Solving Economic Problems with Linear Systems and Matrices: A Beginner’s Guide

Introduction
Linear systems and matrix algebra are powerful tools that help economists and business analysts make sense of complex resource allocation, cost sharing, and market equilibrium problems. In this blog post, we dive into core ideas, explaining how to represent and solve systems of equations using matrices and apply them in practical economic contexts.

1. Linear Systems in Economics
A linear system is a set of equations where each term is either a constant or the product of a constant and a variable. These systems frequently arise in economics, for example:

• Distributing shared costs among multiple departments
• Determining the quantity of raw materials, labor, and transportation to match total capital
• Finding market equilibrium by equating supply and demand

Example: Cost Allocation
Consider the system:

$$\begin{aligned} 4R + 5L + 2T + 9K &= 35 \\ 3R + 2L + 6T + 8K &= 31 \\ 3R + 2L + 2T + 7K &= 27 \\ 7R + 6L + 8T + 9K &= 41 \end{aligned}$$

Where:

• $R =$ Raw materials
• $L =$ Labor
• $T =$ Transportation
• $K =$ Capital

Solving this system helps allocate costs across these resources.

2. Market Equilibrium with Systems of Equations
In multi-product markets, equilibrium is found when supply equals demand in each segment. Suppose we have:

$$\begin{aligned} 4P_1 + 2P_2 – P_3 + P_4 &= 175 \\ 6P_2 + P_1 – 2P_3 + 2P_4 &= 170 \\ 2P_3 + 2P_1 + 2P_2 – 3P_4 &= 205 \\ 10P_4 – 3P_1 + 4P_2 – 3P_3 &= 225 \end{aligned}$$

Solving for $P_1, P_2, P_3, P_4$ gives the market equilibrium prices across four products.

3. Matrix Representation and Operations
A system of linear equations can be written compactly as a matrix equation:
$AX = B$
Where:

• $A$ is the coefficient matrix
• $X$ is the column of variables
• $B$ is the column of constants

Using matrices simplifies computation, particularly with larger systems.

Matrix Basics

• Equality: Two matrices are equal if all corresponding elements are equal.
• Addition: Combine corresponding elements.
• Scalar multiplication: Multiply each element by the scalar.

Matrix Multiplication

• Only defined when the number of columns in the first matrix equals the number of rows in the second.
• Not commutative: $AB \neq BA$

4. Gauss Elimination Method
This step-by-step algorithm simplifies a system to row echelon form, then solves it using back-substitution.

Steps:

1. Use row operations to eliminate variables below the pivot.
2. Continue until the matrix is in upper-triangular form.
3. Solve from the bottom row upward.

This method is ideal for solving systems by hand or programming into spreadsheets.

5. The Identity Matrix and Inverse

• The identity matrix $I$ acts like 1 in multiplication: $AI = IA = A$
• If a matrix $A$ is invertible, then $A^{-1}$ exists and $A^{-1}A = I$
• We solve $AX = B$ by computing $X = A^{-1}B$ when possible

Conclusion
Matrix algebra and linear systems are more than theoretical tools—they power real-world decisions in budgeting, production, and pricing. Whether you’re allocating costs, balancing supply and demand, or modeling economic dependencies, understanding matrices gives you a clear advantage in solving structured problems efficiently.

Further Reading

• Anton & Rorres, Elementary Linear Algebra
• Khan Academy: Matrix Algebra Essentials