Course Information
Course title
Economic Complexity and Nonlinear Dynamics 
Designated for
Curriculum Number
Curriculum Identity Number
Tuesday 3,4(10:20~12:10) Wednesday 3(10:20~11:10) 
Restriction: juniors and beyond OR Restriction: MA students and beyond OR Restriction: Ph. D students
The upper limit of the number of students: 30. 
Ceiba Web Server 
Course introduction video
Table of Core Capabilities and Curriculum Planning
Table of Core Capabilities and Curriculum Planning
Course Syllabus
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Course Description

The economy is a complex system with nonlinear interactions. The complexity modeling paradigm has been strongly sustained since the 1980s by economists and multidisciplinary scientists from various fields, and has also drawn the attention of policy makers dealing with complex phenomena and unpredictable market transitions. Instead, according to the nonlinear viewpoint, the economy may be intrinsically unstable and, even in the absence of external shocks, fluctuations and complex dynamics can arise in the evolution of the economic variables.
This course introduces students to the dynamic analysis of economic phenomena by means of appropriate mathematical methods useful to describe such phenomena. Traditional frontal classes will be complemented with practical lessons, in which the use of some open-source software for the analysis of dynamical systems is presented (e.g. Maxima, E&F chaos). 

Course Objective
The course aims at introducing quantitative and numerical instruments for the formalization and the study of dynamic models that describe complex nonlinear dynamic phenomena in a wide range of economic fields, such as macroeconomics, microeconomics, finance, industrial organization, etc. The course also aims at proposing a critical analysis of a selected literature on economic dynamics, in particular on behavioral models with heterogeneous boundedly rational agents. 
Course Requirement
Grading consists in two parts. The first part regards three assignments, that will be provided every four weeks, and that have to be resubmitted individually or in pairs.
The assignments contribute for 30% of the final grade.
The second part regards the final project. Students need to choose one selected research paper (e.g. among those of the extension readings, or to be agreed among others that deal with the topics addressed in the course), replicate and discuss the main results as their final project. The final project has to be done individually.

Requirements for students after the class:
At the end of the course students should:
1. have acquired the knowledge and the comprehension of the tools of nonlinear systems theory and be able to apply the mathematical methods described in the program in order to analyze economic problems;
2. be able to understand the translation of a real world situation into a mathematical model.
3. be able to deal with complex problems by using the mathematical tools.
4. analyse and eventually perform simulations of periodic and chaotic solutions and provide interpretation of the numerical results;
5. have learned a rigorous and essential language that allows them to communicate the knowledge on economic dynamics clearly and effectively, with a special focus on behavioural models with boundedly rational agents. 
Student Workload (expected study time outside of class per week)
Office Hours
-G.I. Bischi, F. Lamantia, D. Radi, Lecture notes on Dynamical Systems in Economics and Finance. (these lecture notes will be directly provided in class)
Designated reading
Additional readings*:
-Hommes, C.H. (2013). Behavioral rationality and heterogeneous expectations in complex economic systems. Cambridge University Press, Cambridge, UK
-Day, R. H., & Huang, W. (1990). Bulls, bears and market sheep. Journal of Economic Behavior & Organization, 14(3), 299-329.
-Brock, W. A., & Hommes, C. H. (1997). A rational route to randomness. Econometrica: Journal of the Econometric Society, 1059-1095.
-Brock, W. A., & Hommes, C. H. (1998). Heterogeneous beliefs and routes to chaos in a simple asset pricing model. Journal of Economic dynamics and Control, 22(8-9), 1235-1274.
-Agliari, A., Massaro, D., Pecora, N., & Spelta, A. (2017). Inflation targeting, recursive inattentiveness, and heterogeneous beliefs. Journal of Money, Credit and Banking, 49(7), 1587-1619.
-Assenza, T., Bao, T., Hommes, C., & Massaro, D. (2014). Experiments on expectations in macroeconomics and finance. Experiments in macroeconomics, 17(2014), 11-70.
- Wegener, M., Westerhoff, F., & Zaklan, G. (2009). A Metzlerian business cycle model with nonlinear heterogeneous expectations. Economic Modelling, 26(3), 715-720.
-Naimzada, A., & Sodini, M. (2010). Multiple attractors and nonlinear dynamics in an overlapping generations model with environment. Discrete Dynamics in Nature and Society, 2010.
-Bischi, G. I., Merlone, U., & Pruscini, E. (2018). Evolutionary dynamics in club goods binary games. Journal of Economic Dynamics and Control, 91, 104-119.
- Fanti, L., & Gori, L. (2012). The dynamics of a differentiated duopoly with quantity competition. Economic Modelling, 29(2), 421-427.
-Naimzada, A. K., & Tramontana, F. (2012). Dynamic properties of a Cournot–Bertrand duopoly game with differentiated products. Economic Modelling, 29(4), 1436-1439.
-Anufriev, M., Assenza, T., Hommes, C., & Massaro, D. (2013). Interest rate rules and macroeconomic stability under heterogeneous expectations. Macroeconomic Dynamics, 17(8), 1574-1604.
- Naimzada, A., & Pireddu, M. (2014). Dynamics in a nonlinear Keynesian good market model. Chaos: An Interdisciplinary Journal of Nonlinear Science, 24(1), 013142.
-Flaschel, P., Charpe, M., Galanis, G., Proaño, C. R., & Veneziani, R. (2018). Macroeconomic and stock market interactions with endogenous aggregate sentiment dynamics. Journal of Economic Dynamics and Control, 91, 237-256.
-Sordi, S., & Dávila-Fernández, M. J. (2020). Investment behaviour and “bull & bear” dynamics: modelling real and stock market interactions. Journal of Economic Interaction and Coordination, 1-31.
-Wegener, M. (2020). Exchange rate speculation and heterogeneous expectations in a small open economy. Nonlinear dynamics, psychology, and life sciences, 24(1), 105.

*One of these readings can be selected for the final project
Week 4
3/08,3/09  Introduction to dynamical systems and general definitions.
Eigenvalues and eigenvectors. Complex numbers. 
Week 5
3/15,3/16,3/18  One-Dimensional systems in continuous time: from linear to nonlinear differential equations. Applications: the cobweb model, the SI model.
Local bifurcations. Applications: Lux (1995) model, Solow model, Allee effect. 
Week 6
3/22,3/23,3/25   Two-dimensional linear systems of differential equations. Application: the IS-LM model.
2D nonlinear systems of differential equations. Application: the Kaldor business cycle model. 
Week 7
3/29,3/30,4/01  N-dimensional systems in continuous time. Deterministic chaos.
Introduction to difference equations: 1D linear models. Applications: cobweb model, a simple financial market model
Nonlinear 1D discrete time systems: local bifurcations. Logistic map and deterministic chaos. 
Week 8
4/06,4/08  2D linear and nonlinear maps.
Applications: the Cournot model, a financial market model with trend extrapolators 
Week 9
4/12,4/13  Dynamic models of order 2 and higher.
Basins of non-invertible maps. Lyapunov exponents.
Application: duopoly models with gradient dynamics 
Week 10
4/19,4/20  Bounded rationality and economic models.
Applications to a monopolistic and a cobweb model with reference dependent price. 
Week 11
4/26,4/27  Heterogeneous expectations in economics: the case of the cobweb 
Week 12
5/03,5/04  Asset pricing models with heterogeneous beliefs.
A financial market model with market sentiment. 
Week 13
5/10,5/11  Behavioral macro-models and policy analysis 
Week 14
5/17,5/18  Heterogeneous expectations and housing market dynamics 
Week 15
5/24,5/25  Heterogeneous expectations and laboratory experiments 
Week 16
5/31,6/01  Final presentations and discussion