{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Liquidity Constraints and Precautionary Saving\n", "\n", "This notebook generates the figures for the paper\n", "[Liquidity Constraints and Precautionary Saving](https://econ.jhu.edu/people/ccarroll/papers/LiqConstr)\n", "by Carroll, Holm, and Kimball." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "code_folding": [], "lines_to_next_cell": 0 }, "outputs": [], "source": [ "# Some setup stuff\n", "import dashboard.dashboard_widget as LiqConstr\n", "# The warnings package allows us to ignore some harmless but alarming warning messages\n", "from ipywidgets import interactive\n", "import warnings\n", "warnings.filterwarnings(\"ignore\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Counterclockwise Concavification\n", "\n", "The Figure illustrates two examples of counterclockwise concavifications: the introduction of a constraint and the\n", "introduction of a risk. In both cases, we start from the situation with no risk or constraints (solid line).\n", "The introduction of a constraint is a counterclockwise concavification around a kink point $w^{\\#}$.\n", "Below $w^{\\#}$, consumption is lower and the MPC is greater.\n", "The introduction of a risk also generates a counterclockwise concavification of the original consumption function,\n", "but this time around $\\infty$. For all $w < \\infty$, consumption is lower, the MPC is higher,\n", "and the consumption function is strictly more concave." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "code_folding": [], "lines_to_end_of_cell_marker": 2 }, "outputs": [ { "data": { "image/png": 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077z/3ft8+9u3HOEIxppGMMJdGXcxbdo0Ogd15oGgB/Cp6aN3qfbjzTdhzhwoLIS774b//Ee7gFpGAt5sJOSF+Jui4iLWxq/l8M7DfB/zPT8af0Q1qOAGjQobEewTzJP9n+SeV+/Ru1T7cfCgNgXz1FNaI7C77oJJk7Rmg126yAVUC5KQFwLYm7bXtLzxuPtx1OoqfAldq3XlmdBnuDPwTib0myDLGysiM1Pr8rhqFezapfVm79xZ2wt18GDtQ1icWUNeUZQwIAygqaxbFTYsNz+XuO1x/LT1JzYkbOBXw6+AtryxeUlzBtw+gOlR02ndpLXOldqpnJw/N7hu3x4WLdLaC9x2m96V2QNfRVH+2i4gXFXVSt/hZdW2BkLoadvebSyJW0L8iXjOeJ2BVHD72o377rsPn34+jO45mmH3D8PNVf7ArbD9+7URe3o6rF2rPbdkibZdXqdOMh1TAdLWQIhyys7NZkfiDqKjo1met5y8JnkAuLq60qa4DcMGD2PG8hnUqiX9YCrlwgX47DP45BNITgY3N20KpqREezx1qt4VCiTkhQMxGo1E7YpiecJytp/ZTqZHJrwNnu6etBzeklbVWzG5z2RCu4TK8sbKKi7WNrn28NB2UXr+eW2e/Z13tNUxdr5VniOSkBd2LSsriy1btrAkYQnxnvGU1iwFwMPFgwC3AF7e8DID+gygutxMUzW//KKN2Neu1ZY/jh+vffTtCx076l2duAkJeWFXSkpLWBe/jk9/+JTdmbvJjcrFeNSI151e3DbwNoJrB/NE/yfo3qa73qXav9JSbau8FSu0HZXc3WHIkD/bCtStq30ImyYhL2ze2bNn+XLTlyzYt+DP5Y2Al+LF8NHDeTr0ae655x7c3Z18c2pzUFU4fBhat9ba9q5YoS19fP99GD0a6tXTu0JRQRLywuZcLrjM8s3L+Wz3Zxw/dJxTEadAAddprjSjGYZmBp4yPEXbZm31LtVxnD2rTcd8/LG2QqasMVhCgrbDkrBbEvLCJhw7doz5X81n0/FNnPY8DdUAF6hftz7/+c9/MBgMdOrUSS6Ymtv+/TB3rrbZdUkJ3H8/vPKKdmEVJOAdgIS80EVmdiYffvchUb9EkfV1Fod+OwQPgmsLV+4quYshrYfw1MCnaFpfbqozu2PHtEBv0UL7PDERnn1WazNw55361ibMTkJeWIWqquzau4u3ot8i/kw8F2pcAHfAFXq36s3UKVO5P/h+OrftLKN1Sygu1kbrH38MMTHa/PratdCundbiV65nOCwJeWExly5dYsuWLcTExBAdHc3xxsehv7a8sXNpZx5u9zBTB0ylXm25mGdRCxdqbQXOnYNGjeBf/4LHHvvzdQl4hyYhL8zGaDSyZ88eU6j/8MMPlJaW4u3tTUhICP8M/SfterSjWZ1mNGrUSO9yHVdBgTZqf/hhbWXMpUvaBhyTJoHBoD0nnIaEvKiSc+fOERsbS3R0NDExMaZdvwICApg5cyYGg4HfvX+ntW9rujfR1q43bNiQ9PR0Pct2TPv2wbJl2iYcFy9CbCyEhMC//y29Y5yYhLyokJKSEnbu3El0dDTR0dEkJycD4OvrS2hoKKGhofTv35/brnQb/PnMz0z5eAr9WvRj4+iNABLw5nb6tNa+NzFRWxXz0EMweTIEBWmvS8A7NQl5cUvHjx83TcHExcWRnZ2Nq6sr3bt357XXXsNgMBAQEHDNBdOcwhxGRIygnlc9VgxZYXp+0aJFzJgxw9r/Go7l5Ek4dAiCg7V+Ma6uWruBCRPA11fv6oQNkZAX1ygoKGDbtm2m0frBgwcBbSP2ESNGYDAYCA4OxsfnxtvdqarK1G+ncvTiUeLHx+NX48/GVTKSryRV1W5O+uAD+PprqF9f2wvVzQ3i4/WuTtgoCXmBqqqkpqaaQv37778nPz+fatWq0atXLyZNmoTBYODuu+9GKeef/lGpUazbt47X+rxGrzt6XfXaokWLLPGv4dhiYrSOj7/+qvWLef55rZWvXEQVtyAh76Sys7PZunWrKdiPHTsGQOvWrZk8eTIGg4HevXvj5eVVqeMPbDWQVf9Yxdj2Y695rUuXLqa5fHETBw5orQWaNNHm2r28YOVKbf5dumqKcpKdoZyE0Wjkl19+MYX6Dz/8QElJCTVr1iQ4OBiDwUBoaCjNmzev0nlyi3LJLsymoXfDG35NcnIyXbp0qdJ5HFZxsTYV88EH8P338MwzWq92VZULqE5CdoYS5ZaRkWFa3rh582bOnj0LQKdOnXjhhRcwGAx0794dj7I+JWbw9HdPE30kmtRpqdSqdv0dl7y9vc12PoeycCG8+67WIKxZM1iw4M+bliTgRSVJyDuQkpISfvrpJ9Oa9aSkJFRVpV69evTv3x+DwUD//v25/fbbLXL+VT+vYtUvq5jbe+4NAx5g0KBBpKamWqQGu6Kq2rZ5gVcGbamp2gYcS5fCgAEy3y7MQqZr7NyJEyeIiYkhJiaG2NhYLl26hIuLC/feey8Gg8G0vNHVwoHxa8avdF3WlXsa3UPsI7G4ukhA3VBREaxfD2+/re24tHu3FvSlpRLsQqZrnF1BQQGJiYmmdesHDhwAoFGjRgwbNozQ0FBCQkKoU6eO1Wq6XHyZEREjqOlRk7UPrb1lwM+bN4958+ZZpzhbkpsL770HixdrNzC1bavdodqmjfa6BLywAAl5G6eqKocPHzaFenx8PPn5+Xh4eNCrVy8mTpxIaGgobdu2LffyRnMrMZbQxq8NkwMm08C7gS412LT8/D9Xw7z1FnTrpm3Q0a+fzLULi5PpGhuUk5PD1q1bTcH++++/A9CqVSvTKpigoCBq1Kihc6XihlRVWx2zaBGkpWl9ZVxc4MIF2UJP3JRM1zggVVXZu3evaXnjjh07KC4upkaNGgQHB/PCCy8QGhpKi7JNHmxE6oVUwjaGsWLoCvzr+Jf7fa1bt3bcC69FRfDFF9p8+549WsuBJ5/Unvf0lIAXVichr5MLFy5c1b3xzJkzAHTs2JHnnnsOg8FAjx49zLq80ZwKSgoYGTmSE5dO4OFasRqjoqIsVJUN+PpreOQRbZ592TIYO1ZuXBK6kpC3kpKSEnbv3m0are/evRtVValbty79+vUzLW9s2PDGNxHZkhc3v8jPZ35m4+iNNK7VuELvzcnJsVBVOjh8WFvb3rKltoXegw/C5s1ai1+Zbxc2QELegk6dOmWaV4+NjSUrKwsXFxe6devG3LlzMRgMBAYGWnx5o7l9dfArFu9ezHP3Pseg1oMq/P6wsDD7bmugqrBtmzYls3GjtrPSc89pr7m7axdUhbARcuHVjAoLC9m+fbtptL5//34AGjRoYFqzHhISQt26dXWutPJUVaX3J73JL8lnx2M7KjxV4xCee05rNeDrC088oc25W+gGM+F85MKrjTly5Igp1OPj47l8+TLu7u7cf//9LFy4EIPBQLt27XRb3mhuiqIQPS6arIKsSgf8jBkz7KsTZUmJdjH1/vu1ZmEjRkCrVjBxosy3C5snIV9Bubm5xMfHm4I9LS0NgBYtWjBx4kQMBgNBQUHUrFlT50rNL+JABANaDaCmR0283CvXnRKwm+sOFBbCqlVaD5m0NJg/H+bMge7dtQ8h7IBM19yCqqrs27fPFOrbt2+nuLgYLy8v+vbta1q33rJlS71LtaiYIzEY1hqY02sO8/vM17scy3vvPXjjDe3O1K5d4Z//hCFDtLXuQliQTNdYQWZm5lXLG0+fPg1A+/btefbZZ03LG6tVq6ZzpdaRnpPOI189Qrv67Zjdc3aVj2ezG3nn5EBZh8zkZLj7bli9Gvr2lZUywm5JyAOlpaWm5Y0xMTHs2rULo9FInTp1rlre2KhRI71LtbpSYynjvhxHblEuCcMSqO5e9Tlom/sLLz1dWymzdKl2l2pAgLbG3UbvURCiIpw25NPT003dGzdv3szFixdRFIVu3boxZ84cQkND6dq1K25uTvstAuDNH94k/o94lg9ZThu/NmY5Zmpqqm3My6elaT3cV67ULq6OHq3txAQS8MJhOE2CFRYWsmPHDtO69b179wJw++23M3ToUEJDQ+nXrx/15Lbzq4xqN4pSYykTO0002zHnzZtHQkKC2Y5XKQUFWnvfvDxtY44XXwT/8rdmEMJeOPSF16NHj5qmYLZu3UpeXh7u7u707NnTdMG0Q4cODrO80ZzyivLwcvdyrO9Naqq2Wubf/9bm2L/7Djp1Alv4q0KIK+TC603k5eWRkJBgWglz5MgRAJo3b8748eMJDQ2lT58+sv3cLaiqyqj/jaKaazUihkeYPejDwsIIDw836zFvqmz54+rVWpOwceO0i6oDB1qvBiF0Ytchr6oq+/fvN03BJCYmUlRURPXq1enbty/Tp0/HYDDQsmVLxxqRWtg7O98hKjWK9wzvWeT7FhhotkHKzWVlwcyZ2py7m5u2KfasWXDbbdY5vxA2wKzTNYqihAFhAE2bNu1y7Ngxsx27zMWLF4mLizON1suW4rVr147Q0FAMBgM9e/bE09PT7Od2BrtP7abHih480PoBvhzxpX3+ciwu1nrIFBZC+/bQv7+2zl2mZYQdUBTlGHD+L0+Fq6pa6T99bX5OvrS0lOTkZFOo//TTTxiNRnx8fAgJCTHNrTduXLFOiOJalwou0XlpZ0rVUvZM2UPd6pbpsePt7W2ZTpTnzml3p27cCHv3alMzRUWyUkbYFaeYkz99+jSbN28mOjqazZs3k5mZiaIoBAYG8vLLL2MwGOjWrZvTL280tz+y/qDEWML6YestFvCA+W+EunBB21bv/fe1rfYeeURbNePpKQEvnJ7NpeTBgwdpc2Vj49tuu41BgwZhMBjo168fvr6+Olfn2Dre3pHD0w5Tzc2yd/ImJCQwePBg8xzsyBHt5qXcXBg1CubOhTvvNM+xhXAANhfyd911F4sWLaJv37506NABF+kVYnH7zu4j8tdI5vSeY/GABwgPD69ayF++DCkp0LMntGgBTz8NY8ZAu3bmK1IIB2Hzc/LCsvKK8ui6rCsXCy6yd+pe/Gr46V3SjZWWwpo18PLL2sqZEyegTh29qxLCrMw9Jy/DZCc3bdM0fjv/G2seXGO1gB8zZkzF3xQbC126wIQJ2iqZ776TgBeiHGxuukZYz9q9a1n580r+df+/CPYPttp5Bw2q4JaBqanaMshmzeCzz7RNO2QaT4hykekaJ5VblEuzd5rRxq8NW8dvxc3Fxn7fnz6tbYg9frz2+bffaptjO0l7Z+G8nGIJpbC8mh41iRkXQ/0a9a0e8IqicMPBRW6uthzyzTe1OXiDQbtD9YEHrFqjEI5C/uZ1Qr+c+QWALg270KR2E6uf/7oBX1oKH3+s7Z366qtaqB84IC0IhKgiCXkn89XBr+i0tBORv0bqVsO6deuuffLsWZg+XWv3+8MP2sbZLVpYvzghHIyEvBM5lnWMxzY8RpcGXRjc2kw3I1VCVFSU9uC33+Cll0BVtRUzycmwfbtski2EGUnIO4ni0mJG/280pcZSPh/2uVVuerqRdeHhWrh36KBtuXf8uPbC3XfLXqpCmJmEvJN4Jf4Vfjz5I+GDw2lRV6dpEFWFyEgG+/lpjcTGjYNDh+COO/SpRwgnIKtrnETb+m159p5nGdVulH5FXL4MzzxDWIMG2p2r992nXy1COAlZJ+/gVFXVtyd8Xh58+KG2YYeHB6SmkuPnh7fcrSrEdUlbA1FupcZShq4fyso9K61/clWF//1Pm2efOVNrSwDQujUNmza1fj1COCkJeQf2xvY32Ji6ERXL/LV2Q6mpMGAADBum9ZfZvv2qm5kssmGIEOK6JOQdVOKxRF5JeIUx7ccwsdNE651YVbVWBD/+CO++qy2L7NHjqi+x6ibeQjg5CXkHdOHyBcZ8OQb/Ov4seWCJdebkt27V2v8qCqxYoa2amT5d20D7b+RajRDWIyHvgDambiQjL4PPh32OdzVvy54sKwsmTYLgYG1ZJGjz8LfffsO3yEheCOuRkHdAEzpNIHVaKgENAix7og0boG1bWLlSu7j6yivleltQUJBl6xJCmEjIO5CU0yn8eOJHAJrWtvAKlkWLYOhQ8PWFn37SRvHVq5frrfPmzbNsbUIIE7kZysbzfLgAABEhSURBVEFkF2YzImIEKiq/PfUb7q7u5j+Jqmrr3mvWhOHDoaAAXnxRW/9eAa1btzZ/bUKI65KRvANQVZUpUVP4I+sPPv3Hp5YJ+JMnYcgQeOghLeybNtX2Wq1gwAMEBprtPg8hxC1IyDuA5XuWs37/eub3mU+Ppj1u/YaKUFUID9fm3rds0da/V/Eu6fT0dDMVJ4S4FQl5O3ck8wjTN00nxD+El3q+ZN6DnzqlrZqZMgUCA2H/fnjuuSrvr7po0SIzFSiEuBWZk7dzzX2aM7/PfMZ1GIeLYubf2TVrwrlz2kh+0iSztQGWkbwQ1iMNyuzYpYJL1Pasbd6DnjkDb7wBCxdq8+2lpeDqat5zCCFuSBqUCQA+2/cZrd5vxcGMg+Y76MaN0L69tpFH2S9oCwR8ly5dzH5MIcT1ScjboSOZRwiLCqN1vda0qteq6ge8fBmeeEJbPdOokdZvxoK93uWOVyGsR0LezhSWFDIyciTuLu6se3gdbi5muKzy6KOwZAnMmKHd2NSmTdWPeRPe3hZutSCEMJGQtzMzY2eScjqFT/7xSdXuajUaIT9fezxvntbv/a23oJrl934dNGiQxc8hhNBIyNuRUmMpp3JO8cw9zzDkziGVP9CJExASAlOnap+3a6d9biWpqalWO5cQzk5C3o64urgSMTyCt/q/VfmDRERAx46waxf07l3lG5sqQ3rXCGE9EvJ2oMRYwhNRT3Ak8wiKolRuHj4nByZOhBEjoFUr+PlneOwxs619F0LYJgl5OzA3fi5LkpeQlF6F+w4uXtSWSM6Zo23H17Kl+QqsIBnJC2E9EvI2LvZoLK9vf53HOz/OqHajKn6Abdv+bCh25AjMnw/uFmhgVgHShVII65GQt2Fncs8w7qtx3O13N+8NeK9iby4u1vrM9O4Nq1drz/n4mL/ISoiKitK7BCGchll71yiKEgaEATRtauFNK5zAqwmvkl2YzZZHt+Dl7lX+N54+rc29b98OzzwDo0dbrshKyMnJ0bsEIWyZr6Iof52bDVdVtdJ3EErvGht2ufgySelJ9LqjV/nftGMHDBsG2dnw8cc2F/CgtTVITk7WuwwhbJK5e9dIF0ob9GvGrzSt3ZSaHjUrFvAAhYXatExsrLb+3QZJwAthPTInb2My8zMxrDEwImJE+d+UmwtffaU97tsX9u2z2YAHmDFjht4lCOE0ZCRvQ1RVZeI3EzmTe4YvR35Zvjelpmpb8h06BIcPQ7Nm4Gbb/1kbNmyodwlCOA0ZyduQ9356jw2HNrCw30ICG5ZjSm7DBm3HpjNnYNMmLeDtgIzkhbAeCXkbkZSexIuxLzLkziE8c88zt37D22/DP/4Bd94JKSlW7T1TVTKSF8J6JORthJ+XH0PvGsqKIStQytNqwMNDW0WzbZt2o5MdkVVXQliPLKHUWdn3v1zBfvEiHDgAPXuWvdkue88kJCQQFBSkdxlC2CTZ/s/BLN+znCHrh5BTeIsbhI4ehe7dYehQrdkY2GXAg/SuEcKaJOR1dODcAaZvmk5+cf7N72hNTIR77oGMDG2ppJ3vrJSQkKB3CUI4DQl5nVwuvsyIyBF4V/NmzUNrcHW5wYbZn34KwcFQr562NV+vCt4cZYPCwsL0LkEIp2HbC6od2DObnuHXjF+JGRfD7TVvv/EXbt8O998PkZFQp471CrSgwECzTTcKIW5BLrzqICMvg45LOjKh0wT+G/zfa78gPx/OntXWvRcXaxdYPTysXqcQwvqkd40D8Kvhxy9Tf8HH8zqtf8+e1S6unj+vraSxwsba1ubt7S2dKIWwEpmTt6LCkkIW71pMibEEvxp+uLv+bfOOshU0e/fCm286ZMADpKen612CEE5DQt6KXop7iWmbprHt2LZrX9y3T1v/fukSxMfDgw9av0ArkdU1QliPhLyVbDy0kXd+eodp3abRt3nfa7/gpZfAxUW7g/Wee6xfoBWFh1d6/wMhRAXJhVcrOHHpBJ2WdqKZTzN+eOwHqrn9ZRqm7K7Vixe1UbydNBkTQliG3PFqhyZ+M5Gi0iLWP7z+6oCPjISBA6GgQFse6SQBP2bMGL1LEMJpyOoaK1gQsoCT2SdpVa/Vn08uWwZTp2oXWgsLwdNTvwKtbNCgQXqXIITTkOkaC7pw+QL1vOpd+8KCBdoc/IAB2mjeqwKbdAshHJpM19iJs7lnafthW97Y/sbVL7zxhhbwo0bB1187ZcCXq+OmEMIsZLrGAoyqkUe+eoRLhZcY1PpvUxODB0NmJrz+OrjeoF+Ng7PUX49CiGvJSN4CFmxfQGxaLO8a3qVd/XZaa4LVq7WVNG3bwsKFThvwAOvWrdO7BCGchoS8me04voM58XMY0XYEkwMmQ0kJjB0Ljz4KO3fqXZ5NiIqK0rsEIZyGhLyZnck9Q9v6bQkfFI5iNGrhHhEBixZpK2mEjOSFsCIJeTN7uM3DpISlUNu9Jjz2GHz2mTb//vzzepdmMwYPHqx3CUI4DQl5M1matJRlyctQVVXbAGT3bli7FubP11bTCBPZNEQI65GQN4OU0ylMj57OhtQNfz55773wyy8wZ45+hdko2cRbCOuRkK+inMIcRkaOxM/Lj5VDVqDMmKHtwwraShpxjYYNG+pdghBOQ0K+ClRV5YlvnyDtYhqfPbQO3zmvw//9n6yiuQXZMEQI65GQr4Ld6btZu28tr/aex/1LvtMCfvp07a5WcUPSalgI65GQr4JujbqxbcI2/hlboPWjeeIJeOcdrXWwuCFn72kkhDVJyFdCfnE+SelaUN3ftCcueZdh0iRYvFgCvhxkJC+E9UjIV8Kz0c9y3/L7OJH+mxbqb78NS5dqOzuJW5LVNUJYj6RSBX2+/3PCU8JZlW+gSbcQOHJEC3oJ+HKbN2+e3iUI4TQkmSrgaOZRJm+czNM5bRj1VjS0bAmNG+tdlt1p3bq13iUI4TQk5MupqLSIUf8bRcczKu+GH0e56y6tH7wT7ehkLoGBZtsPQQhxCxLy5eSquPJo7d7ErnPDxacObNoEPj56l2WX0tPT9S5BCKchIV8ORtWIq4sr0wyv4NmnH0RHQ6NGepdltxYtWqR3CUI4DbOGvKIoYYqiJCmKkpSRkWHOQ+vmZPZJAt9pw/aDMVCrFnzxBbRpo3dZdk1G8kLclG9Zjl75qFJHP9nI+yZKjCX0W9GHF9/6kT61O1D9h91OvaOTEMLyZCNvK5qf8CpjPtrOwEOlVJ8wWQLeTLp06aJ3CUI4DQn5G9iStgXX+f9mcgrw8staywJhFnLHqxDWIyF/A+nv/5e530Px+Efgtdf0LseheHt7612CEE5DQv4Gxj21lPxJE3Bftlz60ZjZoEGD9C5BCKchIf83EVsXc+TCYZSWLam+bCW4u+tdksNJTU3VuwQhnIaE/F/s+uU7Ah+extHH/qF3KQ5NetcIYT1uehdgKzKzz1I67CEa5ir4vfC+3uUIIYRZyEgebRu/ncO70/1IIafemkvN+/vqXZJDk5G8ENYjIQ/snPs4Azf/TtLo3vg/M1fvchyedKEUwnpkugbo3P1BUvv9QpdP4/QuxSlERUXpXYIQTsOpR/K5eRfJKczBc8BgWm9ORnGT33nWkJOTo3cJQjgNpw15NS+Pcx1b8v7YVhSWFOpdjlMJC6tSvyUhRAU4Z8irKr8/1JdmRzNpHhhCNbdqelfkVJKTk/UuQQin4ZQhf27O8/hv3sXyYf6MmLlK73KczowZM/QuQQin4XQhX/j1//D97zt82bEaDyzfhquLdJa0toYNG+pdghBOw+lCvuCPI/zWrCY1V6+nYS3Z3UkPMpIXwnqcctMQtbgYRXrS6KZhw4ayO5QQNyCbhlTSpSce462XenH+8nkJeJ3Z6i9/IRyRU4R8yScrqL1kJcbdu8gtytW7HKcnXSiFsB7HD/mDBymdOoX4ZtDy3dU082mmd0VOT3rXCGE9jh3yly+TM9TAJdcSYl8dz0PthutdkQASEhL0LkEIp+HQIW9cu4YaR44z93F/XhmzRO9yxBVyx6sQ1uPQzVpcJk0m8+7mvNi2BZ5unnqXI64IDDTbwgEhxC045kj+0CF2xq6kRC2lbs9++Nfx17si8RcykhfCehwv5C9fJnfIAG4b8RiLvn9D72rEdXh7e+tdghBOw+FCPv+pMGqm/s5/xjTmqR7P6l2OuA65EUoI63GokDeu/pTqn6xlYS9Xpr/yLTU9aupdkrgOWV0jhPU4TsgfPkzJlMkkNoVab7xDh9s66F2RuIHw8HC9SxDCaTjO6pomTbgwcRRbu3vwyr1P6V2NuImNGzfqXYIQTsMhRvIl+Xng6UmDD1Yxd9wyFEXRuyRxE2PGjNG7BCGcht2HvPrNN2Q0v43XVkzQuxRRToMGDdK7BCGchn2HfEYG+Y89whkljxot7tK7GlFOMpIXwnrsN+RVlaxHR+J6KYfwZ3rybK+Zelckykmm04SwHrsN+fyPl+ITHc+CAbWYP+1LXBS7/VdxOpbaqEYIcS37TEZV5fLnq9ne3JXe73yFXw0/vSsSFbBu3Tq9SxDCadhnyCsK9WK20X77YXq36Kt3NaKCoqKi9C5BCKdhdyF/cs2HrI19G9XFhdoNm+tdjqgEGckLYT12FfIFPyfjN/Fp3Gb/k8z8TL3LEZU0ePBgvUsQwmnYzx2vRUWce9iAp4eK7+KV1POqp3dFopKk1bAQ1mPWkbyiKGGKoiQpipKUkZFhzkPz67TRNE07z3cvDiX43tFmPbawrqCgIL1LEMKW+Zbl6JWPKo2KFEstZwsMDFSTkpLMcqyc72Px6tOfTd196b/tFB6uHmY5rtCHt7c3OTk5epchhE1SFCVZVVWzbZ9mF3Py3u0CODHmAdqt2yIB7wAk4IWwHpsP+RNZx6FePZqtiaLZHdI+2BFIq2EhrMemQ37np69zqn0zvk9co3cpwozMNY0nhLg1m52TTz91iJL2bVDd3bjt8Gk8a9U1Y3VCCGGbnGJOvtRYSvLYPjTKMsLKTyTgHYysrhHCemwy5D97azyDvz/NwfEDuWOgLJd0NPPmzdO7BCGchk2GfI8vdnKqiQ/tPvqf3qUIC2jdurXeJQjhNGzyjtfmiftQT58GT0+9SxEWEBgYSHp6ut5lCOEUbHIkT/XqKP7+elchLEQCXgjrsc2QFw5t0aJFepcghNOQkBdWJyN5IaxHQl5YnYzkhbAei90MpShKBnCsCofwBc6bqRxzkroqRuqqGKmrYhyxrjtUVTXbnqYWC/mqUhQlyZx3fZmL1FUxUlfFSF0VI3XdmkzXCCGEA5OQF0IIB2bLIW+r/WilroqRuipG6qoYqesWbHZOXgghRNXZ8kheCCFEFUnIC5ulKIq/oigBVx5XaTNjIZyVhLywZSGqqqYoiuIPpOldjBD2SEJe2IMQVVXj9C5CCHtkk62GhfO6MmoPAXyAroqiAPS78s80CXshKkZW1wibVDYXf2W6JkxVVZtZkiaEPZGQFzblLyP5LsDRK0+3AJKRkbwQFSZz8sJqFEUZpijKAkVRfK58HqYoyrArj5cqihKiqmralVF7sqqqC4EsVVWnqKoaLgEvnFF5fm5u9n4JeWEVVwI8EggA6l55ejh/rpqJ0KUwIWyYOX5uJOSFVfxlFF5XVdWy/0H9VVVN+cuXJYFpPl5G7cLpVeTn5kZkdY2wmit/VsZdeRzA1WvffVRVzQLtYmvZk3LBVTi78v7c3IiM5IU1BQC7rzwO5Or/WW/6P6oQTqxKPzeyukZYzZWVM1OA2CtPDS97fGXeUQjxN1X9uZGQF0IIBybTNUII4cAk5IUQwoFJyAshhAOTkBdCCAcmIS+EEA5MQl4IIRyYhLwQQjgwCXkhhHBg/w8USLPEepO1kgAAAABJRU5ErkJggg==\n", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "interactive(LiqConstr.make_concavification_figure,\n", " in_BoroCnstArt=LiqConstr.BoroCnstArt_widget[0],\n", " in_UnempProb=LiqConstr.UnempProb_widget)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Notes:** The solid line shows the linear consumption function in the case with no constraints and no risks.\n", "The two dashed line show the consumption function when we introduce a constraint and a risk, respectively.\n", "The introduction of a constraint is a counterclockwise concavification of the solid consumption function around\n", "$w^{\\#}$, while the introduction of a risk is a counterclockwise concavification around $\\infty$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## How a current constraint can hide a future kink\n", "\n", "The original $\\mathcal{T}$ contains only a single constraint, at the end of period $t+1$, inducing a kink point at\n", "$\\omega_{t,1}$ in the consumption rule $c_{t,1}$. The expanded set of constraints, $\\hat{\\mathcal{T}}$, adds one\n", "constraint at period $t+2$. $\\hat{\\mathcal{T}}$ induces two kink points in the updated consumption rule $\\hat{c}_{t,2}$,\n", "at $\\hat{\\omega}_{t,1}$ and $\\hat{\\omega}_{t,2}$. It is true that imposition of the new constraint causes consumption to\n", "be lower than before at every level of wealth below $\\hat{\\omega}_{t,1}$. However, this does not imply higher prudence\n", "of the value function at every $w <\\hat{\\omega}_{t,1}$. In particular, note that the original consumption function is\n", "strictly concave at $w = \\omega_{t,1}$, while the new consumption function is linear at $\\omega_{t,1}$, so prudence can\n", "be greater before than after imposition of the new constraint at this particular level of wealth.\n", "\n", "The intuition is simple: At levels of initial wealth below $\\hat{\\omega}_{t,1}$, the consumer had been planning to end\n", "period $t+2$ with negative wealth. With the new constraint, the old plan of ending up with negative wealth is no longer\n", "feasible and the consumer will save more for any given level of current wealth below $\\hat{\\omega}_{t,1}$, including\n", "$\\omega_{t,1}$. But the reason $\\omega_{t,1}$ was a kink point in the initial situation was that it was the level of\n", "wealth where consumption would have been equal to wealth in period $t+1$. Now, because of the extra savings induced by\n", "the constraint in $t+2$, the larger savings induced by wealth $\\omega_{t,1}$ implies that the period $t+1$ constraint\n", "will no longer bind for a consumer who begins period $t$ with wealth $\\omega_{t,1}$. In other words, at wealth\n", "$\\omega_{t,1}$ the extra savings induced by the new constraint moves the original constraint and prevents it from being\n", "relevant any more at the original $\\omega_{t,1}$.\n", "\n", "Notice, however, that all constraints that existed in $\\mathcal{T}$ will remain relevant at *some* level of\n", "wealth under $\\hat{\\mathcal{T}}$ even after the new constraint is imposed - they just induce kink points at different\n", "levels of wealth than before, e.g. the first constraint causes a kink at $\\hat{\\omega}_{t,2}$ rather than at $\\omega_{t,1}$." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "code_folding": [] }, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "interactive(LiqConstr.make_future_kink,\n", " in_BoroCnstArt=LiqConstr.BoroCnstArt_widget[1])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Notes:** $c_{t,1}$ is the original consumption function with one constraint that induces a kink point at $\\omega_{t,1}$.\n", "$\\hat{c}_{t,2}$ is the modified consumption function in where we have introduced one new constraint.\n", "The two constraints affect $\\hat{c}_{t,2}$ through two kink points: $\\hat{\\omega}_{t,1}$ and $\\hat{\\omega}_{t,2}$.\n", "Since we introduced the new constraint at a later point in time than the current existing constraint,\n", "the future constraint affects the position of the kink induced by the current constraint and the modified consumption\n", "function $\\hat{c}_{t,2}$ is not a counterclockwise concavification of ${c}_{t,1}$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "## Consumption function with and without a constraint and a risk\n", "\n", "To illustrate the result of Theorem 2 in the paper, The figure shows an\n", "example of optimal consumption rules in period $t$ under different combinations of an immediate risk (realized at the\n", "beginning of period $t+1$) and a future constraint (applying at the end of period $t+1$).\n", "The thinner loci reflect behavior of consumers who face the future constraint, and the dashed loci reflect behavior of\n", "consumers who face the immediate risk. For levels of wealth above $\\omega_{t,1}$ where the future constraint stops\n", "impinging on current behavior for perfect foresight consumers, behavior of the constrained and unconstrained perfect\n", "foresight consumers is the same. Similarly, $\\tilde{c}_{t,1}(w_{t}) = \\tilde{c}_{t,0}(w_{t})$ for levels of wealth above\n", "${\\bar{\\omega}}_{t,1}$ beyond which the probability of the future constraint binding is zero. For both constrained and\n", "unconstrained consumers, the introduction of the risk reduces the level of consumption (the dashed loci are below their\n", "solid counterparts). The significance of Theorem 2 in the paper in this context is that for levels of\n", "wealth below ${\\bar{\\omega}}_{t,1}$, the vertical distance between the solid and the dashed loci is greater for the\n", "constrained (thin line) than for the unconstrained (thick line) consumers, because of the interaction between the\n", "liquidity constraint and the precautionary motive." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "code_folding": [] }, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "interactive(LiqConstr.make_cons_func,\n", " in_BoroCnstArt=LiqConstr.BoroCnstArt_widget[2],\n", " in_TranShkStd=LiqConstr.TranShkStd_widget)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Notes:** $c_{t,0}$ is the consumption function with no constraint and no risk, $\\tilde{c}_{t,0}$ is the consumption\n", "function with no constraint and a risk that is realized at the beginning of period $t+1$, $c_{t,1}$ is the consumption\n", "function with one constraint in period $t+1$ and no risk, and $\\tilde{c}_{t,1}$ is the consumption function with one\n", "constraint in period $t+1$ and a risk that is realized at the beginning of period $t+1$. The figure illustrates that\n", "the vertical distance between $c_{t,1}$ and $\\tilde{c}_{t,1}$ is always greater than the vertical distance between\n", "$c_{t,0}$ and $\\tilde{c}_{t,0}$ for $w < \\bar{\\omega}_{t,1}$." ] } ], "metadata": { "jupytext": { "cell_metadata_json": true, "formats": "ipynb,py:light", "notebook_metadata_filter": "all" }, "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": "3.7.4" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": false }, "varInspector": { "cols": { "lenName": 16, "lenType": 16, "lenVar": 40 }, "kernels_config": { "python": { "delete_cmd_postfix": "", "delete_cmd_prefix": "del ", "library": "var_list.py", "varRefreshCmd": "print(var_dic_list())" }, "r": { "delete_cmd_postfix": ") ", "delete_cmd_prefix": "rm(", "library": "var_list.r", "varRefreshCmd": "cat(var_dic_list()) " } }, "types_to_exclude": [ "module", "function", "builtin_function_or_method", "instance", "_Feature" ], "window_display": false } }, "nbformat": 4, "nbformat_minor": 4 }