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Continuous Time Dynamic Math for Economists

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. Then, 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 or roots of systems of 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 put them into LaTex and PDF.

  • My aim everywhere is "getting to solving" things.  I want people to be able to quickly look something up, understand it, 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 early-stage 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.

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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 equations.  

  • 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)
    • Diff Eqn Application: Rep Individual Constraint (download)​
    • Diff Eqn Application: Government Budget Constraints and Sustainability (download)​

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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.  Here, 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.

  • 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 re-solving everything every time they solve a new problem.  Most people force their problems into a specific form or arrangement that matches a problem they already 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 large

  • 07 Solving Differential Equation Systems with Matrices more Generally (download)

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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 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 has a nice approach that he uses over and over. He 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)​

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Appendices to Differential Equation Section (not finished)

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

  • Representative Agent example (download)

  • Complex roots, repeating roots
  • Fourier Transforms to solve differential equations

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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 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)

    • Stochastic Calculus and Optimization (download)​

    • Stoch Calc Application: Model Part 1 (download)

    • Stoch Calc Application: Model Part 2 (download)

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