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Math4Econ

★ 57 Open GitHub ↗

Mathematics for Economists (Matlab Live Codes)

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This is a work-in-progress [course website](https://fanwangecon.github.io/Math4Econ/) for Mathematics for Economists. Course covers a limited subset of topics from *Mathematics for Economists* (Simon and Blume 1994), and uses various definitions from the book. Applications focus on two period borrowing and savings problems. Matlab's symbolic toolbox is used throughout.

> [**bookdown site**](https://fanwangecon.github.io/Math4Econ/bookdown) and [**bookdown pdf**](https://fanwangecon.github.io/Math4Econ/bookdown/Introductory-Mathematics-for-Economists-with-Matlab.pdf).

Materials are written in matlab livescript files and shown as HTML files. To obtain matlab codes, see [here](docs/gitsetup.md) and [here](docs/gitsetupown.md) for github set up. For HTML files, click on the links below.

From other repositories: For dynamic borrowing and savings problems, see [MEconTools](https://fanwangecon.github.io/MEconTools/) and [Dynamic Asset Repository](https://fanwangecon.github.io/CodeDynaAsset/); For code examples, see also [Matlab Example Code](https://fanwangecon.github.io/M4Econ/), [Stata Example Code](https://fanwangecon.github.io/Stata4Econ/), [Python Example Code](https://fanwangecon.github.io/pyfan/), [R Example Code](https://fanwangecon.github.io/R4Econ/); For intro econ with Matlab, see [Intro Mathematics for Economists](https://fanwangecon.github.io/Math4Econ/), and for intro stat with R, see [Intro Statistics for Undergraduates](https://fanwangecon.github.io/Stat4Econ/). See [here](https://github.com/FanWangEcon) for all of [Fan](https://fanwangecon.github.io/)'s public repositories.

Files are from the [Math4Econ](https://github.com/FanWangEcon/Math4Econ) repository. Please contact [FanWangEcon](https://fanwangecon.github.io/) for issues or problems.

[![](https://img.shields.io/github/last-commit/fanwangecon/Math4Econ)](https://github.com/FanWangEcon/Math4Econ/commits/master) [![](https://img.shields.io/github/commit-activity/m/fanwangecon/Math4Econ)](https://github.com/FanWangEcon/Math4Econ/graphs/commit-activity) [![](https://img.shields.io/github/issues/fanwangecon/Math4Econ)](https://github.com/FanWangEcon/Math4Econ/issues) [![](https://img.shields.io/github/issues-pr/fanwangecon/Math4Econ)](https://github.com/FanWangEcon/Math4Econ/pulls)

# 1  Notations and Functions

1. [Real Number and Intervals](https://fanwangecon.github.io/Math4Econ/calconevar/htmlpdfm/realnumber.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/realnumber.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/htmlpdfm/realnumber.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/htmlpdfm/realnumber.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/calconevar/htmlpdfm/realnumber.html)
	+ Definition and draw a line.
	+ **m**: *linspace() + line() + set(gca, yaxis off) + pbaspect()*
2. [Interval Notations and Examples](https://fanwangecon.github.io/Math4Econ/calconevar/htmlpdfm/interval.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/interval.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/htmlpdfm/interval.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/htmlpdfm/interval.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/calconevar/htmlpdfm/interval.html)
	+ Closed, open intervals.
3. [What is a Function?](https://fanwangecon.github.io/Math4Econ/calconevar/htmlpdfm/whatisfunction.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/whatisfunction.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/htmlpdfm/whatisfunction.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/htmlpdfm/whatisfunction.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/calconevar/htmlpdfm/whatisfunction.html)
	+ Domain, argument, do-domain, image/value, range.
	+ Graph a circle.
	+ **m**: *sin() + plot()*
4. [Function Notations](https://fanwangecon.github.io/Math4Econ/calconevar/htmlpdfm/funcnotations.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/funcnotations.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/htmlpdfm/funcnotations.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/htmlpdfm/funcnotations.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/calconevar/htmlpdfm/funcnotations.html)
	+ Consistent function naming.
5. [Monomials and Polynomial of the 3rd Degree](https://fanwangecon.github.io/Math4Econ/calconevar/htmlpdfm/polynomial.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/polynomial.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/htmlpdfm/polynomial.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/htmlpdfm/polynomial.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/calconevar/htmlpdfm/polynomial.html)
	+ Monomial, polynomial, degree of polynomial.
	+ Graph polynomial of the 3rd degree and monomials of different degrees.
	+ **m**: *syms x + f(x) = a + x + fplot(@(x) f(x,a), [x_low, x_high])*
6. [Local and Global Maximum](https://fanwangecon.github.io/Math4Econ/calconevar/htmlpdfm/localglobal.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/localglobal.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/htmlpdfm/localglobal.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/calconevar/htmlpdfm/localglobal.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/calconevar/htmlpdfm/localglobal.html)
	+ local and global maximum.
	+ **m**: *syms + solve() + diff() + double() + double(solve(diff(f,x),x)), fplot(f,[x_low, x_high])*

# 2  Commonly Used Functions

1. [Exponential and Compounding Interest Rates](https://fanwangecon.github.io/Math4Econ/explog/htmlpdfm/exponential.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/explog/exponential.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/explog/htmlpdfm/exponential.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/explog/htmlpdfm/exponential.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/explog/htmlpdfm/exponential.html)
	+ Exponential function and rules: a^b. Base e exponential, e = 2.71828.
	+ Infinitely compounding interest rate (continuous time).
	+ e^r: borrow 1 dollar, given r, meaning r percent interest, e^r is how much to pay back in principle + interests given infinite compounding.
	+ Log linear equation with a constant term, substraction and division.
	+ **m**: *exp() + fplot() + double(subs())*
2. [Exponential and Log Functions](https://fanwangecon.github.io/Math4Econ/explog/htmlpdfm/exolog.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/explog/exolog.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/explog/htmlpdfm/exolog.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/explog/htmlpdfm/exolog.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/explog/htmlpdfm/exolog.html)
	+ log and natural log (log in matlab base e, log in google base 10).
	+ log rules, and why: log(xy) = log(x) + log(y); log(x^a) = alog(x).
	+ log difference and small rates of change.
	+ **m**: *linspace() + log()*

# 3  Derivatives

1. [Derivative Definition and Rules](https://fanwangecon.github.io/Math4Econ/derivative/htmlpdfm/derivative_rules.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative/derivative_rules.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative/htmlpdfm/derivative_rules.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative/htmlpdfm/derivative_rules.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/derivative/htmlpdfm/derivative_rules.html)
	+ Derivative notations, limit definition, and key rules.
	+ **m**: *syms + diff()*
2. [Continuity and Differentiability](https://fanwangecon.github.io/Math4Econ/derivative/htmlpdfm/continuous_differentiable.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative/continuous_differentiable.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative/htmlpdfm/continuous_differentiable.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative/htmlpdfm/continuous_differentiable.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/derivative/htmlpdfm/continuous_differentiable.html)
	+ Continuous point, set and function, continuously differentiable.
3. [Elasticity and Derivative](https://fanwangecon.github.io/Math4Econ/derivative/htmlpdfm/derivative_elasticity.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative/derivative_elasticity.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative/htmlpdfm/derivative_elasticity.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative/htmlpdfm/derivative_elasticity.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/derivative/htmlpdfm/derivative_elasticity.html)
	+ Elasticity of demand at price p, given h change in p.
	+ Point elasticity of demand at price p.
	+ Elasticity and the limiting definition of derivative.
4. [First Order Taylor Approximation](https://fanwangecon.github.io/Math4Econ/derivative/htmlpdfm/derivative_MPL_first_order_taylor_approximation.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative/derivative_MPL_first_order_taylor_approximation.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative/htmlpdfm/derivative_MPL_first_order_taylor_approximation.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative/htmlpdfm/derivative_MPL_first_order_taylor_approximation.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/derivative/htmlpdfm/derivative_MPL_first_order_taylor_approximation.html)
	+ Differential: change along the tangent line to approximate change in function value.
	+ First order taylor approximation and the limiting definition of derivative.
	+ Differential approximating marginal productivity of labor.
	+ **m**: *syms + f(L) = L^a + sub(f, 1)*
5. [Higher Order Derivatives Cobb Douglas](https://fanwangecon.github.io/Math4Econ/derivative/htmlpdfm/second_derivative.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative/second_derivative.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative/htmlpdfm/second_derivative.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative/htmlpdfm/second_derivative.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/derivative/htmlpdfm/second_derivative.html)
	+ Cobb-Douglas production function, first and second derivatives.
	+ Convex and Concave functions.
	+ **m**: *syms + f(L) = L^a + diff(diff(f, L),L) + fplot() + title({‘title one’ ‘subtitle’}) + ylabel({‘ylab abc’ ‘ylab efg’}) + legend{[‘line a’],[‘lineb’],, ‘Location’,’NW’}*

# 4  Univariate Applications

1. [Marginal Product of Labor](https://fanwangecon.github.io/Math4Econ/derivative_application/htmlpdfm/derivative_MPL_discrete_workers.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative_application/derivative_MPL_discrete_workers.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative_application/htmlpdfm/derivative_MPL_discrete_workers.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative_application/htmlpdfm/derivative_MPL_discrete_workers.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/derivative_application/htmlpdfm/derivative_MPL_discrete_workers.html)
	+ Marginal product for each additional units of workers given different levels of capital.
	+ **m**: *plot() + scatter() + legend(['k=',num2str(K1)], ['k=',num2str(K1)])*
2. [Derivative of Cobb-Douglas Production Function](https://fanwangecon.github.io/Math4Econ/derivative_application/htmlpdfm/derivative_cobb_douglas.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative_application/derivative_cobb_douglas.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative_application/htmlpdfm/derivative_cobb_douglas.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative_application/htmlpdfm/derivative_cobb_douglas.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/derivative_application/htmlpdfm/derivative_cobb_douglas.html)
	+ Marginal product of labor given different levels of capitals.
	+ **m**: *syms + diff() + fplot()*
3. [Derivative Approximation](https://fanwangecon.github.io/Math4Econ/derivative_application/htmlpdfm/derivative_hslope_cobb_douglas.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative_application/derivative_hslope_cobb_douglas.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative_application/htmlpdfm/derivative_hslope_cobb_douglas.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative_application/htmlpdfm/derivative_hslope_cobb_douglas.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/derivative_application/htmlpdfm/derivative_hslope_cobb_douglas.html)
	+ Marginal product and tangent lines.
	+ **m**: *syms + diff() + fplot() + lengend{}*
4. [Household's Savings Problem](https://fanwangecon.github.io/Math4Econ/derivative_application/htmlpdfm/K_save_households.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative_application/K_save_households.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative_application/htmlpdfm/K_save_households.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative_application/htmlpdfm/K_save_households.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/derivative_application/htmlpdfm/K_save_households.html)
	+ Endowments today and tomorrow, borrowing and savings, no shocks.
	+ Grid based or analytical solution.
	+ Supply curve of savings (asset).
	+ **m**: *max() + diff() + solve() + plot() + scatter()*
5. [Firm's Borrowing Problem](https://fanwangecon.github.io/Math4Econ/derivative_application/htmlpdfm/K_borrow_firm.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative_application/K_borrow_firm.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative_application/htmlpdfm/K_borrow_firm.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/derivative_application/htmlpdfm/K_borrow_firm.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/derivative_application/htmlpdfm/K_borrow_firm.html)
	+ Profit maximization choosing capital, with labor fixed.
	+ Grid based or analytical solution.
	+ Demand curve of capital (asset).
	+ Overlay demand and supply curves, visualize interest rate equilibrium
	+ **m**: *max() + diff() + solve() + plot() + scatter()*

# 5  Matrix Basics

1. [Laws of Matrix Algebra](https://fanwangecon.github.io/Math4Econ/matrix_basics/htmlpdfm/matlab_define_matrix.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_basics/matlab_define_matrix.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_basics/htmlpdfm/matlab_define_matrix.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_basics/htmlpdfm/matlab_define_matrix.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/matrix_basics/htmlpdfm/matlab_define_matrix.html)
	+ Scalar: Associative + Communtative + Distributive laws; Matrix: all apply except A times B != B times A.
	+ **m**: *transpose()*
2. [Matrix Addition and Multiplication](https://fanwangecon.github.io/Math4Econ/matrix_basics/htmlpdfm/matrix_algebra_rules.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_basics/matrix_algebra_rules.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_basics/htmlpdfm/matrix_algebra_rules.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_basics/htmlpdfm/matrix_algebra_rules.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/matrix_basics/htmlpdfm/matrix_algebra_rules.html)
	+ Scalar, matrices, and matrix dimensions.
	+ **m**: *dot product*
3. [Creating Matrixes in Matlab](https://fanwangecon.github.io/Math4Econ/matrix_basics/htmlpdfm/matrix_matlab.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_basics/matrix_matlab.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_basics/htmlpdfm/matrix_matlab.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_basics/htmlpdfm/matrix_matlab.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/matrix_basics/htmlpdfm/matrix_matlab.html)
	+ Vectors, matrixes and multiple matrixes.
	+ **m**: *ceil() + eye() + tril() + triu() + rand(N,M,Q)*

# 6  System of Equations

1. [System of Linear Equations](https://fanwangecon.github.io/Math4Econ/matrix_system_of_equations/htmlpdfm/matrix_linear_equations.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_system_of_equations/matrix_linear_equations.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_system_of_equations/htmlpdfm/matrix_linear_equations.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_system_of_equations/htmlpdfm/matrix_linear_equations.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/matrix_system_of_equations/htmlpdfm/matrix_linear_equations.html)
	+ One or multiple linear equations.
	+ Coefficient matrix and augmented form.
2. [Solving for Two Equations and Two Unknowns](https://fanwangecon.github.io/Math4Econ/matrix_system_of_equations/htmlpdfm/matrix_linear_system_2.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_system_of_equations/matrix_linear_system_2.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_system_of_equations/htmlpdfm/matrix_linear_system_2.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_system_of_equations/htmlpdfm/matrix_linear_system_2.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/matrix_system_of_equations/htmlpdfm/matrix_linear_system_2.html)
	+ Two equations and two unknowns matrix form.
	+ Graphical intersection of two lines.
	+ Using linear solver linsolve.
	+ **m**: *linsolve + double(solve(y_1 - y_2 == 0))*
3. [System of Linear Equations Row Echelon Form](https://fanwangecon.github.io/Math4Econ/matrix_system_of_equations/htmlpdfm/matrix_row_echelon_form.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_system_of_equations/matrix_row_echelon_form.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_system_of_equations/htmlpdfm/matrix_row_echelon_form.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_system_of_equations/htmlpdfm/matrix_row_echelon_form.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/matrix_system_of_equations/htmlpdfm/matrix_row_echelon_form.html)
	+ Two equations and two unknowns.
	+ Elementary row operations and row echelon form.
4. [Matrix Inverse](https://fanwangecon.github.io/Math4Econ/matrix_system_of_equations/htmlpdfm/matrix_inverse.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_system_of_equations/matrix_inverse.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_system_of_equations/htmlpdfm/matrix_inverse.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_system_of_equations/htmlpdfm/matrix_inverse.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/matrix_system_of_equations/htmlpdfm/matrix_inverse.html)
	+ Find the inverse of a matrix.

# 7  Matrix Applications

1. [Firm Maximization Problem with Capital and Labor](https://fanwangecon.github.io/Math4Econ/matrix_application/htmlpdfm/KL_borrowhire_firm.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_application/KL_borrowhire_firm.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_application/htmlpdfm/KL_borrowhire_firm.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_application/htmlpdfm/KL_borrowhire_firm.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/matrix_application/htmlpdfm/KL_borrowhire_firm.html)
	+ First order conditions Cobb-Douglas production function with Capital and Labor.
	+ Log linearize first order conditions, matrix form and linsolve Cobb-Douglas production function.
	+ Own and cross price elasticities
	+ **m**: *linsolve() + simplify(exp(linsolve())) + mesh() + meshgrid() + contourf() + clabel() + zlabel()*
2. [Household Maximization with Two Goods and Budget](https://fanwangecon.github.io/Math4Econ/matrix_application/htmlpdfm/twogoods.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_application/twogoods.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_application/htmlpdfm/twogoods.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_application/htmlpdfm/twogoods.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/matrix_application/htmlpdfm/twogoods.html)
	+ Preference over two good, cobb douglas utility.
	+ Indifference curves and budget set.
	+ **m**: *linspace() + meshgrid() + mesh() + contourf() + clabel() + colormap() + zlabel() + plot()*
3. [Capital Demand and Supply Equilibrium Analysis](https://fanwangecon.github.io/Math4Econ/matrix_application/htmlpdfm/demand_supply_taylor_approximate.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_application/demand_supply_taylor_approximate.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_application/htmlpdfm/demand_supply_taylor_approximate.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_application/htmlpdfm/demand_supply_taylor_approximate.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/matrix_application/htmlpdfm/demand_supply_taylor_approximate.html)
	+ Simplified nonlinear form of demand and supply as functions or the interest rate.
	+ First order Taylor linear approximation of nonlinear demand and supply.
	+ **m**: *diff() + subs(S,r,1) + linsolve()*
4. [First Order Taylor Approximation of Demand and Supply Curves](https://fanwangecon.github.io/Math4Econ/matrix_application/htmlpdfm/demand_supply_taylor_approximate_capital.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_application/demand_supply_taylor_approximate_capital.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_application/htmlpdfm/demand_supply_taylor_approximate_capital.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/matrix_application/htmlpdfm/demand_supply_taylor_approximate_capital.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/matrix_application/htmlpdfm/demand_supply_taylor_approximate_capital.html)
	+ Exact solutions for (approximated) equilibrium interest rate and asset supply/demand given linearized demand and supply equations.
	+ Graphical illustration of exact equilibrium and linear approximated equilibrium.
	+ Analyze how productivity, elasticity, wealth, discount factor impact equilibrium prices and quantity given exact solutions to linear approximation.
	+ **m**: *linspace() + subs(diff(S,r), r, r0) + subs(D, {Z,beta}, {Z_num, beta_num}) + fplot() + plot() + line.Color + line.LineStyle*

# 8  Uncertainty

1. [Risky Assets and Different States of the World](https://fanwangecon.github.io/Math4Econ/uncertainty/htmlpdfm/RiskyAsset.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/uncertainty/RiskyAsset.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/uncertainty/htmlpdfm/RiskyAsset.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/uncertainty/htmlpdfm/RiskyAsset.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/uncertainty/htmlpdfm/RiskyAsset.html)
	+ Bad and good states of the world.
	+ Safe savings and risky investments with uncertain returns.
	+ Borrowing to finance risky investments.
	+ **m**: *solve(diff(U, D)==0, diff(U, B)==0, D, B)*

# 9  Equality Constrained Optimization

1. [Profit Maximization and Cost Minimization](https://fanwangecon.github.io/Math4Econ/opti_firm_constrained/htmlpdfm/profit_maximize.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_firm_constrained/profit_maximize.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_firm_constrained/htmlpdfm/profit_maximize.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_firm_constrained/htmlpdfm/profit_maximize.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/opti_firm_constrained/htmlpdfm/profit_maximize.html)
	+ Profit maximization and cost minimization with Cobb Douglas production function given quantity constraint. Constant or decreasing returns to scales, optimal capital and labor given quantity constraint.
	+ **m**: *GRADIENT = subs(GRADIENT, {A,p,w,r,q,alpha,beta},{1,1,1,1,2,0.3,0.7}) + solu = solve(GRADIENT(1)==0, GRADIENT(2)==0, GRADIENT(3)==0, K, L, m, ‘Real’, true)*
2. [Firm Marginal Cost and Profit given Constant Returns to Scale](https://fanwangecon.github.io/Math4Econ/opti_firm_constrained/htmlpdfm/profit_maximize_crs_profit.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_firm_constrained/profit_maximize_crs_profit.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_firm_constrained/htmlpdfm/profit_maximize_crs_profit.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_firm_constrained/htmlpdfm/profit_maximize_crs_profit.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/opti_firm_constrained/htmlpdfm/profit_maximize_crs_profit.html)
	+ Profit maximization over outputs given cost minimization.
	+ Marginal costs and constant returns to scales, perfect competition and zero profits.
3. [Marshallian Constrained Utility Maximization](https://fanwangecon.github.io/Math4Econ/opti_hh_constrained_brsv/htmlpdfm/household_c1_c2_constrained.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_hh_constrained_brsv/household_c1_c2_constrained.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_hh_constrained_brsv/htmlpdfm/household_c1_c2_constrained.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_hh_constrained_brsv/htmlpdfm/household_c1_c2_constrained.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/opti_hh_constrained_brsv/htmlpdfm/household_c1_c2_constrained.html)
	+ Budget constrained intertemporal utility maximization.
	+ Marshallian solutions, indirect utility
	+ Analytical solution, matlab symbolic solution, matlab fminunc numerical solutions
	+ **m**: *diff() + gradient() + fmincon()*
4. [Hicksian Constrained Expenditure Minimization](https://fanwangecon.github.io/Math4Econ/opti_hh_constrained_brsv/htmlpdfm/household_c1_c2_constrained_expmin.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_hh_constrained_brsv/household_c1_c2_constrained_expmin.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_hh_constrained_brsv/htmlpdfm/household_c1_c2_constrained_expmin.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_hh_constrained_brsv/htmlpdfm/household_c1_c2_constrained_expmin.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/opti_hh_constrained_brsv/htmlpdfm/household_c1_c2_constrained_expmin.html)
	+ Optimal expenditure minimization choice given indirect utility.
	+ Hicksian solutions (Dual Problem).
	+ Analytical solution, matlab symbolic solution.
	+ **m**: *diff() + gradient()*
	+ **graph**: *budget + indifference + endowment and optimal choices*
5. [Income and Substitution Effects](https://fanwangecon.github.io/Math4Econ/opti_hh_constrained_brsv/htmlpdfm/household_c1_c2_constrained_r_change.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_hh_constrained_brsv/household_c1_c2_constrained_r_change.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_hh_constrained_brsv/htmlpdfm/household_c1_c2_constrained_r_change.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_hh_constrained_brsv/htmlpdfm/household_c1_c2_constrained_r_change.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/opti_hh_constrained_brsv/htmlpdfm/household_c1_c2_constrained_r_change.html)
	+ Slusky decomposition, expenditure minimization given two prices.
	+ Analytical solution, matlab symbolic solution.
	+ **m**: *diff() + gradient()*

# 10  Inequality Constrained Optimization

1. [Firm Profit Maximization Problem with Borrowing Constraint](https://fanwangecon.github.io/Math4Econ/opti_firm_constrained/htmlpdfm/KL_borrowhire_firm_inqc.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_firm_constrained/KL_borrowhire_firm_inqc.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_firm_constrained/htmlpdfm/KL_borrowhire_firm_inqc.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_firm_constrained/htmlpdfm/KL_borrowhire_firm_inqc.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/opti_firm_constrained/htmlpdfm/KL_borrowhire_firm_inqc.html)
	+ Constrained on capital/borrowing, solve for cases.
	+ If constraint binds, re-optimize labor choice given capital bound.
2. [Borrowing and Savings with Borrowing Constraint](https://fanwangecon.github.io/Math4Econ/opti_hh_constrained_brsv_inequality/htmlpdfm/household_borrow_constrained.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_hh_constrained_brsv_inequality/household_borrow_constrained.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_hh_constrained_brsv_inequality/htmlpdfm/household_borrow_constrained.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_hh_constrained_brsv_inequality/htmlpdfm/household_borrow_constrained.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/opti_hh_constrained_brsv_inequality/htmlpdfm/household_borrow_constrained.html)
	+ Unconstrained and constrained problem.
	+ Analytical solution and fmincon solution.
	+ Optimal borrowing/savings with varying endowments and interests rates.
	+ **m**: *U = @(b) log(z1 - b) + matlabFunction(subs(U, {z1, z2}, {z1v, z2v})); + fmincon(U, b0, A, q); + optimoptions('FMINCON','Display','off');*
3. [Labor and Borrowing/Savings Choices with Borrowing Constraint](https://fanwangecon.github.io/Math4Econ/opti_hh_constrained_brsv_inequality/htmlpdfm/household_asset_labor_constrained.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_hh_constrained_brsv_inequality/household_asset_labor_constrained.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_hh_constrained_brsv_inequality/htmlpdfm/household_asset_labor_constrained.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/opti_hh_constrained_brsv_inequality/htmlpdfm/household_asset_labor_constrained.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/opti_hh_constrained_brsv_inequality/htmlpdfm/household_asset_labor_constrained.html)
	+ Unconstrained work/leisure and borrow/savings problem.
	+ Constrained work/leisure and borrow/savings problem given borrow bound.
	+ Analytical and matlab symbolic solutions.
	+ Numerical solution with fmincon.
	+ **m**: *d_L_b = diff(L, b); + d_L_H = diff(L, H); + GRAD = [d_L_b; d_L_H] + solu = solve(GRAD(1)==0, GRAD(2)==0, b, H, 'Real', true); + solu = simplify(solu) + fmincon(U_neg, b0, A, q) + fmincon(U_neg, b0, A, q, [], [], [], [], [], options) + legendCell = cellstr(num2str(Z2_vec', 'Z2=%-d')) + plot()*

# 11  Equilibrium and Policy

1. [Equilibrium Interest Rate and Tax](https://fanwangecon.github.io/Math4Econ/equilibrium/htmlpdfm/equilibrium_constrainedborrow.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/equilibrium/equilibrium_constrainedborrow.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/equilibrium/htmlpdfm/equilibrium_constrainedborrow.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/equilibrium/htmlpdfm/equilibrium_constrainedborrow.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/equilibrium/htmlpdfm/equilibrium_constrainedborrow.html)
	+ Households supply savings or borrow (with constraint) to smooth consumption.
	+ Firms borrow to finance capital inputs.
	+ Solve for excess demand and supply of assets and equilibrium interest rate.
	+ The effect of a tax on savings and subsidy for borrowing on equilibrium interest rate.
	+ **m**: *U_neg = @(x) -1*(log(z1 - x(1)) + beta_vec(j)*log(z2 + x(1)*r_vec(i)*(1-tau))) + excess_credit_supply = (sum(b_opti_mat, 2) + (-1)*FIRM_K') + min(abs(excess_credit_supply)) + plot(r, excess_credit_supply)*
2. [Equilibrium Interest Rate and Wage](https://fanwangecon.github.io/Math4Econ/equilibrium/htmlpdfm/equilibrium_constrainedborrow_labor.html): [**mlx**](https://github.com/FanWangEcon/Math4Econ/blob/master/equilibrium/equilibrium_constrainedborrow_labor.mlx) \| [**m**](https://github.com/FanWangEcon/Math4Econ/blob/master/equilibrium/htmlpdfm/equilibrium_constrainedborrow_labor.m) \| [**pdf**](https://github.com/FanWangEcon/Math4Econ/blob/master/equilibrium/htmlpdfm/equilibrium_constrainedborrow_labor.pdf) \| [**html**](https://fanwangecon.github.io/Math4Econ/equilibrium/htmlpdfm/equilibrium_constrainedborrow_labor.html)
	+ Households supply labor and enjoy leisure, firms demand labor.
	+ Households borrow with constraints and supply savings, firm demand capital.
	+ Solve for excess supply of assets and labor over wage and interest rates grid.
	+ Solve for market clearing wage and interest rates.
	+ **m**: *U_neg = @(x) -1*(log(z1 + W_vec(j)*x(2) - x(1)) + psi*log(x(3)) + beta_vec(h)*log(z2 + x(1)*(R_vec(i)))) + options = optimoptions('FMINCON','Display','off'); + [x_opti,U_at_x_opti] = fmincon(U_neg, b0, A, q, [], [], [], [], [], options); + KD(i,j) = subs(K_opti,{r,w},{R(i), W(j)}) + LD(i,j) = subs(L_opti,{r,w},{R(i), W(j)}) + jet(numel(chart)) + plot(R, b_opti); + plot(R, -k_opti);*

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