ECON 8000: Math for Economists

Calendar

August 2–22, 2026
Sunday	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday
2	3	4 Lecture 1 Sets, Functions, Logic, and Proofs	5 Lecture 2 Real Analysis in Rn	6 Lecture 3 Linear Algebra PS1 DUE	7 Lecture 4 Differential Calculus	8
9	10 Lecture 5 Curvature, Convexity, and Separation PS2 DUE	11 Lecture 6 General Optimization	12 In-class review PS3 DUE	13 In-class midterm	14 Lecture 7 Constrained Optimization	15
16	17 Lecture 8 Comparative Statics, Envelopes, and Fixed Points	18 Lecture 9 Dynamic Programming and the Bellman Equation PS4 DUE	19 No class: international student orientation	20 Take-home final	21 Take-home final	22

Assessment

Four problem sets: 5% each
In-class midterm: 30% (August 13)
Take-home final: 50% (August 20–21)

Problem sets

PS1 covers Lectures 1 and 2; due at Lecture 3.
PS2 covers Lectures 3 and 4; due at Lecture 5.
PS3 covers Lectures 5 and 6; due at the in-class review.
PS4 covers Lectures 7 and 8; due at Lecture 9.

Topic outline

Lecture 1: Sets, Functions, Logic, and Proofs

Sets: definition and operations; Cartesian product and ℝⁿ
Functions: vocabulary and properties; inverse function
Logic: logical statements and operations; implication and equivalence; properties and quantifiers
Proofs: what a proof is; direct proof and contrapositive; proof by contradiction; proof by induction

Lecture notes (PDF) Slides (PDF)

Lecture 2: Real Analysis in ℝⁿ

Distance and sequences: norm and distance; convergence, subsequences, and limits
Suprema and compactness: suprema and maximizing sequences; closed, bounded, and compact sets
Continuity and existence: continuity; the Weierstrass theorem; intermediate values and fixed points

Lecture notes (PDF) Slides (PDF)

Lecture 3: Linear Algebra

Matrix algebra
Linear maps and linear systems
Square matrices and invertibility; eigenvalues and eigenvectors

Lecture notes (PDF) Slides (PDF)

Lecture 4: Differential Calculus

Functions of one variable
Real-valued functions of several variables
Vector-valued functions

Lecture notes (PDF) Slides (PDF)

Lecture 5: Curvature, Convexity, and Separation

Second-order approximation and curvature
Convex sets and concave functions
Concavity criteria, global optimality, and separation

Lecture notes (PDF) Slides (PDF)

Lecture 6: General Optimization

The problem and existence
Necessary local conditions
Global certification

Lecture notes (PDF) Slides (PDF)

Lecture 7: Constrained Optimization

Equality constraints and the Lagrangian
Inequalities and the KKT conditions
Global certification

Lecture notes (PDF) Slides (PDF)

Lecture 8: Comparative Statics, Envelopes, and Fixed Points

The implicit function theorem
Comparative statics of optimizers and values
Fixed-point theorems

Lecture notes (PDF) Slides (PDF)

Lecture 9: Dynamic Programming and the Bellman Equation

Sequential problems and their recursive form
First-order and envelope conditions; the Euler equation
Finite horizons, fixed points, and a Bellman equation solved by hand

Lecture notes (PDF) Slides (PDF)

References

No textbook is required; the lecture notes are self-contained. The core references for the course are:

Michael Carter, Foundations of Mathematical Economics (MIT Press, 2001).
Angel de la Fuente, Mathematical Methods and Models for Economists (Cambridge University Press, 2000).
Efe A. Ok, Real Analysis with Economic Applications (Princeton University Press, 2007).
Walter Rudin, Principles of Mathematical Analysis, 3rd edition (McGraw-Hill, 1976).
Carl P. Simon and Lawrence Blume, Mathematics for Economists (W. W. Norton, first U.S. edition, 1994).
Nancy L. Stokey and Robert E. Lucas, Jr., with Edward C. Prescott, Recursive Methods in Economic Dynamics (Harvard University Press, 1989).
Rangarajan K. Sundaram, A First Course in Optimization Theory (Cambridge University Press, 1996).

Additional references for particular topics are:

Daron Acemoglu, Introduction to Modern Economic Growth (Princeton University Press, 2009).
Richard Bellman, Dynamic Programming (Princeton University Press, 1957).
Kim C. Border, Fixed Point Theorems with Applications to Economics and Game Theory (Cambridge University Press, 1985).
Dean Corbae, Maxwell B. Stinchcombe, and Juraj Zeman, An Introduction to Mathematical Analysis for Economic Theory and Econometrics (Princeton University Press, 2009).
Ulrich Daepp and Pamela Gorkin, Reading, Writing, and Proving: A Closer Look at Mathematics, 2nd edition (Springer, 2011).
Richard Hammack, Book of Proof, edition 3.2 (2018).
Lars Ljungqvist and Thomas J. Sargent, Recursive Macroeconomic Theory, 4th edition (MIT Press, 2018).
David Romer, Advanced Macroeconomics, 5th edition (McGraw-Hill Education, 2019).
Daniel J. Velleman, How to Prove It: A Structured Approach, 3rd edition (Cambridge University Press, 2019).

Course details

Calendar

Assessment

Problem sets

Topic outline

Lecture 1: Sets, Functions, Logic, and Proofs

Lecture 2: Real Analysis in ℝn

Lecture 3: Linear Algebra

Lecture 4: Differential Calculus

Lecture 5: Curvature, Convexity, and Separation

Lecture 6: General Optimization

Lecture 7: Constrained Optimization

Lecture 8: Comparative Statics, Envelopes, and Fixed Points

Lecture 9: Dynamic Programming and the Bellman Equation

References

Lecture 2: Real Analysis in ℝⁿ