input output lecture edits

Author: maanasee

lectures/input_output.md

@@ -4,28 +4,103 @@ jupytext:
     extension: .md
     format_name: myst
     format_version: 0.13
-    jupytext_version: 1.14.4
+    jupytext_version: 1.14.5
 kernelspec:
   display_name: Python 3 (ipykernel)
   language: python
   name: python3
 ---
 
++++ {"user_expressions": []}
+
 (input_output)=
 
 In this lecture, we will need the following library.
 
-+++
++++ {"user_expressions": []}
 
 # Input-Output Models
 
 ## Overview
 
-We adopt notation in chapters 8 and 9 of the classic book {cite}`DoSSo`.
+The following figure illustrates a network of linkages between 71 sectors obtained from the US Bureau of Economic Analysis’s
+2019 Input-Output Accounts Data.
+
+```{code-cell} ipython3
+---
+jupyter:
+  outputs_hidden: true
+  source_hidden: true
+---
+pip install --upgrade quantecon_book_networks
+pip install quantecon
+```
+
+```{code-cell} ipython3
+import numpy as np
+import pandas as pd
+import networkx as nx
+```
+
+```{code-cell} ipython3
+#hide
+
+import quantecon as qe
+import quantecon_book_networks
+import quantecon_book_networks.input_output as qbn_io
+import quantecon_book_networks.plotting as qbn_plt
+import quantecon_book_networks.data as qbn_data
+ch2_data = qbn_data.production()
+import matplotlib.pyplot as plt
+import matplotlib.cm as cm
+import matplotlib.colors as plc
+from matplotlib import cm
+quantecon_book_networks.config("matplotlib")
+import matplotlib as mpl
+mpl.rcParams.update(mpl.rcParamsDefault)
+
+codes_71 = ch2_data['us_sectors_71']['codes']
+A_71 = ch2_data['us_sectors_71']['adjacency_matrix']
+X_71 = ch2_data['us_sectors_71']['total_industry_sales']
+
+centrality_71 = qbn_io.eigenvector_centrality(A_71)
+color_list_71 = qbn_io.colorise_weights(centrality_71,beta=False)
+
+fig, ax = plt.subplots(figsize=(10, 12))
+plt.axis("off")
+
+qbn_plt.plot_graph(A_71, X_71, ax, codes_71,
+              node_size_multiple=0.0005,
+              edge_size_multiple=4.0,
+              layout_type='spring',
+              layout_seed=5432167,
+              tol=0.01,
+              node_color_list=color_list_71)
+
+plt.show()
+```
+
++++ {"user_expressions": []}
+
+An arrow from $i$ to $j$ implies that sector $i$ supplies some of its output as raw material to sector $j$.
+
+Economies are characterised by many such complex and interdependent multisector production networks.
+
+A basic framework for their analysis is [Leontief's](https://en.wikipedia.org/wiki/Wassily_Leontief) input-output model.
+
+This model's key aspect is its simplicity.
+
+In this lecture, we introduce the standard input-ouput model and approach it as a [linear programming](link to lpp lecture) problem.
+
++++ {"user_expressions": []}
 
-We let 
+## Input Output Analysis
 
- * $X_0$ be the amount of a single exogenous input to production. We'll call this input labor
+We adopt notation from chapters 9 and 10 of the classic book {cite}`DoSSo`.
+
+Let 
+
+ * $X_0$ be the amount of a single exogenous input to production, say labor
  * $X_j, j = 1,\ldots n$ be the gross output of final good $j$
  *  $C_j, j = 1,\ldots n$ be the net output of final good $j$ that is available for final consumption
  * $x_{ij} $ be the quantity of good $i$ allocated to be  an input to producing good $j$ for $i=1, \ldots n$, $j = 1, \ldots n$
@@ -43,9 +118,45 @@ X_j = \min_{i \in \{0, \ldots , n \}} \left( \frac{x_{ij}}{a_{ij}}\right)
 $$
 
 
-To illustrate ideas, we'll begin by setting $n =2$.
+To illustrate ideas, we begin by setting $n =2$.
+
+The following is a simple illustration of this network.
+
+```{code-cell} ipython3
+---
+jupyter:
+  source_hidden: true
+---
+G = nx.DiGraph()
+
+nodes= (1,2)
+edges = ((1,1),(1,2),(2,1),(2,2))
+G.add_nodes_from(nodes)
+G.add_edges_from(edges)
+
+pos_list = ([0,0],[2,0])
+pos = dict(zip(G.nodes(), pos_list))
+labels = (f'a_{11}',f'a_{12}',r'$a_{21}$',f'a_{22}')
+edge_labels = dict(zip(G.edges(), labels))
+
+plt.axis("off")
+
+nx.draw_networkx_nodes(G, pos=pos,node_size=800,node_color = 'white', edgecolors='black')
+nx.draw_networkx_labels(G, pos=pos)
+nx.draw_networkx_edges(G,pos = pos,node_size=300,connectionstyle='arc3,rad=0.2',arrowsize=10,min_target_margin=15)
+#nx.draw_networkx_edge_labels(G, pos=pos, edge_labels = edge_labels)
+plt.text(0.055,0.125, r'$x_{11}$')
+plt.text(1.825,0.125, r'$x_{22}$')
+plt.text(0.955,0.075, r'$x_{21}$')
+plt.text(0.955,-0.05, r'$x_{12}$')
+
+        
+plt.show()
+```
+
++++ {"user_expressions": []}
 
-Feasible allocations must satisfy
+#### Feasible allocations must satisfy
 
 $$
 \begin{aligned}
@@ -182,7 +293,9 @@ $$
 
 where $^*$'s denote optimal choices for the primal and dual problems.
 
-+++
+```{code-cell} ipython3
+
+```
 
 ## Exercise
 
@@ -231,12 +344,6 @@ $$
 
 where $L$ is a vector of labor services used in each industry.
 
-
-
-
-
-
-
 ```{code-cell} ipython3
 
 ```