ISBN: 3540260889
TITLE: Economists' Mathematical...
AUTHOR: Sydsaeter et al.
TOC:

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. Mathematical induction.
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) 21
Limits. Continuity. Uniform continuity. The intermediate value theorem. Differentiable functions. General and special rules for differentiation. Mean value theorems. L'Hpital's rule. Differentials.
4. Partial derivatives 27
Partial derivatives. Young's theorem. C-functions. Chain rules. Differentials. Slopes of level curves. The implicit function theorem. Homogeneous functions. Euler's theorem. Homothetic functions. Gradients and directional derivatives. Tangent (hyper)planes. Supergradients and subgradients. Differentiability of transformations. Chain rule for transformations.
5. Elasticities. Elasticities of substitution 35
Definition. Marshall's rule. General and special rules. Directional elasticities. The passus equation. Marginal rate of substitution. Elasticities of substitution.
6. Systems of equations 39
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. Existente of local and global inverses. Gale-Nikaido theorems. Contraction mapping theorems. Brouwer's and Kakutani's fixed point theorems. Sublattices in Rn. Tarski's fixed point theorem. General results on linear systems of equations.
7. Inequalities 47
Triangle inequalities. Inequalities for arithmetic, geometric, and harmonic means. Bernoulli's inequality. Inequalities of Hlder, Cauchy-Schwarz, Chebyshev, Minkowski, and Jensen.
8. Series. Taylor's formula 49
Arithmetic and geometric series. Convergence of infinite series. Convergence criteria. Absolute convergence. 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 55
Indefinite integrals. General and special rules. 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 63
Solutions of linear equations of first, second, and higher order. Backward and forward solutions. Stability for linear systems. Schur's theorem. Matrix formulations. Stability of first-order nonlinear equations.
11. Differential equations 69
Separable, projective, and logistic equations. Linear first-order equations. Bernoulli and Riccati equations. Exact equations. Integrating factors. Local and global existente theorems. Autonomous first-order equations. Stability. 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. Matrix formulations. Resolvents. Local and global existente and uniqueness theorems. Autonomous Systems. Equilibrium points. Integral curves. Local and global (asymptotic) stability. Periodic solutions. The Poincar-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. Quasilinear equations. Frobenius's theorem.
12. Topology in Euclidean space 83
Basic concepts of point Set topology. Convergence of sequences. Cauchy sequences. 'Cauchy's convergence criterion. Subsequences. Compact sets. HeineBorel's theorem. Continuous functions. Relative topology. Uniform continuity. Pointwise and uniform convergence. Correspondences. Lower and upper hemicontinuity. Infimum and supremum. Lim inf and lim sup.
13. Convexity 89
Convex sets. Convex hull. Carathodory's theorem. Extreme points. KreinMilman's theorem. Separation theorems. Concave and Konvex functions. Hessian matrices. Quasiconcave and quasiconvex functions. Bordered Hessians, Pseudoconcave and pseudoconvex functions.
14. Classical optimization 97
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 105
Basic definitions and results. Duality. Shadow prices. Complementary slackness. Farkas's lemma. Kuhn-Thcker theorems. Saddle point results. Quasiconcave programming. Properties of the value function. An envelope result, Nonnegativity conditions.
16. Calculus of variations and optimal control theory 111
The simplest variational problem. Euler's equation. 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. Properties of the value function. Free terminal time problems. More general terminal conditions. Scrap value functions. Current value formulations. Linear quadratic problems. Infinite horizon. Mixed constraints. Pure state constraints. Mixed and pure state constraints.
17. Discrete dynamic optimization 123
Dynamic programming. The value function. The fundamental equations. A "control parameter free" formulation. Euler's vector differente equation. Infinite horizon. Discrete optimal control theory.
18. Vectors in Rn Abstract spaces 127
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 133
Special matrices. Matrix operations. Inverse matrices and their properties. Trace. Rank. Matrix norms. Exponential matrices. Linear transformations. Generalized inverses. Moore-Penrose inverses. Partitioning matrices. Matrices with complex elements,
20. Determinants 141
2 x 2 and 3 x 3 determinants. General determinants and their properties. Cofactors. Vandermonde and other special determinants. Minors. Cramer's rule.
21. Eigenvalues. Quadratic forms 145
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 151
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 155
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 159
Equilibrium conditions. Reciprocity relations. Monotone comparative statics. Sublattices of Rn. Supermodularity. Increasing differences.
25. Properties of cost and profit functions 163
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 169
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. Laspeyres, Paasche, and general price indices. Fisher's ideal index.
27. Topics from trade theory 175
2 x 2 factor model. No factor intensity reversal. Stolper-Samuelson's theorem. Heckseher-Ohlin-Samuelson's model. Heckseher-Ohlin's theorem.
28. Topics from finance and growth theory . 177
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.
29. Risk and risk aversion theory 181
Absolute and relative risk aversion. Arrow-Pratt risk premium. Stochastic dominante of first and second degree. Hadar-Russell's theorem. RothschildStiglitz's theorem.
30. Finance and stochastic calculus 183
Capital asset pricing model. The single consumption , 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. It's formulas. A stochastic control problem. Hamilton-Jacobi-Bellman's equation.
31. Non-cooperative game theory 187
An n-person game in strategic form. Nash equilibrium. Mixed strategies. Strictly dominated strategies. Two-person games. Zero-sum games. Symmetrie games. Saddle point property of the Nash equilibrium. The classical minimax theorem for two-person zero-sum games. Exchangeability property. Evolutionary game theory. Games of incomplete information. Dominant strategies and Baysesian Nash equlibrium. Pure strategy Bayesian Nash equilibrium.
32. Combinatorics 191
Combinatorial results. Inclusion-exclusion principle. Pigeonhole principle.
33. Probability and statistics 193
Axioms for probability. Rules for calculating probabilities. Conditional probability. Stochastic independence. Bayes's rule. One-dimensional random variables. Probability density functions. Cumulative distribution functions. Expectation. Mean. Variante. 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. Stochastic independence. Conditional expectation and variance. Iterated expectations. Transformations of stochastic variables. Estimators. Bias. Mean square error. Probability limits. Convergence in quadratic mean. Slutsky's theorem. Limiting distribution. 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.
34. Probability distributions 201
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, and Weibull distributions.
35. Method of least squares 207
Ordinary least squares. Linear regression. Multiple regression.
References 211
Index 215
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