structural_decomposition_analysis/structural_decomposition.ipynb

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   "source": [
    "# Practical 11: Structural decomposition analysis\n",
    "\n",
    "Objectives\n",
    "- Construct different SDA methods in Python\n",
    "- Reproduce the SDA examples on page 348-351 of Miller & Blair\n",
    "- Reproduce output of table 8.1 on page 351 of Miller & Blair"
   ]
  },
  {
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   "execution_count": 1,
   "metadata": {
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   "outputs": [],
   "source": [
    "# Import packages\n",
    "import numpy as np\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Exercise 1: Construct the SDA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### 1.1 Define input data for baseline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
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     "slide_type": "fragment"
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   "outputs": [],
   "source": [
    "# t=0 transaction matrix and final demand vector\n",
    "Z_0 = pd.DataFrame([\n",
    "    [10, 20, 25],\n",
    "    [15, 5, 30],\n",
    "    [30, 40, 5]\n",
    "])\n",
    "\n",
    "y_0 = pd.Series([45,30,25])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### 1.2 Define input data for delta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   },
   "outputs": [],
   "source": [
    "# t=1 transaction matrix and final demand vector\n",
    "\n",
    "Z_1 = pd.DataFrame([\n",
    "    [12, 15, 35],\n",
    "    [24, 11, 30],\n",
    "    [36, 50, 8]\n",
    "])\n",
    "\n",
    "y_1 = pd.Series([50,35,26])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### 1.3 Calculate total requirements matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   },
   "outputs": [],
   "source": [
    "# t=0 Baseline year\n",
    "\n",
    "x_0 = Z_0.sum(axis=1) + y_0.values\n",
    "A_0 = Z_0 /x_0.transpose()\n",
    "L_0 = np.linalg.inv(np.eye(3) - A_0)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
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   "outputs": [],
   "source": [
    "# t=1 next year\n",
    "\n",
    "x_1 = Z_1.sum(axis=1) + y_1\n",
    "A_1 = Z_1 /x_1.transpose()\n",
    "L_1 = np.linalg.inv(np.eye(3) - A_1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### 1.4 Calculate your deltas for the decomposition"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "#### 1.4.1 Change in total requirement matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "L_delta = L_1 - L_0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### 1.4.2 Change in final demand"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "y_delta = y_1 - y_0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### 1.4.3 Change in total output to be used as check"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   },
   "outputs": [],
   "source": [
    "x_delta = x_1 - x_0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### 1.5 Decomposition\n",
    "\n",
    "All input data for our SDA is now complete.\n",
    "\n",
    "Start decomposing the effect of technology change and final demand change on the total output. \n",
    "\n",
    "A two factor decomposition."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### 1.5.1 Decomposition (eq 8.3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "tech_change = L_delta @ y_0\n",
    "y_change = L_1 @ y_delta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Decomposition correct\n"
     ]
    }
   ],
   "source": [
    "# Check if result is complete\n",
    "\n",
    "tot_change = tech_change + y_change\n",
    "check = x_delta - tot_change\n",
    "\n",
    "if np.isclose(check, np.zeros(check.shape), atol=1E-6).all():\n",
    "    print('Decomposition correct')\n",
    "else:\n",
    "    print('ERROR - check your decomposition')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
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     "slide_type": "subslide"
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       "      <th></th>\n",
       "      <th></th>\n",
       "      <th>Output_Change</th>\n",
       "      <th>Technology_change_contribution</th>\n",
       "      <th>Final_Demand_change_contribution</th>\n",
       "      <th>Interaction_term</th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.3</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.898419</td>\n",
       "      <td>11.101581</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
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       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
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       "      <td>33.469614</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
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      "text/plain": [
       "                 Output_Change  Technology_change_contribution  \\\n",
       "Eq_8.3 Sector_1           12.0                        0.898419   \n",
       "       Sector_2           20.0                        8.622001   \n",
       "       Sector_3           20.0                        9.009966   \n",
       "       Total              52.0                       18.530386   \n",
       "\n",
       "                 Final_Demand_change_contribution Interaction_term  \n",
       "Eq_8.3 Sector_1                         11.101581              NaN  \n",
       "       Sector_2                         11.377999              NaN  \n",
       "       Sector_3                         10.990034              NaN  \n",
       "       Total                            33.469614                0  "
      ]
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     "execution_count": 11,
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   "source": [
    "# Organize the labels for the table in which you will store your results\n",
    "sectors = [\"Sector_1\", \"Sector_2\", \"Sector_3\", \"Total\"] \n",
    "columns = [\"Output_Change\", \"Technology_change_contribution\", \"Final_Demand_change_contribution\", \"Interaction_term\"] \n",
    "\n",
    "# create a dataframe where you will store your intermediary results\n",
    "step = pd.DataFrame([], columns=columns)\n",
    "\n",
    "## add results to your dataframe\n",
    "step[step.columns[1]] = tech_change\n",
    "step[step.columns[2]] = y_change\n",
    "step[step.columns[0]] = tot_change\n",
    "\n",
    "## Add the row of total\n",
    "step.loc[\"Total\"] = step.sum(0)\n",
    "\n",
    "step.index = [[\"Eq_8.3\"]*4, sectors]\n",
    "\n",
    "step"
   ]
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   "execution_count": 12,
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th>Output_Change</th>\n",
       "      <th>Technology_change_contribution</th>\n",
       "      <th>Final_Demand_change_contribution</th>\n",
       "      <th>Interaction_term</th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.3</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.898419</td>\n",
       "      <td>11.101581</td>\n",
       "      <td>NaN</td>\n",
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       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
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       "      <td>11.377999</td>\n",
       "      <td>NaN</td>\n",
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       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.009966</td>\n",
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       "      <td>NaN</td>\n",
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       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
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       "      <td>0</td>\n",
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      "text/plain": [
       "                 Output_Change  Technology_change_contribution  \\\n",
       "Eq_8.3 Sector_1           12.0                        0.898419   \n",
       "       Sector_2           20.0                        8.622001   \n",
       "       Sector_3           20.0                        9.009966   \n",
       "       Total              52.0                       18.530386   \n",
       "\n",
       "                 Final_Demand_change_contribution Interaction_term  \n",
       "Eq_8.3 Sector_1                         11.101581              NaN  \n",
       "       Sector_2                         11.377999              NaN  \n",
       "       Sector_3                         10.990034              NaN  \n",
       "       Total                            33.469614                0  "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Make a copy of step that you will call results \n",
    "# and which you can use as a basis to assemble all results\n",
    "results = step.copy()\n",
    "results"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### 1.5.2 Decomposition (eq 8.4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "tech_change = L_delta @ y_1\n",
    "y_change = L_0 @ y_delta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Decomposition correct\n"
     ]
    }
   ],
   "source": [
    "# check if our result is complete\n",
    "tot_change = tech_change + y_change\n",
    "\n",
    "check = x_delta - tot_change\n",
    "\n",
    "if np.isclose(check, np.zeros(check.shape), atol=1E-6).all():\n",
    "    print('Decomposition correct')\n",
    "else:\n",
    "    print('ERROR - check your decomposition')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th>Output_Change</th>\n",
       "      <th>Technology_change_contribution</th>\n",
       "      <th>Final_Demand_change_contribution</th>\n",
       "      <th>Interaction_term</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.4</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.784597</td>\n",
       "      <td>11.215403</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.660650</td>\n",
       "      <td>10.339350</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.963899</td>\n",
       "      <td>10.036101</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>20.409146</td>\n",
       "      <td>31.590854</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
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      "text/plain": [
       "                 Output_Change  Technology_change_contribution  \\\n",
       "Eq_8.4 Sector_1           12.0                        0.784597   \n",
       "       Sector_2           20.0                        9.660650   \n",
       "       Sector_3           20.0                        9.963899   \n",
       "       Total              52.0                       20.409146   \n",
       "\n",
       "                 Final_Demand_change_contribution Interaction_term  \n",
       "Eq_8.4 Sector_1                         11.215403              NaN  \n",
       "       Sector_2                         10.339350              NaN  \n",
       "       Sector_3                         10.036101              NaN  \n",
       "       Total                            31.590854                0  "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# create a dataframe where you will store your intermediary results\n",
    "step = pd.DataFrame([], columns=columns)\n",
    "\n",
    "## add results to your dataframe\n",
    "step[step.columns[1]] = tech_change\n",
    "step[step.columns[2]] = y_change\n",
    "step[step.columns[0]] = tot_change\n",
    "\n",
    "## Add the row of total\n",
    "step.loc[\"Total\"] = step.sum(0)\n",
    "\n",
    "step.index = [[\"Eq_8.4\"]*4, sectors]\n",
    "\n",
    "step"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
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   "outputs": [
    {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th>Output_Change</th>\n",
       "      <th>Technology_change_contribution</th>\n",
       "      <th>Final_Demand_change_contribution</th>\n",
       "      <th>Interaction_term</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.3</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.898419</td>\n",
       "      <td>11.101581</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>8.622001</td>\n",
       "      <td>11.377999</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.009966</td>\n",
       "      <td>10.990034</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>18.530386</td>\n",
       "      <td>33.469614</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.4</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.784597</td>\n",
       "      <td>11.215403</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.660650</td>\n",
       "      <td>10.339350</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.963899</td>\n",
       "      <td>10.036101</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>20.409146</td>\n",
       "      <td>31.590854</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                 Output_Change  Technology_change_contribution  \\\n",
       "Eq_8.3 Sector_1           12.0                        0.898419   \n",
       "       Sector_2           20.0                        8.622001   \n",
       "       Sector_3           20.0                        9.009966   \n",
       "       Total              52.0                       18.530386   \n",
       "Eq_8.4 Sector_1           12.0                        0.784597   \n",
       "       Sector_2           20.0                        9.660650   \n",
       "       Sector_3           20.0                        9.963899   \n",
       "       Total              52.0                       20.409146   \n",
       "\n",
       "                 Final_Demand_change_contribution Interaction_term  \n",
       "Eq_8.3 Sector_1                         11.101581              NaN  \n",
       "       Sector_2                         11.377999              NaN  \n",
       "       Sector_3                         10.990034              NaN  \n",
       "       Total                            33.469614                0  \n",
       "Eq_8.4 Sector_1                         11.215403              NaN  \n",
       "       Sector_2                         10.339350              NaN  \n",
       "       Sector_3                         10.036101              NaN  \n",
       "       Total                            31.590854                0  "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Concatenate your results\n",
    "results = pd.concat([results, step], axis=0)\n",
    "results"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### 1.5.3 Decomposition (eq 8.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "tech_change = L_delta @ y_0\n",
    "y_change = L_0 @ y_delta\n",
    "interaction = L_delta @ y_delta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Decomposition correct\n"
     ]
    }
   ],
   "source": [
    "# check if our result is complete\n",
    "tot_change = tech_change + y_change + interaction\n",
    "\n",
    "check = x_delta - tot_change\n",
    "\n",
    "if np.isclose(check, np.zeros(check.shape), atol=1E-6).all():\n",
    "    print('Decomposition correct')\n",
    "else:\n",
    "    print('ERROR - check your decomposition')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
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       "\n",
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       "\n",
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       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th>Output_Change</th>\n",
       "      <th>Technology_change_contribution</th>\n",
       "      <th>Final_Demand_change_contribution</th>\n",
       "      <th>Interaction_term</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.5</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.898419</td>\n",
       "      <td>11.215403</td>\n",
       "      <td>-0.113822</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>8.622001</td>\n",
       "      <td>10.339350</td>\n",
       "      <td>1.038648</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.009966</td>\n",
       "      <td>10.036101</td>\n",
       "      <td>0.953933</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>18.530386</td>\n",
       "      <td>31.590854</td>\n",
       "      <td>1.878759</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                 Output_Change  Technology_change_contribution  \\\n",
       "Eq_8.5 Sector_1           12.0                        0.898419   \n",
       "       Sector_2           20.0                        8.622001   \n",
       "       Sector_3           20.0                        9.009966   \n",
       "       Total              52.0                       18.530386   \n",
       "\n",
       "                 Final_Demand_change_contribution  Interaction_term  \n",
       "Eq_8.5 Sector_1                         11.215403         -0.113822  \n",
       "       Sector_2                         10.339350          1.038648  \n",
       "       Sector_3                         10.036101          0.953933  \n",
       "       Total                            31.590854          1.878759  "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# create a dataframe where you will store your intermediary results\n",
    "step = pd.DataFrame([], columns=columns)\n",
    "\n",
    "## add results to your dataframe\n",
    "step[step.columns[1]] = tech_change\n",
    "step[step.columns[2]] = y_change\n",
    "step[step.columns[3]] = interaction\n",
    "step[step.columns[0]] = tot_change\n",
    "\n",
    "## Add the row of total\n",
    "step.loc[\"Total\"] = step.sum(0)\n",
    "\n",
    "step.index = [[\"Eq_8.5\"]*4, sectors]\n",
    "\n",
    "step"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th>Output_Change</th>\n",
       "      <th>Technology_change_contribution</th>\n",
       "      <th>Final_Demand_change_contribution</th>\n",
       "      <th>Interaction_term</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.3</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.898419</td>\n",
       "      <td>11.101581</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>8.622001</td>\n",
       "      <td>11.377999</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.009966</td>\n",
       "      <td>10.990034</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>18.530386</td>\n",
       "      <td>33.469614</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.4</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.784597</td>\n",
       "      <td>11.215403</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.660650</td>\n",
       "      <td>10.339350</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.963899</td>\n",
       "      <td>10.036101</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>20.409146</td>\n",
       "      <td>31.590854</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.5</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.898419</td>\n",
       "      <td>11.215403</td>\n",
       "      <td>-0.113822</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>8.622001</td>\n",
       "      <td>10.339350</td>\n",
       "      <td>1.038648</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.009966</td>\n",
       "      <td>10.036101</td>\n",
       "      <td>0.953933</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>18.530386</td>\n",
       "      <td>31.590854</td>\n",
       "      <td>1.878759</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                 Output_Change  Technology_change_contribution  \\\n",
       "Eq_8.3 Sector_1           12.0                        0.898419   \n",
       "       Sector_2           20.0                        8.622001   \n",
       "       Sector_3           20.0                        9.009966   \n",
       "       Total              52.0                       18.530386   \n",
       "Eq_8.4 Sector_1           12.0                        0.784597   \n",
       "       Sector_2           20.0                        9.660650   \n",
       "       Sector_3           20.0                        9.963899   \n",
       "       Total              52.0                       20.409146   \n",
       "Eq_8.5 Sector_1           12.0                        0.898419   \n",
       "       Sector_2           20.0                        8.622001   \n",
       "       Sector_3           20.0                        9.009966   \n",
       "       Total              52.0                       18.530386   \n",
       "\n",
       "                 Final_Demand_change_contribution Interaction_term  \n",
       "Eq_8.3 Sector_1                         11.101581              NaN  \n",
       "       Sector_2                         11.377999              NaN  \n",
       "       Sector_3                         10.990034              NaN  \n",
       "       Total                            33.469614                0  \n",
       "Eq_8.4 Sector_1                         11.215403              NaN  \n",
       "       Sector_2                         10.339350              NaN  \n",
       "       Sector_3                         10.036101              NaN  \n",
       "       Total                            31.590854                0  \n",
       "Eq_8.5 Sector_1                         11.215403        -0.113822  \n",
       "       Sector_2                         10.339350         1.038648  \n",
       "       Sector_3                         10.036101         0.953933  \n",
       "       Total                            31.590854         1.878759  "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Concatenate your results\n",
    "results = pd.concat([results, step], axis=0)\n",
    "results"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### 1.5.4 Decomposition (eq 8.6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "tech_change = L_delta @ y_1\n",
    "y_change = L_1 @ y_delta\n",
    "interaction = L_delta @ y_delta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Decomposition correct\n"
     ]
    }
   ],
   "source": [
    "# check if our result is complete\n",
    "tot_change = tech_change + y_change - interaction\n",
    "check = x_delta - tot_change\n",
    "\n",
    "if np.isclose(check, np.zeros(check.shape), atol=1E-6).all():\n",
    "    print('Decomposition correct')\n",
    "else:\n",
    "    print('ERROR - check your decomposition')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th>Output_Change</th>\n",
       "      <th>Technology_change_contribution</th>\n",
       "      <th>Final_Demand_change_contribution</th>\n",
       "      <th>Interaction_term</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.6</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.784597</td>\n",
       "      <td>11.101581</td>\n",
       "      <td>-0.113822</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.660650</td>\n",
       "      <td>11.377999</td>\n",
       "      <td>1.038648</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.963899</td>\n",
       "      <td>10.990034</td>\n",
       "      <td>0.953933</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>20.409146</td>\n",
       "      <td>33.469614</td>\n",
       "      <td>1.878759</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                 Output_Change  Technology_change_contribution  \\\n",
       "Eq_8.6 Sector_1           12.0                        0.784597   \n",
       "       Sector_2           20.0                        9.660650   \n",
       "       Sector_3           20.0                        9.963899   \n",
       "       Total              52.0                       20.409146   \n",
       "\n",
       "                 Final_Demand_change_contribution  Interaction_term  \n",
       "Eq_8.6 Sector_1                         11.101581         -0.113822  \n",
       "       Sector_2                         11.377999          1.038648  \n",
       "       Sector_3                         10.990034          0.953933  \n",
       "       Total                            33.469614          1.878759  "
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# create a dataframe where you will store your intermediary results\n",
    "step = pd.DataFrame([], columns=columns)\n",
    "\n",
    "## add results to your dataframe\n",
    "step[step.columns[1]] = tech_change\n",
    "step[step.columns[2]] = y_change\n",
    "step[step.columns[3]] = interaction\n",
    "step[step.columns[0]] = tot_change\n",
    "\n",
    "## Add the row of total\n",
    "step.loc[\"Total\"] = step.sum(0)\n",
    "\n",
    "step.index = [[\"Eq_8.6\"]*4, sectors]\n",
    "\n",
    "step"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th>Output_Change</th>\n",
       "      <th>Technology_change_contribution</th>\n",
       "      <th>Final_Demand_change_contribution</th>\n",
       "      <th>Interaction_term</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.3</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.898419</td>\n",
       "      <td>11.101581</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>8.622001</td>\n",
       "      <td>11.377999</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.009966</td>\n",
       "      <td>10.990034</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>18.530386</td>\n",
       "      <td>33.469614</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.4</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.784597</td>\n",
       "      <td>11.215403</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.660650</td>\n",
       "      <td>10.339350</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.963899</td>\n",
       "      <td>10.036101</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>20.409146</td>\n",
       "      <td>31.590854</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.5</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.898419</td>\n",
       "      <td>11.215403</td>\n",
       "      <td>-0.113822</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>8.622001</td>\n",
       "      <td>10.339350</td>\n",
       "      <td>1.038648</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.009966</td>\n",
       "      <td>10.036101</td>\n",
       "      <td>0.953933</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>18.530386</td>\n",
       "      <td>31.590854</td>\n",
       "      <td>1.878759</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.6</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.784597</td>\n",
       "      <td>11.101581</td>\n",
       "      <td>-0.113822</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.660650</td>\n",
       "      <td>11.377999</td>\n",
       "      <td>1.038648</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.963899</td>\n",
       "      <td>10.990034</td>\n",
       "      <td>0.953933</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>20.409146</td>\n",
       "      <td>33.469614</td>\n",
       "      <td>1.878759</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                 Output_Change  Technology_change_contribution  \\\n",
       "Eq_8.3 Sector_1           12.0                        0.898419   \n",
       "       Sector_2           20.0                        8.622001   \n",
       "       Sector_3           20.0                        9.009966   \n",
       "       Total              52.0                       18.530386   \n",
       "Eq_8.4 Sector_1           12.0                        0.784597   \n",
       "       Sector_2           20.0                        9.660650   \n",
       "       Sector_3           20.0                        9.963899   \n",
       "       Total              52.0                       20.409146   \n",
       "Eq_8.5 Sector_1           12.0                        0.898419   \n",
       "       Sector_2           20.0                        8.622001   \n",
       "       Sector_3           20.0                        9.009966   \n",
       "       Total              52.0                       18.530386   \n",
       "Eq_8.6 Sector_1           12.0                        0.784597   \n",
       "       Sector_2           20.0                        9.660650   \n",
       "       Sector_3           20.0                        9.963899   \n",
       "       Total              52.0                       20.409146   \n",
       "\n",
       "                 Final_Demand_change_contribution Interaction_term  \n",
       "Eq_8.3 Sector_1                         11.101581              NaN  \n",
       "       Sector_2                         11.377999              NaN  \n",
       "       Sector_3                         10.990034              NaN  \n",
       "       Total                            33.469614                0  \n",
       "Eq_8.4 Sector_1                         11.215403              NaN  \n",
       "       Sector_2                         10.339350              NaN  \n",
       "       Sector_3                         10.036101              NaN  \n",
       "       Total                            31.590854                0  \n",
       "Eq_8.5 Sector_1                         11.215403        -0.113822  \n",
       "       Sector_2                         10.339350         1.038648  \n",
       "       Sector_3                         10.036101         0.953933  \n",
       "       Total                            31.590854         1.878759  \n",
       "Eq_8.6 Sector_1                         11.101581        -0.113822  \n",
       "       Sector_2                         11.377999         1.038648  \n",
       "       Sector_3                         10.990034         0.953933  \n",
       "       Total                            33.469614         1.878759  "
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Concatenate your results\n",
    "results = pd.concat([results, step], axis=0)\n",
    "results"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### 1.5.5 Decomposition (eq 8.7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "tech_change = 0.5 * L_delta @ (y_0 + y_1)\n",
    "y_change = 0.5 * (L_0 + L_1) @ y_delta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Decomposition correct\n"
     ]
    }
   ],
   "source": [
    "# check if our result is complete\n",
    "tot_change = tech_change + y_change\n",
    "check = x_delta - tot_change\n",
    "\n",
    "if np.isclose(check, np.zeros(check.shape), atol=1E-6).all():\n",
    "    print('Decomposition correct')\n",
    "else:\n",
    "    print('ERROR - check your decomposition')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th>Output_Change</th>\n",
       "      <th>Technology_change_contribution</th>\n",
       "      <th>Final_Demand_change_contribution</th>\n",
       "      <th>Interaction_term</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.7</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.841508</td>\n",
       "      <td>11.158492</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.141326</td>\n",
       "      <td>10.858674</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.486933</td>\n",
       "      <td>10.513067</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>19.469766</td>\n",
       "      <td>32.530234</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                 Output_Change  Technology_change_contribution  \\\n",
       "Eq_8.7 Sector_1           12.0                        0.841508   \n",
       "       Sector_2           20.0                        9.141326   \n",
       "       Sector_3           20.0                        9.486933   \n",
       "       Total              52.0                       19.469766   \n",
       "\n",
       "                 Final_Demand_change_contribution Interaction_term  \n",
       "Eq_8.7 Sector_1                         11.158492              NaN  \n",
       "       Sector_2                         10.858674              NaN  \n",
       "       Sector_3                         10.513067              NaN  \n",
       "       Total                            32.530234                0  "
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# create a dataframe where you will store your intermediary results\n",
    "step = pd.DataFrame([], columns=columns)\n",
    "\n",
    "## add results to your dataframe\n",
    "step[step.columns[1]] = tech_change\n",
    "step[step.columns[2]] = y_change\n",
    "step[step.columns[0]] = tot_change\n",
    "\n",
    "## Add the row of total\n",
    "step.loc[\"Total\"] = step.sum(0)\n",
    "\n",
    "step.index = [[\"Eq_8.7\"]*4, sectors]\n",
    "\n",
    "step"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### 1.6 Output table 8.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th>Output_Change</th>\n",
       "      <th>Technology_change_contribution</th>\n",
       "      <th>Final_Demand_change_contribution</th>\n",
       "      <th>Interaction_term</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.3</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.90</td>\n",
       "      <td>11.10</td>\n",
       "      <td>0.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>8.62</td>\n",
       "      <td>11.38</td>\n",
       "      <td>0.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.01</td>\n",
       "      <td>10.99</td>\n",
       "      <td>0.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>18.53</td>\n",
       "      <td>33.47</td>\n",
       "      <td>0.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.4</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.78</td>\n",
       "      <td>11.22</td>\n",
       "      <td>0.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.66</td>\n",
       "      <td>10.34</td>\n",
       "      <td>0.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.96</td>\n",
       "      <td>10.04</td>\n",
       "      <td>0.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>20.41</td>\n",
       "      <td>31.59</td>\n",
       "      <td>0.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.5</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.90</td>\n",
       "      <td>11.22</td>\n",
       "      <td>-0.11</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>8.62</td>\n",
       "      <td>10.34</td>\n",
       "      <td>1.04</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.01</td>\n",
       "      <td>10.04</td>\n",
       "      <td>0.95</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>18.53</td>\n",
       "      <td>31.59</td>\n",
       "      <td>1.88</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.6</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.78</td>\n",
       "      <td>11.10</td>\n",
       "      <td>-0.11</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.66</td>\n",
       "      <td>11.38</td>\n",
       "      <td>1.04</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.96</td>\n",
       "      <td>10.99</td>\n",
       "      <td>0.95</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>20.41</td>\n",
       "      <td>33.47</td>\n",
       "      <td>1.88</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th rowspan=\"4\" valign=\"top\">Eq_8.7</th>\n",
       "      <th>Sector_1</th>\n",
       "      <td>12.0</td>\n",
       "      <td>0.84</td>\n",
       "      <td>11.16</td>\n",
       "      <td>0.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_2</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.14</td>\n",
       "      <td>10.86</td>\n",
       "      <td>0.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sector_3</th>\n",
       "      <td>20.0</td>\n",
       "      <td>9.49</td>\n",
       "      <td>10.51</td>\n",
       "      <td>0.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total</th>\n",
       "      <td>52.0</td>\n",
       "      <td>19.47</td>\n",
       "      <td>32.53</td>\n",
       "      <td>0.00</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                 Output_Change  Technology_change_contribution  \\\n",
       "Eq_8.3 Sector_1           12.0                            0.90   \n",
       "       Sector_2           20.0                            8.62   \n",
       "       Sector_3           20.0                            9.01   \n",
       "       Total              52.0                           18.53   \n",
       "Eq_8.4 Sector_1           12.0                            0.78   \n",
       "       Sector_2           20.0                            9.66   \n",
       "       Sector_3           20.0                            9.96   \n",
       "       Total              52.0                           20.41   \n",
       "Eq_8.5 Sector_1           12.0                            0.90   \n",
       "       Sector_2           20.0                            8.62   \n",
       "       Sector_3           20.0                            9.01   \n",
       "       Total              52.0                           18.53   \n",
       "Eq_8.6 Sector_1           12.0                            0.78   \n",
       "       Sector_2           20.0                            9.66   \n",
       "       Sector_3           20.0                            9.96   \n",
       "       Total              52.0                           20.41   \n",
       "Eq_8.7 Sector_1           12.0                            0.84   \n",
       "       Sector_2           20.0                            9.14   \n",
       "       Sector_3           20.0                            9.49   \n",
       "       Total              52.0                           19.47   \n",
       "\n",
       "                 Final_Demand_change_contribution  Interaction_term  \n",
       "Eq_8.3 Sector_1                             11.10              0.00  \n",
       "       Sector_2                             11.38              0.00  \n",
       "       Sector_3                             10.99              0.00  \n",
       "       Total                                33.47              0.00  \n",
       "Eq_8.4 Sector_1                             11.22              0.00  \n",
       "       Sector_2                             10.34              0.00  \n",
       "       Sector_3                             10.04              0.00  \n",
       "       Total                                31.59              0.00  \n",
       "Eq_8.5 Sector_1                             11.22             -0.11  \n",
       "       Sector_2                             10.34              1.04  \n",
       "       Sector_3                             10.04              0.95  \n",
       "       Total                                31.59              1.88  \n",
       "Eq_8.6 Sector_1                             11.10             -0.11  \n",
       "       Sector_2                             11.38              1.04  \n",
       "       Sector_3                             10.99              0.95  \n",
       "       Total                                33.47              1.88  \n",
       "Eq_8.7 Sector_1                             11.16              0.00  \n",
       "       Sector_2                             10.86              0.00  \n",
       "       Sector_3                             10.51              0.00  \n",
       "       Total                                32.53              0.00  "
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Concatenate your results\n",
    "results = pd.concat([results, step], axis=0).fillna(0)\n",
    "results.round(2)"
   ]
  }
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