Production Networks

Prerequisites

• Introduction to Graphs

• Weighted Graphs

Outcomes

• Recall the key concepts of spectral theory from Linear Algebra

• Desribe the key proprties of the Leintief family of production models

• Explain the difference between “in”-based centrality and “out”-based centrality

• Analyze the impact of sector specific shocks on other sectors of the US economy

References

• QuantEcon Networks chapters 1-2 (especially section 1.2)

Linear Algebra¶

• Linear algebra is the backbone of modern computational algorithms

  • Graphics

  • Statistics

  • Data analysis

  • Optimization

  • Machine Learning

• Workhorse for accelerated computing

  • GPUs work on matrices to efficiently parallellize common computations

  • Specialty hardware like the TPU (tensor processing unit) take this even further

Building Blocks¶

• Vectors: arrays of numbers representing points in multi-dimensional space

• Matrices

  • Rectangular arrays that transform vectors

  • Also used to represent certain datasets/relationships: e.g. adjacency matrix in Graph Theory

• Tensors: higher dimensional collections of numbers that allow high-dimensional

  • Vectors are 1d tensors, matrices 2d tensors, etc.

  • Implemented in numpy as np.array and Julia as the build in array

Some Key Theories¶

1. Linear systems of equations

2. Inner product spaces (length, distance, and angles)

3. Eigenvalues and eigen vectors (spectral theory)

Spectral Theory¶

• Let’s dig into eigenvalues and eigenvectors

• Let $A \in \mathbb{R}^{n \times n}$

• A scalar $\lambda \in \mathbb{C}$ is an eigenvalue of $A$ if there exists a nonzero vector $e \in \mathbb{C}^n$ such that $A e = \lambda e$

• A vector $e$ satisfying this equality is called an eigenvector corresponding to the eigenvalue $\lambda$

• A vector $\epsilon$ sastifying $A ' \epsilon = \lambda \epsilon$ is a left-eigenvector of $A$

Definitions¶

• The set of all eigenvalues of $A$ is called the spectrum of $A$ and is written $\sigma(A)$

• The spectral radius of $A$ is written $r(A) \equiv \max \{ | \lambda | : \lambda \in \sigma(A) \}$

  • The spectral radius important when considering the convergence of a dynamic system driven by $A$

  • If $x_{t+1} = A x_t$ and $r(A)< 1$ then the sequence $\{x\}_t$ is finite and will converge

• Some facts (not proven here)

  • $A$ will have at most $n$ distinct eigenvalues

  • Eigenvectors are only unique to a scalar multiple: if $(\lambda, e)$ is an eigenpair, then so is $(\lambda, \alpha e)$ for any $\alpha \in \mathbb{R}, \alpha \ne 0$

• Sometimes a matrix will have a repeated eigenvalue (the same value works with multiple distinct eigenvectors)

• The algebraic multiplicity of an eigenvalue is the number of times it is repeated with distinct eigenvectors

• An eigenvalue with an algebraic multiplicity of one is called simple

Diagonalization¶

• A matrix $A \in \mathbb{R}^{n \times n}$ is diagonalizable if $A = PDP^{-1}$

  • $D = \text{diag}(\lambda_1, \dots, \lambda_n)$, $P$ is some invertible matrix

• The decomposition is called the eigen decomposition or spectral decomposition of $A$

• The asymptotic properties of $m \mapsto A^m$ are determined by $\sigma(A)$

  • Can be seen when $A$ is diagonalizable

  • If $A = P \text{diag}(\lambda) P^{-1} \Longrightarrow A^m = P \text{diag}(\lambda_i^m) P^{-1}$

Example: Worker Dynamics¶

• Consider a continuum of workers (large number, not counted discretely)

• Workers can be in one of two states: (1) employed and (2) unemployed

• Each month employed workers become unemployed at rate $\alpha$, while unemployed workers are hired at rate $\beta$

• We express these dynamics as a weighted directed graph with adjacency matrix:

• Row 1 gives the probabilities of employment and unemployement for an employed worker

• Row 2 gives the probabilities of employment and unemployement for an unemployed worker

Transitions¶

• Suppose we have a vector $x \in \Delta(2)$ (2 element simplex) representing current fraction of workers that are employed and unemployed

• Question... What matrix operation between $P_w$ and $x$ will give the fraction of workers that start next month as employed and unemployed?

The simulation should reveal the pattern summarized in Figure 1: repeated transitions move the population toward a steady state.

Neumann Series Lemma¶

• We need one more linear alegbra result...

• If $A \in \mathbb{R}^{n \times n}$ and $r(A) < 1$, then $I-A$ is non-singular and

  $$(I-A)^{-1} = \sum_{m=0}^{\infty} A^m$$

  (2)

• This is known as the Neumann series lemma

Production Networks¶

• We will now study a family of economic models that allow us to analyze the economy as a collection of sectors

• These models were proposed by nobel prize winner Wassily Leontief in 1941 and are still commonly used today

• The key idea behind a Leontief model is the input-output table

Input-output tables¶

• Firms (companies) are categorized into $n$ distinct sectors

• Firms use materials produced by other sectors as part of their production process

• The relationship of flows of value are organized into an input/output table

• $a_{ij}$ is called an input-output coefficient and is equal to:

Figure 2 connects this coefficient to both the production network and its matrix representation.

• If $a_{ij}$ is large, sector $i$ is an important supplier of intermediate goods to sector $j$

• The sum of column $j$ is the value of all inputs to sector $j$

• Row $i$ shows how intensively each sector uses good $i$ as an intermediate good

• The input output table can be directly used as the adjacency matrix for a weighted directed graph

Input-output data¶

• In the United States, the Bureau of Economic Analysis is responsible for compiling input-output tables for sectors

• We’ll be studying the Input-Output Accounts Data

• The main set of tables we’ll be using are called the Make-Use tables

• The Make table shows the value of final goods produced by each sector

  • Note this is predominately a diagonal matrix as each sector primarily produces and sells final goods within their sector

  • Sometimes a firm will have secondary outputs as part of their production process

  • “Which industries produce which commodities?”

• The Use tables show the intermediate and final use of commodities across sectors

  • “Who uses or consumes the commodities produced?”

Example: US 15 sector data¶

• Color of nodes is according to their hub-based eigenvector centrality (see below)

• Size of nodes is according to their total outputs (make table)

• Thickness of edges is according to amount of goods

• Represent $a_{ij}$ and point from sector creating the intermediate good (sector $i$) to sector using intermediate good (sector $j$)

Eigenvector Centrality¶

• The node size above shows the hub-based eigenvector centrality

• This is equal to the dominant eigenvector of the adjacency matrix (eigenvector associated with largest eigenvalue)

• This measure of centrality measures the influence of a node in a network

  • When computing this statistic for a node N, the value will be higher if it is connected to other “influential” nodes

  • https://youtu.be/LYyZqlyDEL4?si=x2kYhe3phZxQtQvV&t=202

Example¶

• Let’s compute the eigenvector centrality for the data in the image above

• I have some code to import the make/use files into a helpful struct

• We’ll see what all the fields are as we progress through the lecture

Accounting¶

• Let...

  • $d_i$ be the final consumer demand for good $i$

  • $x_i$ be total sales of sector $i$

  • $z_{ij}$ be inter-industry sales from sector $i$ to sector $j$

Figure 3 shows how these quantities fit together before we write the accounting identity in matrix form.

• For accounts to add up we must have

  $$x_i = \sum_{j=1}^n z_{ij} + d_i$$

  (4)

• Notice that

  $$\frac{z_{ij}}{x_j} = a_{ij} = \text{dollar value of inputs from $i$ per dollar output from $j$}$$

  (5)

• This means

  $$x_i = \sum_{j=1} a_{ij} x_j + d_i$$

  (6)

• Which can be written $x = Ax + d$

Equilibrium¶

• An equilibrium in a Leonteif model with input-output coefficient matrix $A$ and a vector $d \in \mathbb{R}^n$ of final consumer demands for each sector is a vector $x$ for sector-specific output such that $x = Ax + d$ is satisfied

  • Note that $d$ and $A$ are treated as given

• To find this vector requires that we trace the impact of final demand on the intermediate linkages through $A$

• Example:

• Example

  • Suppose $d_3 \uparrow$

  • Will cause 3 to consume more from its suppliers (2 and 4)

  • Which will cause 2 to demand more from 1

  • Which will cause 1 to demand more from 2 and 4

  • ... and so on

• Computing an equilibrium is not entirely straight forward...

Computing an Equilibrium¶

• Define $v_j \equiv x_j - \sum_{i=1}^n z_{ij}$ as the value added by sector $j$

• Assumption: The input-output adjacency matrix $A$ satisfies

  $$\eta_j \equiv \sum_{i=1}^n a_{ij} < 1 \; \forall j \in [n]$$

  (7)

  • This holds whenever $v_j > 0 \forall j$

• When this assumption holds, for each $d \ge 0$ there is a unique, nonnegative output solution given by

  $$x^* = Ld \qquad \text{where } L \equiv (I-A)^{-1}$$

  (8)

• The matrix $L$ is called the Leontief inverse associated with the coefficient matrix $A$

Question

We don’t have a field for d in our SectorData struct.

How could we compute/derive d given the fields we do have?

Demand Shocks¶

• A common form of economic analysis is to consider the impact of external “shocks”

• Often, these are modeled as a one time change in the value of a variable

• Let’s consider a demand shock of size $\Delta d$ such that demand moves from $d_0$ to $d_1 = \Delta d + d_0$

• The equilibrium vector shifts from $x_0 = L d_0$ to $x_1 = L d_1$

• Define $\Delta x = L (d_1 - d_0) = L \Delta d$

What happens between the initial change in demand and the final change in output? Figure 4 traces the response one round at a time.

NSL¶

• We will further assume that $r(A)<1$ so that we can write the expression for $\Delta x$ as an infinite sum using the Neumann Series Lemma:

  $$\Delta x = \Delta d + A (\Delta d) + A^2 (\Delta d) + \cdots$$

  (9)

  1. $\Delta d$ is the initial response in each sector,

  2. $A (\Delta d)$ is the response generated by the first round of backward linkages,

  3. $A^2 (\Delta d)$ is the response generated by the second round, and so on.

• The total response is the sum of the responses at each round

• The typical element $l_{ij}$ of $L = \sum_{m=0}^{\infty} A^m$ shows total impact on sector i of a unit change in the demand for good $j$