firm theory - 1.2 isocost curve and non-linear costs.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "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": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "K, L, r, w, Cbar = symbols('K L r w \\overline{C}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1. Iinear Iso-cost"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Define the iso-cost curves"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, define an expression for cost."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle K r + L w$"
      ],
      "text/plain": [
       "K⋅r + L⋅w"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Kcost = r*K\n",
    "Lcost = w*L\n",
    "cost = Kcost + Lcost\n",
    "cost"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, the iso-cost curve is such that the cost is equal to the fixed level of cost $\\overline{C}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{- L w + \\overline{C}}{r}\\right]$"
      ],
      "text/plain": [
       "⎡-L⋅w + \\overline{C}⎤\n",
       "⎢───────────────────⎥\n",
       "⎣         r         ⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "isocost_sol = solve(cost - Cbar, K)\n",
    "isocost_sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{- L w + \\overline{C}}{r}$"
      ],
      "text/plain": [
       "-L⋅w + \\overline{C}\n",
       "───────────────────\n",
       "         r         "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Retreive the first and only element in the solution, which is our expression for the budget constraint.\n",
    "isocost = isocost_sol[0]\n",
    "isocost"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The slope and intercept of the iso-cost curve is better seen after expanding the expression."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{L w}{r} + \\frac{\\overline{C}}{r}$"
      ],
      "text/plain": [
       "  L⋅w   \\overline{C}\n",
       "- ─── + ────────────\n",
       "   r         r      "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "isocost.expand()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plot the iso-cost"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Let's plot a iso-cost with $\\overline{C}=5, w = 1, r = 1$. \n",
    "\n",
    "Note the use of different brackets in the subs method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "scrolled": true
   },
   "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": [
    "isocost_plot1 = plot(isocost.subs([(Cbar,5), (w,1), (r,1)]), xlim=[0,10], ylim=[0,10], xlabel=\"$L$\", ylabel=\"$K$\", show=False)\n",
    "isocost_plot1.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2. NonIinear Iso-cost (with increasing input cost)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Define the iso-cost curves"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, define an expression for cost."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle K^{2} r + L^{2} w$"
      ],
      "text/plain": [
       " 2      2  \n",
       "K ⋅r + L ⋅w"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Kcost = r*K**2\n",
    "Lcost = w*L**2\n",
    "cost = Kcost + Lcost\n",
    "cost"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, the iso-cost curve is such that the cost is equal to the fixed level of cost $\\overline{C}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\sqrt{\\frac{- L^{2} w + \\overline{C}}{r}}, \\  - \\sqrt{- \\frac{L^{2} w - \\overline{C}}{r}}\\right]$"
      ],
      "text/plain": [
       "⎡     _______________________        _________________________⎤\n",
       "⎢    ╱    2                         ╱  ⎛ 2                 ⎞  ⎥\n",
       "⎢   ╱  - L ⋅w + \\overline{C}       ╱  -⎝L ⋅w - \\overline{C}⎠  ⎥\n",
       "⎢  ╱   ───────────────────── , -  ╱   ─────────────────────── ⎥\n",
       "⎣╲╱              r              ╲╱               r            ⎦"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "isocost_sol = solve(cost - Cbar, K)\n",
    "isocost_sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHkAAAA/CAYAAADE+2c4AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAHbUlEQVR4Ae2cjXHcNhCFdRoVcLE7uHQg2xVE6kCyK4jcgTKuwON04KSCRO7AdgWx3IHUQRR1oLyPwnJIHEhCJMjj0cAMRBI/i9192MUC5Gn18PBwkCKtVqsr0TlLQSvTSKuBo4TkjjVhVgnpZVKJNHCYgo6seC06tyloZRrpNZAEZLH1Uvl7evYyxRQaSAXysZi5ScFQppFeA6lAfiXWvqVnL1NMoYFViuhaa/KN6PycgqFMI70GUlnyfXrWMsVUGhgMsqyY9ThH1qkQGYHOYJDFE5H1PyPwlkkm0kCKw5AX4oXTrpwSakAecvBRpB1ODQ68xMy1ZPtFBPO6nBDklKRSWPJ6TgBr0nH69s4paeOuv86Jx5QAxtAaBLIUihLnFnR9EKBvTXjx+FH3eJsfdos3NPAish79OFNAXSpfKT+4fK3rZ+ULA7NyvVD5SeX5g+43KoPXvU+SY62MPpAfnVgudKFnZEXmMg2yZFHhpGv0yFqW+Tsci3mCkVs9E+w1Jax4kadvkv9SsrEUvVc+ry5Bqjtz4PK6twbygRr2ziL2WXkzhEZsX42DJQIy7jiaZ9orcyIX3SdlW42NV7kYQlP9iTPQNe8HiIGCsqiOpQkd1TAZ6q4hNtWabC4YYaOSZjYTg5ndZvlRtAY0AiDykPRVnTmPeCF937cQYit772PS211LgTDeNmALL72q3tBLAnyJ6S3+CAqx4i7FxJDbWRvngpmsNffcwNCdyv/264ZYMjNryrUPQaOCPANYE+KUma/nY1dWyq9nC2A+6r4WwLn2uL4i6Zn17kZ5qEUayairxmOisg4Th3yK6IRXLfm29kNARulsTUZPEtZcdacVO8Ug6HvdAy58Eowxy6uJtZ2ADvdfD1QegxsUbAmPxTMTe8pkW8Et4EJMMKGVtwyht7vWIETWRHlTpFM3SMx6zMTD4moTUMKbwojSAczqoe3HFUyqUjb1/aI+KM9vp6JRE/EEqXNyPzYL/x0CMkHX1qyxYaQUFE3A8BQXx7oTollYMso2+k1Xtfmpqa5Sfqd2f7jn17qWgDrLh2d/rG/q0wiy+mFtBZ+Orl2ecaP6cpJZha7fRfO88uzfFn1V2DhutQM6F737allxr8JgON5Vrs7XXW1S1WsstgWN46mONbW2bYgZW30ABdrltkT3rIH/+f1V9qStm/VXP6zx0p6fclU/tkzq0o2RG4cvZrfw7LUmu9keNbs0+KCkscw6/mohRIDVhx+sCGuqzn7cdy2gdPLGLBUtLPaqKryJxq/GB02EXkmOkBc86AWyRiEAGf2ky0lj67HvPotqKQDL6wsAyvMnB2W+st5IgcHxCybG+2MBoa3NwZGkA9qVS47f6MgviHye8h1yIWBoljrhADlmHQ6JBpjmKVg3uWcdLC3HlfWdRKExo8sk863Gx9twPk3kbHFEQUNl6KYIgFVX9Ua1MXq9TxZxItPR3iGLPoHPn8oomy0QqbpPrJVLwLbg5bF3w1+NZdsTlPSvMopkbDzVc2WORGvKVVl0ckAQLxTn79EdKw2dPrBW5DYw73R/JbqdHqYvyPnrzAoIbbcpQG6jH1N3GNOo2kZMM5v8dazaJN/XNYDlmfXVayZ66rMm4z79wGQidvdvmBh3OrZUNUt2Vto1Jgv9VJF1Fy+5PkIDJcgCmCjVgpy2rtmS27Qzw7oSZPH2Thkr7UpTvkPu4iXXR2igAFlWzLHgWrnVktWONjsNIiJkyk08DRw64CgmYuYkqy1RXzvya2uc6+ahAaLr12z2BTbHh7wcD7/JeOQXS7dXdI8lgb+iwaF/TjvUgDAt/7XHEQA7XoiYOSbbKDdtkYojNNe+8VIdoLFRrphMA9XAyw442lx26zvkybjOAz1JA1WQzXrbvmwk8MppzzRQgiwXa5aMu95KWmfz/nhLK/tRUILs2MWam9w15fmkaz9wrXHpg4w186lqyC3jxs2l14jkh3lrwAfZLDVkzZTlPfK88Qxy54NslroJtG7bPwea56K5aMAH2Sy1FmHLfQO6BWZz4T3zEamBGsiKsO1c2nfXObKOVOgcm4U+GsBl+y8qFv8O2W0R+VEd3y6fOu/1m56Z+IO+81L/naaaJTtOCpftRdg/giWfCFxA5atNPu57q2f71YN9GutUtF+XkCXbCwhctn0JuOh3yG5C21JF/IG89j9G2E7a+f5+oeu4DYFsARbWyw+9ENIUsJdCRjD9TG3sd72AbBbMT07K+wg6s2wSAtkibPtKZPH7YwFZTGxN6BOHkgE+S9CeytTWmiyBsVoyM5qERZsLLwoW/OdcsvGD70V5ri2QHYBYM+CSWJvMuouCBf/Bkj8tTb4mkM19Yc0vNbPtJGxp8pfyyFUjK3knv3sqGRnhpglkc89Y83qEcedIEoBx1bajmCOPvXgK/hZKs9rW4cJ1SXDWqpz2VANNlmzbqDPJZW+m9lTEzHYQZBddWoS5+PV46dMgCLIT2iJquy5dF4uVrw1kLJhft5tFL1YJSxesDWT+80y24gXMgGB0jVzuzDp/Z70AkP8HJI9j3FzjTAQAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\sqrt{\\frac{- L^{2} w + \\overline{C}}{r}}$"
      ],
      "text/plain": [
       "     _______________________\n",
       "    ╱    2                  \n",
       "   ╱  - L ⋅w + \\overline{C} \n",
       "  ╱   ───────────────────── \n",
       "╲╱              r           "
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Retreive the positive solution\n",
    "isocost = isocost_sol[0]\n",
    "isocost"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plot the iso-cost"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Let's plot a iso-cost with $\\overline{C}=5, w = 1, r = 1$. \n",
    "\n",
    "Note the use of different brackets in the subs method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAX4AAAENCAYAAAAIbA6TAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjEsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy8QZhcZAAAUYElEQVR4nO3de3hU9Z3H8c83yZAJuUHIVSAISQDlomK8FXS33kqLbdWnrdVqL2rdp7b10t3utrvdfdpu7T5Pd2u7tLZdam1l7VrtU7e6SmspigiiEClyByOCXM0NcgFymeS3fyQISsCJMHPOzO/9ep55SOAM58t54D2HM2fOMeecAAD+yAh6AABAchF+APAM4QcAzxB+APAM4QcAzxB+APAM4QcAz7xr+M0sMxmDAACSI549/vsTPgUAIGniCX+mmX37nT9pZvlm9tsEzAQASKCsOJa5WdIfzOxW59z9kmRmZ0l6VNIjCZqL60gAwNBYvAu+a/idczEzu1bSYjPbLWm0pH+SdKtz7s/vfUYAQBDs3S7SZmY/kbRG0j5JcyWtlfQp59ybCZyLPX4AGJpTt8cvabWk6ZKmSYpImijp52a2VtJa59xv3tOIAIBAvOse/zFPMBujIy8EU51zNyVgLvb4AWBo4t7jH3L4kySUQwFAiMUdfj65CwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4BnCDwCeIfwA4JmEh9/Moma2wsxeMbP1ZvatRK8TAHB85pxL7ArMTFKuc67DzCKSlkq60zn34gmeltihACD9WLwLZiVyCkly/a8sHQPfRgYehB0AApKUY/xmlmlmqyU1SFronHtpkGVuM7M6M6ubMmVKMsYCAC8lJfzOuV7n3NmSxkg638ymDrLMPOdcrXOuNicnJxljAYCXknpWj3Nuv6TFkmYnc70AgCOScVZPiZmNGPg6R9LlkjYler0AgMEl/M1dSRWSHjSzTPW/0DzqnHsyCesFAAwiGWf1rJF0TqLXAwCID5/cBQDPEH4A8AzhBwDPEH4A8AzhBwDPEH4A8AzhBwDPEH4A8AzhBwDPEH4A8AzhBwDPEH4A8AzhBwDPEH4A8AzhBwDPEH4A8AzhBwDPEH4A8AzhBwDPEH4A8AzhBwDPEH4A8AzhBwDPEH4A8AzhBwDPEH4A8ExW0AMMprOnV1vebFeGmbIyTJkDj0iGaXh2lqKRTGVmWNBjAkBKMudc0DMcI7uixlV85ofH/Hw0kqHOnr7+ZbIylDMsU+edPlJvNB9SYU5EBTlZKsiJqCAaUUVhVHnRLI3KzVZJ/jCNys1WcV628qKhfK0DgJMV995wKCs4rmi4fnDDDMX6+tTnnGK9Tr19Tl2xPnX29OrQwKOzu1c5wzKVYabWQz3avb9Tm/a2q/VQj0xSW2fsbb/vBeOLtGF3m8oLo6oYkaOKgqgmluerKDeiyqLhqizKVXHeMJnxvwkA6SuU4S/IiWjO9IqT+j16evvUcqBbje1daj7Qrab2LnV0xXRGRYH2tB7SntZObdzTppXbW7S18cBbz8uJZGpWzSgNy8xUVUmuqkrzVFWSp/HFw5WbHTnZPxoABC6U4T8VIpkZKiuIqqwgesLlOnt6tXPfIe1oOag3Bh5vtnVq3a5W/WHdHvUNHAkbX5wr55wmlxdockW+po8p1OTyAlUURvkfAoCUkrbhj1c0kqnq0jxVl+Yd82tdsV5tbz6o1xo69GpDuzbuademve16esNenTeuSCu2tag4L1vTxxQe9Rih4rzsAP4kABCfUL65W1tb6+rq6oIe47gOdse0eW+71u5q1Ss7WrVm537VN3bo8KacVJ6vKRUFqj29SOePH6mqkjz+VwAg0eKODOE/RQ50xbRuV6s27GnVitf3aeW2FjV1dEuSRg6PaFZ1sc6pHKmLa4pVXcoLAYBTLjzhN7OxkuZLKpfUJ2mec+4/T/ScVAz/OznntK35oFa+3qIV21rU1NGlxZsbJUllBdmaWV2si2uKNbO6WKX5J34fAgDiEKrwV0iqcM6tMrN8SS9Luto5t+F4z0mH8A9mR8tBLatv0vP1TXqhvkn7DvZo2phC9fU5XXZGmS6bXKppowuVwYfTAAxdeMJ/zArNHpf0Y+fcwuMtk67hP1pfn9OGPW2q29aiJ9fs0ao39qnPSSX52bp0UolmT6vQzKpiDcviqhoA4hLO8JvZ6ZKWSJrqnGs73nI+hP+dWg50a/HmBi3a2KBtTQe0fk+b8qNZuuLMMs2ZVqFZNcXKzsoMekwA4RW+8JtZnqTnJN3jnHtskF+/TdJtklRZWXnu9u3bkzJXGHX2xPTCa816as1eLdywV22dMeVnZ+mGCys1s6r/fQGuVQTgHcIVfjOLSHpS0tPOuXvfbXkf9/iPpzvWp2WvNWnBmj3a2nRAL2/fp7KCbF199mhdM2O0JpcXBD0igHAIT/it/7zFByW1OOfuiuc5hH9wnT29emZTgx5btUuLNzco1ud04YRRmj2lTNfMGKPCHC4pAXgsVOGfJel5SWvVfzqnJP2jc27B8Z5D+N9dc0dX/5vC2/fp8Vd2KxrJ0Jxpp+mGCyo1o3IEnxMA/BOe8L8XhH9o1u1q1cMr3tDjq3eroyumD5xZpr+aVKprzhmtnGG8IQx4gvD76EBXTP/3ym4t3tyoP67fqxHDI7r+/Ep9+qJxqijMCXo8AIlF+H3mnNPKbfv0y2Wv6+n1e2VmuuH8sbruvEpNHV0Y9HgAEoPwo9+OloOav3yb1u9u0wuvNevimmJ98f3VumB8Ee8DAOmF8OPt2jp79NCL2/XA0tfV1NGtGZUjdPcVEzWrupgXACA9EH4MrrOnV4/W7dADS19XR1dMY4uG6++unKT3VY3iBQBIbYQfJ9Yd69Vjq3Zp7qJXtbu1UxdOKNJXPzBJ544rCno0AO8N4Ud8Ont69fCKN3Tfs6+pqaNLH5parjsur+ETwUDqIfwYmoPdMc1fvl3L6pu0rL5J151Xqa9cMVEl+dxGEkgRhB/vzf6D3Zq7qF7zl29TNJKp299fpZtnjlc0wgfBgJAj/Dg5Wxs79N0Fm/TnjW9qVvUo3TxrvC6dXBb0WACOL+7wc5cPDGpCSZ7u/0ytHv78Bdrb1qWbf1Wn2+bXadf+Q0GPBuAkEX6c0EVVxVpwx8X6h9mTteTVRl3+/ef06xe3K9bb9+5PBhBKHOpB3HbuO6j/eHqzFqzbqzPK8/W9j52lSeX5QY8FoB+HenDqjRk5XD+47mzd+4mztGPfIV31o+c1d9Gr6mHvH0gphB9DYma6avppWnj3Jfrg1ArNXbRFX3joZdU3tAc9GoA4EX68J6PysjX3+nP0s5tqteqN/Zozd6n+e/k2hfHQIYC3I/w4KZefUaY/3nWxLpwwSv/8+Hrd8mCdmju6gh4LwAkQfpy00vyofvW58/TND5+p1oM9mjN3qVZuawl6LADHQfhxSpiZPjtzvL599RRFIxn65LwX9fMlWzn0A4QQ4ccpNeW0Qj3x5Vm64owy3bNgo27/n1VqO9Qd9FgAjkL4ccoVRCP66Y0z9I05Z6ilo1vX/nS5tjcfCHosAAMIPxLCzHTrxRN01+UT1dTRpavvW6YXtzYHPRYAEX4k2EVVo/T722eqKHeYbvrFS3pk5RtBjwR4j/Aj4U4vztVjt8/UhRNG6V8eX6f7nn2VN32BABF+JEVhTkS//Ox5umXWBP3701v09cfWqreP+ANByAp6APgjKzNDX/3AJGVmmH70TL3aO2O697qzlJ3FTV6AZCL8SCoz099eOUmFORF956mNauvs0c9uPFe52fxVBJKFQz0IxK0XT9D3PjZdy+qbdOMvXuJcfyCJCD8C84nasfrJp85VNCtDn35gpTq6YkGPBHiB8CNQs6eW63Mzx2vtrlbd+uBKdfb0Bj0SkPYIPwJ35ZRy3fuJs/TS6y36wkMvqzvGjV2ARCL8CIWPnj1a91w9Tc9ubtTdj6zmVE8ggTiVAqFxwwWVOtgd03ee2qiCaJa+e+00mcV9G1EAcWKPH6HSf32fGq3euV8/fqY+6HGAtET4ETp3XlajyeUF+v7CLfrT+r1BjwOknYSH38weMLMGM1uX6HUhPZiZ/u3aaZo+plB3P7Jam/dyI3fgVErGHv+vJM1OwnqQRqKRTM27qVbDs7P0+fl12neAD3gBp0rCw++cWyKJG7BiyMoLo/qvm87V3tZOfenhVYr1cponcCqE5hi/md1mZnVmVtfY2Bj0OAiJGZUjdc81U7Wsvlnf/9PmoMcB0kJowu+cm+ecq3XO1ZaUlAQ9DkLk47Vj9aX3V+nRup282QucAqEJP3AiX76sRmUFUX3tsbVqaO8MehwgpRF+pITsrEzNvf5sHeiK6au/XcMdvICTkIzTOR+WtFzSJDPbaWa3JHqdSE/Vpfn6xpwz9NyWRj26cmfQ4wApK+GXbHDOXZ/odcAfN144Tuv3tOlfn9qgWROLNXpETtAjASmHQz1IKWamL/51tWJ9ffrmE+uDHgdISYQfKWds0XDdfflELdzwpp7mLB9gyAg/UtLNs8Zrcnm+vvnEeu7cBQwR4UdKimRm6J5rpik/mqWHXtwe9DhASiH8SFnnjhupSWX5mrvoVbVwLR8gboQfKe2Oy2p0qKdX85ZsDXoUIGUQfqS0mrJ8XTX9NM1fvk3NHV1BjwOkBMKPlHfnZdX9e/3Ps9cPxIPwI+VVl+brw9NP0/wXtrPXD8SB8CMt3HFZjTpj7PUD8SD8SAvVpXn6yFn9e/1N7PUDJ0T4kTa+fGmNopEMPf6XXUGPAoQa4UfaqC7N07njRupnS7Zym0bgBAg/0srHa8eqsb1Lz23h9p3A8RB+pJVLJ5dqVO4w/baO6/UDx0P4kVYimRm65pzRWrTpTS7jABwH4Ufa+XjtWPX0Ov2eN3mBQRF+pJ1J5fmaM61Cy7c2Bz0KEEqEH2lpQkmuntnUoNZDPUGPAoQO4UdaumRiiXr7nF6obwp6FCB0CD/S0tljRyg/O0tLXuW0TuCdCD/SUiQzQ++rHqUlW5rknAt6HCBUCD/S1iUTS7Rr/yG91ngg6FGAUCH8SFuX1JSoND9br+zYH/QoQKgQfqStsUXD1eecXnqd0zqBoxF+pLWqkjwO9QDvQPiR1qpL81Tf0MEbvMBRCD/SWlVJnloP9aiZ6/YAbyH8SGvVpXmSpPqGjoAnAcKD8COtVQ2E/7VGwg8cRviR1k4rjGr4sEz2+IGjEH6kNTNTVUmemrkBO/AWwo+0V5KfrfoGTukEDiP8SHt52Vk60B0LegwgNJISfjObbWabzazezL6WjHUCh+VmZ+lAF+EHDkt4+M0sU9J9kj4o6UxJ15vZmYleL3BYfjRLHYQfeEsy9vjPl1TvnNvqnOuW9BtJH03CegFJUu6wLHX29CnW2xf0KEAoZCVhHaMl7Tjq+52SLkjCegFJ0rUzRuvCCUUys6BHAUIhGeEf7F/bMRdOMbPbJN028G2Xma1L6FSpo1gS9w9kOxyNbXEE2+KIdc65qfEsmIzw75Q09qjvx0ja/c6FnHPzJM2TJDOrc87VJmG20GNb9GM7HMG2OIJtcYSZ1cW7bDKO8a+UVGNm481smKRPSnoiCesFAAwi4Xv8zrmYmX1J0tOSMiU94Jxbn+j1AgAGl4xDPXLOLZC0YAhPmZeoWVIQ26If2+EItsURbIsj4t4Wxg0qAMAvXLIBADwTqvBzaYd+ZvaAmTVwSqtkZmPN7Fkz22hm683szqBnCoqZRc1shZm9MrAtvhX0TEEzs0wz+4uZPRn0LEEys21mttbMVsdzdk9oDvUMXNphi6Qr1H8K6EpJ1zvnNgQ6WADM7BJJHZLmx3tebroyswpJFc65VWaWL+llSVd7+vfCJOU65zrMLCJpqaQ7nXMvBjxaYMzsK5JqJRU4564Kep6gmNk2SbXOubg+0xCmPX4u7TDAObdEUkvQc4SBc26Pc27VwNftkjaq/9Pg3nH9Dt9RJjLwCMeeWwDMbIykOZLuD3qWVBOm8A92aQcv/4FjcGZ2uqRzJL0U7CTBGTi0sVpSg6SFzjlvt4WkH0r6e0lchKl/B+BPZvbywFUQTihM4Y/r0g7wk5nlSfqdpLucc21BzxMU51yvc+5s9X8C/nwz8/JQoJldJanBOfdy0LOExEzn3Az1XwX5iwOHi48rTOGP69IO8M/A8ezfSfq1c+6xoOcJA+fcfkmLJc0OeJSgzJT0kYFj27+RdKmZPRTsSMFxzu0e+LFB0v+q/9D5cYUp/FzaAccYeEPzF5I2OufuDXqeIJlZiZmNGPg6R9LlkjYFO1UwnHNfd86Ncc6drv5WPOOcuzHgsQJhZrkDJz7IzHIlXSnphGcEhib8zrmYpMOXdtgo6VFfL+1gZg9LWi5pkpntNLNbgp4pQDMl3aT+PbrVA48PBT1UQCokPWtma9S/o7TQOef1aYyQJJVJWmpmr0haIekp59wfT/SE0JzOCQBIjtDs8QMAkoPwA4BnCD8AeIbwA4BnCD8AeIbwA4BnCD8AeIbwA0CKM7O/MbP74l2e8ANA6psuaW28CxN+AEh90zSE8HPJBgBIcWbWImm8c641nuXZ4weAFGZmYyW1xxt9ifADQKob0vF9ifADQKob0vF9iWP8AJDSzOzX6r8TW/vAT+1xzl10wucQfgDwC4d6AMAzhB8APEP4AcAzhB8APEP4AcAzhB8APEP4AcAzhB8APPP/sIIj/pOHPCgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "isocost_plot1 = plot(isocost.subs([(Cbar,5), (w,1), (r,1)]), (L,0,5), xlim=[0,5], ylim=[0,5], xlabel=\"$L$\", ylabel=\"$K$\", show=False)\n",
    "isocost_plot1.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3. NonIinear Iso-cost (with monopsony power)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Define the iso-cost curves"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, define an expression for cost."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle r \\log{\\left(K \\right)} + w \\log{\\left(L \\right)}$"
      ],
      "text/plain": [
       "r⋅log(K) + w⋅log(L)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Kcost = r*ln(K)\n",
    "Lcost = w*ln(L)\n",
    "cost = Kcost + Lcost\n",
    "cost"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, the iso-cost curve is such that the cost is equal to the fixed level of cost $\\overline{C}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAAAmCAYAAADHjLWDAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAEfElEQVRoBe2bjVHcMBCFOSYF3CQdkA6ADnLpACYdHB3AXAUZ6ABSAnQAHRDo4OggCR2Q9ymSRxY6WwpibIM0o3i9Wu/f08qSL8yenp62SrTZbJalSHZnJey+Bx3bSu4BCQ76Mjd4kp7Tc/W/B3lhcB7gAC7zbRv8VZDgi5cmRcqPvb4UzWQ4f6ne1Odla65+/Vo2pXehvuv7o/tTdRLLtTWGnHjkZO4/49PC4MjHQWOPbvxAxKUGt0p06dpRv+bq69M9du583mvTsrdQPy9tRzoBYBnqtfbWId+/z/FHsn/Um0oS/fJmZwkAncixh0Djje7pb6GtFF9stQG8vhgvlaes18mHwhnD+L0CuA/1ikfpnoT8lHsFRRVSoVT7me6hWRpORJ+KvhV9xXhXkyxJpLqYQOi40XPGV43BZ5yxb+pMNuy1wLC2wwmIKO2r+qWhNvwjfTfSgUxL7wZxwy4NEk6a4LqM/ucYiUE/jYT+MtQ/exvXeSvD+wAZlvXPHu9O/C/iMYF4X+5Bi7cSDYAxMAByrR5r+2IexQYCXq+/vnxpkAiW/qwp8GMFfeYGdM9M7qosEmqWDl2vJE8S3QQALH8T8qxynR3vSgLDpP8WD75bitkMOJuhrFP10RH+1cYzD4EVvxW3fSaqw9fn06VB+i7lP9RbpSxHWa5aa7UNJmXWOX9JpgOV2fzTDrBBael2DwRX5M2S6fG5d3qoDoDZl749TyYkkY/5jU8tPwAo5FllTI7kVhQkBXcvx3hPMMsJmqqitNnib5qZGk5q6GQrj07olWhstBKje9M0hl2SuSN61/p2JJrEOb94r0G7xruC5Y4kUsmtyYaQ1dMCW/JMwhXjVj8k1c4EalYPmBoHzJTKR7xpGMChIlvwKepR/CSu2aqLBmAmwiIWj/hskKJjMXmfh17/vouWrNmCF60kKZ1kU6JYAda2IlwMnHeiVSr+BRWj/iA6eYWw+gE/q1WQbLqU7Nay1JdF5JV0Ki6nsWP0l9ekZ4uBJIezPrAmeTcBIcWd5WUoL9B6FRQDKcVYVjRVuMnAdkNVYrQZSKoku/ayxXSnfE7tfJ/LXl9Hm4kRO9YLkgBayH92JIcCxezvxWPLCmjucClyXE0+sk3mUHpnPTMvecWQtUEYQ1SdIClQDm18iuHQ5x/AAM59ohlDHDEf2BoDDN/jOMQSC5Nrcq0TJEXj9vT8gMZJ/ZON8FqBR88QY8kA/slnKt1Uu+4BLflMM5Y48KMPJCqGTzqTWyJskvkO568Alj2tS8rubpKzT1XEUjfqak+dKn0gbQRISeCb32ibKuhR/XC0DmY41gcSv3qy5DWNGarOu2ryy0gT1MiJzneSZiIfEgEFsNwZia/lsd9TRh7qdN3rBImwBMhUNw3TRSXwvG+5C8Tr7RAZqCANkfVMmxWkzIQNIV5BGiLrmTYrSJkJG0K8gjRE1jNtVpAyEzaEeAVpiKxn2qwgZSZsCPEK0hBZz7RZQcpM2BDiDqTw72az/shpCMffok1+XVBv/n5ZMZr/l/EXKrSxLLpKYhUAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\left[ e^{\\frac{\\overline{C} - w \\log{\\left(L \\right)}}{r}}\\right]$"
      ],
      "text/plain": [
       "⎡ \\overline{C} - w⋅log(L)⎤\n",
       "⎢ ───────────────────────⎥\n",
       "⎢            r           ⎥\n",
       "⎣ℯ                       ⎦"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "isocost_sol = solve(cost - Cbar, K)\n",
    "isocost_sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle e^{\\frac{\\overline{C} - w \\log{\\left(L \\right)}}{r}}$"
      ],
      "text/plain": [
       " \\overline{C} - w⋅log(L)\n",
       " ───────────────────────\n",
       "            r           \n",
       "ℯ                       "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Retreive the first and only element in the solution, which is our expression for the budget constraint.\n",
    "isocost = isocost_sol[0]\n",
    "isocost"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The slope and intercept of the iso-cost curve is better seen after expanding the expression."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle e^{\\frac{\\overline{C}}{r}} e^{- \\frac{w \\log{\\left(L \\right)}}{r}}$"
      ],
      "text/plain": [
       " \\overline{C}  -w⋅log(L) \n",
       " ────────────  ──────────\n",
       "      r            r     \n",
       "ℯ            ⋅ℯ          "
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "isocost.expand()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plot the iso-cost"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Let's plot a iso-cost with $\\overline{C}=5, w = 1, r = 1$. \n",
    "\n",
    "Note the use of different brackets in the subs method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "scrolled": true
   },
   "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": [
    "isocost_plot1 = plot(isocost.subs([(Cbar,5), (w,1), (r,1)]), (L,1,100), xlim=[0,100], ylim=[0,100], xlabel=\"$L$\", ylabel=\"$K$\", show=False)\n",
    "isocost_plot1.show()"
   ]
  },
  {
   "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",
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