consumer theory - 4. optimal consumption, algebraic solution (CD).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": [
    "# 4. Solve for optimal consumption analytically"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the optimization program"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our optimization problem is\n",
    "\\begin{align}\n",
    "    & \\max_{q_1,q_2} \\quad u(q_1,q_2) = q_1^{\\alpha} q_2^{1-\\alpha} \\\\\n",
    "    \\text{s.t.}\\quad & p_1 q_1 + p_2 q_2 = y\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the utility function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle q_{1}^{\\alpha} q_{2}^{1 - \\alpha}$"
      ],
      "text/plain": [
       "  α   1 - α\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",
    "alpha = symbols('alpha')\n",
    "# define parameter beta as 1 - alpha\n",
    "beta = 1 - alpha\n",
    "\n",
    "\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": [
    "## Setup the lagrangian"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\lambda \\left(- p_{1} q_{1} - p_{2} q_{2} + y\\right) + q_{1}^{\\alpha} q_{2}^{1 - \\alpha}$"
      ],
      "text/plain": [
       "                           α   1 - α\n",
       "λ⋅(-p₁⋅q₁ - p₂⋅q₂ + y) + q₁ ⋅q₂     "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lam = symbols('lambda')\n",
    "lagr = u + lam *(y - p1*q1 - p2*q2)\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": [
       "$\\displaystyle \\left[ \\left( \\frac{\\alpha y}{p_{1}}, \\  - \\frac{y \\left(\\alpha - 1\\right)}{p_{2}}, \\  \\frac{\\left(\\frac{\\alpha y}{p_{1}}\\right)^{\\alpha} \\left(- \\frac{y \\left(\\alpha - 1\\right)}{p_{2}}\\right)^{1 - \\alpha}}{y}\\right)\\right]$"
      ],
      "text/plain": [
       "⎡⎛                       α              1 - α⎞⎤\n",
       "⎢⎜                  ⎛α⋅y⎞  ⎛-y⋅(α - 1) ⎞     ⎟⎥\n",
       "⎢⎜                  ⎜───⎟ ⋅⎜───────────⎟     ⎟⎥\n",
       "⎢⎜α⋅y  -y⋅(α - 1)   ⎝ p₁⎠  ⎝     p₂    ⎠     ⎟⎥\n",
       "⎢⎜───, ───────────, ─────────────────────────⎟⎥\n",
       "⎣⎝ p₁       p₂                  y            ⎠⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "foc_sol = solve( ( diff(lagr,q1), diff(lagr,q2), diff(lagr,lam) ), q1,q2,lam)\n",
    "foc_sol"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The solution is a list with one element, because our solution is unique."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{\\alpha y}{p_{1}}, \\  - \\frac{y \\left(\\alpha - 1\\right)}{p_{2}}, \\  \\frac{\\left(\\frac{\\alpha y}{p_{1}}\\right)^{\\alpha} \\left(- \\frac{y \\left(\\alpha - 1\\right)}{p_{2}}\\right)^{1 - \\alpha}}{y}\\right)$"
      ],
      "text/plain": [
       "⎛                       α              1 - α⎞\n",
       "⎜                  ⎛α⋅y⎞  ⎛-y⋅(α - 1) ⎞     ⎟\n",
       "⎜                  ⎜───⎟ ⋅⎜───────────⎟     ⎟\n",
       "⎜α⋅y  -y⋅(α - 1)   ⎝ p₁⎠  ⎝     p₂    ⎠     ⎟\n",
       "⎜───, ───────────, ─────────────────────────⎟\n",
       "⎝ p₁       p₂                  y            ⎠"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "foc_sol[0]"
   ]
  },
  {
   "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": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\alpha y}{p_{1}}$"
      ],
      "text/plain": [
       "α⋅y\n",
       "───\n",
       " p₁"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q1star = foc_sol[0][0]\n",
    "q1star"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{y \\left(\\alpha - 1\\right)}{p_{2}}$"
      ],
      "text/plain": [
       "-y⋅(α - 1) \n",
       "───────────\n",
       "     p₂    "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q2star = foc_sol[0][1]\n",
    "q2star"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Substitute optimal consumption into the direct utility function, to calculate the optimal utility level attainable at the given exogenous variables. \n",
    "\n",
    "The resulting function is our **indirect utility function**, $v(p_1,p_2,y)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(\\frac{\\alpha y}{p_{1}}\\right)^{\\alpha} \\left(- \\frac{y \\left(\\alpha - 1\\right)}{p_{2}}\\right)^{1 - \\alpha}$"
      ],
      "text/plain": [
       "     α              1 - α\n",
       "⎛α⋅y⎞  ⎛-y⋅(α - 1) ⎞     \n",
       "⎜───⎟ ⋅⎜───────────⎟     \n",
       "⎝ p₁⎠  ⎝     p₂    ⎠     "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = u.subs([(q1,q1star), (q2,q2star)])\n",
    "v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question: plotting demand curves (assuming $\\alpha = 1/2$)\n",
    "Given demand function $$ q_1^* = \\frac{\\alpha y}{p_1} $$\n",
    "1. try to plot a demand curve for $d_1^*$, make necessary assumptions about the values of the exogenous variables as you need.\n",
    "1. try to compare two demand curves as $y$ changes. \n",
    "1. try to compare two demand curves as $p_2$ chnages."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question: plotting Engel curves (assuming  $\\alpha=1/2$ )\n",
    "1. try to plot an Engel curve for $d_1^*$, make necessary assumptions about the values of the exogenous variables as you need.\n",
    "1. try to compare two Engel curves as $p_1$ changes. \n",
    "1. try to compare two Engel curves as $p_2$ chnages."
   ]
  },
  {
   "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",
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 },
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}