consumer theory - Bertocchi et al AEJ Policy - individual political candidate problem.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import necessary packages."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "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": [
    "# Class example: Bertocchi et al. AEJ Policy 2020 example---the incumbent politician's choice"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook, we focus on the choice problem of a politician in their paper. We will treat other components of their model as given, and solve the choice problem of one politician. Specifically, we will:\n",
    "1. rely on their solutions on the probability of each politician winning the election;\n",
    "2. take the choices of the political rival as given."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup of a politician's model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A political candidate, $\\zeta \\in \\{I,C\\}$, proposes a policy platform, by choosing three elements $q_{\\zeta} = (\\tau_{\\zeta}, e_{\\zeta}, R_{\\zeta})$, where\n",
    "* $\\tau_{\\zeta}$ is the rate of taxation on wealth;\n",
    "* $e_{\\zeta}$ is public expenditure on education;\n",
    "* $R_{\\zeta}$ is electoral rent, i.e., other public expenditure that benefits the political candidate. \n",
    "\n",
    "The politician is constrained by the government budget constraint\n",
    "\n",
    "$$ e_{\\zeta} + R_{\\zeta} = [\\tau_{\\zeta} - D(\\tau_{\\zeta})]\\omega, $$\n",
    "\n",
    "where \n",
    "* $\\omega$ is total wealth of the jurisdiction; \n",
    "* $D(\\tau_{\\zeta}) = \\frac{1}{2}\\tau_{\\zeta}^2$ is the cost of taxation, that is increasing and convex in $\\tau_{\\zeta}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the optimization program for the incumbent politician"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The incumbent politician I's optimization problem is\n",
    "\\begin{align}\n",
    "    \\max_{\\tau_I, e_I, R_I}\\quad & p_I(q_I, q_C)\\cdot R_I \\\\\n",
    "    \\text{s.t.}\\quad &  \\big(\\tau_I - \\frac{1}{2}\\tau_I^2\\big)\\omega = e_I + R_I\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given the voter's model in the paper, the winning probability of the incumbent is (see more details at the end of this notebook)\n",
    "\n",
    "$$p_I(q_I, q_C) = \\frac{1}{2} + \\phi + (\\tau_C-\\tau_I)\\omega + (e_I-e_C) \\sigma^T, $$\n",
    "where \n",
    "$$ \\sigma^T = \\sigma\\lambda^y + (1-\\sigma)\\lambda^o $$\n",
    "\n",
    "and \n",
    "* $\\phi$ is incumbency advantage\n",
    "* $\\lambda^y$ and $\\lambda^o$ are young and old voters' marginal benefit from education, respectively\n",
    "* $\\sigma$ is independent of politician's policy platform (see more details at the end of this notebook), which is\n",
    "    * increasing in the fraction of the young population, $\\alpha$\n",
    "    * decreasing in the voting cost of the young population\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Thus, the incumbent politician's program becomes\n",
    "\n",
    "\\begin{align}\n",
    "    \\max_{\\tau_I, e_I, R_I} \\quad & \\Big( \\frac{1}{2} + \\phi + (\\tau_C-\\tau_I)\\omega + (e_I-e_C)\\sigma^T \\Big)\\cdot R_I \\\\\n",
    "    \\text{s.t.}\\quad & \\big(\\tau_I - \\frac{1}{2}\\tau_I^2\\big)\\omega = e_I + R_I\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solve the problem using sympy"
   ]
  },
  {
   "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": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "phi, tauC, tauI, omega, eI, eC, sigma, sigmaT, lamY, lamO, RI, RO = symbols('phi, tau_C, tau_I, omega, e_I, e_C, sigma, sigma^T, lambda^y, lambda^o, R_I, R_O')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sigma^{T}$"
      ],
      "text/plain": [
       "σ__T"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sigmaT"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the incumbent's objective function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle R_{I} \\left(\\omega \\left(\\tau_{C} - \\tau_{I}\\right) + \\phi + \\sigma^{T} \\left(- e_{C} + e_{I}\\right) + \\frac{1}{2}\\right)$"
      ],
      "text/plain": [
       "R_I⋅(ω⋅(τ_C - τ_I) + φ + σ__T⋅(-e_C + e_I) + 1/2)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define the winning probability\n",
    "pI = Rational(1,2) + phi + (tauC - tauI)*omega + (eI - eC)*sigmaT\n",
    "# define the objective function\n",
    "uI =  pI * RI\n",
    "# show the defined function\n",
    "uI"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the lagrangian"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Dw4HKGKZzfUcfxRrIXFLSLn1rgcmiz+9rXfMVhn6alS0R2+CzuGmjxRK9+bZEoz2as/rbKxt7OtrHkg/DZqHmMRAqQyPLTsaA4V4iZmybLS5Bf5Zm2wdsSzjfi86NLVtz+ro3Rc/kQKY1w1ZzaE4ZS42bnDz2aeX0CYzuVvpj7e8+1lid8p0a+SJs7hgeKTarqfsQymIExwKGp+60O+e3rWN3tVxaIO1kL5RlzMT2+SX4mAs4o7+LZ1YG4Jk2MTx8HgWhcAyQI+RTWdAy516ngwGjO2YUaHvohmGQ0IaJUzIYW6F9xyPDoVUOo1MyfiHi3NzYI/rn6vPmBkMx6d0LcyWZMNpqE13H4QY74wOdhBf2a5sh1WWREKK/Sxu2x89dfcnD+CSdw0LNYyBUhr4sIfZryzFgvJaKx2xdqfY6utIvdAnbwtg5uZHVNYsJ8Maj9JBFhZg5RSTTgnjbeg5NE+C0dqlxc9pKvqucPoFxtar+WKMHi3P6HVPQpNisvx+2lF+2se2ffalB/ZcOu77hXAfGnxeRKM5k0eWToBCyn+6u5OlvjLPA6u5THbysg/J/Ij54MYlVN5ODLzGc86bsi4DBnXt0jpM8Vebk5uC8BfHB30uHGO3zqjmv52SgrdC+Y6LqO0E5+ZykJRybF+cU0x/gbn86wQREn6MPawduwHhpB+eX8JEOdDJGp5uKC35C9HcB2ebGmPF+q4O+tgnhRtdD+/mqHAPiNUYGcKp+DCzqzPBKjKO34cWzl8SGs3gxZi/JZ7GDlb+5iW+N8dcHwOYem3fWnkP7vCSdFxw3SXxNVM7pE1gza+uPtXukOKffMYZLis1q6jZOsJQ+6LGtyg1NgH3mcJDmyvTL27kZXlZyJ4N4QDmZrE6C0lkdtBXCk7yxC9XhX+9udIy+daw8nJyxwGoFHT0YVJeXpjZfKZ6RwXif7TvRaRw9xXMTkNHMHqttHF0elcLL0I1Y9janCIoP9HGJzk+RDcpT27P6G0RopJDoh4ynqsdAoAyGwC7GgDEr2RgDZruf6Rw7ihPJGIkKqsOEQECfNwni4cKu9xlRPs5x6ItDwXNKv42l5+LN+mGShMrN2YpZHZxsIFOm+AwZ+9bapjyL12w+gQmkeFX96bU7eio5s4330UYyZojfKd/JWlqsO6K/2Gb16z4wTlJjEWV1lnDyGOsuafbXnCo6ee1w8egngoEXKvuZZKcjT4LSWKl+o3izSeWEoYmLiL7D0KeuAoNHDkzQtyW6NoZELr7G6JdKT9HfHDyVHgOr9MvexoD4xVbyBIIbQg6e0uH88mcxF/YooKPN9nar/gF1xoqs0mdjjbfpW84pM6wNZ6vfUubQYaKFUzPzXIPeGGJV6Y9wzj3eTc6xuHhfZNCdFJvV1f3g/fv0z9JJGCZiDC+DmDvI6L/wEw0Y4fHl3J2yiuUNapPHykwkpvjBDajOrQqbY4ji2CSCLFxXHcRjUN+pHEpDv06u1qwlLHzrCFp9WYunrdoRDov1NwfPat/HQA4gI2gIc/ad97+e0NRW+v84UfxhkxD4o/Ls37O97GPbEQKp1VFMMm02p8QiIF6D7HAs3ZLl98hzDB416Y94yTreY3AoUTaH7ojGYpvVr/swh4AimMMZwQHlkd4WgWV7lCzawZPsOLqrO+65QIroO/bhhjzeyMXaHB272Zgrdw35i/U3Bzg+BnKgGE2DRYf/CfsPW/yNAE9HeA8hZN+s1SG2lZHorRR9IpWdbzmnREGh/soxh0a1mVp4jzxHylyT/uQe75FQ5C2eSXdSbFZX90Fe0ZKoYXyNsSRCsZXVIY0jq5g9vB7OEGhxYaW8mpVt8RKzZ+1MomNdtv1C/7j+FuraFtuaxgDOLi8Ej43J20gonlNe9KKfhkW2s2bxzeaUNYX0toohUJP+5B7vxUBbkXCKzerqZlkJziQ0j3RZweCN9DHDnqmpSzKt8bc3ey8LXHGKsHFcKu9/19+yHVTbGBA/rP4PhWZfqfJjV3RZacq5x36It7XTNp1T1hbW28uOQDX6U2C8ZwdrA4IpNqurW9NK8JsWRJjz4Ag4Ao6AIxCBgCZKHGCepkU9Wlc96rByzKR/pOBzypF6c31ZqtafpeN9fRjzt5his87rVuMEizFWLlgB/io/ZE7REXAEHIHDI8B7DbyQG7tVyBYeDvFCnPWyzymGhMdLENiB/iwd70vgqK1Ois06qVuNE9wi/ItiY7A20J0fR8ARcASqREATNi+u8t3oJS/p8kIw+4uPtB/Y+snnFEPC4yUIVKk/ieN9CQ611UmxWSd1a3OCubNhT7A7wrWpnPPjCDgCVSIge9m8EKl4iQOMTHxqCCf6iMHnlCP26noyVac/Gcb7euiVaynFZp3UrcoJVuc2b0AKt7GXPspB6pQdAUfAEdgZArKZGPQnfQdY53we7XGIKCpnXxQ55Muvks/nlBBF8DKDCNSmP+InabwPCrmzxBSbNVS3Kie47Qs+Gv61mI39xM/OutLZdQQcAUdgOQKykbwI91zx+YtwTJRvAylT91fRWP2LPIH85Sjmc0oOFK+XRhX6k2m8H6EXU2zWRd0s/xiXG1V1Nv96xAse58Y9d1NOzxFwBByB3SEg28hKL19zOP+s2SOlPVX+kzmhVIZtZ9BgJfmI+4E7CCSfzykdGn4Si8DW+qP2k8d7rMw1lhcOi23WWN0aV4LBHuf3GzHtq8E1aqLz5Ag4AlsjwF5FJka2M/QPVoFDV3VtVeTQDrDwIPiccoeD/y5DYGv9yTHel0leV60UmzVYt8qVYDCXA/yHIt52hnEPjoAj4Ag4ApkQkF21FZXzv13O1EJ9ZHxOqa9P9sSR68+2vZVis6bq1roSDNovdLA3mNUOD46AI+AIOAL5EGBl6QvZ19BV43wtb0fJ55TtsD9Cy64/2/Ziis0arVutEyzjzJ9nvNIB8x4cAUfAEXAEMiAg24pN5Snbof4cYw4an1PmEPL8KQRcf6bQKZuXYrPm6lbrBAOpmOefj/iIO29nenAEHAFHwBFIQEC2lD3DfELtKj9D6XNKgvJ41RvXn/WVQJgvtlkhdat2goFbQmCsedvZvme5fi94i46AI+AI7BwB2VA+qfZSx6c7FyWJfZ9TkuC7+squP+upQIrNCq1bvRMM3BKGv7lj/5r/k9x6+uctOQKOwEEQkO3kSzuvdXyq82vaBzzYgz6nDMLiiYEIuP4EApVQLMVmxdT9P1g7hsR2NIstAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle R_{I} \\left(\\omega \\left(\\tau_{C} - \\tau_{I}\\right) + \\phi + \\sigma^{T} \\left(- e_{C} + e_{I}\\right) + \\frac{1}{2}\\right) + \\lambda \\left(- R_{I} - e_{I} + \\omega \\left(- \\frac{\\tau_{I}^{2}}{2} + \\tau_{I}\\right)\\right)$"
      ],
      "text/plain": [
       "                                                      ⎛               ⎛     2      ⎞⎞\n",
       "                                                      ⎜               ⎜  τ_I       ⎟⎟\n",
       "R_I⋅(ω⋅(τ_C - τ_I) + φ + σ__T⋅(-e_C + e_I) + 1/2) + λ⋅⎜-R_I - e_I + ω⋅⎜- ──── + τ_I⎟⎟\n",
       "                                                      ⎝               ⎝   2        ⎠⎠"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lam = symbols('lambda')\n",
    "# note: power in sympy is **\n",
    "lagr = uI + lam *( (tauI - Rational(1,2) * tauI**2)*omega  - eI - RI )\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": 16,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left( \\frac{\\sigma^{T} - 1}{\\sigma^{T}}, \\  \\frac{2 e_{C} \\left(\\sigma^{T}\\right)^{2} + \\omega \\left(\\sigma^{T}\\right)^{2} - 2 \\omega \\sigma^{T} \\tau_{C} + 2 \\omega \\sigma^{T} - 3 \\omega - 2 \\phi \\sigma^{T} - \\sigma^{T}}{4 \\left(\\sigma^{T}\\right)^{2}}, \\  - \\frac{2 e_{C} \\left(\\sigma^{T}\\right)^{2} - \\omega \\left(\\sigma^{T}\\right)^{2} - 2 \\omega \\sigma^{T} \\tau_{C} + 2 \\omega \\sigma^{T} - \\omega - 2 \\phi \\sigma^{T} - \\sigma^{T}}{4 \\left(\\sigma^{T}\\right)^{2}}, \\  - \\frac{2 e_{C} \\left(\\sigma^{T}\\right)^{2} - \\omega \\left(\\sigma^{T}\\right)^{2} - 2 \\omega \\sigma^{T} \\tau_{C} + 2 \\omega \\sigma^{T} - \\omega - 2 \\phi \\sigma^{T} - \\sigma^{T}}{4 \\sigma^{T}}\\right), \\  \\left( \\frac{\\omega \\sigma^{T} - \\omega - \\sqrt{\\omega \\left(- 2 e_{C} \\left(\\sigma^{T}\\right)^{2} + \\omega \\left(\\sigma^{T}\\right)^{2} + 2 \\omega \\sigma^{T} \\tau_{C} - 2 \\omega \\sigma^{T} + \\omega + 2 \\phi \\sigma^{T} + \\sigma^{T}\\right)}}{\\omega \\sigma^{T}}, \\  - \\frac{\\sqrt{- \\omega \\left(2 e_{C} \\left(\\sigma^{T}\\right)^{2} - \\omega \\left(\\sigma^{T}\\right)^{2} - 2 \\omega \\sigma^{T} \\tau_{C} + 2 \\omega \\sigma^{T} - \\omega - 2 \\phi \\sigma^{T} - \\sigma^{T}\\right)}}{\\left(\\sigma^{T}\\right)^{2}} + \\frac{2 e_{C} \\left(\\sigma^{T}\\right)^{2} - 2 \\omega \\sigma^{T} \\tau_{C} + 2 \\omega \\sigma^{T} - 2 \\omega - 2 \\phi \\sigma^{T} - \\sigma^{T}}{2 \\left(\\sigma^{T}\\right)^{2}}, \\  0, \\  0\\right), \\  \\left( \\frac{\\omega \\sigma^{T} - \\omega + \\sqrt{\\omega \\left(- 2 e_{C} \\left(\\sigma^{T}\\right)^{2} + \\omega \\left(\\sigma^{T}\\right)^{2} + 2 \\omega \\sigma^{T} \\tau_{C} - 2 \\omega \\sigma^{T} + \\omega + 2 \\phi \\sigma^{T} + \\sigma^{T}\\right)}}{\\omega \\sigma^{T}}, \\  \\frac{\\sqrt{- \\omega \\left(2 e_{C} \\left(\\sigma^{T}\\right)^{2} - \\omega \\left(\\sigma^{T}\\right)^{2} - 2 \\omega \\sigma^{T} \\tau_{C} + 2 \\omega \\sigma^{T} - \\omega - 2 \\phi \\sigma^{T} - \\sigma^{T}\\right)}}{\\left(\\sigma^{T}\\right)^{2}} + \\frac{2 e_{C} \\left(\\sigma^{T}\\right)^{2} - 2 \\omega \\sigma^{T} \\tau_{C} + 2 \\omega \\sigma^{T} - 2 \\omega - 2 \\phi \\sigma^{T} - \\sigma^{T}}{2 \\left(\\sigma^{T}\\right)^{2}}, \\  0, \\  0\\right)\\right]$"
      ],
      "text/plain": [
       "⎡                                                                                                                                                                                                                                           ⎛     \n",
       "⎢⎛                    2         2                                                     ⎛          2         2                                                ⎞    ⎛          2         2                                                ⎞ ⎞  ⎜     \n",
       "⎢⎜σ__T - 1  2⋅e_C⋅σ__T  + ω⋅σ__T  - 2⋅ω⋅σ__T⋅τ_C + 2⋅ω⋅σ__T - 3⋅ω - 2⋅φ⋅σ__T - σ__T  -⎝2⋅e_C⋅σ__T  - ω⋅σ__T  - 2⋅ω⋅σ__T⋅τ_C + 2⋅ω⋅σ__T - ω - 2⋅φ⋅σ__T - σ__T⎠   -⎝2⋅e_C⋅σ__T  - ω⋅σ__T  - 2⋅ω⋅σ__T⋅τ_C + 2⋅ω⋅σ__T - ω - 2⋅φ⋅σ__T - σ__T⎠ ⎟  ⎜ω⋅σ__\n",
       "⎢⎜────────, ───────────────────────────────────────────────────────────────────────, ─────────────────────────────────────────────────────────────────────────, ─────────────────────────────────────────────────────────────────────────⎟, ⎜─────\n",
       "⎢⎜  σ__T                                          2                                                                         2                                                                     4⋅σ__T                                 ⎟  ⎜     \n",
       "⎣⎝                                          4⋅σ__T                                                                    4⋅σ__T                                                                                                             ⎠  ⎝     \n",
       "\n",
       "           _____________________________________________________________________________       ____________________________________________________________________________                                                                      ⎞\n",
       "          ╱   ⎛            2         2                                                ⎞       ╱    ⎛          2         2                                                ⎞              2                                                        ⎟\n",
       "T - ω - ╲╱  ω⋅⎝- 2⋅e_C⋅σ__T  + ω⋅σ__T  + 2⋅ω⋅σ__T⋅τ_C - 2⋅ω⋅σ__T + ω + 2⋅φ⋅σ__T + σ__T⎠     ╲╱  -ω⋅⎝2⋅e_C⋅σ__T  - ω⋅σ__T  - 2⋅ω⋅σ__T⋅τ_C + 2⋅ω⋅σ__T - ω - 2⋅φ⋅σ__T - σ__T⎠    2⋅e_C⋅σ__T  - 2⋅ω⋅σ__T⋅τ_C + 2⋅ω⋅σ__T - 2⋅ω - 2⋅φ⋅σ__T - σ__T      ⎟\n",
       "────────────────────────────────────────────────────────────────────────────────────────, - ─────────────────────────────────────────────────────────────────────────────── + ─────────────────────────────────────────────────────────────, 0, 0⎟\n",
       "                                       ω⋅σ__T                                                                                        2                                                                         2                                 ⎟\n",
       "                                                                                                                                 σ__T                                                                    2⋅σ__T                                  ⎠\n",
       "\n",
       "  ⎛                _____________________________________________________________________________     ____________________________________________________________________________                                                                 \n",
       "  ⎜               ╱   ⎛            2         2                                                ⎞     ╱    ⎛          2         2                                                ⎞              2                                                   \n",
       "  ⎜ω⋅σ__T - ω + ╲╱  ω⋅⎝- 2⋅e_C⋅σ__T  + ω⋅σ__T  + 2⋅ω⋅σ__T⋅τ_C - 2⋅ω⋅σ__T + ω + 2⋅φ⋅σ__T + σ__T⎠   ╲╱  -ω⋅⎝2⋅e_C⋅σ__T  - ω⋅σ__T  - 2⋅ω⋅σ__T⋅τ_C + 2⋅ω⋅σ__T - ω - 2⋅φ⋅σ__T - σ__T⎠    2⋅e_C⋅σ__T  - 2⋅ω⋅σ__T⋅τ_C + 2⋅ω⋅σ__T - 2⋅ω - 2⋅φ⋅σ__T - σ__T \n",
       ", ⎜─────────────────────────────────────────────────────────────────────────────────────────────, ─────────────────────────────────────────────────────────────────────────────── + ─────────────────────────────────────────────────────────────,\n",
       "  ⎜                                            ω⋅σ__T                                                                                      2                                                                         2                            \n",
       "  ⎝                                                                                                                                    σ__T                                                                    2⋅σ__T                             \n",
       "\n",
       "     ⎞⎤\n",
       "     ⎟⎥\n",
       "     ⎟⎥\n",
       " 0, 0⎟⎥\n",
       "     ⎟⎥\n",
       "     ⎠⎦"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "foc_sol = solve(( diff(lagr,tauI), diff(lagr,eI), diff(lagr,RI), diff(lagr,lam) ), tauI,eI,RI,lam)\n",
    "foc_sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\sigma^{T} - 1}{\\sigma^{T}}$"
      ],
      "text/plain": [
       "σ__T - 1\n",
       "────────\n",
       "  σ__T  "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# incumbent's optimal tax rate policy in response to rivals and voters is\n",
    "tauI_sol = foc_sol[0][0]\n",
    "tauI_sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{2 e_{C} \\left(\\sigma^{T}\\right)^{2} + \\omega \\left(\\sigma^{T}\\right)^{2} - 2 \\omega \\sigma^{T} \\tau_{C} + 2 \\omega \\sigma^{T} - 3 \\omega - 2 \\phi \\sigma^{T} - \\sigma^{T}}{4 \\left(\\sigma^{T}\\right)^{2}}$"
      ],
      "text/plain": [
       "          2         2                                                  \n",
       "2⋅e_C⋅σ__T  + ω⋅σ__T  - 2⋅ω⋅σ__T⋅τ_C + 2⋅ω⋅σ__T - 3⋅ω - 2⋅φ⋅σ__T - σ__T\n",
       "───────────────────────────────────────────────────────────────────────\n",
       "                                      2                                \n",
       "                                4⋅σ__T                                 "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# incumbent's optimal education expenditure policy in response to rivals and voters is\n",
    "eI_sol = foc_sol[0][1]\n",
    "eI_sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{2 e_{C} \\left(\\sigma^{T}\\right)^{2} - \\omega \\left(\\sigma^{T}\\right)^{2} - 2 \\omega \\sigma^{T} \\tau_{C} + 2 \\omega \\sigma^{T} - \\omega - 2 \\phi \\sigma^{T} - \\sigma^{T}}{4 \\left(\\sigma^{T}\\right)^{2}}$"
      ],
      "text/plain": [
       " ⎛          2         2                                                ⎞ \n",
       "-⎝2⋅e_C⋅σ__T  - ω⋅σ__T  - 2⋅ω⋅σ__T⋅τ_C + 2⋅ω⋅σ__T - ω - 2⋅φ⋅σ__T - σ__T⎠ \n",
       "─────────────────────────────────────────────────────────────────────────\n",
       "                                       2                                 \n",
       "                                 4⋅σ__T                                  "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# incumbent's optimal political rent policy in response to rivals and voters is\n",
    "RI_sol = foc_sol[0][2]\n",
    "RI_sol"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Comparative statics: Incumbent politician's education policy platform in response to political rival's policy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{2}$"
      ],
      "text/plain": [
       "1/2"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eI_sol.diff(eC)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Appendix: more details on the incumbent's winning probability"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the voter's model (see derivation in Appendix of the paper), the winning probability for the incumbent is\n",
    "\\begin{align}\n",
    "    p_I(q_I,q_C) &= Pr(\\pi_I \\geq \\pi_C) \\\\\n",
    "        &= \\frac{1}{2} + \\phi + \\sigma\\big(U^y(q_I) - U^y(q_C)\\big) + (1-\\sigma) \\big(U^o(q_I) - U^o(q_C)\\big) \\tag{1}\n",
    "\\end{align}\n",
    "where \n",
    "* $\\phi$ is a measure of incumbency advantage; \n",
    "* $\\sigma = \\frac{\\alpha}{\\alpha + (1-\\alpha)\\frac{\\bar{c}^y}{\\bar{c}^o}}$ is independent of politician's polity platform, but is a function of\n",
    "    - $\\alpha$: the share of young in population\n",
    "    - $\\bar{c}^y$ and $\\bar{c}^o$: the upper bounds of each voter group's voting cost"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The paper models voter's utility from politician $\\zeta$'s policy platform as\n",
    "\\begin{align}\n",
    "    U^y(q_{\\zeta}) &= (1-\\tau_{\\zeta})\\omega + \\lambda^y e_{\\zeta} \\\\\n",
    "    U^o(q_{\\zeta}) &= (1-\\tau_{\\zeta})\\omega + \\lambda^o e_{\\zeta}\n",
    "\\end{align}\n",
    "thus \n",
    "\\begin{align}\n",
    "    U^y(q_I) - U^y(q_C) &= (\\tau_C-\\tau_I)\\omega + \\lambda^y (e_I-e_C) \\\\\n",
    "    U^o(q_I) - U^o(q_C) &= (\\tau_C-\\tau_I)\\omega + \\lambda^o (e_I-e_C)\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plugging this into (1), we have\n",
    "\\begin{align}\n",
    "    p_I(q_I,q_C) &= \\frac{1}{2} + \\phi + \\sigma\\big(U^y(q_I) - U^y(q_C)\\big) + (1-\\sigma) \\big(U^o(q_I) - U^o(q_C)\\big) \\\\\n",
    "    &= \\frac{1}{2} + \\phi + \\sigma\\big( (\\tau_C-\\tau_I)\\omega + \\lambda^y (e_I-e_C) \\big) + (1-\\sigma) \\big( (\\tau_C-\\tau_I)\\omega + \\lambda^o (e_I-e_C) \\big) \\\\\n",
    "    &= \\frac{1}{2} + \\phi + (\\tau_C-\\tau_I)\\omega + (e_I-e_C)\\big( \\sigma\\lambda^y + (1-\\sigma)\\lambda^o \\big)\n",
    "\\end{align}"
   ]
  }
 ],
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