maximize_solve.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# Computational Techniques in Python"
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "import scipy\n",
        "import scipy.interpolate\n",
        "import scipy.optimize\n",
        "%matplotlib inline"
      ],
      "outputs": [],
      "execution_count": 101,
      "metadata": {}
    },
    {
      "cell_type": "markdown",
      "source": [
        "Consider a monopolist's problem to maximize its profits\n",
        "$$\n",
        "\\max_p (p-c)D(p)\n",
        "$$\n",
        "Assume the monopolist only knows the demand function at a finite number of points and needs to interpolation between these points to be able to solve the maximization problem.\n",
        "\n",
        "The monopolist knows\n",
        "$$\n",
        "d_i = D(p_i)\n",
        "$$\n",
        "for points $i=1,...,N$."
      ],
      "metadata": {}
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Interpolation\n",
        "First, we want to look at how to interpolation the function $D(p)$ based on the points $p_i$ and $d_i$ for $i=1,...,N$."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "# assume points p_i, d_i for i=1,...,N\n",
        "p = np.linspace(1,5,num=9)\n",
        "d = p**(-2.0)"
      ],
      "outputs": [],
      "execution_count": 102,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "plt.plot(p, d, 'x');"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": [
              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\n"
            ],
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "execution_count": 103,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "D = scipy.interpolate.interp1d(\n",
        "    p, d, fill_value='extrapolate', kind='linear')"
      ],
      "outputs": [],
      "execution_count": 104,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "p_grid = np.linspace(np.min(p), np.max(p))\n",
        "plt.plot(p, d, 'x', label='points $d_i$')\n",
        "plt.plot(p_grid, D(p_grid), label='linear interpolation')\n",
        "plt.legend();"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": [
              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\n"
            ],
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "execution_count": 105,
      "metadata": {}
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Optimization\n",
        "With the interpolated function at hand, we can compute the profit maximizing function by using a maximization routine.\n",
        "\n",
        "`scipy` only has a minimization function, but no maximization function. There is a simple solution: minimize the negative profit function! That is, solving\n",
        "$$\n",
        "\\max_p (p-c)D(p)\n",
        "$$\n",
        "is equivalent to solving\n",
        "$$\n",
        "\\min_p -(p-c)D(p).\n",
        "$$\n",
        "\n",
        "Assume costs are $c=1.3$."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "c = 1.3\n",
        "solution_profit_max = scipy.optimize.minimize_scalar(lambda p: -(p-c)*D(p), \n",
        "                        method='bounded', bounds=(np.min(p), np.max(p)))"
      ],
      "outputs": [],
      "execution_count": 106,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "solution_profit_max"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 107,
          "data": {
            "text/plain": [
              "     fun: -0.19665454545454547\n",
              " message: 'Solution found.'\n",
              "    nfev: 8\n",
              "  status: 0\n",
              " success: True\n",
              "       x: 2.71818181818182"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 107,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "print(f'the profit maximizing price is {solution_profit_max.x}')\n",
        "print(f'the maximal profit is {-solution_profit_max.fun}')"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "the profit maximizing price is 2.71818181818182\n",
            "the maximal profit is 0.19665454545454547\n"
          ]
        }
      ],
      "execution_count": 108,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "plt.plot(p_grid, (p_grid-c)*D(p_grid), label='profit function')\n",
        "plt.plot(solution_profit_max.x, -solution_profit_max.fun, 'x', label='maximum')\n",
        "plt.xlabel('price')\n",
        "plt.ylabel('profit')\n",
        "plt.legend();"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": [
              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\n"
            ],
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "execution_count": 109,
      "metadata": {}
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Solving First-Order Condition\n",
        "An alternative approach to maximization is to solve the first order condition. The first-order condition of the above profit maximization problem is\n",
        "$$\n",
        "p + \\frac{D(p)}{D'(p)} - c = 0\n",
        "$$\n",
        "Assume that the monopolist knows not only the demand at some points \n",
        "$$\n",
        "d_i = D(p_i)\n",
        "$$\n",
        "but also its derivative\n",
        "$$\n",
        "e_i = D'(p_i)\n",
        "$$\n",
        "for points $i=1,...,N$."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "# assume points p_i, e_i for i=1,...,N\n",
        "e = -2*p**(-3.0)"
      ],
      "outputs": [],
      "execution_count": 110,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "plt.plot(p, e, 'x');"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": [
              "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\n"
            ],
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "execution_count": 111,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "D_prime = scipy.interpolate.interp1d(\n",
        "    p, e, fill_value='extrapolate', kind='linear')"
      ],
      "outputs": [],
      "execution_count": 112,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "p_grid = np.linspace(np.min(p), np.max(p))\n",
        "plt.plot(p, e, 'x', label='points $e_i$')\n",
        "plt.plot(p_grid, D_prime(p_grid), label='linear interpolation')\n",
        "plt.legend();"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": [
              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\n"
            ],
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "execution_count": 113,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "# plot first-order condition\n",
        "plt.plot(p_grid, p_grid + D(p_grid)/D_prime(p_grid), label=\"$p+D(p)/D'(p)$\")\n",
        "plt.plot([p_grid[0],p_grid[-1]], [c, c], label='$c$')\n",
        "plt.legend();"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": [
              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\n"
            ],
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "execution_count": 114,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "solution_foc = scipy.optimize.fsolve(lambda p: p+D(p)/D_prime(p)-c, x0=p_grid.mean())\n",
        "p_star = solution_foc[0]"
      ],
      "outputs": [],
      "execution_count": 115,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "# plot first-order condition\n",
        "plt.plot(p_grid, p_grid + D(p_grid)/D_prime(p_grid), label=\"$p+D(p)/D'(p)$\")\n",
        "plt.plot([p_grid[0],p_grid[-1]], [c, c], label='$c$')\n",
        "plt.plot(p_star, p_star + D(p_star)/D_prime(p_star), 'x', label='solution')\n",
        "plt.legend();"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": [
              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\n"
            ],
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "execution_count": 116,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "print(f'optimal price: {p_star}')"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "optimal price: 2.572266151183606\n"
          ]
        }
      ],
      "execution_count": 117,
      "metadata": {}
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Exercise\n",
        "Consider a risk-averse consumer with utility function $U(x)$ who has an initial wealth $W$. With probability 50% he has an accident and loss $L$. He can buy an insurance for a premium $\\pi$, in case there is an accident, the insurance pays him $2\\pi$. Formally, his profit maximization problem is\n",
        "$$\n",
        "\\max_\\pi \\frac{1}{2} U(W-L+\\pi) + \\frac{1}{2} U(W-\\pi)\n",
        "$$\n",
        "Assume $W=100$, $L=20$. We know the utility function for $N$ points:\n",
        "$$\n",
        "v_i = U(x_i)\n",
        "$$\n",
        "for $i=1,...,N$."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "x = np.linspace(1, 100, num=50)\n",
        "v = np.log(x)"
      ],
      "outputs": [],
      "execution_count": 118,
      "metadata": {}
    },
    {
      "cell_type": "markdown",
      "source": [
        "Use points $x_i$ and $v_i$ for $i=1,...,N$ to interpolate function $U$. Plot the interpolated function."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [],
      "outputs": [],
      "execution_count": null,
      "metadata": {}
    },
    {
      "cell_type": "markdown",
      "source": [
        "Use the interpolated function to solve the consumer's utility maximization problem."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [],
      "outputs": [],
      "execution_count": null,
      "metadata": {}
    },
    {
      "cell_type": "markdown",
      "source": [
        "Assume that we also know the derivative of the utility function at $N$ points:\n",
        "$$\n",
        "w_i = U'(x_i)\n",
        "$$\n",
        "for $i=1,...,N$."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "w = 1/x"
      ],
      "outputs": [],
      "execution_count": 119,
      "metadata": {}
    },
    {
      "cell_type": "markdown",
      "source": [
        "Use points $w_i$ for $i=1,...,N$ to interpolate the function $U'$. Solve the consumer's utility maximization problem by solving the first-order condition for the optimal $\\pi$."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [],
      "outputs": [],
      "execution_count": null,
      "metadata": {}
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.7.6"
    },
    "nteract": {
      "version": "0.22.4"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 4
}