Hilbert spaces and narrative economics: A formal operator-theoretic framework for story-driven macroeconomic dynamics

Arizmendi, Luis-Felipe (2025): Hilbert spaces and narrative economics: A formal operator-theoretic framework for story-driven macroeconomic dynamics.

Abstract

Narratives shape beliefs, influence expectations, propagate through populations, and affect macroeconomic dynamics. Behavioral economics has documented systematic deviations from rationality, while narrative economics emphasizes that stories spread epidemically and move asset prices, consumption, investment, and policy. This paper develops a rigorous mathematical framework integrating these insights using Hilbert-space methods and operator theory. Beliefs and narratives are represented as vectors in a complete inner-product space; propagation, distortion, and contagion are modeled by linear and nonlinear operators; and macroeconomic expectations emerge as projections of narrative vectors onto relevant economic directions. We derive conditions for viral amplification, exponential forgetting, the existence and uniqueness of stationary narrative states, and the stability of narrative-driven inflation and output expectations. Behavioral distortions such as availability, anchoring, and herding are formalized as nonlinear perturbations. The framework provides a unified structure for incorporating narratives into modern macroeconomic theory without abandoning mathematical rigor.

• Item Type: MPRA Paper
• Language: English
• Keywords: Hilbert spaces; narrative economics; behavioral macroeconomics; operator theory; expectations; economic methodology.
• Subjects: B - History of Economic Thought, Methodology, and Heterodox Approaches > B4 - Economic Methodology > B41 - Economic Methodology; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C40 - General; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D83 - Search ; Learning ; Information and Knowledge ; Communication ; Belief ; Unawareness; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D84 - Expectations ; Speculations; E - Macroeconomics and Monetary Economics > E6 - Macroeconomic Policy, Macroeconomic Aspects of Public Finance, and General Outlook; E - Macroeconomics and Monetary Economics > E6 - Macroeconomic Policy, Macroeconomic Aspects of Public Finance, and General Outlook > E65 - Studies of Particular Policy Episodes

References

Akerlof, G. A., & Yellen, J. L. (1985). Can small deviations from rationality make significant differences to macroeconomic equilibria? American Economic Review, 75 (4), 708–720. Retrieved from http://www.jstor.org/stable/1821343

Banerjee, A. V. (1992). A simple model of herd behavior. Quarterly Journal of Economics, 107 (3), 797–817. https://doi.org/10.2307/2118364

Bikhchandani, S., Hirshleifer, D., &Welch, I. (1992). A theory of fads, fashion, custom, and cultural change. Journal of Political Economy, 100 (5), 992–1026. https://doi.org/10.1086/261849

DellaVigna, S. (2009). Psychology and economics: Evidence from the field. Journal of Economic Literature, 47 (2), 315–372. https://doi.org/10.1257/jel.47.2.315

Kahneman, D. (2011). Thinking, Fast and Slow. New York, NY: Farrar, Straus and Giroux.

Laibson, D. (1997). Golden eggs and hyperbolic discounting. Quarterly Journal of Economics, 112 (2), 443–477. https://doi.org/10.1162/003355397555253

Shiller, R. J. (2017). Narrative economics. American Economic Review, 107 (4), 967–1004. https: //doi.org/10.1257/aer.107.4.967

Shiller, R. J. (2019). Narrative Economics: How Stories Go Viral and Drive Major Economic Events. Princeton, NJ: Princeton University Press. https://doi.org/10.1515/9780691189970

Sims, C. A. (2003). Implications of rational inattention. Journal of Monetary Economics, 50 (3), 665–690. https://doi.org/10.1016/S0304-3932(03)00029-1

Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185 (4157), 1124–1131. https://doi.org/10.1126/science.185.4157.1124