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(APPENDIX) Appendix {-}

Index and Code Links

Notations and Functions links

  1. Real Number and Intervals: mlx | m | pdf | html
    • Definition and draw a line.
    • m: linspace() + line() + set(gca, yaxis off) + pbaspect()
  2. Interval Notations and Examples: mlx | m | pdf | html
    • Closed, open intervals.
  3. What is a Function?: mlx | m | pdf | html
    • Domain, argument, do-domain, image/value, range.
    • Graph a circle.
    • m: sin() + plot()
  4. Function Notations: mlx | m | pdf | html
    • Consistent function naming.
  5. 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])
  6. 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])

Commonly Used Functions links

  1. 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())
  2. 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()

Derivatives links

  1. Derivative Definition and Rules: mlx | m | pdf | html
    • Derivative notations, limit definition, and key rules.
    • m: syms + diff()
  2. Continuity and Differentiability: mlx | m | pdf | html
    • Continuous point, set and function, continuously differentiable.
  3. 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.
  4. 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)
  5. 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’}

Univariate Applications links

  1. 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)])
  2. Derivative of Cobb-Douglas Production Function: mlx | m | pdf | html
    • Marginal product of labor given different levels of capitals.
    • m: syms + diff() + fplot()
  3. Derivative Approximation: mlx | m | pdf | html
    • Marginal product and tangent lines.
    • m: syms + diff() + fplot() + lengend{}
  4. 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()
  5. 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()

Matrix Basics links

  1. 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()
  2. Matrix Addition and Multiplication: mlx | m | pdf | html
    • Scalar, matrices, and matrix dimensions.
    • m: dot product
  3. Creating Matrixes in Matlab: mlx | m | pdf | html
    • Vectors, matrixes and multiple matrixes.
    • m: ceil() + eye() + tril() + triu() + rand(N,M,Q)

System of Equations links

  1. System of Linear Equations: mlx | m | pdf | html
    • One or multiple linear equations.
    • Coefficient matrix and augmented form.
  2. 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))
  3. System of Linear Equations Row Echelon Form: mlx | m | pdf | html
    • Two equations and two unknowns.
    • Elementary row operations and row echelon form.
  4. Matrix Inverse: mlx | m | pdf | html
    • Find the inverse of a matrix.

Matrix Applications links

  1. 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()
  2. 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()
  3. 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()
  4. 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

Uncertainty links

  1. 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)

Equality Constrained Optimization links

  1. 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)
  2. 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.
  3. 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()
  4. 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
  5. Income and Substitution Effects: mlx | m | pdf | html
    • Slusky decomposition, expenditure minimization given two prices.
    • Analytical solution, matlab symbolic solution.
    • m: diff() + gradient()

Inequality Constrained Optimization links

  1. 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.
  2. 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');
  3. 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()

Equilibrium and Policy links

  1. 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)
  2. 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);*