Oligopoly - Cournot competition.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": [
    "### NOTE: it is important here to further declare these variables to be nonzero and positive (otherwise sympy cannot find solution for the FOCs)\n",
    "q1, q2, c1, c2, a, b, pi1, pi2, q1hat, q2hat = symbols('q1 q2 c1 c2 a b \\pi_1 \\pi_2 \\hat{q}_1 \\hat{q}_2', nonzero=True, positive=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2-firm Cournot competition setup"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Firms and demand"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assume two firms, 1 and 2, competes according to Cournot competition. We assume each firm has constant marginal cost, $c_1$ and $c_2$.\n",
    "\n",
    "The market demand for a product is linear, such that the inverse demand function takes the form of\n",
    "\\begin{align}\n",
    "    p = a - b (q_1 + q_2)\n",
    "\\end{align}\n",
    "\n",
    "We assume $a > c_1$ and $a > c_2$, so that consumer's willingness to pay exceeds the firm's cost to produce."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The firm's profit-maximization problem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Under these modeling assumptions, each firm's profit is\n",
    "\\begin{align}\n",
    "    \\pi_1 &= p\\cdot q_1 - c_1 \\cdot q_1 \\\\\n",
    "            &= (p-c_1)q_1 \\\\\n",
    "    \\pi_2 &= p\\cdot q_2 - c_2 \\cdot q_2 \\\\\n",
    "            &= (p-c_2)q_2 \\\\\n",
    "\\end{align}\n",
    "or, for $i=1,2$,\n",
    "\\begin{align}\n",
    "    \\pi_i &= p\\cdot q_i - c_i \\cdot q_i \\\\\n",
    "            &= (p-c_i)q_i \\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plugging in the inverse demand function, each firm's profit is\n",
    "\\begin{align}\n",
    "    \\pi_1 & = \\big(a - b(q_1 + q_2) - c_1\\big) q_1 \\\\\n",
    "    \\pi_2 & = \\big(a - b(q_1 + q_2) - c_2\\big) q_2\n",
    "\\end{align}\n",
    "or, for $i=1,2$,\n",
    "\\begin{align}\n",
    "    \\pi_i & = \\big(a - b(q_i + q_j) - c_i\\big) q_i\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solve for the Nash equilibrium of the game"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solve the firm's best responses given beliefs about opponent's output level"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assume that firm 1 (2) believes firm 2 (1) would produce an output level of $\\hat{q}_2$ ($\\hat{q}_1$), then each firm would choose their own output levels to maximize:\n",
    "\\begin{align}\n",
    "    \\max_{q_1}\\pi_1 & = \\big(a - b(q_1 + \\hat{q}_2) - c_1\\big) q_1 \\\\\n",
    "    \\max_{q_2}\\pi_2 & = \\big(a - b(\\hat{q}_1 + q_2) - c_2\\big) q_2\n",
    "\\end{align}\n",
    "or, for $i=1,2$,\n",
    "\\begin{align}\n",
    "    \\max_{q_i}\\pi_i & = \\big(a - b(q_i + \\hat{q}_j) - c_i\\big) q_i\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Firm $i$'s FOC is\n",
    "\\begin{align}\n",
    "    \\frac{\\partial \\pi_i}{\\partial q_i} &= \\big(a - b(q_i + \\hat{q}_j) - c_i\\big) -b q_i \\\\\n",
    "    &= a - b\\hat{q}_j - c_i - 2bq_i = 0 \\\\\n",
    "    \\Rightarrow \\qquad q_i^* &= \\frac{a - c_i - b\\hat{q}_j}{2b}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle q_{1} \\left(a - b \\left(q_{1} + q_{2}\\right) - c_{1}\\right)$"
      ],
      "text/plain": [
       "q₁⋅(a - b⋅(q₁ + q₂) - c₁)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define inverse demand function\n",
    "p = a - b*(q1 + q2)\n",
    "# define profit functions for each firm\n",
    "pi1 = (p-c1)*q1\n",
    "pi2 = (p-c2)*q2\n",
    "# check profit function\n",
    "pi1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle q_{2} \\left(a - b \\left(\\hat{q}_1 + q_{2}\\right) - c_{2}\\right)$"
      ],
      "text/plain": [
       "q₂⋅(a - b⋅(\\hat{q}₁ + q₂) - c₂)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# each firm's profit under the beliefs are\n",
    "pi1.subs([(q2, q2hat)])\n",
    "pi2.subs([(q1, q1hat)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{- \\hat{q}_2 b + a - c_{1}}{2 b}\\right]$"
      ],
      "text/plain": [
       "⎡-\\hat{q}₂⋅b + a - c₁⎤\n",
       "⎢────────────────────⎥\n",
       "⎣        2⋅b         ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{- \\hat{q}_1 b + a - c_{2}}{2 b}\\right]$"
      ],
      "text/plain": [
       "⎡-\\hat{q}₁⋅b + a - c₂⎤\n",
       "⎢────────────────────⎥\n",
       "⎣        2⋅b         ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# solve for each firm's best response function\n",
    "q1br_sol = solve(pi1.subs([(q2, q2hat)]).diff(q1), q1)\n",
    "display(q1br_sol)\n",
    "q2br_sol = solve(pi2.subs([(q1, q1hat)]).diff(q2), q2)\n",
    "display(q2br_sol)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{- \\hat{q}_2 b + a - c_{1}}{2 b}, \\  \\frac{- \\hat{q}_1 b + a - c_{2}}{2 b}\\right)$"
      ],
      "text/plain": [
       "⎛-\\hat{q}₂⋅b + a - c₁  -\\hat{q}₁⋅b + a - c₂⎞\n",
       "⎜────────────────────, ────────────────────⎟\n",
       "⎝        2⋅b                   2⋅b         ⎠"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# retrieve the first element of these solutions\n",
    "q1br = q1br_sol[0]\n",
    "q2br = q2br_sol[0]\n",
    "q1br, q2br"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plot the best responses"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "# assume some values on exogenous variables for plots\n",
    "a_value = 10\n",
    "b_value = 1\n",
    "c1_value = 2\n",
    "c2_value = 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "scrolled": false
   },
   "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"
    }
   ],
   "source": [
    "# plot the best response for firm 1\n",
    "q1plot_fun = q1br.subs([(a,a_value),(b,b_value),(c1,c1_value),(c2,c2_value)])\n",
    "q1plot = plot(q1plot_fun, (q2hat,0,10), ylim=[0,10], xlabel=\"$\\hat{q}_2$\", ylabel=\"$q_1$\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 6 - 2 q_{2}$"
      ],
      "text/plain": [
       "6 - 2⋅q₂"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "## plot the best response for firm 2 (inversed graph)\n",
    "# inverse the function\n",
    "q2plot_fun_inv = solve(q2br.subs([(a,a_value),(b,b_value),(c1,c1_value),(c2,c2_value)]) - q2, q1hat)[0]\n",
    "display(q2plot_fun_inv)\n",
    "# plot\n",
    "q2plot = plot(q2plot_fun_inv, (q2,0,10), ylim=[0,10], xlabel=\"$q_2$\", ylabel=\"$\\hat{q}_1$\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "scrolled": true
   },
   "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"
    }
   ],
   "source": [
    "# overlay two graphs by imposing the NE condition that $q1hat = q1$ and $q2hat = q2$\n",
    "\n",
    "# combine curves into one plot\n",
    "Eqplot_combo = q1plot\n",
    "Eqplot_combo.extend(q2plot)\n",
    "\n",
    "# assign colors for each line\n",
    "Eqplot_combo[0].line_color='salmon'\n",
    "Eqplot_combo[1].line_color='orchid'\n",
    "\n",
    "# show plot\n",
    "Eqplot_combo.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solve for the Nash equilibrium output levels"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ q_{1} : \\frac{a - 2 c_{1} + c_{2}}{3 b}, \\  q_{2} : \\frac{a + c_{1} - 2 c_{2}}{3 b}\\right\\}$"
      ],
      "text/plain": [
       "⎧    a - 2⋅c₁ + c₂      a + c₁ - 2⋅c₂⎫\n",
       "⎨q₁: ─────────────, q₂: ─────────────⎬\n",
       "⎩         3⋅b                3⋅b     ⎭"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# solve for NE, imposing that $q1hat = q1$ and $q2hat = q2$\n",
    "NE_sol = solve([q1br.subs([(q2hat, q2)]) - q1, q2br.subs([(q1hat, q1)]) - q2], q1,q2)\n",
    "NE_sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "# retrieve equilibrium output levels of each firm\n",
    "q1star = NE_sol[q1]\n",
    "q2star = NE_sol[q2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{10}{3}$"
      ],
      "text/plain": [
       "10/3"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check solved equilibrium against previous graphs\n",
    "q1star.subs([(a,a_value),(b,b_value),(c1,c1_value),(c2,c2_value)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{- c_{1} + c_{2}}{b}$"
      ],
      "text/plain": [
       "-c₁ + c₂\n",
       "────────\n",
       "   b    "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# compare output levesl of firm 1 vs. firm 2\n",
    "simplify(q1star - q2star)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solve for equilibrium price"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a - b \\left(\\frac{a - 2 c_{1} + c_{2}}{3 b} + \\frac{a + c_{1} - 2 c_{2}}{3 b}\\right)$"
      ],
      "text/plain": [
       "      ⎛a - 2⋅c₁ + c₂   a + c₁ - 2⋅c₂⎞\n",
       "a - b⋅⎜───────────── + ─────────────⎟\n",
       "      ⎝     3⋅b             3⋅b     ⎠"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.subs([(q1,q1star),(q2,q2star)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{a}{3} + \\frac{c_{1}}{3} + \\frac{c_{2}}{3}$"
      ],
      "text/plain": [
       "a   c₁   c₂\n",
       "─ + ── + ──\n",
       "3   3    3 "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pstar = p.subs([(q1,q1star),(q2,q2star)]).simplify()\n",
    "pstar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Comparative statics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### on output levels"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACYAAAArCAYAAAAUo/pwAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACxUlEQVRYCe2Z4VHcMBCF7UwKOEIHpoOb0AHpAEoAOiA1UAIpIUcHkAqSowOuBI4OLu+zVxrZlg+bkfzLO7O2tFrrvVutZFlXHg6HIqeUZVmp/5+G8V33N+rCfTFb/AaxXCpESG3C/lW/lxKNi9DeLWcjBZBkI131QItiL/u+aw/rX3g6o1yo772Gc9XBeFZ9ZcPcaWqquYlBYKdIvEfRRW7AXpSEb25RpF6FWQm7HMLOHbEerkitISV1M7Xng2H2iFm0nhWt2ygjM85KTKQewP2IFD6zDaVI3YwlNRsxkboU2FkYKdkqFBIxyR4xgZPs5yLVTXbI8nqKStYcs4g8CZn1LJRvqqxF9iw0tspqzPZaEtBWykIZ0+0x7KwRa0VgYiV7jk3k490XYj4UIwtLxEYGyrstEfOhGFlgo8Yu8o/dRz5WXGlxPP6VM7anAb9kC6xeP0m3wsmIDfzwT5uX5J8auiViUyP2deoDH/nb5tB9AbEUVdJ7LS9+sygf7Bwf1IcsautvGI9t1qa2CQjAh/A51fkIYSm5DO2UJWwkW/7OJ+nuVSB30hYJ1SGLrbdj7fo6UtxTJz9vg3dT3cRo4NxCw8mBC+KHuKk216Q5JhKAnIQAIsDXEFJ/7DbF+vpD18EDl6wrv0UFQiT/r4BUoTbyayf9a3YmwJP8HqknjZgBAMq3JEN1LmV4/0m9qJ28wwe5ZrjNxlkaH8a7pMmvDnv9CZih7E4IZ2udNprfHf2kTn713RaBMDRMiI1FBQfy60Vt2GsJ2k4xJCWmztdoA9W6uqF0M5GF1dmcIzakzrlkxOwXk9Db4NfXSJEL5PEN5coqzfIRy4vP2tQxp9Gt43P6MjuFOqd0p+yP07FLefbGYaeeld0THWYoiQ7wrUBdTpF3ldQJ783favdLSvJ1TETIIzcsAEMg9hLnjwgOiVvrl+q1/AeljVK+RPfw+QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle - \\frac{2}{3 b}$"
      ],
      "text/plain": [
       "-2 \n",
       "───\n",
       "3⋅b"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# When firm's own cost increases: its output level decreases in equilibrium\n",
    "q1star.diff(c1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{3 b}$"
      ],
      "text/plain": [
       " 1 \n",
       "───\n",
       "3⋅b"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# When a firm's competitor's cost increases: its output level increases in equilibrium\n",
    "q1star.diff(c2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{3 b}$"
      ],
      "text/plain": [
       " 1 \n",
       "───\n",
       "3⋅b"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# When consumer's willingness to pay increases: firm i's equilibrium output level increases\n",
    "q1star.diff(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{a - 2 c_{1} + c_{2}}{3 b^{2}}$"
      ],
      "text/plain": [
       "-(a - 2⋅c₁ + c₂) \n",
       "─────────────────\n",
       "          2      \n",
       "       3⋅b       "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# When demand curve is steeper (demand less elastic): firm i's equilibrium output level ambiguous\n",
    "#     Note: a low-cost firm would decrease output; a high-cost firm may increase output (can you figure out why?)\n",
    "q1star.diff(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### on equilibrium price in the market"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{3}$"
      ],
      "text/plain": [
       "1/3"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# When firm's own cost increases: market price increases in equilibrium\n",
    "pstar.diff(c1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{3}$"
      ],
      "text/plain": [
       "1/3"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# When firm's competitor's cost increases: market price increases in equilibrium\n",
    "pstar.diff(c2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{3}$"
      ],
      "text/plain": [
       "1/3"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# When consumer's willingness to pay increases: market price increases in equilibrium\n",
    "pstar.diff(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAxElEQVQoFYWRjQ2CMBCFbeMABDeADfyZQEZgBndwBGZhBFhBNpANUDco7zV95DRNbHJt7/W7n7YuhLDTcM51ab9grWEdzueoEUzwA0Jr/AL+E1bF8wTdKArSCo0VBgsOEHoBWqG1MPZWeEwcDewVd9/TJ7mNxwXYy79RMmOZKEXbIFXZStvD3P7AjIrKAaq2eNxQJXO9Spt16xHpqkxKZRwF9oDOGfAEbYpVzePyu36/8A3tSGZvsjD6jndVCxf4V0ATmRXq32m5fsF9TwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# When demand curve is steeper (demand less elastic): market price does not change in equilibrium\n",
    "#    Note: this is an artifact of the linear demand function; not generally true\n",
    "pstar.diff(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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