consumer theory - 4. optimal consumption, algebraic solution (numerical).ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import necessary packages."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "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": [
    "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": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle q_{1}^{0.3} q_{2}^{0.7}$"
      ],
      "text/plain": [
       "  0.3   0.7\n",
       "q₁   ⋅q₂   "
      ]
     },
     "execution_count": 4,
     "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",
    "alpha = 0.3\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": 5,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\lambda \\left(- p_{1} q_{1} - p_{2} q_{2} + y\\right) + q_{1}^{0.3} q_{2}^{0.7}$"
      ],
      "text/plain": [
       "                           0.3   0.7\n",
       "λ⋅(-p₁⋅q₁ - p₂⋅q₂ + y) + q₁   ⋅q₂   "
      ]
     },
     "execution_count": 5,
     "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": 6,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left( \\frac{0.3 y}{p_{1}}, \\  \\frac{0.7 y}{p_{2}}, \\  \\frac{0.542881452689825 \\left(\\frac{y}{p_{2}}\\right)^{\\frac{7}{10}}}{p_{1} \\left(\\frac{y}{p_{1}}\\right)^{\\frac{7}{10}}}\\right)\\right]$"
      ],
      "text/plain": [
       "⎡⎛                                    7/10⎞⎤\n",
       "⎢⎜                                ⎛y ⎞    ⎟⎥\n",
       "⎢⎜              0.542881452689825⋅⎜──⎟    ⎟⎥\n",
       "⎢⎜0.3⋅y  0.7⋅y                    ⎝p₂⎠    ⎟⎥\n",
       "⎢⎜─────, ─────, ──────────────────────────⎟⎥\n",
       "⎢⎜  p₁     p₂                 7/10        ⎟⎥\n",
       "⎢⎜                        ⎛y ⎞            ⎟⎥\n",
       "⎢⎜                     p₁⋅⎜──⎟            ⎟⎥\n",
       "⎣⎝                        ⎝p₁⎠            ⎠⎦"
      ]
     },
     "execution_count": 6,
     "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": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{0.3 y}{p_{1}}, \\  \\frac{0.7 y}{p_{2}}, \\  \\frac{0.542881452689825 \\left(\\frac{y}{p_{2}}\\right)^{\\frac{7}{10}}}{p_{1} \\left(\\frac{y}{p_{1}}\\right)^{\\frac{7}{10}}}\\right)$"
      ],
      "text/plain": [
       "⎛                                    7/10⎞\n",
       "⎜                                ⎛y ⎞    ⎟\n",
       "⎜              0.542881452689825⋅⎜──⎟    ⎟\n",
       "⎜0.3⋅y  0.7⋅y                    ⎝p₂⎠    ⎟\n",
       "⎜─────, ─────, ──────────────────────────⎟\n",
       "⎜  p₁     p₂                 7/10        ⎟\n",
       "⎜                        ⎛y ⎞            ⎟\n",
       "⎜                     p₁⋅⎜──⎟            ⎟\n",
       "⎝                        ⎝p₁⎠            ⎠"
      ]
     },
     "execution_count": 7,
     "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": 13,
   "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": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q1star = foc_sol[0][0]\n",
    "q1star"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "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": 14,
     "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": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left(\\frac{\\alpha y}{p_{1}}\\right)^{\\alpha} \\left(- \\frac{y \\left(\\alpha - 1\\right)}{p_{2}}\\right)^{- \\alpha + 1}$$"
      ],
      "text/plain": [
       "     α              -α + 1\n",
       "⎛α⋅y⎞  ⎛-y⋅(α - 1) ⎞      \n",
       "⎜───⎟ ⋅⎜───────────⎟      \n",
       "⎝ p₁⎠  ⎝     p₂    ⎠      "
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = u.subs([(q1,q1star), (q2,q2star)])\n",
    "v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Try general function derivatives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "u = Function('u')(q1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{d}{d q_{1}} u{\\left(q_{1} \\right)}$"
      ],
      "text/plain": [
       " d        \n",
       "───(u(q₁))\n",
       "dq₁       "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u.diff(q1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "p, pb = symbols('p pb')\n",
    "q = Function('q')(p)\n",
    "pstar = Function('p')(pb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3.7.4 64-bit ('anaconda3': virtualenv)",
   "language": "python",
   "name": "python37464bitanaconda3virtualenv4442f6f54c04406894e1eb25c455fa91"
  },
  "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
}