consumer theory - 3. optimal consumption, graphical solution.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": [
    "q1, q2, p1, p2, y = symbols('q1 q2 p1 p2 y')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook, we derive a graphical solution to the consumer's utility maximization problem, i.e., finding the optimal consumption bundles and the highest utility level. \n",
    "\n",
    "We complete this in 3 steps:\n",
    "1. create indifference curves;\n",
    "2. create a budget constraint;\n",
    "3. find the optimal solution graphically."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3.1. Cobb-Douglas (CD) Utility Function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Define a utility function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define a CD utility function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sqrt{q_{1}} \\sqrt{q_{2}}$"
      ],
      "text/plain": [
       "  ____   ____\n",
       "╲╱ q₁ ⋅╲╱ q₂ "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define alpha as a rational number of 1/2\n",
    "alpha = Rational(1,2)\n",
    "# define parameter beta as 1 - alpha\n",
    "beta = 1 - alpha\n",
    "# Note: in sympy syntax: use double star ** for power\n",
    "u = q1**(alpha) * q2**(beta)\n",
    "# show the defined function\n",
    "u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Derive the expression for the indifference curve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{\\bar{u}^{2}}{q_{1}}\\right]$"
      ],
      "text/plain": [
       "⎡  2⎤\n",
       "⎢u̅ ⎥\n",
       "⎢───⎥\n",
       "⎣ q₁⎦"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define a new symbol ubar\n",
    "ubar = symbols('ubar')\n",
    "# explicitly solve for the indifference curve: q2 as a function of ubar and q1\n",
    "indiff_sol = solve(Eq(u, ubar), q2)\n",
    "# show the expression of the indifference curve\n",
    "indiff_sol"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note the square brackets around the above output, this indicates that indiff_sol is a list with one element (not just the element itself). This feature is handy because the solution may not be unique (e.g. solution to a quadratic equation).\n",
    "\n",
    "We can ask python about the type of the object using: type(indiff_sol)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "list"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(indiff_sol)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this case, we only want the element (the solution), we access it using the index.\n",
    "\n",
    "Python index starts with 0, so indiff[0] is the 1st element, indiff[1] is the 2nd, etc."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nice! Now we call this element (the solution we want): indiff"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\bar{u}^{2}}{q_{1}}$"
      ],
      "text/plain": [
       "  2\n",
       "u̅ \n",
       "───\n",
       " q₁"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "indiff = indiff_sol[0]\n",
    "indiff"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3.2. Budget Constraint"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Define the budget constraint"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, define an expression for expenditure."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "exp = p1*q1 + p2*q2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, the budget constraint is such that the expenditure is less than or equal to the budget."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle p_{2} q_{2} \\leq - p_{1} q_{1} + y$"
      ],
      "text/plain": [
       "p₂⋅q₂ ≤ -p₁⋅q₁ + y"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(exp <= y, q2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that the inequality solution is not just in terms of $q_2$. This is because the inequality solver in sympy has limited functionality for now. \n",
    "\n",
    "Our ideal form is\n",
    "\n",
    "\\begin{align}\n",
    "    q_2 \\leq \\frac{-p_1 q_1 + y}{p_2} \\label{eq1}\\tag{1}\n",
    "\\end{align}\n",
    "\n",
    "But this is no big deal, let's treat it as an equality, and keep the sign in mind."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{- p_{1} q_{1} + y}{p_{2}}$"
      ],
      "text/plain": [
       "-p₁⋅q₁ + y\n",
       "──────────\n",
       "    p₂    "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "budget_sol = solve(Eq(exp, y), q2)\n",
    "# Retreive the first and only element in the solution, which is our expression for the budget constraint.\n",
    "bc = budget_sol[0]\n",
    "bc"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The slope and intercept of the budget constraint is better seen after expanding the expression."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{p_{1} q_{1}}{p_{2}} + \\frac{y}{p_{2}}$"
      ],
      "text/plain": [
       "  p₁⋅q₁   y \n",
       "- ───── + ──\n",
       "    p₂    p₂"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bc.expand()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plot the budget constraint"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Let's plot a budget constraint with parameters $y=5, p_1 = 1, p_2 = 1$. \n",
    "\n",
    "Note the use of different brackets in the subs method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# we substitute a list [(y,5), (p1,1), (p2,1)] into bc\n",
    "# the list has three elements, each as a pair: (y,5), (p1,1), (p2,1)\n",
    "bc_plot1 = plot(bc.subs([(y,5), (p1,1), (p2,1)]), xlim=[0,10], ylim=[0,10], xlabel=\"$q_1$\", ylabel=\"$q_2$\", show=False)\n",
    "bc_plot1.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Creat a few budget constraints with different $y$, while holding prices constant."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "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": [
    "# create plots with different parameters\n",
    "bc_plot1 = plot(bc.subs([(y,5), (p1,1), (p2,1)]), xlim=[0,10], ylim=[0,10], xlabel=\"$q_1$\", ylabel=\"$q_2$\", show=False)\n",
    "bc_plot2 = plot(bc.subs([(y,6), (p1,1), (p2,1)]), xlim=[0,10], ylim=[0,10], xlabel=\"$q_1$\", ylabel=\"$q_2$\", show=False)\n",
    "bc_plot3 = plot(bc.subs([(y,7), (p1,1), (p2,1)]), xlim=[0,10], ylim=[0,10], xlabel=\"$q_1$\", ylabel=\"$q_2$\", show=False)\n",
    "\n",
    "# combine plots into one plot\n",
    "bc_plot_combo = bc_plot1\n",
    "bc_plot_combo.extend(bc_plot2)\n",
    "bc_plot_combo.extend(bc_plot3)\n",
    "\n",
    "# assign colors for each line\n",
    "bc_plot_combo[0].line_color='salmon'\n",
    "bc_plot_combo[1].line_color='tomato'\n",
    "bc_plot_combo[2].line_color='orchid'\n",
    "\n",
    "# show plot\n",
    "bc_plot_combo.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3.3. graphically solve for the optimal consumption bundle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Combine an *arbitrary* indifference curves and the budget contraint in the same graph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "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": [
    "# we substitute a list [(y,5), (p1,1), (p2,1)] into bc\n",
    "# the list has three elements, each as a pair: (y,5), (p1,1), (p2,1)\n",
    "bc_plot1 = plot(bc.subs([(y,5), (p1,1), (p2,1)]), xlim=[0,10], ylim=[0,10], xlabel=\"$q_1$\", ylabel=\"$q_2$\", show=False)\n",
    "\n",
    "# plot two indifference curves at arbitrary utility levels\n",
    "indiff_plot1 = plot(indiff.subs(ubar, 1), xlim=[0,10], ylim=[0,10], xlabel=\"$q_1$\", ylabel=\"$q_2$\", show=False)\n",
    "indiff_plot2 = plot(indiff.subs(ubar, 5), xlim=[0,10], ylim=[0,10], xlabel=\"$q_1$\", ylabel=\"$q_2$\", show=False)\n",
    "\n",
    "# let plot_combo[0] be the budget constraint\n",
    "plot_combo = bc_plot1\n",
    "# let plot_combo add on the indifference curves\n",
    "plot_combo.extend(indiff_plot1)\n",
    "plot_combo.extend(indiff_plot2)\n",
    "\n",
    "# assign colors for each line\n",
    "plot_combo[1].line_color='salmon'\n",
    "plot_combo[2].line_color='navy'\n",
    "\n",
    "plot_combo.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Adjust utility levels to find the optimal solution (make your edits here)"
   ]
  },
  {
   "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"
    }
   ],
   "source": [
    "# we substitute a list [(y,5), (p1,1), (p2,1)] into bc\n",
    "# the list has three elements, each as a pair: (y,5), (p1,1), (p2,1)\n",
    "bc_plot1 = plot(bc.subs([(y,5), (p1,1), (p2,1)]), xlim=[0,10], ylim=[0,10], xlabel=\"$q_1$\", ylabel=\"$q_2$\", show=False)\n",
    "\n",
    "# plot two indifference curves at arbitrary utility levels\n",
    "indiff_plot1 = plot(indiff.subs(ubar, 1), xlim=[0,10], ylim=[0,10], xlabel=\"$q_1$\", ylabel=\"$q_2$\", show=False)\n",
    "indiff_plot2 = plot(indiff.subs(ubar, 5), xlim=[0,10], ylim=[0,10], xlabel=\"$q_1$\", ylabel=\"$q_2$\", show=False)\n",
    "\n",
    "# let plot_combo[0] be the budget constraint\n",
    "plot_combo = bc_plot1\n",
    "# let plot_combo add on the indifference curves\n",
    "plot_combo.extend(indiff_plot1)\n",
    "plot_combo.extend(indiff_plot2)\n",
    "\n",
    "# assign colors for each line\n",
    "plot_combo[1].line_color='salmon'\n",
    "plot_combo[2].line_color='navy'\n",
    "\n",
    "plot_combo.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}