Tom added a lecture on input-output analysis; not in toc yet

Author: thomassargent30

lectures/input_output.md

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+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.14.4
+kernelspec:
+  display_name: Python 3 (ipykernel)
+  language: python
+  name: python3
+---
+
+(input_output)=
+
+In this lecture, we will need the following library.
+
++++
+
+# Input-Output Models
+
+## Overview
+
+We adopt notation in chapters 8 and 9 of the classic book {cite}`DoSSo`.
+
+We let 
+
+ * $X_0$ be the amount of a single exogenous input to production. We'll call this input 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 $i=1, \ldots n$, $j = 1, \ldots n$
+ * $x_{0j}$ be the quantity of labor allocated  to produce one unit of good $j$.
+ * $a_{ij}$ be the number of units of good $i$ required to produce one unit of good $j$, $i=0, \ldots, n, j= 1, \ldots n$. 
+ * $w >0$ be the exogenous wage of labor, denominated in dollars per unit of labor
+ * $p$ be an $n \times 1$ vector of prices of produced goods $i = 1, \ldots , n$. 
+ 
+
+
+The production function for goods $j \in \{1, \ldots , n\}$ is the **Leontief** function
+
+$$
+X_j = \min_{i \in \{0, \ldots , n \}} \left( \frac{x_{ij}}{a_{ij}}\right) 
+$$
+
+
+To illustrate ideas, we'll begin by setting $n =2$.
+
+Feasible allocations must satisfy
+
+$$
+\begin{aligned}
+(1 - a_{11}) X_1 - a_{12} X_2 & \geq C_1 \cr 
+-a_{21} X_1 + (1 - a_{22}) X_2 & \geq C_2 \cr 
+a_{01} X_1 + a_{02} X_2 & \leq X_0 
+\end{aligned}
+$$
+
+or more generally
+
+$$
+\begin{aligned}
+(I - a) X &  \geq C \cr 
+a_0^\top X & \leq X_0
+\end{aligned}
+$$ (eq:inout_1)
+
+where $a$ is the $n \times n$ matrix with typical element $a_{ij}$ and $a_0^\top = \begin{bmatrix} a_{01} & \cdots & a_{0n} \end{bmatrix}$.
+
+
+
+If we solve the first block of equations of {eq}`eq:inout_1` for gross output $X$ we get 
+
+$$ 
+X = (I -a )^{-1} C \equiv A C 
+$$ (eq:inout_2)
+
+where $A = (I-a)^{-1}$.  
+
+The coefficient $A_{ij} $ is the amount of good $i$ that is required as an intermediate input to produce one unit of final output $j$.
+
+We assume the **Hawkins-Simon condition** 
+
+$$ 
+\det (I - a) > 0 
+$$
+
+to assure that the solution $X$ of {eq}`eq:inout_2` is a positive vector. 
+
+
+## Production Possibility Frontier
+
+The second equation of {eq}`eq:inout_1` can be written
+
+$$
+a_0^\top X = X_0 
+$$
+
+or 
+
+$$
+A_0^\top C = X_0
+$$ (eq:inout_frontier)
+
+where
+
+$$
+A_0^\top = a_0^\top (I - a)^{-1}
+$$
+
+The $i$th Component $A_0$ is the amount of labor that is required to produce one unit of final output of good $i$ for $i \in \{1, \ldots , n\}$.
+
+Equation {eq}`eq:inout_frontier` sweeps out a  **production possibility frontier** of final consumption bundles $C$ that can be produced with exogenous labor input $X_0$. 
+
+
+## Prices
+
+{cite}`DoSSo` argue that relative prices of the $n$ produced goods must satisfy  
+
+$$ 
+p = a^\top p + a_0 w
+$$
+
+which states that the price of each final good equals the total cost 
+of production, which consists of costs of intermediate inputs $a^\top p$
+plus costs of labor $a_0 w$.
+
+This equation can be written as 
+
+$$
+(I - a^\top) p = a_0 w
+$$ (eq:inout_price)
+
+which implies
+
+$$
+p = (I - a^\top)^{-1} a_0 w
+$$
+
+Notice how  {eq}`eq:inout_price` with {eq}`eq:inout_1` form a
+**conjugate pair**  through the appearance of operators 
+that are transposes of one another.  
+
+This connection surfaces again in a classic linear program and its dual.
+
+
+## Linear Programs
+
+A **primal** problem is 
+
+$$
+\min_{X} w a_0 ^\top X 
+$$
+
+subject to 
+
+$$
+(I -a ) X \geq C
+$$
+
+
+The associated **dual** problem is
+
+$$
+\max_{p} p^\top C 
+$$
+
+subject to
+
+$$
+(I -a)^\top p \leq a_0 w 
+$$
+
+The primal problem chooses a feasible production plan to minimize costs for delivering a pre-assigned vector of final goods consumption $C$.
+
+The dual problem chooses prices to maxmize the value of a pre-assigned vector of final goods $C$ subject to prices covering costs of production. 
+
+Under  sufficient conditions discussed XXXX, optimal value of the primal and dual problems coincide:
+
+$$
+w a_0^\top X^* = p^* C
+$$
+
+where $^*$'s denote optimal choices for the primal and dual problems.
+
+
+
+
+
+
++++
+
+
+## Exercise
+
+{cite}`DoSSo`, chapter 9, carries along an example with the following
+parameter settings:
+
+
+
+$$ 
+a = \begin{bmatrix} .1 & 1.46 \cr
+    .16 & .17 \end{bmatrix}
+$$
+
+$$ 
+a_0 = \begin{bmatrix} .04 & .33 \end{bmatrix}
+$$
+
+$$
+C = \begin{bmatrix} 50 \cr 60 \end{bmatrix}
+$$
+
+$$ 
+X_0 = \begin{bmatrix} 250 \cr 120 \end{bmatrix}
+$$
+
+$$
+X = 50
+$$
+
+
+```{code-cell} ipython3
+
+```
+
+```{code-cell} ipython3
+
+```