et03_itu.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <center>Matching models with imperfectly transferable utility</center>\n",
    "### <center>Alfred Galichon (NYU & Sciences Po)</center>\n",
    "## <center>'math+econ+code' masterclass series</center>\n",
    "#### <center>With python code examples</center>\n",
    "© 2018–2023 by Alfred Galichon. Past and present support from NSF grant DMS-1716489, ERC grant CoG-866274 are acknowledged, as well as inputs from contributors listed [here](http://www.math-econ-code.org/team).\n",
    "\n",
    "**If you reuse material from this masterclass, please cite as:**<br>\n",
    "Alfred Galichon, 'math+econ+code' masterclass series. https://www.math-econ-code.org/"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "# References\n",
    "\n",
    "* Gabrielle Demange and David Gale (1985). \"The Strategy Structure of Two-Sided Matching Markets\". *Econometrica*  53 no. 4, pp. 873-888.\n",
    "* Alvin Roth and Marilda Sotomayor (1990). *Two-sided matching. A Study in Game-Theoretic Modeling and Analysis.* Econometric Society Monographs.\n",
    "* Alfred Galichon, Scott Kominers, and Simon Weber (2019). \"Costly Concessions: An Empirical Framework for Matching with Imperfectly Transferable Utility.\" *Journal of Political Economy* 127 no. 6, pp. 2875-2925.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Matching with nonlinear taxes\n",
    "\n",
    "## Generating the data\n",
    "\n",
    "We will generate the same type of worker/firm data as yesterday."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.spatial.distance import cdist\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "\n",
    "np.random.seed(777)\n",
    "d = 8\n",
    "nbx = 50\n",
    "nby = 30\n",
    "\n",
    "rg = .8\n",
    "rs = .6\n",
    "r  = .7\n",
    "\n",
    "n_x = np.ones(nbx)\n",
    "m_y = np.ones(nby)\n",
    "ξ_x_k = np.random.rand(nbx,d)\n",
    "ζ_y_k = np.random.rand(nby,d)\n",
    "\n",
    "α_x_y = np.zeros((nbx,nby))\n",
    "γ_x_y = np.zeros((nbx,nby))\n",
    "for x in range(nbx):\n",
    "    for y in range(nby):\n",
    "        α_x_y[x,y] = -  np.linalg.norm(ξ_x_k[x,6:7]-ζ_y_k[y,6:7])-5\n",
    "        γ_x_y[x,y] = 10+5*(np.sum( (ξ_x_k[x,0:3]*ζ_y_k[y,0:3])**rg )**(r/rg)+(  np.sum(  ξ_x_k[x,3:6]*ζ_y_k[y,3:6] )**rs )**(r/rs))**r\n",
    "        \n",
    "Φ_x_y = α_x_y + γ_x_y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tax schedule\n",
    "\n",
    "In 2020, the federal tax schedule for a single taxpayer was as follows: \n",
    "\n",
    "|MARGINAL RATE | LOWER BRACKET | UPPER BRACKET  | TAX OWED  |\n",
    "| :----- | :----- | :----- | :------------------------------------------------------------- |\n",
    "| 10\\% | \\$0  | \\$9,700    | 10\\% of taxable income |\n",
    "| 12\\% | \\$9,701 | \\$39,475 | \\$970 plus 12\\% of the excess over \\\\$9,700 |\n",
    "| 22\\% | \\$39,476 | \\$84,200 | \\$4,543 plus 22\\% of the excess over \\\\$39,475 |\n",
    "| 24\\% | \\$84,201 | \\$160,725 | \\$14,382.50 plus 24\\% of the excess over \\\\$84,200 |\n",
    "| 32\\% | \\$160,726 | \\$204,100 | \\$32,748.50 plus 32\\% of the excess over \\\\$160,725 |\n",
    "| 35\\% | \\$204,101 | \\$510,300 | \\$46,638.50 plus 35\\% of the excess over \\\\$204,100 |\n",
    "| 37\\% | \\$510,300 | - | \\$153,798.50 plus 37\\% of the excess over \\\\$510,300 |\n",
    "\n",
    "(Source: https://www.investopedia.com/terms/t/taxbracket.asp)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $N\\left( w\\right) $ be the net wage if gross wage is $w$.\n",
    "\n",
    "Let $\\tau ^{k}$ be the $k$-th tax rate (by increasing order), and $w^{k}$ be the lower bracket associated with $\\tau_k$.\n",
    "\n",
    "Set $w^{0}=0,n^{0}=0,\\tau ^{0}=0.1$.\n",
    "\n",
    "Letting $n^k=N(w^k)$, one has<br>\n",
    "$n^{k+1}=n^{k}+\\left( 1-\\tau ^{k}\\right) \\left( w^{k+1}-w^{k}\\right) $<br>\n",
    "and<br>\n",
    "$N\\left( w\\right) =\\min_{k}\\left\\{ n^{k}+\\left( 1-\\tau ^{k}\\right) \\left(\n",
    "w-w^{k}\\right) \\right\\} $\n",
    "\n",
    "\n",
    "Let $g$ be the gross wage, and $n$ be the net wage, we have that the net wage is a piecewise linear, increasing and concave function of the gross wage, whose slopes are $1-\\tau_k$, so it expresses as<br>\n",
    "$ n = \\min_k \\{N^k+(1-\\tau_k) w\\}$<br>\n",
    "where $N^k = n^k - w^k (1-\\tau^k) = \\tau^k w^k - (n^k - w^k)$.<br> \n",
    "\n",
    "We encode this as:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lower_bracket = np.array([0, 9.701, 39.476, 84.201, 160.726, 204.101, 510.300])/100\n",
    "marginal_tax =  np.array([.1,.12,   .22,    .24,    .32,     .35,     .37    ])\n",
    "tax_on_previous_brackets = np.zeros(lower_bracket.size)\n",
    "for i in range(1,lower_bracket.size):\n",
    "    tax_on_previous_brackets[i] = tax_on_previous_brackets[i-1] + marginal_tax[i-1]*(lower_bracket[i]-lower_bracket[i-1])\n",
    "    \n",
    "N_k = np.append(0,marginal_tax*lower_bracket - tax_on_previous_brackets ) \n",
    "τ_k = np.append(0,marginal_tax)\n",
    "nbk =N_k.size\n",
    "\n",
    "TaxPaid = np.vectorize(lambda g : max(g*τ_k -N_k))\n",
    "NetFromGross = np.vectorize(lambda g : min(N_k+g*(1-τ_k)))\n",
    "\n",
    "gs = np.arange(0,1,0.01)\n",
    "ts = NetFromGross(gs)\n",
    "plt.plot(gs,ts)\n",
    "plt.ylabel('Tax paid')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bargaining whithin a pair\n",
    "In order to study the matching problem, we need to first study the bargaining possibilities within a matched pair. For now we shall drop the subscripts $x$ and $y$, and we will study the set of utilities $(U,V)$ that the worker and the firm can obtain as an outcome of that bargaining. This set is called *feasible utility set*, or *bargaining set*.\n",
    "\n",
    "Most of the material in this lecture is taken from Galichon, Kominers and Weber (2019)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Feasible utilities\n",
    "\n",
    "For each value of $w$ (nominal wage), compute<br>\n",
    "$\\left\\{\n",
    "\\begin{array}{l}\n",
    "U \\text{ = utility of the worker}\\\\\n",
    "V\\text{ = utility of the firm}\n",
    "\\end{array}\n",
    "\\right.$\n",
    "\n",
    "In the transferable utility case (Becker-Shapley-Shubik),<br>\n",
    "$\\left\\{\n",
    "\\begin{array}{l}\n",
    "U(w) = \\alpha + w\\\\\n",
    "V(w) = \\gamma -w,\n",
    "\\end{array}\n",
    "\\right.$<br>\n",
    "but in general, we shall assume that $\\mathcal{U}\\left( w\\right)$ and $\\mathcal{U}\\left( w\\right)$  are nondecreasing (resp. nonincreasing) and continuous, not necessarily linear.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The feasible utility set\n",
    "Given a worker-firm pair, define $\\mathcal{F}$ the feasible utility set as:\n",
    "\n",
    "$\\mathcal{F}=\\left\\{ \\left( U,V\\right) :\\exists w,U\\leq U\\left(\n",
    "w\\right) ,V\\leq V\\left( w\\right) \\right\\} $.\n",
    "\n",
    "equivalently, if $\\mathcal{U}$ is strictly increasing\n",
    "\n",
    "$\\mathcal{F}=\\left\\{ \\left( U,V\\right) :V\\leq V\\left( U\n",
    "^{-1}\\left( U\\right) \\right) \\right\\} .$\n",
    "\n",
    "Note that we are assuming free disposal: if $(U,V)$ is feasible and if $U' \\leq U$ and $V' \\leq V$, then $(U',V')$ is feasible.\n",
    "\n",
    "Also, $w$ does not have to be a wage; it can be the set of terms of a contract."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Frontier of the feasible utility set\n",
    "\n",
    "We define the *frontier* of the feasible utility sets as the set of points $(U,V)\\in \\mathcal{F}$ such that there is no point $(U',V')\\in \\mathcal{F}$ with<br>\n",
    "$U' < U$ and $V' < V$.\n",
    "\n",
    "**Caution here**. This coincides with the *topological frontier* which is defined as the intersection of the closures of the interior and exterior of the set, but not with the *set of Pareto efficient points*, which is in general a subset of the former. Indeed, the set of Pareto efficient points is the set of points $(U,V)\\in \\mathcal{F}$ such that there is no point $(U',V')\\in \\mathcal{F}$ with<br>\n",
    "$U' \\leq U$ and $V' \\leq V$, with at least one strict inequality.<br>\n",
    "In particular, if $\\mathcal{F} = (U,V): U\\leq 1~and~V\\leq 1$, the topological frontier is union of the set $\\{U=1,V\\leq 1\\}$ and $\\{U\\leq 1, V=1\\}$, while the set of Pareto efficient points is $(1,1)$.<br>\n",
    "This distinction will be important when we discuss model with non-transferable utility.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Distance-to-frontier function\n",
    "\n",
    "\n",
    "Given a feasible set $\\mathcal{F}$, compute the distance to the frontier of $\\mathcal{F}$ along the diagonal, with a minus sign if in the interior, and a plus sign if it is not.<br>\n",
    "<br>\n",
    "Define the *distance-to-frontier* function as<br>\n",
    "$D(U,V)=\\min\\{t∈R:(U-t,V-t)∈F\\}$<br>\n",
    "\n",
    "This way:<br>\n",
    "$D(U,V) \\leq 0 $ means that $(U,V)\\in\\mathcal{F},$<br>\n",
    "while<br>\n",
    "$D(U,V) = 0 $ means that $(U,V)$ is on the frontier of $\\mathcal{F}$.\n",
    "\n",
    "**Property**. For $a \\in \\mathbb{R}$, we have $D(U+a,V+a)=D(U,V)+a$.\n",
    "    \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Parameterization of the frontier of the feasible set\n",
    "\n",
    "We can define $\\delta = U-V$ and get an explicit parameterization on the frontier of the feasible set as a function of $\\delta$.<br> \n",
    "$D(\\mathcal{U}(\\delta) , \\mathcal{U}(\\delta) - \\delta) =0$ and<br>\n",
    "$D(\\mathcal{V}(\\delta) + \\delta, \\mathcal{V}(\\delta) ) =0$<br> \n",
    "thus<br>\n",
    "$\\left\\{\n",
    "\\begin{array}[l]\n",
    "~\\mathcal{U}\\left( \\delta \\right) = - D(0,-\\delta)\\\\\n",
    "\\mathcal{V}\\left( \\delta \\right) = - D(\\delta,0)\n",
    "\\end{array}\n",
    "\\right.\n",
    "$<br>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ucalf = lambda D: (lambda w: -D(0,-w) )\n",
    "Vcalf = lambda D: (lambda w: -D(w,0) )\n",
    "\n",
    "def plot_feasible(D,range = (-1,1)):\n",
    "    δs = np.linspace(range[0],range[1],num=500)\n",
    "    Ucal,Vcal = np.vectorize(Ucalf(D)),np.vectorize(Vcalf(D))\n",
    "    us = [Ucal(δ) for δ in δs]\n",
    "    vs = [Vcal(δ) for δ in δs]\n",
    "    umin,vmin = min(us),min(vs)\n",
    "    umax,vmax = max(us),max(vs)\n",
    "    plt.figure(figsize=(4, 4))\n",
    "    plt.xlim([umin, umax+(umax-umin)/3])\n",
    "    plt.ylim([vmin, vmax+(vmax-vmin)/3])\n",
    "    plt.plot(us,vs)\n",
    "    plt.xlabel('U')\n",
    "    plt.ylabel('V')\n",
    "    plt.fill(us+[np.min(us)],vs+[np.min(vs)])\n",
    "    \n",
    "    \n",
    "    plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Some examples\n",
    "### Example 1:  transferable utility (TU)\n",
    " \n",
    "* The Becker model assumes  *additively transferable utility*: it is a model of marriage in which utility can freely be transferred across partners, in a one-to-one manner.<br>\n",
    "$\\left\\{\n",
    "\\begin{array}\n",
    "\\mathcal{U}\\left( w\\right) =\\alpha +w \\\\\n",
    "\\mathcal{V}\\left( w\\right) =\\gamma -w\n",
    "\\end{array}\n",
    "\\right.\n",
    "$<br>\n",
    "and thus, denoting $\\Phi = \\alpha +\\gamma$,<br>\n",
    "$\\mathcal{F}=\\left\\{ \\left( U,V\\right) :u+v\\leq \\Phi \\right\\} $<br>\n",
    "\n",
    "\n",
    "* The distance=to-frontier in the transferable utility case is<br>\n",
    "$D(U,V) = \\frac {U+V - \\Phi} {2}.$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 400x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def create_D_TU(Φ):\n",
    "    return (lambda U,V : (U+V - Φ)/2)\n",
    "\n",
    "plot_feasible(create_D_TU(2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 2: non-transferable utility (NTU)\n",
    "\n",
    "* **Motivation: housing market with price controls.** In the model of *non-transferable utility,* no transfer attempt can be beneficial to the other side of the market. In that case, denote $(\\alpha,\\gamma)$ the only Pareto efficient point, and due to free disposal:<br>\n",
    "$\\mathcal{F=}\\left\\{ \\left( U,V\\right) :U\\leq \\alpha \\text{ and }V\\leq\n",
    "\\gamma \\right\\}. $<br>\n",
    "\n",
    "* The distance-to-frontier is the scalar $t\\in\\mathbb{R}$ such that\n",
    "$\\max \\left( U-\\alpha ,V-\\gamma \\right) -t = 0$, hence:<br>\n",
    "$D\\left( U,V\\right) =\\max \\left( U-\\alpha ,V-\\gamma \\right) $<br>\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 400x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def create_D_NTU(α,γ):\n",
    "    return (lambda U,V : np.maximum(U - α ,V -γ ))\n",
    "\n",
    "plot_feasible(create_D_NTU(1,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 3: linearly transferable utility (LTU)\n",
    "\n",
    "* **Motivation: labor matching with flat tax**. Consider a model of the labor market with proportional taxation, where the tax rate is $\\tau$. We have<br>\n",
    "$ U-\\alpha = (1 - \\tau) (\\gamma - V) $<br>\n",
    "then, setting $\\lambda = 1 /(2 - \\tau)$, one has<br>\n",
    "$\\mathcal{F=}\\left\\{ \\left( U,V\\right) :\\lambda U + (1-\\lambda) V \\leq  \\Phi \\right\\},$<br>\n",
    "where $\\Phi = \\lambda \\alpha + (1-\\lambda) \\gamma $.\n",
    "\n",
    "* As a result, the distance function is<br>\n",
    "$D(U,V) = \\lambda U + (1-\\lambda) V - \\Phi).$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "def create_D_LTU(α,γ,λ):\n",
    "    return (lambda U,V : λ*(U-α)+(1-λ)*(V-γ))\n",
    "gross_wages = lower_bracket / 10\n",
    "net_wages = NetFromGross(lower_bracket)/10\n",
    "D_LTUs = [create_D_LTU(  net_wages[i] , 1 - gross_wages[i], 1/(2-marginal_tax[i]) ) for i in range(len(marginal_tax))]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 4: exponentially transferable utility (ETU)\n",
    "\n",
    "* **Motivation: household bargaining with private consumption.** We consider a model of marriage in which we model intra-household bargaining with private consumption and match amenity.\n",
    "\n",
    "* A match yield amenity $\\alpha$ to man $i$ and utility $\\gamma$ to woman $j$, and private consumptions are denoted $c^{i}$ and $c^{j}$.<br>\n",
    "\n",
    "* The utilities are formed as a combination of the log-consumptions and the match amenities:<br>\n",
    " $\\left\\{\n",
    "\\begin{array}[l]\n",
    "~U = \\tilde{\\alpha} +\\tau \\log c^{i}\\\\\n",
    "V = \\tilde{\\gamma} +\\tau \\log c^{j}\n",
    "\\end{array}\n",
    "\\right.\n",
    "$<br>\n",
    "where the private consumptions are subject to the budget constraint $c^{i}+c^{j}=B$, where $B$ is the joint budget of the household.<br>\n",
    "\n",
    "* We have $c^{i}=\\exp \\left( \\frac{U-\\tilde{\\alpha} }{\\tau }\\right) $ and $c^{j}=\\exp \\left( \\frac{V-\\tilde{\\gamma} }{\\tau }\\right) $ and thus the budget constraint rewrites:<br>\n",
    "$\\exp \\left( \\frac{U- \\tilde{\\alpha} }{\\tau }\\right) +\\exp \\left( \\frac{V-\\tilde{\\gamma} }{\\tau }\\right) \\leq B$<br>\n",
    "and setting $\\alpha = \\tilde{\\alpha} + \\tau \\log B$ and $\\gamma = \\tilde{\\gamma} + \\tau \\log B$, we obtain:<br>\n",
    "$\\mathcal{F=}\\left\\{ \\left( U,V\\right) :\\exp \\left( \\frac{U- \\alpha }{\\tau }%\n",
    "\\right) +\\exp \\left( \\frac{V-\\gamma }{\\tau }\\right) \\leq 2\\right\\} $<br>\n",
    "\n",
    "* The distance function can be obtained as<br> \n",
    "$D(U,V) = \\tau \\log \\left( \\frac {\\exp(\\frac{ U-\\alpha} {\\tau} ) + \\exp(\\frac {V - \\gamma} {\\tau}) } {2} \\right)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 400x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def create_D_ETU(α,γ,τ):\n",
    "    return (lambda U,V : τ * np.log(np.exp( (U - α ) / τ ) + np.exp ((V -γ ) / τ )) )\n",
    "\n",
    "plot_feasible(create_D_ETU(1,1,1))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 4 (continued): limits of the ETU case\n",
    "**Remarks.**\n",
    "* When $\\tau \\rightarrow +\\infty $, we get at first order in $1 / \\tau$<br>\n",
    "\n",
    "    $2+\\frac{U-\\alpha }{\\tau }+\\frac{V-\\gamma }{\\tau }\\leq 2$<br>\n",
    "\n",
    "    which in the limit of a very large $\\tau$, yields $U+V\\leq \\alpha +\\gamma $, the transferable utility case.<br>\n",
    "\n",
    "* When $\\tau \\rightarrow 0$, we get<br>\n",
    "    $\\tau \\log \\left( \\exp \\left( \\frac{U-\\alpha }{\\tau }\\right) +\\exp \\left( \n",
    "\\frac{V-\\gamma }{\\tau }\\right) \\right) \\leq \\tau \\log 2$<br>\n",
    "\n",
    "    which in the limit when $\\tau$ tends to zero, yields $\\max \\left\\{ U-\\alpha ,V-\\gamma \\right\\} \\leq 0$, the non-transferable utility case.<br>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 400x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_feasible(create_D_ETU(1,1,0.05))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 400x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_feasible(create_D_ETU(1,1,5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 5: union of feasible sets\n",
    "\n",
    "* **Motivation: household bargaining with public goods**. Consider a model of marriage where partners make a joint decision on a public good $g\\in G$ e.g. the number of kids; buying a house, etc.<br>\n",
    "\n",
    "* Assume that conditional on $g\\in G$, the utilities are<br>\n",
    " $\\left\\{\n",
    "\\begin{array}[l]\n",
    "~U=\\alpha ^{g}\\left( w\\right)\\text{ increasing}\\\\\n",
    "V=\\gamma ^{g}\\left( w\\right)\\text{ decreasing}\n",
    "\\end{array}\n",
    "\\right.\n",
    "$<br>\n",
    "where $w$ is the term of match - say the share of private consumption that\n",
    "goes to the man.<br>\n",
    "\n",
    "* We can compute the *conditional feasible set*<br>\n",
    "$\\mathcal{F}^{g}\\mathcal{=}\\left\\{ \\left( U,V\\right) :U\\leq \\alpha\n",
    "^{g}\\left( \\left( \\gamma ^{g}\\right) ^{-1}\\left( V\\right) \\right) \\right\\} $<br>\n",
    "\n",
    "* The overall feasible set is the union of the conditional bargaining sets:<br>\n",
    "$\\mathcal{F=\\cup }_{g\\in G}\\mathcal{F}^{g}$.<br>\n",
    "$\\mathcal{F=}\\left\\{ \\left( U,V\\right) :U\\leq \\max_{g}\\alpha ^{g}\\left(\n",
    "\\left( \\gamma ^{g}\\right) ^{-1}\\left( V\\right) \\right) \\right\\} $<br>\n",
    "\n",
    "* The distance function associated with the overall set is the minimum of the distance functions associated with the conditional bargaining sets:<br>\n",
    "$D_{\\mathcal{F}}\\left( U,V\\right) =\\min_{g}D_{\\mathcal{F}^{g}}\\left(\n",
    "U,V\\right) $."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 400x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def create_D_union(Dlist):\n",
    "    return (lambda U,V : min([D(U,V) for D in Dlist]) )\n",
    "\n",
    "D_LTUs_bis  = [create_D_LTU(  a , b, c ) for (a,b,c) in [(0.95,0.95,0.1),(0.95,0.95,0.9),(1.15,1.15,0.5)]]\n",
    "\n",
    "\n",
    "plot_feasible(create_D_union(D_LTUs_bis))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### Example 6: intersection\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 400x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def create_D_inter(Dlist):\n",
    "    return (lambda U,V : max([D(U,V) for D in Dlist]) )\n",
    "\n",
    "\n",
    "D_LTUs_ter  = [create_D_LTU(  a , b, c ) for (a,b,c) in [(1,1,0),(1,1,1),(0.75,0.75,0.5)]]\n",
    "plot_feasible(create_D_inter(D_LTUs_ter))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### Example 6: progressive taxation\n",
    "\n",
    "* **Motivation: labor matching with progressive taxation.**\n",
    "In the case of progressive taxation as introduced above,<br>\n",
    "$\\left\\{\n",
    "\\begin{array}{l}\n",
    "~U\\left( w_{xy}\\right) =\\alpha +N\\left( w\\right) = \\alpha\n",
    "+\\min_{k}\\left\\{ n^{k}+\\left( 1-\\tau _{k}\\right) \\left( w-w^{k}\\right)\n",
    "\\right\\}\\\\\n",
    "V \\left( w_{xy}\\right) =\\gamma _{xy}-w_{xy}\n",
    "\\end{array}\n",
    "\\right.$<br>\n",
    "\n",
    "* We have therefore<br>\n",
    "$\\mathcal{F=\\cap }_{k}\\mathcal{F}^{k}$<br>\n",
    "where<br>\n",
    "$\\mathcal{F}^{k}=\\left\\{ \\left( U,V\\right) : \\frac {U - \\alpha - n^{k}} { 2 -\\tau^k} +  \\frac \n",
    "{1-\\tau ^{k}} {2 - \\tau^k} \\left( V - \\gamma + w^{k}\\right) \\leq 0 \\right\\} $ which is the feasible set associated with a LTU model\n",
    "\n",
    "* Letting $\\tilde{\\alpha}_k = \\alpha + n^{k}$ and $\\tilde{\\gamma}_k=\\gamma - w^{k}$, and $\\lambda^k = \\frac {1} {2 - \\tau^k}$, the distance function can be expressed as:<br>\n",
    "$D_{\\mathcal{F}}\\left( U,V\\right) =\\max_{k} \\left\\{ \\lambda^k (U-\\tilde{\\alpha}^k) + (1-\\lambda^k) (V - \\tilde{\\gamma}^k) \\right\\} $\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 400x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def create_D_inter(Dlist):\n",
    "    return (lambda U,V : max([D(U,V) for D in Dlist]) )\n",
    "\n",
    "plot_feasible(create_D_inter(D_LTUs))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The matching model without heterogeneity\n",
    "\n",
    "Assume $w_{xy}$ is the wage and consider the workers' and the firms' problems. Define the indirect utilities as:<br>\n",
    "$u_{x}=max_{y}\\{U_{xy}(w_{xy}),0\\}$ and $v_{y}=max_{x}\\{V_{xy}(w_{xy}),0\\}$\n",
    "    \n",
    "$(\\mu,u,v,w)$ is an equilibrium matching if the following conditions hold\n",
    "\n",
    "(i) population constraint\n",
    "    \n",
    "$\\sum_{y}\\mu_{xy}+\\mu_{x0}=n_{x}$ and $\\sum_{x}\\mu_{xy}+\\mu_{0y}=m_{y}$\n",
    "    \n",
    "(ii) Stability\n",
    "\n",
    "$u_{x}\\geq U_{xy}(w_{xy})$ and $v_{y}\\geq V_{xy}(w_{xy})$, and\n",
    "\n",
    "$u_{x}\\geq0$ and $v_{y}≥0$\n",
    "\n",
    "(iii) Complementarity\n",
    "\n",
    "$\\mu_{xy}>0$ implies $u_{x}=U_{xy}(w_{xy})$ and $v_{y}=V_{xy}(w_{xy})$<br>\n",
    "$\\mu_{x0}>0$ implies $u_{x}=0$, and  $\\mu_{0y}>0$ implies $v_{y}=0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Equilibrium formulation (model without heterogeneity)\n",
    "Note that we can remove $w_{xy}$ by rewriting stability as $D_{xy}(u_x,v_y) \\geq 0$. Equilibrium can be reexpressed as:<br>\n",
    "(i) population constraint<br> \n",
    "$\\sum_{y}\\mu_{xy}+\\mu_{x0}=n_{x}$ and $\\sum_{x}\\mu_{xy}+\\mu_{0y}=m_{y}$\n",
    "    \n",
    "(ii) Stability<br>\n",
    "$D_{xy}(u_x,v_y) \\geq 0$\n",
    "$u_{x}\\geq0$ and $v_{y}≥0$\n",
    "\n",
    "(iii) Complementarity<br>\n",
    "$\\mu_{xy}>0$ implies  $D_{xy}(u_x,v_y) = 0$<br>\n",
    "$\\mu_{x0}>0$ implies $u_{x}=0$, and  $\\mu_{0y}>0$ implies $v_{y}=0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The matching model with logit heterogeneity\n",
    "\n",
    "* As before, we introduce $(\\varepsilon_y)$ and $(\\eta_x)$ two random utility terms with an i.i.d. Gumbel distribution. Denote $U_{xy}=U_{xy}(w_{xy})$ and $V_{xy}=V_{xy}(w_{xy})$. The average indirect utilities are:<br>\n",
    "$\\left\\{\n",
    "\\begin{array}[l]\n",
    "~u_{x}= \\mathbb{E}[\\max_{y}\\{U_{xy}+T\\epsilon_{y},T\\epsilon_0\\}] = T\\log(1+∑_{y}\\exp(U_{xy}/T))\\\\\n",
    "v_{y}= \\mathbb{E}[\\max_{x}\\{V_{xy}+T\\eta_{x},T\\eta_0\\}] = T\\log(1+\\sum_{x}\\exp(V_{xy}/T))\n",
    "\\end{array}\n",
    "\\right.\n",
    "$\n",
    "\n",
    "* The choice probabilities are<br>\n",
    "$\\Pr(y|x) = \\exp( (U_{xy} - u_x) / T)$ and $\\Pr(0|x) = \\exp(  - u_x / T)$<br>\n",
    "$\\Pr(x|y) = \\exp( (V_{xy} - v_y) / T)$ and $\\Pr(0|y) = \\exp(  - v_y / T)$.\n",
    "\n",
    "* Note that we have $(U_{xy},V_{xy})\\in \\mathcal{F}_{xy}$. Thus, we reexpress\n",
    "$U_{xy}=U_{xy}(w_{xy})$ and $V_{xy}=V_{xy}(w_{xy})$ for some $w_{xy}$ as<br>\n",
    "$D_{xy}(U_{xy},V_{xy})=0$<br>\n",
    "where $D_{xy}$ is the distance function associated with $\\mathcal{F}_{xy}$.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The matching equations\n",
    "\n",
    "* We have<br>\n",
    "$\\left\\{\n",
    "\\begin{array}[l]\n",
    "~U_{xy} = T \\ln \\mu_{xy} - T \\ln  \\mu_{x0}\\\\\n",
    " V_{xy} = T \\ln \\mu_{xy}  - T \\ln  \\mu_{0y},\n",
    "\\end{array}\n",
    "\\right.$<br>\n",
    "and therefore<br>\n",
    "$D_{xy}(T \\ln \\mu_{xy} - T \\ln  \\mu_{x0},T \\ln \\mu_{xy}  - T \\ln  \\mu_{0y})=0$<br>\n",
    "hence<br>\n",
    "$\\mu_{xy}=M_{xy}(\\mu_{x0},\\mu_{0y})$, where $M_{xy}$ is the *matching equation*, given as:<br>\n",
    "$M_{xy}(\\mu_{x0},\\mu_{0y}):=\\exp(- D_{xy}(-T\\ln \\mu_{x0},-T\\ln \\mu_{0y}) / T)$\n",
    "\n",
    "\n",
    "* The equations of the model become:<br>\n",
    "$\\left\\{\n",
    "\\begin{array}[l]\n",
    "~\\mu_{x0}+∑_{y\\in \\mathcal{Y}}M_{xy}(\\mu_{x0},\\mu_{0y}) = n_{x}\\\\\n",
    "\\mu_{0y}+∑_{x\\in\\mathcal{X}}M_{xy}(\\mu_{x0},\\mu_{0y}) = m_{y}\n",
    "\\end{array}\n",
    "\\right.$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solving for the equilibrium using Gauss-Seidel\n",
    "\n",
    "We will verify that we are in the Gross Substitutes / BGH case. Recall\n",
    "\n",
    "$M_{xy}(\\mu_{x0},\\mu_{0y})=\\exp(-D_{xy}(-T\\ln \\mu_{x0},-T\\ln \\mu_{0y}) / T)$\n",
    "\n",
    "Introduce $p_{z}=(p_{x},p_{y})$ with $p_{x}=-\\mu_{x0}$ and $p_{y}=\\mu_{0y}$,\n",
    "\n",
    "$\\left\\{\n",
    "\\begin{array}[l]\n",
    "~Q_{x}(p) = p_{x}-∑_{y∈Y}M_{xy}(-p_{x},p_{y})+n_{x}\\\\\n",
    "Q_{y}(p) = p_{y}+∑_{x∈X}M_{xy}(-p_{x},p_{y})-m_{y}\n",
    "\\end{array}\n",
    "\\right.$\n",
    "\n",
    "Gross substitutes hold.\n",
    "    \n",
    "Law of aggregate supply holds:<br>\n",
    "$\\sum_{x}Q_{x}(p)+∑_{y}Q_{y}(p)=∑_{x}p_{x}+∑_{y}p_{y}+∑_{x}n_{x}-\\sum_{y}m_{y}$ is increasing in all the prices.\n",
    "    \n",
    "Hence $Q$ is inverse isotone.\n",
    "    \n",
    "Now let's see that there is a subsolution and a supersolution.\n",
    "\n",
    "$p_{z}=N$, $N$ large enough yields a supersolution, while $p_{z}=-N$, N large enough yields a subsolution.\n",
    "    \n",
    "Hence there is a solution, and Gauss-Seidel converges."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The class `GKW_model`\n",
    "### Distance function\n",
    "Note that $n_{xy} \\leq \\min_k \\{N_k+(1-\\tau_k) g_{xy}\\}$ is equivalent to $$ \\frac{- N_k - (1-\\tau_k) g_{xy} + n_{xy}} {2 - \\tau_k} \\leq 0~\\forall k.$$\n",
    "As $n_{xy}=U_{xy}-\\alpha_{xy}$ and $g_{xy} = \\gamma_{xy} -V_{xy}$, this is equivalent to $$ \\frac{ (U_{xy} - \\alpha_{xy} ) + (1-\\tau_k) ( V_{xy} - \\gamma_{xy}) -N_k } {2 - \\tau_k} \\leq 0~\\forall k, $$\n",
    "therefore, we get\n",
    "$$ D_{xy}(U,V) = \\max_k \\left\\{ \\frac{ (U - \\alpha_{xy} ) + (1-\\tau_k) ( V - \\gamma_{xy}) -N_k } {2 - \\tau_k} \\right\\} $$\n",
    "or,  setting<br>\n",
    "$\\lambda_k = 1 /  (2 - \\tau_k)$ and<br>\n",
    "$\\nu_k = (1 - \\tau_k) /  (2 - \\tau_k)$ and<br>\n",
    "$\\Phi^k_{xy} = \\lambda_k (\\alpha_{xy}+N_k) + \\nu_k \\gamma_{xy}$<br>\n",
    "$$ D_{xy}(U,V) = \\max_k \\left\\{ \\lambda_k U + \\nu_k V -  \\Phi^k_{xy} \\right\\} $$\n",
    "### Matching function\n",
    "One has $$M_{xy}(\\mu_{x0},\\mu_{0y})=\\exp(-D_{xy}(-\\log \\mu_{x0},-\\log \\mu_{0y}))=\\min_k \\left\\{ \\mu_{x0}^{\\lambda_k}  \\mu_{0y}^ {\\nu_k} e^{ \\Phi^k_{xy}} \\right\\}  $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "λ_k = 1 / (2-τ_k)\n",
    "ν_k = (1-τ_k) / (2-τ_k)\n",
    "Φ_k_x_y = np.zeros((nbk,nbx,nby))\n",
    "for k in range(nbk):\n",
    "    for x in range(nbx):\n",
    "        for y in range(nby):\n",
    "            Φ_k_x_y[k,x,y] = λ_k[k]*(α_x_y[x,y]+N_k[k])+ν_k[k] * γ_x_y[x,y]\n",
    "\n",
    "class GKW_model:    \n",
    "    def __init__(self):\n",
    "        self.n_x = n_x\n",
    "        self.m_y = m_y\n",
    "        self.nbx = n_x.size\n",
    "        self.nby = m_y.size\n",
    "        self.eq_μ_x0 = np.array([])\n",
    "        self.eq_μ_0y = np.array([])\n",
    "        self.eq_μ_x_y = np.array([])\n",
    "        self.eq_deltamarg_x = np.array([])\n",
    "        self.eq_deltamarg_y = np.array([])\n",
    "        self.eq_deltainvmarg_x = np.array([])\n",
    "        self.eq_deltainvmarg_y = np.array([])\n",
    "        self.comp_code = -1\n",
    "        self.comp_nbsteps = -1\n",
    "        self.comp_time = -1.0\n",
    "                \n",
    "\n",
    "    def Mxy(self,x,y,μx0,μ0y):\n",
    "        return(np.min(μx0**λ_k * μ0y**ν_k * np.exp(Φ_k_x_y[:,x,y]) ))\n",
    "        \n",
    "    def M_x_y(self,μ_x0,μ_0y):\n",
    "        μ_x_y = np.zeros((self.nbx,self.nby))\n",
    "        for x in range(self.nbx):\n",
    "            for y in range(self.nby):\n",
    "                μ_x_y[x,y] = self.Mxy(x,y,μ_x0[x],μ_0y[y])\n",
    "        return(μ_x_y)\n",
    "\n",
    "#############\n",
    "mkt = GKW_model()\n",
    "μ_x0 = np.ones(nbx)\n",
    "μ_0y = np.ones(nby)\n",
    "\n",
    "mkt.M_x_y(μ_x0,μ_0y)[0:min(nbx,5),0:min(nby,5)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we compute the mappings<br>\n",
    "$\\mu_{x0} \\to \\mu_{x0}+\\sum_y M_{xy}(\\mu_{x0},\\mu_{0y})$ and<br>\n",
    "$\\mu_{0y} \\to \\mu_{0y}+\\sum_x M_{xy}(\\mu_{x0},\\mu_{0y}).$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def marginx(self,x,μx0,μ_0y):\n",
    "    nx = μx0\n",
    "    for y in range(nby):\n",
    "        nx +=  self.Mxy(x,y,μx0,μ_0y[y])\n",
    "    return(nx)\n",
    "                                             \n",
    "def marginy(self,y,μ_x0,μ0y):\n",
    "    my = μ0y\n",
    "    for x in range(nbx):\n",
    "        my += self.Mxy(x,y,μ_x0[x],μ0y)\n",
    "    return(my)\n",
    "\n",
    "\n",
    "GKW_model.marginx = marginx\n",
    "GKW_model.marginy = marginy\n",
    "\n",
    "def margin_x(self,μ_x0,μ_0y):\n",
    "    μ_x_y = self.M_x_y(μ_x0,μ_0y)\n",
    "    return(μ_x0+np.sum(μ_x_y,axis = 1))\n",
    "\n",
    "def margin_y(self,μ_x0,μ_0y):\n",
    "    μ_x_y = self.M_x_y(μ_x0,μ_0y)\n",
    "    return(μ_0y+np.sum(μ_x_y,axis = 0))\n",
    "\n",
    "GKW_model.margin_x = margin_x\n",
    "GKW_model.margin_y = margin_y\n",
    "\n",
    "#############\n",
    "mkt.marginx(0,μ_x0[0],μ_0y) #[0:min(nbx,5)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we solve in $\\mu_{x0}$ the equation<br>\n",
    "$\\mu_{x0}+\\sum_y M_{xy}(\\mu_{x0},\\mu_{0y})=n_x,$<br>\n",
    "and in $\\mu_{0y}$ the equation<br>\n",
    "$\\mu_{0y}+\\sum_x M_{xy}(\\mu_{x0},\\mu_{0y})=m_y.$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import scipy.optimize as opt\n",
    "\n",
    "def invmarginx(self,x,μ_0y):    \n",
    "    newμx0 = opt.brentq(lambda theμx0 : self.marginx(x,theμx0,μ_0y)-self.n_x[x],0,self.n_x[x])\n",
    "    return(newμx0)\n",
    "\n",
    "def invmarginy(self,y,μ_x0):    \n",
    "    newμ0y = opt.brentq(lambda theμ0y : self.marginy(y,μ_x0,theμ0y)-self.m_y[y],0,self.m_y[y])\n",
    "    return(newμ0y )\n",
    "\n",
    "GKW_model.invmarginx = invmarginx\n",
    "GKW_model.invmarginy = invmarginy\n",
    "\n",
    "#############\n",
    "print(mkt.invmarginx(0,μ_0y)) #[0:min(nbx,5)])\n",
    "print(mkt.invmarginy(0,μ_x0)) #[0:min(nby,5)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from time import time\n",
    "\n",
    "def solveIPFP(self, maxit = 1000,maxvaltol=1e-5,maxsteptol=1e-9,output=0):\n",
    "    start_time = time()\n",
    "    code = 0\n",
    "    deltainvmarg_x = np.zeros(self.nbx)\n",
    "    deltainvmarg_y = np.zeros(self.nby)\n",
    "    μ_x0 = np.copy(self.n_x)\n",
    "    μ_0y = np.zeros(self.nby)\n",
    "    for i in range(maxit):\n",
    "        for y in range(self.nby):\n",
    "            newμ0y = self.invmarginy(y,μ_x0)\n",
    "            deltainvmarg_y[y] = newμ0y - μ_0y[y]\n",
    "            μ_0y[y] = newμ0y\n",
    "        for x in range(self.nbx):\n",
    "            newμx0 = self.invmarginx(x,μ_0y)\n",
    "            deltainvmarg_x[x] = newμx0 - μ_x0[x]\n",
    "            μ_x0[x] = newμx0\n",
    "        \n",
    "        deltamarg_x = self.margin_x(μ_x0,μ_0y) - n_x\n",
    "        deltamarg_y = self.margin_y(μ_x0,μ_0y) - m_y\n",
    "        #print(deltainvmarg_x)\n",
    "        #print(deltainvmarg_y)\n",
    "        #print(np.append(deltainvmarg_x,deltainvmarg_y))\n",
    "        steptol = np.max(np.abs( np.append(deltainvmarg_x,deltainvmarg_y) ) ) \n",
    "        valtol = np.max(np.abs(np.append(deltamarg_x,deltamarg_y)) )\n",
    "        \n",
    "        if output > 1 :\n",
    "            print(\"μ_x0=\",μ_x0)\n",
    "        if valtol < maxvaltol :\n",
    "            code = 0\n",
    "            break\n",
    "        if steptol < maxsteptol :\n",
    "            code = 1\n",
    "            break\n",
    "        code = 2\n",
    "    comp_time = time() - start_time\n",
    "    if output > 0 :\n",
    "        print( 'IPFP converged in', i, 'iterations and ',comp_time,' seconds.')\n",
    "        #print('Value of μ_x0=\",μ_x0)\n",
    "        print('Max absolute margin discrepancy=', valtol)            \n",
    "        print('Max absolute update discrepancy =',steptol)\n",
    "        print('Code =',code)\n",
    "\n",
    "    self.eq_μ_x0 = μ_x0\n",
    "    self.eq_μ_0y = μ_0y\n",
    "    self.eq_μ_x_y = self.M_x_y(μ_x0,μ_0y)\n",
    "    self.eq_deltamarg_x = deltamarg_x\n",
    "    self.eq_deltamarg_y = deltamarg_y\n",
    "    self.eq_deltainvmarg_x = deltainvmarg_x\n",
    "    self.eq_deltainvmarg_y = deltainvmarg_y\n",
    "    self.comp_code = code\n",
    "    self.comp_nbsteps = i\n",
    "    self.comp_time = comp_time\n",
    "    return code\n",
    "\n",
    "    \n",
    "GKW_model.solveIPFP = solveIPFP\n",
    "#############\n",
    "res=mkt.solveIPFP(output=1,maxvaltol=1e-6,maxsteptol=1e-9)  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Recovering the wages\n",
    "\n",
    "Recall that we had<br>\n",
    "$n_{xy}=U_{xy}-\\alpha_{xy}$ and $w_{xy} = \\gamma_{xy} -V_{xy}$,<br>\n",
    "and<br>\n",
    "$U_{xy} = \\log \\frac {\\mu_{xy}} {\\mu_{x0}} $ and $V_{xy} = \\log \\frac {\\mu_{xy}} {\\mu_{0y}},$<br>\n",
    "so as a result, we can compute:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "nw_x_y = np.log(mkt.eq_μ_x_y / np.repeat(mkt.eq_μ_x0.reshape(-1,1),nby,axis=1) ) - α_x_y\n",
    "gw_x_y = γ_x_y - np.log(mkt.eq_μ_x_y / np.repeat(mkt.eq_μ_0y.reshape(1,-1),nbx,axis=0) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "nw_x = np.sum(nw_x_y * mkt.eq_μ_x_y,axis = 1) / n_x\n",
    "gw_x = np.sum(gw_x_y * mkt.eq_μ_x_y,axis = 1) / n_x\n",
    "print('New wages:')\n",
    "print(nw_x)\n",
    "print('Gross wages:')\n",
    "print(gw_x)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.10.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}