Game theory - 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": 7,
   "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, es1, es2, alpha1, alpha2, alphas, beta, gamma, w, theta1, theta2 = symbols('e_1, e_2, e_{s1}, e_{s2}, alpha_1, alpha2, alpha_s, beta, gamma, w, theta_1, theta_2', nonzero=True, positive=True)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3-player team production setup"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Processors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Two processors, 1 and 2, choose their effort levels independently to maximize their payoffs. \n",
    "\n",
    "Under individual pay on their own production, player 1 faces the problem\n",
    "\\begin{align}\n",
    "    \\max_{e_1} 2w(e_1 \\alpha_1)^{\\beta} (e_{s1} \\alpha_s)^{\\gamma} - \\frac{1}{2}e_1^2\n",
    "\\end{align}\n",
    "and player 2 faces the problem\n",
    "\\begin{align}\n",
    "    \\max_{e_2} 2w(e_2 \\alpha_2)^{\\beta} (e_{s2} \\alpha_s)^{\\gamma} - \\frac{1}{2}e_2^2\n",
    "\\end{align}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Supplier"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The supplier of the group chooses efforts towards the cooperation with each player, $e_{s1}$ and $e_{s2}$, so maximize their own payoff of\n",
    "\\begin{align}\n",
    "    \\max_{e_{s1}, e_{s2}} &\\quad \\overbrace{ w \\left( (e_1 \\alpha_1)^{\\beta} (e_{s1} \\alpha_s)^{\\gamma} + (e_2 \\alpha_2)^{\\beta} (e_{s2} \\alpha_s)^{\\gamma} \\right) - \\frac{1}{2}(e_{s1}+e_{s2})^2 }^{\\text{own private payoff}} \\\\\n",
    "        & \\qquad \\underbrace{ + \\theta_1 \\left( 2w(e_1 \\alpha_1)^{\\beta} (e_{s1} \\alpha_s)^{\\gamma} - \\frac{1}{2}e_1^2 \\right) + \\theta_2 \\left( 2w(e_2 \\alpha_2)^{\\beta} (e_{s2} \\alpha_s)^{\\gamma} - \\frac{1}{2}e_2^2 \\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": [
    "### 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": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
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bLcg1WgtpDg8+4tTUhCO+WYXgwPemT7Gt4pPre/vCLsfDqaQJ66Kn+xPCA/1r+maiwU3fTACTiHZj6zRXYBJ4WJQhYAgYAoaAIWAINICAN2AeNMCrsWgIGAKGgCFgCBgChsAAATNgBnDYjSFgCBgChoAhYAi0gIAZMC20kvFoCBgChoAhYAgYAgMEzIAZwGE3hoAhYAgYAoaAIdACAmbAtNBKxqMhYAgYAoaAIWAIDBAwA2YAh90YAoaAIWAIGAKGQAsIeAPmR72WxL9Be893NswZAoaAIWAIGAKGgCFwrwjINnkX2Cf84RofYj37PwJtbS/4meguAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left[ - 0.5 e_{1}^{2} + 2 w \\left(\\alpha_{1} e_{1}\\right)^{\\beta} \\left(\\alpha_{s} e_{s1}\\right)^{\\gamma}, \\  - 0.5 e_{2}^{2} + 2 w \\left(\\alpha_{2} e_{2}\\right)^{\\beta} \\left(\\alpha_{s} e_{s2}\\right)^{\\gamma}\\right]$"
      ],
      "text/plain": [
       "⎡        2              β            γ          2              β            γ⎤\n",
       "⎣- 0.5⋅e₁  + 2⋅w⋅(α₁⋅e₁) ⋅(αₛ⋅e_{s1}) , - 0.5⋅e₂  + 2⋅w⋅(α₂⋅e₂) ⋅(αₛ⋅e_{s2}) ⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define payoff function for the processors (equation 10 in appendix)\n",
    "u1 = 2*w* (e1*alpha1)**beta * (es1*alphas)**gamma - 1/2 * e1**2\n",
    "u2 = 2*w* (e2*alpha2)**beta * (es2*alphas)**gamma - 1/2 * e2**2\n",
    "# check profit function\n",
    "[u1, u2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ROioQaDwAXH9tJa3b447ADgIM2Eo8WoWjwVOZlztgeoIj4AhsEHClc0lwBGZGwJVuZsC9OkcgXdPxMdM6Iqt8YbDeCL92BOZEQHrEi92tTkSUDmdH22M27giZs7e8rmNBoPMVpP8DREI1wW08iTYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left(\\frac{0.5 \\alpha_{1}^{- \\beta} \\alpha_{s}^{- \\gamma} e_{s1}^{- \\gamma}}{\\beta w}\\right)^{\\frac{1}{\\beta - 2}}\\right]$"
      ],
      "text/plain": [
       "⎡                          1  ⎤\n",
       "⎢                        ─────⎥\n",
       "⎢                        β - 2⎥\n",
       "⎢⎛      -β   -γ       -γ⎞     ⎥\n",
       "⎢⎜0.5⋅α₁  ⋅αₛ  ⋅e_{s1}  ⎟     ⎥\n",
       "⎢⎜──────────────────────⎟     ⎥\n",
       "⎣⎝         β⋅w          ⎠     ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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lCrBENr2nxFzuZ+kT0fFNpexG59HhiYrnAo+1lZIKAn0RwFPi0cLVL4TXCZ7VvbQK6J1kcDycWuZ2BkrZd0JAesMzlhjcpuZxofCjGB0VCDQeAI5fWwnrLscFgQME6LB1cbMGh8BjuZcHYJYLBYGCwA6BYnRFEwoCEyNQjG5iwEt1BYFwTsfHTGNEVvnCYCxEOS8ITImA7IgXuxuDiBgdwY6mx2xKIGTK1ip1bQWB1leQ/g8742lD6dXwuQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left(\\frac{0.5 \\alpha_{2}^{- \\beta} \\alpha_{s}^{- \\gamma} e_{s2}^{- \\gamma}}{\\beta w}\\right)^{\\frac{1}{\\beta - 2}}\\right]$"
      ],
      "text/plain": [
       "⎡                          1  ⎤\n",
       "⎢                        ─────⎥\n",
       "⎢                        β - 2⎥\n",
       "⎢⎛      -β   -γ       -γ⎞     ⎥\n",
       "⎢⎜0.5⋅α₂  ⋅αₛ  ⋅e_{s2}  ⎟     ⎥\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)\n",
    "e2br_sol = solve(u2.diff(e2), e2)\n",
    "display(e2br_sol)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left(\\frac{0.5 \\alpha_{1}^{- \\beta} \\alpha_{s}^{- \\gamma} e_{s1}^{- \\gamma}}{\\beta w}\\right)^{\\frac{1}{\\beta - 2}}, \\  \\left(\\frac{0.5 \\alpha_{2}^{- \\beta} \\alpha_{s}^{- \\gamma} e_{s2}^{- \\gamma}}{\\beta w}\\right)^{\\frac{1}{\\beta - 2}}\\right)$"
      ],
      "text/plain": [
       "⎛                          1                              1  ⎞\n",
       "⎜                        ─────                          ─────⎟\n",
       "⎜                        β - 2                          β - 2⎟\n",
       "⎜⎛      -β   -γ       -γ⎞       ⎛      -β   -γ       -γ⎞     ⎟\n",
       "⎜⎜0.5⋅α₁  ⋅αₛ  ⋅e_{s1}  ⎟       ⎜0.5⋅α₂  ⋅αₛ  ⋅e_{s2}  ⎟     ⎟\n",
       "⎜⎜──────────────────────⎟     , ⎜──────────────────────⎟     ⎟\n",
       "⎝⎝         β⋅w          ⎠       ⎝         β⋅w          ⎠     ⎠"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# retrieve the first element of these solutions\n",
    "e1br = e1br_sol[0].simplify()\n",
    "e2br = e2br_sol[0].simplify()\n",
    "e1br, e2br"
   ]
  },
  {
   "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_{s1} \\alpha_s)^{\\gamma} \\alpha_1^{\\beta} \\right)^{\\frac{1}{2-\\beta}} \\tag{P1} \\\\\n",
    "    e_2 &= \\left( 2 w \\beta (e_{s2} \\alpha_s)^{\\gamma} \\alpha_2^{\\beta} \\right)^{\\frac{1}{2-\\beta}} \\tag{P2}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Plot the processor's best responses"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "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, (es1,0,10), ylim=[0,10], xlabel=\"$\\hat{e}_{s1}$\", ylabel=\"$e_1$\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Supplier's problem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\theta_{1} \\left(- 0.5 e_{1}^{2} + 2 w \\left(\\alpha_{1} e_{1}\\right)^{\\beta} \\left(\\alpha_{s} e_{s1}\\right)^{\\gamma}\\right) + \\theta_{2} \\left(- 0.5 e_{2}^{2} + 2 w \\left(\\alpha_{2} e_{2}\\right)^{\\beta} \\left(\\alpha_{s} e_{s2}\\right)^{\\gamma}\\right) + w \\left(\\left(\\alpha_{2} e_{2}\\right)^{\\beta} \\left(\\alpha_{s} e_{s2}\\right)^{\\gamma} + \\left(\\alpha_{1} e_{1}\\right)^{\\beta} \\left(\\alpha_{s} e_{s1}\\right)^{\\gamma}\\right) - 0.5 \\left(e_{s1} + e_{s2}\\right)^{2}$"
      ],
      "text/plain": [
       "   ⎛        2              β            γ⎞      ⎛        2              β     \n",
       "θ₁⋅⎝- 0.5⋅e₁  + 2⋅w⋅(α₁⋅e₁) ⋅(αₛ⋅e_{s1}) ⎠ + θ₂⋅⎝- 0.5⋅e₂  + 2⋅w⋅(α₂⋅e₂) ⋅(αₛ⋅\n",
       "\n",
       "       γ⎞     ⎛       β            γ          β            γ⎞                 \n",
       "e_{s2}) ⎠ + w⋅⎝(α₂⋅e₂) ⋅(αₛ⋅e_{s2})  + (α₁⋅e₁) ⋅(αₛ⋅e_{s1}) ⎠ - 0.5⋅(e_{s1} + \n",
       "\n",
       "       2\n",
       "e_{s2}) "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define payoff function for the supplier (equation 12 in appendix)\n",
    "us = w*( (e1*alpha1)**beta * (es1*alphas)**gamma + (e2*alpha2)**beta * (es2*alphas)**gamma) - 1/2 * (es1 + es2)**2 + theta1*u1 + theta2*u2\n",
    "# check profit function\n",
    "us"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{- 1.0 e_{s1} \\left(e_{s1} + e_{s2}\\right) + 2 \\gamma \\theta_{1} w \\left(\\alpha_{1} e_{1}\\right)^{\\beta} \\left(\\alpha_{s} e_{s1}\\right)^{\\gamma} + \\gamma w \\left(\\alpha_{1} e_{1}\\right)^{\\beta} \\left(\\alpha_{s} e_{s1}\\right)^{\\gamma}}{e_{s1}}$"
      ],
      "text/plain": [
       "                                                β            γ              β \n",
       "-1.0⋅e_{s1}⋅(e_{s1} + e_{s2}) + 2⋅γ⋅θ₁⋅w⋅(α₁⋅e₁) ⋅(αₛ⋅e_{s1})  + γ⋅w⋅(α₁⋅e₁) ⋅\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                          e_{s1}                              \n",
       "\n",
       "           γ\n",
       "(αₛ⋅e_{s1}) \n",
       "────────────\n",
       "            "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{- 1.0 e_{s2} \\left(e_{s1} + e_{s2}\\right) + 2 \\gamma \\theta_{2} w \\left(\\alpha_{2} e_{2}\\right)^{\\beta} \\left(\\alpha_{s} e_{s2}\\right)^{\\gamma} + \\gamma w \\left(\\alpha_{2} e_{2}\\right)^{\\beta} \\left(\\alpha_{s} e_{s2}\\right)^{\\gamma}}{e_{s2}}$"
      ],
      "text/plain": [
       "                                                β            γ              β \n",
       "-1.0⋅e_{s2}⋅(e_{s1} + e_{s2}) + 2⋅γ⋅θ₂⋅w⋅(α₂⋅e₂) ⋅(αₛ⋅e_{s2})  + γ⋅w⋅(α₂⋅e₂) ⋅\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                          e_{s2}                              \n",
       "\n",
       "           γ\n",
       "(αₛ⋅e_{s2}) \n",
       "────────────\n",
       "            "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# derive supplier's FOCs (equation 13 in appendix)\n",
    "display(us.diff(es1).simplify())\n",
    "display(us.diff(es2).simplify())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "## solve for processors' best response functions (sympy cannot do it)\n",
    "# esbr_sol = solve( (us.diff(es1), us.diff(es2)), es1, es2)\n",
    "# display(esbr_sol)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solve for supplier's best responses"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "    e_{s1} + e_{s2} &= (1+2\\theta_1)w(e_1 \\alpha_1)^{\\beta} \\gamma e_{s1}^{\\gamma-1}\\alpha_s^{\\gamma} \\tag{1} \\\\\n",
    "    e_{s1} + e_{s2} &= (1+2\\theta_2)w(e_2 \\alpha_2)^{\\beta} \\gamma e_{s2}^{\\gamma-1}\\alpha_s^{\\gamma} \\tag{2}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Equating (1) and (2), we have\n",
    "\\begin{align}\n",
    "    && (1+2\\theta_1)w(e_1 \\alpha_1)^{\\beta} \\gamma e_{s1}^{\\gamma-1}\\alpha_s^{\\gamma} &= (1+2\\theta_2)w(e_2 \\alpha_2)^{\\beta} \\gamma e_{s2}^{\\gamma-1}\\alpha_s^{\\gamma} \\\\\n",
    "    \\Rightarrow && (1+2\\theta_1) (e_1 \\alpha_1)^{\\beta} e_{s1}^{\\gamma-1} &= (1+2\\theta_2) (e_2 \\alpha_2)^{\\beta} e_{s2}^{\\gamma-1} \\\\\n",
    "    \\Rightarrow && \\left( \\frac{e_{s2}}{e_{s1}} \\right)^{\\gamma-1} &= \\frac{(1+2\\theta_1) (e_1 \\alpha_1)^{\\beta}}{(1+2\\theta_2) (e_2 \\alpha_2)^{\\beta} } \\\\\n",
    "    \\Rightarrow && \\frac{e_{s2}}{e_{s1}}  &= \\left( \\frac{(1+2\\theta_1) (e_1 \\alpha_1)^{\\beta}}{(1+2\\theta_2) (e_2 \\alpha_2)^{\\beta} } \\right)^{\\frac{1}{\\gamma-1}} \\\\\n",
    "    \\Rightarrow && \\frac{e_{s2}}{e_{s1}}  &= \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{1}{\\gamma-1}}  \\left( \\frac{e_1 \\alpha_1}{e_2 \\alpha_2} \\right)^{\\frac{\\beta}{\\gamma-1}} \\\\\n",
    "    \\Rightarrow && e_{s2}  &= \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{1}{\\gamma-1}}  \\left( \\frac{e_1 \\alpha_1}{e_2 \\alpha_2} \\right)^{\\frac{\\beta}{\\gamma-1}} e_{s1}  \\tag{3}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plugging (3) into (1), we have\n",
    "\\begin{align}\n",
    "    e_{s1} + e_{s2} &= (1+2\\theta_1)w(e_1 \\alpha_1)^{\\beta} \\gamma e_{s1}^{\\gamma-1}\\alpha_s^{\\gamma} \\\\\n",
    "    e_{s1} + \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{1}{\\gamma-1}}  \\left( \\frac{e_1 \\alpha_1}{e_2 \\alpha_2} \\right)^{\\frac{\\beta}{\\gamma-1}} e_{s1} &= (1+2\\theta_1)w(e_1 \\alpha_1)^{\\beta} \\gamma e_{s1}^{\\gamma-1}\\alpha_s^{\\gamma} \\\\\n",
    "    \\left( 1 + \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{1}{\\gamma-1}}  \\left( \\frac{e_1 \\alpha_1}{e_2 \\alpha_2} \\right)^{\\frac{\\beta}{\\gamma-1}} \\right) e_{s1} &= (1+2\\theta_1)w(e_1 \\alpha_1)^{\\beta} \\gamma e_{s1}^{\\gamma-1}\\alpha_s^{\\gamma} \\\\\n",
    "    e_{s1} &= \\frac{ (1+2\\theta_1)w(e_1 \\alpha_1)^{\\beta} \\gamma e_{s1}^{\\gamma-1}\\alpha_s^{\\gamma} }{  1 + \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{1}{\\gamma-1}}  \\left( \\frac{e_1 \\alpha_1}{e_2 \\alpha_2} \\right)^{\\frac{\\beta}{\\gamma-1}}  } \\\\\n",
    "    \\frac{e_{s1}}{e_{s1}^{\\gamma-1}} &= \\frac{ (1+2\\theta_1)w(e_1 \\alpha_1)^{\\beta} \\gamma \\alpha_s^{\\gamma} }{  1 + \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{1}{\\gamma-1}}  \\left( \\frac{e_1 \\alpha_1}{e_2 \\alpha_2} \\right)^{\\frac{\\beta}{\\gamma-1}}  } \\\\\n",
    "    e_{s1}^{1-(\\gamma-1)} &= \\frac{ (1+2\\theta_1)w(e_1 \\alpha_1)^{\\beta} \\gamma \\alpha_s^{\\gamma} }{  1 + \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{1}{\\gamma-1}}  \\left( \\frac{e_1 \\alpha_1}{e_2 \\alpha_2} \\right)^{\\frac{\\beta}{\\gamma-1}}  } \\\\\n",
    "    e_{s1}^{2-\\gamma} &= \\frac{ (1+2\\theta_1)w(e_1 \\alpha_1)^{\\beta} \\gamma \\alpha_s^{\\gamma} }{  1 + \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{1}{\\gamma-1}}  \\left( \\frac{e_1 \\alpha_1}{e_2 \\alpha_2} \\right)^{\\frac{\\beta}{\\gamma-1}}  } \\\\\n",
    "    e_{s1} &= \\left( \\frac{ (1+2\\theta_1)w(e_1 \\alpha_1)^{\\beta} \\gamma \\alpha_s^{\\gamma} }{  1 + \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{1}{\\gamma-1}}  \\left( \\frac{e_1 \\alpha_1}{e_2 \\alpha_2} \\right)^{\\frac{\\beta}{\\gamma-1}}  } \\right)^{\\frac{1}{2-\\gamma}} \\tag{S1}\n",
    "\\end{align}\n",
    "which is equation (14) in the paper appendix (I believe there might be a typo in the appendix equation)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, we can find the best response for $e_{s2}$\n",
    "\\begin{align}\n",
    "    e_{s2} &= \\left( \\frac{ (1+2\\theta_2)w(e_2 \\alpha_2)^{\\beta} \\gamma \\alpha_s^{\\gamma} }{  1 + \\left( \\frac{1+2\\theta_2} {1+2\\theta_1} \\right)^{\\frac{1}{\\gamma-1}}  \\left( \\frac{e_2 \\alpha_2}{e_1 \\alpha_1} \\right)^{\\frac{\\beta}{\\gamma-1}}  } \\right)^{\\frac{1}{2-\\gamma}}   \\tag{S2}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Nash Equilibrium"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Nash equilibrium of this model is determined by the following four 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_{s1} \\alpha_s)^{\\gamma} \\alpha_1^{\\beta} \\right)^{\\frac{1}{2-\\beta}} \\tag{P1} \\\\\n",
    "    e_2 &= \\left( 2 w \\beta (e_{s2} \\alpha_s)^{\\gamma} \\alpha_2^{\\beta} \\right)^{\\frac{1}{2-\\beta}} \\tag{P2} \\\\\n",
    "    e_{s1} &= \\left( \\frac{ (1+2\\theta_1)w(e_1 \\alpha_1)^{\\beta} \\gamma \\alpha_s^{\\gamma} }{  1 + \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{1}{\\gamma-1}}  \\left( \\frac{e_1 \\alpha_1}{e_2 \\alpha_2} \\right)^{\\frac{\\beta}{\\gamma-1}}  } \\right)^{\\frac{1}{2-\\gamma}} \\tag{S1} \\\\\n",
    "    e_{s2} &= \\left( \\frac{ (1+2\\theta_2)w(e_2 \\alpha_2)^{\\beta} \\gamma \\alpha_s^{\\gamma} }{  1 + \\left( \\frac{1+2\\theta_2} {1+2\\theta_1} \\right)^{\\frac{1}{\\gamma-1}}  \\left( \\frac{e_2 \\alpha_2}{e_1 \\alpha_1} \\right)^{\\frac{\\beta}{\\gamma-1}}  } \\right)^{\\frac{1}{2-\\gamma}}   \\tag{S2}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solve for equilibrium efforts"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dividing (P1) by (P2), we have\n",
    "\\begin{align}\n",
    "    && \\frac{e_1}{e_2} &= \\left( \\frac{2 w \\beta (e_{s1} \\alpha_s)^{\\gamma} \\alpha_1^{\\beta}}{2 w \\beta (e_{s2} \\alpha_s)^{\\gamma} \\alpha_2^{\\beta} } \\right)^{\\frac{1}{2-\\beta}} \n",
    "    \\\\\n",
    "    &&   &= \\left( \\frac{e_{s1}^{\\gamma} \\alpha_1^{\\beta}}{ e_{s2} ^{\\gamma} \\alpha_2^{\\beta} } \\right)^{\\frac{1}{2-\\beta}} \n",
    "    \\\\\n",
    "    \\Rightarrow && \\frac{e_1 \\alpha_1 }{e_2 \\alpha_2 } &= \\left( \\frac{e_{s1}^{\\gamma} \\alpha_1^{\\beta}}{ e_{s2} ^{\\gamma} \\alpha_2^{\\beta} } \\right)^{\\frac{1}{2-\\beta}} \\frac{\\alpha_1}{\\alpha_2}\n",
    "    \\\\\n",
    "    &&   &= \\left( \\frac{e_{s1}}{e_{s2}} \\right)^{\\frac{\\gamma}{2-\\beta}} \\left( \\frac{\\alpha_1}{\\alpha_2} \\right)^{\\frac{\\beta}{2-\\beta}} \\frac{\\alpha_1}{\\alpha_2}\n",
    "    \\\\\n",
    "    \\Rightarrow &&  \\frac{e_1 \\alpha_1 }{e_2 \\alpha_2 }  &= \\left( \\frac{e_{s1}}{e_{s2}} \\right)^{\\frac{\\gamma}{2-\\beta}} \\left( \\frac{\\alpha_1}{\\alpha_2} \\right)^{\\frac{2}{2-\\beta}}  \n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, plug (3) into this equation above, we have\n",
    "\\begin{align}\n",
    "    && \\frac{e_1 \\alpha_1 }{e_2 \\alpha_2 }  &= \\left( \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{1}{1-\\gamma}}  \\left( \\frac{e_1 \\alpha_1}{e_2 \\alpha_2} \\right)^{\\frac{\\beta}{1-\\gamma}} \\right)^{\\frac{\\gamma}{2-\\beta}} \\left( \\frac{\\alpha_1}{\\alpha_2} \\right)^{\\frac{2}{2-\\beta}}  \n",
    "    \\\\\n",
    "    &&   &= \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{\\gamma}{(1-\\gamma)(2-\\beta)}}  \\left( \\frac{e_1 \\alpha_1}{e_2 \\alpha_2} \\right)^{\\frac{\\beta \\gamma}{(1-\\gamma)(2-\\beta)}} \\left( \\frac{\\alpha_1}{\\alpha_2} \\right)^{\\frac{2}{2-\\beta}}  \n",
    "    \\\\\n",
    "    \\Rightarrow && \\frac{e_1 \\alpha_1 }{e_2 \\alpha_2 } \\left( \\frac{e_1 \\alpha_1}{e_2 \\alpha_2} \\right)^{- \\frac{\\beta \\gamma}{(1-\\gamma)(2-\\beta)}}  &= \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{\\gamma}{(1-\\gamma)(2-\\beta)}}  \\left( \\frac{\\alpha_1}{\\alpha_2} \\right)^{\\frac{2}{2-\\beta}}  \n",
    "    \\\\\n",
    "    \\Rightarrow && \\left( \\frac{e_1 \\alpha_1}{e_2 \\alpha_2} \\right)^{ \\frac{2 - \\beta - 2\\gamma}{(1-\\gamma)(2-\\beta)}}  &= \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{\\gamma}{(1-\\gamma)(2-\\beta)}}  \\left( \\frac{\\alpha_1}{\\alpha_2} \\right)^{\\frac{2}{2-\\beta}} \n",
    "    \\\\\n",
    "    \\Rightarrow && \\frac{e_1 \\alpha_1}{e_2 \\alpha_2}  &= \\left( \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{\\gamma}{(1-\\gamma)(2-\\beta)}}  \\left( \\frac{\\alpha_1}{\\alpha_2} \\right)^{\\frac{2}{2-\\beta}}  \\right)^{ \\frac{(1-\\gamma)(2-\\beta)}{2 - \\beta - 2\\gamma} }\n",
    "    \\\\\n",
    "    \\Rightarrow && \\frac{e_1 \\alpha_1}{e_2 \\alpha_2}  &=  \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{\\gamma}{2 - \\beta - 2\\gamma}}  \\left( \\frac{\\alpha_1}{\\alpha_2} \\right)^{\\frac{2(1-\\gamma)}{2 - \\beta - 2\\gamma}}  \\tag{E12}\n",
    "\\end{align}\n",
    "\n",
    "which is entirely an expression of exogenous variables. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Expression (E12) allows us to rewrite (S1) and (S2) in simpler forms as\n",
    "\\begin{align}\n",
    "    e_{s1} &= \\underbrace{ \\left( \\frac{ (1+2\\theta_1)w \\alpha_1^{\\beta} \\gamma \\alpha_s^{\\gamma} }{  1 + \\left( \\frac{1+2\\theta_1} {1+2\\theta_2} \\right)^{\\frac{1}{\\gamma-1}}  \\left( \\frac{e_1 \\alpha_1}{e_2 \\alpha_2} \\right)^{\\frac{\\beta}{\\gamma-1}}  } \\right)^{\\frac{1}{2-\\gamma}} }_{\\text{define as: }D_1} e_1^{\\frac{\\beta}{2-\\gamma}} \n",
    "    = D_1 e_1^{\\frac{\\beta}{2-\\gamma}} \\tag{S1'}\n",
    "    \\\\\n",
    "    e_{s2} &= \\underbrace{ \\left( \\frac{ (1+2\\theta_2)w \\alpha_2^{\\beta} \\gamma \\alpha_s^{\\gamma} }{  1 + \\left( \\frac{1+2\\theta_2} {1+2\\theta_1} \\right)^{\\frac{1}{\\gamma-1}}  \\left( \\frac{e_2 \\alpha_2}{e_1 \\alpha_1} \\right)^{\\frac{\\beta}{\\gamma-1}}  } \\right)^{\\frac{1}{2-\\gamma}} }_{\\text{define as: }D_2}  e_2^{\\frac{\\beta}{2-\\gamma}} \n",
    "    = D_2 e_2^{\\frac{\\beta}{2-\\gamma}} \\tag{S2'}\n",
    "\\end{align}\n",
    "\n",
    "Note that $D_1$ and $D_2$ are entirely exogenous expressions after plugging in (E12) for $\\frac{e_2 \\alpha_2}{e_1 \\alpha_1}$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now use (S1') and (S2') to solve for equilibrium processor effort $e_1$ and $e_2$.\n",
    "\n",
    "Plugging in (S1') into (P1), we have\n",
    "\\begin{align}\n",
    "    &&   e_1 &= \\left( 2 w \\beta (e_{s1} \\alpha_s)^{\\gamma} \\alpha_1^{\\beta} \\right)^{\\frac{1}{2-\\beta}} \n",
    "    \\\\\n",
    "    &&   &= \\left( 2 w \\beta \\alpha_s^{\\gamma} \\alpha_1^{\\beta} \\right)^{\\frac{1}{2-\\beta}}  e_{s1} ^{\\frac{\\gamma}{2-\\beta}}\n",
    "    \\\\\n",
    "    &&   &= \\left( 2 w \\beta \\alpha_s^{\\gamma} \\alpha_1^{\\beta} \\right)^{\\frac{1}{2-\\beta}}  \\left( D_1 e_1^{\\frac{\\beta}{2-\\gamma}} \\right) ^{\\frac{\\gamma}{2-\\beta}}\n",
    "    \\\\\n",
    "    &&   &= \\left( 2 w \\beta \\alpha_s^{\\gamma} \\alpha_1^{\\beta} \\right)^{\\frac{1}{2-\\beta}}  D_1^{\\frac{\\gamma}{2-\\beta}}   e_1^{\\frac{\\beta \\gamma}{(2-\\beta)(2-\\gamma)}}  \n",
    "    \\\\\n",
    "    \\Rightarrow && e_1 e_1^{-\\frac{\\beta \\gamma}{(2-\\beta)(2-\\gamma)}}  &= \\left( 2 w \\beta \\alpha_s^{\\gamma} \\alpha_1^{\\beta} \\right)^{\\frac{1}{2-\\beta}}  D_1^{\\frac{\\gamma}{2-\\beta}}   \n",
    "    \\\\\n",
    "    \\Rightarrow && e_1^{\\frac{4 - 2\\beta - 2\\gamma}{(2-\\beta)(2-\\gamma)}}  &= \\left( 2 w \\beta \\alpha_s^{\\gamma} \\alpha_1^{\\beta} \\right)^{\\frac{1}{2-\\beta}}  D_1^{\\frac{\\gamma}{2-\\beta}}   \n",
    "    \\\\\n",
    "    \\Rightarrow && e_1^*  &= \\left( 2 w \\beta \\alpha_s^{\\gamma} \\alpha_1^{\\beta} \\right)^{\\frac{(2-\\gamma)}{4 - 2\\beta - 2\\gamma}}  D_1^{\\frac{\\gamma (2-\\gamma)}{4 - 2\\beta - 2\\gamma}}   \n",
    "\\end{align}\n",
    "\n",
    "This is the equilibrium effort by processor 1 that is entirely determined by exogenous variables."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, we can solve for $e_2^*$ by plugging in (S2') into (P2), yielding\n",
    "\\begin{align}\n",
    "    e_2^*  &= \\left( 2 w \\beta \\alpha_s^{\\gamma} \\alpha_2^{\\beta} \\right)^{\\frac{(2-\\gamma)}{4 - 2\\beta - 2\\gamma}}  D_2^{\\frac{\\gamma (2-\\gamma)}{4 - 2\\beta - 2\\gamma}}  \n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, plugging in $e_1^*$ into (S1') solves for the equilibrium effort level $e_{s1}^*$ for supplier\n",
    "\\begin{align}\n",
    "    e_{s1}^* &= D_1 (e_1^*)^{\\frac{\\beta}{2-\\gamma}} \n",
    "        \\\\\n",
    "        &= D_1 \\left( \\left( 2 w \\beta \\alpha_s^{\\gamma} \\alpha_1^{\\beta} \\right)^{\\frac{(2-\\gamma)}{4 - 2\\beta - 2\\gamma}}  D_1^{\\frac{\\gamma (2-\\gamma)}{4 - 2\\beta - 2\\gamma}}   \\right) ^{\\frac{\\beta}{2-\\gamma}} \n",
    "        \\\\\n",
    "        &= D_1 \\left( 2 w \\beta \\alpha_s^{\\gamma} \\alpha_1^{\\beta} \\right)^{\\frac{\\beta}{4 - 2\\beta - 2\\gamma}}  D_1^{\\frac{\\beta \\gamma }{4 - 2\\beta - 2\\gamma}}   \n",
    "        \\\\\n",
    "        &= \\left( 2 w \\beta \\alpha_s^{\\gamma} \\alpha_1^{\\beta} \\right)^{\\frac{\\beta}{4 - 2\\beta - 2\\gamma}}  D_1^{\\frac{4 - 2\\beta - 2\\gamma + \\beta \\gamma }{4 - 2\\beta - 2\\gamma}}\n",
    "        \\\\\n",
    "        &= \\left( 2 w \\beta \\alpha_s^{\\gamma} \\alpha_1^{\\beta} \\right)^{\\frac{\\beta}{4 - 2\\beta - 2\\gamma}}  D_1^{\\frac{(2-\\beta)(2-\\gamma) }{4 - 2\\beta - 2\\gamma}}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, plugging in $e_1^*$ into (S1') solves for the equilibrium effort level $e_{s1}^*$ for supplier\n",
    "\\begin{align}\n",
    "    e_{s2}^* &= D_2 (e_2^*)^{\\frac{\\beta}{2-\\gamma}} \n",
    "        \\\\\n",
    "        &= \\left( 2 w \\beta \\alpha_s^{\\gamma} \\alpha_2^{\\beta} \\right)^{\\frac{\\beta}{4 - 2\\beta - 2\\gamma}}  D_2^{\\frac{(2-\\beta)(2-\\gamma) }{4 - 2\\beta - 2\\gamma}}\n",
    "\\end{align}"
   ]
  },
  {
   "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": "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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