Caltech seal California Institute of Technology
Division of the Humanities and Social Sciences
Spring 2009

Professor: KC Border
Email: kcb@caltech.edu

Ec 181: Convex Analysis and Economic Theory

•  Lecture Notes •  Exams •  Homework •  Topics •  Theory •  Applications • 

This is a new course that is designed to introduce you to convex analysis and its applications in economics. It is still under development. This year I am planning on asking students to make presentations of some of the material.

Convex analysis is the study of the properties of convex sets and convex and concave functions. The fundamental results in the field are the separating hyperplane theorems. These have results interpretations as the existence of prices, so they are fundamental in many areas of general economic equilibrium theory.

Another class of theorems goes by the name of the Theorem of the Alternative. These theorems give conditions for the existence of solutions to linear inequalities in terms of the existence of solutions to an alternative set of inequalities. This may not seem especially useful, but these results are at the heart of fundamental results in decision theory and asset pricing theory.

The goal of this course is to present the useful results from convex analysis in a way that you understand their proofs and can use them in economics. There will be a lot of proofs in this course, and you will be expected to prove things. If you do not like proving theorems, you should not take this course.

The course will concentrate on convex analysis in finite dimensional spaces, but I will also discuss infinite dimensional spaces (which are necessary in mathematical finance) whenever possible. In particular, I will try to avoid making use of the dimensionality of the space when possible. But some things that are true for finite dimensional spaces are not true for infinite dimensional spaces.

Exams

Text

There is no required textbook for the class. I will make my own notes available via the web. I expect these to change over the course of the term based on feedback that I receive from you. For those of you who like having a textbook, I recommend Optima and Equilibria: An Introduction to Nonlinear Analysis by J.-P. Aubin, Springer-Verlag, 1993; Fundamentals of Convex Analysis by J.-B. Hiriart-Urruty and C. Lemaréchal, Springer--Verlag, 2001; and Convex Analysis and Nonlinear Optimization: Theory and Examples by J. M. Borwein and A. S. Lewis, Springer, 2006.

Lecture Notes

These will be revised over the course of the term.

Additional Readings

Convex Analysis

  1. Charalambos D. Aliprantis and Kim C. Border. 2006. Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer--Verlag, Berlin.
  2. Achim Bachem and Walter Kern. 1992. Linear Programming Duality: An Introduction to Oriented Matroids. Springer--Verlag, Berlin.
  3. Errett Bishop and Robert R. Phelps. The Support Functionals of a Convex Set. In Klee1963.
  4. Kim C. Border. 1985. Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press, New York.
  5. Jonathan M. Borwein and Chris H. Hamilton. 2009. Symbolic Fenchel Conjugation. Mathematical Programming 116: 17–35.
  6. J. M. Borwein and A. S. Lewis. 1991. Duality Relationships for Entropy-Like Minimization Problems. SIAM Journal on Control and Optimization 29: 325-338. On-line.
  7. Jonathan M. Borwein and Adrian S. Lewis. 2006. Convex Analysis and Nonlinear Optimization: Theory and Examples. Springer, New York.
  8. Jonathan M. Borwein and Qiji J. Zhu. 2006. Variational Methods in Convex Analysis. Journal of Global Optimization 35: 197–213. On-line.
  9. Charles Castaing and Michel Valadier. 1977. Convex Analysis and Measurable Multifunctions. Springer--Verlag, Berlin.
  10. George Bernard Dantzig. A Proof of the Equivalence of the Programming Problem and the Game Problem. In Koopmans1951. On-line.
  11. George Bernard Dantzig. 1963. Linear Programming and Extensions. Princeton University Press, Princeton.
  12. Gerard Debreu. 1964. Nonnegative Solutions of Linear Inequalities. International Economic Review 5: 178–184. JSTOR
  13. Ivar Ekeland and Roger Temam. 1976. Convex Analysis and Variational Problems. North Holland, Amsterdam.
  14. Ivar Ekeland and Thomas Turnbull. 1983. Infinite-Dimensional Optimization and Convexity. University of Chicago Press, Chicago.
  15. Ky Fan, Irving Glicksberg, and A. J. Hoffman. 1957. Systems of Inequalities Involving Convex Functions. Proceedings of the American Mathematical Society 8: 617–622. JSTOR
  16. Werner Fenchel. 1953. Convex Cones, Sets, and Functions. Princeton University, Department of Mathematics. Lecture Notes. From notes taken by D. W. Blackett, Spring 1951.
  17. Monique Florenzano and Cuong Le Van. 2001. Finite Dimensional Convexity and Optimization. Springer--Verlag, New York and Heidelberg.
  18. Komei Fukuda. 2004. Frequently Asked Questions in Polyhedral Computation. Swiss Federal Institute of Technology. On-line.
  19. Komei Fukuda and A. Prodon. 1996. Double Description Method Revisited. In Combinatorics and Computer Science. M. Deza, R. Euler, and I. Manoussakis, ed. Springer--Verlag, Berlin. On-line.
  20. Jerry W. Gaddum. 1952. A Theorem on Convex Cones with Applications to Linear Inequalities. Proceedings of the American Mathematical Society 3: 957–960. JSTOR
  21. David Gale. Convex Polyhedral Cones and Linear Inequalities. In Koopmans1951. On-line.
  22. David Gale. 1960. Theory of Linear Economic Models. McGraw-Hill, New York.
  23. David Gale. 1969. How to Solve Linear Inequalities. American Mathematical Monthly 76: 589–599.
  24. David Gale, Victor Klee, and R. Tyrrell Rockafellar. 1968. Convex Functions on Convex Polytopes. Proceedings of the American Mathematical Society 19: 867–873. JSTOR
  25. David Gale, Harold W. Kuhn, and Albert W. Tucker. Linear Programming and the Theory of Games. In Koopmans1951. On-line.
  26. John R. Giles. 1982. Convex Analysis with Application in Differentiation of Convex Functions. Pitman Advanced Publishing Program, Boston.
  27. A. J. Goldman and Albert W. Tucker. Polyhedral Convex Cones. In KuhnTucker1956.
  28. Neil E. Gretsky, Joseph M. Ostroy, and William R. Zame. 2002. Subdifferentiability and the Duality Gap. Positivity 6: 261–274. On-line.
  29. Hubert Halkin. 1966. Necessary and Sufficient Condition for a Convex Set to be Closed. American Mathematical Monthly 73: 628–630. JSTOR
  30. Hubert Halkin. 1966. A Property of Nonseparated Convex Sets. Proceedings of the American Mathematical Society 17: 1389–1395. JSTOR
  31. G. H. Hardy, J. E. Littlewood, and G. Pólya. 1929. Some Simple Inequalities Satisfied by Convex Functions. Messenger of Mathematics 58: 145–152.
  32. Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal. 1993. Convex Analysis and Minimization Algorithms I. Springer--Verlag, Berlin.
  33. Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal. 1993. Convex Analysis and Minimization Algorithms II. Springer--Verlag, Berlin.
  34. Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal. 2001. Fundamentals of Convex Analysis. Springer--Verlag, Berlin.
  35. Lars Hörmander. 1954. Sur La Fonction D'Appui Des Ensembles Convexes Dans Un Espace Localement Convexe. Arkiv för Matematik 3: 181–186.
  36. Ralph Howard. 1998. Alexandrov's Theorem on the Second Derivatives of Convex Functions Via Rademacher's Theorem on the First Derivatives of Lipschitz Functions. Department of Mathematics, University of South Carolina. Lecture notes. On-line.
  37. Victor L. Klee, Jr. 1948. The Support Property of a Convex Set. Duke Mathematical Journal 15: 767–772. On-line.
  38. Victor L. Klee, Jr. 1949. A Characterization of Convex Sets. American Mathematical Monthly 56: 247–249.
  39. Victor L. Klee, Jr. 1951. Convex Sets in Linear Spaces. Duke Mathematical Journal 18: 443–466. On-line.
  40. Victor L. Klee, Jr. 1956. Strict Separation of Convex Sets. Proceedings of the American Mathematical Society 7: 735–737. JSTOR
  41. Victor L. Klee, Jr. 1963. Convexity. American Mathematical Society, Providence, RI.
  42. Victor L. Klee, Jr. 1963. On a Question of Bishop and Phelps. American Journal of Mathematics 85: 95–98. JSTOR
  43. Tjalling C. Koopmans, ed. 1951. Activity Analysis of Production and Allocation: Proceedings of a Conference. John Wiley and Sons, New York. On-line.
  44. Harold W. Kuhn and Albert W. Tucker, ed. 1950. Contributions to the Theory of Games, I. Princeton University Press, Princeton.
  45. Harold W. Kuhn and Albert W. Tucker, ed. 1953. Contributions to the Theory of Games, II. Princeton University Press, Princeton.
  46. Harold W. Kuhn and Albert W. Tucker, ed. 1956. Linear Inequalities and Related Systems. Princeton University Press, Princeton.
  47. Massimo Marinacci and Luigi Montrucchio. 2006. On Concavity and Supermodularity. Collegio Carlo Alberto. Carlo Alberto Notebooks. On-line.
  48. Theodore S. Motzkin. 1951. Two Consequences of the Transposition Theorem on Linear Inequalities. Econometrica 19: 184–185.
  49. Theodore S. Motzkin, Harold Raiffa, G. L. Thompson, and R. M. Thrall. The Double Description Method. In KuhnTucker1953.
  50. Hukukane Nikaidô. 1968. Convex Structures and Economic Theory. Academic Press, New York.
  51. Robert R. Phelps. 1993. Convex Functions, Monotone Operators and Differentiability. Springer--Verlag, Berlin.
  52. Josip E. Pečarić, Frank Proschan, and Y. L. Tong. 1992. Convex Functions, Partial Orderings, and Statistical Applications. Academic Press, New York.
  53. J. Ponstein. 1967. Seven Kinds of Convexity. SIAM Review 9: 115–119. JSTOR
  54. A. W. Roberts and D. E. Varberg. 1973. Convex Functions. Academic Press, New York.
  55. A. W. Roberts and D. E. Varberg. 1974. Another Proof that Convex Functions Are Locally Lipschitz. American Mathematical Monthly 81: 1014–1016. JSTOR
  56. R. Tyrrell Rockafellar. 1970. Convex Analysis. Princeton University Press, Princeton.
  57. R. Tyrrell Rockafellar. 1964. Duality Theorems for Convex Functions. Bulletin of the American Mathematical Society 70: 189-192. On-line.
  58. R. Tyrrell Rockafellar. 1968. Integrals Which Are Convex Functionals. Pacific Journal of Mathematics 24: 525–539. On-line.
  59. R. Tyrrell Rockafellar. 1971. Integrals Which Are Convex Functionals. II. Pacific Journal of Mathematics 39: 439–469. On-line.
  60. Josef Stoer and Christoph Witzgall. 1970. Convexity and Optimization in Finite Dimensions I. Springer--Verlag, Berlin.
  61. John W. Tukey. 1942. Some Notes on the Separation of Convex Sets. Portugaliae Mathematicae 3: 95–102.
  62. Frederick Valentine. 1964. Convex Sets. McGraw-Hill, New York.
  63. Hermann Weyl. The Elementary Theory of Convex Polyhedra. In KuhnTucker1950.
  64. Günter M. Ziegler. 1995. Lectures on Polytopes. Springer--Verlag, New York.

Applications

  1. Sydney N. Afriat. 1973. On a System of Inequalities in Demand Analysis: An Extension of the Classical Method. International Economic Review 14: 460–472. JSTOR
  2. Kenneth J. Arrow, Leonid Hurwicz, and Hirofumi Uzawa, ed. 1958. Studies in Linear and Non-Linear Programming. Stanford University Press, Stanford, California.
  3. David Blackwell. 1953. Equivalent Comparisons of Experiments. Annals of Mathematical Statistics 24: 265–272. JSTOR
  4. Kim C. Border. 1985. More on Harsanyi's Cardinal Welfare Theorem. Social Choice and Welfare 1: 279–281. On-line.
  5. Kim C. Border. 1991. Implementation of Reduced Form Auctions: A Geometric Approach. Econometrica 59: 1175–1187. JSTOR
  6. Kim C. Border. 1992. Revealed Preference, Stochastic Dominance, and the Expected Utility Hypothesis. Journal of Economic Theory 56: 20–42. On-line.
  7. Kim C. Border. 2007. Reduced Form Auctions Revisited. Economic Theory 31: 167–181. On-line.
  8. Donald J. Brown and Rosa L. Matzkin. 1996. Testable Restrictions on the Equilibrium Manifold. Econometrica 64: 1249–1262. JSTOR
  9. Donald J. Brown and Jan Werner. 1995. Arbitrage and Existence of Equilibrium in Infinite Asset Markets. Review of Economic Studies 62: 101–114. JSTOR
  10. Guillaume Carlier. A general existence result for the principal-agent problem with adverse selection. Journal of Mathematical Economics 35: 129 - 150. On-line.
  11. John S. Chipman, Daniel L. McFadden, and Marcel K. Richter, ed. 1990. Preferences, Uncertainty, and Optimality: Essays in Honor of Leonid Hurwicz. Westview Press, Boulder, Colorado.
  12. Jakša Cvitanić and Ioannis Karatzas. 1992. Convex Duality in Constrained Portfolio Optimization. Annals of Applied Probability 2: 767–818. JSTOR
  13. W. Erwin Diewert. 1971. An Application of the Shephard Duality Theorem: A Generalized Leontief Production Function. Journal of Political Economy 79: 481–507. JSTOR
  14. W. Erwin Diewert. 1973. Afriat and Revealed Preference Theory. Review of Economic Studies 40: 419–425. JSTOR
  15. W. Erwin Diewert and Alan D. Woodland. 1977. Frank Knight's Theorem in Linear Programming Revisited. Econometrica 45: 375–398. JSTOR
  16. Peter C. Fishburn. 1974. Convex Stochastic Dominance with Continuous Distribution Functions. Journal of Economic Theory 7: 143–158.
  17. Peter C. Fishburn. 1975. Separation Theorems and Expected Utility. Journal of Economic Theory 11: 16–34. On-line.
  18. David A. Freedman and Roger A. Purves. 1969. Bayes' Method for Bookies. Annals of Mathematical Statistics 40: 1177–1186. JSTOR
  19. David Gale. 1960. Theory of Linear Economic Models. McGraw-Hill, New York.
  20. David Gale. 1967. A Geometric Duality Theorem with Economic Applications. Review of Economic Studies 34: 19–24. JSTOR
  21. David Gale. 1973. On the Theory of Interest. American Mathematical Monthly 80: 853–868. JSTOR
  22. Arlo D. Hendrickson and Robert J. Buehler. 1971. Proper Scores for Probability Forecasters. Annals of Mathematical Statistics 42: 1916–1921. JSTOR
  23. David C. Heath and William D. Sudderth. 1972. On a Theorem of De Finetti, Oddsmaking, and Game Theory. Annals of Mathematical Statistics 43: 2072–2077. JSTOR
  24. Leonid Hurwicz. Programming in Linear Spaces. In ArrowHurwiczUzawa1958.
  25. Dale W. Jorgenson and Lawrence J. Lau. 1974. The Duality of Technology and Economic Behaviour. Review of Economic Studies 41: 181–200. JSTOR
  26. Tjalling C. Koopmans, ed. 1951. Activity Analysis of Production and Allocation: Proceedings of a Conference. John Wiley and Sons, New York. On-line.
  27. Tjalling C. Koopmans. 1953. Activity Analysis and Its Applications. American Economic Review 43: 406–414. JSTOR
  28. Tjalling C. Koopmans. 1961. Convexity Assumptions, Allocative Efficiency, and Competitive Equilibrium. Journal of Political Economy 69: 478–479. JSTOR
  29. Tjalling C. Koopmans. 1977. Concepts of Optimality and Their Uses. American Economic Review 67: 261–274. JSTOR
  30. David M. Kreps. 1981. Arbitrage and Equilibrium in Economies with Infinitely Many Commodities. Journal of Mathematical Economics 8: 15–35.
  31. Harold W. Kuhn and Albert W. Tucker, ed. 1950. Contributions to the Theory of Games, I. Princeton University Press, Princeton.
  32. John O. Ledyard. 1986. The Scope of the Hypothesis of Bayesian Equilibrium. Journal of Economic Theory 39: 59–82.
  33. Edmond Malinvaud. 1953. Capital Accumulation and Efficient Allocation of Resources. Econometrica 21: 233–268. JSTOR
  34. Edmond Malinvaud. 1962. Efficient Capital Accumulation: A Corrigendum. Econometrica 30: 570–573. JSTOR
  35. Rosa L. Matzkin and Marcel K. Richter. 1991. Testing Strictly Concave Rationality. Journal of Economic Theory 53: 287–303. On-line.
  36. Daniel L. McFadden and Marcel K. Richter. Stochastic Rationality and Revealed Preference. In ChipmanMcFaddenRichter1990.
  37. Paul R. Milgrom. 1981. Good News and Bad News: Representation Theorems and Applications. Bell Journal of Economics 12: 380–391. JSTOR
  38. Michael Mussa and Sherwin Rosen. 1978. Monopoly and Product Quality. Journal of Economic Theory 18: 301–317.
  39. Hukukane Nikaidô. 1968. Convex Structures and Economic Theory. Academic Press, New York.
  40. Bezalel Peleg and Menachem E. Yaari. 1970. Efficiency Prices in Infinite-Dimensional Space. Journal of Economic Theory 2: 41–85.
  41. Marcel K. Richter and Kam-Chau Wong. 2004. Concave Utility on Finite Sets. Journal of Economic Theory 115: 341–357. On-line.
  42. Jean-Charles Rochet. 1987. A Necessary and Sufficient Condition for Rationalizability in a Quasi-Linear Context. Journal of Mathematical Economics 16: 191–200.
  43. Dana Scott. 1964. Measurement Structures and Linear Inequalities. Journal of Mathematical Psychology 1: 233–247. On-line.
  44. Arja H. Turunen-Red and Alan D. Woodland. On Economic Applications of the Kuhn–Fourier Theorem. In Wooders1999.
  45. Hirofumi Uzawa. The Kuhn–Tucker Conditions in Concave Programming. In ArrowHurwiczUzawa1958.
  46. Hal R. Varian. 1987. The Arbitrage Principle in Financial Economics. Journal of Economic Perspectives 1: 55–72. JSTOR
  47. Hermann Weyl. Elementary Proof of a Minimax Theorem Due to von Neumann. In KuhnTucker1950.
  48. Myrna H. Wooders, ed. 1999. Topics in Mathematical Economics and Game Theory: Essays in Honor of Robert J. Aumann. American Mathematical Society, Providence, RI.
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Updated October 14, 2009 by KC Border.