Literature

-neighbourhood of point 387 n-dimensional space 231

nth derivative 163 nth partial sum 71

absolute value 37

amortization installment 90, 93 amortization table 93

amount of annuity 85 annuity 85, 90; ordinary 85 antiderivative 197

apex 129

approximation: by rectangles 215; by trapeziums 215

Argand diagram 49 argument 117 artificial variable 338 augmented matrix 291

auxiliary objective function 350

back substitution 301 basic solution 293

basis of vector space 245 Bernoulli–l’Hospital’s rule 178 binomial coefficient 28 bordered Hessian 426 break-even point 172

canonical form 293, 339 Cartesian product 24, 25 Cauchy–Schwarz inequality 237 chain rule 160, 398 characteristic equation 369

Cobb–Douglas production function 383, 404 Cobb–Douglas utility function 431

cobweb model 476 coefficient 288

coefficient of the polynomial 126 cofactor 265

column vector 230 complex number 47 component of vector 230

composite mapping 114, 272

composition 114 composition of relations 108 conclusion 3, 9

conjunction 2 constant-factor rule 199

constrained optimization problem 424 consumer surplus 225

continuous future income flow 222 contradiction 1, 5

convex combination 240, 310 convex polyhedron 334 coordinate of vector 230 cosine function 141 cotangent function 141 Cramer’s rule 269

criterion: Leibniz 77; quotient 78; root 79 critical point 170

debt 90

definite solution 448 degeneration case 351 degree of freedom 292 demand-price-function 172 dependent variable 117 deposits 85

depreciation: arithmetic-degressive 101; degressive 101; digital 102; geometric-degressive 102; linear 101; table 101

derivative 156; directional 399; partial 387; second 163

determinant: Casorati’s 473; Wronski’s 452 determinant of matrix 264

difference equation: linear 472, of the first order 474, of second order 478

difference of vectors 233 difference quotient 155 difference set 17

differential equation 444; homogeneous 451; non-homogeneous 451; ordinary 444; with separable variables 447

differential of function 164 differential quotient 156

514 Index

dimension of a vector space 245 dimension of matrix 255 direction field 445

disjunction 2

domain of the function 110, 117 domain of the mapping 110 double integral 436

downward parabola 128 dual problem 358

duality 357; economic interpretation 361

effective rate of interest 83 eigenvalue 368 eigenvalue equation 369 eigenvector 368

elasticity 166, 183

elementary transformation 292 empty set 16

entering variable 342 equal matrices 256 equal vectors 232 equivalence 4

equivalent transformation 292 Euclidean distance 236 Euler’s theorem 404

extreme point 310

factor of the polynomial 131 Falk’s scheme 260

feasible region 309 first-derivative test 170

first-order differential equation 445 first-order partial derivative 390 forcing term 451

function 110; algebraic 136; antisymmetric 124; arccosine 143; arccotangent 143; arcsine 143;

arctangent 143; bounded 123; bounded from: above 123, below 123; circular 140; complementary 453; concave 125, 175; constant 126; continuous 151, 387; continuously differentiable 156; convex 125, 175; cubic 126; decreasing 121, 168; differentiable 156; elastic 167; even 124; exponential 137; homogeneous of

degree k 403; implicitly defined 405; increasing 121, 168; inelastic 167;

inside 120; left-continuous 154; linear 126; logarithmic 138; non-decreasing 121, 168; non-increasing 121, 168; odd 124;

outside 120; periodic 125; propositional 7; quadratic 126; rational 134, improper 134, proper 134; right-continuous 154; strictly concave 125, 175; strictly convex 125, 175; strictly decreasing 121, 168; strictly increasing 121, 168; symmetric 124; trigonometric 140

function of a real variable 117 fundamental system of the differential

equation 453

fundamental theorem of algebra 130

Gauss–Jordan elimination 293 Gaussian elimination 293, 299

general solution of the differential equation 445, 448, 453

general solution of the system of linear equations 289

generalized chain rule 398

geometric interpretation of an LPP 330 Gini coefficient 224

global maximum 169, 410 global maximum point 169, 410 global minimum 169, 410 global minimum point 169, 410 global sufficient conditions 434 gradient of function 392

Hessian matrix 411 higher-order derivative test 171

higher-order partial derivative 391 Horner’s scheme 132

hypothesis 3

identity matrix 257

imaginary part of the complex number 47 implication 3

implicit-function theorem 408 independent variable 117 indeterminate form 178 inflection point of function 176 initial value problem 448

inner product 235 input–output model 277

integral: definite 210; improper 219, 221; indefinite 198

integrand 198 integration by parts 204

integration by substitution 200 interest 80; compound 81; simple 80 inverse 273

inverse demand function 172 inverse element 244 investment project 97 isoquant 384, 445

Jacobian determinant 407

Kepler’s formula 217 kernel 271

Lagrange multiplier method 425

Lagrange’s theorem 425

Lagrangian function 425

Lagrangian multiplier 425

law: associative 6, 19, 235, 259;

commutative 6, 19, 235, 259; distributive 6, 19, 235, 259; of de Morgan 6

leading principal minor 379 leaving variable 342

length of vector 236 Leontief model 277 limit of sequence 65

limit of function 148; left-side 149; right-side 149

linear combination 240

linear differential equation of order n 451 linear objective function 329

linear programming problem 329 linear space 244

linear substitution 200 linearly dependent vectors 241

linearly independent vectors 241 loan: amortized 90

loan repayments 90 local maximum 169, 410

local maximum point 169, 410, 423 local minimum 169, 410

local minimum point 169, 410, 423 local sufficient condition 426 logarithmic differentiation 162 Lorenz curve 224

mapping 110; bijective 112; identical 116, 273; injective 112; inverse 114; linear 271; surjective 112

marginal 156 marginal cost 213, 389 marginal function 156

marginal propensity to consume 159 market price 225

matrix 255; antisymmetric 256; diagonal 257; indefinite 377; inverse 273; invertible 273; lower triangular 257; negative definite 377; negative semi-definite 377; orthogonal 263; positive definite 377; positive

semi-definite 377; symmetric 257; upper triangular 257

matrix difference 258 matrix product 260 matrix representation 288

matrix representation of an LPP 330 matrix sum 258

mean-value theorem 184, 214

method: of undetermined coefficients 456 minor 265

mixed LPP 360 monopolist 172

monotonicity of function 168 mortgage 94 multiplier-accelerator-model 480

Index 515

necessary first-order conditions 411 negation 2

negative integer 32

neutral element 244, 258, 261 Newton’s method 189

Newton’s method of second order 190 Newton–Leibniz’s formula 210 non-negativity constraint 309, 329 norm 236

number: irrational 32; natural 32; rational 32; real 32

objective row 340

one-parametric set of solutions 299 one-to-one mapping 112 onto-mapping 112

operation: logical 1 optimal solution 330 optimality criterion 341

optimization by substitution 425 order of matrix 255

order of the differential equation 444 orthogonal vectors 239

parabola 128

partial differential 394 partial elasticity 403 partial rate of change 402 particular solution 448 Pascal’s triangle 29

payment: annual 85; periodic 85 period of function 125

periods for interest 85 permutation 26

pivot 300, 342 pivot column 342

pivot element 300, 342 pivot row 342

pivoting procedure 294, 339 polar form of complex number 49 pole of second order 222 polynomial 126

polynomial function 126 power function 136 power set 16

premises 9

present value of annuity 86 price–demand function 195 primal problem 358 principal 80

producer surplus 226 production function 383 profit function 416

proof: by induction 13; direct 10; indirect 10, of contradiction 10, of contrapositive 10

proportional rate of change 166

516 Index

proposition: compound 1, 5; existential 8; false 1; open 7; true 1; universal 8

Pythagorean theorem 238

quadratic form 376; indefinite 377; negative definite 377; negative semi-definite 377; positive definite 377; positive semi-definite 377

radian 141

range of the function 110, 117 range of the mapping 110 rank of matrix 290

rate of interest 80

real part of complex number 47 rectangle formula 248 rectangle rule 297

redemption table 90, 94 regula falsi 191

relation: binary 107; inverse 108; reflexive 107; symmetric 107

remainder 130 remainder theorem 130 rentability 98

return to scale 403; decreasing 404; increasing 404

Riemann integral 210 right-hand side 288 right-hand side vector 330 Rolle’s theorem 184

root function 136

root of the function 128 row vector 230

saddle point 412 Sarrus’ rule 265

scalar multiplication 233, 258 scalar product 235

sequence 61; arithmetic 62; bounded 65; decreasing 64; geometric 63; increasing 64; strictly decreasing 64; strictly increasing 64

series 73; alternating 77; geometric 75; harmonic 74

set 15; cardinality 15; complement 17; convex 310; disjoint 17; finite 15; infinite 15; intersection 17; union 16

set of feasible solutions 309 set of solutions 289 shadow price 361

short form of the tableau 341 simplex algorithm 343 simplex method 339 Simpson’s formula 218

sine function 141 slack variable 315, 337

smallest subscript rule 344

solution of systems of linear equations 289 solution of systems of linear inequalities 309;

degenerate 314; feasible 309; non-degenerate 314

solution of the differential equation 445 solutions: linearly dependent 452, 473; linearly

independent 452, 473

square matrix: non-singular 269; regular 269; singular 269

standard form of an LPP 336 stationary point 170, 411 Steinitz’s procedure 248 straight line 127

subset 16

sufficient second-order conditions 411 sum of the infinite series 73

sum of vectors 233 sum–difference rule 199 surface 384

surplus variable 337

system: consistent 291; homogeneous 289, 302; inconsistent 291; non-homogeneous 289

system of linear equations 287 system of linear inequalities 308

tangent function 141 tautology 5

Taylor polynomial 187 Taylor’s formula 187 total differential 394 transition matrix 262 transpose of matrix 256 transposed vector 230 trivial solution 292 truth table 2

unit vector 232 unknown 288 upward parabola 129 utility function 431

variable 288; basic 293; non-basic 293 vector 230

vector space 244, 259 Venn diagram 17 Vieta’s theorem 133

withdrawals 86

Young’s theorem 392

zero element 244 zero matrix 258

zero of the function 128

As the intersection between economics and mathematics continues to grow in both theory and practice, a solid grounding in mathematical concepts is essential for all serious students of economic theory.

In this clear and entertaining volume, Rakesh V. Vohra sets out the basic concepts of mathematics as they relate to economics. The book divides the mathematical problems that arise in economic theory into three types: feasibility problems, optimality problems and fixed-point problems.

Of particular salience to modern economic thought are the sections on lattices, supermodularity, matroids and their applications. In a departure from the prevailing fashion, much greater attention is devoted to linear programming and its applications.

Of interest to advanced students of economics as well as those seeking a greater understanding of the influence of mathematics on ‘the dismal science’,

Advanced Mathematical Economics follows a long and celebrated tradition of the application of mathematical concepts to the social and physical sciences.

Series: Routledge Advanced Texts in

Economics and Finance

November 2004: 208pp Hb: 0-415-70007-8: £65.00 Pb: 0-415-70008-6: £22.99 eB: 0-203-79995-X: £22.99

Available as an inspection copy