KC Border's notes on selected topics in mathematical economics

Colleagues and students:

A few years ago, I began to put my course “handouts” on the web instead of handing them out. Here is a list of some of them. I offer no warranty as to the correctness of these notes, but I am not aware of any errors. However, many of them are incomplete. Feel free to use them in your courses or studies.

I will add to this list as I have time. The file sizes and dates reflect the latest uploads. The revision date in the text reflects only major revisions. If I find typos, reformat a few equations, or simply play around with different fonts, I may upload the new version without changing the revision date in the text. Sorry, but I'm just not compulsive enough to assign version numbers and keep them up to date. (As I make revisions, I am now adding a time stamp that indicates the last time the file was converted to pdf from LaTeX. This is usually an indication that something has changed, even if only the format.)

I'd appreciate it if you let me know if you find any mistakes, or if you just find something too obscure. Corrections are also greatly appreciated.

— KCB

Mathematical Economics

Cost and production

• Notes on support functions, or profit and cost functions. 118K, August 21, 2009.
• Traditional approach to cost minimization. 120K, November 25, 2008.
• The Cobb--Douglas Production Function. 64K, May 6, 2008.
• Production possibility frontier and marginal costs. 96K, November 4, 2003.
• Worked examples. 103K, October 14, 2004.

Rational choice and demand theory

• Rational choice as I interpret Ket Richter. 236K, November 10, 2009.
• Introductory notes on stochastic rationality. 184K, January 14, 2009.
• Preference relations and their properties. 179K, October 2, 2008.
• From the point of view of competitive demand theory, local nonsatiation is no more general than monotonicity. 272K, June 28, 2009.
• Neoclassical demand theory, as you might find in Samuelson's Foundations. 145K, November 11, 2009.
• Duality and demand theory. 74K, November 9, 2004.
• The weak axiom of revealed preference implies negative semidefiniteness of the Slutsky matrix. 97K, November 29, 2001.
• Partial notes on Hurwicz and Uzawa's approach to the integrability problem. 225K, August 10, 2005.
• Review of demand theory. A crib sheet for qualifying exams. 70K, October 31, 2009.
• Worked examples of deriving demand functions from utility functions in the full n-good case. Includes Cobb–Douglas (logarithmic and exponential versions), linear, Leontieff, quasi-linear Cobb–Douglas; with accurate pictures of indifference curves. 118K, February 24, 2008.

Arrow–Debreu–McKenzie–Nikaidô Model

• Introduction to the Arrow–Debreu–McKenzie–Nikaidô Model. 109K, February 25, 2009.
• Notes on boundedness of the set of allocations. 103K, April 3, 2003.
• Statement and proof of the First Welfare Theorem. 78K, February 25, 2009.
• Statement and proof of the Second Welfare Theorem. 115K, February 27, 2002.
• Hurwicz's approach to the Second Welfare Theorem via Saddlepoints. 111K, November 5, 2004.
• Notes on the core of an economy, based on Debreu and Scarf and Richter. 111K, February 21, 2006.
• (Non-)Existence of Walrasian Equilibrium. Examples of what can go wrong when standard assumptions are violated, plus a sketch of an existence proof based on the excess demand approach. 197K, April 28, 2009.
• Worked examples of computing Walrasian equilibria for Robinson Crusoe economies. 68K, April 10, 2003.

Decision under undertainty

• Desultory notes on Decision Theory including Dutch book arguments. 235K, January 16, 2009.
• Note on the de Finetti–Arrow–Pratt coefficient of risk aversion. 228K, November 12, 2003.

Financial economics

• Investment and risk aversion. 90K, September 11, 2003.
• Sargent on Lucas “tree” models. 94K, April 30, 2008.
• Notes on Arbitrage. 242K, December 16, 2008.

Search

• Sargent on job search. 62K, May 5, 2005.
• Sargent on job matching. 133K, December 4, 2003.

Miscellany

• A short proof of Arrow's General (Im)Possibility Theorem. 81K, February 26, 2002.
• Notes on Common Knowledge. 158K, August 26, 2008.
• A clarification of Holmstrom's appendix from moral hazard in teams. 124K, February 25, 2009.
• An introduction to trade with private information. 114K, December 6, 2006.

Mathematics

Calculus

• The first-order linear differential equation and its economic interpretation. 107K, October 30, 2007.
• Maximization of a function of one variable. 214K, September 29, 2008.
• Maximization of a function of several variables. 229K, October 14, 2004.
• Envelope theorem. 161K, October 23, 2008.
• Comparative statics. 86K, September 28, 2001.
• Increasing differences and comparative statics. 92K, October 12, 2001.
• Euler's theorem on homogeneous functions. 96K, September 28, 2004.
• Leibniz' rule for differentiating an integral. 1.2M, September 23, 2008.
• Advanced integration by parts. A bit sloppy in places. 149K, May 25, 2005.

Linear algebra

• What you should remember about linear algebra and matrices. 347K, September 10, 2008.
• More than you want to know about quadratic forms. 181K, July 23, 2009.
• Notes on various versions of the Theorem of the Alternative. 261K, September 25, 2008.

Convex analysis

• Separating Hyperplane Theorems. 368K, June 19, 2009.
• Supergradients and subgradients. 237K, October 15, 2003.
• Theorem of the Alternative. 261K, September 25, 2008.
• Saddlepoints and constrained maximization. 155K, October 16, 2005.
• Notes on the theory of Linear Programming and saddlepoints. 138K, February 25, 2009.
• Notes on the Simplex Algorithm and how to use it. 296K, February 25, 2009.
• An example of a pointed, generating, finite cone in R⁴ whose dual cone has a different number of extreme rays, which I prepared at the behest of Roko Aliprantis. 129K, December 15, 2004.

Metric spaces and topology

• What you should remember about metric spaces. 199K, October 14, 2009.
• Correspondences. 156K, April 23, 2009.

Probability

• Stochastic Dominance. 133K, October 31, 2001.
• Kolmogorov extension theorem. 221K, July 19, 2004.

Miscellany

• Asymptotic Cones. 136K, January 29, 2002.
• Fixed point theorems. 195K, April 26, 2005.
• Lattice comparative statics. 197K, March 19, 2003.

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