Mathematical Methods - Foundations Of Economics
Mathematical MethodsFoundations of EconomicsJosef LeydoldInstitute for Statistics and Mathematics · WU WienWinter Semester 2020/21
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IntroductionJosef Leydold – Mathematical Methods – WS 2020/21Introduction – 1 / 27
LiteratureI A LPHA C. C HIANG , K EVIN WAINWRIGHTFundamental Methods of Mathematical EconomicsMcGraw-Hill, 2005.I K NUT S YDSÆTER , P ETER H AMMONDEssential Mathematics for Economics AnalysisPrentice Hall, 3rd ed., 2008.I K NUT S YDSÆTER , P ETER H AMMOND, ATLE S EIERSTAD, A RNES TRØMFurther Mathematics for Economics AnalysisPrentice Hall, 2005.I J OSEF L EYDOLDMathematik für Ökonomen3. Auflage, Oldenbourg Verlag, München, 2003 (in German).Josef Leydold – Mathematical Methods – WS 2020/21Introduction – 2 / 27
Further ExercisesBooks from Schaum’s Outline Series (McGraw Hill) offer many exampleproblems with detailed explanations. In particular:I S EYMOUR L IPSCHUTZ , M ARC L IPSONLinear Algebra, 4th ed., McGraw Hill, 2009.I R ICHARD B RONSONMatrix Operations, 2nd ed., McGraw Hill, 2011.I E LLIOT M ENDELSONBeginning Calculus, 3rd ed., McGraw Hill, 2003.I ROBERT W REDE , M URRAY R. S PIEGELAdvanced Calculus, 3rd ed., McGraw Hill, 2010.I E LLIOTT M ENDELSON3,000 Solved Problems in Calculus, McGraw Hill, 1988.Josef Leydold – Mathematical Methods – WS 2020/21Introduction – 3 / 27
Über die mathematische MethodeMan kann also gar nicht prinzipieller Gegner dermathematischen Denkformen sein, sonst müßte man dasDenken auf diesem Gebiete überhaupt aufgeben. Was manmeint, wenn man die mathematische Methode ablehnt, istvielmehr die höhere Mathematik. Man hilft sich, wo es absolutnötig ist, lieber mit schematischen Darstellungen undähnlichen primitiven Behelfen, als mit der angemessenenMethode.Das ist nun aber natürlich unzulässig.Joseph Schumpeter (1906)Über die mathematische Methode der theoretischen Ökonomie, Zeitschrift fürVolkswirtschaft, Sozialpolitik und Verwaltung Bd. 15, S. 30–49 (1906).Josef Leydold – Mathematical Methods – WS 2020/21Introduction – 4 / 27
About the Mathematical MethodOne cannot be an opponent of mathematical forms of thoughtas a matter of principle, since otherwise one has to stopthinking in this field at all. What one means, if someonerefuses the mathematical method, is in fact highermathematics. One uses a schematic representation or otherprimitive makeshift methods where absolutely required ratherthan the appropriate method.However, this is of course not allowed.Joseph Schumpeter (1906)Über die mathematische Methode der theoretischen Ökonomie, Zeitschrift fürVolkswirtschaft, Sozialpolitik und Verwaltung Bd. 15, S. 30–49 (1906).Translation by JL.Josef Leydold – Mathematical Methods – WS 2020/21Introduction – 5 / 27
Static Analysis of EquilibriaI At which price do we have market equilibrium?Find a price where demand and supply function coincide.I Which amounts of goods have to be produced in a nationaleconomy such that consumers’ needs are satisfied?Find the inverse of the matrix in a Leontief input-output model.I How can a consumer optimize his or her utility?Find the absolute maximum of a utility function.I What is the optimal production program for a company?Find the absolute maximum of a revenue function.Josef Leydold – Mathematical Methods – WS 2020/21Introduction – 6 / 27
Comparative-Statistic AnalysisI When market equilibrium is distorted, what happens to the price?Determine the derivative of the price as a function of time.I What is the marginal production vector when demand changes in aLeontief model?Compute the derivative of a vector-valued function.I How does the optimal utility of a consumer change, if income orprices change?Compute the derivative of the maximal utility w.r.t. exogenousparameters.Josef Leydold – Mathematical Methods – WS 2020/21Introduction – 7 / 27
Dynamic AnalysisI Assume we know the rate of change of a price w.r.t. time.How does the price evolve?Solve a difference equation or differential equation, resp.I Which political program optimizes economic growth of a state?Determine the parameters of a differential equation, such that theterminal point of a solution curve is maximal.I What is the optimal investment and consumption strategy of aconsumer who wants to maximize her intertemporal utility?Determine the rate of savings (as a function of time) whichmaximizes the sum of discounted consumption.Josef Leydold – Mathematical Methods – WS 2020/21Introduction – 8 / 27
Learning Outcomes – Basic ConceptsI Linear Algebra:matrix and vector · matrix algebra · vector space · rank and lineardependency · inverse matrix · determinant · eigenvalues ·quadratic form · definiteness and principle minorsI Univariate Analysis:function · graph · one-to-one and onto · limit · continuity ·differential quotient and derivative · monotonicity · convex andconcaveI Multivariate Analysis:partial derivative · gradient and Jacobian matrix · total differential ·implicit and inverse function · Hessian matrix · Taylor seriesJosef Leydold – Mathematical Methods – WS 2020/21Introduction – 9 / 27
Learning Outcomes – OptimizationI Static Optimization:local and global extremum · saddle point · convex and concave ·Lagrange function · Kuhn-Tucker conditions · envelope theoremI Dynamic Analysis:integration · differential equation · difference equation · stable andunstable equilibrium point · difference equations · cobweb diagram· control theory · Hamiltonian and transversality conditionJosef Leydold – Mathematical Methods – WS 2020/21Introduction – 10 / 27
CourseI Reading and preparation of new chapters (handouts) in self-study.I Presentation of new concepts by the course instructor by means ofexamples.I Homework problems.I Discussion of students’ results of homework problems.I Final test.Josef Leydold – Mathematical Methods – WS 2020/21Introduction – 11 / 27
Prerequisites Knowledge about fundamental concepts and tools (like terms, sets,equations, sequences, limits, univariate functions, derivatives,integration) is obligatory for this course. These are (should have been)already known from high school and mathematical courses in yourBachelor program.For the case of knowledge gaps we refer to the Bridging CourseMathematics. A link to learning materials of that course can be foundon the web page.Some slides still cover these topics and are marked by symbol in thetitle of the slide.However, we will discuss these slide only on request.Josef Leydold – Mathematical Methods – WS 2020/21Introduction – 12 / 27
Prerequisites – Issues The following problems may cause issues:I Drawing (or sketching) of graphs of functions.I Transform equations into equivalent ones.I Handling inequalities.I Correct handling of fractions.I Calculations with exponents and logarithms.I Obstructive multiplying of factors.I Usage of mathematical notation.Presented “solutions” of such calculation subtasks are surprisinglyoften wrong.Josef Leydold – Mathematical Methods – WS 2020/21Introduction – 13 / 27
Table of Contents – I – PropedeuticsLogic, Sets and MapsLogicSetsBasic Set OperationsMapsSummaryJosef Leydold – Mathematical Methods – WS 2020/21Introduction – 14 / 27
Table of Contents – II – Linear AlgebraMatrix AlgebraPrologMatrixComputations with MatricesVectorsEpilogSummaryLinear EquationsSystem of Linear EquationsGaussian EliminationGauss-Jordan EliminationSummaryVector SpaceVector SpaceRank of a MatrixJosef Leydold – Mathematical Methods – WS 2020/21Introduction – 15 / 27
Table of Contents – II – Linear Algebra / 2Basis and DimensionLinear MapSummaryDeterminantDefinition and PropertiesComputationCramer’s RuleSummaryEigenvaluesEigenvalues and EigenvectorsDiagonalizationQuadratic FormsPrinciple Component AnalysisSummaryJosef Leydold – Mathematical Methods – WS 2020/21Introduction – 16 / 27
Table of Contents – III – AnalysisReal FunctionsReal FunctionsGraph of a FunctionBijectivitySpecial FunctionsElementary FunctionsMultivariate FunctionsIndifference CurvesPathsGeneralized Real FunctionsLimitsSequencesLimit of a SequenceSeriesLimit of a FunctionJosef Leydold – Mathematical Methods – WS 2020/21Introduction – 17 / 27
Table of Contents – III – Analysis / 2ContinuityDerivativesDifferential QuotientDerivativeThe DifferentialElasticityPartial DerivativesGradientDirectional DerivativeTotal DifferentialHessian MatrixJacobian MatrixL’Hôpital’s RuleSummaryInverse and Implicit FunctionsJosef Leydold – Mathematical Methods – WS 2020/21Introduction – 18 / 27
Table of Contents – III – Analysis / 3Inverse FunctionsImplicit FunctionsSummaryTaylor SeriesTaylor SeriesConvergenceCalculations with Taylor SeriesMultivariate FunctionsSummaryIntegrationAntiderivativeRiemann IntegralFundamental Theorem of CalculusImproper IntegralDifferentiation under the Integral SignJosef Leydold – Mathematical Methods – WS 2020/21Introduction – 19 / 27
Table of Contents – III – Analysis / 4Double IntegralSummaryJosef Leydold – Mathematical Methods – WS 2020/21Introduction – 20 / 27
Table of Contents – IV – Static OptimizationConvex and ConcaveMonotone FunctionsConvex SetConvex and Concave FunctionsUnivariate FunctionsMultivariate FunctionsQuasi-Convex and Quasi-ConcaveSummaryExtremaExtremaGlobal ExtremaLocal ExtremaMultivariate FunctionsEnvelope TheoremSummaryJosef Leydold – Mathematical Methods – WS 2020/21Introduction – 21 / 27
Table of Contents – IV – Static Optimization / 2Lagrange FunctionConstraint OptimizationLagrange ApproachMany Variables and ConstraintsGlobal ExtremaEnvelope TheoremSummaryKuhn Tucker ConditionsGraphical SolutionOptimization with Inequality ConstraintsKuhn-Tucker ConditionsKuhn-Tucker TheoremSummaryJosef Leydold – Mathematical Methods – WS 2020/21Introduction – 22 / 27
Table of Contents – V – Dynamic AnalysisDifferential EquationA Simple Growth ModelWhat is a Differential Equation?Simple MethodsSpecial Differential EquationsLinear Differential Equation of Second OrderQualitative AnalysisSummaryDifference EquationWhat is a Difference Equation?Linear Difference Equation of First OrderA Cobweb ModelLinear Difference Equation of Second OrderQualitative AnalysisSummaryJosef Leydold – Mathematical Methods – WS 2020/21Introduction – 23 / 27
Table of Contents – V – Dynamic Analysis / 2Control TheoryThe Standard ProblemSummaryJosef Leydold – Mathematical Methods – WS 2020/21Introduction – 24 / 27
Science TrackI Discuss basics of mathematical reasoning.I Extend our tool box of mathematical methods for staticoptimization and dynamic optimization.I For more information see the corresponding web pages for thecourses Mathematics I and Mathematics II.Josef Leydold – Mathematical Methods – WS 2020/21Introduction – 25 / 27
Computer Algebra System (CAS)Maxima is a so called Computer Algebra System (CAS), i.e., one canIIIIIImanipulate algebraic expressions,solve equations,differentiate and integrate functions symbolically,perform abstract matrix algebra,draw graphs of functions in one or two variables,.wxMaxima is an IDE for this system:http://wxmaxima.sourceforge.net/You find an Introduction to Maxima for Economics on the web page ofthis course.Josef Leydold – Mathematical Methods – WS 2020/21Introduction – 26 / 27
May you do well!Josef Leydold – Mathematical Methods – WS 2020/21Introduction – 27 / 27
Chapter 1Logic, Sets and MapsJosef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 1 / 30
PropositionWe need some elementary knowledge about logic for doingmathematics. The central notion is “proposition”.A proposition is a sentence with iseither true (T) or false (F).IIII“Vienna is located at river Danube.” is a true proposition.“Bill Clinton was president of Austria.” is a false proposition.“19 is a prime number.” is a true proposition.“This statement is false.” is not a proposition.Josef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 2 / 30
Logical ConnectivesWe get compound propositions by connecting (simpler) propositions byusing logical connectives.This is done by means of words “and”, “or”, “not”, or “if . . . then”, knownfrom everyday language.ConnectiveSymbolNamenot P PP QP QP QP QnegationP and QP or Qif P then QP if and only if QJosef Leydold – Mathematical Methods – WS e1 – Logic, Sets and Maps – 3 / 30
Truth TableTruth values of logical connectives.PQ PP QP QP QP QTTFTTTTTFFFTFFFTTFTTFFFTFFTTLet P “ x is divisible by 2” and Q “ x is divisible by 3”.Proposition P Q is true if and only if x is divisible by 2 and 3(i.e., by 6).Josef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 4 / 30
Negation and DisjunctionI Negation P is not the “opposite” of proposition P.Negation of P “all cats are black”is P “Not all cats are black”(And not “all cats are not black” or even “all cats are white”!)I Disjunction P Q is in a non-exclusive sense:P Q is true if and only ifI P is true, orI Q is true, orI both P and Q are true.Josef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 5 / 30
ImplicationThe truth value of implication P Q seems a bit mysterious.Note that P Q does not make any proposition about the truth valueof P or Q!Which of the following propositions is true?I “If Bill Clinton is Austrian citizen, then he can be elected forAustrian president.”I “If Karl (born 1970) is Austrian citizen, then he can be elected forAustrian president.”I “If x is a prime number larger than 2, then x is odd.”Implication P Q is equivalent to P Q:( P Q) ( P Q)Josef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 6 / 30
A Simple Logical ProofWe can derive the truth value of proposition ( P Q) ( P Q) bymeans of a truth table:PQ P( P Q)( P Q)( P Q) ( P Q)TTFTTTTFFFFTFTTTTTFFTTTTThat is, proposition ( P Q) ( P Q) is always trueindependently from the truth values for P and Q.It is a so called tautology.Josef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 7 / 30
TheoremsMathematics consists of propositions of the form: P implies Q,but you never ask whether P is true.(Bertrand Russell)A mathematical statement (theorem, proposition, lemma, corollary ) isa proposition of the form P Q.P is called a sufficient condition for Q.A sufficient condition P guarantees that proposition Q is true. However,Q can be true even if P is false.Q is called a necessary condition for P,Q P.A necessary condition Q must be true to allow P to be true. It does notguarantee that P is true.Necessary conditions often are used to find candidates for validanswers to our problems.Josef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 8 / 30
QuantorsMathematical texts often use the expressions “for all” and “there exists”,resp.In formal notation the following symbols are used:QuantorSymbolfor all !@there exists athere exists exactly onethere does not existsJosef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 9 / 30
Set The notion of set is fundamental in modern mathematics.We use a simple definition from naïve set theory:A set is a collection of distinct objects.An object a of a set A is called an element of the set. We write:a ASets are defined by enumeratingor a description of their elements within curly brackets . . . .A {1, 2, 3, 4, 5, 6}B { x x is an integer divisible by 2}Josef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 10 / 30
Important Sets SymbolDescription NZQR[ a, b ]( a, b )[ a, b )Cempty set sometimes: {}natural numbers {1, 2, 3, . . .}a{. . . , 3, 2, 1, 0, 1, 2, 3, . . .}rational numbers { nk k, n Z, n 6 0}integersreal numbers{ x R a x b}open interval{ x R a x b}half-open interval { x R a x b}complex numbers { a bi a, b R, i2 1}closed intervalaalso: ] a, b [Josef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 11 / 30
Venn Diagram We assume that all sets are subsets of some universal superset Ω.Sets can be represented by Venn diagrams where Ω is a rectangleand sets are depicted as circles or ovals.AΩJosef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 12 / 30
Subset and Superset Set A is a subset of B, A B , if all elements of A also belong to B,x A x B.BA BΩVice versa, B is then called a superset of A, B A .Set A is a proper subset of B, A B(or: A B),if A B and A 6 B.Josef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 13 / 30
Basic Set Operations SymbolDefinitionNameA BA BA\BA{ x x A and x B}{ x x A or x B}{ x x A and x 6 B}Ω\Aintersectionaunionset-theoretic differenceacomplementalso: A BTwo sets A and B are disjoint if A B .Josef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 14 / 30
Basic Set Operations BABAA BA BΩABA\BΩJosef Leydold – Mathematical Methods – WS 2020/21ΩAAΩ1 – Logic, Sets and Maps – 15 / 30
Rules for Basic Operations RuleNameA A A A AIdempotenceA A and A Identity( A B) C A ( B C ) and( A B) C A ( B C )AssociativityA B B ACommutativityandA B B AA ( B C ) ( A B) ( A C ) andA ( B C ) ( A B) ( A C )A A ΩandA A Josef Leydold – Mathematical Methods – WS 2020/21andDistributivityA A1 – Logic, Sets and Maps – 16 / 30
De Morgan’s Law ( A B) A BABand( A B) A BAΩBΩA union B complemented is theequivalent of A complementedintersected with B complemented.Josef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 17 / 30
Cartesian Product The setA B {( x, y) x A, y B}is called the Cartesian product of sets A and B.Given two sets A and B the Cartesian product A B is the set of allunique ordered pairs where the first element is from set A and thesecond element is from set B.In general we haveA B 6 B A.Josef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 18 / 30
Cartesian Product The Cartesian product of A {0, 1} and B {2, 3, 4} isA B {(0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4)}.A B01Josef Leydold – Mathematical Methods – WS 2020/21234(0, 2) (0, 3) (0, 4)(1, 2) (1, 3) (1, 4)1 – Logic, Sets and Maps – 19 / 30
Cartesian Product The Cartesian product of A [2, 4] and B [1, 3] isA B {( x, y) x [2, 4] and y [1, 3]}.3B [1, 3]A B2101234A [2, 4]Josef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 20 / 30
Map A map (or mapping) f is defined by(i) a domain D f ,(ii) a codomain (target set) W f and(iii) a rule, that maps each element of D to exactly one element of W .f : D W,IIIIx 7 y f ( x )x is called the independent variable, y the dependent variable.y is the image of x, x is the preimage of y.f ( x ) is the function term, x is called the argument of f .f ( D ) {y W : y f ( x ) for some x D }is the image (or range) of f .Other names: function, transformationJosef Leydold – Mathematical Methods – WS 2020/211 – Logic, Sets and Maps – 21 / 30
Injective · Surjective · Bijective Each argument has exactly one image.Each y W , however, may have any number of preimages.Thus we can characterize maps by their possible number of preimages.I A map f is called one-to-one (or injective), if each element in thecodomain has at most one preimage.I It is called onto (or surjective), if each element in the codomainhas at least one preimage.I It is called bijective, if it is both one-to-one and onto, i.e., if eachelement in the codomain has exactly one preimage.Injections have
Computer Algebra System (CAS) Maxima is a so called Computer Algebra System (CAS), i.e., one can Imanipulate algebraic expressions, . Mathematical Methods – WS 2020/21Introduction –26 / 27. May you do well! Josef Leydold – Mathematical Meth
2. Dowling E.T, Introduction to Mathematical Economics, 2nd Edition, Schaum’s Series, McGraw-Hill, New York, 2003(E TD) 3. R.G.D Allen, Mathematical Economics 4. Mehta and Madnani -Mathematics for Economics 5. Joshi and Agarwal-Mathematics for Economics 6. Taro Yamane-Mathematics for Economics 7. Damodar N.Gujarati, Basic Econometrics, McGraw .
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Lecture notes based mostly on Chiang and Wainwright, Fundamental Methods of Mathematical Economics. 1 Mathematical economics Why describe the world with mathematical models, rather than use verbal theory and logic? After all, this was the state of economics until not too long ago (say, 1950s
Managerial Economics Klein Mathematical Methods for Economics Krugman/Obstfeld/Melitz International Economics: Theory & Policy* Laidler The Demand for Money Lynn Economic Development: Theory and Practice for a Divided World Miller Economics Today* Miller/Benjamin The Economics of Macro Issues Miller/Benjamin/North The Economics of Public Issues .
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Labor Economics, Public Economics, Applied Econometrics, and Economics of Education Tia Hilmer, Professor O ce: NH-317, Email: chilmer@sdsu.edu Econometrics, Natural Resources, Environmental Economics . Mathematical Economics (3) Prerequisite: Mathematics 124 or 150. Recommended: Economics 320 or 321. .
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