consumer theory - 6. optimal consumption, quasi-linear.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": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "q1, q2, p1, p2, y = symbols('q1 q2 p1 p2 y', nonnegative=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 6. Quasi-linear utility function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For a general quasi-utility funciton $$ U(q_1,q_2) = u(q_1) + q_2, $$ we calculate its MRS, which is $$ MRS = \\frac{U_1}{U_2} = u'(q_1) $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u{\\left(q_{1} \\right)}$"
      ],
      "text/plain": [
       "u(q₁)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "### In sympy we can define a function without specifying its functional form\n",
    "u = Function('u')(q1)\n",
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": true
   },
   "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": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# we can take derivative with respect to this defined function\n",
    "u.diff(q1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle q_{2} + u{\\left(q_{1} \\right)}$"
      ],
      "text/plain": [
       "q₂ + u(q₁)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# we can define a quasi-linear utility function using this u function \n",
    "#  (notice that we don't use u(q1) here to ask sympy to calculate any thing, otherwise there will be an error)\n",
    "U = u + q2\n",
    "U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "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": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# we can then calculate the MRS of this quasi-linear utility funciton\n",
    "U.diff(q1) / U.diff(q2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the optimization program (general quasi-linear utility)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our optimization problem is\n",
    "\\begin{align}\n",
    "    & \\max_{q_1,q_2}\\quad U(q_1,q_2) = u(q_1) + q_2 \\\\\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": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle q_{2} + u{\\left(q_{1} \\right)}$"
      ],
      "text/plain": [
       "q₂ + u(q₁)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define the non-linear part of the function\n",
    "u = Function('u')(q1)\n",
    "\n",
    "# define the quasi-linear utility function U\n",
    "U = u + q2\n",
    "# show the defined function\n",
    "U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the lagrangian"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\lambda \\left(- p_{1} q_{1} - p_{2} q_{2} + y\\right) + q_{2} + u{\\left(q_{1} \\right)}$"
      ],
      "text/plain": [
       "λ⋅(-p₁⋅q₁ - p₂⋅q₂ + y) + q₂ + u(q₁)"
      ]
     },
     "execution_count": 8,
     "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": 9,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAVCAYAAAB7R6/OAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAARUlEQVQoFWNgYGAoA+K7UGyEhc/QCRR0+f//PwMyBoqBxZmADLxgVAEkeEbDgchwYIEmp5mMjIwgZigQuwBxOhALAfFuAP68FOR8/BSoAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\right]$"
      ],
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# sympy may not be able to find a solution for us\n",
    "foc_sol = solve( ( diff(lagr,q1), diff(lagr,q2), diff(lagr,lam) ), q1,q2,lam)\n",
    "foc_sol"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the optimization program (a specific u function form)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our optimization problem is\n",
    "\\begin{align}\n",
    "    & \\max_{q_1,q_2}\\quad U(q_1,q_2) = u(q_1) + q_2 = ln(q_1) + q_2 \\\\\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": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle q_{2} + \\log{\\left(q_{1} \\right)}$"
      ],
      "text/plain": [
       "q₂ + log(q₁)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define the non-linear part of the function\n",
    "u = ln(q1)\n",
    "\n",
    "# define the quasi-linear utility function U\n",
    "U = u + q2\n",
    "# show the defined function\n",
    "U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the lagrangian"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\lambda \\left(- p_{1} q_{1} - p_{2} q_{2} + y\\right) + q_{2} + \\log{\\left(q_{1} \\right)}$"
      ],
      "text/plain": [
       "λ⋅(-p₁⋅q₁ - p₂⋅q₂ + y) + q₂ + log(q₁)"
      ]
     },
     "execution_count": 11,
     "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": 12,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left( \\frac{p_{2}}{p_{1}}, \\  \\frac{- p_{2} + y}{p_{2}}, \\  \\frac{1}{p_{2}}\\right)\\right]$"
      ],
      "text/plain": [
       "⎡⎛p₂  -p₂ + y  1 ⎞⎤\n",
       "⎢⎜──, ───────, ──⎟⎥\n",
       "⎣⎝p₁     p₂    p₂⎠⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# sympy may not be able to find a solution for us\n",
    "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": 13,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{p_{2}}{p_{1}}, \\  \\frac{- p_{2} + y}{p_{2}}, \\  \\frac{1}{p_{2}}\\right)$"
      ],
      "text/plain": [
       "⎛p₂  -p₂ + y  1 ⎞\n",
       "⎜──, ───────, ──⎟\n",
       "⎝p₁     p₂    p₂⎠"
      ]
     },
     "execution_count": 13,
     "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": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{p_{2}}{p_{1}}$"
      ],
      "text/plain": [
       "p₂\n",
       "──\n",
       "p₁"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q1star = foc_sol[0][0]\n",
    "q1star"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{- p_{2} + y}{p_{2}}$"
      ],
      "text/plain": [
       "-p₂ + y\n",
       "───────\n",
       "   p₂  "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q2star = foc_sol[0][1]\n",
    "q2star"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " $q_1^*$ is clearly positive. We can discuss whether $q_2^*$ is positive."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "# further declare that the exogenous variables are positive\n",
    "p1, p2, y = symbols('p1 p2 y', positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{- p_{2} + y}{p_{2}}$"
      ],
      "text/plain": [
       "-p₂ + y\n",
       "───────\n",
       "   p₂  "
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q2star"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle p_{2} < y$"
      ],
      "text/plain": [
       "p₂ < y"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# find condition that q2* > 0 (interior solution)\n",
    "solve( (y-p2)/p2 > 0, p2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACUAAAAzCAYAAAAQEcGFAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACyklEQVRYCe2ZjU0bQRCFfVEKQJTgdJAoFWA6SNJBoAMiV4CgA0MFJOkg6SCKO4AOQHTgvG+1Y53HiB3vnhGRbqX13s7t7Xv3Zm7/3E0mkwPlrbRarR63jAMbuq57EhuYW+WVywuRmuw7C/OXw4UHtkRqtm8Ckf7F5QRSb2D12tJIKuqRUam9K8UYo3ymvKDsA+Z7P/q2Xa5b3DfXZ34pMMaVuQP9ovonZwtXq0hJiakQ/mSUY5V3DhHbb2cLV6tIqfcHqfQzo6DKjUOcqZ5GZmcPVatIiVCaF6UY4MxfV4Ym2/tse3GljMNnXSyNZDZC9FG2pTXataxSqgdCbA0aT/TdSmqDUHZnUzxB6i0/Dembnr0WmQuV98rvcl/V8cTzTaRyLBFXKWVyxNiGgnY/Wla7TwQYyf8akK75ClkPnZuttqwmJUBipz8+Xat+JZVs/Krl1OQ+4mkqhZj3iCWW0E2xZG9RHVNDKGIkfNniPt/XYPWRVFTKYkwpkNmLDZYUi12psyKpSCclkF3vjzEVVWxUalQqqkC03RhT/7VSxRGdt9NUM1PBrpc93akyq0xspI/KXzXyp71gsjT+RGPqWKAs6gBm5ztV/ZKsa9bj1YcZenYrFUlJpamesnMDrtkY9Je8HOSaaqjKacyJ8nr9voVaMETc1z83wH1+Y5C2VZDRPUgbQepVqUhKqqRYEaiB+XU4RNmm046t+lJtq4+BeIui+2iUUzryMZLYsjqQ/Z7bDFLsQgpwrxL7PBTiIxgsFd0HUlYENz0YsmzUOcE76qtn91vKECkBoBLpQmTsfJMA/yBCTVv01Kv7iZKyeMJ93oWuy/ZqNKZQau9k7HWKSuV4YvwpnmGqLe2YhngJtvQLlbdyMSN/OD1LSp0y3thx9Fz1QwGszzc9So6v5i/RSPFWGyOwABgQmU76U4rn0VT3mOpszeHV/Qn5D8EbwbI9irhGAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{y}{p_{1}}\\right]$"
      ],
      "text/plain": [
       "⎡y ⎤\n",
       "⎢──⎥\n",
       "⎣p₁⎦"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# find corner solution, i.e., what is q1* given q2* = 0 (when the condition above fails)\n",
    "q1star_corner_sol = solve((y - p1*q1 - p2*q2).subs(q2,0),q1)\n",
    "q1star_corner_sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{y}{p_{1}}$"
      ],
      "text/plain": [
       "y \n",
       "──\n",
       "p₁"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# retrieve the first element of the list\n",
    "q1star_corner = q1star_corner_sol[0]\n",
    "q1star_corner"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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
}