consumer theory - Ottoni-Wilhelm et al AER 2017 - CD example.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import necessary packages."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# import all functions from python labrary: sympy\n",
    "from sympy import *\n",
    "# import python library: matplotlib (for nice plots)\n",
    "import matplotlib\n",
    "# set sympy output to print pretty math expressions\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Declare variables we are going to use as 'symbols'. We will add more along the way, if necessary."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "xi, gi, G, Gi, wi, alpha, beta = symbols('x_i g_i G G_{-i} w_i alpha beta')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Class example: Ottoni-Wilhelm et al. AER 2017 example"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the optimization program"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our optimization problem is\n",
    "\\begin{align}\n",
    "    & \\max_{x_i,g_i}\\quad u(x_i, G, g_i) = u(x_i, g_i + G_{-i}, g_i) \\\\\n",
    "    \\text{s.t.}\\quad & x_i + g_i = w_i\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For 3 goods, $(q_1,q_2,q_3)$, CD utility function would be something like\n",
    "$$u(q_1, q_2, q_3) = q_1^{\\alpha} q_2^{\\beta} q_3^{1-\\alpha-\\beta}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, for $u(x_i, g_i + G_{-i}, g_i)$, its CD form would be similar to\n",
    "$$u(x_i, g_i + G_{-i}, g_i) = x_i^{\\alpha} (g_i + G_{-i})^{\\beta} g_i^{1-\\alpha-\\beta}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Its ln form would be something like\n",
    "$$u(x_i, g_i + G_{-i}, g_i) = \\alpha\\ln(x_i) + \\beta\\ln(g_i + G_{-i}) + (1-\\alpha-\\beta)ln(g_i)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the utility function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle \\alpha \\log{\\left(G_{-i} + g_{i} \\right)} + \\beta \\log{\\left(g_{i} \\right)} + \\left(- \\alpha - \\beta + 1\\right) \\log{\\left(x_{i} \\right)}$"
      ],
      "text/plain": [
       "α⋅log(G_{-i} + gᵢ) + β⋅log(gᵢ) + (-α - β + 1)⋅log(xᵢ)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Note: in sympy syntax: use double star ** for power\n",
    "u = (1-alpha-beta) * ln(xi) + alpha * ln(gi + Gi) + beta * ln(gi)\n",
    "# show the defined function\n",
    "u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the lagrangian"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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tma5xAgfCcVcnr1R4pDTdhUzyqv7oE72W8BHoF8nh+eaxVy+o9ksY97qb+qaJjYS72j6exyIbWRlmQUe16E10vIUJydFLj8l5vEWGnfom+bXwawtbNPLLnVS8nYzkZa1oGnMt7LImmWiYxMU1OmqvXiOMfWxxXk11cidDuOZdxBS7fLwXyBdhj3UMzsf7LTjjXNKi6mtAksg7AWu7hNcGWlYgj/DBSy8g4Xyrwoc2PKoNOsQxRpDBK7h6yXJftK/xPBKg4QW6UbHwS7ic+qaVjXrax1r7Ux3V4rfScS0fzcdnzOPmPGwhkMHvkfza2i+3qGrvvl1jrqGwu/iP4Rph4mMr82qkk8O882dldAlPxv9Ax+KdKitebjoe6ZAvbNnJcC9o33R5evNv7ZvnaB9rHfW2+S397Ro4ol/f+uV2O/YacUT/ydFFSx+b08n/AYQNyLgD5fW2AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\alpha \\log{\\left(G_{-i} + g_{i} \\right)} + \\beta \\log{\\left(g_{i} \\right)} + \\lambda \\left(- g_{i} + w_{i} - x_{i}\\right) + \\left(- \\alpha - \\beta + 1\\right) \\log{\\left(x_{i} \\right)}$"
      ],
      "text/plain": [
       "α⋅log(G_{-i} + gᵢ) + β⋅log(gᵢ) + λ⋅(-gᵢ + wᵢ - xᵢ) + (-α - β + 1)⋅log(xᵢ)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lam = symbols('lambda')\n",
    "lagr = u + lam *(wi - xi - gi)\n",
    "lagr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solve the first-order conditions. \n",
    "\n",
    "Note the syntax of the solve function, especially the use of the brackets:\n",
    "solve( (eq1, eq2, eq3), var1, var2, var3)\n",
    "\n",
    "The equations are by default treated as equal to 0 by the solver."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left( - \\frac{G_{-i} \\alpha}{2} + \\frac{G_{-i}}{2} - \\frac{\\alpha w_{i}}{2} - \\frac{\\beta w_{i}}{2} + w_{i} - \\frac{\\sqrt{G_{-i}^{2} \\alpha^{2} - 2 G_{-i}^{2} \\alpha + G_{-i}^{2} + 2 G_{-i} \\alpha^{2} w_{i} + 2 G_{-i} \\alpha \\beta w_{i} - 2 G_{-i} \\alpha w_{i} + 2 G_{-i} \\beta w_{i} + \\alpha^{2} w_{i}^{2} + 2 \\alpha \\beta w_{i}^{2} + \\beta^{2} w_{i}^{2}}}{2}, \\  \\frac{G_{-i} \\alpha}{2} - \\frac{G_{-i}}{2} + \\frac{\\alpha w_{i}}{2} + \\frac{\\beta w_{i}}{2} + \\frac{\\sqrt{G_{-i}^{2} \\alpha^{2} - 2 G_{-i}^{2} \\alpha + G_{-i}^{2} + 2 G_{-i} \\alpha^{2} w_{i} + 2 G_{-i} \\alpha \\beta w_{i} - 2 G_{-i} \\alpha w_{i} + 2 G_{-i} \\beta w_{i} + \\alpha^{2} w_{i}^{2} + 2 \\alpha \\beta w_{i}^{2} + \\beta^{2} w_{i}^{2}}}{2}, \\  \\frac{- G_{-i} \\alpha + G_{-i} - \\alpha w_{i} - \\beta w_{i} + 2 w_{i} + \\sqrt{G_{-i}^{2} \\alpha^{2} - 2 G_{-i}^{2} \\alpha + G_{-i}^{2} + 2 G_{-i} \\alpha^{2} w_{i} + 2 G_{-i} \\alpha \\beta w_{i} - 2 G_{-i} \\alpha w_{i} + 2 G_{-i} \\beta w_{i} + \\alpha^{2} w_{i}^{2} + 2 \\alpha \\beta w_{i}^{2} + \\beta^{2} w_{i}^{2}}}{2 w_{i} \\left(G_{-i} + w_{i}\\right)}\\right), \\  \\left( - \\frac{G_{-i} \\alpha}{2} + \\frac{G_{-i}}{2} - \\frac{\\alpha w_{i}}{2} - \\frac{\\beta w_{i}}{2} + w_{i} + \\frac{\\sqrt{G_{-i}^{2} \\alpha^{2} - 2 G_{-i}^{2} \\alpha + G_{-i}^{2} + 2 G_{-i} \\alpha^{2} w_{i} + 2 G_{-i} \\alpha \\beta w_{i} - 2 G_{-i} \\alpha w_{i} + 2 G_{-i} \\beta w_{i} + \\alpha^{2} w_{i}^{2} + 2 \\alpha \\beta w_{i}^{2} + \\beta^{2} w_{i}^{2}}}{2}, \\  \\frac{G_{-i} \\alpha}{2} - \\frac{G_{-i}}{2} + \\frac{\\alpha w_{i}}{2} + \\frac{\\beta w_{i}}{2} - \\frac{\\sqrt{G_{-i}^{2} \\alpha^{2} - 2 G_{-i}^{2} \\alpha + G_{-i}^{2} + 2 G_{-i} \\alpha^{2} w_{i} + 2 G_{-i} \\alpha \\beta w_{i} - 2 G_{-i} \\alpha w_{i} + 2 G_{-i} \\beta w_{i} + \\alpha^{2} w_{i}^{2} + 2 \\alpha \\beta w_{i}^{2} + \\beta^{2} w_{i}^{2}}}{2}, \\  \\frac{- G_{-i} \\alpha + G_{-i} - \\alpha w_{i} - \\beta w_{i} + 2 w_{i} - \\sqrt{G_{-i}^{2} \\alpha^{2} - 2 G_{-i}^{2} \\alpha + G_{-i}^{2} + 2 G_{-i} \\alpha^{2} w_{i} + 2 G_{-i} \\alpha \\beta w_{i} - 2 G_{-i} \\alpha w_{i} + 2 G_{-i} \\beta w_{i} + \\alpha^{2} w_{i}^{2} + 2 \\alpha \\beta w_{i}^{2} + \\beta^{2} w_{i}^{2}}}{2 w_{i} \\left(G_{-i} + w_{i}\\right)}\\right)\\right]$"
      ],
      "text/plain": [
       "⎡⎛                                            ________________________________\n",
       "⎢⎜                                           ╱       2  2           2         \n",
       "⎢⎜  G_{-i}⋅α   G_{-i}   α⋅wᵢ   β⋅wᵢ        ╲╱  G_{-i} ⋅α  - 2⋅G_{-i} ⋅α + G_{-\n",
       "⎢⎜- ──────── + ────── - ──── - ──── + wᵢ - ───────────────────────────────────\n",
       "⎣⎝     2         2       2      2                                             \n",
       "\n",
       "______________________________________________________________________________\n",
       "  2             2                                                         2   \n",
       "i}  + 2⋅G_{-i}⋅α ⋅wᵢ + 2⋅G_{-i}⋅α⋅β⋅wᵢ - 2⋅G_{-i}⋅α⋅wᵢ + 2⋅G_{-i}⋅β⋅wᵢ + α ⋅wᵢ\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                2                                             \n",
       "\n",
       "_______________________                                       ________________\n",
       "2           2    2   2                                       ╱       2  2     \n",
       "  + 2⋅α⋅β⋅wᵢ  + β ⋅wᵢ    G_{-i}⋅α   G_{-i}   α⋅wᵢ   β⋅wᵢ   ╲╱  G_{-i} ⋅α  - 2⋅\n",
       "───────────────────────, ──────── - ────── + ──── + ──── + ───────────────────\n",
       "                            2         2       2      2                        \n",
       "\n",
       "______________________________________________________________________________\n",
       "      2           2             2                                             \n",
       "G_{-i} ⋅α + G_{-i}  + 2⋅G_{-i}⋅α ⋅wᵢ + 2⋅G_{-i}⋅α⋅β⋅wᵢ - 2⋅G_{-i}⋅α⋅wᵢ + 2⋅G_{\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                2                             \n",
       "\n",
       "_______________________________________                                       \n",
       "            2   2           2    2   2                                        \n",
       "-i}⋅β⋅wᵢ + α ⋅wᵢ  + 2⋅α⋅β⋅wᵢ  + β ⋅wᵢ    -G_{-i}⋅α + G_{-i} - α⋅wᵢ - β⋅wᵢ + 2⋅\n",
       "───────────────────────────────────────, ─────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "        ______________________________________________________________________\n",
       "       ╱       2  2           2           2             2                     \n",
       "wᵢ + ╲╱  G_{-i} ⋅α  - 2⋅G_{-i} ⋅α + G_{-i}  + 2⋅G_{-i}⋅α ⋅wᵢ + 2⋅G_{-i}⋅α⋅β⋅wᵢ\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                           2⋅wᵢ⋅(G_{-i} + wᵢ)                 \n",
       "\n",
       "_______________________________________________________________⎞  ⎛           \n",
       "                                    2   2           2    2   2 ⎟  ⎜           \n",
       " - 2⋅G_{-i}⋅α⋅wᵢ + 2⋅G_{-i}⋅β⋅wᵢ + α ⋅wᵢ  + 2⋅α⋅β⋅wᵢ  + β ⋅wᵢ  ⎟  ⎜  G_{-i}⋅α \n",
       "───────────────────────────────────────────────────────────────⎟, ⎜- ──────── \n",
       "                                                               ⎠  ⎝     2     \n",
       "\n",
       "                                 _____________________________________________\n",
       "                                ╱       2  2           2           2          \n",
       "  G_{-i}   α⋅wᵢ   β⋅wᵢ        ╲╱  G_{-i} ⋅α  - 2⋅G_{-i} ⋅α + G_{-i}  + 2⋅G_{-i\n",
       "+ ────── - ──── - ──── + wᵢ + ────────────────────────────────────────────────\n",
       "    2       2      2                                                          \n",
       "\n",
       "______________________________________________________________________________\n",
       "   2                                                         2   2           2\n",
       "}⋅α ⋅wᵢ + 2⋅G_{-i}⋅α⋅β⋅wᵢ - 2⋅G_{-i}⋅α⋅wᵢ + 2⋅G_{-i}⋅β⋅wᵢ + α ⋅wᵢ  + 2⋅α⋅β⋅wᵢ \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                   2                                                          \n",
       "\n",
       "__________                                       _____________________________\n",
       "    2   2                                       ╱       2  2           2      \n",
       " + β ⋅wᵢ    G_{-i}⋅α   G_{-i}   α⋅wᵢ   β⋅wᵢ   ╲╱  G_{-i} ⋅α  - 2⋅G_{-i} ⋅α + G\n",
       "──────────, ──────── - ────── + ──── + ──── - ────────────────────────────────\n",
       "               2         2       2      2                                     \n",
       "\n",
       "______________________________________________________________________________\n",
       "     2             2                                                         2\n",
       "_{-i}  + 2⋅G_{-i}⋅α ⋅wᵢ + 2⋅G_{-i}⋅α⋅β⋅wᵢ - 2⋅G_{-i}⋅α⋅wᵢ + 2⋅G_{-i}⋅β⋅wᵢ + α \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                   2                                          \n",
       "\n",
       "__________________________                                               _____\n",
       "   2           2    2   2                                               ╱     \n",
       "⋅wᵢ  + 2⋅α⋅β⋅wᵢ  + β ⋅wᵢ    -G_{-i}⋅α + G_{-i} - α⋅wᵢ - β⋅wᵢ + 2⋅wᵢ - ╲╱  G_{-\n",
       "──────────────────────────, ──────────────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "______________________________________________________________________________\n",
       "  2  2           2           2             2                                  \n",
       "i} ⋅α  - 2⋅G_{-i} ⋅α + G_{-i}  + 2⋅G_{-i}⋅α ⋅wᵢ + 2⋅G_{-i}⋅α⋅β⋅wᵢ - 2⋅G_{-i}⋅α\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                              2⋅wᵢ⋅(G_{-i} + wᵢ)                              \n",
       "\n",
       "__________________________________________________⎞⎤\n",
       "                       2   2           2    2   2 ⎟⎥\n",
       "⋅wᵢ + 2⋅G_{-i}⋅β⋅wᵢ + α ⋅wᵢ  + 2⋅α⋅β⋅wᵢ  + β ⋅wᵢ  ⎟⎥\n",
       "──────────────────────────────────────────────────⎟⎥\n",
       "                                                  ⎠⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "foc_sol = solve(( diff(lagr,xi), diff(lagr,gi), diff(lagr,lam) ), xi,gi,lam)\n",
    "foc_sol"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The solution is a list. We have two solutions here. Let's check."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first solution is positive in $x_i$ and $g_i$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 29.1937515251343$"
      ],
      "text/plain": [
       "29.1937515251343"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "foc_sol[0][0].subs([(alpha,0.4),(beta,0.1),(wi,40),(Gi,40)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 10.8062484748657$"
      ],
      "text/plain": [
       "10.8062484748657"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "foc_sol[0][1].subs([(alpha,0.4),(beta,0.1),(wi,40),(Gi,40)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The second solution can be negative in $x_i$ and $g_i$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 54.8062484748657$"
      ],
      "text/plain": [
       "54.8062484748657"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "foc_sol[1][0].subs([(alpha,0.4),(beta,0.1),(wi,40),(Gi,40)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -14.8062484748657$"
      ],
      "text/plain": [
       "-14.8062484748657"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "foc_sol[1][1].subs([(alpha,0.4),(beta,0.1),(wi,40),(Gi,40)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We take the first solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To access the first element in our solution, which is a list by itself, we use the list index again."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{G_{-i} \\alpha}{2} + \\frac{G_{-i}}{2} - \\frac{\\alpha w_{i}}{2} - \\frac{\\beta w_{i}}{2} + w_{i} - \\frac{\\sqrt{G_{-i}^{2} \\alpha^{2} - 2 G_{-i}^{2} \\alpha + G_{-i}^{2} + 2 G_{-i} \\alpha^{2} w_{i} + 2 G_{-i} \\alpha \\beta w_{i} - 2 G_{-i} \\alpha w_{i} + 2 G_{-i} \\beta w_{i} + \\alpha^{2} w_{i}^{2} + 2 \\alpha \\beta w_{i}^{2} + \\beta^{2} w_{i}^{2}}}{2}$"
      ],
      "text/plain": [
       "                                            __________________________________\n",
       "                                           ╱       2  2           2           \n",
       "  G_{-i}⋅α   G_{-i}   α⋅wᵢ   β⋅wᵢ        ╲╱  G_{-i} ⋅α  - 2⋅G_{-i} ⋅α + G_{-i}\n",
       "- ──────── + ────── - ──── - ──── + wᵢ - ─────────────────────────────────────\n",
       "     2         2       2      2                                               \n",
       "\n",
       "______________________________________________________________________________\n",
       "2             2                                                         2   2 \n",
       "  + 2⋅G_{-i}⋅α ⋅wᵢ + 2⋅G_{-i}⋅α⋅β⋅wᵢ - 2⋅G_{-i}⋅α⋅wᵢ + 2⋅G_{-i}⋅β⋅wᵢ + α ⋅wᵢ  \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                              2                                               \n",
       "\n",
       "_____________________\n",
       "          2    2   2 \n",
       "+ 2⋅α⋅β⋅wᵢ  + β ⋅wᵢ  \n",
       "─────────────────────\n",
       "                     "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xistar = foc_sol[0][0]\n",
    "xistar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{G_{-i} \\alpha}{2} - \\frac{G_{-i}}{2} + \\frac{\\alpha w_{i}}{2} + \\frac{\\beta w_{i}}{2} + \\frac{\\sqrt{G_{-i}^{2} \\alpha^{2} - 2 G_{-i}^{2} \\alpha + G_{-i}^{2} + 2 G_{-i} \\alpha^{2} w_{i} + 2 G_{-i} \\alpha \\beta w_{i} - 2 G_{-i} \\alpha w_{i} + 2 G_{-i} \\beta w_{i} + \\alpha^{2} w_{i}^{2} + 2 \\alpha \\beta w_{i}^{2} + \\beta^{2} w_{i}^{2}}}{2}$"
      ],
      "text/plain": [
       "                                     _________________________________________\n",
       "                                    ╱       2  2           2           2      \n",
       "G_{-i}⋅α   G_{-i}   α⋅wᵢ   β⋅wᵢ   ╲╱  G_{-i} ⋅α  - 2⋅G_{-i} ⋅α + G_{-i}  + 2⋅G\n",
       "──────── - ────── + ──── + ──── + ────────────────────────────────────────────\n",
       "   2         2       2      2                                                 \n",
       "\n",
       "______________________________________________________________________________\n",
       "       2                                                         2   2        \n",
       "_{-i}⋅α ⋅wᵢ + 2⋅G_{-i}⋅α⋅β⋅wᵢ - 2⋅G_{-i}⋅α⋅wᵢ + 2⋅G_{-i}⋅β⋅wᵢ + α ⋅wᵢ  + 2⋅α⋅β\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                       2                                                      \n",
       "\n",
       "______________\n",
       "   2    2   2 \n",
       "⋅wᵢ  + β ⋅wᵢ  \n",
       "──────────────\n",
       "              "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gistar = foc_sol[0][1]\n",
    "gistar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculate *unfunded crowd-out* $$ \\frac{d g_i^*}{d G_{-i}} |_{d w_i = 0} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{G_{-i} \\alpha^{2} - 2 G_{-i} \\alpha + G_{-i} + \\alpha^{2} w_{i} + \\alpha \\beta w_{i} - \\alpha w_{i} + \\beta w_{i} + \\left(\\alpha - 1\\right) \\sqrt{G_{-i}^{2} \\alpha^{2} - 2 G_{-i}^{2} \\alpha + G_{-i}^{2} + 2 G_{-i} \\alpha^{2} w_{i} + 2 G_{-i} \\alpha \\beta w_{i} - 2 G_{-i} \\alpha w_{i} + 2 G_{-i} \\beta w_{i} + \\alpha^{2} w_{i}^{2} + 2 \\alpha \\beta w_{i}^{2} + \\beta^{2} w_{i}^{2}}}{2 \\sqrt{G_{-i}^{2} \\alpha^{2} - 2 G_{-i}^{2} \\alpha + G_{-i}^{2} + 2 G_{-i} \\alpha^{2} w_{i} + 2 G_{-i} \\alpha \\beta w_{i} - 2 G_{-i} \\alpha w_{i} + 2 G_{-i} \\beta w_{i} + \\alpha^{2} w_{i}^{2} + 2 \\alpha \\beta w_{i}^{2} + \\beta^{2} w_{i}^{2}}}$"
      ],
      "text/plain": [
       "                                                                            __\n",
       "        2                          2                                       ╱  \n",
       "G_{-i}⋅α  - 2⋅G_{-i}⋅α + G_{-i} + α ⋅wᵢ + α⋅β⋅wᵢ - α⋅wᵢ + β⋅wᵢ + (α - 1)⋅╲╱  G\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                         _____________________________________\n",
       "                                        ╱       2  2           2           2  \n",
       "                                    2⋅╲╱  G_{-i} ⋅α  - 2⋅G_{-i} ⋅α + G_{-i}  +\n",
       "\n",
       "______________________________________________________________________________\n",
       "     2  2           2           2             2                               \n",
       "_{-i} ⋅α  - 2⋅G_{-i} ⋅α + G_{-i}  + 2⋅G_{-i}⋅α ⋅wᵢ + 2⋅G_{-i}⋅α⋅β⋅wᵢ - 2⋅G_{-i\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "______________________________________________________________________________\n",
       "           2                                                         2   2    \n",
       " 2⋅G_{-i}⋅α ⋅wᵢ + 2⋅G_{-i}⋅α⋅β⋅wᵢ - 2⋅G_{-i}⋅α⋅wᵢ + 2⋅G_{-i}⋅β⋅wᵢ + α ⋅wᵢ  + 2\n",
       "\n",
       "_____________________________________________________\n",
       "                          2   2           2    2   2 \n",
       "}⋅α⋅wᵢ + 2⋅G_{-i}⋅β⋅wᵢ + α ⋅wᵢ  + 2⋅α⋅β⋅wᵢ  + β ⋅wᵢ  \n",
       "─────────────────────────────────────────────────────\n",
       "__________________                                   \n",
       "       2    2   2                                    \n",
       "⋅α⋅β⋅wᵢ  + β ⋅wᵢ                                     "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CO_unfunded = gistar.diff(Gi).simplify()\n",
    "CO_unfunded"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{0.025 \\left|{- 0.36 G_{-i} + 12.0 \\sqrt{0.0009 G_{-i}^{2} - 0.02 G_{-i} + 1} + 4.0}\\right|}{\\left|{\\sqrt{0.0009 G_{-i}^{2} - 0.02 G_{-i} + 1}}\\right|}$"
      ],
      "text/plain": [
       "      │                       __________________________________      │\n",
       "      │                      ╱              2                         │\n",
       "0.025⋅│-0.36⋅G_{-i} + 12.0⋅╲╱  0.0009⋅G_{-i}  - 0.02⋅G_{-i} + 1  + 4.0│\n",
       "───────────────────────────────────────────────────────────────────────\n",
       "                │   __________________________________│                \n",
       "                │  ╱              2                   │                \n",
       "                │╲╱  0.0009⋅G_{-i}  - 0.02⋅G_{-i} + 1 │                "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CO_unfunded_plot = abs( CO_unfunded.subs([(alpha,0.4),(beta,0.1),(wi,40)]) )\n",
    "CO_unfunded_plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x2743b2e5d08>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(CO_unfunded_plot, (Gi,0,100), xlabel=\"$G_{-i}$\", ylabel=\"unfunded crowd-out\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculate *balanced-budget crowd-out* $$ \\frac{d g_i^*}{d G_{-i}} |_{d w_i = - d G_{-i}} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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SymSnqexu+d+cOBlGlNzuL/1b+/6SkMNgPvcE9mhZlP9m9xzJ7bZ4uCTmrHZO8Vc5yZC1zCp9968ftB/jf0j2E7qawTOh2F3fSHKPldOrcpdJ7yYfpV1lObqC8KdHlnosDq6vD9qX6NeM1VnnvaYcd4SUzp5susae0fdoMSlp31O2/V0BvDyYbd/CtqIuXtqryjplEVUuUttqdtm8RDV+lqr8Kp2o/kErRQY+bdLNXunfuq+sYVm8zFXENlr3C2NMnEivJPe5W3yeTOQ919sDY+60ext94ywkaEAPCtMfXAuX5u2Uhjt3Htz/m44v3r7WuSubt+GAnbw65+t+MG4gd769tRZ7fu2hlyyZmmhow+TYH4lvDj79NKti1ROsW/+s59ceUq5aEhn3K9sFSzbXhm6Xfhu0Mba1l1nrnK67HfNav43T+WTb1obJuV/JaZc6B55z7RtrhrMx2bO+7r+4bzDlcpWRqfxy+lvPVzEZqE14pnBfaPNEwr3JhFv9wOL3555+z3Xst+val1usvzfLNOV8fayfOBV+M/9go/e17x6uddx8jaC9yzXs461TDat4kdOErLEcSabJPpGulajHe76nPezJpgnsueQejX0fm4/sYzQJ7IutCtmqd0c5RZ1KUDZ7yCYPF5XfXmoX/dief/HDW/cVCfNK25r+dbEy1wNXC9sluvfUmDzMVvbensxy3oX2M0U/cI46FTo/1HjQLNVgvAflPCnxLDQAOnx0On8dDv0Hg1PObwePPRS703rlFNay+0bb6noVJqNHm+cR+GbE1CRdFauespSrRxiT9bbHKsvhynbBMs21odu84eSq/YaDV59Ktn67uBkjK7mS0y51tt5yc+37GOv+79mY7Flfd4At/5TLVUam8svp/44SH7ZPV/mpPXDfx87t1qj+zVX9KKz7KGP9qjbIFvfnJm/J9qW27gs+Ha/pJ7b6VLGXLraRX6xp+bZy2R4u03bttZL9yCZjy6VtT+yrYdUALPRTcTma7BNJ5hL1eLf3tL3ZNIE9R+9RgcOHvXawX6uwb6ExmgT2xVaFbNWrIKeoUwnKZg/Z5OHS8tskKBkv+lKTuWS+ENrTyT5nApYly1xVbCXMEt1jLJ6t7L0dk/u9MCo0fiD1jIf/qbz55KUfJxQ6fyLvh9HuDdx+mDnHSs9AvtZm4L/rvO1M9pPxRIM7oBeuJ99TX9B5+6DsY/s5vQsddO58/Oci/p+Iuw/fCpfUhTx3zzcplInEamQlmShXspc4xNTbCcsm857dLjjnhTZ0e2edG6c03I54AG/o17WJlTCyvIs4Kd4udV5oX3OKcWdjskt9gyEt+9/GjJq5jIxlmdvPS0R1bc+NzNoXL/w/CE/FoeszjcT5VH4ON+oUt2hfcVSIS8+l7dxlKhueianvJ+6f2jYvepv9vKRo0z/WHvaRdqqQVaTky4NJ5yrLUZAr+fPeTFK7vKft2KaL2mXpu/RZC/tq/EZ1whxi+gQ3q09EnV1k35Aptipoq7PVKZWxNWXzZr1YWn5jEi0ZJva+IpkWsSxZ5kpyi8lrhe4xyc9uO2MSdZi33rx5Exv2brgAwTckd0r9APMq7P+ja9+F6w/aNw83utY4nfvh1PHuOT8YdYP5OvZN7yftuzee+gnIv03Tn7a3D1Qu3A+65v9vsN8fbXzt/caW07f83uze1daFaXw2+pF8T5X1bvmWxFYTq5zlKqS9dF3gxiQ1sSpRRqTvrHbBMgXOs9sGxXM7Y9e0Ndq7/fle27+1uW1p2kbtL5zi3WzbLgJnOpEMszlZlD3qLJmztv1bMgm6zW4j1jDZUt811UFye8LPA7LvD9NZw2OY1th5SH+2ndq0FL8ZUNa53750X8HtlTuPXqbnqo8kP3+56c9puz6Vziedwrmf9qu2b3Xcvrl9Ed5partidxFIJwqzqC8zTCfmXHnZppb9aklMx9f1Re2c49bgJL9tMjXB42X2Lv5XROenYC893aYvrk+2bU2sLE+MW6q34hUpR8qniue9GJb9MJJ7k77cUntadsXdrU0l++x2ObCa3U8PrDaxb7+MzT0+m31bPtJ7E1uF8rXonrJXWwWdd1enVtpqdtvTls2YvWSbXX4V52Y/NibffpiVfKLvK8pnNssgW7EyVxnb6sYiYvgknUzoF9Tcx1LOD0T/1eZ/sXbj7ofmVU5p+MuK0YfkVQnvMHIOvjvEECXyPVY5y5XSXv3AHKVkokD3WCXKJnkyOW04FHavjIZ6zDnfWueS9o3lkoqJ0lnURpRmkkrfWL5T4SSHB9n9FeJFnyI3D6W/yE5TetzyV15+WcIvYtwd+O+no/Duc/nrzYsBaoeRnx92zK1bs9/+WzvJ44cZv9nc9Bd1/nprmc6S/5bslXex+pTCnqlY7U3vGHbSKfnzXky+a8OkkPuI9jTXFGyG9lGaRZ/hc+gw1Gmv57WxSSGP0tjVPSW27Nxio2u7rFNHs5XtIHsm68fWzGeDMlcN29K6x7QRMWXv7ZiEKg3jB9f2z039aT0uLQH4xvPckpUHRvY0OLIlq3iLbhvyjIzOqPO9UpaKyV7aiFT63uN677o7llsMiJe0k1+asJ5znf/b5QN1Lp+ORPRSjP46ohonOT1p4gcwl62pJTGrkfdIglTAvmR9WmW6xKx2o/cMaLXcG2aI3ARNIfcR7Wk4KdjMtUfq8EfQITWTNr3a2KSQh7rYWjffPoWdLN1hbJW4f9Ba7jB8WoWW7GF7m1osnye3k6n6qh9q/celX2jffjK0WGCl81SR/1icwPEiJuV7PDwXGk2yyl2ulL7z3pObZFWrErltOKL37hiN6DDXazOdN7BvLJskTJa0ERsxSaJvLNypcNLdSzL6rX1/8dgsAaR99v6B8rD+2Z31UibWbcmEyUvF9dt5fuAb/tHyRwvTVFJxTunPWiJF4Vv7WdfV/cQ4KY8Zam/sJW+R+jRm7S1Zban3GItEflXcGxboslrug9rTKFez6dtDnLLfo/v5heOkOoykv2ev2tisludsdXHPdepItpIuyfuxtfIpXeZqYlta95ibSyyf3S5zFANhThgB88PxK+0vljaYkwZhITAkQLkaEtnfOTbcn83mSIx9r2mdnYn098DED9p/bDpH4iFd/H8sXo7otXWb4xTHXLxc0I869pcIjdOxB/mf9/3Cpc13PZmTLYm5uVI7EQD28YaCVTwrQkJgSED1h2f4IRTOIbCCAHVqBbzEUc/SP9iizNXCdgvdY4ppDJ89f5kQwyA6jGB51hoHgaQEKFdJcW6SGDbcBHuxTLHvNeqzM5H+r7V9ru0zbf5S4RD9A+sja1uv2RMJLiWOp80vXHjgpu+81KS/WqjRWdZmSUzJbjl5YaSclWAfzxpW8awICYELAmrbD3GPvlCKEwhsSIA6tSH866xP0T/YqMxVwXYj3a9L2rXPXT57/s+Ea3XxgQAEIAABCEAAAisJqGPnNfa/W5lMVdGtj/VaKZQnDZ4qHX+N0LpnOh8ue9Re23rvQSbbMsmSmFsrs7P8YR9vMFjFsyIkBCAAAQhA4CwE6B/kszRsb7O9y4dljm4D5CoEIAABCEAAAhCAgAiESYRfdfitjr/U5v9geKF9syQUkCAAAQhAAAIQgAAEIAABCEDg2AT4MuHY9kU7CEAAAhCAAAQgkISAJg28TJCXSfKnr3b8ufEjB34hAAEIQAACEIAABCAAAQicggCTCacwM0pCAAIQgAAEIACBJAT82esHmljwHzJ7UsHnOAhAAAIQgAAEIAABCEAAAhA4AQEmE05gZFSEAAQgAAEIQAACiQi0f7b8XOm90qTCoj90TiQLyUAAAhCAAAQgAAEIQAACEIBAQQJMJhSETVYQgAAEIAABCEBg5wTaLxH8p8Y/7FwXxIcABCAAAQhAAAIQgAAEIACBGQSYTJgBi6AQgAAEIAABCEDgzATClwj+7wS7fz3u+IUABCAAAQhAAAIQgAAEIACBMxB4cgYl0RECEIAABCAAAQhAIBmBZqkjljhKxpOEIAABCEAAAhCAAAQgAAEI7IIAkwm7MBNCQgACEIAABCAAgWoIfCdJ2uWOqhEKQSAAAQhAAAIQgAAEIAABCEAgL4G33rx5kzcHUocABCAAAQhAAAIQgAAEIAABCEAAAhCAAAQgAAEIQGDXBPjPhF2bD+EhAAEIQAACEIAABCAAAQhAAAIQgAAEIAABCEAAAvkJvPXPf/6TTxPycyYHCEAAAhCAAAQgAAEIQAACEIAABCAAAQhAAAIQgMBuCbDM0W5Nh+AQgAAEIAABCEAAAhCAAAQgAAEIQAACEIAABCAAgTIEWOaoDGdygQAEIAABCEAAAhCAAAQgAAEIQAACEIAABCAAAQjslgCTCbs1HYJDAAIQgAAEIAABCEAAAhCAAAQgAAEIQAACEIAABMoQeFImG3KBAAQgAIGvvvrqB1F4BgkIQAACEIAABCAAAQhAAAIQ2D8BPeO9tX8t0AACEIBAPAEmE+JZERICEIDAYgLqZD5V5A/obC5GSEQIQAACEIAABCAAAQhAAAIQgAAEIACBDQmwzNGG8MkaAhA4FYGPpO2Pp9IYZSEAAQhAAAIQgAAEIAABCEAAAhCAAAQOQ4DJhMOYEkUgAIHKCXwq+V5WLiPiQQACEIAABCAAAQhAAAIQgAAEIAABCEBglACTCaNY8IQABCCQnICXOPoteaokCAEIQAACEIAABCAAAQhAAAIQgAAEIACBAgSYTCgAmSwgAIFzE9Akgpc4+vncFNAeAhCAAAQgAAEIQAACEIAABCAAAQhAYM8EmEzYs/WQHQIQ2AuBzyXoD3sRFjkhAAEIQAACEIAABCAAAQhAAAIQgAAEIDAk8GTowTkEIAABCCQn4CWOkn2ZoLSeSsLnQcr3wv4f8n+dXHISvCAA+wscN09gdRMPFyEAAQhAAAIQmEGAfsUMWDOCwvU2LPjc5sNVCEDgnASYTDin3dEaAhAoREAdUC9xlPq/Er5Ruv7aoXE6fqGDX7W9/+hT5veknesq2Jex8OpcYLUaYd4ETlqH80IldQhAAAInJFDofkK/Ik/Z2jXXAmVv13zyFBlShQAEzk6AZY7OXgLQHwIQyE3gE2XwMnEmn6nj7EmK1n2jg/fk90HrUWjvzvWXYbOer7R5UuPIrhb2e2AMq/qtdMY6XL9VkBACEIDA/giUuJ/Qr8hTLvbONXfZ2zufPKWGVCEAgVMT4MuEU5sf5SFwbgIaBPcSQV8GCh9q78FwD46n/JLgI6XXfUUQ8rraKYyXLvpM28fhYrtk0U+69p02y/q59pbX6f0Swm25c+f6B23tEk6e1LCfl3VKyTBKR+W5yJ6Kt0f2UUxSBzoLq0JlKbV5lqRXVR1eokBMnBPZcxRH0N/3jT+0tV+w+V7X3mdG4+3F8+z23YudLOdRbBX0OGydWlimStxPaun/Li7LC9m2+eUqc9VwHeMTUd9yl72q+Ywxww8CEIBAbgJMJuQmTPoQgECVBELH1G+y+I36xunYg+G/av+xtnaAPFydv1Ma/lLg7qC6wn2hcP4PhK+1faLzboBHx8+0Wa5n2rx/0Pl33vecO7m/y/9uXr04KQ6r6VxLd08kzLan4u2VfQr7zUrjLKxKlaVZ8PMFrqYO51LxZPa8whj096R2O3H+oGNPXH+vrbv/XUXciUfQL3vbvxMcVYt5FFsFPQ5bp1YUouz3E7Gvov+7tCwvZZu7zNXCdYxPpO5Zy17NfMaY4QcBCECgBIG3S2RCHhCAAAQqJOCB+X/05VJn0YMtHsj/oe+/4vhTxX15K77y/EnX3Qn+q46/1dZNJDiezn/U7qk2D5ZfTXDouicsPNHwV21FnfL2FxN9ea3HFpMa1nu2PSX7btnHGlo6esDDA4er3N5YrdS7eFlaZZwVkcWppjo8qQn2nERz84K4+d7hCenhANwr+ZdeFu+mrCsunqa+rmCULOrZ6+Ke6tRKW80uM25ntBXrEyqvVf3flXxmtzuzgYYIkrNoO76Wq8VeybZDFcdkW/UAAA/4SURBVKu7whUre8prVbnrlOMAAhCAwM4JMJmwcwMiPgQgsJiA/3Pgv6Gj2k/EA/ZP5e/B+7XOg/xXEwBtosrDg9mWwxMJv7f+I3tPbrwehgky+oHG8fsPcCNJ5PVS/lt3rmfZU/Iehv0dy/oh1Ntit1NWa/QuWpYWGyZxxArq8C2NsOctOtPXvPTDtyOXPdFd+ku2ETGSeJ2yviYhtyyRs9fFPdWpNbZaVjpCrNz3E6XvPvra/u8aPrPanVUwH5cPLdKOJ+Jqddew7eOaXd9ylr2EfPo6cgwBCEBglwSe7FJqhIYABCCwnoAH+b22/+uJpNwRXuxCh9Nv6Y+mL38/BPlhxP+DMBqml7nfIv1X7/xBcZoHKe2b/1jQ3oP5VxMO/Ti5jltZlP6WkxrR9pS8h2Gfy6ZtuidlVawstZy33ldSh3NhOJ09b4GUrX3f8f2i+Ndst+RacQ37roBXOOohbXXAOrWqWOS+n7Tpa79l/ze6LK+CORE5R5mrhOuExn9639K91UGhkz+PtGlrv2W5+xMERxCAAAQ2JsBkwsYGIHsIQGAbAuoMTq0V7UGWB11f+9amv0oYXS5JaXsiwGv1e7JhuPyEvK+cv1p40fqG+D73H2g28urYSwx1a2K3YZfslaYH2+3aP+t8Ib9RHkEWr1e9aeda+UfZM8hbJXvJFs390Tx5f2tmlVPzUmUpsw7RZSnY+WYdVhhPrnq5LP95r/9Xpmu3dOw2yJOiboMetG//3yX5w7zTn+skT5G2Ya5cc8NLj2ibOm2F96TBz9r7fuP7hSetfWx3YRuFwb6PXPq/F/fd/oXUx+IfbVtsNUq/iK3E/hR1qk94Ttl0PIW/+aKLrq9qa0L62fq/fd1vHUuOqPtKPw3Fia7nbTzFKVLmauHa6u39HN1D+Gxlr0Y+fVYcQwACENiCAJMJW1AnTwhAoEoC6ix6UMyd0RSD8l5G4n8mFG0G3XStmyCYCNd4Sy4PAvUH83/VuR/IvO+cwrXpdn5zDkJnufkfAR37rasH7Z3P/2nvwSdPfpiPv+j4MRxv/lBnOcec5BuzZ8uoGvaBYzT3MV0z+VXHKpOed5PNVJbu5jsVQPK4Xr6j/cXyaHPLUggfU4c92eAJA08UfK+tm0zQ8XNtlqd1r3XgduJDbU074guKOyqzr5V2kiVH27BYjVtsdM0sl7QPHytucy/Tvp3Y/kVp+dhp9u8p2FdA+k7MXI77jPqXZx0rrdGyL/8ltsVWA/olbBWyPE2dWlI2Q5x795O15TdL/3dQpBadSv+x+8rDEpY9AUqVuRq5RuteoOzVyKdXTDiEAAQgUJ4AkwnlmZMjBCBQLwEPsvhPvMbWJo2WWvE9cPCgvQcjxpzfNLLrBtoeT+N+le5f4kLODuXOsvXv5LIO2jwA5UEpDy57MLEdSKy9cz1mzxrZz+UuExRxNbIqovhIJsnL0kgec7w8gP+HtmFbNbcs3a3Dqv8e8HQ4O3+BdDGBoXOXk6+1Nc7thzYPwg7DTckcYhbd7cWehjLXplcgZY+2zX7Qsd+O9YRQs8yRzrHvFbHkHlNlf5ZtsVVyu4wlOGWri7CyxdHr1KyyGeA4jvu/3ndOrJoXE1KUX6WRq//bybviYOy+4uSWsLwSI2eZq5zrwy3dA6isZa92PleFBQ8IQAACBQg8KZAHWUAAAhConoA6ii8kpAfB2rex18j8d0V+eSMBD97YDQfbGk/J4GV4vJxI33lQP8UXE/00u+Ogvx8Cu0HB7uKjnH7L2O5dhW0mSbRf9FCneM7nf7V5H+s+Ubzot0QVdsqeVbEPcs7iPgYspNMO/veDvOMTXR8r17/Jf+pTfUeripUFGrpMel9kE/IYaxs24yOZrtqCIOessqQ4MXX4lcK1A2du27o2Qv5+E9N5dhOQAd4vunbRvun8SuYQttspjOtt6nLcpe+DkEf19uzJOsumIZ7j/OHjEWe72G6tK2lfy3W6tn+s7IdyONe2JW1FXWxriPayV5V1yiKGspSs3QzpzS2bluPe/SRJ+e2ZJeowNZ+xTEMeV/eV4D+bpfNQ3OJlbky3W35Bv2Rlr81rpu5mVWXZa/VhDwEIQOCIBJ4cUSl0ggAEIDCHgDqhXgvcndGxAdcuKV33G53/1v7HznP8wAO0t9J6petPlc7rsejyb9421t4P85bNy4u0g3ljUaL9lI4fTq6WR5GfBwn9IDQmk/3e0zXr3w0k6niRC3lk+/NPpX/LnrWxT8JdOo+WN/n7SxLbbvgGe4ztamN1JXMmvbt8lH6VZakT8PIgSVm6TLJpF5s2QSw8YOD2o98W2c8TncOJvrF2ZJj01Tn2vEKy1KYXS0wNUvVEQmcfMS9pX+d1urZ/wL89nW3bwrbKcU9pdX+QLntqWy13lXXKgollalvNLpuW456TnEnaGqXzVHmN9WNHRcjA5yIfpX+rLK9hWbzMVcQ2WvcLY0ycSK9Nyt6EOHhDAAIQOAQBJhMOYUaUgAAElhJQB/OZ4r6vffcwpuPmjWPtuzdrQ7gvFNYTCfcmEzx428Udkc0Dbw5zL5w70w8K1x+8G0lultdzhfYbq93gstL3g5m34YCgvDrn655IaTrknW9lB5Lvnj2rYb8D7tWw2qKY1ViWJJMH7z0Y/Dcdd1+VFCpLzu835dVvA7zskZdB65yuWz6v8d84nY/K3F4vtZcc1bUNt9jomttcb0va5W6t6RG+ttnwSxIHw76PsJLdd6fsu9K22OrRTv7Nbqs/s3o4RZ1KUDZ7yCYPF7U1vdSu+rG9a0UPxWvyviJBXmlb2oZbj2JlzpkFVwvbJbq3OtzaH6bs3VKSaxCAAARKEHi7RCbkAQEIQKBGAnoIaAflhstv+OHADwGNC+F87EEdD4xNuvBgcW+yoc3P+Yw6peMJDcs3NugzGifGU+l+qa2bSHAcnbeDg/+5kYbfPr6n143o+S9Jvhh7VsN+B9yrYZW/9FzmUHFZ8kCwJxefSUYPUjSuUFlymzScJLXfcLD7U8nTb7dGZX6UvMzv3uxpKittOnqfChw8ANvW7b4BsG/6++5o2V9pW9sMWxWyVa+CnKJOJSibPWSTh0vLb5OgZLzqx07mlPFCaE89sT9sT5tniAQsS5a5qthKmCW6x1j7EGUvRlHCQAACEMhN4O3cGZA+BCAAgRoJqJPvDqX/LM3LDb3obfbzskLtALvFbwfS/T8IDu9B6yn3qS7c+r+EB8X3gJzfjvlGx83n0f3EQvpeUsiDdN0bvv0wGY49QOmBjwvXk68ZuAyyXYSp4URyRdlT4WpjXy33ClkVKWq1lqUg19eC4Ifs33Xeb6PMJndZcnvketY45W853tE29OvarAiZHxPL+BtkuNvWK1zRtiGSzWybKl231S4fF/cWndte5uD/nxlOCtkC2Ff35MChK8MGs8SJsetFrvqKrQraSrY8VZ1SuZ3d7sysI7PL78z0swcP9fvufWUpy5JlLjusmRms0D0mp92XvRglCQMBCECgBIG33rx5UyIf8oAABCBQFQF1Vn+VQFOTAl7K42pdZ/k5vON9q+Phm0iNfvL/j7bhnyc314Y/CucHVA9eeNDBA4Ovwt5pfBeuP2jva6uc0vBAkuW/WB6ln6jCWBY7L4PU5Ck/P1Q+aO//b7DfHzq++LLB17d2kmmWPRX+qWSugr1kycZdafsNuaX/mdCYtSZWseVsjd6KW6QsKR/Xx9b2t1Tz5GY3+KtjD3T+pP1VPZRfm16WOqz03Q7YNW2B9m4fvtf2b23vamvaLu0v3C2ZLwJOnCj+4nKsuLu1p3FI/lk2DaxcXnxfaSd6bJu7bbfibmJfybbYSebd2neubfuQtrKV8j1dXQw6765OrbTVrHanXzZjjpeUX8W524+NybsNs5JPdLujfGazDLIVK3OVsXUbs0j31ra39tJ19n0uNZ9b8nENAhCAwF4IMJmwF0shJwQgUAUBdSj/K0H81ufYZIMfdPzWZ/f/C1UILSEkkx9m/Jak5f+LzpvJAh3jMhPYkr3yXjzwkxnLaPKpWO1N71EYI57Sy5NgTR3W3pNEfsuuapdCZuwZb2Kx8hdvo5Pd8anEh0xh3/jc9hWyNjYp5FEau7qnxJaYW2x0bZd16mi2sh1kz2T92Jr5bFDmqmFbWveYNiJ12YvJkzAQgAAEaifwdu0CIh8EIACBygh4LfAP1LH0oN7Qeekif/ZclZOsfjv11nILVcl7JGEqYO9Jo11MHCVmtRu9Z5Z3T1i2f4LsJdX24FLIjD3rtXQK+9ar3TrJamOTQh7q4royERM7hZ2cz2Fslbh/0NrgMHxahZbsYXubWiY+tzPlKgQgAIEdEHiyAxkREQIQgEBNBF5KGL+Z54e94R8Sf6ROZ/avEpTHrOVRFN6fCz8E2drPe+2Fm0lgb+wlb/+PcGdquy74lqy21HsdtbuxbU//wfEX2u+lLq+WGXveLRdNAHF6qgMvcVXSrbZvSWEL51Ubm9XynK0u7rlOHclW0iV5P7ZWPqXLXE1sS+secz/IwScmX8JAAAIQqJ0AyxzVbiHkgwAEqiIQOrpeZuRHHftLhMbp2AP8z/t+4VIVu57cf5FAu1gepQpwCYSAfTxEWMWzIiQEhgRUfzzJ/Ur76pe/GsrOOQRqJECdqscqZ+kfbFHmamG7he4xJbwWPjGyEgYCEIBAKQIsc1SKNPlAAAKHIKAOpT+L9kCNB236zkuO+KuFWp3l3dvyKLWynCsX7OOJwSqeFSEhcEFA96eftTGRcEGFEwgsJ0CdWs4uQ8xT9A82KnNVsN1I95iiWgWfGEEJAwEIQKAUASYTSpEmHwhA4EgEPGnwVJ1ef43Qumc6Hy571F6rYe/lDfzH0XtaHqUGbilkgH08RVjFsyIkBCAAAQhA4CwE6B/kszRsb7OFz20+XIUABE5IgGWOTmh0VIYABNYRCJMIvyqVb3X8pTb/wfEL7T9elzKxIQABCEAAAhCAAAQgAAEIQAACEIAABCBQJwG+TKjTLkgFAQhUTECTBl5Gwssd+bNXO//p8l7+DLURmB8IQAACEIAABCAAAQhAAAIQgAAEIAABCMwhwGTCHFqEhQAEIPAnAX/y+oEmFp5q70kFn+MgAAEIQAACEIAABCAAAQhAAAIQgAAEIHBIAkwmHNKsKAUBCBQg0P7Z8nPl9UqTCv5SAQcBCEAAAhCAAAQgAAEIQAACEIAABCAAgUMSYDLhkGZFKQhAoACB9ksE/6HxDwXyIwsIQAACEIAABCAAAQhAAAIQgAAEIAABCGxGgMmEzdCTMQQgsGcC4UsE/3eC3b8ed/xCAAIQgAAEIAABCEAAAhCAAAQgAAEIQOCYBJ4cUy20ggAEIFCEQLPUEUscFWFNJhCAAAQgAAEIQAACEIAABCAAAQhAAAIbEmAyYUP4ZA0BCOyewHfSoF3uaPfKoAAEIAABCEAAAhCAAAQgAAEIQAACEIAABKYI/D+tKRzGGu7/TgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\frac{- G_{-i} \\alpha \\beta - G_{-i} \\alpha - G_{-i} \\beta + G_{-i} - \\alpha \\beta w_{i} - \\alpha w_{i} - \\beta^{2} w_{i} + \\beta w_{i} - \\left(\\beta + 1\\right) \\sqrt{G_{-i}^{2} \\alpha^{2} - 2 G_{-i}^{2} \\alpha + G_{-i}^{2} + 2 G_{-i} \\alpha^{2} w_{i} + 2 G_{-i} \\alpha \\beta w_{i} - 2 G_{-i} \\alpha w_{i} + 2 G_{-i} \\beta w_{i} + \\alpha^{2} w_{i}^{2} + 2 \\alpha \\beta w_{i}^{2} + \\beta^{2} w_{i}^{2}}}{2 \\sqrt{G_{-i}^{2} \\alpha^{2} - 2 G_{-i}^{2} \\alpha + G_{-i}^{2} + 2 G_{-i} \\alpha^{2} w_{i} + 2 G_{-i} \\alpha \\beta w_{i} - 2 G_{-i} \\alpha w_{i} + 2 G_{-i} \\beta w_{i} + \\alpha^{2} w_{i}^{2} + 2 \\alpha \\beta w_{i}^{2} + \\beta^{2} w_{i}^{2}}}$"
      ],
      "text/plain": [
       "                                                                              \n",
       "                                                              2               \n",
       "-G_{-i}⋅α⋅β - G_{-i}⋅α - G_{-i}⋅β + G_{-i} - α⋅β⋅wᵢ - α⋅wᵢ - β ⋅wᵢ + β⋅wᵢ - (β\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                              ________________________________\n",
       "                                             ╱       2  2           2         \n",
       "                                         2⋅╲╱  G_{-i} ⋅α  - 2⋅G_{-i} ⋅α + G_{-\n",
       "\n",
       "         _____________________________________________________________________\n",
       "        ╱       2  2           2           2             2                    \n",
       " + 1)⋅╲╱  G_{-i} ⋅α  - 2⋅G_{-i} ⋅α + G_{-i}  + 2⋅G_{-i}⋅α ⋅wᵢ + 2⋅G_{-i}⋅α⋅β⋅w\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "______________________________________________________________________________\n",
       "  2             2                                                         2   \n",
       "i}  + 2⋅G_{-i}⋅α ⋅wᵢ + 2⋅G_{-i}⋅α⋅β⋅wᵢ - 2⋅G_{-i}⋅α⋅wᵢ + 2⋅G_{-i}⋅β⋅wᵢ + α ⋅wᵢ\n",
       "\n",
       "________________________________________________________________\n",
       "                                     2   2           2    2   2 \n",
       "ᵢ - 2⋅G_{-i}⋅α⋅wᵢ + 2⋅G_{-i}⋅β⋅wᵢ + α ⋅wᵢ  + 2⋅α⋅β⋅wᵢ  + β ⋅wᵢ  \n",
       "────────────────────────────────────────────────────────────────\n",
       "_______________________                                         \n",
       "2           2    2   2                                          \n",
       "  + 2⋅α⋅β⋅wᵢ  + β ⋅wᵢ                                           "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CO_balanced = ( gistar.diff(Gi) + gistar.diff(wi) * (-1) ).simplify()\n",
    "CO_balanced"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{0.025 \\left|{- 0.46 G_{-i} + 22.0 \\sqrt{0.0009 G_{-i}^{2} - 0.02 G_{-i} + 1} + 14.0}\\right|}{\\left|{\\sqrt{0.0009 G_{-i}^{2} - 0.02 G_{-i} + 1}}\\right|}$"
      ],
      "text/plain": [
       "      │                       __________________________________       │\n",
       "      │                      ╱              2                          │\n",
       "0.025⋅│-0.46⋅G_{-i} + 22.0⋅╲╱  0.0009⋅G_{-i}  - 0.02⋅G_{-i} + 1  + 14.0│\n",
       "────────────────────────────────────────────────────────────────────────\n",
       "                │   __________________________________│                 \n",
       "                │  ╱              2                   │                 \n",
       "                │╲╱  0.0009⋅G_{-i}  - 0.02⋅G_{-i} + 1 │                 "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CO_balanced_plot = abs( CO_balanced.subs([(alpha,0.4),(beta,0.1),(wi,40)]) )\n",
    "CO_balanced_plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x2743b171cc8>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(CO_balanced_plot, (Gi,0,100), ylim=[0,1], xlabel=\"$G_{-i}$\", ylabel=\"balanced-budget crowd-out\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}