{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "notes" } }, "source": [ "# 2018-02-04 Econ 113 Analyzing Growth" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Setting up the Python/Jupyter environment" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "\n", "IPython.OutputArea.prototype._should_scroll = function(lines) {\n", " return false;}" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "%%javascript\n", "\n", "IPython.OutputArea.prototype._should_scroll = function(lines) {\n", " return false;}" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# keep output cells from shifting to autoscroll: little scrolling\n", "# subwindows within the notebook are an annoyance..." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Tools: Analyzing Growth\n", "\n", "* Economics gives us numbers: prices and quantities over time\n", "* Proportional growth—compound interest\n", " * The time over which some growth process takes place\n", " * The rate at which growth (or shrinkage) takes place\n", " * The amount that the variable grows to\n", " * T, g, y… with little t standing in for any potential moment\n", "* Don’t get snowed! Accurately assess how important or consequential things are!\n", "\n", "----\n", "\n", " \n", "\n", "## The Usefulness of Math\n", "\n", "**Listen to Richard Feynman**:\n", "\n", "\"Richard\n", "\n", ">To make calculations, the Maya had invented a \n", "system of bars and dots to represent numbers… \n", "rules… to calculate and predict… the risings \n", "and settings of Venus…. Only a few Maya priests \n", "could do such elaborate calculations…. \n", "\n", ">Suppose we were to ask one of them how to do \n", "just one step in the process of predicting when \n", "Venus will next rise as a morning star—subtracting \n", "two numbers….. How would the priest explain?….\n", "\n", ">He could either teach us the… bars and dots and \n", "the rule… or he could tell us what he was really doing: \n", "\n", ">>Suppose we want to subtract 236 from 584. \n", "First, count out 584 beans and put them in \n", "a pot. Then take out 236 beans and put them \n", "to one side. Finally, count the beans left \n", "in the pot. That number is the result…\n", "\n", ">You might say, ‘My Quetzalcoatl! What \n", "tedium… what a job!’ \n", "\n", ">To which the priest would reply:\n", "\n", ">>That’s why we have the rules…. The rules \n", "are tricky, but they are a much more efficient \n", "way of getting the answer…. We can predict \n", "the appearance of Venus by counting beans \n", "(which is slow, but easy to understand) or \n", "by using the tricky rules (which is much \n", "faster, but you must spend years in school to \n", "learn them)…\n", "\n", "----\n", "\n", " \n", "\n", "\"Muhammad\n", "\n", "**Muḥammad ibn Mūsā al-Khwārizmī (c. 780-850)**:\n", "\n", "* _Al-Kitāb al-Mukhtaṣar fī Hisāb al-Jabr wa’l-Muḳābala_\n", " * The Compendious Book on Calculation by \n", "Restoration and Balancing\n", "* Worked in Baghdad at the House of Wisdom established \n", "by the Kalif Al-Mamun\n", "* Why \"al_Khwārizmī\"?\n", " * Muhammed = \"the praiseworthy, the glorified\"\n", " * Mose = \"is born\" (Thutmose = \"Thoth is born\")\n", " * How many Muhammed ibn Mūsās were wandering \n", "around ninth century Baghdad?\n", "* Algorithm...\n", "\n", "----\n", "\n", " \n", "\n", "\"Isaac\n", "\n", "**Isaac Newton (1642-1727)**: \n", "\n", "* _Philosophiæ Naturalis Principia Mathematica_\n", " * _Mathematical Principles of Natural Philosophy_\n", "* Worked in Cambridge at the university there\n", " * Except when he fled the plague...\n", "* Arithmetic and accounting\n", "* Algebra and calculus\n", "* What-if machines—ways of doing a huge number \n", "of potential calculations all at once…\n", "\n", "----\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exponential Growth\n", "\n", "$ \\frac{dy}{dt} = g(y - a) $\n", "\n", "* g = 0.01\n", "* a = 10\n", "* Start at t = -500 with y = a + 0.00674\n", "* Does nothing for a long time—stays very near 10— \n", "then explodes\n", " * and keeps on exploding...\n", "* Let's see:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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2GW3btgXA29ubxx9/PNuxffr0oXfv3kBGi9eCBQt44oknmDRpEpDxfri6ujJp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\nvI9NSUlpiohNLisrK2SfW3PA/AU35i94Xczc7T/h0luZR7T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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# make sure graphs are displayed in main notebook\n", "%matplotlib inline \n", "\n", "import numpy as np # make sure libraries are\n", "import pandas as pd # in the namespace\n", "import matplotlib as mpl\n", "import matplotlib.pyplot as plt\n", "from IPython.display import Image\n", "\n", "plt.style.use('seaborn-whitegrid') # graphics setup\n", "figure_size = plt.rcParams[\"figure.figsize\"]\n", "figure_size[0] = 10\n", "figure_size[1] = 7\n", "plt.rcParams[\"figure.figsize\"] = figure_size\n", "\n", "a = 10 # set the initial asymptote\n", "g = 0.01 # set the growth rate\n", "array = np.zeros((1001,2)) # initialize an array\n", "exponential_function_df = pd.DataFrame(array, \n", " columns=[\"time\",\"value\"]) # create a dataframe\n", "\n", "# calculate values for the exponential function\n", "\n", "for t in range(-500,501, 1):\n", " exponential_function_df.time[t+500] = t\n", " exponential_function_df.value[t+500] = a + np.exp(g*t)\n", " \n", "exponential_function_df = exponential_function_df.set_index(\"time\")\n", "exponential_function_df.plot() # and graph\n", "\n", "plt.ylabel(\"Value\", fontsize=20) # set labels\n", "plt.xlabel(\"Time\", fontsize=20)\n", "plt.suptitle(\"Exponential Growth\", fontsize=28)\n", "plt.title(\"asymptote = 10; growth rate = 0.01\", fontsize=16)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Math presentation can be ugly inline**:\n", "\n", "* dy/dt = g(y - a)\n", "* y = 10 + exp(0.01(t-10))\n", "\n", " \n", "\n", "**Or prettier**:\n", "\n", "(1)          \n", "$ \\frac{dy}{dt} = d(y - a) $\n", "\n", "(2)          \n", "$ y = 10 + e^{0.01(t-10} $\n", "\n", "----\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Rule of 72 for Exponential Growth**:\n", "\n", "* Rules of thumb for a growth rate g:\n", " * (y-a) compounded continuously doubles every 0.693/g periods\n", " * (y-a) grows a thousandfold every 6.91/g periods\n", "* But we use the _Rule of 72_:\n", " * Why? 72 has many divisors\n", " * Why? 69.3—ln(2)—is for continuous compounding\n", " * Let's see:\n", "* Why is the thousandfold time ten times the doubling time?\n", " * 1024 = 2^10\n", " * ln(1024) = 10 * ln(2)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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6cUtMTCQyMpLmzZsD0L59ezw8PLJ+njp1qjXjiYiIFBq/no5l4voQIq4m0/HR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\nncvo0aNZsmQJhmGwceNGS5ZY6n47dJGHPgjjq21xPN2mDht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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "array = np.zeros((21,2)) # initialize an array\n", "rule_of_72_df = pd.DataFrame(array, \n", " columns=[\"rate\",\"doubling_time_times_rate\"]) # create a dataframe\n", "\n", "rule_of_72_df.rate[0] = 0\n", "rule_of_72_df.doubling_time_times_rate[0] = np.log(2)*100\n", "\n", "for i in range(1, 21):\n", " rule_of_72_df.rate[i] = i*.01\n", " rule_of_72_df.doubling_time_times_rate[i] = (i * \n", " np.log(2)/np.log(1 + i*.01))\n", "\n", "rule_of_72_df = rule_of_72_df.set_index(\"rate\")\n", "rule_of_72_df.plot()\n", "plt.ylabel(\"Doubling Time Times Rate\", fontsize=20) # set labels\n", "plt.xlabel(\"Rate\", fontsize=20)\n", "plt.xlim(0,)\n", "plt.suptitle(\"Rule of 72\", fontsize=28)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For compounded growth rates of between 6.5% and 9.5% per period, \n", "the product of the doubling time and the growth rate is closer \n", "to 72 than to any other integer...\n", "\n", "----\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**\"exp\" \"E\"**\n", "\n", "* It’s a function: we feed it a number x, and get out exp(x) (or E^x)\n", "* We can calculate it:\n", " * exp(x) = 1+x+(x^2)/2+(x^3)/6+(x^4)/24+…\n", " * exp(0)=1\n", " * exp(1)=2.71828182846…\n", "* But we don’t have to…\n", " * Over 1618-1731, Napier, Oughtred, Bernoulli, Leibnitz, \n", "and Euler did it for us…" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exponential Convergence\n", "\n", "(dy/dt) = g(k - y)\n", "\n", "* start at t = 0 with y = 0\n", "* g = 0.01; k = 100\n", "* heads rapidly for k\n", "* and then stays there\n", "* (k-y) halves in… \n", " * …guess what? 72/g\n", "* (k-y) shrinks to a thousandth of its initial value in… \n", " * …guess what? 7.2/g\n", "* Let's see:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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m6tSpTy2GDRs2MH/+/Ke2vvslJCTg4+ODh4cHjRs3ZsKECSQnJz+w3oYNG3jj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\nXD311FNKSkrSu+++q6ZNm0qSpkyZovHjx+ull17Stddeqy5dupj5KwJ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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "k = 10 # set the final asymptote\n", "g = 0.01 # set the growth rate\n", "array = np.zeros((1001,2)) # initialize an array\n", "exponential_function_df = pd.DataFrame(array, \n", " columns=[\"time\",\"value\"]) # create a dataframe\n", "\n", "# calculate values for the exponential function\n", "\n", "for t in range(0, 1001, 1):\n", " exponential_function_df.time[t] = t\n", " exponential_function_df.value[t] = k + (0 - k) * np.exp(-g*t)\n", " \n", "exponential_function_df = exponential_function_df.set_index(\"time\")\n", "exponential_function_df.plot() # and graph\n", "\n", "plt.ylabel(\"Value\", fontsize=20) # set labels\n", "plt.xlabel(\"Time\", fontsize=20)\n", "plt.suptitle(\"Exponential Convergence\", fontsize=28)\n", "plt.title(\"asymptote = 10; growth rate = 0.01\", fontsize=16)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "----\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Logistic Growth\n", "\n", "* Suppose we want to combine:\n", " * (dy/dt) = g(y - a) when y is small…\n", " * (dy/dt) = g(k - y) when y is big…\n", "* (dy/dt) = g(y-a)(1-y/k), for k >> a\n", "\n", "(3)          \n", "$ \\frac{dy}{dt} = g(y-a)\\left(1-\\frac{y}{k}\\right) $" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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ToGeXJapv21CN7d3a7DgA4DYoeABM88I3e5VfXKrnbu3GfWYBoBZR8ACYYmfK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\n6phhGPr8x6OKbhOqliENzI4DAPAAFDy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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "array = np.zeros((1001,2)) # initialize an array\n", "logistic_df = pd.DataFrame(array, \n", " columns=[\"time\",\"value\"]) # create a dataframe\n", "\n", "a = 100\n", "g = 0.03\n", "k = 1000\n", "\n", "logistic_df.time[0] = 0\n", "logistic_df.value[0] = a + 0.1\n", "\n", "for t in range(1,1001):\n", " logistic_df.time[t] = t\n", " logistic_df.value[t] = (logistic_df.value[t-1] + g * \n", " (logistic_df.value[t-1]-a)*(1-logistic_df.value[t-1]/k))\n", "\n", "logistic_df = logistic_df.set_index(\"time\")\n", "logistic_df.plot()\n", "plt.ylabel(\"Value\", fontsize=20) # set labels\n", "plt.xlabel(\"Time\", fontsize=20)\n", "plt.suptitle(\"Logistic Curve\", fontsize=28)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "----\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Growth: Things to Remember\n", "\n", "* Asymptote: a (in the negative direction, for growth and logistic)\n", "* Asymptote: k (in the positive direction, for convergence and logistic)\n", "* **Rule of 72**: 72 divided by the growth rate gives you:\n", " * the doubling time (for growth) \n", " * or halving (for convergence) time\n", " * Why 72? Why not 0.693?\n", " * 72 is easier to do in your head: \n", " * 72 = 36x2 = 24x3 = 18x4 = 12x6 = 9x8\n", " * If things aren’t continuous but come in steps…\n", "* **Rule of 720**: 1000-fold time is 10 times doubling time" ] }, { "cell_type": "markdown", "metadata": {}, "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": "3.6.1" } }, "nbformat": 4, "nbformat_minor": 2 }