A Index and Code Links
A.1 Notations and Functions links
- Real Number and Intervals: mlx | m | pdf | html
- Definition and draw a line.
- m: linspace() + line() + set(gca, yaxis off) + pbaspect()
- Interval Notations and Examples: mlx | m | pdf | html
- Closed, open intervals.
- What is a Function?: mlx | m | pdf | html
- Domain, argument, do-domain, image/value, range.
- Graph a circle.
- m: sin() + plot()
- Function Notations: mlx | m | pdf | html
- Consistent function naming.
- Monomials and Polynomial of the 3rd Degree: mlx | m | pdf | 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])
- Local and Global Maximum: mlx | m | pdf | html
- local and global maximum.
- m: syms + solve() + diff() + double() + double(solve(diff(f,x),x)), fplot(f,[x_low, x_high])
A.2 Commonly Used Functions links
- Exponential and Compounding Interest Rates: mlx | m | pdf | 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())
- Exponential and Log Functions: mlx | m | pdf | 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()
A.3 Derivatives links
- Derivative Definition and Rules: mlx | m | pdf | html
- Derivative notations, limit definition, and key rules.
- m: syms + diff()
- Continuity and Differentiability: mlx | m | pdf | html
- Continuous point, set and function, continuously differentiable.
- Elasticity and Derivative: mlx | m | pdf | 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.
- First Order Taylor Approximation: mlx | m | pdf | 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)
- Higher Order Derivatives Cobb Douglas: mlx | m | pdf | 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’}
A.4 Univariate Applications links
- Marginal Product of Labor: mlx | m | pdf | html
- Marginal product for each additional units of workers given different levels of capital.
- m: plot() + scatter() + legend([‘k=,’num2str(K1)], [‘k=,’num2str(K1)])
- Derivative of Cobb-Douglas Production Function: mlx | m | pdf | html
- Marginal product of labor given different levels of capitals.
- m: syms + diff() + fplot()
- Derivative Approximation: mlx | m | pdf | html
- Marginal product and tangent lines.
- m: syms + diff() + fplot() + lengend{}
- Household’s Savings Problem: mlx | m | pdf | 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()
- Firm’s Borrowing Problem: mlx | m | pdf | 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()
A.5 Matrix Basics links
- Laws of Matrix Algebra: mlx | m | pdf | html
- Scalar: Associative + Communtative + Distributive laws; Matrix: all apply except A times B != B times A.
- m: transpose()
- Matrix Addition and Multiplication: mlx | m | pdf | html
- Scalar, matrices, and matrix dimensions.
- m: dot product
- Creating Matrixes in Matlab: mlx | m | pdf | html
- Vectors, matrixes and multiple matrixes.
- m: ceil() + eye() + tril() + triu() + rand(N,M,Q)
A.6 System of Equations links
- System of Linear Equations: mlx | m | pdf | html
- One or multiple linear equations.
- Coefficient matrix and augmented form.
- Solving for Two Equations and Two Unknowns: mlx | m | pdf | 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))
- System of Linear Equations Row Echelon Form: mlx | m | pdf | html
- Two equations and two unknowns.
- Elementary row operations and row echelon form.
- Matrix Inverse: mlx | m | pdf | html
- Find the inverse of a matrix.
A.7 Matrix Applications links
- Firm Maximization Problem with Capital and Labor: mlx | m | pdf | 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()
- Household Maximization with Two Goods and Budget: mlx | m | pdf | html
- Preference over two good, cobb douglas utility.
- Indifference curves and budget set.
- m: linspace() + meshgrid() + mesh() + contourf() + clabel() + colormap() + zlabel() + plot()
- Capital Demand and Supply Equilibrium Analysis: mlx | m | pdf | 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()
- First Order Taylor Approximation of Demand and Supply Curves: mlx | m | pdf | 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
A.8 Uncertainty links
- Risky Assets and Different States of the World: mlx | m | pdf | 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)
A.9 Equality Constrained Optimization links
- Profit Maximization and Cost Minimization: mlx | m | pdf | 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)
- Firm Marginal Cost and Profit given Constant Returns to Scale: mlx | m | pdf | html
- Profit maximization over outputs given cost minimization.
- Marginal costs and constant returns to scales, perfect competition and zero profits.
- Marshallian Constrained Utility Maximization: mlx | m | pdf | html
- Budget constrained intertemporal utility maximization.
- Marshallian solutions, indirect utility
- Analytical solution, matlab symbolic solution, matlab fminunc numerical solutions
- m: diff() + gradient() + fmincon()
- Hicksian Constrained Expenditure Minimization: mlx | m | pdf | 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
- Income and Substitution Effects: mlx | m | pdf | html
- Slusky decomposition, expenditure minimization given two prices.
- Analytical solution, matlab symbolic solution.
- m: diff() + gradient()
A.10 Inequality Constrained Optimization links
- Firm Profit Maximization Problem with Borrowing Constraint: mlx | m | pdf | html
- Constrained on capital/borrowing, solve for cases.
- If constraint binds, re-optimize labor choice given capital bound.
- Borrowing and Savings with Borrowing Constraint: mlx | m | pdf | 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’);
- Labor and Borrowing/Savings Choices with Borrowing Constraint: mlx | m | pdf | 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()
A.11 Equilibrium and Policy links
- Equilibrium Interest Rate and Tax: mlx | m | pdf | 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)*
- Equilibrium Interest Rate and Wage: mlx | m | pdf | 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)) + psilog(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);*
Simon, Carl P, and Lawrence Blume. 1994. Mathematics for Economists. 2nd ed. New York City, New York: W. W. Norton & Company. https://wwnorton.com/books/9780393957334/.
The MathWorks Inc. 2019. MATLAB. https://www.mathworks.com/products/matlab.html.
Xie, Yihui. 2020. Bookdown: Authoring Books and Technical Documents with r Markdown. https://CRAN.R-project.org/package=bookdown.