Business cycle theory concerns the description and explanation of the observed ups and downs of main macroeconomic variables. After the early attempts to provide informal and non mathematical explanations, based on verbal arguments and empirical observations, the business cycle theory has been mainly considered a problem of mathematical economics after the works by Samuelson [28] and Hicks [16]. So, a business cycle model is now considered a dynamic model, usually formulated in the framework of the theory of dynamical systems, whose mathematical structure allows for fluctuations in major macroeconomic variables. A broad, and purely formal, classification of business cycle models distinguishes between linear and nonlinear, continuous and discrete time models.
Linear models, which include the classical linear multiplier accelerator models (see, for instance, [28], [29], [16]) are usually expressed by two0dimensional linear dynamical systems, characterized by damped oscillations converging to the unique steady state, with the superposition of random variables whose presence prevents convergence and causes the occurrence of persistent irregular oscillations. So, in these models the economy is endogenously stable and fluctuations emerge as a consequence of exogenous, possibly non economic, disturbances.
Instead, nonlinear models, such as those proposed by Kaldor [18] and Goodwin [12] allow for endogenously generated persistent oscillations, i.e. stable fluctuations which are driven by deterministic processes entirely related to internal economic mechanisms embodied in the structure of the model (see [22] or [9] for historical overviews).
The majority of the models proposed in the period between 1940 and 1960, both linear and nonlinear, were formulated in continuous time, because the theory of two0dimensional ordinary differential equations was very developed in that period, especially in connection with nonlinear oscillators in physics. However, after the eighties, several authors proposed discrete time models, following the general interest related to the discovery that chaotic dynamics can be easily obtained even with low dimensional discrete time models. Of course, the choice between these two kinds of time concepts is very important also from the point of view of the economic meaning of the model and the interpretation of the results, and many discussions on this topic appeared in the literature (see e.g. [10], [9], [17], [23]).
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Multistability and role of non invertibility in a discrete time business cycle model
