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Mathematical Economic Modelling for Aspiring PhD Students

These are the notes for a crash course on mathematical economic modelling for students wanting to go on to PhD's in economics.  My intention is to help these students "keep things clear" in their heads during their first year in graduate school and then hope it's a resources for them later in their program or even later when they are professors.

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The page and approach reflects my bias as a macroeconomist. I get to dynamic models pretty quickly, for example. But the heart of my approach is keeping the basic approach simple.  90% of the problems you'll solve in grad school are maximizing utility subject to budget constraints.   The rest of the time we seem to just be working on mathematical tools to deal with the complications resulting from your utility maximation problem.

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I try to focus on the core optimization problem in the least technical way possible in order to not get lost and help you learn the process you will use over and over.  I then add appendices to cover a range of topics.

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Finally, I assume you know multivariate calculus and basic integration and that's about it.

MAIN COURSE

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Introduction​

  • How modern economists model the world

  • The basic process/path: optimization subject to budget constraints

  • DONWLOAD

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Static Optimization: Unconstrained​​

  • Marginal Benefit = Marginal Cost

  • Firms maximizing profit

  • Utility maximization

  • DONWLOAD

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Static Optimization: Constrained​

  • Generally microeconomic problems

  • Firms maximizing profit as cost minimization

  • Utility maximization

  • DONWLOAD

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Static Optimization Examples and Complications

  • Some common problems for practice

  • A common complication: Handling Uncertainty

  • DONWLOAD

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Dynamic (Discrete-time) Optimization: 2-Period Problems​

  • Generally these are macroeconomic problems.

  • Lagrangians and utility maximization subject to budget constraints

  • The Euler equation

  • DONWLOAD

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Dynamic (Discrete-time) Optimization: Longer-Horizon Problems​

  • Generally these are macroeconomic problems.

  • Lagrangians and utility maximization subject to budget constraints

  • Getting more from FOCS: Equations of Motion

  • DONWLOAD

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Discrete-time Dynamic Examples and Complications

  • Ramsey-Cass-Koopmans Growth

  • Open Economies

  • DONWLOAD

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Dynamic (Continuous-time) Optimization

  • Generally these are macroeconomic problems.

  • Lagrangians and Hamiltonians

  • DONWLOAD

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Continuous-time Dynamic Examples and Complications

  • Ramsey-Cass-Koopmans Growth

  • Open Economies

  • DONWLOAD

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APPENDICES

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Creating Utility Functions​

  • Preferences and basic rationality assumptions

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Commonly Used Functions​

  • Common utility functions

    • Arrow-Pratt risk aversion​

  • Common production functions

  • DONWLOAD

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Optimality Conditions and Larger Problems​

  • Hessians and Bordered Hessians (download)

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Squeezing More from Static Problems​

  • Slutsky

  • Marginal Rates of Transformation

  • Indirect Utility, Expenditure Functions

  • Hicksian Compensated Demand

  • DONWLOAD

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Handling Corner Solutions

  • Khun-Tucker conditions

  • DONWLOAD

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Alternative Optimization Method

  • Optimizing with Separating Hyperplanes

  • DONWLOAD

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Time Series: Basics​

  • Time series, AR(1), etc. (download)

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Time Series: Solving Difference Equations​

  • Iteration (download)

  • Method of Undetermined Coefficients (dowload)

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Integration​

  • Integration Review (handwritten download)​

  • Integration Application: Probabilities (download)​

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Solving Differential Equations​

  • Forcing the solution (download)

  • Method of Undetermined Coefficients (download)​

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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 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)

    • Stochastic Calculus and Optimization (download)​

    • Stoch Calc Application: Model Part 1 (download)

    • Stoch Calc Application: Model Part 2 (download)

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Cleve Moler (MATLAB) Teaches Solving ODEs in MATLAB

These are all short.  Usually 5-15 minutes in length

Gilbert Strang's Highlights of Calculus

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.

  • INTRODUCTION

  • 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 3-D

  • 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

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

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