Verfügbarkeit: Mathematics for economists

Mathematics for economists:

Gespeichert in:

Bibliographische Detailangaben
Hauptverfasser:	Simon, Carl P. 1945- (VerfasserIn), Blume, Lawrence E. (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	New York [u.a.] Norton 1994
Ausgabe:	1. ed., international student ed.
Schlagworte:	Wirtschaftsmathematik Lehrbuch
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	Hier auch später erschienene, unveränderte Nachdrucke
Beschreibung:	XXIV, 930 S. graph. Darst.
ISBN:	9780393117523 0393957330

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Datensatz im Suchindex

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adam_text	Contents Preface xxi part ι Introduction 1 Introduction 3 1.1 MATHEMATICS IN ECONOMIC THEORY 3 1.2 MODELS OF CONSUMER CHOICE 5 Two-Dimensional Model of Consumer Choice 5 Multidimensional Model of Consumer Choice 9 2 One-Variable Calculus: Foundations 10 2.1 FUNCTIONS ON R1 10 Vocabulary of Functions 10 Polynomials 11 Graphs 12 Increasing and Decreasing Functions 12 Domain 14 Interval Notation 15 2.2 LINEAR FUNCTIONS 16 The Slope of a Line in the Plane 16 The Equation of a Line 19 Polynomials of Degree One Have Linear Graphs 19 Interpreting the Slope of a Linear Function 20 2.3 THE SLOPE OF NONLINEAR FUNCTIONS 22 2.4 COMPUTING DERIVATIVES 25 Rules for Computing Derivatives 27 VI CONTENTS 2.5 DIFFERENTIABILITY AND CONTINUITY 29 A Nondifferentiable Function 30 Continuous Functions 31 Continuously Differentiable Functions 32 2.6 HIGHER-ORDER DERIVATIVES 33 2.7 APPROXIMATION BY DIFFERENTIALS 34 3 One-Variable Calculus: Applications 39 3.1 USING THE FIRST DERIVATIVE FOR GRAPHING 39 Positive Derivative Implies Increasing Function 39 Using First Derivatives to Sketch Graphs 41 3.2 SECOND DERIVATIVES AND CONVEXITY 43 3.3 GRAPHING RATIONAL FUNCTIONS 47 Hints for Graphing 48 3.4 TAILS AND HORIZONTAL ASYMPTOTES 48 Tails of Polynomials 48 Horizontal Asymptotes of Rational Functions 49 3.5 MAXIMA AND MINIMA 51 Local Maxima and Minima on the Boundary and in the Interior 51 Second Order Conditions 53 Global Maxima and Minima 55 Functions with Only One Critical Point 55 Functions with Nowhere-Zero Second Derivatives 56 Functions with No Global Max or Min 56 Functions Whose Domains Are Closed Finite Intervals 56 3.6 APPLICATIONS TO ECONOMICS 58 Production Functions 58 Cost Functions 59 Revenue and Profit Functions 62 Demand Functions and Elasticity 64 4 One-Variable Calculus: Chain Rule 70 4.1 COMPOSITE FUNCTIONS AND THE CHAIN RULE 70 Composite Functions 70 Differentiating Composite Functions: The Chain Rule 72 4.2 INVERSE FUNCTIONS AND THEIR DERIVATIVES 75 Definition and Examples of the Inverse of a Function 75 The Derivative of the Inverse Function 79 The Derivative of .ť 7 80 CONTENTS VII Exponents and Logarithms 82 5.1 EXPONENTIAL FUNCTIONS 82 5.2 THE NUMBER e 85 5.3 LOGARITHMS 88 Base 10 Logarithms 88 Base e Logarithms 90 5.4 PROPERTIES OF EXP AND LOG 91 5.5 DERIVATIVES OF EXP AND LOG 93 5.6 APPLICATIONS 97 Present Value 97 Annuities 98 Optimal Holding Time 99 Logarithmic Derivative 100 part m Linear Algebra 6 Introduction to Linear Algebra 107 6.1 LINEAR SYSTEMS 107 6.2 EXAMPLES OF LINEAR MODELS 108 Example 1: Tax Benefits of Charitable Contributions 108 Example 2: Linear Models of Production 110 Example 3: Markov Models of Employment 113 Example 4: IS-LM Analysis 115 Example 5: Investment and Arbitrage 117 7 Systems of Linear Equations 122 7.1 GAUSSIAN AND GAUSS-JORDAN ELIMINATION 122 Substitution 123 Elimination of Variables 125 72 ELEMENTARY ROW OPERATIONS 129 73 SYSTEMS WITH MANY OR NO SOLUTIONS 134 7.4 RANK —THE FUNDAMENTAL CRITERION 142 Application to Portfolio Theory 147 7.5 THE LINEAR IMPLICIT FUNCTION THEOREM 150 VU! CONTENTS 8 Matrix Algebra 153 8.1 MATRIX ALGEBRA 153 Addition 153 Subtraction 154 Scalar Multiplication 155 Matrix Multiplication 155 Laws of Matrix Algebra 156 Transpose 157 Systems of Equations in Matrix Form 158 8.2 SPECIAL KINDS OF MATRICES 160 8.3 ELEMENTARY MATRICES 162 8.4 ALGEBRA OF SQUARE MATRICES 165 8.5 INPUT-OUTPUT MATRICES 174 Proof of Theorem 8.13 178 8.6 PARTITIONED MATRICES (optional) 180 8.7 DECOMPOSING MATRICES (optional) 183 Mathematical Induction 185 Including Row Interchanges 185 9 Determinants: An Overview 188 9.1 THE DETERMINANT OF A MATRIX 189 Defining the Determinant 189 Computing the Determinant 191 Main Property of the Determinant 192 9.2 USES OF THE DETERMINANT 194 9.3 IS-LM ANALYSIS VIA CRAMER S RULE 197 10 Euclidean Spaces 199 10.1 POINTS AND VECTORS IN EUCLIDEAN SPACE 199 The Real Line 199 The Plane 199 Three Dimensions and More 201 10.2 VECTORS 202 10.3 THE ALGEBRA OF VECTORS 205 Addition and Subtraction 205 Scalar Multiplication 207 10.4 LENGTH AND INNER PRODUCT IN Rn 209 Length and Distance 209 The Inner Product 213 CONTENTS IX 10.5 LINES 222 10.6 PLANES 226 Parametric Equations 226 Nonparametric Equations 228 Hyperplanes 230 10.7 ECONOMIC APPLICATIONS 232 Budget Sets in Commodity Space 232 Input Space 233 Probability Simplex 233 The Investment Model 234 IS-LM Analysis 234 11 Linear Independence 237 11.1 LINEAR INDEPENDENCE 237 Definition 238 Checking Linear Independence 241 11.2 SPANNING SETS 244 11.3 BASIS AND DIMENSION IN R 247 Dimension 249 11.4 EPILOGUE 249 part їм Calculus of Several Variables 1 2 Limits and Open Sets 253 12.1 SEQUENCES OF REAL NUMBERS 253 Definition 253 Limit of a Sequence 254 Algebraic Properties of Limits 256 12.2 SEQUENCES IN Rm 260 12.3 OPEN SETS 264 Interior of a Set 267 12.4 CLOSED SETS 267 Closure of a Set 268 Boundary of a Set 269 12.5 COMPACT SETS 270 12.6 EPILOGUE 272 CONTENTS 13 Functions of Several Variables 273 13.1 FUNCTIONS BETWEEN EUCLIDEAN SPACES 273 Functions from Rn to R 274 Functions from Rk to R™ 275 13.2 GEOMETRIC REPRESENTATION OF FUNCTIONS 277 Graphs of Functions of Two Variables 277 Level Curves 280 Drawing Graphs from Level Sets 281 Planar Level Sets in Economics 282 Representing Functions from Rk to R1 for k > 2 283 Images of Functions from R1 to Rm 285 13.3 SPECIAL KINDS OF FUNCTIONS 287 Linear Functions on Rk 287 Quadratic Forms 289 Matrix Representation of Quadratic Forms 290 Polynomials 291 13.4 CONTINUOUS FUNCTIONS 293 13.5 VOCABULARY OF FUNCTIONS 295 Onto Functions and One-to-One Functions 297 Inverse Functions 297 Composition of Functions 298 14 Calculus of Several Variables 300 14.1 DEFINITIONS AND EXAMPLES 300 14.2 ECONOMIC INTERPRETATION 302 Marginal Products 302 Elasticity 304 14.3 GEOMETRIC INTERPRETATION 305 14.4 THE TOTAL DERIVATIVE 307 Geometric Interpretation 308 Linear Approximation 310 Functions of More than Two Variables 311 14.5 THE CHAIN RULE 313 Curves 313 Tangent Vector to a Curve 314 Differentiating along a Curve: The Chain Rule 316 14.6 DIRECTIONAL DERIVATIVES AND GRADIENTS 319 Directional Derivatives 319 The Gradient Vector 320 CONTENTS XI 14.7 EXPLICIT FUNCTIONS FROM Rn TO Rm 323 Approximation by Differentials 324 The Chain Rule 326 14.8 HIGHER-ORDER DERIVATIVES 328 Continuously Differentiable Functions 328 Second Order Derivatives and Hessians 329 Young s Theorem 330 Higher-Order Derivatives 331 An Economic Application 331 14.9 Epilogue 333 1 5 Implicit Functions and Their Derivatives 334 15.1 IMPLICIT FUNCTIONS 334 Examples 334 The Implicit Function Theorem for R2 337 Several Exogenous Variables in an Implicit Function 341 1 5.2 LEVEL CURVES AND THEIR TANGENTS 342 Geometric Interpretation of the Implicit Function Theorem 342 Proof Sketch 344 Relationship to the Gradient 345 Tangent to the Level Set Using Differentials 347 Level Sets of Functions of Several Variables 348 15.3 SYSTEMS OF IMPLICIT FUNCTIONS 350 Linear Systems 351 Nonlinear Systems 353 15.4 APPLICATION: COMPARATIVE STATICS 360 15.5 THE INVERSE FUNCTION THEOREM (opţionali 364 15.6 APPLICATION: SIMPSON S PARADOX 368 part iv Optimization 1 6 Quadratic Forms and Definite Matrices 375 16.1 QUADRATIC FORMS 375 16.2 DEFINITENESS OF QUADRATIC FORMS 376 Definite Symmetric Matrices 379 XII CONTENTS Application: Second Order Conditions and Convexity 379 Application: Conic Sections 380 Principal Minors of a Matrix 381 The Definiteness of Diagonal Matrices 383 The Definiteness of 2 X 2 Matrices 384 16.3 LINEAR CONSTRAINTS AND BORDERED MATRICES 386 Definiteness and Optimality 386 One Constraint 390 Other Approaches 391 16.4 APPENDIX 393 1 7 Unconstrained Optimization 396 17.1 DEFINITIONS 396 17.2 FIRST ORDER CONDITIONS 397 17.3 SECOND ORDER CONDITIONS 398 Sufficient Conditions 398 Necessary Conditions 401 17.4 GLOBAL MAXIMA AND MINIMA 402 Global Maxima of Concave Functions 403 17.5 ECONOMIC APPLICATIONS 404 Profit-Maximizing Firm 405 Discriminating Monopolist 405 Least Squares Analysis 407 18 Constrained Optimization I: First Order Conditions 411 18.1 EXAMPLES 412 18.2 EQUALITY CONSTRAINTS 413 Two Variables and One Equality Constraint 413 Several Equality Constraints 420 18.3 INEQUALITY CONSTRAINTS 424 One Inequality Constraint 424 Several Inequality Constraints 430 18.4 MIXED CONSTRAINTS 434 18.5 CONSTRAINED MINIMIZATION PROBLEMS 436 18.6 KUHN-TUCKER FORMULATION 439 CONTENTS XIII 18.7 EXAMPLES AND APPLICATIONS 442 Application: A Sales-Maximizing Firm with Advertising 442 Application: The Averch-Johnson Effect 443 One More Worked Example 445 19 Constrained Optimization II 448 19.1 THE MEANING OF THE MULTIPLIER 448 One Equality Constraint 449 Several Equality Constraints 450 Inequality Constraints 451 Interpreting the Multiplier 452 19.2 ENVELOPE THEOREMS 453 Unconstrained Problems 453 Constrained Problems 455 19.3 SECOND ORDER CONDITIONS 457 Constrained Maximization Problems 459 Minimization Problems 463 Inequality Constraints 466 Alternative Approaches to the Bordered Hessian Condition 467 Necessary Second Order Conditions 468 19.4 SMOOTH DEPENDENCE ON THE PARAMETERS 469 19.5 CONSTRAINT QUALIFICATIONS 472 19.6 PROOFS OF FIRST ORDER CONDITIONS 478 Proof of Theorems 18.1 and 18.2: Equality Constraints 478 Proof of Theorems 18.3 and 18.4: Inequality Constraints 480 20 Homogeneous and Homothetic Functions 483 20.1 HOMOGENEOUS FUNCTIONS 483 Definition and Examples 483 Homogeneous Functions in Economics 485 Properties of Homogeneous Functions 487 A Calculus Criterion for Homogeneity 491 Economic Applications of Euler s Theorem 492 20.2 HOMOGENIZING A FUNCTION 493 Economic Applications of Homogenization 495 20.3 CARDINAL VERSUS ORDINAL UTILITY 496 XIV CONTENTS 20.4 HOMOTHETIC FUNCTIONS 500 Motivation and Definition 500 Characterizing Homothetic Functions 501 21 Concave and Quasiconcave Functions 505 21.1 CONCAVE AND CONVEX FUNCTIONS 505 Calculus Criteria for Concavity 509 21.2 PROPERTIES OF CONCAVE FUNCTIONS 51 7 Concave Functions in Economics 521 21.3 QUASICONCAVE AND QUASICONVEX FUNCTIONS 522 Calculus Criteria 525 21.4 PSEUDOCONCAVE FUNCTIONS 527 21.5 CONCAVE PROGRAMMING 532 Unconstrained Problems 532 Constrained Problems 532 Saddle Point Approach 534 21.6 APPENDIX 537 Proof of the Sufficiency Test of Theorem 21.14 537 Proof of Theorem 21.15 538 Proof of Theorem 21.17 540 Proof of Theorem 21.20 541 22 Economic Applications 544 22.1 UTILITY AND DEMAND 544 Utility Maximization 544 The Demand Function 547 The Indirect Utility Function 551 The Expenditure and Compensated Demand Functions 552 The Slutsky Equation 555 22.2 ECONOMIC APPLICATION: PROFIT AND COST 557 The Profit-Maximizing Firm 557 The Cost Function 560 22.3 PARETO OPTIMA 565 Necessary Conditions for a Pareto Optimum 566 Sufficient Conditions for a Pareto Optimum 567 22.4 THE FUNDAMENTAL WELFARE THEOREMS 569 Competitive Equilibrium 572 Fundamental Theorems of Welfare Economics 573 CONTENTS XV part v Eigenvalues and Dynamics 23 Eigenvalues and Eigenvectors 579 23.1 DEFINITIONS AND EXAMPLES 579 23.2 SOLVING LINEAR DIFFERENCE EQUATIONS 585 One-Dimensional Equations 585 Two-Dimensional Systems: An Example 586 Conic Sections 587 The Leslie Population Model 588 Abstract Two-Dimensional Systems 590 k-Dimensional Systems 591 An Alternative Approach: The Powers of a Matrix 594 Stability of Equilibria 596 23.3 PROPERTIES OF EIGENVALUES 597 Trace as Sum of the Eigenvalues 599 23.4 REPEATED EIGENVALUES 601 2x2 Nondiagonalizable Matrices 601 3x3 Nondiagonalizable Matrices 604 Solving Nondiagonalizable Difference Equations 606 23.5 COMPLEX EIGENVALUES AND EIGENVECTORS 609 Diagonalizing Matrices with Complex Eigenvalues 609 Linear Difference Equations with Complex Eigenvalues 611 Higher Dimensions 614 23.6 MARKOV PROCESSES 615 23.7 SYMMETRIC MATRICES 620 23.8 DEFINITENESS OF QUADRATIC FORMS 626 23.9 APPENDIX 629 Proof of Theorem 23.5 629 Proof of Theorem 23.9 630 24 Ordinary Differential Equations: Scalar Equations 633 24.1 DEFINITION AND EXAMPLES 633 24.2 EXPLICIT SOLUTIONS 639 Linear First Order Equations 639 Separable Equations 641 24.3 LINEAR SECOND ORDER EQUATIONS 647 Introduction 647 XVI CONTENTS Real and Unequal Roots of the Characteristic Equation 648 Real and Equal Roots of the Characteristic Equation 650 Complex Roots of the Characteristic Equation 651 The Motion of a Spring 653 Nonhomogeneous Second Order Equations 654 24.4 EXISTENCE OF SOLUTIONS 657 The Fundamental Existence and Uniqueness Theorem 657 Direction Fields 659 24.5 PHASE PORTRAITS AND EQUILIBRIA ON R1 666 Drawing Phase Portraits 666 Stability of Equilibria on the Line 668 24.6 APPENDIX: APPLICATIONS 670 Indirect Money Metric Utility Functions 671 Converse of Euler s Theorem 672 25 Ordinary Differential Equations: Systems of Equations 674 25.1 PLANAR SYSTEMS: AN INTRODUCTION 674 Coupled Systems of Differential Equations 674 Vocabulary 676 Existence and Uniqueness 677 25.2 LINEAR SYSTEMS VIA EIGENVALUES 678 Distinct Real Eigenvalues 678 Complex Eigenvalues 680 Multiple Real Eigenvalues 681 25.3 SOLVING LINEAR SYSTEMS BY SUBSTITUTION 682 25.4 STEADY STATES AND THEIR STABILITY 684 Stability of Linear Systems via Eigenvalues 686 Stability of Nonlinear Systems 687 25.5 PHASE PORTRAITS OF PLANAR SYSTEMS 689 Vector Fields 689 Phase Portraits: Linear Systems 692 Phase Portraits: Nonlinear Systems 694 25.6 FIRST INTEGRALS 703 The Predator-Prey System 705 Conservative Mechanical Systems 707 25.7 LIAPUNOV FUNCTIONS 7П 25.8 APPENDIX: LINEARIZATION 715 CONTENTS XVII part vi Advanced Linear Algebra 26 Determinants: The Details 719 26.1 DEFINITIONS OF THE DETERMINANT 719 26.2 PROPERTIES OF THE DETERMINANT 726 26.3 USING DETERMINANTS 735 The Adjoint Matrix 736 26.4 ECONOMIC APPLICATIONS 739 Supply and Demand 739 26.5 APPENDIX 743 Proof of Theorem 26.1 743 Proof of Theorem 26.9 746 Other Approaches to the Determinant 747 2 7 Subspaces Attached to a Matrix 750 27.1 VECTOR SPACES AND SUBSPACES 750 R as a Vector Space 750 Subspaces of R 751 27.2 BASIS AND DIMENSION OF A PROPER SUBSPACE 755 27.3 ROW SPACE 757 27.4 COLUMN SPACE 760 Dimension of the Column Space of A 760 The Role of the Column Space 763 27.5 NULLSPACE 765 Affine Subspaces 765 Fundamental Theorem of Linear Algebra 767 Conclusion 770 27.6 ABSTRACT VECTOR SPACES 771 27.7 APPENDIX 774 Proof of Theorem 27.5 774 Proof of Theorem 27.10 775 28 Applications of Linear Independence 779 28.1 GEOMETRY OF SYSTEMS OF EQUATIONS 779 Two Equations in Two Unknowns 779 Two Equations in Three Unknowns 780 Three Equations in Three Unknowns 782 XVIII CONTEXTS 28.2 PORTFOLIO ANALYSIS 783 28.3 VOTING PARADOXES 784 Three Alternatives 785 Four Alternatives 788 Consequences of the Existence of Cycles 789 Other Voting Paradoxes 790 Rankings of the Quality of Firms 790 28.4 ACTIVITY ANALYSIS: FEASIBILITY 791 Activity Analysis 791 Simple Linear Models and Productive Matrices 793 28.5 ACTIVITY ANALYSIS: EFFICIENCY 796 Leontief Models 796 part vii Advanced Analysis 29 Limits and Compact Sets 803 29.1 CAUCHY SEQUENCES 803 29.2 COMPACT SETS 807 29.3 CONNECTED SETS 809 29.4 ALTERNATIVE NORMS 811 Three Norms on R 811 Equivalent Norms 813 Norms on Function Spaces 815 29.5 APPENDIX 816 Finite Covering Property 816 Heine-Borel Theorem 817 Summary 820 30 Calculus of Several Variables II 822 30.1 WEIERSTRASS S AND MEAN VALUE THEOREMS 822 Existence of Global Maxima on Compact Sets 822 Rolle s Theorem and the Mean Value Theorem 824 30.2 TAYLOR POLYNOMIALS ON R1 827 Functions of One Variable 827 30.3 TAYLOR POLYNOMIALS IN Rn 832 30.4 SECOND ORDER OPTIMIZATION CONDITIONS 836 Second Order Sufficient Conditions for Optimization 836 Indefinite Hessian 839 CONTENTS XIX Second Order Necessary Conditions for Optimization 840 30.5 CONSTRAINED OPTIMIZATION 841 part viii Appendices A1 Sets, Numbers, and Proofs 847 A1.1 SETS 847 Vocabulary of Sets 847 Operations with Sets 847 Al .2 NUMBERS 848 Vocabulary 848 Properties of Addition and Multiplication 849 Least Upper Bound Property 850 A1.3 PROOFS 851 Direct Proofs 851 Converse and Contrapositive 853 Indirect Proofs 854 Mathematical Induction 855 A2 Trigonometric Functions 859 A2.1 DEFINITIONS OF THE TRIG FUNCTIONS 859 A2.2 GRAPHING TRIG FUNCTIONS 863 A2.3 THE PYTHAGOREAN THEOREM 865 A2.4 EVALUATING TRIGONOMETRIC FUNCTIONS 866 A2.5 MULTIANGLE FORMULAS 868 A2.6 FUNCTIONS OF REAL NUMBERS 868 A2.7 CALCULUS WITH TRIG FUNCTIONS 870 A2.8 TAYLOR SERIES 872 A2.9 PROOF OF THEOREM A2.3 873 A3 Complex Numbers 876 A3.1 BACKGROUND 876 Definitions 877 Arithmetic Operations 877 A3.2 SOLUTIONS OF POLYNOMIAL EQUATIONS 878 XX CONTENTS АЗ.З GEOMETRIC REPRESENTATION 879 A3.4 COMPLEX NUMBERS AS EXPONENTS 882 A3.5 DIFFERENCE EQUATIONS 884 A4 Integral Calculus 887 A4.1 ANTIDERIVATIVES 887 Integration by Parts 888 A4.2 THE FUNDAMENTAL THEOREM OF CALCULUS 889 A4.3 APPLICATIONS 890 Area under a Graph 890 Consumer Surplus 891 Present Value of a Flow 892 A5 Introduction to Probability 894 A5.1 PROBABILITY OF AN EVENT 894 A5.2 EXPECTATION AND VARIANCE 895 A5.3 CONTINUOUS RANDOM VARIABLES 896 A6 Selected Answers 899 Index 921
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spelling	Simon, Carl P. 1945- Verfasser (DE-588)170767981 aut Mathematics for economists Carl P. Simon and Lawrence Blume 1. ed., international student ed. New York [u.a.] Norton 1994 XXIV, 930 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Wirtschaftsmathematik (DE-588)4066472-7 gnd rswk-swf Wirtschaftsmathematik 1\p (DE-588)4123623-3 Lehrbuch gnd-content Wirtschaftsmathematik (DE-588)4066472-7 s DE-604 Blume, Lawrence E. Verfasser (DE-588)170098389 aut Digitalisierung UB Bamberg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024970271&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk
spellingShingle	Simon, Carl P. 1945- Blume, Lawrence E. Mathematics for economists Wirtschaftsmathematik (DE-588)4066472-7 gnd
subject_GND	(DE-588)4066472-7 (DE-588)4123623-3
title	Mathematics for economists
title_auth	Mathematics for economists
title_exact_search	Mathematics for economists
title_full	Mathematics for economists Carl P. Simon and Lawrence Blume
title_fullStr	Mathematics for economists Carl P. Simon and Lawrence Blume
title_full_unstemmed	Mathematics for economists Carl P. Simon and Lawrence Blume
title_short	Mathematics for economists
title_sort	mathematics for economists
topic	Wirtschaftsmathematik (DE-588)4066472-7 gnd
topic_facet	Wirtschaftsmathematik Lehrbuch
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024970271&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT simoncarlp mathematicsforeconomists AT blumelawrencee mathematicsforeconomists

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