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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Introductionâ€‹

How modern economists model the world

The basic process/path: optimization subject to budget constraints
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Static Optimization: Unconstrainedâ€‹â€‹

Marginal Benefit = Marginal Cost

Firms maximizing profit

Utility maximization

DONWLOAD
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Static Optimization: Constrainedâ€‹

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 (Discretetime) Optimization: 2Period Problemsâ€‹

Generally these are macroeconomic problems.

Lagrangians and utility maximization subject to budget constraints

The Euler equation

DONWLOAD
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Dynamic (Discretetime) Optimization: LongerHorizon Problemsâ€‹

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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Discretetime Dynamic Examples and Complications

RamseyCassKoopmans Growth

Open Economies

DONWLOAD
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Dynamic (Continuoustime) Optimization

Generally these are macroeconomic problems.

Lagrangians and Hamiltonians

DONWLOAD
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Continuoustime Dynamic Examples and Complications

RamseyCassKoopmans Growth

Open Economies

DONWLOAD
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APPENDICES
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Creating Utility Functionsâ€‹

Preferences and basic rationality assumptions
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Commonly Used Functionsâ€‹

Common utility functions

ArrowPratt risk aversionâ€‹


Common production functions

DONWLOAD
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Optimality Conditions and Larger Problemsâ€‹

Hessians and Bordered Hessians (download)
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Squeezing More from Static Problemsâ€‹

Slutsky

Marginal Rates of Transformation

Indirect Utility, Expenditure Functions

Hicksian Compensated Demand

DONWLOAD
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Handling Corner Solutions

KhunTucker conditions

DONWLOAD
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Alternative Optimization Method

Optimizing with Separating Hyperplanes

DONWLOAD
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Time Series: Basicsâ€‹

Time series, AR(1), etc. (download)
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Time Series: Solving Difference Equationsâ€‹

Iteration (download)

Method of Undetermined Coefficients (dowload)
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Integrationâ€‹
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Solving Differential Equationsâ€‹

Forcing the solution (download)

Method of Undetermined Coefficients (download)â€‹
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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)
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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) "â€‹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 Inputâ€‹

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 Equationsâ€‹

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 METHODSâ€‹

Pictures of Solutionsâ€‹

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 SUBSPACESâ€‹

The Column Space of a Matrixâ€‹

Independence, Basis, and Dimension

The Big Picture of Linear Algebra

Graphs

Incidence Matrices of Graphs


EIGENVALUES AND EIGENVECTORSâ€‹

Eigenvalues and Eigenvectorsâ€‹

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 ATAâ€‹

Positive Definite Matricesâ€‹

Singular Value Decomposition (the SVD)

Boundary Conditions Replace Initial Conditions

Laplace Equation


FOURIER AND LAPLACE TRANSFORMSâ€‹

Fourier Seriesâ€‹

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