{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Measuring the Welfare Cost of Inflation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*Author: Mohammed Aït Lahcen, University of Basel*" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this notebook I present several methods and results related to measuring the cost of inflation in terms of aggregate social welfare. This topic is an old and enduring one in the monetary economics literature.\n", "\n", "In particular, I replicate some of the results presented in Lucas (2000), Lagos and Wright (2005) and Craig and Rocheteau (2006)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Fitting money demand data" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First, we do some imports as usual..." ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# Computations imports\n", "import numpy as np\n", "from scipy.integrate import quad as integrate\n", "import scipy.optimize as opt\n", "from scipy.optimize import curve_fit\n", "\n", "# Data imports\n", "import pandas as pd\n", "from lmfit import Model\n", "\n", "# Graphics imports\n", "%matplotlib notebook\n", "from matplotlib import rcParams\n", "rcParams['figure.figsize'] = (12, 8) # Sets the size of the figures in the notebook\n", "import matplotlib.pyplot as plt\n", "import seaborn as sns # Better quality figures\n", "from cycler import cycler\n", "import matplotlib.style\n", "matplotlib.style.use('ggplot')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "... and then have a look at the money demand data used by Lucas (2000)." ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('
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');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "mpl.figure.prototype._root_extra_style = function(el){\n", " var fig = this\n", " el.on(\"remove\", function(){\n", "\tfig.close_ws(fig, {});\n", " });\n", "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('tabindex', 0)\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager)\n", " manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " // select the cell after this one\n", " var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n", " IPython.notebook.select(index + 1);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] == html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "}\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel != null) {\n", " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.figure()\n", "plt.scatter(data['Nominal interest rates']*100,data['Real balances']*100, label = 'Data')\n", "r = np.linspace(0,0.15)\n", "plt.plot(r*100, log_md(r, *log_est)*100, 'r-', label='log-log fit')\n", "plt.plot(r*100, slog_md(r, *slog_est)*100, 'g-', label='semi-log fit')\n", "plt.xlim(0,15)\n", "plt.ylim(10,60)\n", "plt.xlabel('Nominal interest rate, percent')\n", "plt.ylabel('Real balances (M1 / NGDP), percent')\n", "plt.legend(loc='best')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## A reduced-form measure of the welfare cost of inflation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In order to measure the cost of inflation, we consider the welfare cost of increasing the nominal interest rate starting from $r_0 = 3\\%$ assumed to correspond to an inflation rate of 0. Following Lucas (2000), this is done by measuring the area under the fitted money demand curve for the log-log and semi-log specifications.\n", "\n", "Define the function measuring the welfare cost of an increase in interest rates from r0 to r1 in terms of the decrease in real balances from m0 to m1. Since we first approximate the money demand and we need to measure the cost using instead the inverse money demand a little geometric manipulation is needed." ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# Welfare cost using the log-log specification\n", "def log_wc(r1,r0,*p): \n", " integral, err = integrate(lambda r: log_md(r, *p),r0,r1)\n", " return integral - (r1 - r0) * log_md(r1,*p) + (log_md(r0,*p) - log_md(r1,*p)) * r0\n", "\n", "# Welfare cost using the semi-log specification\n", "def slog_wc(r1,r0,*p):\n", " integral, err = integrate(lambda r: slog_md(r, *p),r0,r1)\n", " return integral - (r1 - r0) * slog_md(r1,*p) + (slog_md(r0,*p) - slog_md(r1,*p)) * r0" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. 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