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Mathematics and Mathematica for Economists,9781577180340
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Mathematics and Mathematica for Economists

by ;
ISBN13:

9781577180340

ISBN10:
1577180348
Format:
Hardcover
Pub. Date:
11/1/1999
Publisher(s):
Wiley-Blackwell
List Price: $121.95
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Summary

Mathematics and Mathematica for Economists integrates the computer algebra system, Mathematica, with an array of mathematical topics that are used to analyze systems that arise in business and economics.

Table of Contents

Preface xiii
1 Introduction to Mathematica
1(37)
1.1 Doing Arithmetic with Mathematica
1(5)
1.1.1 The In and Out tags
2(1)
1.1.2 Numbers, symbols, and assignment
3(3)
1.2 Functions
6(10)
1.2.1 Built-in Mathematica functions
7(3)
1.2.2 User-defined functions
10(3)
1.2.3 Algebraic manipulations
13(2)
1.2.4 Lists and tables
15(1)
1.3 Algebra and Calculus
16(9)
1.3.1 Sums and products
16(3)
1.3.2 Solving equations
19(3)
1.3.3 Calculus
22(3)
1.4 Graphics in Mathematica
25(10)
1.4.1 Plot
25(2)
1.4.2 Plot3D
27(2)
1.4.3 ParametricPlot and ParametricPlot3D
29(2)
1.4.4 ContourPlot
31(1)
1.4.5 Graphics primitives and the Show function
32(3)
1.5 Modules and Packages
35(3)
2 A Review of Calculus
38(43)
2.1 Limits and Continuity of Functions
38(13)
2.1.1 Limits
38(10)
2.1.2 Continuity
48(3)
2.2 Differentiation of Functions
51(18)
2.2.1 Derivatives
51(5)
2.2.2 General rules of differentiation
56(6)
2.2.3 Higher-order derivatives
62(2)
2.2.4 Relative maxima and minima
64(5)
2.3 Integration
69(12)
2.3.1 Indefinite integral
69(3)
2.3.2 The definite integral
72(9)
3 Vectors
81(32)
3.1 Vectors
81(9)
3.1.1 Some terminology
81(2)
3.1.2 Mathematica representation of vectors
83(2)
3.1.3 Geometric interpretation of a vector
85(2)
3.1.4 Displaying vectors with Mathematica
87(3)
3.2 Vector Operations
90(8)
3.2.1 Addition
90(2)
3.2.2 Scalar multiplication
92(2)
3.2.3 Subtraction
94(1)
3.2.4 Dot product
95(3)
3.3 Projection Vectors
98(8)
3.3.1 Geometric view
98(5)
3.3.2 Orthogonal vector
103(3)
3.4 Vector Spaces
106(7)
3.4.1 Definition
106(2)
3.4.2 Linear dependence
108(2)
3.4.3 Basis and rank of a vector space
110(3)
4 Matrices
113(56)
4.1 Matrices
113(6)
4.1.1 Definition
113(3)
4.1.2 Creating matrices with Mathematica
116(3)
4.2 Matrix Operations
119(13)
4.2.1 Matrix addition and subtraction
119(2)
4.2.2 Matrix multiplication
121(6)
4.2.3 Matrix inversion
127(5)
4.3 Rank, Elementary Operations and the Inverse Matrix
132(14)
4.3.1 Rank of a matrix
132(1)
4.3.2 Elementary operations
133(10)
4.3.3 Gauss elimination method and inverse matrix
143(3)
4.4 Determinants
146(16)
4.4.1 The determinant
146(7)
4.4.2 Submatrices, minors, and cofactors
153(4)
4.4.3 The cofactor, adjoint, and inverse matrices
157(5)
4.5 Applications of Matrix Algebra to Economics
162(7)
4.5.1 Leontief input-output models
162(4)
4.5.2 Regression equation and estimation
166(3)
5 Systems of Linear Equations
169(33)
5.1 Systems of Linear Equations
169(8)
5.1.1 Definition and solutions
169(6)
5.1.2 Existence of solutions
175(2)
5.2 Nonhomogeneous Linear Systems of Equations
177(11)
5.2.1 Case of n equations in n unknowns
177(5)
5.2.2 Case of m equations in n unknowns
182(6)
5.3 Homogeneous Systems of Equations
188(8)
5.3.1 Solutions to homogeneous systems
188(3)
5.3.2 Fundamental sets of solutions
191(5)
5.4 Economic Applications
196(6)
5.4.1 Linear programming problems
196(3)
5.4.2 Leontief input-output closed model
199(3)
6 Eigenvalues and Eigenvectors
202(29)
6.1 Linear Transformations
202(7)
6.1.1 Definition
202(3)
6.1.2 Orthogonal transformations
205(4)
6.2 The Eigensystem of a Matrix
209(10)
6.2.1 Eigenvalues and eigenvectors
209(3)
6.2.2 Finding eigenvalues and eigenvectors with Mathematica
212(3)
6.2.3 An economic example
215(4)
6.3 Diagonalization of a Square Matrix
219(12)
6.3.1 Similar matrices
219(4)
6.3.2 Diagonalization of a real-symmetric matrix
223(3)
6.3.3 Orthogonal transformation and change of basis
226(5)
7 Real Quadratic Forms
231(30)
7.1 Definitions
231(8)
7.1.1 Linear, bilinear, and quadratic forms
231(3)
7.1.2 Geometric view of quadratic forms
234(5)
7.2 Signs of Quadratic Forms
239(12)
7.2.1 Definitions
239(1)
7.2.2 Eigenvalues and the sign of Q(x) = x(T)Ax
240(3)
7.2.3 Determinants and the sign of Q(x) = x(T)Ax
243(8)
7.3 Constrained Quadratic Forms
251(10)
7.3.1 Geometric view
251(3)
7.3.2 Sign of quadratic forms with constraints
254(7)
8 Multivariable Differential Calculus
261(58)
8.1 Limits and Continuity
261(9)
8.1.1 Limits
261(6)
8.1.2 Continuity
267(3)
8.2 Partial Differentiation
270(9)
8.2.1 Partial derivatives
270(4)
8.2.2 Computing partial derivatives with Mathematica
274(3)
8.2.3 Geometric interpretation of partial derivatives
277(2)
8.3 Gradient Vector and Hessian Matrix
279(11)
8.3.1 Gradient vector and directional derivatives
279(6)
8.3.2 Hessian matrix
285(5)
8.4 Differentials
290(12)
8.4.1 Total differentials
290(4)
8.4.2 Higher-order differentials
294(2)
8.4.3 Total differentials with constraints
296(6)
8.5 Derivatives of a Composite Function
302(7)
8.5.1 Total derivative of a composite function
302(4)
8.5.2 Partial derivatives of composite functions
306(3)
8.6 Homogeneous Functions
309(10)
8.6.1 Definition
309(2)
8.6.2 Geometric view
311(3)
8.6.3 Some properties
314(3)
8.6.4 Homothetic function
317(2)
9 Taylor Series and Implicit Functions
319(26)
9.1 Taylor Series
319(9)
9.1.1 Taylor polynomials
319(4)
9.1.2 Computing Taylor polynomials with Mathematica
323(5)
9.2 Implicit Functions
328(17)
9.2.1 Definitions
328(4)
9.2.2 Implicit Function Theorem
332(5)
9.2.3 Simultaneous equation systems and implicit functions
337(8)
10 Concave and Quasiconcave Functions
345(47)
10.1 Concave and Convex Functions
345(10)
10.1.1 Convex sets
345(3)
10.1.2 Concave and convex functions
348(5)
10.1.3 Epigraph and hypograph
353(2)
10.2 Differentiable Concave and Convex Functions
355(12)
10.2.1 Differentiable functions
355(6)
10.2.2 Twice differentiable functions
361(6)
10.3 Some Properties of Concave and Convex Functions
367(4)
10.4 Quasiconcave and Quasiconvex Functions
371(16)
10.4.1 Definitions
371(9)
10.4.2 Level sets, quasiconcavity, and quasiconvexity
380(7)
10.5 Differentiable Quasiconcave and Quasiconvex Functions
387(5)
11 Optimization
392(50)
11.1 Unconstrained Extreme Values
392(23)
11.1.1 Some definitions
392(7)
11.1.2 Necessary and sufficient conditions for relative extrema
399(9)
11.1.3 Finding relative extrema with Mathematica
408(4)
11.1.4 Concavity, convexity, and absolute extrema
412(3)
11.2 Constrained Extreme Values
415(27)
11.2.1 Method of direct substitution
415(4)
11.2.2 Lagrangian function
419(8)
11.2.3 Sufficient conditions
427(7)
11.2.4 Interpretation of the Lagrange multiplier
434(4)
11.2.5 Quasiconcavity, quasiconvexity, and absolute extrema
438(4)
12 Mathematical Programming
442(50)
12.1 Extrema with Inequality Constraints
442(10)
12.1.1 Feasible region and inequality constraints
442(4)
12.1.2 The geometry of optimization problems
446(6)
12.2 Fritz John Necessary Conditions
452(15)
12.2.1 Feasible and descent directions sets
452(6)
12.2.2 Fritz John necessary condition
458(9)
12.3 Kuhn-Tucker Necessary and Sufficient Conditions
467(21)
12.3.1 Kuhn-Tucker necessary conditions
467(10)
12.3.2 Kuhn-Tucker sufficient conditions
477(5)
12.3.3 Optimization with equality and inequality constraints
482(6)
12.4 Linear Programming
488(4)
13 Ordinary Differential Equations
492(61)
13.1 Introduction
492(14)
13.1.1 Basic terminology
492(2)
13.1.2 Solutions of differential equations
494(5)
13.1.3 General and particular solutions
499(7)
13.2 First-Order Differential Equations
506(18)
13.2.1 Separable differential equation
506(4)
13.2.2 Linear differential equation
510(4)
13.2.3 Tangent fields and graphical solutions
514(10)
13.3 Higher-Order Linear Differential Equations
524(20)
13.3.1 Theory of L[y] = 0
524(5)
13.3.2 Theory of L[y] = g(x)
529(3)
13.3.3 Finding the general solution of L[y] = 0
532(7)
13.3.4 Finding the general solution of L[y] = g(x)
539(5)
13.4 Solving Differential Equations with Mathematica
544(9)
13.4.1 The DSolve function
544(3)
13.4.2 The DSolve package
547(2)
13.4.3 The NDSolve function
549(4)
14 Systems of Differential Equations
553(47)
14.1 Systems of First-Order Differential Equations
553(22)
14.1.1 First-order systems and fundamental solutions
553(7)
14.1.2 Linear constant coefficient homogeneous systems
560(12)
14.1.3 Inhomogeneous linear systems
572(3)
14.2 The DSolve and NDSolve Functions for Systems
575(5)
14.2.1 The DSolve function
575(3)
14.2.2 The NDSolve function
578(2)
14.3 Autonomous Systems and the Phase Plane
580(20)
14.3.1 Phase space for linear systems
580(6)
14.3.2 Linearization of nonlinear systems
586(5)
14.3.3 Two economic examples
591(9)
15 Difference Equations
600(45)
15.1 The Difference Calculus
600(9)
15.1.1 Forward difference operators
600(3)
15.1.2 Difference equations
603(2)
15.1.3 Solutions of difference equations
605(4)
15.2 Solving Linear Difference Equations
609(22)
15.2.1 First-order linear difference equations
609(3)
15.2.2 Higher-order homogeneous difference equations
612(6)
15.2.3 Inhomogeneous difference equations
618(7)
15.2.4 Solving difference equations with RSolve
625(3)
15.2.5 Stability of linear difference equations
628(3)
15.3 Systems of Difference Equations
631(14)
15.3.1 Systems of linear difference equations
631(9)
15.3.2 Systems of nonlinear difference equations
640(1)
15.3.3 An economic example
641(4)
Appendix: MathEcon Package 645(14)
Index 659


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