Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T21:59:15.846Z Has data issue: false hasContentIssue false
  • English
  • Français

DISCRETE CHOICE AND COMPLEX DYNAMICS IN DETERMINISTIC OPTIMIZATION PROBLEMS

Published online by Cambridge University Press:  13 February 2012

Takashi Kamihigashi
Takashi Kamihigashi*
Affiliation:
RIEB, Kobe University
*
Address correspondence to: Takashi Kamihigashi, RIEB, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper shows that complex dynamics arises naturally in deterministic discrete choice problems. In particular, it shows that if the objective function of a maximization problem can be written as a function of a sequence of discrete variables, and if the (maximized) value function is strictly increasing in an exogenous variable, then for almost all values of the exogenous variable, any optimal path exhibits aperiodic dynamics. This result is applied to a maximization problem with indivisible durable goods, as well as to a Ramsey model with an indivisible consumption good. In each model, it is shown that optimal dynamics is almost always complex. These results are illustrated with various numerical examples.

Keywords

Discrete ChoiceComplex DynamicsChaosIndivisible Goods
Type
Articles
Information
Macroeconomic Dynamics , Volume 16 , Supplement S1: Nonlinear Dynamics in Equilibrium Models , April 2012 , pp. 52 - 69
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aguirregabiria, Victor and Mira, Pedro (2010) Dynamic discrete choice structural models: A survey. Journal of Econometrics 156, 38–67.CrossRefGoogle Scholar
Aoki, Masanao (1998) Simple model of asymmetrical business cycles: Interactive dynamics of a large number of agents with discrete choices. Macroeconomic Dynamics 2, 427–442.CrossRefGoogle Scholar
Bischi, Gian-Italo, Gallegati, Mauro, Gardini, Laura, Leombruni, Roberto, and Palestrini, Antonio (2006) Herd behavior and nonfundamental asset price fluctuations in financial markets. Macroeconomic Dynamics 10, 502–528.CrossRefGoogle Scholar
Chabrillac, Yves and Crouzeix, J.-P. (1987) Continuity and differentiability properties of monotone real functions of several variables. Mathematical Programming Study 30, 1–16.CrossRefGoogle Scholar
Dechert, Davis W. and Nishimura, Kazuo (1983) A complete characterization of optimal growth paths in an aggregate model with a non-concave production function. Journal of Economic Theory 31, 332–354.CrossRefGoogle Scholar
Dudley, R.M. (2002) Real Analysis and Probability. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Kamihigashi, Takashi (2000a) Indivisible labor implies chaos. Economic Theory 15, 585–598.CrossRefGoogle Scholar
Kamihigashi, Takashi (2000b) The policy function of a discrete-choice problem is a random number generator. Japanese Economic Review 51, 51–71.CrossRefGoogle Scholar
Kamihigashi, Takashi and Roy, Santanu (2006) Dynamic optimization with a nonsmooth, nonconvex technology: The case of a linear objective function. Economic Theory 29, 325–340.CrossRefGoogle Scholar
Kamihigashi, Takashi and Roy, Santanu (2007) A nonsmooth, nonconvex model of optimal growth. Journal of Economic Theory 132, 435–460.CrossRefGoogle Scholar
Keane, Michael P. and Wolpin, Kenneth I. (2009) Empirical applications of discrete choice dynamic programming models. Review of Economic Dynamics 12, 1–22.CrossRefGoogle Scholar
Nishimura, Kazuo, Sorger, Gerhard, and Yano, Makoto (1994) Ergodic chaos in optimal growth models with low discount rates. Economic Theory 4, 705–717.CrossRefGoogle Scholar
Nishimura, Kazuo and Yano, Makoto (1995) Non-linear dynamics and chaos in optimal growth: An example. Econometrica 63, 981–1001.CrossRefGoogle Scholar
Verbrugge, Randal (2003) Interactive-agent economies: An elucidative framework and survey of results. Macroeconomic Dynamics 7, 424–472.CrossRefGoogle Scholar