Self-similar measures in multi-sector endogenous growth models. (English) Zbl 1354.91100
Summary: We analyze two types of stochastic discrete time multi-sector endogenous growth models, namely a basic Uzawa-Lucas [H. Uzawa, Int. Econ. Rev. 6, 18–31 (1965; Zbl 0142.17207); R. E. Lucas jun., “On the mechanics of economic development”, J. Monet. Econ. 22, No. 1, 3–42 (1988; doi:10.1016/0304-3932(88)90168-7)] model and an extended three-sector version as in [the first and second author, “Endogenous technological progress in a multi-sector growth model”, Econ. Model 27, No. 5, 1017–1028 (2010; doi:10.1016/j.econmod.2010.04.008)]. As in the case of sustained growth the optimal dynamics of the state variables are not stationary, we focus on the dynamics of the capital ratio variables, and we show that, through appropriate log-transformations, they can be converted into affine iterated function systems converging to an invariant distribution supported on some (possibly fractal) compact set. This proves that also the steady state of endogenous growth models – i.e., the stochastic balanced growth path equilibrium – might have a fractal nature. We also provide some sufficient conditions under which the associated self-similar measures turn out to be either singular or absolutely continuous (for the three-sector model we only consider the singularity).
MSC:
| 91B62 | Economic growth models |
| 91B66 | Multisectoral models in economics |
| 37N40 | Dynamical systems in optimization and economics |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 28A78 | Hausdorff and packing measures |
Keywords:
multi-sector model; Uzawa-Lucas model; three-sector model; iterated function system; self-similar measure; endogenous growth modelCitations:
Zbl 0142.17207References:
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