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MATHEMATICS FOR ECONOMISTSpdf电子书版本下载
- CARL P.SIMON LAWRENCE BLUME 著
- 出版社: W.W.NORTON & COMPANY
- ISBN:9780393957334
- 出版时间:1994
- 标注页数:930页
- 文件大小:111MB
- 文件页数:955页
- 主题词:
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图书目录
PARTⅠ Introduction 3
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
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
Deffnition and Examples of the Inverse of a Function 75
The Derivative of the Inverse Function 79
The Derivative of x m/n 80
5 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
PARTⅡ Linear Algebra 107
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
7.2 ELEMENTARY ROW OPERATIONS 129
7.3 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
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
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 Rn 247
Dimension 249
11.4 EPILOGUE 249
PARTⅢ Calculus of Several Variables 253
12 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
13 Functions of Several Variables 273
13.1 FUNCTIONS BETWEEN EUCLIDEAN SPACES 273
Functions from Rn to R 274
Functions from Rk to Rm 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 Rl 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
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
15 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
15.2 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 (optional) 364
15.6 APPLICATION:SIMPSON’S PARADOX 368
PARTⅣ Optimization 375
16 Quadratic Forms and Definite Matrices 375
16.1 QUADRATIC FORMS 375
16.2 DEFINITENESS OF QUADRATIC FORMS 376
Definite Symmetric Matrices 379
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 × 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
17 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
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
2 0 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
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 517
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
PART Ⅴ Eigenvalues and Dynamics 579
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
2 × 2 Nondiagonalizable Matrices 601
3 × 3 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
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 711
25.8 APPENDIX:LINEARIZATION 715
PART Ⅵ Advanced Linear Algebra 719
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
27 Subspaces Attached to a Matrix 750
27.1 VECTOR SPACES AND SUBSPACES 750
Rn as a Vector Space 750
Subspaces of Rn 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
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 803
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 Rn 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
Second Order Necessary Conditions for Optimization 840
30.5 CONSTRAINED OPTIMIZATION 841
PARTⅧ Appendices 847
A1 Sets,Numbers,and Proofs 847
A1.1 SETS 847
Vocabulary of Sets 847
Operations with Sets 847
A1.2 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
Denitions 877
Arithmetic Operations 877
A3.2 SOLUTIONS OF POLYNOMIAL EQUATIONS 878
A3.3 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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