{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Revenues and costs \n", "## Prof. Rabanal [Econ 133]\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A firm's revenues is defined as $R= p \\times q$. We know that inverse demand is $p = a - b q$ where $a,b>0$. Therofore $R$ is a function of $q$ or R(q).\n", "\n", "### marginal revenue\n", "We are mostly interested in marginal analysis. Therefore, we compute how revenues changes, or \n", "\n", "\\begin{align}\n", "\\Delta R = \\Delta q \\times p + \\Delta p \\times q \n", "\\end{align}\n", "\n", "The equation above tell us that revenues changes via q and p! Now, given that we are interesed in figuring out how our change of production affects the change of revenue, we divide everything by $\\Delta q$\n", "\n", "\\begin{align}\n", "\\frac{\\Delta R}{\\Delta q } = p + \\frac{\\Delta p}{\\Delta q} \\times q \n", "\\end{align}\n", "\n", "Given our demand function, we know that the slope or $\\frac{\\Delta p}{\\Delta q}= -b$, \n", "\n", "\\begin{align}\n", "\\frac{\\Delta R}{\\Delta q } = p - b \\times q = a - 2bq\n", "\\end{align}\n", "\n", "***Assuming a linear demand function, then the slope of MR is twice as the demand***\n", "\n", "### revenues and elastiticities ($\\epsilon$)\n", "\n", "we will see that when $\\Delta q >0 $, MR is positive when $|\\epsilon| = |\\frac{\\Delta \\% q }{\\Delta \\% p }| >1$. The idea is that you will increase revenues if and only if the change of prices ($\\Delta p$) is NOT greater than the change of production ($\\Delta q$). More formally, we can use our expression before\n", "\n", "\\begin{align}\n", "\\frac{\\Delta R}{\\Delta q } = p + \\frac{\\Delta p}{\\Delta q} \\times q = p \\left ( 1 - \\frac{1}{|\\epsilon|}\\right)\n", "\\end{align}\n", "\n", "Note that $MR = 0$ when $\\epsilon = 0$. That is, when the (percentage) change of prices is the same as the (percentage) change of quantity, then the revenues do not change! Perhaps a graph can help to illustrate the relationship between revenues and elasticities. \n" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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Tg1CtiATSsu2/8NjEZaz58QDdG1fkz93rUbZoAb/LCltmtijL1P45+bYF4Zzb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\nGDPrQObWwwiFg/hJWxAiIpI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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "## Call libraries\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "%matplotlib inline\n", "fig1, ax = plt.subplots()\n", "q = np.arange(0.,101, 1)\n", "pd = 200 -2*q\n", "ax.plot(q,pd)\n", "plt.ylabel('price')\n", "plt.xlabel('quantity')\n", "plt.text(50, 100, r'A')\n", "fig2, ax2 = plt.subplots()\n", "ax2.plot(q,pd*q)\n", "plt.text(50, 5000, r'A')\n", "plt.ylabel('revenues')\n", "plt.xlabel('quantity')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " Note that at the left of A (q=50), the revenues are increasing, but a decreasing rate. This is due to the effect of cutting prices for the units the firm was selling before. The only way to convince people to buy more of the product is by cutting prices. Therefore, every extra unit the firm wants to sell has a negative effect on revenues. This negative effect becomes more important as the firm produces more units. Thus, at some point, the revenues decrease. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### costs\n", "\n", "- We decompose the costs in two types: one that **does** depend on the units produced, and other that **does not** depend on the units products. The former is called *fixed* (FC) and the latter *variable* (VC)\n", "\n", "\\begin{align}\n", "C = FC + VC \n", "\\end{align}\n", "\n", "- Similar to the case of revenues, a key variable to determine the optimal units of production is the change of costs, or \n", "\n", "\\begin{align}\n", "\\Delta C = \\Delta FC + \\Delta VC = \\Delta VC\n", "\\end{align}\n", "\n", "\\begin{align}\n", "MC \\equiv \\Delta C / \\Delta q = \\Delta VC / \\Delta q \n", "\\end{align}\n", "\n", "\n", "- Therefore, assuming the change of costs is due to the change of variable costs. \n", "\n", "- Sometimes, it is handy to work in average terms (or unit costs). That is if for example the total costs are 1000 and the firm produces 200 units, then in average the cost is 5 or the unit cost is 5. This does not imply that the extra cost ($\\Delta C$) to produce every single unit was 5. 5 is just the average cost!\n", "\n", "\\begin{align}\n", " AC \\equiv C/q = FC/q + VC/q = AFC + AVC\n", "\\end{align}\n", "\n", "- Note that AFC is decreasing on q. AVC will depend on the shape of MC \n", "\n", "- For example is MC is constant then AVC is constant. If MC is increasing (decreasing), then AVC is increasing (decreasing). \n", "\n", "- Perhaps a graph can help to grasp the different shapes of costs " ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/Users/jrabanal/nota/lib/python2.7/site-packages/ipykernel_launcher.py:7: RuntimeWarning: divide by zero encountered in divide\n", " import sys\n", "/Users/jrabanal/nota/lib/python2.7/site-packages/ipykernel_launcher.py:8: RuntimeWarning: invalid value encountered in divide\n", " \n", "/Users/jrabanal/nota/lib/python2.7/site-packages/ipykernel_launcher.py:9: RuntimeWarning: divide by zero encountered in divide\n", " if __name__ == '__main__':\n", "/Users/jrabanal/nota/lib/python2.7/site-packages/ipykernel_launcher.py:14: RuntimeWarning: divide by zero encountered in divide\n", " \n", "/Users/jrabanal/nota/lib/python2.7/site-packages/ipykernel_launcher.py:16: RuntimeWarning: invalid value encountered in divide\n", " app.launch_new_instance()\n", "/Users/jrabanal/nota/lib/python2.7/site-packages/ipykernel_launcher.py:17: RuntimeWarning: divide by zero encountered in divide\n" ] }, { "data": { "text/plain": [ "" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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P5eL6WZclm8sW1mdz2cI067IH1Mm67AHz+XX5/eVcrlCen8+6LLlcrnBCytfP\nl+dcj3U5f+pwBywftK3LHfTKuizOuQOnR9GdF2Y9g//jsz7OwrMWHtu+dNdN/10y+zhOajyPm/7z\nZRYufpGHX3qTr1wzW1f30ifPPP+7gEi81E0Zdpxz3ScDcmRz2cJ88Ukhf6IACicYhzvwZII/dc75\n8z3q5Ofz+yw+2Rxw4nF076t4nXP+CS7ntyN/sstfGBfXybejMN9bWTB/0LJzvd5NN9DKO+h/czO8\ntXJQ32I68OAox/bjOtmyoZ1N3zZStUkm1FUQ0dW9yBEzIBK8pMiOt2Di+YP6FuUd9EPEMCbUVjCm\nMs7m3e1s29NB075OJtZVMLYmqcAXkWGtvIP+0q8P6dslgZOBl7fs4fbH1/Dn11uo3hPlI+84nuvO\nncKEOg0/LCLDj76M7YcXNu/m7mfe4DertgNwwcmNfOjsyVw8cyyJqP5AFZHBpbFuhtCWXe0s+etm\nfrV8G2/t66QmGeXimeO4bPZ4zp/eMOLuwxeRoaGgL4FszvHs+p08+OI2nny1idbODJXxCO88qYEL\nZzRy4fRGjq8/9mfUiogU0+2VJRDxjAtObuSCkxtJZXL8+fWdPLm6iT+ua+Z3rzYBMLGugrdPHcM7\npo7h7BNGc1JjFZ6nL3NFZPAo6AdJPOoxb8ZY5s0Yi3OON3a28cxrO3nujRaeea2ZX7+4DYDqRJQz\nJtcxe2ItsybUMGtCDVPqRyn8RWTAqOumBJxzbNjZxoub9/DSlt28uHkP65paSWf9z6IiFmH6uCpO\nHlfNtLFVnNRYxUmNo5g8ppJYRCM9iohPffRlJpXJsa6pldVv7mNtUyvrmlpZ81Yrza1dhToRz5hY\nV8EJ9ZUcP6aSyWMqmTy6komjK5hQl6RhVEJ/CYiMIOqjLzPxqMfsibXMnlh7QPnejjRv7Gzj9R37\n2djSxsaWdja1tPHoyu3saU8fuI+Ix3G1SY6rSTKuNslxNQnGVicZW5OgsTpBY5U/ra3QEL0iI4mC\nfpirrYgxZ3IdcybXHbSutTPN1t0dbNvdwZt7O9i2p4O39nayfW8nK7bu4bd7O+nKHDyaYtQzxoyK\nU1+VoH5UnNGj4oypjFFXGWd0ZYzRo+LUVvjLdRUxaitiVCejRNVtJFKWFPRlrDoZ49TxMU4dX9Pr\neucc+zozNLd2smNfF837u9i5P8XO/V3s2p+ipa2LlrYUW3e3s7s9zd6OdK/7yatKRKlJRqlO+sFf\nHcxXJaP6QvJ1AAAJxUlEQVRUJ6JUJaJUJYNpIkplIkpVIkJlPMqoeJTKRITKeISKmMZPFxlKCvoQ\nMzNqgyvyaWOrD1s/k82xtyPNno40e9pT/ny7/2rtzLC3wz8ZtHb6y837u3hjZxutnRlaOzOkskc2\nFruZ/4VzZTxCRTxCZSxKMh6hMuYvV8QiJGMRkjGvMF8Rj5CIeiRj3dP8fCLqkegxH494JGKeP416\nOrHIiKagl4JoxPO7c6oSx7R9KpOjrSvD/q4MbalMMJ+lvStDWypLW1eG9lSWjpQ/bUtl6UxnaQ+W\nu9I5mlu7aE9l6Ezn6Exn6Uj7dXL9vGcgHvGIR4NXxCMWtaAsEpQZsYhHLFJUJ19WtBwN6sQ8Ixb1\niHrd20Uj5tfxvMI0Guwj6hnRoCwSbONP/X1GPQteHpGIPx8JynSSkv4atKA3s0uAf8MflfTHzrmh\nHYFMhpwfpH6f/0ByzpHOOjozfuh3BSeBrkyOrkyWzrQ/7Urn6MxkSWVypDK5YH33fCqTI5XtXp/O\nOroyOdLZ7jptXRlSWVcoy2RzpLKOVCZLJueX52+DHSqRotDvOR/1PDyPwgkk6hmeddcrvMw/0eTX\neda9D88zIgYRzyPiUVh/4JSg3uHLPfP3Ydb93mbdx2H5OtZdJ78fr8e6nuX+1N+fZ4bn9ahbvN7r\nLrOidZ5RqF+8XW91wmJQgt7MIsCdwN8AW4G/mtnDzrnVg/F+Em5mRjxqxKMeNclYqZuDc45MzpHJ\nOlJZ/2SQDk4Ofrm/nMkF02yObM6RDtblt83kcmSyLljn1ymU5xzZrL9NLucK+806v35+ffGyX8/f\nTzbnyDrI5veby9GZ8ev421Con3N0bxOs767ncPn1ReXD4K7sIdEd/gSPBeyxbP44+/mTkVF8EuHA\nMi//dCm/nGA6/22T+dS7ThzU4xisK/q3A+udcxsAzGwJcBWgoJeyZ2ZBtw5UjNDHaPhPgvJPADnX\nfYJwOYpOEMHJwfknlXzdXL6slxOJ61FemMefd86Ry9HrfiA/T/BUKQ7YZ67H/inss7uey2+Pv29X\nvI3zn0jloNA+/ylR3f8eOee3Nd/O/H5ywbb5NuePBwcNx9hVejQGK+gnAluKlrcC7xik9xKRIeY/\n7NvvjpHhr2Q3RpvZ9Wa2zMyWNTc3l6oZIiKhN1hBvw2YXLQ8KSgrcM4tcs7Ndc7NbWxsHKRmiIjI\nYAX9X4HpZjbVzOLAfODhQXovERE5hEHpo3fOZczsRuAJ/Nsrf+Kce2Uw3ktERA5t0O6jd849Bjw2\nWPsXEZEjo1GqRERCTkEvIhJyCnoRkZAbFk+YMrNmYNNRbNIA7Byk5gxnI/G4R+Ixw8g87pF4zNC/\n4z7BOXfY+9OHRdAfLTNbdiSPzwqbkXjcI/GYYWQe90g8Zhia41bXjYhIyCnoRURCrlyDflGpG1Ai\nI/G4R+Ixw8g87pF4zDAEx12WffQiInLkyvWKXkREjlDZBb2ZXWJma81svZndXOr2DAYzm2xmS81s\ntZm9YmafDcrHmNmTZvZaMB1d6rYOBjOLmNmLZvZIsDzVzJ4LPvNfBgPlhYaZ1ZnZ/Wa2xsxeNbNz\nR8JnbWafD/77XmVmi80sGbbP2sx+YmY7zGxVUVmvn635vh8c+wozO2ug2lFWQV/0iMJLgZnAAjOb\nWdpWDYoM8EXn3EzgHOCG4DhvBp5yzk0HngqWw+izwKtFy7cD33XOTQN2A58sSasGz78BjzvnTgHO\nwD/2UH/WZjYRWAjMdc7Nxh/8cD7h+6zvAS7pUdbXZ3spMD14XQ/cNVCNKKugp+gRhc65FJB/RGGo\nOOe2O+deCOZb8f/Hn4h/rPcG1e4Fri5NCwePmU0CLgd+HCwbcBFwf1AlVMdtZrXABcDdAM65lHNu\nDyPgs8YfVLHCzKJAJbCdkH3WzrmngV09ivv6bK8Cfup8fwHqzGz8QLSj3IK+t0cUTixRW4aEmU0B\nzgSeA8Y557YHq94CxpWoWYPpe8D/AXLBcj2wxzmXCZbD9plPBZqBfw+6q35sZqMI+WftnNsGfAvY\njB/we4HlhPuzzuvrsx20fCu3oB9RzKwK+BXwOefcvuJ1zr9dKlS3TJnZFcAO59zyUrdlCEWBs4C7\nnHNnAm306KYJ6Wc9Gv8KdiowARjFwV0coTdUn225Bf1hH1EYFmYWww/5nzvnHgiKm/J/ygXTHaVq\n3yA5D/hbM9uI3y13EX7/dV3w5z2E7zPfCmx1zj0XLN+PH/xh/6wvBt5wzjU759LAA/iff5g/67y+\nPttBy7dyC/oR8YjCoF/6buBV59x3ilY9DFwXzF8HPDTUbRtMzrlbnHOTnHNT8D/b3zvn/gewFPhg\nUC1Ux+2cewvYYmYzgqL3AKsJ+WeN32VzjplVBv+95487tJ91kb4+24eBa4O7b84B9hZ18fSPc66s\nXsBlwDrgdeDWUrdnkI7xfPw/51YALwWvy/D7q58CXgN+B4wpdVsH8d9gHvBIMH8i8DywHvhPIFHq\n9g3wsc4BlgWf94PA6JHwWQNfBtYAq4D/ABJh+6yBxfjfQaTx/3r7ZF+fLWD4dxW+DqzEvyNpQNqh\nX8aKiIRcuXXdiIjIUVLQi4iEnIJeRCTkFPQiIiGnoBcRCTkFvchRMrPPmVll0fJjwQiUdWb2mVK2\nTaQ3ur1S5CgFv9yd65zb2aN8Cv69/7NL0CyRPumKXkLHzG41s3Vm9qdgnPObzOwPZjY3WN8QhDVm\nNsXMnjGzF4LXO4PyecE2+XHifx78YnEh/tgsS81saVB3o5k1AF8HTjKzl8zsm2b2UzO7uqhdPzez\n0I22KsNf9PBVRMqHmZ2NP3zCHPz/vl/AHxWxLzuAv3HOdZrZdPxfMs4N1p0JzALeBJ4FznPOfd/M\nvgC8u+cVPf5gZLOdc3OCtlwIfB54MBiO+J10//RdZMjoil7C5l3Ar51z7c4f8fNwYyHFgB+Z2Ur8\nn9wXP8jmeefcVudcDn8YiilH0xDn3B/xx2ZqBBYAv3LdQ/CKDBld0ctIkaH7wiZZVP55oAn/yU4e\n0Fm0rqtoPsux/f/yU+Cj+H9lfOIYthfpN13RS9g8DVxtZhVmVg1cGZRvBM4O5j9YVL8W2B5ctX8M\n/5F2h9MKVB9h+T3A5wCcc6uPYN8iA05BL6Hi/Ecw/hJ4GfgN/tDW4D/N6NNm9iLQULTJD4DrzOxl\n4BT8B38cziLg8fyXsUXv3QI8Gzzs+ptBWRP+oyD//diPSqR/dHulhJqZ3Qbsd859q0TvX4k/5OxZ\nzrm9pWiDiK7oRQaJmV2MfzV/h0JeSklX9CIiIacrehGRkFPQi4iEnIJeRCTkFPQiIiGnoBcRCTkF\nvYhIyP1/X6NkRKyCkB4AAAAASUVORK5CYII=\n", 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YcT/krYM+U+DKVyCor95RuTW3S/QOowmhRvSK0vE47LDhNUj/Mxi9YM5rMGq+WhPfDtxq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\nsROjExkfPl41xlaUTqwlQ0I78IiUcqs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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig3, ax = plt.subplots()\n", "q = np.arange(0.,101, 1)\n", "FC = 200 \n", "VC = 20*q \n", "VC1 = q**2\n", "MC1 = 2*q\n", "ax.plot(q,FC/q,label='AFC')\n", "ax.plot(q,VC/q,label='AVC = MC')\n", "ax.plot(q, (FC+VC)/q,label='AC')\n", "plt.xlabel('quantity')\n", "plt.legend()\n", "\n", "fig4, ax = plt.subplots()\n", "ax.plot(q,FC/q,label='AFC')\n", "ax.plot(q,MC1,label='MC')\n", "ax.plot(q,VC1/q,label='AVC')\n", "ax.plot(q, (FC+VC1)/q,label='AC')\n", "plt.xlabel('quantity')\n", "plt.legend()\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### optimal decision\n", "\n", "- A firms seeks to maximize profits ($\\pi$). We want to solve that problem using marginal analysis. \n", "\n", "\\begin{align}\n", "\\pi = REV - COST\n", "\\end{align}\n", "\n", "\\begin{align}\n", "\\Delta \\pi = \\Delta REV - \\Delta COST\n", "\\end{align}\n", "\n", "Note that if $ \\Delta \\pi >0$ then the firm is making money and make sense to increase production! Similarly, if $ \\Delta \\pi <0$, then the firm is losing money and the firm should cut production. A firm is indiffirent when $\\Delta REV = \\Delta COST$, or **MR = MC**\n", "\n", "\\begin{align}\n", "\\frac{ \\Delta REV}{\\Delta q } = \\frac{ \\Delta COST} {\\Delta q }\n", "\\end{align}\n", "\n", "- For example if $p=200 - 2q$ and $C = 20*q$. Then we know that $MR = 200 - 4q$ and $MC = 20$ . Optimal production is when $MR=MC$, or \n", "\\begin{align}\n", "200-4q = 20\\rightarrow q = 45\n", "\\end{align}\n", "\n", "- The optimal price that firm sets is when $p=200 - 2\\times 45 = 110$ and the firms profits are $(110-20)\\times 45= 4050$\n", "\n", "- In this example, we assume that there were not fixed costs. Our analysis of optimal production does not change but we need to make sure that the firm is making money. Otherwise the firm should stop operating. For example if the fixed cost are 500, then profits shrinks to 3550.\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### competitive markets vs. monopoly \n", "\n", "- A market is competitive when the firms do not have market power. That is, they take prices as given. In our formula of MR, the value of $\\Delta p$ for a competitive firm is equal to zero because that firm cannot alter the market prices. Thus, $MR = p$. \n", "- An optimal production for a typical firm is when $MR=MC$. However, now we know that in a **competitive** case $MR\\equiv p = MC$\n", "- Going back to our previous example, $p$ is determined by the market demand, \n", " \n", " \\begin{align}\n", " 200-2q = 20\\rightarrow q = 90\n", " \\end{align}\n", " \n", " - The quantitive produced in competitive markets is higher than the quantity offered by a monopolist. We will see that this translates in a DWL. \n", " " ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/Users/jrabanal/nota/lib/python2.7/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in divide\n", " \n" ] }, { "data": { "text/plain": [ "" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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+jDvusmd/KXU9WiAKqYfrPky/Ov2Yu30uPx34yeo4TsWnuAdvPNCQ74a2w9e7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\nZMr6KVbHUXaUPU3i+y3HCQy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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig6, ax = plt.subplots()\n", "q = np.arange(0.,101, 1)\n", "pd = 200 -2*q\n", "VC = 20*q \n", "ax.plot(q,pd,label='demand')\n", "ax.plot(q,VC/q,label='MC')\n", "ax.plot(q,pd-2*q,label='MR')\n", "conds = pd-2*q<=20\n", "ax.fill_between(q,pd,20,where=conds)\n", "ax.set_ylim(0, 200)\n", "ax.set_xlim(0, 95)\n", "plt.ylabel('price')\n", "plt.xlabel('quantity')\n", "plt.legend()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " - Recall that under monopoly p, q = 110, 45 . That is the CS is $(200-110) *45/2$ and PS = $(110-20) *45$. \n", " - Under competitive markets, we have PS = 0 (due to MC=AC = P no fixed cost!) and CS = $(200-20)*90$. Compared to the competitive case, the monopoly generates a DWL equal to region in blue. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### a natural monopoly \n", "\n", "- we say that a monopoly is natural if the costs of production (in particular fixed) are too expensive so that makes sense to have fewer firms in the market. For example going back to our previous example if we add fixed costs equal to 1000. Our new graph becomes \n" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/Users/jrabanal/nota/lib/python2.7/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in divide\n", " \n", "/Users/jrabanal/nota/lib/python2.7/site-packages/ipykernel_launcher.py:8: RuntimeWarning: divide by zero encountered in divide\n", " \n", "/Users/jrabanal/nota/lib/python2.7/site-packages/ipykernel_launcher.py:8: RuntimeWarning: invalid value encountered in divide\n", " \n" ] }, { "data": { "text/plain": [ "" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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PDydrx84K+4YbGR5JixotmLhjIgcTD1bIMbUSXMOJn4a2YspjDYi7kk73qM1M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\nAWGMMSau/wPW4WJaXwbCjQAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig7, ax = plt.subplots()\n", "q = np.arange(0.,101, 1)\n", "pd = 200 -2*q\n", "VC = 20*q \n", "ax.plot(q,pd,label='demand')\n", "ax.plot(q,VC/q,label='MC')\n", "ax.plot(q,pd-2*q,label='MR')\n", "ax.plot(q,1000/q+VC/q,label='AC')\n", "\n", "ax.set_ylim(0, 200)\n", "ax.set_xlim(0, 95)\n", "plt.ylabel('price')\n", "plt.xlabel('quantity')\n", "plt.legend()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- A the competitive equilibrium (q=90) we observe that the unit cost is higher that the price that a competitive firm is selling. The firm then shuts down and the only way to have a market is to have fewer producers in the market. A monopolist is still making money. In this case, $(110 - AC) * 45$. Recall that before the $AC=MC= 20$ now $AC = 1000/q + 20= 1000/45+20=42$ Therefore profits are $(110-42)*45= 3060. \n", "\n", "- A regulator perhaps want to have a situation in which the producer breaks even. This translates in having a price ceiling of $P=AC$. That is, \n", "\n", "\\begin{align}\n", "200 - 2q = 1000/q + 20 \\rightarrow 2q^2 - 180q + 1000 \n", "\\end{align}\n", "\n", "Solving the quadratic formulat we get $q\\approx 84$ Therefore the price that regulator wants to set is $200- 2*84$ (note that AC also crosses P at $q\\approx 6$ but this generates lower welfare)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([ 84.05124838, 5.94875162])" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.roots([2,-180,1000])" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/plain": [ "" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.core.display import HTML\n", "def css_styling():\n", " styles = open(\"custom.css\", \"r\").read()\n", " return HTML(styles)\n", "css_styling()" ] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.10" } }, "nbformat": 4, "nbformat_minor": 2 }