Monopoly - 1. quantity vs price.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import necessary packages."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "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": 3,
   "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",
    "Q, c, a, b, pi = symbols('Q c a b \\pi', nonzero=True, positive=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# monopoly setup"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Firm and demand"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assume one monopolistic firm sells a product in a given market. We assume the firm has a constant marginal cost, $c$.\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\n",
    "\\end{align}\n",
    "\n",
    "We assume $a > c$, so that consumer's willingness to pay exceeds the firm's cost to produce."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Graph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "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"
    }
   ],
   "source": [
    "# Plotting inverse market demand\n",
    "DInverse = a - b*Q\n",
    "Dplot0 = plot(DInverse.subs([(a,100),(b,Rational(1,2))]), (Q,0,300), xlim=[0,260], ylim=[0,130], xlabel=\"$Q$\", ylabel=\"$p$\", show=False)\n",
    "\n",
    "# Plotting MB curve\n",
    "MRcurve = diff(DInverse*Q, Q)\n",
    "MRplot0 = plot(MRcurve.subs([(a,100),(b,Rational(1,2))]), (Q,0,300), xlim=[0,200], ylim=[0,130], xlabel=\"$Q$\", ylabel=\"$p$\", show=False)\n",
    "\n",
    "# Plotting MC curve\n",
    "MCcurve = c\n",
    "MCplot0 = plot(MCcurve.subs([(c,20)]), (Q,0,300), xlim=[0,200], ylim=[0,130], xlabel=\"$Q$\", ylabel=\"$p$\", show=False)\n",
    "\n",
    "\n",
    "# combine plots into one plot\n",
    "plot_combo = Dplot0\n",
    "plot_combo.extend(MRplot0)\n",
    "plot_combo.extend(MCplot0)\n",
    "\n",
    "# assign colors for each line\n",
    "plot_combo[0].line_color='blue'\n",
    "plot_combo[1].line_color='darkblue'\n",
    "plot_combo[2].line_color='orchid'\n",
    "\n",
    "# show plot\n",
    "plot_combo.show()"
   ]
  },
  {
   "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 &= p\\cdot Q - c \\cdot Q \\\\\n",
    "            &= (p-c)Q \\tag{1}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plugging in the inverse demand function, each firm's profit is\n",
    "\\begin{align}\n",
    "    \\pi(Q) & = \\big(a - bQ - c\\big) Q   \\tag{2}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From here, we can try to find the FOC w.r.t. Q for the firm. But we will explore an alternative development before going this route."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The firm's profit-maximization problem (re-write in terms of p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Instead, from equation (1), we can plug in the demand function (not inverse demand) and rewrite $\\pi$ in terms of $p$. Specifically, the firm's demand function is \n",
    "$$ Q = \\frac{a}{b} - \\frac{p}{b}.$$\n",
    "Plugging into (1), and we have\n",
    "\\begin{align}\n",
    "    \\pi(p) & = (p-c)\\big(\\frac{a}{b} - \\frac{p}{b}\\big)  \\tag{3}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Monopolist chooses Q to maximize profit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the firm chooses $Q$ to maximize its profit, the firm maximizes (2), i.e.,\n",
    "\\begin{align}\n",
    "    \\max_{Q}\\pi(Q) & = \\big(a - bQ - c\\big) Q\n",
    "\\end{align}\n",
    "which implies the FOC\n",
    "\\begin{align}\n",
    "    \\big(a - bQ - c\\big) - bQ = 0 \n",
    "    \\quad\\Rightarrow\\quad \n",
    "    Q^* = \\frac{a-c}{2b}\n",
    "\\end{align}\n",
    "Plugging back into the inverse demand funciton, the monopolist would price at\n",
    "\\begin{align}\n",
    "    p^* &= a - bQ^* \\\\\n",
    "        &= a - \\frac{a-c}{2} \\\\\n",
    "        &= \\frac{a+c}{2}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Monopolist chooses p to maximize profit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The firm maximizes\n",
    "\\begin{align}\n",
    "    \\max_p \\pi(p) & = (p-c)\\big(\\frac{a}{b} - \\frac{p}{b}\\big) \n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "which leads to FOC\n",
    "\\begin{align}\n",
    "    \\frac{\\partial \\pi(p)}{\\partial p} & = \\big(\\frac{a}{b} - \\frac{p}{b}\\big) - \\frac{p-c}{b} \\\\\n",
    "        &= \\frac{a}{b} - \\frac{2p}{b} + \\frac{c}{b} =0 \\quad\\Rightarrow\\quad p^* = \\frac{a+c}{2}\n",
    "\\end{align}\n",
    "\n",
    "the same profit-maximizing price as the case when the monopolist chooses quantity."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Challenge"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Q1**: solve two cases of the monopoly profit-maximization problem using sympy\n",
    "* if the firm chooses output level to maximize profit\n",
    "* if the firm chooses price level to maximize profit\n",
    "* does it matter what the firm choose? why or why not?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Q2**: (Drew Fallon challenge) Why isn't the firm's profit maximized when $p - AC$ is maximized?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "    \\max_{Q}\\pi &= pQ - c(Q) \\\\\n",
    "                &= pQ - Q* \\underbrace{\\frac{c(Q)}{Q}}_{AC(Q)} \\\\\n",
    "                &= Q \\big( p - AC(Q) \\big)\n",
    "\\end{align}\n",
    "Let's take the FOC w.r.t. Q\n",
    "\\begin{align}\n",
    "    \\frac{\\partial \\pi}{\\partial Q} &= (p-AC) + Q \\frac{\\partial \\big( p-AC(Q) \\big)}{\\partial Q}\n",
    "\\end{align}\n",
    "If the firm chooses $Q$ where $p-AC(Q)$ is maximized, then we would have \n",
    "$$\\frac{\\partial \\big( p-AC(Q) \\big)}{\\partial Q} = 0$$ \n",
    "which means $ \\frac{\\partial \\pi}{\\partial Q} > 0 $ as long as $p - AC > 0$, in other words, the firm is not maximizing profit at such quantity if the profit is positive."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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