Labor_supply.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Load needed libraries\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "OECD = pd.read_csv('data/OECD_Productivity.csv')\n",
    "OECD.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df = OECD.copy()\n",
    "\n",
    "countries = OECD['Country'].unique()\n",
    "locs = OECD['LOCATION'].unique()\n",
    "\n",
    "df = df.loc[df['Subject'].isin(('GDP per hour worked', 'Average hours worked per person employed'))]\n",
    "\n",
    "df = df.loc[df['Measure'].isin(('USD, constant prices, 2010 PPPs', 'Persons/Hours'))]\n",
    "\n",
    "df = df[['LOCATION', 'Subject', 'Time', 'Value']]\n",
    "\n",
    "# Set MultiIndex; can choose ordering\n",
    "df.set_index(['Subject', 'LOCATION', 'Time'], inplace=True)\n",
    "\n",
    "# Create nested columns\n",
    "df = df.unstack(0)\n",
    "\n",
    "# Remove one unnecessary level of naming\n",
    "df = df['Value']\n",
    "\n",
    "df.columns = ['h', 'y_h']\n",
    "\n",
    "df['cntr'] = df.index.get_level_values(0)\n",
    "\n",
    "# Show prepared dataset\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = df['h']\n",
    "y = df['y_h']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.scatter(x, y, alpha=0.2, label='All countries')\n",
    "plt.scatter(x['KOR'], y['KOR'], color='C2', alpha=0.3, label='Korea')\n",
    "plt.scatter(x['POL'], y['POL'], color='C3', alpha=0.3, label='Poland')\n",
    "\n",
    "plt.legend(frameon=False)\n",
    "\n",
    "plt.title('GDP per hour vs average hours worked')\n",
    "plt.xlabel('Average hours worked per employee')\n",
    "plt.ylabel('GDP per hour (constant 2010 PPP USD)')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for loc in locs[:9]:\n",
    "    plt.scatter(np.log(x[loc]), np.log(y[loc]), alpha=0.2, label=loc)\n",
    "\n",
    "plt.legend(frameon=False, ncol=2)\n",
    "\n",
    "plt.title('GDP per hour vs average hours worked')\n",
    "plt.ylabel('Average hours worked per employee')\n",
    "plt.xlabel('GDP per hour (constant 2010 PPP USD)')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Utility maximization problem\n",
    "\\begin{align*}\n",
    "\\max_{c,\\,h}\\quad & U=\\frac{c^{1-\\sigma}}{1-\\sigma}-\\phi\\frac{h^{1+\\kappa}}{1+\\kappa}\\\\\n",
    "\\text{subject to}\\quad & y=Ah^{1-\\alpha}\\\\\n",
    " & c=y\n",
    "\\end{align*}\n",
    "\n",
    "Lagrangian\n",
    "\\begin{align*}\n",
    "\\mathcal{L}&=\\frac{c^{1-\\sigma}}{1-\\sigma}-\\phi\\frac{h^{1+\\kappa}}{1+\\kappa}+\\lambda\\left[Ah^{1-\\alpha}-c\\right]\n",
    "\\end{align*}\n",
    "\n",
    "FOCs\n",
    "\\begin{eqnarray*}\n",
    "c & : & c^{-\\sigma}-\\lambda=0\\\\\n",
    "h & : & -\\phi h^{\\kappa}+\\lambda\\left(1-\\alpha\\right)Ah^{-\\alpha}=0\n",
    "\\end{eqnarray*}\n",
    "\n",
    "Optimality condition\n",
    "\\begin{align*}\n",
    "\\left(1-\\alpha\\right)Ah^{-\\alpha}c^{-\\sigma}=\\phi h^{\\kappa}\n",
    "\\end{align*}\n",
    "\\begin{align*}\n",
    "c&=\\left[\\frac{\\phi}{\\left(1-\\alpha\\right)A}h^{\\alpha+\\kappa}\\right]^{-\\frac{1}{\\sigma}}&=\\left[\\frac{\\left(1-\\alpha\\right)A}{\\phi}\\right]^{\\frac{1}{\\sigma}}h^{-\\frac{\\alpha+\\kappa}{\\sigma}}\n",
    "\\end{align*}\n",
    "Plug into budget constraint and solve for optimal $h$\n",
    "\\begin{align*}\n",
    "\\left[\\frac{\\left(1-\\alpha\\right)A}{\\phi}\\right]^{\\frac{1}{\\sigma}}h^{-\\frac{\\alpha+\\kappa}{\\sigma}}&=Ah^{1-\\alpha}\n",
    "\\end{align*}\n",
    "\\begin{align*}\n",
    "h^{-\\frac{\\alpha+\\kappa+\\sigma\\left(1-\\alpha\\right)}{\\sigma}}&=A^{\\frac{\\sigma-1}{\\sigma}}\\left(\\frac{\\phi}{1-\\alpha}\\right)^{\\frac{1}{\\sigma}}\n",
    "\\end{align*}\n",
    "\\begin{align*}\n",
    "h&=A^{\\frac{1-\\sigma}{\\alpha+\\kappa+\\sigma\\left(1-\\alpha\\right)}}\\left(\\frac{1-\\alpha}{\\phi}\\right)^{\\frac{1}{\\alpha+\\kappa+\\sigma\\left(1-\\alpha\\right)}}\n",
    "\\end{align*}\n",
    "\\begin{align*}\n",
    "\\ln h&=\\frac{1-\\sigma}{\\alpha+\\kappa+\\sigma\\left(1-\\alpha\\right)}\\ln A+\\frac{1}{\\alpha+\\kappa+\\sigma\\left(1-\\alpha\\right)}\\ln\\left(\\frac{1-\\alpha}{\\phi}\\right)\n",
    "\\end{align*}\n",
    "\n",
    "Rewrite above in terms of labor productivity $y/h$\n",
    "\\begin{align*}\n",
    "\\frac{y}{h}&=Ah^{-\\alpha}\n",
    "\\end{align*}\n",
    "\\begin{align*}\n",
    "\\ln\\left(y/h\\right)&=\\ln A-\\alpha\\ln h\n",
    "\\end{align*}\n",
    "\\begin{align*}\n",
    "\\ln A&=\\ln\\left(y/h\\right)+\\alpha\\ln h\n",
    "\\end{align*}\n",
    "\\begin{align*}\n",
    "\\ln h&=\\frac{1-\\sigma}{\\alpha+\\kappa+\\sigma\\left(1-\\alpha\\right)}\\left[\\ln\\left(y/h\\right)+\\alpha\\ln h\\right]+\\frac{1}{\\alpha+\\kappa+\\sigma\\left(1-\\alpha\\right)}\\ln\\left(\\frac{1-\\alpha}{\\phi}\\right)\n",
    "\\end{align*}\n",
    "\\begin{align*}\n",
    "\\left(1-\\frac{\\alpha-\\alpha\\sigma}{\\alpha+\\kappa+\\sigma\\left(1-\\alpha\\right)}\\right)\\ln h&=\\frac{1-\\sigma}{\\alpha+\\kappa+\\sigma\\left(1-\\alpha\\right)}\\ln\\left(y/h\\right)+\\frac{1}{\\alpha+\\kappa+\\sigma\\left(1-\\alpha\\right)}\\ln\\left(\\frac{1-\\alpha}{\\phi}\\right)\n",
    "\\end{align*}\n",
    "\\begin{align*}\n",
    "\\left(\\frac{\\kappa+\\sigma}{\\alpha+\\kappa+\\sigma\\left(1-\\alpha\\right)}\\right)\\ln h&=\\frac{1-\\sigma}{\\alpha+\\kappa+\\sigma\\left(1-\\alpha\\right)}\\ln\\left(y/h\\right)+\\frac{1}{\\alpha+\\kappa+\\sigma\\left(1-\\alpha\\right)}\\ln\\left(\\frac{1-\\alpha}{\\phi}\\right)\n",
    "\\end{align*}\n",
    "\n",
    "End result\n",
    "\\begin{align*}\n",
    "\\ln h&=\\frac{1-\\sigma}{\\kappa+\\sigma}\\ln\\left(y/h\\right)+\\frac{1}{\\kappa+\\sigma}\\ln\\left(\\frac{1-\\alpha}{\\phi}\\right)\n",
    "\\end{align*}\n",
    "\\begin{align*}\n",
    "\\frac{\\partial\\ln h}{\\partial\\ln\\left(y/h\\right)}&=\\frac{1-\\sigma}{\\kappa+\\sigma}\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import statsmodels.formula.api as smf\n",
    "\n",
    "# Exclude Korea as outlier\n",
    "df_no_Korea = df[df['cntr'] != 'KOR']\n",
    "\n",
    "results = smf.ols('np.log(h) ~ np.log(y_h) + C(cntr, Treatment(\"OECD\"))', data=df_no_Korea).fit()\n",
    "\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "elasticity = results.params[-1]\n",
    "intercept = results.params[0]\n",
    "\n",
    "print(elasticity)\n",
    "\n",
    "country_effects = results.params[1:-1]\n",
    "country_effects.index = [e[29:-1] for e in country_effects.index]\n",
    "\n",
    "# country_effects.index"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "country_effects.sort_values().plot(kind='bar', color='C0', figsize=(10,5))\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "country_effects.sort_values()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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