Continuous Time Dynamic Math for Economists
This page started with my own notes and developed into a version I can eventually offer as a Master's level course or a specialty PhD course. Since I'm at an undergraduate teachingoriented university, during the school year I often do less theoretical work. Then whenever I return to certain research topics, I have to refresh my memory on integrals, differential equations and some key elements in optimization. So I started collecting that material. I realized that there is an intuitive path through this material that I think would be the best way for people to learn it anyway, at least its the logical path through my thinking. Essentially it runs like this: you need to know basic integration, then treat differential equations as an application. Once you get that far, you can focus on the intuition behind differential equations, but you need some key details. For example, if you understand quadratic equations and "finding the roots" of an equation, it's very obvious why you might want to find roots of differential equations. Then it's clear why you might focus on roots in systems of differential equations. Once you understand solving systems, you realize that it all turns on the coefficient matrix (the "A" matrix) and you can take whatever collection of equations you are working on and use the same basic techniques over and over. Having that consistent process helps you keep clear in your head where you are in the problem and where you need to go. Finally, you're ready to solve optimization problems. I always have Grad Students in mind. I wish I'd found a page like this when I was in Grad School.
Notes and disclaimers:

Currently these notes are handwritten. I plan to slowly put them into LaTex and PDF with a bit more explanation.

My aim everywhere is "getting to solving" things. I too plan to use this page as a resource when I forget stuff. I want to quickly look it up, see an example or walkthrough, then get back to work using that material.

The math required is: multivariate calculus, integration, and some basic linear algebra.

I draw heavily on a few sources and I encourage any PhD economics student to buy these.

Feguson and Lim  "" This is a great and simple resource. They do an amazing job explaining this material. And they go all the way to continuous time stochastic models. This is accessible to Masters students and earlystage PhDs.

Turnovsky  This is a PhD level book and pretty dense. I found that with the material I show here (Integration then basic Diff Eqs) you can access all the material.

Schaum's Outline  This is accessible to undergrads. I taught myself a lot of the math I know from this book between my undergraduate and graduate studies. It's not enough, but I find it an amazing, quick reference.

Integration

Every time I work in this area I personally have to refresh my basic integration techniques. So I collected here some key rules we use in economics and then some applications for practice.

01 Integration Review (handwritten download)

Basic Rules

L'Hopital's Rule

Leibnitz's Rule

Partial Integration

Exact Differential Equations

Integrating Factors



02 Integration Application: Probabilities (download)
Differential Equations Part I: Solving the Equations

I try to show these as an application from Integration. My focus is on solving differential equations. I ignore the complex cases and focus on getting you to solving them. Then I hope to provide some better understanding and intuition.

03 Differential Equation Basics (download)

Jumping in with direct solutions and steps to solving

General Solutions and Exact Differential Equations (nearly identical to the material covered under integrals but I wanted diff eqns all in one place so I added it here too).

Brute Force (just apply the general solution and work through it)

Separation of Variables: Proportional Growth example (probably most common basic Econ application of diff equations)

Steps to Solving First Order Diff Equations: 1) rewrite, 2) integrating factor, 3) multiply through and integrate, 4) rearrange and solve

Bernoulli Equations (shows up in Solow growth models already)

 04 Differential Equations Application: Term Structure of Interest Rates (download)
Differential Equations Part II: Intuition and Building up to Systems

When I first learned to solve for the characteristic roots (also called "eigenvalues") and such, I just did it and didn't care why. But later (actually when working through Ferguson and Lim), I understood why I cared. But math isn't all compartmentalized like we teach it. So I start with why and when we use the quadratic equation to solve for the roots of a quadratic equation and build the intuition from there. I can't wait to teach this material and see if that insight helps students advance to eigenvalues and eigenvectors for systems. I hope so.

05 Characteristic Equations, Roots and Solving Systems (download)

Refresh on roots and quadratic equations

Roots and basic differential equations

Example: Dynamic Supply and Demand Model

Stability

The Discriminant

Solving Systems of Differential Equations

I try to emphasize that you just need to understand the "A" matrix and why. What you slowly realize over time is that people aren't usually resolving everything every time they solve a new problem. Most people force their problems into a specific form or arrangement that matches a problem they know how to solve. So, if you learn what to do with a generic A matrix, then you learned 100% of what you need. Now when you hit a problem, rewrite your problem to fit this structure and you know what to do.

06 Solving Differential Equations Systems with Matrices (download)

Overview

The A Matrix, Eigenvalues and Eigenvectors

Making Eigenvectors unique with normalizations

Solving for constants with initial values

solving a small system by hand"

solving the same system with Cramer's Rule which helps if the system is larger


 07 Solving Differential Equation Systems with Matrices more Generally (download)
Saddlepaths and Stable Branches and Solving Everything for Phase Diagrams

This is the key to solving dynamic macroeconomic models in continuous time. Period. Whether you draw a phase diagram in the end or simulate your model (which I don't do here), this section shows you how to understand and solve for everything. It shows how to do announced/anticipated cases in perfect foresight models and solve for every piece of the model. I follow Turnovsky (2000) really closely here. He just has a nice approach he uses over and over that convinced me it's a great way to think of things.

08 Saddlepaths and Stable Branches (download)

Restructuring your system to get the "A" matrix you want
 Deviations from steady state
 Anticipated future shocks: initial jumps, time paths and final convergence.
 Review and connecting the math to phase diagrams

 09 Saddlepaths Application: Cagan Money Model with Sluggish Wage Adjustment (download)
 10 Saddlepaths and Stable Branches Deeper Dive: photocopy from Turnovsky textbook for now (download)
Appendices to Differential Equation Section (not finished)

Linear Homogeneity and Normality Conditions that often help sign things: (download)

Example with a Ramsey Model (from Carlos Vegh's textbook)
 Complex roots, repeating roots
 Fourier Transforms to solve differential equations
Dynamic Optimization

Optimal Control Lagrangians, Hamiltonians and All That (download)
Stochastic Calculus

Basics and Introduction (download)

John Cochrane's page for Asset Pricing. He covers Stochastic Calculus at the beginning of his course and you can find both videos and PDF notes. I highly recommend the videos and the notes. They really are the best Stochastic Calculus notes I've found. It's from these that I finally understood the stochastic integral only in 2020!! and it's super simple!
Continuous Time Stochastic Optimization

Stochastic Optimal Control Basics (download)

Stochastic Optimal Control Solutions (download)

Stochastic Optimal Control with Poisson and General Ito (download)

More complicated material (currently copies from Turnovsky, 2000)
Simulation Methods (not finished)

Excel and Euler Approximations (based heavily on Shone)

Solving and approximating differential equations in R and Matlab (anyone who wants to convert my code to python is invited!)

Discretize your model and use Dynare.
Cleve Moler (MATLAB) Teaches Solving ODEs in MATLAB
These are all short. Usually 515 minutes in length
Gilbert Strang's Highlights of Calculus

Prof Gilbert Strang's short collection of videos and why he made these (click here)

MIT Courseware: click here

Prof Strang's Free Calculus textbook (download here).

Highlights and the Big Picture of Calculus

Derivatives
Gilbert Strang and Dr. Cleve Moler (MATLAB) "Learn Differential Equations: Up Close" (click here), MIT OpenCourseware, 2015
There are a lot here, but each are relatively short, about 15 minutes each.

FIRST ORDER EQUATIONS

Response to Exponential Input

Response to Oscillating Input

Solution for Any Input

Step Function and Delta Function

Response to Complex Exponential

Integrating Factor for Constant Rate

Integrating Factor for a Varying Rate

The Logistic Equation

The Stability and Instability of Steady States

Separable Equations


SECOND ORDER EQUATIONS

Second Order Equations

Forced Harmonic Motion

Unforced Damped Motion

Impulse Response and Step Response

Exponential Response – Possible Resonance

Second Order Equations with Damping

Electrical Networks: Voltages and Currents

Method of Undetermined Coefficients

An Example of Undetermined Coefficients

Variation of Parameters

Laplace Transform: First Order Equation

Laplace Transform: Second Order Equation

Laplace Transforms and Convolution


GRAPHICAL AND NUMERICAL METHODS

Pictures of Solutions

Phase Plane Pictures: Source, Sink, Saddle

Phase Plane Pictures: Spirals and Centers

Two First Order Equations: Stability

Linearization at Critical Points

Linearization of two nonlinear equations

Eigenvalues and Stability: 2 by 2 Matrix, A

The Tumbling Box in 3D


VECTOR SPACES AND SUBSPACES

The Column Space of a Matrix

Independence, Basis, and Dimension

The Big Picture of Linear Algebra

Graphs

Incidence Matrices of Graphs


EIGENVALUES AND EIGENVECTORS

Eigenvalues and Eigenvectors

Diagonalizing a Matrix

Powers of Matrices and Markov Matrices

Solving Linear Systems

The Matrix Exponential

Similar Matrices

Symmetric Matrices, Real Eigenvalues, Orthogonal Eigenvectors

Second Order Systems


APPLIED MATHEMATICS AND ATA

Positive Definite Matrices

Singular Value Decomposition (the SVD)

Boundary Conditions Replace Initial Conditions

Laplace Equation


FOURIER AND LAPLACE TRANSFORMS

Fourier Series

Examples of Fourier Series

Fourier Series Solution of Laplace's Equation

Heat Equation

Wave Equation

Prof Gilbert Strang's 2008 MIT Courseware Course MIT 18.085 Computational Science and Engineering: click here