Background
The term “Curse of Dimensionality” refers to the increasing complexity and computational difficulty encountered when expanding the number of dimensions (variables) in a mathematical model. Originated from problems in multi-dimensional space, this term has relevance across various fields, including economics.
Historical Context
First recognized in the context of numerical analysis and data processing in the mid-20th century, the term “Curse of Dimensionality” was introduced by Richard Bellman in 1961 during his work on dynamic programming. The curse reflects the exponential increase in volume associated with adding extra dimensions to a mathematical space, leading to higher computational costs and difficulties in data analysis.
Definitions and Concepts
The Curse of Dimensionality is the phenomenon where the demand for data and computation increases exponentially as the dimensionality of the space increases. In economic modeling, this problem arises as models attempt to incorporate an increasing number of consumers, firms, time periods, or other variables, making the models difficult to analyze and solve.
Major Analytical Frameworks
Classical Economics
In classical economics, models typically aim for simplicity; therefore, the curse of dimensionality is less prominent. However, as models evolve to capture real-world complexities, they encounter this problem.
Neoclassical Economics
Neoclassical economics often involves multiple variables such as consumer preferences, technology, and constraints. Economists using large-scale neoclassical models frequently face the curse of dimensionality as they try to aggregate individual behaviors into market outcomes.
Keynesian Economics
In Keynesian models, the focus on aggregate demand and supply could be extended into more complex model setups, potentially encountering the curse of dimensionality when adding variables like policy influence over several periods.
Marxian Economics
While Marxian economics often emphasizes socio-economic relationships, the analytical models used may still fall into the trap of dimensionality when trying to model too many external and internal variables.
Institutional Economics
As institutional economics takes into account a broad array of non-market factors, incorporating numerous institutional parameters can lead to complex systems struggling with dimensionality issues.
Behavioral Economics
Behavioral models, which integrate psychological and social factors, can include extensive variables covering cognitive biases, risk assessments, and consumer behavior, thus being prone to the curse of dimensionality.
Post-Keynesian Economics
Post-Keynesian models generally add complexities like financial market imperfections and endogenous money, making them susceptible to dimensionality challenges in detailed multi-variable setups.
Austrian Economics
Austrian economics, with its focus on individual actor decisions, encounters these issues particularly in dynamic models attempting to predict behavior in settings with numerous influencing factors.
Development Economics
Development economists constructing intricate growth and development models with a plethora of affecting variables— from education to technology — face significant dimensionality concerns.
Monetarism
While monetarist models often use a few key variables, adding additional dimensions to account for varying monetary behaviors and external influences increases complexity and can trigger dimensionality-related difficulties.
Comparative Analysis
The curse of dimensionality is closely aligned no matter the framework due to the rise in computational demands and decreasing availability of data points per function as dimensionality rises, thus challenging prediction accuracy and model robustness across different economic schools of thought.
Case Studies
Suggested Books for Further Studies
- “Dynamic Programming” by Richard Bellman
- “Applied Multivariate Statistical Analysis” by Richard A. Johnson and Dean W. Wichern
- “Econometrics” by Fumio Hayashi
- “Computational Economic Analysis: Tools and Techniques” by David Kendrick, Volker Wieland, and others.
Related Terms with Definitions
- Dimensional Analysis: A method in mathematics and engineering examining relationships in physical quantities by identifying their fundamental dimensions.
- Computational Complexity: Study of the amount of resources needed for the execution of algorithms.
- Dynamic Programming: A method for solving complex problems by breaking them down into simpler sub-problems, utilized commonly in computer science, economics, operational research, etc.