9.3 Linear algebra in economics and optimization

Linear algebra plays a crucial role in economics and optimization. It provides powerful tools for modeling complex systems, from input-output analysis to portfolio management. These applications showcase how matrix operations and vector spaces can solve real-world problems in finance and economics.

This section explores specific uses of linear algebra in economic equilibrium, game theory, and portfolio optimization. We'll see how matrices represent economic relationships, payoffs in games, and asset correlations, demonstrating the versatility of linear algebraic techniques across different domains.

Linear Programming with Matrices and Simplex

Formulating Linear Programming Problems

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• Linear programming optimizes a linear objective function subject to linear constraints expressed as inequalities or equations
• Standard form maximizes or minimizes a linear function subject to linear equality constraints and non-negative variables
• Matrix notation represents problems compactly using vectors and matrices for objective function coefficients, constraint coefficients, and decision variables
• Slack variables convert inequality constraints to equality constraints for simplex method application

Simplex Method and Tableau

• Simplex method iteratively solves linear programming problems by moving between basic feasible solutions along feasible region edges
• Simplex tableau provides tabular representation facilitating simplex method application
• Algorithm steps involve selecting entering and leaving variables, performing pivot operations, and updating tableau until optimal solution reached
• Reduced cost coefficients in tableau indicate potential for improvement in objective function value

Duality and Advanced Concepts

• Duality theory reveals relationship between primal and dual problems in linear programming
• Shadow prices represent marginal value of resources in optimal solution
• Complementary slackness conditions link primal and dual solutions
• Sensitivity analysis examines effects of parameter changes on optimal solution
• Interior point methods offer alternative approach to solving large-scale linear programming problems

Linear Algebra for Economic Equilibrium

Input-Output Models

• Input-output models represent interdependencies between economic sectors showing how output of one sector becomes input for another
• Input-output matrix (technology matrix) describes relationship between inputs and outputs in economy
• Leontief inverse matrix calculates total (direct and indirect) effects of final demand changes on output
• Matrix algebra techniques solve for equilibrium output levels and analyze economic system stability
• Multiplier effects calculated using matrix operations determine overall impact of sector changes on entire economy (employment multipliers, income multipliers)

Economic Equilibrium Analysis

• Economic equilibrium achieved when total supply equals total demand for each sector represented by system of linear equations
• Matrix inversion solves for equilibrium output levels given final demand vector
• Eigenvalue analysis assesses stability of economic systems and identifies key sectors
• Structural decomposition analysis uses matrix techniques to study changes in economic structure over time
• Input-output price models employ linear algebra to analyze price propagation through economy

Advanced Applications

• Dynamic input-output models incorporate time dimension using difference equations and matrix exponentials
• Regional input-output models use block matrices to represent interregional trade flows
• Environmentally extended input-output analysis incorporates pollution emissions and resource use into economic models
• Social Accounting Matrices (SAMs) extend input-output framework to include income distribution and institutional accounts

Linear Algebra in Game Theory

Matrix Representation of Games

• Game theory uses matrices to represent payoffs in strategic interactions between rational decision-makers
• Payoff matrix in two-player game shows outcomes for each combination of player strategies
• Bimatrix games represent non-zero-sum games with separate payoff matrices for each player
• Extensive form games converted to normal form using matrix representation

Equilibrium Concepts and Computation

• Nash equilibrium identified using linear algebra techniques to solve systems of equations representing best response strategies
• Mixed strategies represented as probability vectors analyzed using linear algebra methods
• Linear complementarity problems arising in certain game-theoretic models solved using specialized algorithms (Lemke-Howson algorithm)
• Correlated equilibrium computed using linear programming techniques

Advanced Game Theory Applications

• Evolutionary game theory uses eigenvalue analysis to study stability of strategies and dynamics of population games
• Repeated games analyzed using matrix powers and limiting distributions
• Coalition formation in cooperative games studied using characteristic function represented as hypercube
• Shapley value calculation in cooperative games involves permutation matrices and weighted averaging

Linear Algebra for Portfolio Optimization

Modern Portfolio Theory

• Modern Portfolio Theory models relationship between risk and return in investment portfolios using linear algebra
• Covariance matrix of asset returns represents relationships between different assets' price movements
• Efficient frontier calculated using quadratic programming techniques based on matrix operations
• Mean-variance optimization formulated as quadratic program with linear constraints

Risk Management and Factor Models

• Capital Asset Pricing Model (CAPM) employs linear regression to determine expected asset return based on systematic risk
• Principal Component Analysis (PCA) identifies main factors driving asset returns and reduces dimensionality of large datasets
• Matrix factorization methods (Cholesky decomposition) simulate correlated asset returns for Monte Carlo simulations
• Factor models decompose asset returns into systematic and idiosyncratic components using matrix algebra (Fama-French three-factor model)

Advanced Portfolio Techniques

• Black-Litterman model combines investor views with market equilibrium using Bayesian updating and matrix operations
• Risk parity portfolios constructed using optimization techniques involving covariance matrices
• Robust portfolio optimization incorporates parameter uncertainty using matrix norm constraints
• Machine learning techniques in portfolio management often rely on linear algebra operations (support vector machines, neural networks)