THE UNIVERSITY OF MARYLAND
COLLEGE PARK MARYLAND
Math Camp
July 27 – August 26, 2010
Sushant Acharya
Jeffrey Borowitz
Office: TYD 3115M
Office: MOR 0105
Email: acharya@econ.umd.edu
Email: borowitz@econ.umd.edu
Office Hours: TBA
Office Hours: TBA
Lecture:
M-F 9:00am-12:00pm
Discussion: M-F 1:00pm-3:00pm
Room:
Computer Science Instructional Center (CSI) - Room 1122
Mathematical Economics Outline
The purpose of this course is to develop basic mathematical tools and intuition that will be
valuable in analyzing a wide variety of economic problems. Through the first part, we will cover
static optimization problems and related topics. Following the course on Probability and
Statistics, we will continue with dynamic economic analysis.
The course textbooks are Simon, C. and L. Blume: Mathematics for economists. Norton and Co.
New York. 1994 (referred to in the readings as (SB)) and De La Fuente, A. (2000): Mathematical
methods and models for economists, Cambridge: Cambridge University Press (referred to as
(DLF)). Most of the topics covered in the course are treated in these books though we will
deviate from the treatment of the texts in the lectures. Other useful resources are:
Static Optimization:
Edlin, A. and C. Shannon. “Strict monotonicity in comparative statics”, Journal of Economic
Theory 81,no.1, July 1998. Pp. 201-219. (ES)
Franklin, J. Methods of mathematical economics: Linear and nonlinear programming, fixed
point theorems. Springer-Verlag. New York. 1980. (F)
Kolmogorov, A. and S.V. Fomin. Introductory real analysis. Dover Publications. New York.
1970. (KF)
Luenberger, D. Optimization by vector space methods. John Wiley. New York. 1969. (L)
Mas-Colell, A., Whinston, M. and J. Green. Microeconomic theory. Oxford University Press.
Oxford. 1995. (MWG)
Milgrom, P. and C. Shannon. “Monotone comparative statics.” Econometrica 64. 1994. Pp. 151181. (MS)
Rockafellar, R.T. Convex analysis. Princeton University Press. Princeton. Princeton. 1970. (R)
Rudin, W. (1964) Principles of Mathematical Analysis, New York: McGraw-Hill
Sundaram, R.K. (1996): A First Course in Optimization Theory, Cambridge: Cambridge
University Press. (S)
Takayama, A. Mathematical economics. Second Edition. Cambridge University Press. 1985 (T)
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Dynamic Optimization:
Chiang, A. (1992): Elements of Dynamic Optimization, New York: McGraw Hill,
Dixit, A. (1990): Optimization in Economic Theory, 2nd Ed., Oxford: Oxford University Press
Farmer, R. (1999): The Macroeconomics of Self-Fulfilling Prophecies, 2nd Ed. Cambridge, MA:
MIT-Press.
Stokey, N. and R. Lucas with E. Prescott (1989): Recursive Methods in Economic Dynamics,
Cambridge, MA: Harvard University Press.
Merton, R. (1990): Continuous-Time Finance, Oxford: Blackwell Publishers,
Perko, L. (2001): Differential Equations and Dynamical Systems, 3rd Ed., New York: Springer
Verlag.
Sargent, T. (1987): Dynamic Macroeconomic Theory, Cambridge, MA: Harvard University
Press.
Takayama, A. (1994): Analytical Methods in Economics, Ann Arbor: University of Michigan
Press.
There will be one test worth 100% of the grade. Problem sets along with solutions in PDF will be
provided at www.econ.umd.edu/vincent. You are strongly recommended to try these problems
(in groups, if so desired) before the review sessions and before looking at the solutions. Note,
however, the suggested solutions do not come with a guarantee! We also offer some crude lecture
notes at the same site. There may well be typos on them and you use them at your own risk!
The order of material in lectures may vary slightly from the list below.
Static Optimization
I) Preliminary Concepts
i) Some Examples.
ii) Continuity and Linearity.(SB 13; DLF 1.5.b, 2.1-2.8)
iii) Vector geometry. (SB 10.1- 10.4)
iv) Hyperplanes – Definition. Supporting and separating hyperplanes. (DLF 6.1.c-6.1.d)
v) Derivatives and Gradients. (SB 14.4, 14.6, 14.8; DLF 4.1-4.2)
vi) Homogeneous and Homothetic Functions (SB 20.1, 20.4; DLF 4.5)
vii) Some More Geometry of Vectors in Rn
II) Concepts and Problems in Unconstrained Optimization
i) Convexity, concavity and quasi-concavity. (SB 16, 21.1-21.3; DLF 6.1.a, 6.2-6.3)
ii) Necessary conditions for an Optimum. (SB 17.1- 17.4; R 23, 25, 27)
iii) Sufficient conditions for an Optimum. (SB 17.1- 17.4)
iv) Minimizing versus maximizing.
III) Constrained Optimization I– Representing Constraint Sets
i) Some Examples.
ii) Functional Representation of Constraint Sets.
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iii) Open, Closed, Bounded Sets. (SB 12;DLF 2.4)
iv) Convex sets.
IV) Constrained Optimization II – Kuhn Tucker Theory
i) Examples.
ii) Lagrange’s Theorem (SB 18.2; DLF 7.1.b)
iii) Kuhn-Tucker Theorem and Differentiability (Necessary Conditions) (SB 18.3, 18.6;
DLF 7.1.c)
iv) The Constraint Qualification.
v) Complementary Slackness.
vi) The Saddle-Value Theorem and the Linear Programming Theorem. (SB 21.5).
V) Applications and Examples
i) Consumer Choice: Non-convex choice sets. (DLF 8.1-8.2)
ii) Cost Functions and Shephard’s Lemma
VI) Comparative Statics
i) Implicit function theorem. (SB 15.1- 15.3.)
ii) The Theorem of the Maximum. (S 9.2)
iii) The Envelope Theorem (SB 19.1, 19.2)
iv) Correspondences and Fixed Point Theorems (S 9.1, 9.4)
v) Monotone Comparative Statics. Readings: MWG MK, ML, ES, MS.
Dynamic Optimization
VII) Discrete Time Intertemporal Optimization
i) Alternative Methods of Discrete Time Intertemporal Optimization
ii) The Maximum Principle
iii) Dynamic Programming (DLF, 12.1)
 DLF, Chapter 13
 Dixit, A.K. (1990): Optimization in Economic Theory, 2nd Ed., Oxford: Oxford University Press,
Chapters 10 and 11.
Sundaram, R.K. (1996): A First Course in Optimization Theory, Cambridge: Cambridge University Press,
Chapters 11 and 12, Appendix C.
VIII) Difference Equations
i) Basic Concepts for Univariate equations (DLF, 9.1, 9.2, 9.4, 9.5)
ii) Linear Systems (DLF, 10.1, 10.2)
iii) Elements of Nonlinear Systems (DLF, 10.3)
 DLF, Chapter 11
 Sargent, T.J. (1987): Macroeconomic Theory, 2nd Ed., New York: Academic Press, Chapter 9.
 Simon, C.P., and L. Blume (1994): Mathematics for Economists, New York: W.W. Norton, Chapter
23.
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VIII) Differential Equations (If time allows)
i) Basic Concepts for Univariate equations (DLF, 9.1, 9.2, 9.3, 9.5)
ii) Linear Systems (DLF, 10.1, 10.2)
iii) Elements of Nonlinear Systems (DLF, 10.3)
 DLF, Chapter 11
 Perko, L. (2001): Differential Equations and Dynamical Systems, 3rd Ed., New York: Springer Verlag,
Chapters 1 and 2.
 Simon, C.P., and L. Blume (1994): Mathematics for Economists, New York: W.W. Norton, Chapters
24 and 25.
X) Continuous Time Intertemporal Optimization (Not Covered)
i) The Maximum Principle (DLF, 12.2)
ii) Dynamic programming
 Chiang, A.C. (1992): Elements of Dynamic Optimization, New York: McGraw Hill, Chapters 7 to 10.
 DLF, Chapter 13
 Dixit, A.K. (1990): Optimization in Economic Theory, 2nd Ed., Oxford: Oxford University Press,
Chapters 10 and 11.
XI) Extensions to Stochastic Setting (Not covered)
i) Stochastic Difference Equations
ii) Stochastic Discrete Time Intertemporal Optimization
iii) Stochastic Differential Equations
iv) Stochastic Continuous Time Intertemporal Optimization
 Dixit, A.K. (1990): Optimization in Economic Theory, 2nd Ed., Oxford: Oxford University Press,
Chapter 11.
 Dixit, A.K., and R.S. Pindyck (1994): Investment Under Uncertainty, Princeton: Princeton University
Press, Chapters 3 and 4.
 Farmer, R.E.A. (1999): The Macroeconomics of Self-Fulfilling Prophecies, 2nd Ed., Cambridge, MA:
MIT-Press, Chapters 2 and 3.
 Merton, R.C. (1990): Continuous-Time Finance, Oxford: Blackwell Publishers, Chapters 3 to 5.
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