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. 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’Hˆopital’s rule. Differentials.

4. Partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Partial derivatives. Young’s theorem. Ck-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. Existence 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.

viii

7. Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Triangle inequalities. Inequalities for arithmetic, geometric, and harmonic

means. Bernoulli’s inequality. Inequalities of H¨older, 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 existence 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 existence and uniqueness theorems. Autonomous systems. Equilibrium

points. Integral curves. Local and global (asymptotic) stability. Periodic solutions.

The Poincar´e–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. Heine–

Borel’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.

ix

13. Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Convex sets. Convex hull. Carath´eodory’s theorem. Extreme points. Krein–

Milman’s theorem. Separation theorems. Concave and convex 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–Tucker 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 difference 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.

x

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 × 2 and 3 × 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.

xi

27. Topics from trade theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

2 × 2 factor model. No factor intensity reversal. Stolper–Samuelson’s theorem.

Heckscher–Ohlin–Samuelson’s model. Heckscher–Ohlin’s theorem.

28. Topics from finance and growth theory . . . . . . . . . . . . . . . . . . . . . . . . . 177

Compound interest. Effective rate of interest. Present value calculations. Internal

rate of return. Norstrøm’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

dominance of first and second degree. Hadar–Russell’s theorem. Rothschild–

Stiglitz’s theorem.

30. Finance and stochastic calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

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. Itˆo’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. Symmetric

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. 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. Stochastic independence. Conditional expectation

and variance. Iterated expectations. Transformations of stochastic variables.

Estimators. Bias. Mean square error. Probability limits. Convergence in

xii

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),

35. Method of least squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Ordinary least squares. Linear regression. Multiple regression.

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Index . . . . . . . . . . . . . . . . . . . . . . .

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