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)?
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
K. Arrow and D. Starrett, Cost-theoretical and demand-theoretical approaches to the theory of price determination, in: Carl Menger and the Austrian School of Economics, ed. J. R. Hicks and W. Weber, Oxford Clarendon Press, Oxford (1973), 129–148.
L. Boggio, Stability of production prices in a model of general interdependence, in [33], 83–114.
M. Caminati and F. Petri (eds.), Convergence to long-period positions, Special issue of Political Economy, 6 (1990).
P. Chander, The nonlinear input-output model, J. Econ. Theory, 30 (1983), 219–229.
R.A. Dana, M. Florenzano, C. Le Van and D. Lévy, Production prices and general equilibirium prices, J. Math. Econ., 18 (1989), 263–280.
G. Duménil and D. Lévy, The classicals and the neoclassicals: a rejoinder to Frank Hahn, Cambridge J. Econ., 9 (1985), 327–345.
T. Fujimoto, Non-linear generalization of the Frobenius theorem, J. Math. Econ., 6 (1979), 17–21.
T. Fujimoto, Non-linear Leontief models in abstract spaces, J. Math. Econ., 15 (1986), 151–156.
T. Fujimoto and U. Krause, Strong ergodicity for strictly increasing nonlinear operators, Lin. Alg. Appl., 71 (1985), 101–112.
T. Fujimoto and U. Krause, Ergodic price setting with technical progress, in [33], 115–124.
T. Fujimoto and U. Krause, Asymptotic properties for inhomogeneous iterations of nonlinear operators, SIAM J. Math. Anal., 19 (1988), 841–853.
T. Fujimoto and U. Krause, Stable inhomogeneous iterations of nonlinear positive operators on Banach spaces,to appear in SIAM J. Math. Anal.
E. Kohlberg, The Perron-Frobenius theorem without additivity, J. Math. Econ., 10 (1982), 299–303.
U. Krause, Minimal cost pricing leads to prices of production, Cahiers de la R. C. P. Systemes de Prix de Production 2, 3 (“La Gravitation”), Paris 1984.
U. Krause, Perron’s stability theorem for nonlinear mappings, J. Math. Econ., 15 (1986), 275–282.
U. Krause, A nonlinear extension of the Birkhoff-Jentzsch theorem, J. Math. Anal. Appl., 114 (1986), 552–568.
U. Krause, Gravitation processes and technical change: convergence to fractal patterns and path stability,in [3], 317–327.
U. Krause, Path stability of prices in a nonlinear Leontief model, Ann. Oper. Res., 37 (1992), 141–148.
U. Krause, Relative stability for ascending and positively homogeneous opera-tors on Banach spaces,to appear in J. Math. Anal. Appl.
U. Krause, Path stability in positive discrete dynamical systems,to appear.
U. Krause, Positive nonlinear systems: Some results and applications, to appear in the Proceedings of the First World Congress of Nonlinear Analysts, Tampa 1992.
U. Krause and P. Ranft, A limit set trichotomy for monotone nonlinear dynamical systems, Nonlin. Anal. TMA, 19 (1992), 375–392.
U. Krause and R. Nussbaum, A limit set trichotomy for self-mappings of normal cones in Banach spaces, Nonlin. Anal. TMA, 20 (1993), 855–870.
M. Morishima, Generalizations of the Frobenius-Wielandt theorems for non-negative square matrices, J. London Math. Soc., 36 (1961), 211–220.
M. Morishima, Equilibrium, Stability, and Growth, Oxford University Press, London (1964).
M. Morishima and T. Fujimoto, The Frobenius theorem, its SolowSamuelson extension and the Kuhn-Tucker theorem, J. Math. Econ., 1 (1974), 199–205.
H. Nikaido, Balanced growth in multi-sectoral income propagation under autonomous expenditure schemes, Rev. Econ. Studies, 31 (1964), 25–42.
H. Nikaido, Convex Structures and Economic Theory, Academic Press, New York (1968).
R. Nussbaum, Some nonlinear weak ergodic theorems, SIAM J. Math. Anal., 21 (1990), 436–460.
Y. Oshime, An extension of Morishima’s nonlinear Perron-Frobenius theorem, J. Math. Kyoto Univ., 23 (1983), 803–830.
Y. Oshime, Perron-Frobenius problem for weakly sublinear maps in a Euclidean positive orthant, J.pan J. Ind. Appl. Math., 9 (1992), 313–350.
T. T. Read, Balanced growth without constant returns to scale, J. Math. Econ., 15 (1986), 171–178.
W. Semmler (ed.), Competition, Instability, and Nonlinear Cycles, Springer-Verlag, Berlin etc. (1986).
R. M. Solow and P. A. Samuelson, Balanced growth under constant returns to scale, Econometrica, 21 (1953), 412–424.
I. Steedman, Natural prices, differential profit rates and the classical competitive process, The Manchester School, June ed. (1984), 123–140.
B. P. Stigum, Balanced growth under uncertainty, J. Econ. Th., 5 (1972), 42–68.
D. Weller, Hilbert’s metric, part metric, and self-mappings of a cone,Ph. D. diss., Universität Bremen (1987).
C. C. von Weizsäcker, Steady State Capital Theory, Springer-Verlag, Berlin etc. (1971).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-642-48719-4_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58767-5
Online ISBN: 978-3-642-48719-4
eBook Packages: Springer Book Archive