In some previous papers the present authors reassembled the Hicksian trade cycle model in a new way. The floor was tied to depreciation on capital, itself the cumulative sum of past net investments, for which the principle of acceleration provided an explanation. Hence no alien elements were needed to include capital, and so close the system. The resulting model created a growth trend along with growth rate cycles, which could be periodic or quasiperiodic. The model with floor only, retained the unrealistic feature that with high accelerators it still created pure exponential growth. Therefore a ceiling along Hicks’s original line of thought was added. The simplest argument behind the principle of acceleration is a fixed coefficient limitational production technology that makes producers use capital and other inputs in strict proportion to the output. The same idea could, however, also be used for an upper limit (ceiling) to possible output (real income), again making use of the capital variable. In the current paper, the ceiling, using capital stock as a capacity limit for production, is added. To carry out such a mathematical analysis is the objective of the present paper. It then turns out that pure growth no longer exists, and chaos and multistability become possible, which were not in the previous model. Growing models are not readily analyzed by standard methods, so the model is converted to new “relative” variables, the growth factor and the capital to income ratio. This makes the piecewise linear model non-linear, but also has the advantage of reducing its dimension from three to two. A variety of bifurcation scenarios are explored, and a full understanding of the two dimensional (originally three-dimensional), piecewise smooth map defined via four-pieces, is attained, especially using a reduction to a one-dimensional return map.

Regular and Chaotic Growth Cycles in a Hicksian Floor/Roof Model

GARDINI, LAURA;
2010

Abstract

In some previous papers the present authors reassembled the Hicksian trade cycle model in a new way. The floor was tied to depreciation on capital, itself the cumulative sum of past net investments, for which the principle of acceleration provided an explanation. Hence no alien elements were needed to include capital, and so close the system. The resulting model created a growth trend along with growth rate cycles, which could be periodic or quasiperiodic. The model with floor only, retained the unrealistic feature that with high accelerators it still created pure exponential growth. Therefore a ceiling along Hicks’s original line of thought was added. The simplest argument behind the principle of acceleration is a fixed coefficient limitational production technology that makes producers use capital and other inputs in strict proportion to the output. The same idea could, however, also be used for an upper limit (ceiling) to possible output (real income), again making use of the capital variable. In the current paper, the ceiling, using capital stock as a capacity limit for production, is added. To carry out such a mathematical analysis is the objective of the present paper. It then turns out that pure growth no longer exists, and chaos and multistability become possible, which were not in the previous model. Growing models are not readily analyzed by standard methods, so the model is converted to new “relative” variables, the growth factor and the capital to income ratio. This makes the piecewise linear model non-linear, but also has the advantage of reducing its dimension from three to two. A variety of bifurcation scenarios are explored, and a full understanding of the two dimensional (originally three-dimensional), piecewise smooth map defined via four-pieces, is attained, especially using a reduction to a one-dimensional return map.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11576/2502391
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