Economists’ Mathematical Manual · · Knut Sydsæter Arne Strøm Peter Berck Economists’ Mathematical Manual Fourth Edition 123 ProfessorKnutSydsæter AssociateProfessorArneStrøm UniversityofOslo UniversityofOslo DepartmentofEconomics DepartmentofEconomics P.O.Box10955Blindern P.O.Box10955Blindern NO-0317Oslo NO-0317Oslo Norway Norway [email protected] [email protected] ProfessorPeterBerck UniversityofCalifornia,Berkeley DepartmentofAgriculturaland ResourceEconomics Berkeley,CA94720-3310 USA [email protected] ISBN978-3-540-26088-2 e-ISBN978-3-540-28518-2 DOI10.1007/978-3-540-28518-2 SpringerHeidelbergDordrechtLondonNewYork LibraryofCongressControlNumber:2009937018 (cid:2)c Springer-VerlagBerlinHeidelberg 1991, 1993, 1999, 2005,CorrectedSecond Printing2010 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective lawsandregulationsandthereforefreeforgeneraluse. Coverdesign:WMXDesignGmbH,Heidelberg Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface to the fourth edition The fourth edition is augmented by more than 70 new formulas. In particular, we have included some key concepts and results from trade theory, games of incomplete information and combinatorics. In addition there are scattered additions of new formulas in many chapters. Again we are indebted to a number of people who has suggested corrections, im- provements and new formulas. In particular, we would like to thank Jens-Henrik Madsen, Larry Karp, Harald Goldstein, and Geir Asheim. Inareferencebook,errorsareparticularlydestructive. Wehopethatreaderswho find our remaining errors will call them to our attention so that we may purge them from future editions. Oslo and Berkeley, May 2005 Knut Sydsæter, Arne Strøm, Peter Berck From the preface to the third edition Thepracticeofeconomicsrequiresawide-rangingknowledgeofformulasfrommathe- matics,statistics,andmathematicaleconomics. Withthisvolumewehopetopresent aformularytailoredtotheneedsofstudentsandworkingprofessionalsineconomics. In addition to a selection of mathematical and statistical formulas often used by economists, this volume contains many purely economic results and theorems. It containsjusttheformulasandtheminimumcommentaryneededtorelearnthemath- ematics involved. We have endeavored to state theorems at the level of generality economists might find useful. In contrast to the economic maxim, “everything is twice more continuously differentiable than it needs to be”, we have usually listed the regularity conditions for theorems to be true. We hope that we have achieved a level of explication that is accurate and useful without being pedantic. During the work with this book we have had help from a large group of peo- ple. It grew out of a collection of mathematical formulas for economists originally compiled by Professor B. Thalberg and used for many years by Scandinavian stu- dents and economists. The subsequent editions were much improved by the sugges- tions and corrections of: G. Asheim, T. Akram, E. Biørn, T. Ellingsen, P. Frenger, I. Frihagen, H. Goldstein, F. Greulich, P. Hammond, U. Hassler, J. Heldal, Aa. Hylland, G. Judge, D. Lund, M. Machina, H. Mehlum, K. Moene, G. Nord´en, A. Rødseth, T. Schweder, A. Seierstad, L. Simon, and B. Øksendal. As for the present third edition, we want to thank in particular, Olav Bjerkholt, Jens-Henrik Madsen, and the translator to Japanese, Tan-no Tadanobu, for very useful suggestions. Oslo and Berkeley, November 1998 Knut Sydsæter, Arne Strøm, Peter Berck Contents 1. Set Theory. Relations. Functions ................................. 1 Logical operators. Truth tables. Basic concepts of set theory. Cartesian prod- ucts. 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. Proper- ties of functions. Asymptotes. Newton’s approximation method. Tangents and normals. Powers, exponentials, and logarithms. Trigonometric and hyperbolic functions. Complexnumbers. DeMoivre’sformula. Euler’sformulas. nthroots. 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 Ho¨lder, Cauchy–Schwarz, Cheby- shev, Minkowski, and Jensen. 8. Series. Taylor’s formula ......................................... 49 Arithmeticandgeometricseries. Convergenceofinfiniteseries. Convergencecri- teria. Absoluteconvergence. First-andsecond-orderapproximations. Maclaurin and Taylor formulas. Series expansions. Binomial coefficients. Newton’s bino- mialformula. Themultinomialformula. Summationformulas. Euler’sconstant. 9. Integration ...................................................... 55 Indefinite integrals. General and special rules. Definite integrals. Convergence ofintegrals. Thecomparisontest. Leibniz’sformula. Thegammafunction. Stir- ling’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 formu- lations. Stability of first-order nonlinear equations. 11. Differential equations ............................................ 69 Separable, projective, and logistic equations. Linear first-order equations. Ber- noulli and Riccati equations. Exact equations. Integrating factors. Local and globalexistencetheorems. Autonomousfirst-orderequations. 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 so- lutions. The Poincar´e–Bendixson theorem. Liapunov theorems. Hyperbolic equilibrium points. Olech’s theorem. Liapunov functions. Lotka–Volterra mod- els. Alocalsaddlepointtheorem. Partialdifferentialequationsofthefirstorder. Quasilinear equations. Frobenius’s theorem. 12. Topology in Euclidean space ..................................... 83 Basic concepts of point set topology. Convergence of sequences. Cauchy se- quences. 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 hemi- continuity. 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 slack- ness. 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 ............... 111 The simplest variational problem. Euler’s equation. The Legendre condition. Sufficient conditions. Transversality conditions. Scrap value functions. More generalvariationalproblems. Controlproblems. Themaximumprinciple. Man- gasarian’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 con- straints. 17. Discrete dynamic optimization ................................. 123 Dynamic programming. The value function. The fundamental equations. A “controlparameterfree”formulation. Euler’svectordifferenceequation. Infinite horizon. Discrete optimal control theory. 18. Vectors in Rn. Abstract spaces ................................. 127 Lineardependenceandindependence. Subspaces. Bases. Scalarproducts. 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 condi- tions for a contraction. Inner-product spaces. Hilbert spaces. Cauchy–Schwarz’ and Bessel’s inequalities. Parseval’s formula.