firm theory - 1.6 cost minimization to profit maximization.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import necessary packages."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "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": [
    "### NOTE: it is important here to further declare these variables to be nonzero and positive (otherwise sympy cannot find solution for the FOCs)\n",
    "#    Thanks to Abhi Agarwal for discovering this trick\n",
    "L, K, w, r, qbar = symbols('L K w r \\overline{q}', nonzero=True, positive=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1. Solve for cost-minimizing input bundle analytically"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the optimization program"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our optimization problem is\n",
    "\\begin{align}\n",
    "    & \\min_{L,K} \\quad w L + r K \\\\\n",
    "    \\text{s.t.}\\quad & L^{\\alpha} K^{\\beta} = \\overline{q}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the production function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle K^{\\beta} L^{\\alpha}$"
      ],
      "text/plain": [
       " β  α\n",
       "K ⋅L "
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "### Note that it is important here to further declare these variables to be nonzero and positive\n",
    "#    Thanks to Abhi Agarwal for discovering this trick\n",
    "alpha, beta = symbols('alpha beta',nonzero=True, positive=True)\n",
    "\n",
    "# Note: in sympy syntax: use double star ** for power\n",
    "y = L**(alpha) * K**(beta)\n",
    "# show the defined function\n",
    "y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the lagrangian"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle K r + L w + \\lambda \\left(- K^{\\beta} L^{\\alpha} + \\overline{q}\\right)$"
      ],
      "text/plain": [
       "              ⎛   β  α               ⎞\n",
       "K⋅r + L⋅w + λ⋅⎝- K ⋅L  + \\overline{q}⎠"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lam = symbols('lambda')\n",
    "lagr = w*L + r*K + lam *(qbar - y)\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": [
       "$\\displaystyle \\left[ \\left( \\frac{\\overline{q} \\alpha \\left(\\frac{\\overline{q}^{\\frac{- \\alpha - \\beta + 1}{\\beta}} \\alpha^{- \\frac{\\alpha}{\\beta}} r w^{\\frac{\\alpha}{\\beta}}}{\\beta}\\right)^{\\frac{\\beta}{\\alpha + \\beta}}}{w}, \\  \\overline{q}^{\\frac{1}{\\beta}} \\left(\\frac{w \\left(\\alpha^{\\frac{\\alpha}{\\beta}} \\beta\\right)^{\\frac{\\beta}{\\alpha + \\beta}} \\left(\\overline{q}^{\\frac{- \\alpha - \\beta + 1}{\\beta}} r w^{\\frac{\\alpha}{\\beta}}\\right)^{- \\frac{\\beta}{\\alpha + \\beta}}}{\\overline{q} \\alpha}\\right)^{\\frac{\\alpha}{\\beta}}, \\  \\left(\\frac{\\overline{q}^{\\frac{- \\alpha - \\beta + 1}{\\beta}} \\alpha^{- \\frac{\\alpha}{\\beta}} r w^{\\frac{\\alpha}{\\beta}}}{\\beta}\\right)^{\\frac{\\beta}{\\alpha + \\beta}}\\right)\\right]$"
      ],
      "text/plain": [
       "⎡⎛                                                                            \n",
       "⎢⎜                                                   β                        \n",
       "⎢⎜                                                 ─────                      \n",
       "⎢⎜                                                 α + β                     ⎛\n",
       "⎢⎜               ⎛            -α - β + 1  -α     α⎞                          ⎜\n",
       "⎢⎜               ⎜            ──────────  ───    ─⎟                          ⎜\n",
       "⎢⎜               ⎜                β        β     β⎟                          ⎜\n",
       "⎢⎜               ⎜\\overline{q}          ⋅α   ⋅r⋅w ⎟                          ⎜\n",
       "⎢⎜\\overline{q}⋅α⋅⎜────────────────────────────────⎟                          ⎜\n",
       "⎢⎜               ⎝               β                ⎠       beta______________ ⎜\n",
       "⎢⎜──────────────────────────────────────────────────────,   ╲╱ \\overline{q} ⋅⎜\n",
       "⎣⎝                          w                                                ⎝\n",
       "\n",
       "                                                 α                            \n",
       "                                                 ─                            \n",
       "                                                 β                            \n",
       "          β                                 -β  ⎞                             \n",
       "        ─────                              ─────⎟                             \n",
       "        α + β                              α + β⎟                             \n",
       "  ⎛ α  ⎞      ⎛            -α - β + 1    α⎞     ⎟   ⎛            -α - β + 1  -\n",
       "  ⎜ ─  ⎟      ⎜            ──────────    ─⎟     ⎟   ⎜            ──────────  ─\n",
       "  ⎜ β  ⎟      ⎜                β         β⎟     ⎟   ⎜                β        \n",
       "w⋅⎝α ⋅β⎠     ⋅⎝\\overline{q}          ⋅r⋅w ⎠     ⎟   ⎜\\overline{q}          ⋅α \n",
       "────────────────────────────────────────────────⎟ , ⎜─────────────────────────\n",
       "                 \\overline{q}⋅α                 ⎠   ⎝               β         \n",
       "\n",
       "             ⎞⎤\n",
       "             ⎟⎥\n",
       "             ⎟⎥\n",
       "          β  ⎟⎥\n",
       "        ─────⎟⎥\n",
       "        α + β⎟⎥\n",
       "α     α⎞     ⎟⎥\n",
       "──    ─⎟     ⎟⎥\n",
       "β     β⎟     ⎟⎥\n",
       "  ⋅r⋅w ⎟     ⎟⎥\n",
       "───────⎟     ⎟⎥\n",
       "       ⎠     ⎠⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "foc_sol = solve( ( diff(lagr,L), diff(lagr,K), diff(lagr,lam) ), [L,K,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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vftwf879oGXUxkssOXZt2Adc1jmp0AEQHXVfyzvuihy7d1FKSTmZnPMj4Tan7a9TjqZKtXf9pa9ksaeo036lyOp0RMALXR+DMLpEuWPr747RPOMJpRusDnbydcXiV/wf/idnBaynUMJac1kXDQ0jG/91p/6/4DQWfHJ6cojnC1k+obIwtrtUAp8r5y+JSP7j/LF/P/ZsiNwZojIR15U/lrhul8FwfMf1HR8Ip/HSQOrkdecNX5VyfxPDF0gjQ4UJIo+13p9f9lS7CeaROLDJ6mukXd3C6UMPxUZ6MpPLu1TsAu5LQaVPagQbPYlOSSpeR6NFuda7n1LPvhWPoiC91Xe8k5D7TqJPDqn/wG0OPJK/5qQdoZKc13dY1o3TozAg4KHV9Wucx4mzqX/xSrybV/Sn1eKrwmc8pTnRXlkXrvzJh5hFt7tRpvl18+r4RMAIbRkC6ife+8gl1nXyueluxEPuRZ8oPBzWd6P+s8epjQMyWdDr7sqs/D57rjmb9eeu5+I1Na9JzgY5CpyAuGQ2ttAreHI17wby7SOGcgnHCVzhhsNBI3csY4szN5XtU+vyCkG8YcLDTFqCbKrnSYKgknvnXNaPnXU7KKH6gNzBEHb3mGueBrB42WjhM9ane1xYWR+CrAu9Rlxy9Dk5Xwhn3kYlR1BOnp41efjcnORFt9OKe8OR9KzVCWLSMMm+Vgxk8X/g/00l1/TeGnvAeZHCI5klbqHQvdG8MzxfEWe3R3Lo/uB5PlWgn9T/aWuqOgxEwAreDAPb/NWZ2Rpv3EKhxUNMJzHBjJ+GtzGcY/2PYZgSOjXRY+JsMGTXAOGRXCcp7izt3BkapkgoYDE4we6iDaWbg3uXoDcJxgtzwMiRPDCymMlK+1GnOk8Ghc6Z4ttaZCfyIXH8QXTpFiBh1tj+RY5RGIBlYKoswuErTH0VP9YGOMepzqdkIXfmHsq+eL4mBaDOKCsY4NCeOTsXAnye9TkR+b6N9ImXfaN6ZvKLR6yz/ydKfZyuW0Z+Znp4xeoru+kn/6A701o+6ppPtO/SK/seE4gZHxoh3C/6e6/oH8TVER4/he2rcM3yG1v2R9Xgqf5uu/wglHNgAjVP0lYMRMAK3g8BSMwqHIkibkhzUdDI01UbihdFy1gj18ZcbqWSo5sb+as5pH6/Xek4DrbzPes11/+zeWjzWy+1SnkPjXaKx0LOoswuRN9k2BPSOgztHa8dEW5oV7mGcLv25m9IOzlBYmIZKh1Cfgwq9M/2d31+eYSCf0FBZdo7m6RmOUtXJqmscMmhMLfc1yggWW4P4pl1qtk1zOliKGxw1HieviW0Vfv7NEnV/TD2eyvGW63/IRB18n/dL5T2n/gU9/6+AgMqLNo+lLejE1Lk/QxeuwLGz2BgCZ7ppJf5iCR71Nzmoke8shrJBgEERdBgdoLe3uWMuUy0xYAgww9RHDI9mY8zzrjB53WyDIFP/rHQboPiyOAK8E3vsCCoOxBUIptFT5TvVUVmCZaaeL9rZU3Me6vyvoevIg5kWj8TDpdG0Ek5EXbbSI4SLl1Gd+RXOo10em9XUdGPzKRa/UN0fWo+n8r31+h9y/awTRlA51tAfke/q/9JZL5Tpb6o/W+twmYIFu5mnDj7JRRuIPuuUC5s84rdlpufoV/DZ7feO2+Ta+r2MO0uT2ORwsi4WncH7BQUmSkN5Tw7id8oS0BMZHyj3h5M5yAklCC8APY6AmAxB3eMaI6ViUveo5IDNx6uTstM98v83/ae0OQ67xl0ybkrwjJMM3z8rT+9SJyAcFkOAXszZdXYx7o5NmE4owpgOsLsUC/yiazLZxYw95TGoMarr5i5Rsz7GmY7ORKI+VdrXbWl0n2m+tAHg3qnDFY/yaJbJHEyKjRAuUUZDy6QN01L3xMMkg0PpYvfgQawMqVeDCE2INBTnPh6H1uMJLKYkW67/DZlYKkP46O7vmL+qNwym8D3qRTsO10BPsmDXflvLiyVbfe9+p30ierRZb+nAfndYEYGshyg7/KmwZUZzIDqVHzY6cU8C1Q/qTunReurbyQhqDK32sHP6ODMHeM8A8/TpmQGCsuOzCpUhonPW6NFLR28PyoGenr51mYl50irupKC05Jd6mCYRcCIjMA6BVGfHJXHsAgh8mGn8VIBWCRKJH+mfpq4sQTvREO0ijZH0MgYJU/sqPal7GHLf6MDo6Qro98C9K86U+5f0/aVnY/MqXkalymSsILcWvzDOS9XjqcVyqY5fejY1P9JhmxFweg4ZpNPel2AMWKQ9SdYWUvk/Up7oWsqwxOfAkKfYbuZ6p1JHovgUWYe1ERD+Xwn7T3RcHOVegq+BeY4are/h81V+/pD/BzrCcJ6q4Hix6b1uTh9AoVWOpgSlF4BMn+loBoy1MGjeFq2pvDTpnlyLh0EjCyeJfGEEWhBQHbW2bsFlg7ei13cxh3CkzPSCNkcOR5IYHl06D507unczp2MGTFOv04BgAF0KfP4LR7ZoEC+Vo9wkrGdVW9N8NuG6t4zGtCXX0BVTy30CVptNUgCDRerxVMBWrP91FkNvokcOF3Id+bsEY1ZIyLqanMof53Tru5mvhseUjDKGJR38KWxcTFOARzqEmWn6k+pp58yki0xMe3jxvRc/U0brB3OCg3qRgQGUMH6qEVHii2kqC3Trn3VI8QTua+I0AvfeVTqc3TYHthH9bjMMxccxbqPXjJ+uFddORSsyvrkCAtEztEJWzqKGAHqIsLrxcZft2S8K/URfnsUoe2Nq7+bn0pdN5xTOHuvoc7AT1tLPe1060VtGO2hLppZ72dp3XWpzMdh7PZ6NPvaV3uNER/+TdsSezcSyBLA52d2/WAcXOIlmZ2eantW/Jfylrj/TEYG2gRmJ1VTjkfSgc2YTN8tO12lgicg59O2QHvE29S85FnHwSwpZgsf8HlJPvhG9yRssZl5KOvNFR+sl30ndxUGN8B/iZOi/hMX446g7oiSngWcHxWSI1eJdMmygQ+/ACYPQaQkRh9HfOG+J5ltGYBMIUE+34iBtApA1mJDeQRGnMFCvRPQl/9Fzq9QFyY8e/rYmzJC1SLXop6eZHg1S31S4kA/8L+n80wy2c7VaGS0hculyX4LHpWkWwmDv9bgUzODAu8wRmJSifTU6qiPoMmZ69OmzUTyqrQGjysEckPjMhg3bmbRj6OV6X3WG6zq1gZlGxYquTxxoxXuhe2N4rmhd+aTXwb8yf2RfhEeVD9O26VjgGF1WSruIM5/rVnqHlEfTF0T+MeH/RWTRelh3UP9vPJjw31RaGEKVYSIBoheOKTNd4TUF0PWwcT/Wy2JIOBRCQBUCZZY6FzLJtqnbhXK7OTLNd+TmALiCwBgghEnYoyCVFgOGncZ/kX6qetn1DNps9pYaCl2j/Gk4WEN0ZnDofj1c0oP1eOlctKFLQO+N2fUcHkevRVJ+6AC+uYk+YGkG8iRDR/9D5Au831L8vYbBZZRxwuCLcn+mc+rN2Q72K4ExqdyX4k34wA8j74yos5aKuhR4sWtqeq90v+T7VgKDI9TjEsUa9Tp0QAmaW6DBWvqXqn+VrXoFphgV8/eO5wEf9bOiojJdc5ZSle+Fk1I8oje/l65kPWropwvZnjzqdZShqxTo4Qh9I+tncqHfJ/AW+Z38z3JQxQSOJxWhUly6pmHGwGlOD6MRYm1PZeTpnOnAxCckUHQ9ZFrYGSh3JPw7FQFwV1p2savKR/f+qYNNrYz3VGDv0lG3q17NeaScegQCoZem1l8aAYwHnE+Mmerd0DnKnnKNQB7k96GO1sZRdIKfQXUhx6dHcvKu5+KfRmxs7yZromgI0c9MNyOwWQrnyHDRoENfKJ2ineDD9ebDhDKirQMXHPdkMGTMTnawX1vwieW+FJtpoy3h8oOO1OEh/nivMIY44r0q+r7NxUDpd1uPCxck9Rr7gM6xQwTVPXQ6Ml3a7G1xWVXH0KVNfdrafgxkhkGFeJ8GJrkcrcZj06a/nHCdp6Ud/CW4LsajyoIOZ+oLehR/amw4s4VEs6pvOk/tfhBFX+te62itntH2varFTfaN4o91nIME//9ev6g7qPX7Y855wZkX/SQnCuZOhnoRUnGecygeIwEJKN1PL5PuP6w9a76wmXT1F6C8Vd3xyVwEHqssqsqpsqDy/ax7ZxV6bka3lJ56neU1jusX/Ns5y9AXgzlQuaFs2XWcQEMQei3d0A/vByNlKeg9YcQRvdWMl2Okv6gL9XuXzsl/zq7nZ3UOucTrJR5P+FHcythRWnQ3jvrQKXGB/wnNjV8MLiPhQVyc0yE72K8p9uxyL8Vsxij44Z2i/oWjA37R/pd+3yLPSpSxdb9KeO/eHutxjf3Zp6E/kwE6m9o2CKDLmOEwWBdug+1eLs7qfW+KuwhT0w0kv0w0lR9tLkc9VA5X/ea1zhfgEbuDUdQhg3l1sYs5ypko+uCV+GCQER3BddEOn9kOqsCnYldMZWar9ad6VgXFrRyg6mY+0bNWL70ZL1+HUjmSwuwQdfnbNNzK5Sf908lAY0zvIlOuhxqiiu7QgUDUUTplHNZFILCf0vi+Uv0P54wN3ipnVO8JPe8PdTQbQjp0Qje1SRodar38KA96SMmjyrdGkDwYqSW07nqu9DjQYVjGxnWxniklbPtROvLsqqvki+xDA7T2FgaXkQTDyG1bBgH2UXdWlX9quS/IJHh+l+nzPlbtfKPNL/a+LYDBHutxySJ9nYkdAgfVD3QYdbHTHp0Knmhf/UsR4oG2Y3RQuhhkGpRW72+RTUdzeaBL+wIziar2dSjWpfjsY67t+RI8Sh5GUXkn0aWVPm3Lv35P6Uo788VH6+v8cj7bQW0S1DWjDc0ejZZo028BtAoIAnagpsNYT4lBhRGBMcE5lZ7R7lGjLUrjcI5AOEmVYj2P4jsLIRAGVeWoDc1HOiYZZXoHeB+gU3c4uEcHTlPPpTQX8gg+gq8LUdOnYbp26yOfvl3PqXdTejc7pyiLHoZdn4yKsuswuozq0qIzdU35nswgqsdZ+HxquS/Clt6RpPfye0Qe4aye5Ff4fdsUBieC7vsC++AI4UuEUJ3r3fNE9Xawwyl69znWBEj8oWvoQEVvUe9ZU7srW0P80o6Onq7axHqLWNR5LMwfepSRy8EOquJOCZfa+0vPpuR1lubB2Z35NzDe6sbcfIrtFHgJYxShPYbvDkXgPb1IVDaOVHZ6mb7VOZW/eC+jaN5SoNEgoIQd1kWghEHF7JBf8/sR3NOYsiazCnpfcN6KOCW5IXsoepfqDM8v7Xo+tXezWn9aCffnCXI3R43/fJrPMv9cDd5o6IzIxm/UyqhZ5rR/rTOIVhJparkvzR7vEZ/zoI25FEq8b0UwuIV6fKkgas9iRgU6Z9chlykzxAbZqHUHY6OCz/2U0kbFmsTW1rEoyR9LSz5XfeZzcIPq8hRERbvT/l8y3+D1jTiZ+y+gGHFjzRRK7BHXc2n2pMd4C+O/J6ofT0CAqb67b5AmyF06yUcQ1Mu8q17N0iBcmV6fUXyJPXRMs+y413QeWcPd57wFHxcdZ9GJeJccPEZwL40ABI1LsrU9Sw5W84H0OQ44HYKdDVYtTcg3lYcaqdVPg+eQoY+BZt1IDlZfogWfB/8LZjGJNPXqUn0NoiXet1IYRB0oRS9k3Nv/keRntJEwaRrsXdJt/Eon804xkBAB3dPXBkXcQ/1vHYvS/GVbg/dy6RHUq9aTN0rlLsCe6mAnQ6Y5MCI3xJCZkz0912ys9HAOkVtPm/GLHtI6HCg/emkc5iEAjjfZaMyDrUjqErrhpCMsNzQYrhjSKeR7zZG0eFz/f5UvhvBFr+jZtCfldbbreT2DOJf+Hd2rmnXB77U8ErksH7rgU9FtOmSRZf0/5At568+2fh48hwyt/AqH13rAe12vB5TNVd/3KeXeKmDBm6o/YMQx5B2Z/b4VxCDqQNSJgqjsklQ47LtkPjONE0eHMfVsVKAe63ihI232qX/szyc63h9FqFxk8mWn6V84OJ8iVzl2/qQkfiHQ2G4AABvBSURBVAIj8AGzpTEajAW8ZP6SPtJ5vVyjfWXfhihf+K/uI6WuoVF1cuicQbnfdITO+BOMu7PB/DUTXrhmmi98VG3Qhbi7fPRgl1zfMR1T7DAIhvTM7ljURVlnVCR2VUwZqcKzWJ4NX+xYzYA+Kw4U1hDDbEZOTtqDQFsHTE+Su8d6B56qHGmgaIxwSqD1jo7YuZyZBtW3HHXeGUQLYwIaJ+9bWwLFLbnreVsWbffQpc900GDHhhnI9w/x08tzjeBb+RxZdxXGlJEEw9iNeoCc4bz7fQeNPwMGFNN7e9sTxSn2vv2Z/eSz3dbjyRK3JwwHnbZs72FSB5L0IenopPPnpC7UAOEUHVGTP412gfzFR9Id6N+0L434uKSDd/nZqxbhkTE6RUd3SLfQ29ytB5vjaCBDqoxslIQB9FiHHdSBuLVEo2en3kNEI4Qx8VVLXN8ahwCNGsH18w6H3t/cwOEMMlo318EpYmCKj7ZpNKknvleg8wg0ooN6PJVv5yyUDp7Ocxt35yPRpa6OHl1oZBPy/dy4v5fLQWUkrKifVT1Q3U297Lrf64jtBYgSfGY8BndwKH7J922OCHuvx3NkP1xavZ/RHo/Sb0qHTYRz6s9J9dcKRnLnfBqtP4f2GGe2AraEdEl0GqZUuSwjLu/3Hj97FQhEPWamlR3UQGVD/wxxx5qCDbG1L1b0ElPRo7Lvi/ltc4viwNk/UZLbZvl63KnxoLMEpzIMw7nMYFhsLeC0HV1n0Ys9ZEOcrZVN8DO1jHjfrUcDxf3/770elyqBMOhL0bsWnehMujS61sYb+6mw8Vaz0x6H9yqOgdpK8n4VzOKMcX5NW0M80LFMm8ssnGbABopNTVs/jdZMMPR6JBbYF/gNBDCrOsOEXXWu+1v+7BW8p7KW7Lyb0fGS7h/p542dC0OvFvPED1tAOy+fW2ef3QKrdQq3Dkaf/Gog2C13idGnqiHv42GF5/Qwo7M25TxnfiZPhW7ghj7es6M2tYyQe4n624DXlyshsPd6XAqmSn9uTW+NFDAcpLEzO+hQPHmvhQPODTp8rLM7kuXO6MkhFR/s5IqdwREOeGeihR8knNSG4zQ1A/f6Po3WTDP0ejAW4i11nAqz8BnCWT3JK2TI8SjnekcEadvW+7bJDd3B/BF5ZKAub86eGClDZ/QHegLw/6JjdwYFxqwqED0zvJgnCkTXDj0ICDtGrHZX7j1ibeKxsI1F9XXFtgneboiJ/yFZ/6OO/70hmWkQ6bSgkXu5Ib4w3mbrUNV7GnMa5KV3cV8SulFlJJmRlfJEdjbLoHe7c3r2koybdhkEDlKPy4BxN1L3PyEWhnspwivTwd4ZNbMj1wPe66YjyvsOHrN1JnQmBEZ0N2Nb1HC6ZE+C46VPo02AISWZggU+w5C6QLy5n5mbwt9QLKiX1EWOLdkTQ/lvxsOf+1duomse6D8M6f+u8z1ORcQ4YA0luwi/1rnDQASE1yVlMpCKo3UggIHKR7NdJzsAWuH2f8t5/Bf9/68V8uvNgvogXYVRw3TQzTQo4quUoUVDSShF747air9jy0jx7YyuWD4rZbX7elwQJ6ZF/lfo4YjwfhSkvQop8Z1GsZTZVBu3mS45Lqsw357Jpsog60w4nfNptHZJ++9OwYL3e0j7S71plj33mmn5zFxXOzCFv36p72IEb5uyJ4Yy3xIPbP/KfXTNGy0RdnVLlYJeJCrAl7ti3MweFgG9WCg/XrQuhXVY2S3YIARYmnDUdahpHab0cjScgwDZYKQjl9EG4d4cS0epx5sD9koMMXpKGKWXcLyUhs422vMU1L4zqEMbf7VOuGz33jG0nV9scd6bk5Dx4t5DfnQdZcHl7DAWC+VPWXI0R8XbeGEQp172lDsdNs17nbTG8tfGxIV7UZ8rfi7E3d2jB7vjuJ1hHAFGUdllDYXiYASuiQD1kdHTUB7X5GU3eeeGKxoAvqP2gzBs9lTuRp4LjKYppJKP7e6vZuRc4G/OIxzvz+YQ2EjaI5fRRiDeNBtHqcebBnlF5sKAvzTC18UOo6X+nFQXOvm+2rJrfBqth6vWx9QFpvf2tr2Ks6XPXrUJEzZm1O+2OLu9d1+c/5G557MOuzUGZeyxscWPVKjdloYZ3z0COB0Sgt60N1UX3VlyxRJVWWxWt4m3tG5RdSR9t+2KMBXLWjI9EbEvJdObxYhekdARy+iKcO4m66PV47nACw9GvLCvCLts1yQD6/4Z+Zxt54oWdF5Iz2E/OxiBqyKg+pjsnCPUx6aueeOqyJbNnF57djQ7ZE9CWahMbUEEmBpY4hueC7K4LdIo2L5jWxzP50aNCR1p7GpYdLrTfM5mUWCZxRFGTxMIBy2jWQV8I4kPVY9vpMz6xAy7MEac+uJfes40Vu/fcQkhP1sTgTQQIlsiTaFeM+Ol8zqMgypjAoXB95dwEByMwOoISEFQ9xjF3+1MhBKg9Tmb9efkJ7zux6FLRt/4HhnTqnDi3uOZ/o8YkO+bIwimMmX0lGlTR6v7hymjI9SzpWU4cD1eGrqt02fdIKHErCZmSfVOD025+ccILI9AdLpEHV8+x5VyOIyDCl4yjr7S3+9qZPb8iYOVit7ZlERAdY7vkL2rOohjddMBh3Lo0QIUU0S/1oGjQ6cTuB4yICeCZaN4tzKKf0YnGHU6XN0/ShnttnKtyPiR6/GKMG41qxhBfTWVQdUP9kVI30gWDT4nZTtzKphOVxKBqNNRx0vSviqtQzmoICmDAiPpfSkP1gk4GIHFEVBdY5omBvpfFs/swBkIR3qmv62JyLt89J5q6gybS+yycRHfTCtizfVn0r3Rk6vLQ4Vdl9GhSmIhYW6kHi+E3i7IpumP0lGTR1CVlk8ZfqCDDlhm9jC7wsEIXBuBqNOpjl+bmZL5H85BBRwpDtYIfJoN3pJ4mZYROEEgGzZM0/yL6l0oipM4vhiMAI4+3wn9hYNzYcoo6mFDrjPoq+9zXdqbrExrfy45jja1tyqHA5RRJYtPOhE4fD3ulNwPjIARMAIbRODBBnkqwpKMCow+ByOwKALZeD3MTqylwZLTxcggPc3hvLNOnNkNrNU9cz51j1G4hKfSMjJ3+IDMkpXRYrA6w2SrAGSHGuf06KPcdHrusoy2Wne2xNct1eMt4W5ejIARKIJATPE93BrUwzqoRYrdRIyAEZiMgAw/puwyMsG0qDT9U/e4fqTrto2Pwomt8lR81vUedepoJeceZRTPlNfhndMopD2WUfDu/24Ebq0edyNxzCe5A+KYwlkqI3BgBOpTfP/zgeW0aEbACKyIQDYKcEaftRj2Z6OEio8zGz2BbBzEaGIaueJ/ZrBumwmgkxsBI3CzCPyng0h+1gF6ELksxm0jEPX64QFgONE19RHU/zNWOBmR6QOxY9M5vhEwAttDQI5k26jmVEbZ4fChaLKzdj3giKbda+s3dY5D+ko6hem/OKpcl9oV1rpNYDoYASNgBEBgpK7/91tFzTburZb8deUe+X5el9myuZ/omrqDOjqbGwZxNFZOYARuDIG/Sd6T6Z95VJRevra1pTizbY7rVWCzbrsK7M7UCBgBI7AZBNwObKYozMgNIlCf4nuD4ltkI2AESiMgRxQntM0RZfSU3vsTxzXn/zr/+88IGAEjYASMQBEE1N5E20Kb5GAEjoZA1Ouo54eRzw7qYYrSghiBsghkR3MO0ebmRkzZ/bWNoIyIzYyetvHne0bACBgBI7AeAgXan/WYdU5GwAgUR8AOanFITdAI7B+BbBz8U/98j3RUyD3WjJKyjjQF0WFtKSOobaOnd5H8awSMgBEwAjePwJz25+bBMwC3hkB8Xoa9Ow4V7KAeqjgtjBEog0B2MhkBfZiNhbGEGS39SGmfcOg8lGfb+tOxtB3fCBgBI2AEDopAgfaniczhpj82BfT1zSIQU3wPB8CsTZIOh4YFMgJGoEJARsJ7ci4fVTdGnGQDo9qFV3QYQe1afzqCsqMaASNgBIzA0RGY0/60YEMHaepszW1TSxTfKoWA2ntmT32h4x863st0nxr7jMQyf80lVcvksiJVj6CuCLazMgI7ROCjQo3KJ5K9df3pDjExy0bgIgIy0P4Yelwk5IdG4LYRKNX+hPEe0yFvG9UFpc/O6ceyG3BIv9KBo8pSoW8WzPaWScdSqsPNErCDesvV2rIbgQsIqKFh6gg9oCWC15+WQNE0doGAjLL7Q49dCGQmjcDKCBRuf8JBDWN+ZWluI7tcZo+k+5qbHjKC/f5toLC6lKnTRZhHHV+dgaUytIO6FLKmawT2j8DfpPS+miOGGqznOug9xdl9xPUcek5rBIyAETACN4HA7PanhlKMLtlBrYGywOnnHTbDY+XlGVQLAC6S2FZRv5fJ4UpUvQb1SsA7WyOwdQRaekFHsywaT0cncgIjYASMgBG4aQRKtD81AH/L57EesvbIp0sioE5pZk8xevrBkvncIm1hGx0ujFAfLthBPVyRThNIFZ1eGDayQYH/Um8c9Azl8oXusZbgnq7ZOIeRsA9075A9N8jpYASMgBGYi0A2IuioCV35TOfo2h+lP09GFRQ3ZhjE5iIvmnHm8uP0RuAGEYjpj2HQ3yAEy4os3ZWW8WR990K5oe8Cb9uKy8Af+J60I8tktT5VHFTmiv+LjkMKuD6ku83xOQ6olAvOJ4vZ62sIvtQ1DmyEUDwf6oa/axmo+H9rCNDZ8tAG/taK5Xb4yUbb95IYAy0ZybrHNeu07gcS2ajjE0x0BCadqnvo3H/Tf0qb47yv5y8jnf+NwNIIqL79Sh0kH53T9u8x/JyZDoN+jzJsnedPVD/SjCnVF3QcAdw5B3f7GAKhcIj6/FNhutciRxv5r2SOrnlD//Tk/lWHFzALhFsM2fBhnSCB3VajtzHd0A89Y9X3K1VxMKBQNs14uuVgBDaDANO6flH9pv46GIFVEVC9w8HEOHsmndnUlU1jDf37MuvWxCcNtE4w8GKaPJ2H7hBM6PhnLQRUj7ENqZ/oUur07kJ+l3gHbeeuUHrC++t8oOeYFeIdfJfBPaZNH6VdwOHGH/0rugYH1cEIvEKZZBj+pv9vA5LcONEoNV+An5WmaXRFMv8bgS0hsEujaksAmpdJCGCYMYLf3GgsTYULitKxTIejjjL1txnQscxUIbwtWq/vTv1rBNZHYOf1L9kwet9i1Gl9AA+aI86ERGNZQltwx0AbKmXupbZB72Wzw7MM9StTsYN65QLYQvbR6EjJYDihaMJZhT3uvW55AWwogY7DlhFwHd1y6RyfNzr7Tjr2snGMjq1mpOg8xQs93ICFOvyu0uHstjmwjei+NALFEXirOMXrEIxZYp5RUx7/S8u9GLV2W1wecyiC7WEHiuygLlNp9kr1UzH+a8NQYspvrN9IcslY4qWoG1jpvn+MwMYQeJX5OYqBtTF4zU4XAtKROKFNR5ToyTiWjo3RnIh3qQecOD819DK0HIyAERiOQHQWxbTI4Skdsw8B1p926TBsyMC+j46fD0Qg2+HEfjkwye6i2UHdXZEtyjBTX5q9MdxrKp7HYWAtyo2JGwEjYAT2jUBTn6ZOwBCp5nTGZzDiUf2fGSyHNULqgvrcCCyFgN4h3sXXOjyCWh7kVkyzE8XoaqyjL5/z7VIMzKsleUeDwg7q0Up0njw4ojikKUi58AIw+tS859HTO4j8u20EMEYcjMDqCGTHk1GDuu5kQ0J0anM0gSUVjDKcBOlf4hMYQeXzXsxccTACayOQ6p8yPYI+5V1jynzItDaWh8svY/l7TV8lGXWNrmOTuE9z58DhZL+yQLQZbcvvrsxWuewflCNlSntHQErkqZTKCw7JQmPEovd3dHyje0/0/7aO3xSvvkZVtxyMwCYReJW5sjGyyeI5PFOMlobuRNgYTT3p4JM+5fNezzkUB52bHIHQs7r/sPasOZsFug5GYEkEjrREAtsGW4bOn+bmZUtieGTaOKKsj8fxB1sCtuI/pMP41JtDYQRoE0QS3A89s8YOauGKs3dyGEstMmBoORiBvSJAY+lgBFZFQLoUR7PSnTIq0oio7jdHUPnmW+cUuA6dvKoszuymEYgOvujw2y0YepcY6aOT57EOO6hlSvIj4Yqj5M6zMngOocLGeoRDb5znKb53hexfI2AEjodAGomSWGFgHU9CS7QnBJiSZSNuTyVmXkEgOvhCn+4dFUZR35ejWk2/37tA5v/mEGAg6Xd1DBy6PbGDenP12gIbgZtB4FWW9EhT1G6m8A4oKFOyzkZPDyinRToWAtHBF/p079J9lwVomy22d9lW5T9PNe36/umqvNxKZrljhf0IWBJy6GAH9dDFa+GMwE0jEDujhoF102BY+OsgIIOC9aV8g5F6+Ijr63DiXI3AJASigy/WUE8ispVEGnViJJh9NGITsq2wtkc+Ln3/dI/y7IHn1LGienz4vWC8BnUP1dE8GgEjMAWBMKjCwJpCw2mMwCwEZEh0ri+dRdiJjcA6CEQHX3T4rZPrsrnwTn6uzqInej+9FnUi1sLOM0ImYjclmeor7yIbUd1Em1IfQaVXycEIGAEjcBQEwkH1WqOjlKjlMAJGYG0EooMv9Ona+RfPT44V9i5G/pfFiZugEVgOAWbfsPb0JjpV6g5q9JItB60pGwEjYATWQyAMKuu29TB3TkbACBwLgejgC316COmykf+KUdRDCGQhDo2A6invIdPSb2LtNJ1IOKgeOT10tbZwRuA2Eci95En4rNxvEwhLbQSMgBGYgECeUhgdfIdyUDMcGPusEQ8nfAJKTmIEVkGA0dOXsmuOPK06ZmskQHFQY2e2kwerwO1MjIARMALLIhBGlQ2QZXE2dSNgBI6HQOjN1/UOv6OImY19pkt+fxSZLMfxEFAHCiOn7AL/2fGk65YIB9XBCBgBI3BUBOI7YWFoHVVOy2UEjIARKI0Au7QSoqPv7upAv3JSWYvqqb4HKtMjiZJH9xk9/csRO4kulRUOqqf4XkLIz4yAEdgzAj9l5j/YsxDm3QgYASNwBQTey3keeVohIn6q4ws5A+9nef1nBLaCAKP7T+WcRmf7Vvhago9YTpD8Uj4zE1N848ESmZqmETACRuAaCIRhFSMB1+DBeRoBI2AE9ohAOGzR0bdHGXp5ZmRKzukniuiZNr1oOcJaCKhO4pfhnIYds1bW18rnZKlp/Tuob1+LI+drBIyAEVgCAXodpeQhHYbWEtmYphEwAkbgiAiEw3Z4A1ltBdOYDzuV+YiV8+gy0XEiGQ//7tXKMQZK08BpfYpvPKjF9akRMAJGYPcIJKNDjmoYW7sXyAIYASNgBJZEII/eoDMPuUHSktiZthEwApMQiIFSHPN7OKgxxfdkaHUSaScyAkbACGwPgZeZJXbBczACRsAIGIF+BEJf3tIITj8qjmEEjMBSCMRAaTWC+lvOKR4slbHpGgEjYASugcAPOVPWGDkYASNgBIxAPwKPc5Rv+6M6hhEwAkZgNgIxUJpmvTGCGnPu48HsHEzACBgBI7AVBPIGA0wZiRGBrbBmPoyAETACW0Ug6Uvpz5iBslU+zZcRMALHQCAGStPAad1BZa2BgxEwAkbgiAh8J6Eeal2VN0s6YulaJiNgBIohkPUkxqKd02KompARMAI9CMRA6dkIaniuPen92AgYASOwOwT4lhghpq3dXfnXCBgBI2AEmgjEbBNP720i42sjYASWQiAGSu82ttT0jXvqLfsj5/aermPK71IMmK4RMAJGYHUEpOf+SabScW+unrkzNAJGwAjsBAHpyl/E6rvWlTspMLNpBHaOgHQOg6TJRtP/m9I9r5niSwinNLzXu7v+NQJGwAgcB4FnEoVpvjE6cBzJLIkRMAJGoAAC0o8sg+D4ugA5kzACRsAIDEEg/M/qs1bhoP6aU0eEIcQcxwgYASOwGwTUI/dVZvaL3TBtRo2AETAC6yLwZc6ODj0HI2AEjMAaCHyYM4kB0/QdVO79lB98kP/9ZwSMgBE4IgI4qY/ydJIjymeZjIARMAKTEMh68ZESv2SK3SQiTmQEjIARGI/AezlJ9d3lGEGNG+HBjiftFEbACBiB7SMQowIxSrB9js2hETACRmAdBD7P2TxdJzvnYgSMgBFICMQXFmLA9N599ZKlJ+o5Sye6vm+wjIARMAJHRUC67olke64jLcQ/qpyWywgYASMwBgHpRjYp+U52oJdBjAHOcY2AEZiFgHQP3z5lmWlll8UIKoTvtvW9f9/rUEHDwQgYgUMiIOOLab5MX/Mo6iFL2EIZASMwFoHcccdOmh49HQue4xsBIzAZAeke9A6+Z7VBEsTqDmp8kNk7XIKMgxEwAkdG4DMJ90SK0R1yRy5ly2YEjEAvAtlAZFbJF+rA89rTXsQcwQgYgYIIhN8Zy00T6bqD+kPO7JOCmZqUETACRmBzCMgIo0OO48XmmDNDRsAIGIF1EfhG2f0ovehPy6yLu3MzAkbg3r3HGYRv62BUa1C5qV40f8i+jo7PjYARODQC0nmse3huw+zQxWzhjIAR6EBAOpDRi+91vOPR0w6QfNsIGIHFEMi+50Ppn5M9kOojqGT+nQ4+ZB+7KS3GkAkbASNgBDaAwKfi4bl0HmsgHIyAETACt4YAs0g+tXN6a8VueY3A9RHI/ib2VywzrZhqOqj0ohFiuPXuyr9GwAgYgQMiIKPsV4mFk/rWAcWzSEbACBiBTgRyxxzrTk/WfnUm8AMjYASMQFkEYv3pyfResjiZ4ptueJpvWehNzQgYASNgBIyAETACRsAIGAEjYAQqBNRJ9osu3lUn2ZvVzXzSHEHlNh+yZ5pveLU5qv+MgBEwAkbACBgBI2AEjIARMAJGwAhMRyBP72VJaevmbGcjqGSlRH/o76U8Wqa+ORgBI2AEjIARMAJGwAgYASNgBIyAEZiNgHxNlpU+0vGm/M2zz1u1jaCSKR+yf6TE3jgENByMgBEwAkbACBgBI2AEjIARMAJGYBYC2b/EOWUw9Mw5hXiXg8o0X8KXd3/+NQJGwAgYASNgBIyAETACRsAIGAEjMAuBz3Pqp11UWqf4Elne7RP9PdfROvRKHAcjYASMgBEwAkbACBgBI2AEjIARMAJDEJCP+U/F+06jp190xe8aQb2nREzzZdjVo6hd6Pm+ETACRsAIGAEjYASMgBEwAkbACPQikAdAWULaOXoKkc4R1PTw/n3mB7OI9T05rL9zz8EIGAEjYASMgBEwAkbACBgBI2AEjMBQBPLaU0ZP+f5y6+69Qeuig0okEcNBfShCn0Qi/xsBI2AEjIARMAJGwAgYASNgBIyAERiCwBifsnOKb2Qkx5RPzbwrorGgNR753wgYASNgBIyAETACRsAIGAEjYASMQCcC8iM/1kOOQZ8w7R1BJScR5UOqf9fxjhzW1u2AiedgBIyAETACRsAIGAEjYASMgBEwAkYgEJAv+ZvOmdr7Y9y79D/IQYVA9nx/F2GvRb2EqJ8ZASNgBIyAETACRsAIGAEjYASMAD4kmyJ9ONQ5BbL/Dyz9MBiYERMQAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{\\overline{q} \\alpha \\left(\\frac{\\overline{q}^{\\frac{- \\alpha - \\beta + 1}{\\beta}} \\alpha^{- \\frac{\\alpha}{\\beta}} r w^{\\frac{\\alpha}{\\beta}}}{\\beta}\\right)^{\\frac{\\beta}{\\alpha + \\beta}}}{w}, \\  \\overline{q}^{\\frac{1}{\\beta}} \\left(\\frac{w \\left(\\alpha^{\\frac{\\alpha}{\\beta}} \\beta\\right)^{\\frac{\\beta}{\\alpha + \\beta}} \\left(\\overline{q}^{\\frac{- \\alpha - \\beta + 1}{\\beta}} r w^{\\frac{\\alpha}{\\beta}}\\right)^{- \\frac{\\beta}{\\alpha + \\beta}}}{\\overline{q} \\alpha}\\right)^{\\frac{\\alpha}{\\beta}}, \\  \\left(\\frac{\\overline{q}^{\\frac{- \\alpha - \\beta + 1}{\\beta}} \\alpha^{- \\frac{\\alpha}{\\beta}} r w^{\\frac{\\alpha}{\\beta}}}{\\beta}\\right)^{\\frac{\\beta}{\\alpha + \\beta}}\\right)$"
      ],
      "text/plain": [
       "⎛                                                                             \n",
       "⎜                                                   β                         \n",
       "⎜                                                 ─────                       \n",
       "⎜                                                 α + β                     ⎛ \n",
       "⎜               ⎛            -α - β + 1  -α     α⎞                          ⎜ \n",
       "⎜               ⎜            ──────────  ───    ─⎟                          ⎜ \n",
       "⎜               ⎜                β        β     β⎟                          ⎜ \n",
       "⎜               ⎜\\overline{q}          ⋅α   ⋅r⋅w ⎟                          ⎜ \n",
       "⎜\\overline{q}⋅α⋅⎜────────────────────────────────⎟                          ⎜ \n",
       "⎜               ⎝               β                ⎠       beta______________ ⎜w\n",
       "⎜──────────────────────────────────────────────────────,   ╲╱ \\overline{q} ⋅⎜─\n",
       "⎝                          w                                                ⎝ \n",
       "\n",
       "                                                α                             \n",
       "                                                ─                             \n",
       "                                                β                             \n",
       "         β                                 -β  ⎞                              \n",
       "       ─────                              ─────⎟                              \n",
       "       α + β                              α + β⎟                              \n",
       " ⎛ α  ⎞      ⎛            -α - β + 1    α⎞     ⎟   ⎛            -α - β + 1  -α\n",
       " ⎜ ─  ⎟      ⎜            ──────────    ─⎟     ⎟   ⎜            ──────────  ──\n",
       " ⎜ β  ⎟      ⎜                β         β⎟     ⎟   ⎜                β        β\n",
       "⋅⎝α ⋅β⎠     ⋅⎝\\overline{q}          ⋅r⋅w ⎠     ⎟   ⎜\\overline{q}          ⋅α  \n",
       "───────────────────────────────────────────────⎟ , ⎜──────────────────────────\n",
       "                \\overline{q}⋅α                 ⎠   ⎝               β          \n",
       "\n",
       "            ⎞\n",
       "            ⎟\n",
       "            ⎟\n",
       "         β  ⎟\n",
       "       ─────⎟\n",
       "       α + β⎟\n",
       "     α⎞     ⎟\n",
       "─    ─⎟     ⎟\n",
       "     β⎟     ⎟\n",
       " ⋅r⋅w ⎟     ⎟\n",
       "──────⎟     ⎟\n",
       "      ⎠     ⎠"
      ]
     },
     "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": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Eji7QJ9CJNEGCGT6bI+HFEiO8JtzcJo0Pxrf//yqUIYOH00DABfo0+jFtBV+r5L+YUmHFBX+fZopWm7m2hxNCwAX6hDozNuVCx7YbTVrbOvPd87YLfnpoXLEWuUCfXocjuAhwCNEFv6GLdppb54jRKR38iyWn1JuxLRJie96M2/2PIvNpPlH7TvGm4kdZ53SOrSQPp4DAv9Ry4m8jbparAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\overline{q} \\alpha \\left(\\frac{\\overline{q}^{\\frac{- \\alpha - \\beta + 1}{\\beta}} \\alpha^{- \\frac{\\alpha}{\\beta}} r w^{\\frac{\\alpha}{\\beta}}}{\\beta}\\right)^{\\frac{\\beta}{\\alpha + \\beta}}}{w}$"
      ],
      "text/plain": [
       "                                                   β  \n",
       "                                                 ─────\n",
       "                                                 α + β\n",
       "               ⎛            -α - β + 1  -α     α⎞     \n",
       "               ⎜            ──────────  ───    ─⎟     \n",
       "               ⎜                β        β     β⎟     \n",
       "               ⎜\\overline{q}          ⋅α   ⋅r⋅w ⎟     \n",
       "\\overline{q}⋅α⋅⎜────────────────────────────────⎟     \n",
       "               ⎝               β                ⎠     \n",
       "──────────────────────────────────────────────────────\n",
       "                          w                           "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Lstar = foc_sol[0][0]\n",
    "Lstar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\overline{q}^{\\frac{1}{\\beta}} \\left(\\frac{w \\left(\\alpha^{\\frac{\\alpha}{\\beta}} \\beta\\right)^{\\frac{\\beta}{\\alpha + \\beta}} \\left(\\overline{q}^{\\frac{- \\alpha - \\beta + 1}{\\beta}} r w^{\\frac{\\alpha}{\\beta}}\\right)^{- \\frac{\\beta}{\\alpha + \\beta}}}{\\overline{q} \\alpha}\\right)^{\\frac{\\alpha}{\\beta}}$"
      ],
      "text/plain": [
       "                                                                     α\n",
       "                                                                     ─\n",
       "                                                                     β\n",
       "                   ⎛          β                                 -β  ⎞ \n",
       "                   ⎜        ─────                              ─────⎟ \n",
       "                   ⎜        α + β                              α + β⎟ \n",
       "                   ⎜  ⎛ α  ⎞      ⎛            -α - β + 1    α⎞     ⎟ \n",
       "                   ⎜  ⎜ ─  ⎟      ⎜            ──────────    ─⎟     ⎟ \n",
       "                   ⎜  ⎜ β  ⎟      ⎜                β         β⎟     ⎟ \n",
       "beta______________ ⎜w⋅⎝α ⋅β⎠     ⋅⎝\\overline{q}          ⋅r⋅w ⎠     ⎟ \n",
       "  ╲╱ \\overline{q} ⋅⎜────────────────────────────────────────────────⎟ \n",
       "                   ⎝                 \\overline{q}⋅α                 ⎠ "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Kstar = foc_sol[0][1]\n",
    "Kstar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Substitute optimal input into the cost, to calculate the minimum cost level necessary at the given output variables. \n",
    "\n",
    "The resulting function is our **cost function**, $C(w,r,\\overline{q})$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left(\\frac{\\alpha^{- \\frac{\\alpha}{\\beta}}}{\\beta}\\right)^{\\frac{\\beta}{\\alpha + \\beta}} \\left(\\frac{\\left(\\overline{q}^{\\frac{- \\alpha - \\beta + 1}{\\beta}} r w^{\\frac{\\alpha}{\\beta}}\\right)^{- \\frac{\\beta}{\\alpha + \\beta}}}{\\overline{q} \\alpha}\\right)^{\\frac{\\alpha}{\\beta}} \\left(\\overline{q} \\alpha \\left(\\overline{q} \\alpha \\left(\\overline{q}^{\\frac{- \\alpha - \\beta + 1}{\\beta}} r w^{\\frac{\\alpha}{\\beta}}\\right)^{\\frac{\\beta}{\\alpha + \\beta}}\\right)^{\\frac{\\alpha}{\\beta}} \\left(\\overline{q}^{\\frac{- \\alpha - \\beta + 1}{\\beta}} r w^{\\frac{\\alpha}{\\beta}}\\right)^{\\frac{\\beta}{\\alpha + \\beta}} + \\overline{q}^{\\frac{1}{\\beta}} r \\left(\\alpha^{\\frac{\\alpha}{\\beta}} \\beta\\right)^{\\frac{\\beta}{\\alpha + \\beta}} \\left(w \\left(\\alpha^{\\frac{\\alpha}{\\beta}} \\beta\\right)^{\\frac{\\beta}{\\alpha + \\beta}}\\right)^{\\frac{\\alpha}{\\beta}}\\right)$"
      ],
      "text/plain": [
       "                                                α                             \n",
       "                                                ─ ⎛                           \n",
       "                                                β ⎜                           \n",
       "        β   ⎛                              -β  ⎞  ⎜                           \n",
       "      ───── ⎜                             ─────⎟  ⎜               ⎛           \n",
       "      α + β ⎜                             α + β⎟  ⎜               ⎜           \n",
       "⎛ -α ⎞      ⎜⎛            -α - β + 1    α⎞     ⎟  ⎜               ⎜           \n",
       "⎜ ───⎟      ⎜⎜            ──────────    ─⎟     ⎟  ⎜               ⎜           \n",
       "⎜  β ⎟      ⎜⎜                β         β⎟     ⎟  ⎜               ⎜           \n",
       "⎜α   ⎟      ⎜⎝\\overline{q}          ⋅r⋅w ⎠     ⎟  ⎜               ⎜           \n",
       "⎜────⎟     ⋅⎜──────────────────────────────────⎟ ⋅⎝\\overline{q}⋅α⋅⎝\\overline{q\n",
       "⎝ β  ⎠      ⎝          \\overline{q}⋅α          ⎠                              \n",
       "\n",
       "                                                                              \n",
       "                                       α                                      \n",
       "                                       ─                                      \n",
       "                                       β                                      \n",
       "                                   β  ⎞                                 β     \n",
       "                                 ─────⎟                               ─────   \n",
       "                                 α + β⎟                               α + β   \n",
       "    ⎛            -α - β + 1    α⎞     ⎟  ⎛            -α - β + 1    α⎞        \n",
       "    ⎜            ──────────    ─⎟     ⎟  ⎜            ──────────    ─⎟        \n",
       "    ⎜                β         β⎟     ⎟  ⎜                β         β⎟        \n",
       "}⋅α⋅⎝\\overline{q}          ⋅r⋅w ⎠     ⎠ ⋅⎝\\overline{q}          ⋅r⋅w ⎠      + \n",
       "                                                                              \n",
       "\n",
       "                                                  \n",
       "                                                α⎞\n",
       "                                                ─⎟\n",
       "                                                β⎟\n",
       "                             β   ⎛          β  ⎞ ⎟\n",
       "                           ───── ⎜        ─────⎟ ⎟\n",
       "                           α + β ⎜        α + β⎟ ⎟\n",
       "                     ⎛ α  ⎞      ⎜  ⎛ α  ⎞     ⎟ ⎟\n",
       "                     ⎜ ─  ⎟      ⎜  ⎜ ─  ⎟     ⎟ ⎟\n",
       "beta______________   ⎜ β  ⎟      ⎜  ⎜ β  ⎟     ⎟ ⎟\n",
       "  ╲╱ \\overline{q} ⋅r⋅⎝α ⋅β⎠     ⋅⎝w⋅⎝α ⋅β⎠     ⎠ ⎠\n",
       "                                                  "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = (w*L + r*K).subs([(L,Lstar), (K,Kstar)])\n",
    "# simplify function does not work well here, but we can further simplify by hand\n",
    "C.simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2. Solve Profit-maximization problem under this cost function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The firm chooses output level $q$ to maximize its profit\n",
    "$$\\max_{q} \\pi(q,p,w,r) = p*q - C(w,r,q)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# define symbols p and q\n",
    "p, q = symbols('p q', nonzero=True, positive=True)\n",
    "# define profit function\n",
    "profit = p*q - C.subs([(qbar, q)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\alpha^{- \\alpha \\left(\\frac{1}{\\alpha + \\beta} + \\frac{1}{\\beta}\\right)} \\beta^{- \\frac{\\beta}{\\alpha + \\beta}} q^{- \\frac{\\alpha}{\\alpha + \\beta} - \\frac{\\alpha}{\\beta} - \\frac{\\alpha}{\\beta \\left(\\alpha + \\beta\\right)} - \\frac{\\beta}{\\alpha + \\beta} - 1} r^{- \\frac{\\alpha}{\\alpha + \\beta}} w^{- \\frac{\\alpha^{2}}{\\beta \\left(\\alpha + \\beta\\right)}} \\left(\\alpha^{\\alpha \\left(\\frac{1}{\\alpha + \\beta} + \\frac{1}{\\beta}\\right)} \\beta^{\\frac{\\beta}{\\alpha + \\beta} + 1} p q^{\\frac{\\alpha}{\\alpha + \\beta} + \\frac{\\alpha}{\\beta} + \\frac{\\alpha}{\\beta \\left(\\alpha + \\beta\\right)} + \\frac{\\beta}{\\alpha + \\beta} + 1} r^{\\frac{\\alpha}{\\alpha + \\beta}} w^{\\frac{\\alpha^{2}}{\\beta \\left(\\alpha + \\beta\\right)}} - \\alpha^{\\frac{\\alpha \\left(\\frac{\\alpha}{\\beta} + 1\\right)}{\\alpha + \\beta}} \\beta q^{\\frac{\\alpha^{2}}{\\beta \\left(\\alpha + \\beta\\right)} + \\frac{2 \\alpha}{\\alpha + \\beta} + \\frac{\\beta}{\\alpha + \\beta} + \\frac{1}{\\beta}} r w^{\\frac{\\alpha}{\\beta}} - \\alpha^{\\frac{\\alpha}{\\beta} + 1} q^{\\frac{\\alpha}{\\beta} + \\frac{\\alpha}{\\beta \\left(\\alpha + \\beta\\right)} + 1 + \\frac{1}{\\alpha + \\beta}} r w^{\\frac{\\alpha \\left(\\frac{\\alpha}{\\beta} + 1\\right)}{\\alpha + \\beta}} + \\alpha^{\\frac{\\alpha}{\\alpha + \\beta} + \\frac{\\alpha}{\\beta} + 1} \\beta^{\\frac{\\beta}{\\alpha + \\beta}} p q^{\\frac{\\alpha}{\\alpha + \\beta} + \\frac{\\alpha}{\\beta} + \\frac{\\alpha}{\\beta \\left(\\alpha + \\beta\\right)} + \\frac{\\beta}{\\alpha + \\beta} + 1} r^{\\frac{\\alpha}{\\alpha + \\beta}} w^{\\frac{\\alpha^{2}}{\\beta \\left(\\alpha + \\beta\\right)}}\\right)}{\\alpha + \\beta}$"
      ],
      "text/plain": [
       "                                                                              \n",
       "                                                                         2    \n",
       "    ⎛  1     1⎞   -β        α     α       α         β         -α       -α     \n",
       " -α⋅⎜───── + ─⎟  ─────  - ───── - ─ - ───────── - ───── - 1  ─────  ───────── \n",
       "    ⎝α + β   β⎠  α + β    α + β   β   β⋅(α + β)   α + β      α + β  β⋅(α + β) \n",
       "α              ⋅β     ⋅q                                   ⋅r     ⋅w         ⋅\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "⎛                                                                             \n",
       "⎜                                                                            2\n",
       "⎜   ⎛  1     1⎞    β            α     α       α         β          α        α \n",
       "⎜ α⋅⎜───── + ─⎟  ───── + 1    ───── + ─ + ───────── + ───── + 1  ─────  ──────\n",
       "⎜   ⎝α + β   β⎠  α + β        α + β   β   β⋅(α + β)   α + β      α + β  β⋅(α +\n",
       "⎝α             ⋅β         ⋅p⋅q                                 ⋅r     ⋅w      \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "         ⎛α    ⎞                                                              \n",
       "       α⋅⎜─ + 1⎟         2                                                    \n",
       "         ⎝β    ⎠        α        2⋅α      β     1    α    α      α       α    \n",
       "───    ─────────    ───────── + ───── + ───── + ─    ─    ─ + 1  ─ + ─────────\n",
       " β)      α + β      β⋅(α + β)   α + β   α + β   β    β    β      β   β⋅(α + β)\n",
       "    - α         ⋅β⋅q                             ⋅r⋅w  - α     ⋅q             \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "           α + β                                                              \n",
       "\n",
       "                  ⎛α    ⎞                                                     \n",
       "                α⋅⎜─ + 1⎟                                                     \n",
       "         1        ⎝β    ⎠      α     α        β        α     α       α        \n",
       " + 1 + ─────    ─────────    ───── + ─ + 1  ─────    ───── + ─ + ───────── + ─\n",
       "       α + β      α + β      α + β   β      α + β    α + β   β   β⋅(α + β)   α\n",
       "            ⋅r⋅w          + α             ⋅β     ⋅p⋅q                         \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "                          ⎞\n",
       "                      2   ⎟\n",
       " β          α        α    ⎟\n",
       "──── + 1  ─────  ─────────⎟\n",
       " + β      α + β  β⋅(α + β)⎟\n",
       "        ⋅r     ⋅w         ⎠\n",
       "───────────────────────────\n",
       "                           "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# FOC is\n",
    "profit.diff(q).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we impose $\\alpha = 1/4$, $\\beta = 1/4$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle q \\left(p - 2 q \\sqrt{r} \\sqrt{w}\\right)$"
      ],
      "text/plain": [
       "q⋅(p - 2⋅q⋅√r⋅√w)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "profit = ( p*q - C.subs([(qbar, q), (alpha, Rational(1,4)), (beta, Rational(1,4))]) ).simplify()\n",
    "profit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solve for the optimal output level that maximizes profit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{p}{4 \\sqrt{r} \\sqrt{w}}\\right]$"
      ],
      "text/plain": [
       "⎡   p   ⎤\n",
       "⎢───────⎥\n",
       "⎣4⋅√r⋅√w⎦"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qstar_sol = solve(profit.diff(q), q)\n",
    "qstar_sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{p}{4 \\sqrt{r} \\sqrt{w}}$"
      ],
      "text/plain": [
       "   p   \n",
       "───────\n",
       "4⋅√r⋅√w"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qstar = qstar_sol[0]\n",
    "qstar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## comparative statics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{4 \\sqrt{r} \\sqrt{w}}$"
      ],
      "text/plain": [
       "   1   \n",
       "───────\n",
       "4⋅√r⋅√w"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qstar.diff(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{p}{8 r^{\\frac{3}{2}} \\sqrt{w}}$"
      ],
      "text/plain": [
       "   -p    \n",
       "─────────\n",
       "   3/2   \n",
       "8⋅r   ⋅√w"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qstar.diff(r)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{p}{8 \\sqrt{r} w^{\\frac{3}{2}}}$"
      ],
      "text/plain": [
       "   -p    \n",
       "─────────\n",
       "      3/2\n",
       "8⋅√r⋅w   "
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qstar.diff(w)"
   ]
  },
  {
   "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
}