demand and supply - 1. demand.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notebook created by: Leshui He; Department of Economics, Bates College (for econ 260)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import necessary packages."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "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."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "p, pb, pk, y = symbols('p pb pk y')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1.1. Demand function and demand curve"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Define a demand function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define a demand function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle - 20 p + 20 pb + 3 pk + 2 y + 171$"
      ],
      "text/plain": [
       "-20⋅p + 20⋅pb + 3⋅pk + 2⋅y + 171"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define linear demand function as in class notes\n",
    "Qd = 171 - 20*p + 20*pb + 3*pk + 2*y\n",
    "# show the defined function\n",
    "Qd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculate derivative of the demand function with respect to $p$ by hand. \n",
    "\n",
    "$\\LaTeX$ Note: \n",
    "* mark your inline LaTeX math expression with `$` at its beginning and the end\n",
    "* the partial derivative sign in LaTeX math is: `\\partial`\n",
    "* the fraction in LaTeX math is: `\\frac{numerator}{denominator}`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The derivative is: $\\frac{\\partial Q_d}{\\partial p} = -20$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\LaTeX$ Note:\n",
    "* to type a centered equation (called math display) use `$$ (math here) $$`, for example, `$$\\frac{\\partial Q_d}{\\partial p} = -20$$` produces:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{\\partial Q_d}{\\partial p} = -20$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\LaTeX$ Note:\n",
    "* to type multi-lined equations, use `\\begin{align} and \\end{align}` to surround your equations\n",
    "* use `\\\\` for linebreak between each equation\n",
    "* use `&`  on each line to tell LaTeX where you'd like the equations to align"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "\\frac{\\partial Q_d}{\\partial p} &= \\frac{\\partial}{\\partial p} (171 - 20 p + 20 p_b + 3 p_k + 2 y) \\\\\n",
    "  &= \\frac{\\partial}{\\partial p} (171 + 20 p_b + 3 p_k + 2 y) + \\frac{\\partial}{\\partial p}(-20 p) \\\\\n",
    "  &= -20\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "\\frac{\\partial Q_d}{\\partial p} &= \\frac{\\partial}{\\partial p} (171 - 20p + 20p_b + 3p_k + 2y) \\\\\n",
    "&= \\frac{\\partial}{\\partial p}(171 + 20 p_b + 3 p_k + 2y) + \\frac{\\partial}{\\partial p} (-20p) \\\\\n",
    "&= -20\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculate derivative of the demand function with respect to $p$ using sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -20$"
      ],
      "text/plain": [
       "-20"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Note: we have seen the use of diff() function in sympy, which requires two inputs\n",
    "diff(Qd , p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -20$"
      ],
      "text/plain": [
       "-20"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Note: in sympy, you can calculate derivative of a sympy object using the \"diff\" method\n",
    "#     it trails behind the object you want to take derivative of\n",
    "#     you specify the variable you want to take derivative w.r.t. in the parenthsis\n",
    "Qd.diff(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Understand a sympy function Rational()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAOCAYAAADaOrdAAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABHklEQVQ4Ea2UjW3CMBCFE9QBItiAbgDtBM0IsAIjVGIE2IAZGAE2oGEDsgE/G4TvVX5SiIwEciydzn6+e+/OiZ03TZN55Hk+Yb7FpuA34694clch7oz/xFZw1M4tmIh4g1WYVAuJv2ohb+Z45WMnbCzsgQjwF3tLhPgFdrKAPZg622k9YJI65hAcIyQHsJJjLPoQKSG7RET8TcskEVUZIe9CwyQR2IaB0VW3BdxdL8fVJo7NR6mduNoYubs8J4nwe/qYYt/GWJ0kEsrf48eRVtzJvg8RvRZfEZEp2PG/W99QecbTG8+e2ldQ1c4JeXpCus/KFWyi/Q8mGf+7qhGJK6rAdIsPBK3xen5uYHrw/rTuDFW9ZN/H9s36hxxxZHd0YM1+jR3cSQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle 1.0$"
      ],
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = 1/13\n",
    "x*13"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOCAYAAAASVl2WAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAWUlEQVQYGWP4//8/AwwzMDAYAfFdIBaAibEwMjIKAAVmA/E7IDYBYiUgRgCYShANBGVADGLATWACieIDowogoUMwHBhBQQyMj9VA9aA4AcUFiL4HxOeA+DQAT1cpCApe64MAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rational(1,13)*13"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Replace specific values into $p_b, p_k, y$, and call it Dcurve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 286 - 20 p$"
      ],
      "text/plain": [
       "286 - 20⋅p"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Notice the use of () and [] in the subs method in sympy\n",
    "# Notice the use of funciton Rational()\n",
    "Demand = Qd.subs([ (pb,4) , (pk,Rational(10,3)) , (y,Rational(25,2)) ])\n",
    "Demand"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plot the demand curve with the above specific values for $p_b, p_k, y$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Step 1: inverse the demand. \n",
    "\n",
    "Recall that the y-axis of the demand curve is p, not Qd! We need to inverse the Dcurve function first!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{143}{10} - \\frac{q}{20}\\right]$"
      ],
      "text/plain": [
       "⎡143   q ⎤\n",
       "⎢─── - ──⎥\n",
       "⎣ 10   20⎦"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define a new symbol q\n",
    "q = symbols('q')\n",
    "# solve the equation Demand = q, which is equivalent to (Demand - q) = 0. \n",
    "#   Notice how the syntax required by sympy takes equations without the equal sign.\n",
    "#   Notice how the solution of sympy is stored in a list (with square brackets around it)\n",
    "Dplot = solve(Demand - q, p)\n",
    "Dplot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{143}{10} - \\frac{q}{20}$"
      ],
      "text/plain": [
       "143   q \n",
       "─── - ──\n",
       " 10   20"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Dplot[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{143}{10} - \\frac{q}{20}$"
      ],
      "text/plain": [
       "143   q \n",
       "─── - ──\n",
       " 10   20"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# retrieve the solution (p as a function of q) from the list\n",
    "DInverseSimple = solve(Demand - q, p)[0]\n",
    "DInverseSimple"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Step 2: plot the (inverse) demand curve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7f8b60939490>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# use plot function\n",
    "plot(DInverseSimple, (q,0,286), xlabel=\"$q$\", ylabel=\"$p$\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plot the demand curve with $p_b, p_k, y$ explicitly substituted later on"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Step 1: inverse the demand. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle pb + \\frac{3 pk}{20} - \\frac{q}{20} + \\frac{y}{10} + \\frac{171}{20}$"
      ],
      "text/plain": [
       "     3⋅pk   q    y    171\n",
       "pb + ──── - ── + ── + ───\n",
       "      20    20   10    20"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define a new symbol q\n",
    "q = symbols('q')\n",
    "# solve the equation Demand = q, which is equivalent to (Demand - q) = 0. \n",
    "#   Notice how we solve for the equation and retrieve the first element of the list in one step\n",
    "DInverse = solve(Qd - q, p)[0]\n",
    "DInverse"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Step 2: plot the demand curve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "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"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7f8b90a40450>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# use plot function\n",
    "plot(DInverse.subs([(pb,4),(pk,Rational(10,3)),(y,Rational(25,2))]), (q,0,286), xlabel=\"$q$\", ylabel=\"$p$\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1.2 Changes of other factors in the demand function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1.2.1 change in $p_b$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Take derivative of $Q_d$ with respect to $p_b$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 20$"
      ],
      "text/plain": [
       "20"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Qd.diff(pb)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q: what is the interpretation of this derivative?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A: for each dollar increase in $p_b$, $Q_d$ increases by $20$ units"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot a demand curve with $p_b=5, p_k=3\\frac{1}{3}, y=12.5$ explicitly substituted in."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "scrolled": false
   },
   "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"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7f8b80b97c50>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# use plot function\n",
    "plot(DInverse.subs([(pb,5),(pk,Rational(10,3)),(y,Rational(25,2))]), (q,0,300), xlabel=\"$q$\", ylabel=\"$p$\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Useful trick for plots in sympy: store, hide and show plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "# plots can be stored as an object by itself too!\n",
    "#    Notice we used option \"show=False\" to hide the plot for now\n",
    "Dplot1 = plot(DInverse.subs([(pb,5),(pk,Rational(10,3)),(y,Rational(25,2))]), (q,0,300), xlabel=\"$q$\", ylabel=\"$p$\", show=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "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": [
    "# show the plot when we need it\n",
    "Dplot1.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Compare demand curves"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot two demand curves on the same graph, both with $p_k=3\\frac{1}{3}, y=12.5$. \n",
    "\n",
    "Let the first demand curve with $p_b=4$, the second with $p_b = 5$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Lpfo5DQr1y0Rc5P33cPZshdYYZvyrWCvWYlVWe56duJ2gZWMTTjwNrmQK7ymlWngr1X9pUKhHk1QK93gD7okGcFysiVOwlryOKSn1NleEzg86CG1sItmUpGBUIcH11RSO1MJbqX5Ig0I9mdvRjhzchXvqCOTkYi1YijVvMSY3z9NccYTWEzHC21pw2hzK63z4FvnJDXqbq5TqVRoUquck3JIpvC+/D6Xl2MtWYWpnYixvhbfb7RLZG6LtVCvpeIqKhX789UHsYi28leoHNCjU03NvXsPduQm5cxPzyhSsmfOwxkzwPDcdTxHe1kLriThWgYV/ZZDyhT6sHL0HhlJ9SINCPRsRQS6exT15GLl5HTP2FewVazHVQz3P7r6bILSpmc7LHdi+HCrfrKakVgtvpfqIBoXyRtJp3BOHcBt2Q3cic0mQpaswpeWeZ3dcaSd+JEbH+TYKRhRkCu/R3i6VrpR6av07KIwxPwTWAs0P3ArVD/wdMBK4AfwLEYk+aZYGxfMlXZ24B3fhnjgMto21dBXWjLmYvHxvc12h7VSc0NYWnHia4imlBNdWklflba5Sqsf6fVDUAe3Ajx8Iim8DERH5c2PMvwN8IvJvnzRLg+LFkEgIp2Evcv4UFBVhL1mFeW0WxvJWTLtJl9j+CJG9YbCgfHY5/hVB7BJv99ZQSj1R/w4KAGPMSGDzA0FxBVgiIveMMUOB/SLyxCZVg+LFcm834u7ciNy6AZVDsOvXYsa+4v0eGG1pYociRHeHsfIsfCsCVCzyY+Vp4a3Uc5KVQRETkYoH/h4VEd8jnrsB2ABQU1Mzo7Gx8bmvV/2ciCCX38fZvQUiIcyU6djzl2CGDPM8u/uTbsKbmum41E5ORQ6BNVWUTi/DWFp4K9XLBnZQPEj3KPqOOGncU0dxL56FW42YqTOwl72BKat48pOfoPOjDkIbm+m+naB4cgkVdX6KxhX3wqqVUvdlZVDooacsJYku3IY9uMcbwIA1bwnWgqWY/AJvc12h/VwboY1NpGNpiieVEFxXRV61Ft5K9YKsDIrvAOEHymy/iPzxk+ZoUPQfEovg7N2KvP8eFBVjLV+NNXUWxu6Fwrsh01+4SZfyeRX4X68kp1QLb6U86N9BYYz5W2AJEASagG8CPwP+HqgBbgL/XEQiT5qlQdH/uHdv4R5vQM6fhkBlpvAeP8l74d2eJrIjRPxIlILRRRSPL6ZisRbeSj2j/h0UvUmDon8SEeTDizi7NkO4BTNiNFb9OqxhNZ5nJ5u7iewO03YyTk55DoHVlZTOLNfCW6mno0Gh+gdxHNwzx3H374DOdsyU17CWrsby+T3P7rraScvGJrpvJsh7KZ/K9dUUTdDCW6ke0qBQ/Yt0J3AP78M9cQgKi7BercVatAJT4O1eFeIK7WdbCW1pIR1JUTqrHN9SP/lDvRXpSg0CGhSqf3LjUdx925Fzp6GwEGvxSqyZ8zC2t2LaTbvEG6K0n2slcTNB2ZwKAquC5JTn9tLKlRpwNChU/yaf3MHZuQm5/hH4g9jL12AmTvFceDsdaSI7w8QORzC2wbc0gG9pACtfC2+lPkeDQvV/IoJ8fDlTeLd8gpk6E2vmfKzhIzzPToaShDc3036uDduXQ2BlJWWztfBW6gEaFCp7iOvgnjuVKbxb45hXp2IvX43xBz3P7rrRSfxIlLaTreQNySe4voqiV4r1HhhKaVCobCTJbtwj+3GP7AfHwZq9AKuuHlPo7V4VIkL7+TbCm5tJhVIUjiui8s1q8odp4a0GNQ0Klb2kLY6zbwdy9gTkF2AtX4M1bRYmx1vhLWkhfiRKeEcIt8uhdEY5gdWV5Pq08FaDkgaFyn7SdI/0vm3QeA0KCjOHoyZN8154dzlEd4eJHYyQX1NA4egifMsD2AXeLjWiVJbRoFADh3v1Cs6uTdB0DzOsJnOG94jRnuemIikie0K0Holhl9j4Xw9SPs+HsbW/UIOCBoUaWMR1kfOncPZuh7Y4ZvZC7NkLMYFKz7MTN7sIbWym62onuVV5BNdVUTypRAtvNdBpUKiBSVJJnBOHkEN7IdmNNWMe1uKVmOISb3NF6LjYTmhTM6nmJIVjigiur6KgxtuZ40r1YxoUamCT9jbcAztxTx+DvDyshcuw5tRhcr0V0+II8WMxIttbyA3kkhvII7Cmklx/Xi+tXKl+Q4NCDQ7S0oSzezNy9UOoqsaeU4epnY4x3s7EdhIOsYNRortDIFBR58e3IoBdqIW3GjA0KNTg4jZexd2xEbl3G4YOx65fhzVqrOe5qWiK8LYW2k7FsYpsAiuDlM/3YXK0v1BZL3uDwhjzvwG/CwjwPvDbIpJ41OM1KNSnRFzk/fdw9m6DeBQz/lXsFWsxldWeZyduJwhtbKLro05yg7kE36zWwltlu+wMCmPMMOAQ8KqIdBlj/h7YKiJ//ajnaFCoz5NUCvd4A+6hPVDhxxo+AmvJ65iSUm9zRei83EFkd4juWwnyhxcQXF9N4UgtvFVW8hQUfX0j4hyg0BiTAoqAu328HpVlTG4u9sJlWK/NzgTG4b2475/Bmr8Ua14dJi//2eYaQ/HEEorGF9N6IkZ4Wwu3v3uDkmmlBNdUkRvUwlsNHn196OkPgD8DuoCdIvKlxz1e9yjUk0i4BWfPFuSD96G0DHvpKqidiWV7K6bdbpfo3jDR/WHEESoW+vHVB8gp7uv/r6VUj2TtoScf8A/ArwEx4H8CPxWRv/nc4zYAGwBqampmNDY2vuilqizk3ryGu3MT0tUJuXnY9WuxxkzwPDcdzxTeidsJ0pEU/pVByhf6sHL0HhiqX8vaoPjnwCoR+Z37P/8WMFdEvvao5+gehXoaIoJ7+X3cnZsgFsGMmYBdvxZT/ZLn2d13E4Q2NdN5uYMcfy7BtVWUTCvVwlv1V1kbFHOAHwKzyBx6+mvglIj85aOeo0GhnoWk07gnD+Me3AWJBGbaLKylr2OVVXie3XE5c4Z38m43BSMKCKyvomh0cS+sWqlelZ1BAWCM+VMyh57SwHvA74pI96Mer0GhvJCuTtyDu3GvXITOdqw5i7DmL8Hke7tXhbhC26k4oa0t5AVysYpzCK6tJK/q2Yp0pZ6D7A2Kp6VBoXqDGwnj7t2KXDwLxaXYS1/HvDYbY3ksvJMu0YMRorvDSMqlfIGPwMogdokW3qrPaVAo9Szc2424uzYhN6/D8BFYC5djjX/Vc8+QbksT2d5C/FgMK8/CtyJAxSI/Vp4W3qrPaFAo9axEBLl8Aff8aeTy+5hRY7Hr12GGDvc8u/uTbsKbmum41E7xpBJKppVROr0MY2nhrV44DQqlvBInjXvqKO6BndDVhZk6A3vpKky5z/Pszo86CG1qfuAM7yqKxmnhrV4oDQqleoskunAP7cE91gC2jTVvMda8xb1TeJ9pJby1mXQ0TemMMnwrguQP0cJbvRAaFEr1NolFcE8cxj26H4pKsJasxJo+F+P1DO+kS6whQuflDrqudlI2t4LAG5XklGrhrZ4rDQqlnhf37q3MGd6NVyFQmbmHdy8U3k57mvDOEPHDUUyuhW+ZH9/iAFa+Ft7qudCgUOp5EhHkw0s4uzZBceZy41b9OqxhNZ5nJ1uShDY303G+jcJxRZTOKKdsVrkW3qq3aVAo9SKI6+CeOY67bwd0tmMmv4a17A0sX8Dz7K5rnUT3hem40E7eS/kE11dRPMHbvcGVeoAGhVIvknQncA/vwz16APPScMzwEViLVmAKvN2rQkRoP9tGaHMz6UiKoleKCa6rIv8lb0W6UmhQKNU33HgMd9825NxpKCzEqqvHmjUfY3srpt20S/xQlMjOENiGksmlBFYFySnP7aWVq0FIg0KpviSf3MHZtQm59hH4Atgr1mAm1novvDscYocjRHaGMLbBtySAb5kW3uqZaFAo1ddEBLl6BWfnJmj5JHM4auV6rJdHep6dCiUJbWmm/WwbdqlN4I1KymZXYGwtvFWPaVAo1V+I6yBnT+Ls2w7d3ZhXJmMveR3jD3qe3XWji9DGJhLXuyieXEL5PB9FE4v1HhiqJzQolOpvJNmNe+YY7t7t4DhYsxZg1a3AFHm7dIeI0PF+G7GDEbqudlE4rojg+moKhmvhrR5Lg0Kp/kraWnH2bUfOnoC8fKy6FVizF2JyvBXTkhbiR6KEd4ZwOx1KppcRXFNFrk8Lb/VQGhRK9XfSfA9n12bk48tQ7sOqX4v1ai3GeCumnS6H6O4wnVc7Sd5NULHYj295ALvA26VG1ICTvUFhjKkAfgBMBgT4iogcfdTjNShUtnOvfYizcyMYg7FsrJXrsEaM8Tw3GU4S2dZC2+lW7BIb/+tByuf5tPBWn8rqoPgR0CAiPzDG5AFFIhJ71OM1KNRA4LoOvP8ezp6t0BbHTJiEvWItJljleXbiZhehjc10Xe2kaGIx5fN9FE8q0cJbZWdQGGPKgHPAaOnhIjQo1EAiqSTusYO4h/ZCKoU1cx6mrh6rpNTbXBE6LrYTa4jQ9WEnhWOKCK6voqDG25njKqtlbVBMA94BLgFTgdPAH4hIx6Oeo0GhBiJpb8M9sBP3zHGorMKa/BrWnDpMrsfC2xHix2JEtrfgtDuUTi8jsKaSXH9eL61cZZGsDYqZwDFggYgcN8Z8F2gVkf/rc4/bAGwAqKmpmdHY2PjiF6vUC+CGmnB3b0GuXISyCuxlb2Bqp3svvBMO0b1hYvsjFIwspGB4Ab76IHahFt6DSNYGxRDgmIiMvP/zIuDficiaRz1H9yjUYODe+DhzD4x7t2HIMOyV67BGjfM8NxVLEdkRovV4DKvQxr8ySMUCHyZH+4tBIDuDAsAY0wD8rohcMcb8CVAsIt941OM1KNRgIeIiF85mCu94FDNuYuYeGJXVnmd330nQsrGZrg87yA3mElhTRcnUUi28B7asDoppZL4emwdcA35bRKKPerwGhRpsJJ3CPd6A++EluHUDa/pcrCUrMSVl3uaK0Hm5g9DGZpKfdFM6s4zyBT4KRxb1zsJVf5O9QfG0NCjUYCWd7bgHduOeOgx2DtaCZVjz6jB5+d7mOkL8RJT4wSjJT5KUTC0lsLaKvKAW3gOMBoVSg4WEW3D2bEU+OA+lZVjLVmPVzsBY3gpvt9slui9MdF8YcYSKBX78KwPYxd7uraH6DQ0KpQYb9+Z13J0bEduG7gR2/TqsMRM8z03HU4S3ZwrvwvHFFE8opnyhDytX74GR5TQolBqMRAT30jncPVshGsaMmYBdvxZT/ZLn2Ym7CcJbW+i82E6OP5fgmkpKXivTwjt7aVAoNZhJOo178jDuwV2QSGCmzcJeugpTVu55dueVDlo2NpG8201+TQHBX6mmaJQW3llIg0IpBdLViduwG/fEIXipBmvUWKz5SzD53u5VIa7QdipO/Hjss5smBddWkVflrUhXL5QGhVLq5yQaxjm8Dzl9FIpLsZe+jnltNsbydia2m3SJHYgQ2RNGUi7l8334Xw+SU6KFdxbQoFBK/TL3diPurk3IzesQrM70F+Mmeu4Z0m1pIjtaiB+NYeVa+N/IXNLcytPCux/ToFBKPZyIIJcv4OzeDJEQZsp07HmLMUOHe56dbOomtLmZdCyN054msLqK0hllGEsL735Ig0Ip9XjiOLinj+K+/x7cbsTUTs9cdLDc53l258eZM7y7byXIH5ZP8M1qisZ5uze46nUaFEqpnpFEF+6hPbjHGsCANXcx1sJlvVN4v9dKeEsz6Wia0pll+JYHyR+ihXc/oUGhlHo6Eovg7NuOnD8NRcVYy9dgTZ2JsT0W3imX1uMxwltacLtdyuZWEFhVSU6ZFt59TINCKfVs5O4t0js3QmcnuE7mlqwTJnkuvJ32NJGdIWKHo5gcg68+iG+RHytfC+8+okGhlHp2IoJ8eClTeIeaMTWjsVauwxpW43l2siVJaHMzTrtDKpwk8EYlZbPKtfB+8TQolFLeievgnjmOu38HdLRjJr+GtewNLF/A8+zOax2ENzaTaEyQ91I+wfVVFE8o6YVVqx7SoFBK9R7pTuAe3od7vCFzhdrxk7AWLccUert0h4jQfq6N0KZm0pEUZXPKqajzk/+StyJd9YgGhVKq97nxGO7+7cjZU1BYiFVXjzVrPsb2Vky7acK0m6wAABP2SURBVJfWYzHCW1twEy5ls8sJvFFJTnluL61cPUR2B4UxxgZOAXdEZO3jHqtBodSLJ5/cxdm1Ebn2EfgC2CvX907h3ekQ2RUi3hAFG3xLA/iW+LEKvH3zSj1U1gfF14GZQJkGhVL9k4ggV6/gHNqbOWFv6LBM4f3yKM+zU6Ekoa0tpONpUi3dBFZVUjanAmNr4d2LsjcojDHDgR8BfwZ8XYNCqf5NXBc5exJn33Zob8W8Wou9fA3GH/Q8u+tGJ6GNzSSud5FXnUdwXRVFr5boPTB6R1YHxU+Bt4FS4I8eFhTGmA3ABoCampoZjY2NL3aRSqlfIslu3KMHcA/vA8fBWrQca/ZCTJG3S3eICB3vZwrvVChF2dxyyhf4KRiuhbdH2RkUxpi1wGoR+ZoxZgmPCIoH6R6FUv2LtLXiHDuAHD0AeflYdSsygZHjrZiWtBA7EiF2IEo6mqJ0RjmB1ZXk+rTwfkZZGxRvA/8SSAMFQBnwjyLym496jgaFUv2TNH+Cs3sz8tEHUO7DXr4aM3kaxng7E9vpcojuDhM7GAEDFYv9+JYHsLXwflrZGRS/sAjdo1BqQHCvfYizaxN8chfz0stYr7+JVdMLhXckRXhrM22nW7GKLYJrqiibrYX3U9CgUEr1HyIucv4MzrmTcP1jzIRJmWtIBas8z07c6iLWEKXtZJzcykzhXTxZC+8eyP6g6CkNCqWyh6SSuMcO4h7aC6kU1sx5WItXYoq9XbpDROi42J4pvJuTFI4pIri+ioKawl5a+YCkQaGU6r+kow13/07c08cgNxezYBnWvEVYud7uVSGOED8WI7K9BafdoWxOOf6VQXL9eb208gFFg0Ip1f9JqAln9xakNQ4d7Zk77NVO91x4uwmH6N4wXde7SFzvorzOh39FELtIC+8HaFAopbKH23gVd+cm5O4tGDIMe+U6rFHjPM9NxVKEt7XQdjKOVWjjXxmkYoEPk6P9BRoUSqlsI+IiF87i7NkK8Shm3ESsFWuwqoZ6nt19J0HLxma6PuygcFwR5fN9lEwtHeyFtwaFUio7STqFe7wBt2EPDHkJK1CFtfR1TEmZt7kidF7uIH44SsfFdgpGFhJ8s4rCkd4ulZ7FNCiUUtnN7WjDbdiDnDwMdg7WgqVY8xZj8jwW3q7QeiJOeFsLTmuakqmlBNZUkVc56ApvDQql1MAg4RacPVuRD85DSVnmcNSU6RjLY+Hd7RLdHya6N4y4gm9xAN9SP3axt3trZBENCqXUwOLevI57/BBy6SxUDcWuX4s19hXPc9PxFLFDUaJ7wlj5Fv6VQcoX+rByvQVRFtCgUEoNPCKCXDqPs2cLRMOYMeOx69dhql/yPLv7XjehTU10ftBBji+X4NpKSqaVYawBW3hrUCilBi5Jp3FPHcE9sBMSCayFS7FmLcSUlXue3Xmlg9CmJrrvdlM0sQT/sgCFYwZk4a1BoZQa+KSrE+fQXuTMMUinseYvwZq/BJPv7V4V4gpt59oIv9tEOp6meHIJwXVV5FV5K9L7GQ0KpdTgIdEwzt6tyIWzUFyKteR1rOmzMZa3M7HdpEvsQITonjBuyqVioQ9ffZCckgFReGtQKKUGH/d2I+6uTcjN6zB0ONbilVjjX/V8Yl26LU1kR4hkUzfdtxL4VgSoqPNj5WV14a1BoZQanEQEuXIB9/xp5IP3MSPHYq9chxk63PPsZFM3oc3NdFxoJ6ciB//qSspmlGdr4a1BoZQa3MRxcE8fzRTenR2Y2hlYS1dhVfg9z+78uIPQxmaMbZCUS/DNaorGebs3eB/IzqAwxrwM/BgYArjAOyLy3cc9R4NCKfU4kujCPbQX9+gBGDoMa+RYrIXLMAXe7lUhrtB+tpXQ5hbS0RRFr2YK7/whWVN4Z21QDAWGisgZY0wpcBr4FRG59KjnaFAopXrCjUVw921Hzp+GouJMfzFjHsb2WHinXOINUSK7QpBrKJ1Siv/1SnLK+n3hnZ1B8XnGmHeB74nIrkc9RoNCKfU05O4tnF2bkBtXIVCJvWINZsJkz4W3054mdjhKZGcIk2PwLQvgWxLAyu+3hXf2B4UxZiRwEJgsIq2PepwGhVLqaYkI8tEHOLs2gStQWopdvw5rWI3n2cmWJOHNzbSfb8MuyyHwRiVls/tl4Z3dQWGMKQEOAH8mIv/4kL9vADYA1NTUzGhsbHzBK1RKDQTiOrjnT+Pu3gId7ZjJ07CXrcb4Ap5nd13vJPRuM90t3eRV5hFYVUnxK97uDd7LsjcojDG5wGZgh4j85yc9XvcolFJeSXcC9/C+TOEtLmZuHfaCZZhCb5fuEBHaL7QTfreJVDhF0YRiguuryH/J25njvSQ7g8JkDhL+CIiIyP/ak+doUCileou0xnH2bYNoGGm6lym8Z83H2N6KaTftEj8cI7KzBbfLpWxWOYE3KsmpyO2llT+TrA2KhUAD8D6Zr8cC/O8isvVRz9GgUEr1NveTO7i7NiPXPgRfAHv5Gsyrtd4L706HyK4Q8YYoVrFF+VwfvqV+rAJv37x6RtkZFM9Cg0Ip9by4H1/OFN7Nn2RO2Js5D+vlUZ7npsJJYoeixPZHsEvsTOE9pwJjv9DCW4NCKaV6g7gu7vlTuHu3Q1scM7E285Vaf9Dz7ERjFy0bm0hc6yKvOo/AuiqKXy3xvOfSQxoUSinVmyTZjXv0AO7hfeA4WHUrMv1FkbdvMokIHRfaCW1qJtWSpHRGGRWL/RS87O3M8R7QoFBKqedB2ltx9u1ArlyEdApr0QqsOQsxOd6KaXGE+MkYkc0tOB0OpTPLCayuJNf33ApvDQqllHqepPkTnN2bkY8+gHIf9vLVmMnTMMbbmdhOl0N0T5jYgQgAFYv9VCzzk1PU65cE0aBQSqkXwb32Yabw/uQuZsp0rOlzsUaO8Tw3FU0R3tpC29k4eZX5lM+roHy+rzcLbw0KpZR6UURc3HOncY/sh5ZPMBMmZQrvYLXn2YnbCULvNtH1cSe5lXkE11VRPLlXCm8NCqWUetEklcI9dhD30B5IpbBmzMVashJTXOptrgidlzKFd7IpSdGkYgL1lRSM8FR4a1AopVRfkY423P07cU8fw4wcgxk1FmtuHSY3z9tcR2g9HqP9/TY6L3dQ8loZwTWV5Aaeaa4GhVJK9TUJNeE07EbOn4Gycuylb2CmzvBceLsJh+i+CNF9YXChfJEPf30Qu+ipzvDWoFBKqf7CbbyKu3MTcvcWZmJt5gzv0eM9z03HUoS3tdB6Mk7+ywWUTi+nYoEPk9OjDNCgUEqp/kTExb10Hnf3ZohFMWNfwa5fi6ka6nl2950E0QMR2k7GyQ3mElhTRcnU0icV3hoUSinVH0k6hXviEG7DHuhOYGbMw66rx5SWeZ7dcbmd0MZmkve6KRhZSHB9FYWjHnmpdA0KpZTqz6SzA/fgbqTpLnLnJtb8JVjzl2Dy8r3NdYXWE3HC21pwWtOUL/RRUecnr/KXCm8NCqWUygYSCeHs2YJcOg8lZdhLV2GmzcJYHgvvbpfY4QiRHSHEESoW+PCvDGIXf3aGtwaFUkplE/fW9UzhfbsRqoZirVyLPeYVz3PT8RThHSFaj8Ww8i389UHKF/mwcr3dxFuDQiml+oCIIB+cxz1/BrlyATN6PHb9OsyQlzzP7r7XTWhTE50fdJBblcfIfz8me4PCGLMK+C5gAz8QkT9/3OM1KJRSA404adyTR3AP7oKuLsy0mZlzMMrKPc/u/LCDZEuSigW+7AwKY4wNfAjUA7eBk8Cvi8ilRz1Hg0IpNVBJVyduwx7cEw2YmtGY4SOwFizF5Bf0xnhPQeGtQfFmNvCxiFwTkSTwE+DNPlyPUkr1GVNYhL1yHfbX/hhKy3AbdpP+y7dxTh1BXKdP19aXQTEMuPXAz7fv/04ppQYtyx8k5wu/gf27f4AJVOFu+QfS//3/wfnwEn11BKjX747xFB62K/RL/ysYYzYAGwByc3OZOXPm815Xn2lpaaGysrKvl/HcDOTtG8jbBrp9fSrRhbTG4dv/FSw701085QUHT58+fUFEJj/rEvoyKG4DLz/w83Dg7ucfJCLvAO8AFBcXD+iOYubMmej2ZaeBvG2g29fX3HQaee847v4d0NmBqZ2BvewNTLmvR883xiS8vH5fBsVJYJwxZhRwB3gL+I0+XI9SSvVLVk4OzFqANWU67uF9uMcOkL54DjNnEdai5VgFnu5V8UR9FhQikjbG/BtgB5mvx/5QRC721XqUUqq/MwWF2MtXY82ch7N3G3LrGs5fnkAWr8SaMQ9jP9Wlx3usL/coEJGtwNaePj4YDD7H1fS9DRs29PUSnquBvH0DedtAt6+/MeU+cr7wG7h3b+Hu2oy77Z9wjzdgr1iLeWXyw64k+46n19Mzs5VSKnuJCPLRBzi7NkOoCTN5GtacOqzhIx58mKfzKPp0j0IppZQ3xhjM+FcxYyfgnj+Du3sLzoWzuJOmYS9fjfEFPL9GX55H0WPGmFXGmCsXLlzgz//8sVf5yBojR45kypQpTJs27bOv/EYiEerr6xk3bhz19fVEo9E+XmXPfeUrX6GqqorJk3/+DbxHbY+I8Pu///uMHTuW2tpazpw501fL7rGHbd+f/MmfMGzYMKZNm8a0adPYuvXnR1Hffvttxo4dy4QJE9ixY0dfLLnHbt26xdKlS5k4cSKTJk3iu9/9LjBw3r9Hbd9Aef8SiQSzZ89m2mvTqf3Sv+LbbS5WXT3O5Qt0/pdv8c6vvYkx5u+MMXkAxpj8+z9/bIw5bowZ+cQXEZF+/Y9M0X0VGF1fXy+1tbVy8eJFyXYjRoyQlpaWX/jdN77xDXn77bdFROTtt9+WP/7jP+6LpT2TAwcOyOnTp2XSpEmf/e5R27NlyxZZtWqVuK4rR48eldmzZ/fJmp/Gw7bvm9/8pnznO9/5pcdevHhRamtrJZFIyLVr12T06NGSTqdf5HKfyt27d+X06dMiItLa2irjxo2TixcvDpj371HbN1DeP9d1pa2tTUREksmkzJ49W44ePSobfv0tufJf/qOk/uYdAf4K+KpkPlO/BvzV/f9+C/g7ecLncDbsUXx2qY+dO3fy1ltv8e677/b1mp6Ld999ly9/+csAfPnLX+ZnP/tZH6+o5+rq6vD7/b/wu0dtz7vvvstv/dZvYYxh7ty5xGIx7t2798LX/DQetn2P8u677/LWW2+Rn5/PqFGjGDt2LCdOnHjOK3x2Q4cOZfr06QCUlpYyceJE7ty5M2Dev0dt36Nk2/tnjKGkpASAVCpFKpXCGMM/7NzF6H/zDexf/wrAj4Bfuf+UN+//DPBTYLl5wn1UsyEofuFSH8OHD3/sm5wtjDGsXLmSGTNm8M47mS8kNDU1MXRo5p66Q4cOpbm5uS+X6NmjtufOnTu8/PLPz7XM5vf0e9/7HrW1tXzlK1/57NBMNm/fjRs3eO+995gzZ86AfP8e3D4YOO+f4zhMmzaNqqoq6uvrGTNmDBUVFeTk5GAsG37xEkmffaaKSBqIA48tMrIhKH4p6Z4Qflnh8OHDnDlzhm3btvH973+fgwcP9vWSXhh5yDftsvE9/epXv8rVq1c5e/YsQ4cO5Q//8A+B7N2+9vZ2vvjFL/IXf/EXlJU9+p7OA2X7BtL7Z9s2Z8+e5fbt25w4cYIPPvjgYQ/7dMN6dPmkB2VDUPzCpT5u377NSy95v7FHX/t0G6qqqvjCF77AiRMnqK6u/mwX/t69e1RVVfXlEj171PYMHz6cW7d+fj3IbH1Pq6ursW0by7L4vd/7vc8OT2Tj9qVSKb74xS/ypS99iV/91V8FBtb796jtGyjv36cqKipYsmQJx44dIxaLkU6nP/3Tg5dI+uwz1RiTA5QDkcfNzYag+OxSH8lkkp/85CesX7++r9fkSUdHB21tbZ/9986dO5k8eTLr16/nRz/KHDr80Y9+xJtvZvdV1x+1PevXr+fHP/4xIsKxY8coLy//7BBHNnnwuPw//dM/ffaNqPXr1/OTn/yE7u5url+/zkcffcTs2bP7aplPJCL8zu/8DhMnTuTrX//6Z78fKO/fo7ZvoLx/LS0txGIxALq6uti9ezcTJ05k6dKl/PSnP/30YV8GPi13N97/GeCfAXvlYbtRD3pS290f/gGrgQ9Hjx4t3/rWt7x+SaDPXb16VWpra6W2tlZeffXVz7YpFArJsmXLZOzYsbJs2TIJh8N9vNKee+utt2TIkCGSk5Mjw4YNkx/84AeP3B7XdeVrX/uajB49WiZPniwnT57s49U/2cO27zd/8zdl8uTJMmXKFFm3bp3cvXv3s8d/61vfktGjR8v48eNl69atfbjyJ2toaBBApkyZIlOnTpWpU6fKli1bBsz796jtGyjv37lz52TatGkyZcoUmTRpkvzpn/6piGQ+Z2bNmiVjxowR4H8C+ZL5PC24//PHwAlgtDzhMzirzszmCcfRlFJKPVTW3uFOKaVUFtCgUEop9VjZdq2n/v0dNaWUGoB0j0IppdRjaVAopZR6LA0KpZRSj6VBoZRS6rE0KJRSaoAyxowzxuw3xpwyxnzbGPPxs8zRoFBKqQHIGGMDPwa+LiIzgULg4rPMyravxyqllOqZXwEuicintyD8AIg9yyDdo1BKqYHpNeDsAz9PBc49yyANCqWUGpjCwCsAxpg5wG8B559lULZdFFAppVQPGGOCwBagCNgKfAmoERH3qWdpUCil1MBmjHkZ+KmIzHmW5+uhJ6WUGvim8oyHnUD3KJRSSj2B7lEopZR6LA0KpZRSj6VBoZRS6rE0KJRSSj2WBoVSSqnH0qBQSin1WBoUSimlHkuDQiml1GP9/+cb0/WJ8JC5AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# create plots with different parameters\n",
    "Dplot0 = plot(DInverse.subs([(pb,4),(pk,Rational(10,3)),(y,Rational(25,2))]), (q,0,300), xlabel=\"$q$\", ylabel=\"$p$\", show=False)\n",
    "Dplot1 = plot(DInverse.subs([(pb,5),(pk,Rational(10,3)),(y,Rational(25,2))]), (q,0,300), xlabel=\"$q$\", ylabel=\"$p$\", show=False)\n",
    "\n",
    "# combine plots into one plot\n",
    "Dplot_combo = Dplot0\n",
    "Dplot_combo.extend(Dplot1)\n",
    "\n",
    "# assign colors for each line\n",
    "Dplot_combo[0].line_color='salmon'\n",
    "Dplot_combo[1].line_color='orchid'\n",
    "\n",
    "# show plot\n",
    "Dplot_combo.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Questions and challenges"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q1: Add a 3rd demand curve to the graph above, similarly with $p_k=3\\frac{1}{3}, y=12.5$, but $p_b=6$.\n",
    "\n",
    "Make its color \"blue\"."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "# code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q2: Do all three demand curves have the same slope? Explain the reason behind it in terms of the demand function and its derivatives.\n",
    "\n",
    "Write up your answer below in markdown."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(answer here)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q3: \n",
    "* Is the gap between the first and second demand curves the same as the gap between the second and the third curves? \n",
    "* How much is the vertical distance between each two curves? \n",
    "* How much is the horizontal distance between each two curves? \n",
    "\n",
    "Explain the reason behind your observations in terms of the demand function and its derivatives.\n",
    "\n",
    "Write up your answer below in markdown."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(answer here)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q4: By hand\n",
    "\n",
    "* calculate the derivative of the demand function with respect to $p_k$\n",
    "* calculate the derivative of the demand function with respect to $y$\n",
    "\n",
    "Write up your answer below in markdown."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(answer here)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q5: Verify your answers using sympy\n",
    "\n",
    "* calculate the derivative of the demand function with respect to $p_k$\n",
    "* calculate the derivative of the demand function with respect to $y$\n",
    "\n",
    "Write up your answer below in code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "# code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q6: What are the interpretations of each of the derivatives above? Interpret both their magnitudes and their signs. \n",
    "\n",
    "Write up your answer below in markdown."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(answer here)"
   ]
  }
 ],
 "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"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}