Chapter 69: Linear Algebra in Economics¶

Prerequisites: Non-negative Matrices & Perron-Frobenius (Ch17) · Matrix Equations (Ch20) · Basics of Probability and Statistics

Chapter Outline: Algebraic Balance of Social Production → The Leontief Input-Output Model → Consumption Matrix $A$ and the Production Equation $(I-A)x=d$ → Economic Meaning of Matrix Inversion: The Multiplier Effect → Price Evolution and Spectral Analysis → Portfolio Optimization: Covariance Matrices in the Markowitz Model → Linear Representations in Arbitrage Pricing Theory (APT) → Applications: National Industrial Planning, Financial Risk Assessment, and Inflation Prediction

Extension: Economics is the "resource mapping" of linear algebra; it abstracts complex transactions and supply chains into flow matrices. It proves that long-term economic equilibrium is essentially the principal eigenvector of a positive operator—the algebraic engine for understanding modern macro-regulation and financial mathematics.

In modern economics, industries are interdependent. Agriculture needs machinery, machine shops need steel, and steel mills need electricity. Linear Algebra provides the perfect framework for describing this complex web of supply. Through the Leontief Input-Output Model, we can precisely calculate the total output required from each sector to satisfy a specific set of societal demands. This chapter introduces this Nobel Prize-winning theory of economic algebra.

69.1 The Leontief Input-Output Model¶

Definition 69.1 (Consumption Matrix $A$)

In an economic system with $n$ sectors, $a_{ij}$ represents the value of sector $i$'s output required to produce 1 unit of sector $j$'s output. Linear System: Let $\mathbf{x}$ be the vector of total outputs and $\mathbf{d}$ be the vector of external demands. The system satisfies: $$(I - A)\mathbf{x} = \mathbf{d}$$

The Multiplier Effect

If $(I-A)$ is invertible, then $\mathbf{x} = (I-A)^{-1}\mathbf{d}$. Using the power series $(I-A)^{-1} = I + A + A^2 + \cdots$, each term represents successive rounds of indirect demand triggered by the initial external consumption.

69.2 Portfolio Optimization (Markowitz)¶

Technique: Minimum Variance Portfolio

Given the covariance matrix $\Sigma \succ 0$ of $n$ assets' returns, find the weight vector $\mathbf{w}$ that minimizes risk for a target return. This is a Quadratic Programming problem under linear constraints, whose analytic solution involves the calculation of $\Sigma^{-1}$.

69.3 Price Equilibrium and Spectral Theory¶

Theorem 69.1 (Price Equilibrium)

In long-term equilibrium, the price vector $\mathbf{p}$ satisfies $\mathbf{p}^T = \mathbf{p}^T A + \mathbf{v}^T$, where $\mathbf{v}$ is the vector of value-added. This essentially involves finding the left eigenvector associated with the production process.

Exercises¶

1. [Basics] Given consumption matrix $A = \begin{pmatrix} 0.5 & 0.4 \\ 0.2 & 0.5 \end{pmatrix}$ and external demand $\mathbf{d} = (1, 1)^T$, find the total output $\mathbf{x}$.

Solution

Steps: 1. Calculate $I - A = \begin{pmatrix} 0.5 & -0.4 \\ -0.2 & 0.5 \end{pmatrix}$. 2. Inversion: $\det = 0.25 - 0.08 = 0.17$. 3. $(I-A)^{-1} = \frac{1}{0.17} \begin{pmatrix} 0.5 & 0.4 \\ 0.2 & 0.5 \end{pmatrix}$. 4. $\mathbf{x} = \frac{1}{0.17} \begin{pmatrix} 0.9 \\ 0.7 \end{pmatrix} \approx \begin{pmatrix} 5.29 \\ 4.12 \end{pmatrix}$.

2. [Property] Why must the spectral radius $\rho(A)$ be less than 1 in the Leontief model?

Solution

Economic Reasoning: 1. $\rho(A) < 1$ ensures $(I-A)^{-1}$ exists and is non-negative. 2. Physically, this means the total input (direct + indirect) required to produce 1 unit of output is less than 1 unit. 3. If $\rho(A) \ge 1$, the system consumes resources faster than it produces them, making it impossible to satisfy any positive external demand. Conclusion: This is the algebraic criterion for a "productive" or sustainable economy.

3. [Calculation] Two assets have correlation 0.5 and standard deviations 10% and 20%. Write the covariance matrix $\Sigma$.

Solution

Steps: 1. Variances: $\sigma_1^2 = 0.01$, $\sigma_2^2 = 0.04$. 2. Covariance: $\sigma_{12} = \rho \sigma_1 \sigma_2 = 0.5 \cdot 0.1 \cdot 0.2 = 0.01$. Conclusion: $\Sigma = \begin{pmatrix} 0.01 & 0.01 \\ 0.01 & 0.04 \end{pmatrix}$.

4. [Portfolio] Prove that the positive definiteness of the covariance matrix determines a unique risk minimum.

Solution

Reasoning: Portfolio variance is $\mathbf{w}^T \Sigma \mathbf{w}$. If $\Sigma \succ 0$, this is a strictly convex hyper-paraboloid. Optimization theory states that a strictly convex function over a convex feasible set has a unique global minimum.

5. [Application] What is "Forward Linkage" in industrial analysis?

Solution

Algebraic manifestation: It corresponds to the row sums of the matrix $(I-A)^{-1}$. It describes how much a particular sector's output is utilized as input for other sectors. Linear algebra reveals the "bottleneck" sectors of a national economy through these row sum calculations.

6. [Calculation] Is the economic system described by $A = \begin{pmatrix} 0.1 & 0.9 \\ 0.8 & 0.1 \end{pmatrix}$ feasible?

Solution

Steps: Characteristic equation: $(0.1-\lambda)^2 - 0.72 = 0 \implies \lambda = 0.1 \pm \sqrt{0.72} \approx 0.1 \pm 0.85$. The maximum eigenvalue $\rho(A) \approx 0.95 < 1$. Conclusion: Feasible. Despite high cross-consumption, the system remains productive.

7. [Duality] What do "Shadow Prices" in economics correspond to in Linear Programming?

Solution

Conclusion: The dual variables $\mathbf{y}$. Shadow prices represent the marginal increase in the objective function (total value) for a unit increase in a constrained resource ($b_i$). This captures the marginal utility of matrix constraints.

8. [Arbitrage] What is the linear factor model in Arbitrage Pricing Theory (APT)?

Solution

It assumes returns $R$ can be modeled as $R = E + B\mathbf{f} + \epsilon$, where $B$ is the loading matrix (Betas) and $\mathbf{f}$ is the vector of common factors. This essentially decomposes asset fluctuations into a systemic subspace component and an idiosyncratic component.

9. [Property] Prove: If $A \ge 0$ is a consumption matrix with $\rho(A) < 1$, then $(I-A)^{-1} \ge 0$.

Solution

Reasoning: The series $(I-A)^{-1} = \sum_{k=0}^\infty A^k$ converges. Since powers of non-negative matrices are non-negative, and their sum is non-negative, the result follows. This is an application of M-matrix theory (Ch38A), ensuring that increased demand leads to increased production.

10. [Application] Describe the role of linear algebra in analyzing "Global Value Chains" (GVC).

Solution

By constructing Multi-Regional Input-Output (MRIO) tables, the matrix entries cross borders. Using Schur complements and matrix inversion on these partitioned matrices, researchers can trace how many times a product crosses borders during production and precisely quantify each nation's value-added contribution.

Chapter Summary¶

Linear algebra is the "universal scale" of quantitative economics:

Production Closure: The Leontief model proves that complex social production can be described via fixed points of linear operators, establishing the mathematical basis for sectoral coordination.
Geometrization of Risk: Portfolio theory transforms uncertain financial fluctuations into optimization paths over convex surfaces using covariance matrices, powering modern quantitative finance.
Multiplier Effect of Value: Inversion theory reveals how local economic shocks are amplified through industrial chains, providing the logic engine for government policy evaluation.