Abstract
In many cases an economic system can be modelled by a mapping transforming a state of the economic system at a certain period of time into the state of the system at the next period. If the transformation under consideration can be assumed to be linear then the well-established theory of linear operators can be applied; thereby spectral theory, including Perron-Frobenius theory for positive matrices and positive linear operators, is of particular importance. Very often, however, linearity is not an appropriate idealization, in which case a rigorous analysis may become very difficult or even impossible. It is this state of affairs which brings positive nonlinear systems into play, this not only in economics. Positivity and related mathematical properties are quite natural assumptions in economics. The state space is often given, e.g., if states are described by quantities or prices, by the positive orthant (or some more general convex cone) in Euclidean space. The transformation of such a state space may possess additional properties related to positivity as various forms of monotonicity. This is the case for the two economic problems considered in this paper: Balanced growth in a nonlinear multisectoral framework and price setting among several production units which depend on each other by technology. Given the transformation T mapping the state space K, a convex cone, into itself, the following questions will be addressed: Does there exist a unique equilibrium, that is does the fixed point equation Tx ⋆ = x ⋆ possess a unique solution x ⋆ є K (up to a positive scalar)?
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Krause, U. (1995). Positive Nonlinear Systems in Economics. In: Maruyama, T., Takahashi, W. (eds) Nonlinear and Convex Analysis in Economic Theory. Lecture Notes in Economics and Mathematical Systems, vol 419. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-48719-4_14
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DOI: https://doi.org/10.1007/978-3-642-48719-4_14
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