ipynb/barry_ke-applied_computational_economics_and_finance.ipynb

{
    "cells": [
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "# Applied Computational Economics and Finance"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "This notebook contains notes and python implementation of some coding examples in \"Applied Computational Economics and Finance\" by Mario J. Miranda and Paul L. Fackler. "
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 1,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Warning: Function psi has the same name as a MATLAB builtin. We suggest you rename the function to avoid a potential name conflict.\n",
                        "> In path (line 109)\n",
                        "  In addpath (line 86)\n",
                        "Warning: Function psi has the same name as a MATLAB builtin. We suggest you rename the function to avoid a potential name conflict.\n",
                        "> In path (line 109)\n",
                        "  In addpath (line 86)\n",
                        "Warning: Function psi has the same name as a MATLAB builtin. We suggest you rename the function to avoid a potential name conflict.\n",
                        "Warning: Function psi has the same name as a MATLAB builtin. We suggest you rename the function to avoid a potential name conflict.\n",
                        "\n"
                    ]
                }
            ],
            "source": [
                "addpath('C:\\Users\\barry\\Documents\\Academics\\Quantecon-Notes\\functions')\n",
                "addpath('C:\\Users\\barry\\Documents\\Academics\\Quantecon-Notes\\functions\\CEtools')\n",
                "cd 'C:\\Users\\barry\\Documents\\Academics\\Quantecon-Notes\\functions'"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Chapter 3. Nonlinear equations"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "### Bisection method\n",
                "\n",
                "Bisection method is a root finding algorithm based on Intermediate Value Theorem. If $f$ is continuous, and $f(a)f(b)<0$, then $f$ must have at least one root in $[a,b]$\n",
                "\n",
                "Algorithm: WLOG, start with $f(a)<0, f(b)>0$. Compute $f(x) = f(\\frac{a+b}{2})$. If $f(x)>0$, then set $b = x$ and continue the iteration. \n",
                "\n",
                "Greatest strength: robustness. Guaranteed to compute a root to a precision $\\epsilon$ with no more than $\\log{((b-a)/\\epsilon)}/\\log(2)$ iterations. \n",
                "\n",
                "### Function iteration:\n",
                "\n",
                "Function iteration is used to compute a fixed-point, $x=g(x)$ of a function. For root finding, it is equivalent to solve $x=x-f(x)$.\n",
                "\n",
                "Algorithm: pick an initial guess $x^{0}$. Updating scheme: $x^{(k+1)} = x^{k} - f(x^{k})$\n",
                "\n",
                "### Newton's Method\n",
                "\n",
                "Based on the principle of successive linearization: replace nonlinear problem with a sequence of simpler linear problems whose solutions converge to the solution of the nonlinear problem. \n",
                "\n",
                "Algorithm: $x^{(k+1)} = x^{k}-[f'(x^k)]^{-1}f(x^k)$\n",
                "\n",
                "In theory, Newton's method converges if $f$ is continuously differentiable ($C^2$ function) and if the initial value of $x$ is sufficiently close to a root. \n",
                "\n",
                "### Quasi-Newton method\n",
                "\n",
                "Similar to Newton's method but replace $f'$ with something easier to compute. \n",
                "\n",
                "1. Secant method: $x^{k+1} = x^k-\\frac{x^k-x^{k-1}}{f(x^k)-f(x^{k-1})}f(x^k)$. It requires two starting value.\n",
                "2. Broyden's method: generates a sequence of vectors $x^k$ and matrices $B^k$ that approximate the root of $f$ and the inveser of the Jacobian matrix. $x^{k+1} = x^k - B^k f(x^k)$ and $B^{k+1} = B^k + ((d^k)-u^k)d^{k^T}B^k)/(d^{k^T}u^k)$ where $u^k = B^k(f(x^{k+1}-f(x^k))$ and $d^k = x^k - x^{k-1}$\n",
                "\n",
                "\n",
                "\n",
                "**Example: Cournot Duopoly.**\n",
                "\n",
                "The inverse demand function is $p=q^{-1/\\eta}$ and the two firms faces cost function $C_i(q_i)=\\frac{1}{2}c_iq_i^2$ for $i=1,2$. The profit of the firm is $\\pi_i(q_1, q_2) = P(q_1+q_2)q_i - C(q_i)$ and it solves the FOC: $\\frac{\\partial \\pi_i}{\\partial q_i} = 0$"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 17,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Created file 'C:\\Users\\barry\\Documents\\Academics\\Quantecon-Notes\\functions\\cournot.m'.\n"
                    ]
                }
            ],
            "source": [
                "%%file functions/cournot.m\n",
                "\n",
                "function fval = cournot(q)\n",
                "    c = [0.6;0.8]; eta = 1.6;\n",
                "    e = -1/eta;\n",
                "    fval = sum(q)^e + e*sum(q)^(e-1)*q - diag(c)*q;"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 18,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n",
                        "Equation solved.\n",
                        "\n",
                        "fsolve completed because the vector of function values is near zero\n",
                        "as measured by the value of the function tolerance, and\n",
                        "the problem appears regular as measured by the gradient.\n",
                        "\n",
                        "\n",
                        "ans =\n",
                        "\n",
                        "    0.8396\n",
                        "    0.6888\n",
                        "\n",
                        "\n"
                    ]
                }
            ],
            "source": [
                "fsolve(@cournot,[0.2;0.2])"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": null,
            "metadata": {},
            "outputs": [],
            "source": []
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "### Complementarity problem\n",
                "\n",
                "For two n-vectors $a$ and $b$, and a function $f$ from $R^n$ to $R^n$, one must find an n-vector s.t. $x_i < a_i \\Rightarrow f_i(x)\\geq 0$ and $x_i < b_i \\Rightarrow f_i(x) \\leq 0$ for all $i = 1,2,\\cdots n$. Denote the problem to be $CP(f,a,b)$\n",
                "\n",
                "Root finding is special case with $a_i = -\\infty, b_i =\\infty$\n",
                "\n",
                "**Example: simple commodity competitive spatial price equilibrium**\n",
                "\n",
                "Suppose there are $n$ distinct regions and that excess demand for the commodity is $E_i(p_i)$ of the price $p_i$ in region $i$. In absence of trade, equilibrium is solved by solving $E_i(p_i)=0$ for each $i$. \n",
                "\n",
                "Suppose the trade can take place among regions, and the cost of transferring good from $i$ to $j$ is $c_{ij}$. Suppose $x_{ij}$ is the amount shipped from $i$ to $j$ and it cannot exceed capacity $b_{ij}$. Spatial arbitrage profit: $p_j-p_i-c_{ij}$. Equilibrium obtains only if all spatial arbitrage profits are zero. This requires that $0\\leq x_{ij}\\leq b_{ij}$ and \n",
                "1. $x_{ij} > 0 \\Rightarrow p_j-p_i-c_{ij} \\geq 0$\n",
                "2. $x_{ij} < b_{ij} \\Rightarrow p_j -p_i - c_{ij} \\leq 0$\n",
                "\n",
                "where $p_i$ can be solved using $\\sum_k[x_{ki} - x_{ik}] = E_i(p_i)$\n",
                "\n",
                "Let $f_{ij}(x) = E^{-1}_j(\\sum_k[x_{kj} - x_{jk}]) - E^{-1}_i(\\sum_k[x_{ki} - x_{ik}]) - c_{ij}$, then $x$ is equilibrium iff it solves $CP(f,0,b)$\n",
                "\n",
                "### Complementarity Methods\n",
                "\n",
                "$x$ solves $CP(f,a,b)$ iff it solves $\\hat{f}(x) = \\min(\\max(f(x), a-x), b-x) = 0$, which is a root finding problem."
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "# Chapter 5: Numerical Integration and Differentiation\n",
                "\n",
                "## Differentiation\n",
                "Finite difference: $f'(x_0) \\simeq \\frac{f(x_0+h)-f(x_0)}{h}$\n",
                "\n",
                "Two-sided derivative: $f'(x_0) \\simeq \\frac{f(x_0+h)-f(x_0-h)}{2h}$. It is better as long as the fourth order residual on both sides of $x_0$ are of similar magnitude (prove using Taylor expansion)\n",
                "\n",
                "In practice, compute TS as $f'(x_0) \\simeq \\frac{f(x_0+h)-f(x_0-h)}{x_0+h - (x_0-h)}$ so both have same argument. \n",
                "\n",
                "Miranda-Fackler: set $h = max(|x_0|,1)\\sqrt[3]{\\epsilon} \\simeq \\max(|x_0,1|)\\times 6 \\times 10^{-6}$, where $\\epsilon$ is margion epsilon (for 64-bit machine, $\\epsilon = 2^{-52}$).\n",
                "\n",
                "**Example: Two-sided finite difference for single variable function**"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 25,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Created file 'C:\\Users\\barry\\Documents\\Academics\\Quantecon-Notes\\TS_diff.m'.\n"
                    ]
                }
            ],
            "source": [
                "%%file TS_diff.m\n",
                "function TS = TS_diff(f,x0)\n",
                "    \n",
                "    h = max(abs(x0),1)*6.0*1e-6;\n",
                "    \n",
                "    TS = (f(x0+h)-f(x0-h))/(2*h);\n"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 26,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n",
                        "ans =\n",
                        "\n",
                        "    7.3891\n",
                        "\n",
                        "\n"
                    ]
                }
            ],
            "source": [
                "TS_diff(@exp, 2)"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "**Example: Two-sided finite difference for Jacobian of an arbitrary function**"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 19,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Created file 'C:\\Users\\barry\\Documents\\Academics\\Quantecon-Notes\\functions\\TS_jac.m'.\n"
                    ]
                }
            ],
            "source": [
                "%%file functions/TS_jac.m\n",
                "function fjac = TS_jac(f,x)\n",
                "    \n",
                "    h = max(abs(x),1)*6.0*1e-6;\n",
                "    xh1 = x+h;\n",
                "    xh0 = x-h;\n",
                "    h = xh1-xh0;\n",
                "    \n",
                "    for j =1:length(x)\n",
                "        xx = x; xx(j) = xh1(j); f1 = feval(f,xx);\n",
                "        xx(j) = xh0(j); f0 = feval(f,xx);\n",
                "        fjac(:,j) = (f1-f0)/h(j);\n",
                "    end\n",
                "        "
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 20,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n",
                        "ans =\n",
                        "\n",
                        "    0.5403         0         0         0         0\n",
                        "         0   -0.4161         0         0         0\n",
                        "         0         0   -0.9900         0         0\n",
                        "         0         0         0    0.2837         0\n",
                        "         0         0         0         0    0.5403\n",
                        "\n",
                        "\n"
                    ]
                }
            ],
            "source": [
                "x = [1,2,3,5,1];\n",
                "TS_jac(@sin, x)"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Integration\n",
                "\n",
                "Three methods: Newton-Cato, Gaussian Quadrature, Monte-Carlo (see NYU note)\n",
                "\n",
                "**Example: matlab built-in function for integration**"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 25,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n",
                        "q =\n",
                        "\n",
                        "    1.9475\n",
                        "\n",
                        "\n"
                    ]
                }
            ],
            "source": [
                "fun = @(x) exp(-x.^2).*log(x).^2;\n",
                "q = integral(fun,0,Inf)\n",
                "\n"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "**Example: symbolic integral**\n",
                "Integrate $x^2 + xy + 3$ with respect to x from $x=0$ to $x=y$"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 26,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        " \n",
                        "q =\n",
                        " \n",
                        "(5*y^3)/6 + 3*y\n",
                        " \n",
                        "\n"
                    ]
                }
            ],
            "source": [
                "syms x y\n",
                "q = int(x^2 + y*x +3, x,0,y)"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "**Example: Monte-Carlo**\n",
                "Integrate $x^2 + y^3 + 3z$ on $x^2 + y^2 + z^2 \\leq 1$"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 40,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n",
                        "I =\n",
                        "\n",
                        "    0.6130\n",
                        "\n",
                        "\n"
                    ]
                }
            ],
            "source": [
                "Npoints = 10000;\n",
                "X = randn(Npoints, 3);\n",
                "X = X./sqrt(sum(X.^2, 3));\n",
                "\n",
                "R = nthroot(rand(Npoints, 1), 3);\n",
                "X = X.*R;\n",
                "\n",
                "I = sum(X(:,1).^2 + X(:,2).^3 + 3*X(:,3))/Npoints"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Initial Value Problems##\n",
                "\n",
                "Difference between IVP and BVP:\n",
                "\n",
                "Initial Value Problem (IVP): can be solved in a recursive or evolutionary fashion. \n",
                "\n",
                "Boundary Value Problem (BVP): require the entire solution to be computed simultaneously because the solution at one point depends on the solution everywhere else.\n",
                "\n",
                "The most common IVP is to find a function $x: [0,T]\\rightarrow R^2$ whose initial value $x(0)$ is known and over its domain it satisfies $x'(t) = f(t,x(t))$. Usually in economics time is autonomous, and thus the equation is of form $x'(t) = f(x(t))$\n",
                "\n",
                "Euler's method: discretize time. Given $x$ at time $t_i$ and a step size $h$, we approximate $x_{i+1} = x_i + hf(t_i, x_i)$. \n",
                "\n",
                "Other refinements: Runge-Kutta method\n",
                "\n",
                "**Example: Commercial fishery**\n",
                "\n",
                "Consider the following model:\n",
                "$$\\begin{aligned}\n",
                "p &= \\alpha - \\beta Ky \\\\\n",
                "\\pi &= py - cy^2/{2S} - f \\\\\n",
                "\\frac{dS}{dt} &= (a-bS)S-Ky\\\\\n",
                "\\frac{dK}{dt} &= \\delta \\pi\n",
                "\\end{aligned}$$\n",
                "\n",
                "where $p$ is the price for fish, $K$ is the size of industry, $y$ is the catch rate of the firm, $\\pi$ is the profit, and $S$ is the fish population.\n",
                "\n",
                "Model solution: industry is competitive so marginal profit = marginal cost: $\\frac{\\partial \\pi}{\\partial y} = 0\\Rightarrow p = cy/S$, which is the inverse supply function in the short run. \n",
                "\n",
                "Market clearing: demand = supply: $\\alpha - \\beta K y = cy/S$ gives short-run equilibrium catch rate $y = \\alpha S/(c+\\beta SK)$ and price $p = ac/(c+\\beta SK)$ and profit function $\\pi = \\frac{c\\alpha^2 S}{2(c+\\beta SK)^2} - f$\n",
                "\n",
                "The dynamics is solved by a 2-dimensional IVP:\n",
                "\n",
                "$$\\begin{aligned}\n",
                " S' &= (a-bS)S - \\frac{\\alpha SK}{c+\\beta SK} \\\\ \n",
                " K' &= \\delta (\\frac{c\\alpha^2 S}{2(c+\\beta SK)^2}-f)\n",
                "\\end{aligned}$$"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 50,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n"
                    ]
                },
                {
                    "data": {
                        "image/png": 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\n",
                        "text/plain": [
                            "<IPython.core.display.Image object>"
                        ]
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "a = 1;\n",
                "b = 1;\n",
                "alpha = 1;\n",
                "c = 1;\n",
                "beta = 2.75;\n",
                "f = 0.06;\n",
                "delta = 10;\n",
                "\n",
                "\n",
                "f = @(t,x) [(a-b*x(1))*x(1) - (alpha * x(1)*x(2))/(c+beta*x(1)*x(2)); delta*((c*alpha^2*x(1))/(2*(c+beta*x(1)*x(2))^2)-f)];\n",
                "\n",
                "[t, xa] = ode45(f, [0 10], [1 1]);\n",
                "plot(t, xa(:,1))\n",
                "title('S(t)')\n",
                "xlabel('t'), ylabel('S')"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 53,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n"
                    ]
                },
                {
                    "data": {
                        "image/png": 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\n",
                        "text/plain": [
                            "<IPython.core.display.Image object>"
                        ]
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "plot(t, xa(:,2))\n",
                "title('K(t)')\n",
                "xlabel('t'), ylabel('K')"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "# Chapter 6: Function Approximation\n",
                "\n",
                "In functional equation problem, one want to find $f$ such that $Tf = 0$. In functional fixed-point problem, one want to find $f$ such that $f = Tf$\n",
                "\n",
                "Example: differential equations $g(x,f(x), f'(x)) = 0$, integral equations $f(x) - \\int g(x)f(x)dx = 0$. In these cases, $g$ is known and we need to find $f$ satisfying the equation. \n",
                "\n",
                "## Interpolation principle\n",
                "\n",
                "Use of an approximating function $\\hat{f}$ that is easy to evaulate in place of the function of interest $f$. Steps:\n",
                "\n",
                "1. Choose $\\hat{f}(x) = \\sum_{j=1}^n \\phi_j(x)c_j$ where $\\phi_j$ are basis functions chosen. \n",
                "2. Choose $n$ conditions (usually the value of $f$ at n points, or the derivatives of $f$ at these $n$ points) and solve for undetermined coefficients by the conditions.\n",
                "\n",
                "## Polynomial Interpolation\n",
                "\n",
                "According to Weierstrass Theorem, any continuous real-valued function $f$ on interval $[a,b]$ can be approximated to any degree of accuracy using a polynomial.\n",
                "\n",
                "1. The basis functions are $\\{x^i\\}$ for $i = 0,1, 2 \\cdots n-1$ (Vandermonde matrices)\n",
                "2. Solve the undetermined coefficients using values at $n$ points. \n",
                "\n",
                "Evenly spaced interpolation nodes: $x_i = a + \\frac{i-1}{n-1}(b-a)$\n",
                "\n",
                "Chebychev nodes: $x_i = \\frac{a+b}{2} + \\frac{b-a}{2}\\cos{(\\frac{n-i+0.5}{n}\\pi)}$\n",
                "\n",
                "Chebychev-nodes have good theoretical properties. \n",
                "1. Rivlin's Theorem: the approximation error associated with the $n^{th}$ degree Chebychev-node polynomial approximation cannot exceeds $2\\pi \\log(n)+2$ times the lowest error attainable at the same polynomial degree\n",
                "2. Jackson's theorem: the approximation error of $n^{th}$ degree Chebychev-node polynomial approximation can be bounded by \n",
                "\n",
                "$$\n",
                "||f-p_n|| \\leq \\frac{6}{n}||f'||(b-a)(\\log(n)/\\pi + 1)\n",
                "$$\n",
                "\n",
                "Other polynomial basis functions: Chebychev basis. Good basis for interpolate function values at the Chebychev nodes. Also orthogonal. \n",
                "\n",
                "## Other basis\n",
                "\n",
                "Piecewize Polynomial Splines: a series of $k^th$ order polynomial segments spliced together so as to preserve continuity of derivatives of order $k-1$ or less. \n",
                "\n",
                "Piecewise Linear basis: good for non-smooth or discontinuous function. Derivatives at breakpoints are not continuous. \n",
                "\n",
                "## Matlab Examples\n",
                "\n",
                "Matlab functions for approximation: FUNDEFN, FUNFITF, FUNFITXY, FUNEVAL, FUNNODE, FUNBAS, **from CEtools**\n",
                "\n",
                "fundefn: create structured Matlab variable containing numerous fields of intformation  necessary for forming approximations in the chosen function space\n",
                "\n",
                "funfitf: determines the basis coefficients of a member from the specified function space that approximates the function defined in an m-file\n",
                "\n",
                "funfitxy: compute coefficients for function value given at any arbitray points\n",
                "\n",
                "funeval: evaluate the approximated function at one or more points"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 21,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n",
                        "fspace = \n",
                        "\n",
                        "  struct with fields:\n",
                        "\n",
                        "          d: 2\n",
                        "          n: [10 15]\n",
                        "          a: [-1 2]\n",
                        "          b: [4 9]\n",
                        "    bastype: {'spli'  'spli'}\n",
                        "      parms: {{1x3 cell}  {1x3 cell}}\n",
                        "\n",
                        "\n"
                    ]
                }
            ],
            "source": [
                "%10th degree Chebychev approximant for univariate functions on interval [-1,2]\n",
                "fspace = fundefn('cheb', 10, -1,2);\n",
                "%cubic spline approximants for bivariate functions defined on 1<=x1<=2, 4<=x2<=9. \n",
                "%10 points in x1 and 15 points in x2\n",
                "fspace = fundefn('spli', [10 15], [-1 2],[4 9],3)"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Example: interpolate $f(x) = x^2exp(\\alpha x)$"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 51,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Created file 'C:\\Users\\barry\\Documents\\Academics\\Quantecon-Notes\\functions\\nexp.m'.\n"
                    ]
                }
            ],
            "source": [
                "%%file functions\\nexp.m\n",
                "function fx = nexp(x, alpha)\n",
                "    fx = x.^2.*exp(alpha.*x);"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 53,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n"
                    ]
                },
                {
                    "data": {
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\n",
                        "text/plain": [
                            "<IPython.core.display.Image object>"
                        ]
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "alpha = 2;\n",
                "space = fundefn('cheb', 10, -1,1);\n",
                "c = funfitf(space,'nexp', alpha);\n",
                "x = nodeunif(1001, -1,1);\n",
                "yact = nexp(x, alpha);\n",
                "yapp = funeval(c, space, x);\n",
                "plot(x, yact-yapp)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 57,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n"
                    ]
                }
            ],
            "source": [
                "%return the value of the matrix of the basis\n",
                "funbas(space, x);\n",
                "%return the value of the matrix of the derivative of the basis function\n",
                "funbas(space, x, 3);"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Solving functional equations\n",
                "\n",
                "One class of functional equation is in the form of $g(x, f(x)) = 0$ for $x\\in [a,b]$. We want to find $f(x)\\simeq \\hat{f}(x) =\\sum_{j=1}^n \\phi_j(x)c_j = \\phi(x)c$ that satisfies $g(x,\\phi(x)c)\\simeq 0$ for $x\\in [a,b]$\n",
                "\n",
                "General approach: collocatoin: choose $c$ in such a way as to make the residual zero at $n$ prescribed nodes: $g(x_i, \\phi(x_i)c) = 0$ for $i = 1\\cdots n$, which can be solved using root finding methods. \n",
                "\n",
                "Similarly, for differential equations $g(x, f(x), f'(x)) = 0$, subject to $b(f(x_b)) = 0$, we solve for $g(x_i, \\phi(x_i)c, \\phi'(x_i)c)=0$ and the boundary function $b(\\phi(x_b)c)=0$\n",
                "\n",
                "### Example: Cournot Oligopoly\n",
                "\n",
                "An oligopolistic firm's action will affect price, so the marginal revenue is $p+q\\frac{dp}{dq}$. The Cournot assumption is that the firm acts as if any output change it makes will be unmatched by its competitors. It implies $\\frac{dp}{dq} = \\frac{1}{D'(p)}$ where $D(p)$ is the market demand.\n",
                "\n",
                "The supply for this firm is given by finding the function $q = S(p)$ such that \n",
                "\n",
                "$$\n",
                "MR = MC \\Rightarrow p + \\frac{S(p)}{D'(p)} - MC(S(p)) = 0\n",
                "$$\n",
                "\n",
                "Suppose the market demand $q = D(p) = p^{-\\eta}$ and marginal cost $MC(q) = \\alpha \\sqrt{q} + q^2$. Then we want $q = S(p)$ such that \n",
                "\n",
                "$$\n",
                "(p - \\frac{qp^{\\eta + 1}}{\\eta}) - (\\alpha \\sqrt{q} + q^2) = 0\n",
                "$$"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 15,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Created file 'C:\\Users\\barry\\Documents\\Academics\\Quantecon-Notes\\functions\\cournot_resid.m'.\n"
                    ]
                }
            ],
            "source": [
                "%%file functions\\cournot_resid.m\n",
                "function resid = cournot_resid(c, p, alpha, eta, Phi);\n",
                "    dp = (-1./eta)*p.^(eta+1);\n",
                "    q = Phi*c;\n",
                "    resid = p + q.*dp - alpha*sqrt(q) - q.^2;"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 32,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n"
                    ]
                }
            ],
            "source": [
                "alpha = 1; eta = 1.5;\n",
                "n = 25; a = 0.3; b = 3;\n",
                "fspace = fundefn('cheb', n, a,b);\n",
                "p = funnode(fspace);\n",
                "Phi = funbas(fspace, p);\n",
                "c = Phi\\sqrt(p);\n",
                "c = broyden('cournot_resid', c, p, alpha, eta, Phi);"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 33,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n"
                    ]
                },
                {
                    "data": {
                        "image/png": 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\n",
                        "text/plain": [
                            "<IPython.core.display.Image object>"
                        ]
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "%supply curves for different number of firms\n",
                "plot(funeval(c, fspace, p),p)\n",
                "hold on\n",
                "plot(3*funeval(c, fspace, p),p)\n",
                "plot(5*funeval(c, fspace, p),p)\n",
                "plot(10*funeval(c, fspace, p),p)\n",
                "plot(20*funeval(c, fspace, p),p)\n",
                "%demand curve\n",
                "plot(p.^(-eta),p)"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "### Function Inverse\n",
                "\n",
                "Suppose we want to approximate function $f$ that satisfies $f(f(x)) = x$ on some interval $a \\leq x \\leq b$. Suppose $f(x) = 2\\exp(x - 3)$ on $5\\leq x \\leq 10$, then we want to solve $2\\exp(f(x) - 3)-x = 0$"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 111,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n"
                    ]
                }
            ],
            "source": [
                "fspace = fundefn('cheb', 10 ,5,10);\n",
                "x = funnode(fspace);\n",
                "Phi = funbas(fspace, x);\n",
                "c = funfitxy(fspace, x, x);\n",
                "c = broyden('inverse_ex',c,x, Phi);"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 107,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Created file 'C:\\Users\\barry\\Documents\\Academics\\Quantecon-Notes\\functions\\inverse_ex.m'.\n"
                    ]
                }
            ],
            "source": [
                "%%file functions\\inverse_ex.m\n",
                "function resid = inverse_ex(c, x, Phi)\n",
                "    resid = 2*exp((Phi*c) -3) - x;\n",
                "    "
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 112,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n",
                        "inverse =\n",
                        "\n",
                        "    3.9224\n",
                        "    3.9694\n",
                        "    4.0530\n",
                        "    4.1577\n",
                        "    4.2682\n",
                        "    4.3726\n",
                        "    4.4627\n",
                        "    4.5334\n",
                        "    4.5818\n",
                        "    4.6064\n",
                        "\n",
                        "\n"
                    ]
                }
            ],
            "source": [
                "inverse = funeval(c, fspace, x)"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "### Boundary value problem\n",
                "\n",
                "Initial value problem: $x'(t) = f(t, x(t))$ subject to $x(t_0) = x_0$\n",
                "\n",
                "Boundary value problem: one seeks a solution function $x(t): [a,b]\\rightarrow R^d$ such that $r(t, x(t), x'(t)) = 0$ subject to $b_i(t_i^b, x(t_i^b), x'(t_i^b)) = 0$ for $i = 1\\cdots d$. \n",
                "\n",
                "The BVP can be nonlinear in $x'(t)$. And the solution function need not be known at any specific point. \n",
                "\n",
                "Boundary value problem arise most often in deterministic optimal control problem, the solutions to which can be expressed as a set of differential equations defining the dynamic behavior of a set of $d_s$ state variables and $d_x$ control variables, along with initial conditions of $d_s$ state variables and $d_x$ transversality conditions. \n",
                "\n",
                "Example: $r(t, x(t), x'(t)) = x'(t) - x(t)A$ where $A = \\begin{bmatrix} -1 & -0.5 \\\\ 0 & -0.5 \\end{bmatrix}$ with boundary conditions $x_1(0) = 1$ and $x_2(1)=1$. Closed form solution: $x_1(t) = e^{-t}$, $x_2(t) = ce^{-t/2} + e^{-t}$ where $c = e^{0.5}(1-e^{-1})$\n",
                "\n",
                "Need to define a structure variable as following:"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 35,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n",
                        "model = \n",
                        "\n",
                        "  struct with fields:\n",
                        "\n",
                        "      func: 'pbvp'\n",
                        "        tb: [2x1 double]\n",
                        "    params: {[2x2 double]}\n",
                        "\n",
                        "\n"
                    ]
                }
            ],
            "source": [
                "A = [-1 -0.5; 0 -0.5];\n",
                "model.func = 'pbvp'; %name of the function\n",
                "model.tb = [0;1]; %position of the boundary condition: 0 for x1 and 1 for x2\n",
                "model.params = {A}"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 36,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Created file 'C:\\Users\\barry\\Documents\\Academics\\Quantecon-Notes\\functions\\pbvp.m'.\n"
                    ]
                }
            ],
            "source": [
                "%%file functions\\pbvp.m\n",
                "function out1 = pbvp(flag, t, x, dx, A);\n",
                "switch flag\n",
                "case 'r'\n",
                "    out1 = dx - x*A;\n",
                "case 'b'\n",
                "    out1 = [x(1,1) - 1; x(2,2)-1];\n",
                "end"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "It uses the bvpsolve function in CEMatlab module"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 39,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n"
                    ]
                },
                {
                    "data": {
                        "image/png": 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\n",
                        "text/plain": [
                            "<IPython.core.display.Image object>"
                        ]
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "n = 5;\n",
                "a = 0;\n",
                "b = 2;\n",
                "fspace = fundefn('cheb', n-1, a,b);\n",
                "tnodes = funnode(fspace);\n",
                "fspace = fundefn('cheb', n, a, b);\n",
                "\n",
                "% initial condition\n",
                "c = zeros(fspace.n, 2);\n",
                "\n",
                "t = linspace(a,b, 201)';\n",
                "\n",
                "optset('broyden', 'defaults')\n",
                "[c,x,r] = bvpsolve(model, fspace, tnodes, c, t);\n",
                "\n",
                "figure(1)\n",
                "C = exp(0.5)*(1-exp(-1));\n",
                "plot(t, exp(-t)-x(:,1), '-', t, C*exp(-t/2)+exp(-t) - x(:,2), '--');\n",
                "title('BVP approximation error');\n",
                "xlabel('t')\n",
                "legend('x_1','x_2')"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "### Example: Commodity market equilibrium\n",
                "\n",
                "At time $t=0$ there are $S_0$ units of periodically produced commodity available. No more goods will be produced until $t=1$, at which all of currently available good must be consumed. The change in stocks is negative of demand, which we assume $s'(t) = -q = -D(p) = -p^{-\\eta}$. \n",
                "\n",
                "To prevent arbitrage and to induce storage, the price must rise at a rate that covers the cost of capital, $r$, and the phisical storage charge $C$: $p'(t) = rp + C$. \n",
                "\n",
                "Define $x = [p,s]$, then we have residual function $r(t,x,x') = x' - [rx_1+C, -x_1^{-\\eta}]$ with boundary condition $x_2(0) - S = 0, x_2(1) = 0$"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 56,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n"
                    ]
                }
            ],
            "source": [
                "S = 1;\n",
                "r = 0.5;\n",
                "C = 0.1;\n",
                "eta = 2;\n",
                "\n",
                "clear model\n",
                "model.func = 'commodity';\n",
                "model.tb = [0;1];\n",
                "model.params = {r, C, eta, S};"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 57,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Created file 'C:\\Users\\barry\\Documents\\Academics\\Quantecon-Notes\\functions\\commodity.m'.\n"
                    ]
                }
            ],
            "source": [
                "%%file functions\\commodity.m\n",
                "\n",
                "function out1 = commodity(flag, t, x, dx, r,C,eta,S);\n",
                "    switch flag\n",
                "    case 'r'\n",
                "        out1 = dx - [r*x(:,1)+C -x(:,1).^eta];\n",
                "    case 'b'\n",
                "        out1 = x(:,2) - [S; 0];\n",
                "    end"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 72,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n"
                    ]
                },
                {
                    "data": {
                        "image/png": 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\n",
                        "text/plain": [
                            "<IPython.core.display.Image object>"
                        ]
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "n = 5;\n",
                "a = 0;\n",
                "b = 1;\n",
                "fspace = fundefn('cheb', n-1, a,b);\n",
                "tnodes = funnode(fspace);\n",
                "fspace = fundefn('cheb', n, a, b);\n",
                "\n",
                "% initial condition\n",
                "\n",
                "\n",
                "t = linspace(a,b, 201)';\n",
                "\n",
                "c = ones(fspace.n, 2);\n",
                "\n",
                "optset('broyden', 'defaults')\n",
                "[c,x,r] = bvpsolve(model, fspace, tnodes, c, t);\n",
                "\n",
                "close all\n",
                "figure(1)\n",
                "plot(t, x(:,1), '-', t, x(:,2), '--');\n",
                "title('Commodity market equilibrium');\n",
                "xlabel('t')\n",
                "legend('price','stock')"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": null,
            "metadata": {},
            "outputs": [],
            "source": []
        },
        {
            "cell_type": "code",
            "execution_count": null,
            "metadata": {},
            "outputs": [],
            "source": []
        },
        {
            "cell_type": "code",
            "execution_count": null,
            "metadata": {},
            "outputs": [],
            "source": []
        }
    ],
    "metadata": {
        "kernelspec": {
            "display_name": "Matlab",
            "language": "matlab",
            "name": "matlab"
        },
        "language_info": {
            "codemirror_mode": "octave",
            "file_extension": ".m",
            "help_links": [
                {
                    "text": "MetaKernel Magics",
                    "url": "https://metakernel.readthedocs.io/en/latest/source/README.html"
                }
            ],
            "mimetype": "text/x-octave",
            "name": "matlab",
            "version": "0.16.7"
        },
        "latex_envs": {
            "LaTeX_envs_menu_present": true,
            "autoclose": false,
            "autocomplete": true,
            "bibliofile": "biblio.bib",
            "cite_by": "apalike",
            "current_citInitial": 1,
            "eqLabelWithNumbers": true,
            "eqNumInitial": 1,
            "hotkeys": {
                "equation": "Ctrl-E",
                "itemize": "Ctrl-I"
            },
            "labels_anchors": false,
            "latex_user_defs": false,
            "report_style_numbering": false,
            "user_envs_cfg": false
        }
    },
    "nbformat": 4,
    "nbformat_minor": 2
}