Economists' mathematical manual / Knut Sydsæter, Arne Strøm, Peter Berck.

Format:

Book

Author/Creator:

Sydsæter, Knut.

Contributor:

Strøm, Arne.

Berck, Peter, 1950-

Language:

English

Subjects (All):

Economics, Mathematical.

Physical Description:

xi, 206 pages : illustrations ; 24 cm

Edition:

Third, revised and enlarged edition.

Place of Publication:

Berlin ; New York : Springer, [1999]

Summary:

This volume presents mathematical formulas and theorems common to economics. It includes both formulas like Roy's identity that are peculiar to economics and formulas like Leibniz's rule that are common to many areas of applied mathematics. The volume is meant to be a reference work, to be used by the student in conjunction with a textbook and by the researcher in need of exact statements of mathematical results. The volume is the first grouping of this material for a specifically economist audience. This third edition is extensively revised and contains more than 250 new formulas, as well as new figures.

Contents:

1. Set Theory. Relations. Functions 1

Logical operators

Truth tables

Basic concepts of set theory

Cartesian products

Relations

Different types of orderings

Zorn's lemma

Functions

Inverse functions

Finite and countable sets

2. Equations. Functions of one variable. Complex numbers 7

Roots of quadratic and cubic equations

Cardano's formulas

Polynomials

Descartes's rule of signs

Classification of conics

Graphs of conics

Properties of functions

Asymptotes

Newton's approximation method

Tangents and normals

Powers, exponentials, and logarithms

Trigonometric and hyperbolic functions

Complex numbers

De Moivre's formula

Euler's formulas

nth roots

3. Limits. Continuity. Differentiation (one variable) 19

Limits

Continuity

Uniform continuity

The intermediate value theorem

Differentiable functions

General and special rules for differentiation

Mean value theorems

L'Hopital's rule

Differentials

4. Partial derivatives 25

Partial derivatives

Young's theorem

C[superscript k]-functions

Chain rules

Slopes of level curves

The implicit function theorem

Homogeneous functions

Euler's theorem

Homothetic functions

Gradients and directional derivatives

Tangent (hyper) planes

5. Elasticities. Elasticities of substitution 31

Definition

Marshall's rule

General and special rules

Directional elasticities

The passus equation

Marginal rate of substitution

Elasticities of substitution

6 Systems of equations 35

General systems of equations

Jacobian matrices

The general implicit function theorem

Degrees of freedom

The "counting rule"

Functional dependence

The Jacobian determinant

The inverse function theorem

Existence of local and global inverses

Gale-Nikaido theorems

The contraction mapping theorem

Brouwer's and Kakutani's fixed point theorems

Sublattices in R[superscript n]

Tarski's fixed point theorem

General results on linear systems of equations

7. Inequalities 41

Triangle inequalities

Inequalities for arithmetic, geometric, and harmonic means

Inequalities of Holder, Cauchy-Schwarz, Chebyshev, Minkowski, and Jensen

8. Series. Taylor's formula 43

Arithmetic and geometric series

Convergence of infinite series

Convergence criteria

First- and second-order approximations

Maclaurin and Taylor formulas

Series expansions

Binomial coefficients

Newton's binomial formula

The multinomial formula

Summation formulas

Euler's constant

9. Integration 47

Indefinite integrals

Definite integrals

Convergence of integrals

The comparison test

Leibniz's formula

The gamma function

Stirling's formula

The beta function

The trapezoid formula

Simpson's formula

Multiple integrals

10. Difference equations 55

Solutions of linear equations of first, second, and higher order

Backward and forward solutions

Stability

Schur's theorem

Matrix formulations

11. Differential equations 61

Separable, projective, and logistic equations

Linear first-order equations

Bernoulli and Riccati equations

Exact equations

General linear equations

Variation of parameters

Second-order linear equations with constant coefficients

Euler's equation

General linear equations with constant coefficients

Stability of linear equations

Routh-Hurwitz's stability conditions

Normal systems

Linear systems

Resolvents

Local and global existence and uniqueness theorems

Autonomous systems

Equilibrium points

Integral curves

Local and global (asymptotic) stability

Periodic solutions

The Poincare-Bendixson theorem

Liapunov theorems

Hyperbolic equilibrium points

Olech's theorem

Liapunov functions

Lotka-Volterra models

A local saddle point theorem

Partial differential equations of the first order

Quasi-linear equations

Frobenius's theorem

12. Topology in Euclidean space 73

Basic concepts of point set topology

Convergence of sequences

Cauchy sequences

Continuous functions

Relative topology

Pointwise and uniform convergence of sequences of functions

Correspondences

Lower and upper hemicontinuity

Infimum and supremum

13. Convexity 79

Convex sets

Convex hull

Caratheodory's theorem

Extreme points

Krein-Milman's theorem

Separation theorems

Concave and convex functions

Hessian matrices

Quasi-concave and quasi-convex functions

Bordered Hessians

Pseudo-concave and pseudo-convex functions

14. Classical optimization 87

Basic definitions

The extreme value theorem

Stationary points

First-order conditions

Saddle points

One-variable results

Inflection points

Second-order conditions

Constrained optimization with equality constraints

Lagrange's method

Value functions and sensitivity

Properties of Lagrange multipliers

Envelope results

15. Linear and nonlinear programming 95

Basic definitions and results

Duality

Shadow prices

Complementary slackness

Farkas's lemma

Kuhn-Tucker theorems

Saddle point results

Quasi-concave programming

Properties of the value function

An envelope result

Nonnegativity conditions

16. Calculus of variations and optimal control theory 101

The simplest variational problem

The Legendre condition

Sufficient conditions

Transversality conditions

Scrap value functions

More general variational problems

Control problems

The maximum principle

Mangasarian's and Arrow's sufficient conditions

Free terminal time problems

More general terminal conditions

Current value formulations

Linear quadratic problems

Infinite horizon

Mixed constraints

Pure state constraints

Mixed and pure state constraints

17. Discrete dynamic optimization 113

Dynamic programming

The value function

The fundamental equations

A "control parameter free" formulation

Euler's vector difference equation

Discrete optimal control theory

18. Vectors in R[superscript n]. Abstract spaces 117

Linear dependence and independence

Subspaces

Bases

Scalar products

Norm of a vector

The angle between two vectors

Vector spaces

Metric spaces

Normed vector spaces

Banach spaces

Ascoli's theorem

Schauder's fixed point theorem

Fixed points for contraction mappings

Blackwell's sufficient conditions for a contraction

Inner-product spaces

Hilbert spaces

Cauchy-Schwarz' and Bessel's inequalities

Parseval's formula

19. Matrices 123

Special matrices

Matrix operations

Inverse matrices and their properties

Trace

Rank

Matrix norms

Exponential matrices

Linear transformations

Generalized inverses

Moore-Penrose inverses

Partitioned matrices

Matrices with complex elements

20. Determinants 131

2 [times] 2 and 3 [times] 3 determinants

General determinants and their properties

Cofactors

Vandermonde and other special determinants

Minors

Cramer's rule

21. Eigenvalues. Quadratic forms 135

Eigenvalues and eigenvectors

Diagonalization

Spectral theory

Jordan decomposition

Schur's lemma

Cayley-Hamilton's theorem

Quadratic forms and criteria for definiteness

Singular value decomposition

Simultaneous diagonalization

Definiteness of quadratic forms subject to linear constraints

22. Special matrices. Leontief systems 141

Properties of idempotent, orthogonal, and permutation matrices

Nonnegative matrices

Frobenius roots

Decomposable matrices

Dominant diagonal matrices

Leontief systems

Hawkins-Simon conditions

23. Kronecker products and the vec operator.

Differentiation of vectors and matrices 145

Definition and properties of Kronecker products

The vec operator and its properties

Differentiation of vectors and matrices with respect to elements, vectors, and matrices

24. Comparative statics 149

Equilibrium conditions

Reciprocity relations

Monotone comparative statics

Sublattices of R[superscript n]

Supermodularity

Increasing differences

25. Properties of cost and profit functions 153

Cost functions

Conditional factor demand functions

Shephard's lemma

Profit functions

Factor demand functions

Supply functions

Hotelling's lemma

Puu's equation

Elasticities of substitution

Allen-Uzawa's and Morishima's elasticities of substitution

Cobb-Douglas and CES functions

Law of the minimum, Diewert, and translog cost functions

26. Consumer theory 159

Preference relations

Utility functions

Utility maximization

Indirect utility functions

Consumer demand functions

Roy's identity

Expenditure functions

Hicksian demand functions

Cournot, Engel, and Slutsky elasticities

The Slutsky equation

Equivalent and compensating variations

LES (Stone-Geary), AIDS, and translog indirect utility functions

Laspeyre, Paasche, and general price indices

Fisher's ideal index

27. Topics from finance and growth theory 165

Compound interest

Effective rate of interest

Present value calculations

Internal rate of return

Norstrom's rule

Continuous compounding

Solow's growth model

Ramsey's growth model

28. Risk and risk aversion theory 169

Absolute and relative risk aversion

Arrow-Pratt risk premium

Stochastic dominance of first and second degree

Hadar-Russell's theorem

Rothschild-Stiglitz's theorem

29. Finance and stochastic calculus 171

Capital asset pricing model

The single consumption β asset pricing equation

The Black-Scholes option pricing model

Sensitivity results

A generalized Black-Scholes model

Put-call parity

Correspondence between American put and call options

American perpetual put options

Stochastic integrals

Ito's formulas

A stochastic control problem

Hamilton-Jacobi-Bellman's equation

30. Non-cooperative game theory 175

An n-person game in strategic form

Nash equilibrium

Mixed strategies

Strictly dominated strategies

Two-person games

Zero-sum games

Symmetric games

Saddle point property of the Nash equilibrium

The classical minimax theorem for two-person zero-sum games

Exchangeability property

Evolutionary game theory

31. Probability and statistics 179

Axioms for probability

Rules for calculating probabilities

Conditional probability

Stochastic independence

Bayes's rule

Random variables (one-dimensional)

Probability density functions

Cumulative distribution functions

Expectation

Mean

Variance

Standard deviation

Central moments

Coefficients of skewness and kurtosis

Chebyshev's and Jensen's inequalities

Moment generating and characteristic functions

Two-dimensional random variables and distributions

Covariance

Cauchy-Schwarz's inequality

Correlation coefficient

Marginal and conditional density functions

Conditional expectation and variance

Iterated expectations

Transformations of stochastic variables

Estimators. Bias

Mean square error

Probability limits

Consistency

Testing

Power of a test

Type I and type II errors

Level of significance

Significance probability (P-value)

Weak and strong law of large numbers

Central limit theorem

32. Probability distributions. Method of least squares 187

Beta, binomial, binormal, chi-square, exponential, extreme value (Gumbel), F-, gamma, geometric, hypergeometric, Laplace, logistic, lognormal, multinomial, multivariate normal, negative binomial, normal, Pareto, Poisson, Student's t-, uniform, Weibull distributions

Method of least squares

Multiple regression.

Notes:

Berck's name appears first on the earlier edition.

Includes bibliographical references (pages [193]-196) and index.

ISBN:

354065447X

OCLC:

40857113