THE UNIVERSITY OF MARYLAND
COLLEGE PARK, MARYLAND
Math Camp
Static Optimization: July 25 - August 7, 2012
Dynamic Optimization: August 20 - August 24, 2012
Jonathan Kreamer
Office: Tydings 4118B
Email: kreamer@econ.umd.edu
Brian Quistorff
Office: Tydings 4118L
Email: quistorff@econ.umd.edu
Office Hours: M-F 3:00 - 3:30pm Office Hours: M-F 8:30 - 9:00am
Lecture: M-F 9:00am - 12:00pm
Discussion: M-F 1:00pm - 3:00pm
Room: Tydings 2110
Mathematical Economics Outline
The purpose of this course is to develop mathematical tools and intuition that will be valuable in analyzing a wide variety of economic problems. Through 8 lectures, we will cover static optimization problems and related topics. Following an 8 day course on Probability and
Statistics, we will continue with 5 lectures on 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.
151-181. (MS)
Rockafellar, R.T. Convex Analysis . Princeton University Press. Princeton. Princeton. 1970. (R)
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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)
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 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 R n
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.
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III) Constrained Optimization I– Representing Constraint Sets i) Some Examples. ii) Functional Representation of Constraint Sets. 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 , 2 nd
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
)
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
DLF, Chapter 11

Sargent, T.J. (1987): Macroeconomic Theory , 2 nd
Ed., New York: Academic Press, Chapter 9.

Simon, C.P., and L. Blume (1994): Mathematics for Economists , New York: W.W. Norton, Chapter
23.
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 , 3 rd
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):
Chapters 10 and 11.

Optimization in Economic Theory , 2 nd
Ed., Oxford: Oxford University Press,
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 , 2 nd
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 , 2 nd
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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