Game theory - team production (simplified problem for Hjort QJE 2014).ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import necessary packages."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# import all functions from python labrary: sympy\n",
    "from sympy import *\n",
    "# import python library: matplotlib (for nice plots)\n",
    "import matplotlib\n",
    "# set sympy output to print pretty math expressions\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Declare variables we are going to use as 'symbols'. We will add more along the way, if necessary."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "### NOTE: it is important here to further declare these variables to be nonzero and positive (otherwise sympy cannot find solution for the FOCs)\n",
    "e1, e2, es, alpha1, alpha2, alphas, beta, gamma, w, theta1, theta2 = symbols('e_1, e_2, e_{s}, alpha_1, alpha2, alpha_s, beta, gamma, w, theta_1, theta_2', nonzero=True, positive=True)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2-player team production setup"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Processors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One processors, player 1, chooses their effort level to maximize their payoff. \n",
    "\n",
    "Under individual pay on their own production, the processor faces the problem\n",
    "\\begin{align}\n",
    "    \\max_{e_1} 2w(e_1 \\alpha_1)^{\\beta} (e_s \\alpha_s)^{\\gamma} - \\frac{1}{2}e_1^2\n",
    "\\end{align}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Supplier"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The supplier chooses effort towards the cooperation with the processor, $e_{s}$, so maximize their own payoff of\n",
    "\\begin{align}\n",
    "    \\max_{e_{s}} &\\quad \\overbrace{ 2w \\left( (e_1 \\alpha_1)^{\\beta} (e_s \\alpha_s)^{\\gamma} \\right) - \\frac{1}{2} e_{s}^2 }^{\\text{own private payoff}} \\\\\n",
    "        & \\qquad \\underbrace{ + \\theta_1 \\left( 2w(e_1 \\alpha_1)^{\\beta} (e_s \\alpha_s)^{\\gamma} - \\frac{1}{2}e_1^2 \\right) }_{\\text{social incentives}}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solve for the Nash equilibrium of the game"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each player chooses efforts simultaneously and independently, holding beliefs about each others' choices. \n",
    "\n",
    "The processor, holding a belief about the supplier's effort $\\hat{e}_s$, solves\n",
    "\\begin{align}\n",
    "    \\max_{e_1} 2w(e_1 \\alpha_1)^{\\beta} (\\hat{e}_s \\alpha_s)^{\\gamma} - \\frac{1}{2}e_1^2\n",
    "\\end{align}\n",
    "\n",
    "The supplier, holding a belief about the processor's effort $\\hat{e}_1$, solves\n",
    "\\begin{align}\n",
    "    \\max_{e_{s}} &\\quad \\overbrace{ 2w \\left( (\\hat{e}_1 \\alpha_1)^{\\beta} (e_s \\alpha_s)^{\\gamma} \\right) - \\frac{1}{2} e_{s1}^2 }^{\\text{own private payoff}} \\\\\n",
    "        & \\qquad \\underbrace{ + \\theta_1 \\left( 2w(\\hat{e}_1 \\alpha_1)^{\\beta} (e_s \\alpha_s)^{\\gamma} - \\frac{1}{2}\\hat{e}_1^2 \\right) }_{\\text{social incentives}}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solve the players' best responses given beliefs about team members' effort levels"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note: we omit the hats here to simplify the notations. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Processor's problem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - 0.5 e_{1}^{2} + 2 w \\left(\\alpha_{1} e_{1}\\right)^{\\beta} \\left(\\alpha_{s} e_{s}\\right)^{\\gamma}$"
      ],
      "text/plain": [
       "        2              β           γ\n",
       "- 0.5⋅e₁  + 2⋅w⋅(α₁⋅e₁) ⋅(αₛ⋅e_{s}) "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define payoff function for the processors (equation 10 in appendix)\n",
    "u1 = 2*w* (e1*alpha1)**beta * (es*alphas)**gamma - 1/2 * e1**2\n",
    "# check profit function\n",
    "u1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left(\\frac{0.5 \\alpha_{1}^{- \\beta} \\alpha_{s}^{- \\gamma} e_{s}^{- \\gamma}}{\\beta w}\\right)^{\\frac{1}{\\beta - 2}}\\right]$"
      ],
      "text/plain": [
       "⎡                         1  ⎤\n",
       "⎢                       ─────⎥\n",
       "⎢                       β - 2⎥\n",
       "⎢⎛      -β   -γ      -γ⎞     ⎥\n",
       "⎢⎜0.5⋅α₁  ⋅αₛ  ⋅e_{s}  ⎟     ⎥\n",
       "⎢⎜─────────────────────⎟     ⎥\n",
       "⎣⎝         β⋅w         ⎠     ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# solve for processors' best response functions (equation 11 in appendix)\n",
    "e1br_sol = solve(u1.diff(e1), e1)\n",
    "display(e1br_sol)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(\\frac{0.5 \\alpha_{1}^{- \\beta} \\alpha_{s}^{- \\gamma} e_{s}^{- \\gamma}}{\\beta w}\\right)^{\\frac{1}{\\beta - 2}}$"
      ],
      "text/plain": [
       "                         1  \n",
       "                       ─────\n",
       "                       β - 2\n",
       "⎛      -β   -γ      -γ⎞     \n",
       "⎜0.5⋅α₁  ⋅αₛ  ⋅e_{s}  ⎟     \n",
       "⎜─────────────────────⎟     \n",
       "⎝         β⋅w         ⎠     "
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# retrieve the first element of these solutions\n",
    "e1br = e1br_sol[0].simplify()\n",
    "e1br"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In markdown, we can type up the processors' best response functions\n",
    "\\begin{align}\n",
    "    e_1 &= \\left( 2 w \\beta (e_{s} \\alpha_s)^{\\gamma} \\alpha_1^{\\beta} \\right)^{\\frac{1}{2-\\beta}} \\tag{P} \n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Plot the processor's best responses"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "# assume some values on exogenous variables for plots\n",
    "alpha1_value = 0.5\n",
    "alphas_value = 0.5\n",
    "beta_value = 0.5\n",
    "gamma_value = 0.5\n",
    "w_value = 10\n",
    "theta1_value = 0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot the best response for processor 1\n",
    "e1plot_fun = e1br.subs([(alpha1,alpha1_value),(alphas,alphas_value),(beta,beta_value),(gamma,gamma_value),(w,w_value)])\n",
    "e1plot = plot(e1plot_fun, (es,0,10), ylim=[0,10], xlabel=\"$\\hat{e}_{s}$\", ylabel=\"$e_1$\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Supplier's problem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle - 0.5 e_{s}^{2} + \\theta_{1} \\left(- 0.5 e_{1}^{2} + 2 w \\left(\\alpha_{1} e_{1}\\right)^{\\beta} \\left(\\alpha_{s} e_{s}\\right)^{\\gamma}\\right) + 2 w \\left(\\alpha_{1} e_{1}\\right)^{\\beta} \\left(\\alpha_{s} e_{s}\\right)^{\\gamma}$"
      ],
      "text/plain": [
       "           2      ⎛        2              β           γ⎞              β       \n",
       "- 0.5⋅e_{s}  + θ₁⋅⎝- 0.5⋅e₁  + 2⋅w⋅(α₁⋅e₁) ⋅(αₛ⋅e_{s}) ⎠ + 2⋅w⋅(α₁⋅e₁) ⋅(αₛ⋅e_\n",
       "\n",
       "    γ\n",
       "{s}) "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define payoff function for the supplier\n",
    "us = 2*w*( (e1*alpha1)**beta * (es*alphas)**gamma ) - 1/2 * (es)**2 + theta1*u1\n",
    "# check profit function\n",
    "us"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{- 1.0 e_{s}^{2} + 2 \\gamma \\theta_{1} w \\left(\\alpha_{1} e_{1}\\right)^{\\beta} \\left(\\alpha_{s} e_{s}\\right)^{\\gamma} + 2 \\gamma w \\left(\\alpha_{1} e_{1}\\right)^{\\beta} \\left(\\alpha_{s} e_{s}\\right)^{\\gamma}}{e_{s}}$"
      ],
      "text/plain": [
       "           2                   β           γ                β           γ\n",
       "- 1.0⋅e_{s}  + 2⋅γ⋅θ₁⋅w⋅(α₁⋅e₁) ⋅(αₛ⋅e_{s})  + 2⋅γ⋅w⋅(α₁⋅e₁) ⋅(αₛ⋅e_{s}) \n",
       "─────────────────────────────────────────────────────────────────────────\n",
       "                                  e_{s}                                  "
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# derive supplier's FOCs \n",
    "us.diff(es).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left(\\frac{0.5 \\alpha_{1}^{- \\beta} \\alpha_{s}^{- \\gamma} e_{1}^{- \\beta}}{\\gamma w \\left(\\theta_{1} + 1.0\\right)}\\right)^{\\frac{1}{\\gamma - 2}}\\right]$"
      ],
      "text/plain": [
       "⎡                      1  ⎤\n",
       "⎢                    ─────⎥\n",
       "⎢                    γ - 2⎥\n",
       "⎢⎛      -β   -γ   -β⎞     ⎥\n",
       "⎢⎜0.5⋅α₁  ⋅αₛ  ⋅e₁  ⎟     ⎥\n",
       "⎢⎜──────────────────⎟     ⎥\n",
       "⎣⎝  γ⋅w⋅(θ₁ + 1.0)  ⎠     ⎦"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "## solve for processors' best response functions (sympy cannot do it)\n",
    "esbr_sol = solve( us.diff(es), es)\n",
    "esbr_sol\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left(\\frac{0.5 \\alpha_{1}^{- \\beta} \\alpha_{s}^{- \\gamma} e_{1}^{- \\beta}}{\\gamma w \\left(\\theta_{1} + 1.0\\right)}\\right)^{\\frac{1}{\\gamma - 2}}$"
      ],
      "text/plain": [
       "                      1  \n",
       "                    ─────\n",
       "                    γ - 2\n",
       "⎛      -β   -γ   -β⎞     \n",
       "⎜0.5⋅α₁  ⋅αₛ  ⋅e₁  ⎟     \n",
       "⎜──────────────────⎟     \n",
       "⎝  γ⋅w⋅(θ₁ + 1.0)  ⎠     "
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# retrieve the first element of these solutions\n",
    "esbr = esbr_sol[0].simplify()\n",
    "esbr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In markdown, we can type up the supplier's best response functions\n",
    "\\begin{align}\n",
    "    e_s &= \\left( (1+\\theta_1) 2 w \\gamma (e_1 \\alpha_1)^{\\beta} \\alpha_s^{\\gamma} \\right)^{\\frac{1}{2-\\gamma}} \\tag{P} \n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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xaSCc+Mifn8ovPuU5vt64Rlsj8H+z4f2R2r2XIAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle 2.92401773821287 e_{1}^{0.333333333333333} \\left(\\theta_{1} + 1.0\\right)^{0.666666666666667}$"
      ],
      "text/plain": [
       "                   0.333333333333333           0.666666666666667\n",
       "2.92401773821287⋅e₁                 ⋅(θ₁ + 1.0)                 "
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "esplot_fun"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot the best response for supplier\n",
    "esplot_fun = esbr.subs([(alpha1,alpha1_value),(alphas,alphas_value),(beta,beta_value),(gamma,gamma_value),(w,w_value),(theta1,theta1_value)])\n",
    "esplot = plot(esplot_fun, (e1,0,10), ylim=[0,10], xlabel=\"$\\hat{e}_{1}$\", ylabel=\"$e_s$\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Nash Equilibrium"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Nash equilibrium of this model is determined by the two first order conditions, where each player chooses their best responses according the correct beliefs about each others' strategies\n",
    "\\begin{align}\n",
    "    e_1 &= \\left( 2 w \\beta (e_{s} \\alpha_s)^{\\gamma} \\alpha_1^{\\beta} \\right)^{\\frac{1}{2-\\beta}} \\tag{P} \\\\\n",
    "    e_{s} &= \\left( (1+\\theta_1) 2 w \\gamma (e_1 \\alpha_1)^{\\beta} \\alpha_s^{\\gamma} \\right)^{\\frac{1}{2-\\gamma}} \\tag{S}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solve for equilibrium efforts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "# calculate equilibrium efforts\n",
    "equil_sol = solve( (u1.diff(e1), us.diff(es)) , e1, es)\n",
    "equil_sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left(\\frac{0.5 \\alpha_{1}^{- \\beta} \\alpha_{s}^{- \\gamma} \\left(1.4142135623731 \\gamma^{0.5} w^{0.5} \\left(\\theta_{1} + 1.0\\right)^{0.5}\\right)^{- \\frac{\\beta \\gamma}{\\beta + \\gamma - 2.0}}}{\\beta w}\\right)^{\\frac{1}{\\beta - 2}} \\left(2.0^{0.5 \\beta + 1.0} \\alpha_{1}^{\\beta} \\alpha_{s}^{\\gamma} \\beta^{0.5 \\beta} \\gamma w^{0.5 \\beta + 1.0} \\left(\\theta_{1} + 1.0\\right)\\right)^{\\frac{\\gamma}{\\left(\\beta - 2\\right) \\left(\\beta + \\gamma - 2\\right)}}, \\  \\left(1.4142135623731 \\gamma^{0.5} w^{0.5} \\left(\\theta_{1} + 1.0\\right)^{0.5}\\right)^{\\frac{\\beta}{\\beta + \\gamma - 2}} \\left(\\frac{2.0^{- 0.5 \\beta - 1.0} \\alpha_{1}^{- \\beta} \\alpha_{s}^{- \\gamma} \\beta^{- 0.5 \\beta} w^{- 0.5 \\beta - 1.0}}{\\gamma \\left(\\theta_{1} + 1.0\\right)}\\right)^{\\frac{1}{\\beta + \\gamma - 2}}\\right)$"
      ],
      "text/plain": [
       "⎛                                                                      1      \n",
       "⎜                                                                    ─────    \n",
       "⎜                                                                    β - 2    \n",
       "⎜⎛                                                          -β⋅γ    ⎞         \n",
       "⎜⎜                                                       ───────────⎟         \n",
       "⎜⎜                                                       β + γ - 2.0⎟         \n",
       "⎜⎜      -β   -γ ⎛                 0.5  0.5           0.5⎞           ⎟         \n",
       "⎜⎜0.5⋅α₁  ⋅αₛ  ⋅⎝1.4142135623731⋅γ   ⋅w   ⋅(θ₁ + 1.0)   ⎠           ⎟      ⎛  \n",
       "⎜⎜──────────────────────────────────────────────────────────────────⎟     ⋅⎝2.\n",
       "⎝⎝                               β⋅w                                ⎠         \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                               γ              \n",
       "                                                      ───────────────────     \n",
       "                                                      (β - 2)⋅(β + γ - 2)     \n",
       " 0.5⋅β + 1.0   β   γ  0.5⋅β    0.5⋅β + 1.0           ⎞                     ⎛  \n",
       "0           ⋅α₁ ⋅αₛ ⋅β     ⋅γ⋅w           ⋅(θ₁ + 1.0)⎠                   , ⎝1.\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                          β                                   \n",
       "                                      ─────────                               \n",
       "                                      β + γ - 2 ⎛   -0.5⋅β - 1.0   -β   -γ  -0\n",
       "               0.5  0.5           0.5⎞          ⎜2.0            ⋅α₁  ⋅αₛ  ⋅β  \n",
       "4142135623731⋅γ   ⋅w   ⋅(θ₁ + 1.0)   ⎠         ⋅⎜─────────────────────────────\n",
       "                                                ⎝                  γ⋅(θ₁ + 1.0\n",
       "\n",
       "                            ⎞\n",
       "                            ⎟\n",
       "                            ⎟\n",
       "                       1    ⎟\n",
       "                   ─────────⎟\n",
       "                   β + γ - 2⎟\n",
       ".5⋅β  -0.5⋅β - 1.0⎞         ⎟\n",
       "    ⋅w            ⎟         ⎟\n",
       "──────────────────⎟         ⎟\n",
       ")                 ⎠         ⎠"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1_star = equil_sol[0][0].simplify()\n",
    "es_star = equil_sol[0][1].simplify()\n",
    "e1_star, es_star"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Equilibrium output level"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plugging the equilibrium effort levels into the production function yields the equilibrium output of the group."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left(\\alpha_{s} \\left(1.4142135623731 \\gamma^{0.5} w^{0.5} \\left(\\theta_{1} + 1.0\\right)^{0.5}\\right)^{\\frac{\\beta}{\\beta + \\gamma - 2}}\\right)^{\\gamma} \\left(0.5^{\\frac{1}{\\beta - 2}} \\alpha_{1} \\left(2.0^{0.5 \\beta + 1.0} \\alpha_{1}^{\\beta} \\alpha_{s}^{\\gamma} \\beta^{0.5 \\beta} \\gamma w^{0.5 \\beta + 1.0} \\left(\\theta_{1} + 1.0\\right)\\right)^{\\frac{\\gamma}{\\left(\\beta - 2\\right) \\left(\\beta + \\gamma - 2\\right)}}\\right)^{\\beta} \\left(\\frac{\\alpha_{1}^{- \\beta} \\alpha_{s}^{- \\gamma} \\left(1.4142135623731 \\gamma^{0.5} w^{0.5} \\left(\\theta_{1} + 1.0\\right)^{0.5}\\right)^{- \\frac{\\beta \\gamma}{\\beta + \\gamma - 2.0}}}{\\beta w}\\right)^{\\frac{\\beta}{\\beta - 2}} \\left(\\frac{2.0^{- 0.5 \\beta - 1.0} \\alpha_{1}^{- \\beta} \\alpha_{s}^{- \\gamma} \\beta^{- 0.5 \\beta} w^{- 0.5 \\beta - 1.0}}{\\gamma \\left(\\theta_{1} + 1.0\\right)}\\right)^{\\frac{\\gamma}{\\beta + \\gamma - 2}}$"
      ],
      "text/plain": [
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                       γ                      \n",
       "⎛                                                β    ⎞  ⎛                    \n",
       "⎜                                            ─────────⎟  ⎜     1              \n",
       "⎜                                            β + γ - 2⎟  ⎜   ─────            \n",
       "⎜   ⎛                 0.5  0.5           0.5⎞         ⎟  ⎜   β - 2    ⎛   0.5⋅\n",
       "⎝αₛ⋅⎝1.4142135623731⋅γ   ⋅w   ⋅(θ₁ + 1.0)   ⎠         ⎠ ⋅⎝0.5     ⋅α₁⋅⎝2.0    \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                     β ⎛      \n",
       "                                                          γ         ⎞  ⎜      \n",
       "                                                 ───────────────────⎟  ⎜      \n",
       "                                                 (β - 2)⋅(β + γ - 2)⎟  ⎜  -β  \n",
       "β + 1.0   β   γ  0.5⋅β    0.5⋅β + 1.0           ⎞                   ⎟  ⎜α₁  ⋅α\n",
       "       ⋅α₁ ⋅αₛ ⋅β     ⋅γ⋅w           ⋅(θ₁ + 1.0)⎠                   ⎠ ⋅⎜──────\n",
       "                                                                       ⎝      \n",
       "\n",
       "                                                           β                  \n",
       "                                                         ─────                \n",
       "                                                         β - 2                \n",
       "                                                -β⋅γ    ⎞                     \n",
       "                                             ───────────⎟                     \n",
       "                                             β + γ - 2.0⎟                     \n",
       " -γ ⎛                 0.5  0.5           0.5⎞           ⎟      ⎛   -0.5⋅β - 1.\n",
       "ₛ  ⋅⎝1.4142135623731⋅γ   ⋅w   ⋅(θ₁ + 1.0)   ⎠           ⎟      ⎜2.0           \n",
       "────────────────────────────────────────────────────────⎟     ⋅⎜──────────────\n",
       "                       β⋅w                              ⎠      ⎝              \n",
       "\n",
       "                                           \n",
       "                                           \n",
       "                                           \n",
       "                                      γ    \n",
       "                                  ─────────\n",
       "                                  β + γ - 2\n",
       "0   -β   -γ  -0.5⋅β  -0.5⋅β - 1.0⎞         \n",
       " ⋅α₁  ⋅αₛ  ⋅β      ⋅w            ⎟         \n",
       "─────────────────────────────────⎟         \n",
       "    γ⋅(θ₁ + 1.0)                 ⎠         "
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define production function\n",
    "output = (e1*alpha1)**beta * (es*alphas)**gamma\n",
    "# calculate equilibrium output\n",
    "output_equil = output.subs([(e1, e1_star), (es, es_star)]).simplify()\n",
    "output_equil"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Plot equilibrium output as a function of social preferences"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYwAAAEbCAYAAADJWrOxAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjEsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy8QZhcZAAAfc0lEQVR4nO3deXzV9Z3v8dcHQthDCEmALIQ17JsEBQVF0Frrgtatm90vj7nd7DYd2+kdp9Pxdpleb/uY2+m0ta22tjqtWrc6iiIIyL5vYQ8EQvYA2cj+uX+cKMhD5Yck53dO8n4+HjzI+eUsb4/JefP7fX+/79fcHRERkQvpEXYAERGJDyoMEREJRIUhIiKBqDBERCQQFYaIiASiwhARkUBUGCIiEogKQ0REAglUGGZ2X5BtIiLSdQXdw/jUO2z7dAfmEBGRGJfwXt80s48CHwNGmdlz53xrIFDZmcFERCS2vGdhAGuAYiAV+D/nbK8BdnRWKBERiT2myQdFRCSIC+1hAGBmNcCbzZII9ALq3D2ps4KJiEhsCVQY7j7w3NtmdhtweackEhGRmPS+D0mZ2Tp3n9PBeUREJEYFPST14XNu9gDyOHuISkREuoFAhQHccs7XLcARYHGHpxERkZils6RERCSQoFODjDaz582s3MzKzOxZMxt9gcf8tv2+u87ZlmJmr5jZgfa/B1/qf4CIiERH0KlB/gT8GRgOZAB/AR6/wGMeAT543rb7gWXuPg5Y1n5bRETiQNDCMHf/g7u3tP95jAsMerv7SqDqvM2LgUfbv34UuO2i0oqISGiCDnovN7P7gSeIFMU9wN/MLAXA3c8vhncz1N2L2x9TbGbpFxtYRETCEWjQ28wK3uPb7u7vOJ5hZiOBF9x9SvvtU+6efM73T7r7O45jmNkSYAnApEmTZu3evfuCOUVE5G2sI58s6B7GRHdveFsKsz7nbwug1MyGt+9dDAfK3u2O7v4r4FcAeXl5OpVLRCRkQccw1gTcdiHPcXZtjU8Bz76P5xARkRBcaD2MYUAm0NfMZnJ29yYJ6HeBxz4OLABSzew48ADwQ+DPZvY5oBC465LSi4hI1FzokNQNRFbWywIeOmd7DfCd93qgu3/0Xb61KGg4ERGJHe9ZGO7+KPComd3h7k9FKZOIiMSgoIPeU8xs8vkb3f1fOjiPiIjEqKCFUXvO132Am4H8jo8jIiKxKugCSueu542Z/YTIGU8iItJNBD2t9nz9gPecfFBERLqWoAso7eTs3FE9gHTg+50VSkREYk/QMYybgcHAfCAZeNHdN3daKhERiTlBD0ktBv4ApAK9gN+Z2Zc7LZWIiMScoHsYnwfmuHsdgJn9CFgL/HtnBRMRkdgSeD0MoPWc26108CyIIiIS24LuYfwOWG9mf22/fRvwm86JJCIisSjodRgPmdkKYB6RPYvPuPvWzgwmIiKxJegeBu6+BdjSiVlERCSGvd8L90REpJtRYYiISCAqDBERCUSFISIigagwREQkEBWGiIgEosIQEZFAVBgiIhKICkNERAJRYYiISCAqDBERCUSFISIigagwREQkEBWGiIgEosIQEZFAVBgiIhKICkNERAJRYYiIdEHu3uHPGXiJVhERiV2nzzSzsaCK9QWVrC+oIi8nhX+6ZVKHvoYKQ0QkDp2sa2LDkSrWH66ioraB53cU4w6JPXswIzuZUWn9O/w1VRgiInGgsraRDQVVrC+oorCqntf2lgHQO6EHN08bzn2LxnHFqCHMHJFMn149OyVDKIVhZl8DPg84sBP4jLs3hJFFRCQWVdY2sr4gsgexvqCSvSU1APTt1ZPbZmbwjetzmTNmCNOyBtE7oXMK4nxRLwwzywS+Akxy9zNm9mfgI8Aj0c4iIhIrquqa2HikkjUHK1l3uIp9pWcL4gOThnLL9AzmjI4URK+e4ZyvFNYhqQSgr5k1A/2AEyHlEBEJxan6JjYWVPHGoUrWHY7sQfTqYfTvk8DUzEHcOiP8gjhf1AvD3YvM7CdAIXAGWOruS6OdQ0Qkmk6faWZDQRVrD1Wy9nAle0uqmZGdTH5xNXk5Kfz9DRnMGZ3C1MxkEhNioyDOF8YhqcHAYmAUcAr4i5l9wt0fO+9+S4AlACNGjIh2TBGRS1LX2MLGI5GCKD7dwAs7TtDmkJjQg1kjBvO163K5auyQmC6I81lnXNzxni9odhfwQXf/XPvtTwJz3P0L7/aYvLw837RpU7QiiohctIbmVrYfO8XqgxWsOVTJ9mOnaGlzevU0rs5NY3LGIOaO7tyzmN6BdeSThTGGUQjMMbN+RA5JLQLUBiISV5pb29hx/DRrD0UKYtPRk4wc0o9D5XVMzRzEkqtHM3fMEPJyUuibGLWC6FRhjGGsN7MngS1AC7AV+FW0c4iIXIy2NmdvSQ1rDlVwsKyWF3YUU9vYAsDE4UncOyeHuWOGcMWoFAb26RVy2s4R9UNS74cOSYlItLk7RyvreONQ5FTXtYcrqaprAuDGKcNI6Z/IVWNTmTN6CCn9E0NO+67i/pCUiEhMKqtpYO2hSlYfiBxmSu7Xi90nqhmW1IcF49O4ckwqV40dwvBBfcOOGgoVhoh0W7WNLWwoqGTbsdO8vKvkrYvlkvokcOWYVBZNTOeynMGMTu2PWYf+Yz0uqTBEpNtobm1760ymNw5WsLUwciZTct8EpmYls3hmBvPGpjI5YxA9e6ggzqfCEJEuy905VF7LqgORgjh9ppmNR05iBtPaz2SaNzaVy3IGR/NU17ilwhCRLqW8ppFNR6p4Nb+MNw5WUFIdmdc0Z0g/FuSm8dmrRjF3zBCS+8XsQHXMUmGISFxraG5lQ0EVqw9WsOpABfnF1YxK7c+p+iauHJvKvPY/2Sn9wo4a91QYIhJX3J384hpWHSjnSEUdT20toqmljV49rX1OpvHMH5vKlMxB9NA4RIdSYYhIzCurbmD1wQpWH6hg5YEKKmobAbh81GA+cUUO83NTuWJUCv0S9ZHWmfTuikjMaWxpZfORk6w8UMHK/eXsKa4mZ0g/ahpamDc2lfnjUpk/Lo1hg/qEHbVbUWGISOjcnYKKOjYdreKlXaWsPVTJmeZWEnoYs3IG8/c3jOea3DQmDU/SYaYQqTBEJBS1jS2sOVjB6/vLWXmgnGNVZxiV2h935668LK4el8acMUMY0FsfU7FC/ydEJCrcnT3F1by+v5x9JTX8bUcxLW1Ov8SeXDkmlSXzRzN/XCojUweEHVXehQpDRDrNqfomVh+sYMW+cl7fX055TWSw+pZpw/n8/NFck5vGrJzBcbOAUHenwhCRDtPW5uw+Uc2KfWWs2F9O9ZlmDpTVktQngfm5aSzITeOa3DTSkzRYHY9UGCJySU7XN/PGwQqW7S3j9f3lb53yOi1rEB+cMowf5KYxIzuZhJ7ai4h3KgwRuShvXji3fF8ZK/aVsfnoSWZkJ3O4oo7549K4dnwaV+emkTqgd9hRpYOpMETkgmobW3jjYAUr9pWxfG/5W/MzTclM4ovXjuXa8elMz07WDK9dnApDRN5RQUUdr+0tY/neMppb21hfUMXA3gnMz01lwfh0FmgsottRYYgIAI3NrWw8cjJSEvvKKKioA2Bs+gAWT8/ga9fnMitnML00FtFtqTBEurGK2kaW7y3jtb1lHC6vZV9pLYkJPZg7egifvnIkCyeka5ZXeYsKQ6QbcXf2ltSwLL+UZXvL2HbsFO4wNKk3145P55s3jOeqsamaxE/ekX4qRLq4xpZW1h6qZFl+GcvySxncP5HdJ6qZnjWIry7KZdHEdCZnJGnNarkgFYZIF1RR28iq/eW8tLuEVQcqqG9qpW+vnswbl8p1E9O5dkI66QM1YC0XR4Uh0gVE1q6u49X8Ul7dU8rmwpPMHplCYWU9t8/M5LqJQ5k7ZojWrZZLosIQiVOtbc7moyfZUFDJU1uK3jqraUpmEvctGsf1E9OZlDFIh5qkw6gwROJIfVMLqw5U8MqeUl7bW0ZVXRMzsgaRndKPz84bxaIJ6WQk9w07pnRRKgyRGFdZ28iy/DKW7inhSGU9B9sn81s4IZ3rJg3lmtw0BvbpFXZM6QZUGCIxqLCyjpd3l/LKnlI2Ha2izSEzuS93z85idk4Ks0el6AI6iToVhkgMeHNxoaW7S3l5dwl7S2oYmz6AhB7GlxaO4wOThurUVwmdCkMkJG1tzpbCk7y0q4RjJ+t5eXcpZpCXM5jv3jSR6ycNJWdI/7BjirxFhSESRU0tbWw4UsmLO0t4ZU8p5TWNJPbswc3ThvPDD09l0cShpA3UtOASm1QYIp2sobmVlfvLeWlXCa/mlzJxeBI7i05z7fh0bpgyjGvHa9Ba4oMKQ6QT1DW2sPpgBc9tP8HyvWXUN7WS1CeB6yYN5aYpw7lqXKouopO4o8IQ6SA1Dc28treMF3cWs2JfOZMzkiisqmfxjExunDKMOaOHkJigM5skfqkwRC7B6TPNvLqnlO3HTvHExmM0tbYxNKk3H718BDdOGUreyCFahU66jFAKw8ySgYeBKYADn3X3tWFkEblYp+ubWbqnhBd3FrP6YAXNrc7skYO5d24OH5o6jJnZg+mhkpAuKKw9jJ8BL7n7nWaWCGiFFolpp88088qeUpbuKWbV/krONLeSmdyXT80dyYemDWdGVrJKQrq8qBeGmSUBVwOfBnD3JqAp2jlELqSmoZlX80v5245iVu6voKm1jczkvvzdgtFck5vO9CxN7CfdSxh7GKOBcuB3ZjYd2Azc5+51IWQReZv6phZezS/jtfxSXtxVQlNLG8MH9eGTc3O4adpwZmQnqySk2zJ3j+4LmuUB64Cr3H29mf0MqHb3/3Xe/ZYASwBGjBgx6+jRo1HNKd1HQ3MrK/aV8/yOEyzLL6WhuY2FE9LJGdKPm6cN15iExLMO/cENozCGAevcfWT77fnA/e5+07s9Ji8vzzdt2hSlhNIdNLe2sfpgBesPV/HYuqPUNraQ0j+RD00dxi3TMpg9MkUlIV1Bh/4QR/2QlLuXmNkxMxvv7vuARcCeaOeQ7qe1tY2NR0/y3PYT/PfOYk7WNzMmbUCkJKZnMHf0EBI0A6zIuwrrLKkvA39sP0PqMPCZkHJIF+fu7D5RzbPbinhh+wma25y6xlaumzSUW6dncHVuKr0TdMW1SBChFIa7bwPywnht6R6OVtbx7LYTbCioYvXBChJ6GAvGp3HHrCyuyU2jX6KuWRW5WPqtkS6jsraRV/PLeGJjIVsLTwFw16ws/vftU7lxyjAG908MOaFIfFNhSFw709TK0j0lPLvtBK/vL2da5iDONLdy/40TuHV6hta3FulAKgyJO61tzrrDlTy9pYiy6gZWHawgY1Afllw9mttmZDB+WFLYEUW6JBWGxI19JdU8s/UEf91aREl1AwN7J3DHZZl8ceFYLtdpsCKdToUhMa2itpHntp3g6a3H2VVUzYzsZCZnJPHdmydy3cShWlNCJIpUGBJzGltaeS2/jKe2HKfo1Bnyi2uYmjmIB26ZxC3TM0gdoCVMRcKgwpCY4O7sLDrNs9tO8NSW45yqbyZ9YG8+deVIfnrPUMYPGxh2RJFuT4UhoSqvaeSZrUU8ufk4+0prmDM6hfnj0rjjskzmjU3VldciMUSFIVHX3NrG8r1lvJpfylNbimhtc2ZkJ/Ovt03h5mnDSe6n6yVEYpEKQ6LmQGkNf9l8nKe3HKeitonUAYl8ccEYbpmewbihOuQkEutUGNKp6hpb+NuOYv5r0zFO1jVRWFXPoonp3J2XzTW5aTrkJBJHVBjS4dydbcdO8dSW4/x1SxF1Ta2MSevPPbOz+PBl2aQN1FlOIvFIhSEd5nR9M3/depwnNh5jb0lkAPtDU4fzkcuzuWzEYK1UJxLnVBhySdydTUdP8uqeUh5Zc4TGljamZg7iwduncOv0DAb26RV2RBHpICoMeV9O1zfz9Nbj/Gl9IQfKahmR0pe7ZmXxkctHMCVzUNjxRKQTqDAkMHdn67FTLN1dyu/eKKCxpY3p2cn8+I5p3Dx9uNaYEOni9BsuF1TX2MIz24p4bF0h+cXVjEjpy52zsvjYFSOYnKG9CZHuQoUh72p/aQ0vbD/Bb984Qm1jCxOHJ/Hg7VNYPCOTAb31oyPS3ei3Xt6mubWNpbtL+f3aI6wvqKJfrx7cMiOTu/OyuWxEss50EunGVBgCQFl1A8/vOMEvXz9MWU0jWYP7cv+NE7g7L5sULW0qIqgwujV3Z0vhKR5dc4T/3lXMoL69mJo5iB/OzeGa3HR6akEiETmHCqMbamxp5fntxSzdXcLSPaUM7J3AvXNG8sm5OYxM7R92PBGJUSqMbqSspoHH1hXyp/VHqaht4gOThvKvt03h9pmZ9NcgtohcgD4luoFdRad5ZmsRj649Qkubs3B8Op++aiTzxqZqEFtEAlNhdFGtbc6r+aX8ZnUBGwqqGNC7J5+5ahQfu3yEDjuJyPuiwuhi6hpbeGHHCf5jxSGOVtaTmdyX7940kbtnZ5OkeZ1E5BKoMLqIsuoGHllzhMfWHWVwv0SGDEjkWzdM4IbJQ7XmhIh0CBVGnNtfUs2vVxXwzLYiWtqcD04exufnj2ZWzuCwo4lIF6PCiEPuzoaCKv7z9UOsP1zJgD69+OjlI/jcvFHkDNH4hIh0DhVGHGlrc5buKeU/Xz/EtmOnSOmfyN8tGMsn5owgpb9WsRORzqXCiANNLW08s62IF7afYOWBCrJT+vL9xZO5c1Y2fRN7hh1PRLoJFUYMO9PUyhMbC/n1ysOcON3AxOFJ/MfHL+MDkzSQLSLRp8KIQTUNzfzXhmP84vVDVNY1cfnIFB788FQW5KbpQjsRCY0KI4acqm/it28c4ZE3CqhuaOGevCzuzMtm9siUsKOJiIRXGGbWE9gEFLn7zWHliAUVtY38ZnUBv19zhLqmVm6YPJQvLxyntbFFJKaEuYdxH5APJIWYIVQVtY38euVhdhadZu3hSm6aOpwvLRzLhGHd9i0RkRgWSmGYWRZwE/Ag8PUwMoSpsraRX608zO/XHqWxpZXbZmTyyteuZmz6wLCjiYi8q7D2MH4KfAvoVp+QlbWN/GrVYX6/JlIUi2dk8qWFYxmTNiDsaCIiFxT1wjCzm4Eyd99sZgve435LgCUAI0aMiFK6znH6TDMPrzrMXzYeo645MkbxpYXjGJuuohCR+GHuHt0XNPsBcC/QAvQhMobxtLt/4t0ek5eX55s2bYpSwo5T19jCI2uO8MvXD1Hd0MJN04bztevG6dCTiERLh56HH/U9DHf/NvBtgPY9jG++V1nEo8aWVp7YcIxfrDhESXUDiyak8/UP5DI5Q2c9iUj80nUYHai1zXlmaxEPvbKfolNnuGd2NvfMzuayEZo5VkTiX6iF4e4rgBVhZugI7s6y/DL+7eV97CutYUpmEj+8Yyrzx6WFHU1EpMNoD+MSbT92igdfzKetzWlsaeX/fWwmH5oynB49NIWHiHQtKoz3qbCynh+/vJcXdhSTOiCRr1+fy1152fTSpIAi0kWpMC7S6TPN/Puy/SzfV86JUw18ZeFYllwzhgG99VaKSNemT7mAWlrbeHxDIf/31QOcrG9iyfzRfHbeKIYm9Qk7mohIVKgwAlixr4wH/5bPgbJa5oxO4bs3TdLEgCLS7agw3sORijq+/8IeTp1ppqm1jV/eO4sPTBqqNSlEpFtSYbyDusYWfr78IA+vKqBXT+Pr14/nE3NH0DtBy6GKSPelwjiHu/PSrhK+9/weSqob+PDMTO6/cQLpGqcQEVFhvOlQeS3/9OwujlXVk57Um59/fCazcrTSnYjIm7p9YTQ0t/Lz5Qf55euH6d2rB9+6YTwfvXwECbqeQkTkbbp1YSzfV8YDz+6msKqe22dm8u0PTSB9oA4/iYi8k25ZGBW1jXzv+T0UVtaR0NP40+ev4MqxqWHHEhGJad2qMNydv24t4l9e2EN9YytfvW4cn5s/Smc/iYgE0G0K4/jJer7z112s3F/OrJzB/PDDUxk3VAsZiYgE1eULo7W1jd+vO8q/vbwPgO/dOpl75+RoNlkRkYvUpQujsLKeb/5lO63uzB6ZwoO3TyFrcL+wY4mIxKUuWRjuzp82FPLg3/LpacYDt07ijsuyNKWHiMgl6HKFUXK6gX94agev7y9n3thUfnznNDKS+4YdS0Qk7nWpwnhx5wkeeHYPtY0tfH/xZD5+hcYqREQ6SpcojLrGFh54bjdPbj7O7TMz+MqiXEal9g87lohIlxL3hbGr6DRffnwrRyrr+PLCsdy3aJym9RAR6QRxWxhtbc5vVhfw45f3kjqgN4//jznMGT0k7FgiIl1WXBZGVV0TD72yj8fWFXLD5KH86I5pJPdLDDuWiEiXFneFsf3YKb7wxy2U1zTwozumcXeeTpcVEYmGuCkMd+eJjcd44NndpA3szZP/80qmZSWHHUtEpNuIi8Jwh289uYO/bD7O1blp/OyeGQzur0NQIiLRFBeFcbC8lsrNx/nKonHct2gcPXVthYhI1MVFYTS3tvG7T8/m2gnpYUcREem24uKChQnDBqosRERCFheF0UNnQYmIhC4uCkNERMKnwhARkUBUGCIiEogKQ0REAlFhiIhIIFEvDDPLNrPlZpZvZrvN7L5oZxARkYsXxoV7LcA33H2LmQ0ENpvZK+6+J4QsIiISUNT3MNy92N23tH9dA+QDmdHOISIiFyfUMQwzGwnMBNaHmUNERC4stMIwswHAU8BX3b36Hb6/xMw2mdmmM2fORD+giIi8jbl79F/UrBfwAvCyuz8U4CHRDykiEv86dF6lqBeGRZbHexSocvevBnyYCkNE5OLFfWHMA1YBO4G29s3fcfcX3+NhKgwRkYsX34XxPsVFSBGRGNOhhaErvUVEJBAVhoiIBKLCEBGRQFQYIiISiApDREQCUWGIiEggKgwREQlEhSEiIoGoMEREJBAVhoiIBKLCEBGRQFQYIiISiApDREQCUWGIiEggKgwREQlEhSEiIoGoMEREJBAVhoiIBKLCEBGRQFQYIiISiApDREQCUWGIiEggKgwREQlEhSEiIoGoMEREJBAVhoiIBKLCEBGRQFQYIiISiApDREQCUWGIiEggKgwREQlEhSEiIoGoMEREJBAVhoiIBKLCEBGRQFQYIiISiApDREQCUWGIiEgg5u5hZ7ggM9sFNISdI0akAhVhh4gRei/O0ntxlt6Ls/q4+5SOerKEjnqiTtbg7nlhh4gFZrZJ70WE3ouz9F6cpffiLDPb1JHPp0NSIiISiApDREQCiZfC+FXYAWKI3ouz9F6cpffiLL0XZ3XoexEXg94iIhK+eNnDEBGRkMV0YZjZB81sn5kdNLP7w84TFjPLNrPlZpZvZrvN7L6wM4XNzHqa2VYzeyHsLGEys2Qze9LM9rb/fMwNO1NYzOxr7b8fu8zscTPrE3amaDGz35pZWfslCG9uSzGzV8zsQPvfgy/1dWK2MMysJ/Bz4EZgEvBRM5sUbqrQtADfcPeJwBzgi934vXjTfUB+2CFiwM+Al9x9AjCdbvqemFkm8BUgr/26g57AR8JNFVWPAB88b9v9wDJ3Hwcsa799SWK2MIDLgYPuftjdm4AngMUhZwqFuxe7+5b2r2uIfChkhpsqPGaWBdwEPBx2ljCZWRJwNfAbAHdvcvdT4aYKVQLQ18wSgH7AiZDzRI27rwSqztu8GHi0/etHgdsu9XViuTAygWPn3D5ON/6QfJOZjQRmAuvDTRKqnwLfAtrCDhKy0UA58Lv2w3MPm1n/sEOFwd2LgJ8AhUAxcNrdl4abKnRD3b0YIv/oBNIv9QljuTDsHbZ161O6zGwA8BTwVXevDjtPGMzsZqDM3TeHnSUGJACXAb9w95lAHR1w2CEetR+fXwyMAjKA/mb2iXBTdT2xXBjHgexzbmfRjXYxz2dmvYiUxR/d/emw84ToKuBWMztC5DDlQjN7LNxIoTkOHHf3N/c2nyRSIN3RdUCBu5e7ezPwNHBlyJnCVmpmwwHa/y671CeM5cLYCIwzs1FmlkhkAOu5kDOFwsyMyHHqfHd/KOw8YXL3b7t7lruPJPIz8Zq7d8t/Sbp7CXDMzMa3b1oE7AkxUpgKgTlm1q/992UR3fQEgHM8B3yq/etPAc9e6hPG7OSD7t5iZl8CXiZyxsNv3X13yLHCchVwL7DTzLa1b/uOu78YYiaJDV8G/tj+j6rDwGdCzhMKd19vZk8CW4icVbiVbnTFt5k9DiwAUs3sOPAA8EPgz2b2OSKFetclv46u9BYRkSBi+ZCUiIjEEBWGiIgEosIQEZFAVBgiIhKICkNERAJRYYiISCAqDBERCUSFISISB8zsOjP7w0XcP8vM7unIDCoMEZH4MJ3IFexBLaKD5xZTYYiIxIfpwDAzW2VmJWZ2HUD7fHvPmtkmM9tgZuPNbB7wEHCnmW1rv8+dZrbOzLab2WozS7vYACoMEZH4MB2ocPf5wBeAj7fPYv0w8HV3zwP+Gbjf3VcTmcB1sbvPcPcCYLm7z3H36cArwN0XGyBmJx8UEZGI9mJIIbJIFEQ+u08RWUVvMvBUZJJeEoBV7fcZD+w752k+3T6m0RsYBnzHzEYD/wgMcvc7L5RDhSEiEvsmAdvd/c1VJqcBu4jsdfyju//m3Dub2RAiqw42t9/+JJFlrxe6e62ZrQR2u/th4HPtM/1ekA5JiYjEvunA9nNuTwN2EFmO9gYz6wFgZlPb1wMZxdsXnJsKrGkvizuILC6182JDqDBERGLfdCIF8aYpRPYwfkvkczy/fa2cf/DImhV7iayNscvMrgQeBb5iZquAXOCwu9ddbAithyEi0k21H7p6ELgeeNjdf/Ce91dhiIhIEDokJSIigagwREQkEBWGiIgEosIQEZFAVBgiIhKICkNERAJRYYiISCAqDBERCUSFISIigfx/ZNX8crqBTvUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot the production function\n",
    "output_equil_plot_fun = output_equil.subs([(alpha1,alpha1_value),(alphas,alphas_value),(beta,beta_value),(gamma,gamma_value),(w,w_value)])\n",
    "output_equil_plot = plot(output_equil_plot_fun, (theta1,0,10), ylim=[0,10], xlabel=\"$\\theta_{1}$\", ylabel=\"output\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Comparative Statics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q: what comparative statics can you think of analyzing in this model?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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