Mathematical Programming for Economic Applications

Course Overview

This course provides a rigorous foundation in mathematical programming techniques for economic modeling. The focus is on applying mathematical tools from set theory, topology, calculus, and optimization to solve economic problems. By the end of the course, students will be able to analyze unconstrained and constrained optimization problems in microeconomics, macroeconomics, and finance.

Course Outline

Lecture 1: Mathematical Preliminaries

📌 Objective: Introduce essential mathematical tools for optimization and economic modeling.

1.1 Basic Set Theory

• Sets, subsets, and Cartesian products

• Open, closed, compact, and convex sets

• Relations and functions

1.2 Topological Structure in (at most) 3-Dimensional Euclidean Space

• Metric spaces and Euclidean topology

• Convergence and continuity

• Compactness and connectedness

• Economic applications: Existence of equilibrium, Weierstrass theorem

1.3 Multivariable Calculus

• Limits, continuity, and differentiability

• Partial derivatives and directional derivatives

• Taylor’s theorem and approximation

• Economic applications: Marginal analysis, production functions

1.4 Quadratic Forms

• Positive definite, semi-definite, and negative definite matrices

• Eigenvalues and their economic interpretation

• Economic applications: Second-order conditions for optimization

1.5 Convex Analysis

• Convex sets and convex functions

• Jensen’s inequality

• Economic applications: Risk aversion, cost minimization, utility maximization

Lecture 2: Unconstrained Optimization

📌 Objective: Develop techniques to solve optimization problems without constraints.

2.1 Introduction to Optimization Problems

• Definition of optimization: Maximization vs. Minimization

• Economic motivation: Profit maximization, cost minimization, consumer utility maximization

• Graphical intuition: Tangency conditions and critical points

2.2 First-Order Necessary Conditions (FOCs)

• Definition of stationary points: ∇𝑓(𝑥) = 0

• Interpretation: Economic meaning of marginal conditions

• Existence of local extrema

2.3 Second-Order Conditions (SOCs) and Sufficient Conditions

• Hessian matrix: Definition and intuition

• Positive definite, negative definite, and their economic implications

• Second derivative test for local optima

• Application to profit maximization and cost minimization

2.4 Newton’s Method for Optimization

• Iterative approach to finding local optima

• Convergence properties

• Applications in economic modeling

2.5 Economic Applications

• Consumer theory: Maximizing utility subject to budget constraints (Lagrangian method preview)

• Production theory: Profit maximization and cost minimization

• Dynamic optimization preview: Intertemporal consumption choice

Lecture 3: Optimization with Equality Constraints

📌 Objective: Solve constrained optimization problems using the Lagrange method.

3.1 The Concept of Constrained Optimization

• Why constraints? Economic relevance

• Geometric intuition: Tangency conditions

3.2 The Method of Lagrange Multipliers

• Setting up the Lagrangian: L(x,λ)=f(x)+λg(x)L(x, \lambda) = f(x) + \lambda g(x)L(x,λ)=f(x)+λg(x)

• First-order necessary conditions (FOCs): Stationarity, feasibility, and interpretation of multipliers

• Economic meaning of Lagrange multipliers: Shadow prices

3.3 Second-Order Conditions and the Bordered Hessian

• Hessian bordered by constraints

• Positive and negative definiteness conditions

• Interpretation of curvature in constrained spaces

3.4 Envelope Theorem and Comparative Statics

• Differentiating the value function

• Sensitivity analysis of constraints

• Economic applications: Taxation, consumer surplus, and producer surplus

3.5 Economic Applications

• Consumer theory: Utility maximization subject to budget constraints

• Firm theory: Cost minimization subject to production constraints

• General equilibrium: Walrasian equilibrium and constrained optimization

Lecture 4: Optimization with Inequality Constraints

📌 Objective: Introduce the Kuhn-Tucker conditions for non-linear constrained optimization.

4.1 The Challenge of Inequality Constraints

• When equality constraints are not enough

• Economic motivation: Nonlinear pricing, labor supply, and firm production decisions

4.2 Kuhn-Tucker Necessary and Sufficient Conditions

• Definition of the Karush-Kuhn-Tucker (KKT) conditions

• Complementary slackness: Economic interpretation

• Feasibility and Lagrange multipliers for inequalities

4.3 Convex Programming and Duality Theory

• Convexity in optimization problems

• Strong duality theorem and its economic significance

• Slater’s condition and its implications

4.4 Economic Applications

• Nonlinear pricing: Price discrimination in monopoly

• Taxation: Optimal income taxation under constraints

• General equilibrium with constraints: Existence and welfare analysis

Reading Materials

Prescribed Textbooks:

• Simon & Blume (1994) – Mathematics for Economists (Optimization & Economic Applications)

• Sundaram (1996) – A First Course in Optimization Theory (Mathematical Foundation of Optimization)

Recommended Readings:

• Amir Beck (2022) – Introduction to Nonlinear Optimization: Theory, Algorithms, and Applications with Python and MATLAB (Numerical methods for optimization)

• Stephen Boyd & Lieven Vandenberghe (2004) – Convex Optimization (Advanced convex programming and duality)

• Nocedal & Wright (2006) – Numerical Optimization (Optimization algorithms and computational methods)