💰Intro to Mathematical Economics Unit 2 – Linear Algebra in Economics

Key Concepts and Definitions

• Linear algebra studies linear equations, matrices, and vector spaces
• Matrices represent systems of linear equations in a compact form
• Vector spaces consist of a set of vectors that can be added together and multiplied by scalars
• Scalars are real numbers that can be used to multiply vectors
• Linear independence means a set of vectors cannot be expressed as linear combinations of each other
• Span refers to the set of all possible linear combinations of a given set of vectors
• Basis is a linearly independent set of vectors that spans a vector space
• Dimension of a vector space equals the number of vectors in its basis

Matrices and Vector Spaces

• Matrices are rectangular arrays of numbers arranged in rows and columns
• Elements of a matrix are denoted by $a_{ij}$, where $i$ represents the row and $j$ represents the column
• Square matrices have an equal number of rows and columns
• Identity matrix has 1s on the main diagonal and 0s elsewhere (e.g., $I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$)
• Transpose of a matrix $A$, denoted as $A^T$, is obtained by interchanging the rows and columns of $A$
• Symmetric matrices are equal to their transpose ($A = A^T$)
• Vectors are matrices with only one column or one row
  • Column vectors are denoted as $\vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$
  • Row vectors are denoted as $\vec{v} = \begin{bmatrix} v_1 & v_2 & \cdots & v_n \end{bmatrix}$

Linear Equations in Economics

• Linear equations represent economic relationships between variables
• Supply and demand equations are examples of linear equations in economics
  • Supply equation: $Q_s = a + bP$, where $Q_s$ is quantity supplied, $P$ is price, and $a$ and $b$ are constants
  • Demand equation: $Q_d = c - dP$, where $Q_d$ is quantity demanded, $P$ is price, and $c$ and $d$ are constants
• Equilibrium price and quantity can be found by solving the system of supply and demand equations
• Leontief input-output model uses linear equations to analyze interdependencies between industries
• Maximization and minimization problems in economics can be formulated as linear programming problems

Matrix Operations and Economic Applications

• Matrix addition and subtraction are performed element-wise ($C_{ij} = A_{ij} + B_{ij}$ or $C_{ij} = A_{ij} - B_{ij}$)
• Scalar multiplication of a matrix is performed by multiplying each element by the scalar ($B_{ij} = kA_{ij}$)
• Matrix multiplication is defined for matrices $A$ and $B$ if the number of columns in $A$ equals the number of rows in $B$
  • Element $c_{ij}$ of the product matrix $C = AB$ is given by $c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$
• Matrix multiplication is used in input-output analysis to calculate the total output required to meet a given final demand
• Cramer's rule uses determinants of matrices to solve systems of linear equations

Eigenvalues and Eigenvectors in Economic Models

• Eigenvalues and eigenvectors are important in analyzing the long-term behavior of dynamic economic models
• Eigenvalue $\lambda$ and eigenvector $\vec{v}$ of a square matrix $A$ satisfy the equation $A\vec{v} = \lambda\vec{v}$
• Characteristic equation $\det(A - \lambda I) = 0$ is used to find the eigenvalues of a matrix
• Eigenvectors corresponding to each eigenvalue can be found by solving $(A - \lambda I)\vec{v} = \vec{0}$
• Dominant eigenvalue determines the long-term growth rate in economic models (e.g., Leontief input-output model)
• Stability of equilibrium points in dynamic economic models can be analyzed using eigenvalues

Input-Output Analysis

• Input-output analysis studies the interdependencies between industries in an economy
• Leontief input-output model represents the economy as a system of linear equations
• Input-output table shows the flow of goods and services between industries
  • Rows represent the distribution of an industry's output to other industries and final demand
  • Columns represent the inputs required by an industry from other industries and value added
• Technical coefficients matrix $A$ represents the input requirements per unit of output for each industry
• $(I - A)^{-1}$ is the Leontief inverse matrix, which shows the total output required to meet a unit increase in final demand
• Output multipliers can be calculated from the Leontief inverse matrix to analyze the impact of changes in final demand on total output

Linear Programming and Optimization

• Linear programming is a method for optimizing a linear objective function subject to linear constraints
• Objective function represents the goal of the optimization problem (e.g., maximizing profit or minimizing cost)
• Constraints are linear inequalities or equations that limit the feasible region of the problem
• Feasible region is the set of all points that satisfy the constraints
• Optimal solution is the point in the feasible region that maximizes or minimizes the objective function
• Simplex method is an algorithm for solving linear programming problems
  • Iteratively moves from one vertex of the feasible region to another until the optimal solution is found
• Duality in linear programming relates the primal and dual problems, which have the same optimal value

Practical Applications in Economic Analysis

• Portfolio optimization uses linear programming to find the optimal asset allocation that maximizes return or minimizes risk
• Production planning problems can be formulated as linear programming problems to determine the optimal production levels
• Transportation problems aim to minimize the cost of shipping goods from supply points to demand points
• Resource allocation problems involve optimally distributing limited resources among competing activities
• Environmental economics uses linear programming to analyze the trade-offs between economic activities and environmental impacts
• Game theory models strategic interactions between economic agents and can be analyzed using linear algebra
• Econometrics employs linear regression models to estimate the relationships between economic variables
• Macroeconomic models, such as the IS-LM model, use systems of linear equations to analyze the interactions between different sectors of the economy