Mathematics and Mathematica for Economists

by Cliff Huang (Vanderbilt University, Tennessee); Philip S. Crooke (Vanderbilt University, Tennessee)

• ISBN13: 9781577180340
• ISBN10: 1577180348
• Format: Hardcover
• Copyright: 1999-11-01
• Publisher: Wiley-Blackwell

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