Numerical Methods and Analysis

Content

Objective

Most real-world economic models cannot be solved analytically and lack closed-form solutions. ECON 254 bridges the gap between economic theory and computational implementation. This course equips advanced undergraduate students with the skills to solve static and dynamic economic models using fundamental numerical methods and the Python programming language.

The content progresses from basic numerical techniques (linear algebra, interpolation, root-finding) to advanced optimization (direct NLP solvers, indirect Lagrange/KKT approaches), and from there to dynamic programming and stochastic methods (backward induction, Value Function Iteration, Markov processes). Throughout the course, theoretical concepts are anchored in core economic frameworks; students learn not only to code algorithms but also to understand the underlying economic mechanisms, to perform convergence and error analysis, and to build reproducible analytical workflows. The last two weeks are devoted to group project presentations, in which student teams numerically formulate, solve, and defend a chosen economic optimization problem.

Course Content

The course begins with the setup of the Python scientific computing ecosystem (NumPy, SciPy, Matplotlib) and vectorized operations, followed by interpolation and function approximation. It then covers linear equation systems (Ax=b) and matrix decomposition techniques, applied to multi-sectoral economic networks such as Leontief input-output analysis. Fixed-point iteration, root-finding methods (bisection, Newton-Raphson), and nonlinear equation systems (multivariate Newton-Raphson, Jacobian) are used to solve market-clearing prices and multi-good consumer optimization. In the second half of the course, unconstrained optimization (gradient, Hessian, BFGS), constrained optimization (direct NLP solvers and the indirect Lagrange/KKT approach), finite-horizon dynamic problems (Cake-eating, Life-Cycle), and the stochastic Value Function Iteration (VFI) along with the Income Fluctuation Problem are covered. The last two weeks are reserved for project presentations in which student groups present a chosen optimization problem.

Course Learning Outcomes

Upon successful completion of this course, the student will be able to:

- Configure the Python scientific computing stack (NumPy, SciPy, Matplotlib) to numerically formulate economic models; organize analyses in reproducible Jupyter Notebook format integrating code, model, and interpretation.
- Apply linear equation systems ($Ax=b$) and matrix decomposition to Leontief input-output analysis; compute market-clearing prices via root-finding methods (bisection, Newton-Raphson); utilize nonlinear equation systems (multivariate Newton-Raphson, Jacobian) for multi-good consumer optimization and cost minimization.
- Integrate unconstrained optimization algorithms (gradient, Hessian, BFGS) into firm profit maximization and factor demand derivation; solve constrained optimization problems via both \textbf{direct methods} (penalty, barrier, SQP) and the \textbf{indirect approach} (Lagrangian, KKT).
- Numerically solve finite-horizon dynamic problems (Cake-eating, Life-Cycle) using backward induction and the Bellman principle; formulate life-cycle models under time-varying deterministic income streams and code retirement/savings planning applications.
- Formulate an economic optimization or equilibrium problem from scratch, select an appropriate numerical method, and implement it in Python; explain and defend the approach, code, findings, and limitations through a written report and an oral presentation.

Teaching and Learning Methods

- Computer-assisted applied instruction: live coding, algorithm development, error analysis, and visualization with Jupyter Notebook/Lab.
- Conceptual discussion and intuitive learning: blending economic theory with Python-based numerical methods.
- Application on real economic models: IS-LM, Leontief, consumer/firm optimization, life-cycle, VFI.
- Short, application-oriented learning activities with instant feedback; students may use AI during coding, but must be able to explain and defend every step used.
- Reproducible research production: preparing dynamic reports and term projects with Jupyter and Markdown integration.

References

1. QuantEcon: \url{https://quantecon.org/} (Primary platform for Python-based economics education)
2. Iskhakov, F. \textit{Foundations of Computational Economics}. \\ \url{https://fedor.iskh.me/compecon}
3. Additional resources and tools you may find useful:
- SciPy documentation: https://docs.scipy.org/doc/scipy/
- Matplotlib documentation: https://matplotlib.org/stable/
- Jupyter: https://jupyter.org/

Weekly Contents

Assessment System

Relation of Proficiency

ECTS