanism,” The Quarterly,Journal of Economics, 92 (May 1978), pp.355-359Chapter 191)Linear programming2)/otc/Guide/OptWeb/3)Global Optimization4)http://www.fi.uib.no/~antonych/glob.html5)Tom Cavalier's Optimization Links/faculty/t/m/tmc7/tmclinks.html6)Optimization Center at Northwestern Univ.7)/OTC/8)Yinyu ye at Stanfod9)/~yyye/Chapter 201)J. J. H. Forrest and J. A. Tomlin,Implementing the simplex method for the OptimizationSubroutine Library, IBM SYSTEMS JOURNAL, VOL 31,NO 1, 1992ATTENDANCESince exams will test your knowledge of materials covered in class, you are expected to attend class. You are responsible for material covered in class (including changes in assignment or rescheduling the test) even if you are unavoidably absent.GRADING POLICYYour final grade will be based on your performance on two exams and homework and it will be determined as follows:Mid exam 15%Final exam 60%Homework 15%Class participation 10%Total 100%Your total will be converted into final course grade and plus and minus grading will be used in this course.。