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  • 1.
    Agliari, Anna
    et al.
    Catholic University of Milano, Italy.
    Gardini, Laura
    University of Urbino, Italy.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Global Bifurcations in Duopoly when the Cournot Point is Destabilized through a Subcritical Neimark Bifurcation2003Report (Other academic)
    Abstract [en]

    An adaptive oligopoly model, where the demand function is isoelastic and the competitors operate under constant marginal costs, is considered. The Cournot equilibrium point then loses stability through a subcritical Neimark bifurcation. The present paper focuses some global bifurcations, which precede the Neimark bifurcation, and produce other attractors which coexist with the still attractive Cournot fixed point.

  • 2.
    Agliari, Anna
    et al.
    Dip. Scienze Economiche e Sociali, Catholic University.
    Gardini, Laura
    Dip. Scienze Economiche, University of Urbino .
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics. Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Global bifurcations in duopoly when the Cournot Point is destabilized through a subcritical Neimark bifurcation2006In: International Game Theory Review, ISSN 0219-1989, Vol. 8, no 1, p. 1-20Article in journal (Refereed)
  • 3.
    Agliari, Anna
    et al.
    Univ Cattolica Sacro Cuore, Ist Econometria & Matematica, Italy.
    Gardini, Laura
    University of Urbino, Dept of Economics, Italy.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM). Umeå University, Faculty of Social Sciences, Department of Economics.
    Global bifurcations of basins in a triopoly game2002In: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, ISSN 0218-1274, Vol. 12, no 10, p. 2175-2207Article in journal (Refereed)
    Abstract [en]

    A Cournot model based on bounded inverse demand function and constant marginal production costs is studied. The case of three producers is considered and the adjustment process reduces to a three-dimensional noninvertible map in the output of competitors. The analysis of the dynamical behavior of the map is performed by the "critical curve method", extended to the critical surfaces in 3D. By this method, we explain the different bifurcations in the basins of attraction and in the attracting sets. In particular, given the economic application, feasible trajectories are focused, starting from the simple situation of two identical producers and extending the results to the generic case.

  • 4.
    Agliari, Anna
    et al.
    Dipartimento di Scienze Economiche e Sociali, Catholic University.
    Gardini, Laura
    Dept. Scienze Economiche, University of Urbino.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics. Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Some global bifurcations related to the apperance of closed invariant curves2005In: Mathematics and Computers in Simulations, ISSN 0378-4754, Vol. 68, no 3, p. 201-219Article in journal (Refereed)
    Abstract [en]

    In this paper, we consider a two-dimensional map (a duopoly game) in which the fixed point is destabilized via a subcritical Neimark–Hopf (N–H) bifurcation. Our aim is to investigate, via numerical examples, some global bifurcations associated with the appearance of repelling closed invariant curves involved in the Neimark–Hopf bifurcations. We shall see that the mechanism is not unique, and that it may be related to homoclinic connections of a saddle cycle, that is to a closed invariant curve formed by the merging of a branch of the stable set of the saddle with a branch of the unstable set of the same saddle. This will be shown by analyzing the bifurcations arising inside a periodicity tongue, i.e., a region of the parameter space in which an attracting cycle exists.

  • 5.
    Agliari, Anna
    et al.
    Catholic University in Milan, Italy/University of Parma, Italy.
    Gardini, Laura
    University of Parma, Italy/Istituto di Scienze Economiche, University of Urbino, Italy.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics.
    The dynamics of a triopoly Cournot game2000In: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 11, no 15, p. 2531-2560Article in journal (Refereed)
    Abstract [en]

    This paper reconsiders the Cournot oligopoly (noncooperative) game with iso-elastic demand and constant marginal costs, one of the rare cases where the reaction functions can be derived in closed form. It focuses the case of three competitors, and so also extends the critical line method for non-invertible maps to the study of critical surfaces in 3D. By this method the various bifurcations of the attractors and their basins are studied. As a special case the restriction of the map to an invariant plane when two of the three firms are identical is focused.

  • 6.
    Agliari, Anna
    et al.
    Catholic University, Faculty of Economics, Milan, Italy.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    A Cournot duopoly model with bounded demand2002Book (Other academic)
  • 7.
    Agliari, Anna
    et al.
    Univ Sacred Heart, Ist Econometria & Matemat, Milan.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics.
    A Cournot duopoly with bounded inverse demand function2002In: OLIGOPOLY DYNAMICS: MODELS AND TOOLS, BERLIN: SPRINGER-VERLAG BERLIN , 2002, p. 171-194Conference paper (Other academic)
  • 8.
    Ahmed, E
    et al.
    Department of Mathematics, Faculty of Science, Mansoura University, Egypt.
    Elsadany, AA
    Department of Basic Science, Faculty of Computers and Informatics, Ismailia, Suez Canal University, Egypt; Department of Mathematics, Shanghai University, Shanghai 200444, China.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    On Bertrand duopoly game with differentiated goods2015In: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 251, p. 169-179Article in journal (Refereed)
    Abstract [en]

    The paper investigates a dynamic Bertrand duopoly with differentiated goods in which boundedly rational firms apply a gradient adjustment mechanism to update their price in each period. The demand functions are derived from an underlying CES utility function. We investigate numerically the dynamical properties of the model. We consider two specific parameterizations for the CES function and study the Nash equilibrium and its local stability in the models. The general finding is that the Nash equilibrium becomes unstable as the speed of adjustment increases. The Nash equilibrium loses stability through a period-doubling bifurcation and the system eventually becomes chaotic either through a series of period-doubling bifurcations or after a Neimark–Sacker bifurcation.

  • 9. Canovas, Jose S.
    et al.
    Panchuk, Anastasiia
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Asymptotic dynamics of a piecewise smooth map modelling a competitive market2015In: Mathematics and Computers in Simulation, ISSN 0378-4754, E-ISSN 1872-7166, Vol. 117, p. 20-38Article in journal (Refereed)
    Abstract [en]

    In the present work we study asymptotic dynamics of a multi-dimensional piecewise smooth map which models an oligopoly market where competitors use adaptive scheme for reaction choice. Each competitor also defines the moment for renewing the capital equipment depending on how intensively the latter is used. Namely, the larger output is produced, the quicker the capital exhausts. It is shown then that the asymptotic dynamics of the map allows coexistence of different metric attractors in which case it is sensitive to initial conditions. We also investigate stability of trajectories representing Cournot equilibria which are here not fixed but periodic points. In particular, it is shown that several such Cournot equilibria, belonging to different invariant manifolds, may coexist some of them being locally asymptotically stable and some being unstable. (C) 2015 International Association for Mathematics and Computers in Simulation (IMACS).

  • 10.
    Canovas, Jose S
    et al.
    Univ Politecn Cartagena, Dpto Matemat Aplicada Estadist, Spain.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Marin, Manuel Ruiz
    Univ Politecn Cartagena, Dpto Metodos Cuantitat Informat, Spain.
    Detecting Chaos in a Duopoly Model via Symbolic Dynamics2010In: Discrete and continuous dynamical systems. Series B, ISSN 1531-3492, E-ISSN 1553-524X, Vol. 13, no 2, p. 269-278Article in journal (Refereed)
    Abstract [en]

    This paper considers a Cournot duopoly model assuming isoelastic demand and smooth cost functions with built-in capacity limits. When the firms cannot obtain positive profits they are assumed to choose small "stand-by" outputs rather than closing down, in order to avoid substantial fitting up costs when market conditions turnout more favorable. It is shown that the model provides chaotic behavior. In particular, the system has positive topological entropy and hence the map is chaotic in the Li-Yorke sense. Moreover, chaos is not only topological but also physically observable.

  • 11.
    Cánovas, José S
    et al.
    Dpto. Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Ruiz Marín, Manuel
    Dpto. Métodos Cuantitativos e Informáticos, Universidad Politécnica de Cartagena.
    A Method for Studying Non-Autonomous Affine Maps: An Application to a Two-Region Business Cycle Model2007In: Chaos, Solitons & Fractals, ISSN 0960-0779, Vol. 34, no 4, p. 1285-1298Article in journal (Refereed)
    Abstract [en]

    This paper is devoted to the study of the dynamics of a non-autonomous affine system. We apply the results to a two-region business cycle model of the Samuelson [Interactions between the multiplier analysis and the principle of acceleration. Rev Econ Statist 1939;21:75-8] multiplier-accelerator type.

  • 12.
    Cánovas, José S
    et al.
    Departamento Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM). Umeå University, Faculty of Social Sciences, Department of Economics.
    Ruíz, Manuel
    Departamento Métodos Cuantitativos e Informáticos, Universidad Politécnica de Cartagena.
    The Cournot-Theocharis Problem Reconsidered2008In: Chaos, Solitons & Fractals, ISSN 0960-0779, Vol. 37, no 4, p. 1025-1039Article in journal (Refereed)
    Abstract [en]

    In 1959 Theocharis [Theocharis RD. On the stability of the Cournot solution on the oligopoly problem. Review of economic Studies 1959;27:133–4] showed that with linear demand and constant marginal costs Cournot equilibrium is destabilized when the competitors become more than three. With three competitors the Cournot equilibrium point becomes neutrally stable, so, even then, any perturbation throws the system into an endless oscillation. Theocharis’s argument was in fact proposed already in 1939 by Palander [Palander T. Konkurrens och marknadsjämvikt vid duopol och oligopol. Ekonomisk Tidskrift 1939;41:124–45, 222–50]. None of these authors considered the global dynamics of the system, which necessarily becomes nonlinear when consideration is taken of the facts that prices, supply quantities, and profits of active firms cannot be negative. In the present paper, we address the global dynamics.

  • 13.
    Gallegati, Mauro
    et al.
    Department of Economics, Universita Politecnica delle Marche, Italy.
    Gardini, Laura
    Department of Economics, University of Urbino, Italy.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Sushko, Irina
    Institute of Mathematics, National Academy of Sciences of Ukraine, Ukraine.
    Hicks’ trade cycle revisited: cycles and bifurcations2003In: Mathematics and Computers in Simulation, ISSN 0378-4754, E-ISSN 1872-7166, Vol. 63, no 6, p. 505-527Article in journal (Refereed)
    Abstract [en]

    In the Trade Cycle, Hicks introduced the idea that endogenous fluctuations could be coupled with a growth process via nonlinear processes. To argue for this hypothesis, Hicks used a piecewise-linear model. This paper shows the need for a reinterpretation of Hicks’ contribution in the light of a more careful mathematical investigation. In particular, it will be shown that only one bound is needed to have non explosive outcome if the equilibrium point is an unstable focus. It will also be shown that when the fixed point is unstable the attracting set has a particular structure: It is a one-dimensional closed invariant curve, made up of a finite number of linear pieces, on which the dynamics are either periodic or quasi-periodic. The conditions under which the model produces periodic or quasi-periodic trajectories and the related bifurcations as a function of the main economic parameters are determined.

  • 14. Gardini, Laura
    et al.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Sushko, Irina
    A Goodwin-Type Model with Piecewise Linear Investment Function2006In: Business Cycle Dynamics: Models and Tools, Springer-Verlag , 2006Chapter in book (Other academic)
  • 15. Gardini, Laura
    et al.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Sushko, Irina
    The Hicksian Model with Investment Floor and Income Ceiling2006In: Business Cycle Dynamics: Models and Tools, Springer-Verlag , 2006Chapter in book (Other academic)
  • 16. Panchuk, A.
    et al.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Oligopoly model with recurrent renewal of capital revisited2015In: Mathematics and Computers in Simulation, ISSN 0378-4754, E-ISSN 1872-7166, Vol. 108, p. 119-128Article in journal (Refereed)
    Abstract [en]

    The aim of the present paper is to investigate an oligopoly market, modelled by using CES production function in combination with the isoelastic demand function. It is supposed that the competitors act not under constant, but eventually decaying returns, and thus, from time to time they need to renew their capital equipment, choosing its optimal amount according to the current market situation. It is shown that the asymptotic trajectories depend essentially on the value of the global capital durability, and are also sensitive to the initial choice of individual inactivity times. In particular, the firms may merge into different groups renewing their capitals simultaneously, which lead to distinct dynamical patterns. It is also studied how the capital wearing out rate influences the system behaviour.

  • 17. Panchuk, Anastasiia
    et al.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Cournot equilibrium stability in a non-autonomous system modeling the oligopoly market2009In: Iteration Theory (ECIT 08): proceedings of the European Conference on Iteration Theory, Yalta, Crimea, Ukraine, September 7 - 13, 2008 / [ed] Alexandr Sharkovsky, Irina Sushko, Graz: Karl-Franzens-Universität Graz , 2009, in printChapter in book (Other academic)
  • 18.
    Panchuk, Anastasiia
    et al.
    Institute of Mathematics, National Academy of Sciences, Ukraine.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Stability in a non-autonomous iterative system: an application to oligopoly2009In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 58, no 10, p. 2022-2034Article in journal (Refereed)
    Abstract [en]

    This paper reconsiders the relation between oligopoly and perfect competition, more specifically the problem of emergent instability when the number of competitors increases, as pointed out by several authors. A process of mixed short and long run dynamics is set up. In the short run the competitors are subject to capacity limits due to fixed capital stocks, in the long run they may renew these stocks and so in the moments of reinvestment have access to a constant returns technology. The evolution of the system depends on the number of competitors, the interval between their entry on the market, and the durability of capital. The main result is a theorem showing that if capital has a durability of more periods than the spacing of reinvestment times among the firms, multiplied with their total number, then the system always contracts to the Cournot equilibrium state.

  • 19.
    Puu, Tonu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Good Old Economic Geography2016In: COMPLEX NETWORKS AND DYNAMICS: SOCIAL AND ECONOMIC INTERACTIONS / [ed] Commendatore, P.; MatillaGarcia, M.; Varela, L.M.; Canovas, J.S., Springer, 2016, p. 329-359Conference paper (Refereed)
    Abstract [en]

    This chapter discusses classical economic modelling in continuous two-dimensional geographical space, focusing some ingenious models due to Harold Hotelling and Martin Beckmann concerning population growth and migration, and spatial market equilibrium, respectively. It also adds some modelling of business cycles, and discusses the issue the shape of market areas. Focus is on transversality and stability of structure.

  • 20.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    A century of oligopoly theory 1838-19412002In: OLIGOPOLY DYNAMICS: MODELS AND TOOLS, BERLIN: SPRINGER-VERLAG BERLIN , 2002, p. 1-14Conference paper (Other academic)
  • 21.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics.
    A CHAOTIC PROCESS WITH SLOW FEED BACK - THE CASE OF BUSINESS CYCLES1992In: ECONOMIC EVOLUTION AND DEMOGRAPHIC CHANGE: FORMAL MODELS IN SOCIAL SCIENCES, BERLIN: SPRINGER-VERLAG BERLIN , 1992, p. 3-34Conference paper (Other academic)
  • 22.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics.
    A chaotic process with slow feedback: the case of business cycles1992In: Lecture notes in economics and mathematical systems, ISSN 0075-8442, E-ISSN 2196-9957, Vol. 395, p. 3-34Article in journal (Refereed)
  • 23.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    A mathematics refresher for students of economics2004Book (Other academic)
  • 24.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    A New Approach to Modeling Bertrand Duopoly2017In: Review of Behavioral Economics, ISSN 2326-6198, E-ISSN 2326-6201, Vol. 4, no 1, p. 51-67Article in journal (Refereed)
    Abstract [en]

    Bertrand oligopoly needs global demand functions which apply to close substitutes. This is a problem, because economic theory never supplied anything but local definitions for substitutes. Lancaster's "new theory of demand" is therefore invoked to supply one. In its format one can also quantify closeness of substitutes and incorporate optimisation of design. The present study focuses the pure price dynamics for Bertrand oligopoly when the design of the competing products is given, though quantified through Lancaster's approach. Resulting is some complex dynamics, including high periodicity and chaos.

  • 25.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics.
    A simplified model of spatiotemporal population dynamics.1985In: Environment and planning A, ISSN 0308-518X, E-ISSN 1472-3409, Vol. 17, no 9, p. 263-269Article in journal (Refereed)
    Abstract [en]

    This paper is an extension of the model of population growth and migration originally developed by H. Hotelling in 1921. This model consists of two ingredients, a logistic growth function and a linear spatial diffusion term. The author notes that the saturation population can be affected by the development of new technology and that improvements in transportation have increased the possibilities for migration. "Basic nonlinearities are introduced by use of a production technology with increasing-decreasing returns to scale. It is demonstrated how industrial takeoffs, population transitions, and agglomerative spatial patterns can emerge by changing the model parameters.

  • 26.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Arts, Sciences, and Economics: A Historical Safari2006Book (Other academic)
    Abstract [en]

    This book deals with the economic aspects of changing attitudes in arts and sciences. The effects of the public good character of culture, along with the very long production period and lifetime for its products, are emphasized, since both contribute to the failure of normal market solutions. Embodiment of ideas, and the consequences of modern reproduction technology for protection of property rights are closely examined. The evolution within arts and sciences, which often seems to return to previously scrapped ideals, is illustrated by detailed case studies, in which the importance of changing tastes, rather than progress proper, is emphasized. The author attempts an understanding for this using Darwinian evolution in combination with modern mathematical complexity theory, expressed in terms accessible to the general reader.

  • 27.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Attractors, Bifurcations, & Chaos: Nonlinear Phenomena in Economics2003 (ed. 2)Book (Other academic)
  • 28.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics.
    Bertrand oligopoly revisited2001In: Discrete dynamics in nature and society, ISSN 1026-0226, E-ISSN 1607-887X, ISSN 1026-0226, Vol. 6, no 1, p. 1-10Article in journal (Refereed)
    Abstract [en]

    This paper reconsiders Bertrand duopoly and oligopoly in the spatial formulation due to Hotelling, 1929. The equilibrium configurations of price and location structure are considered, given elastic demand, and a full dynamics is formulated in order to check for stability of equilibrium and the possibilities of complex dynamics, such as occurs easily with Cournot oligopoly. The main discussion concerns Hotelling's original case of two sellers on a given interval, though results for different cases, such as three firms on a circle, and lattices in 2D are indicated.

  • 29.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Capacity Limits2011In: Oligopol: Old Ends - New Mean / [ed] Tönu Puu, Berlin: Springer Berlin/Heidelberg, 2011, p. 101-125Chapter in book (Other academic)
    Abstract [en]

    In Chap. 2 the Cournot–Theocharis stability issue, which will be called so, though it was raised 20 years earlier by Palander, was discussed. The problem was that in certain model families increasing competition through adding new competitors destabilised Cournot equilibrium. Accordingly, the idea of a straight path leading from monopoly, over oligopoly, to perfect competition, where profits were eliminated and marginal cost pricing was used, became subject to doubt. What interest do the properties of an equilibrium have, after all, if it becomes unstable?

  • 30.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics.
    Catastrophe theory applied to the refraction of traffic1988In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 22, no 2-3, p. 315-318Article in journal (Refereed)
  • 31.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM). Umeå University, Faculty of Social Sciences, Department of Economics.
    Catastrophic structural change in a continuous regional model2004In: Complexity in Economics, Edward Elgar Publishing , 2004Chapter in book (Other academic)
  • 32.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics.
    Chaos in business cycles1991In: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 1, no 5, p. 457-473Article in journal (Refereed)
    Abstract [en]

    The business cycle is studied in terms of the mapping

    Zt = λZt−1 − (λ+1)Z3t−1 − σYt−1

    Yt = Zt−1 + Y−1

    where the variables Y, Z denote income and rate of income change respectively, and λ, σ are two structural parameters. The model produces chaotic or periodic output for income differences. For small σ income acts as a slow feed back causing bifurcations between periodic and chaotic behaviour over the cycle. Typically, transitions between prosperity and depression set in with chaos after which there follows a period halving route to order.

  • 33.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM). Umeå University, Faculty of Social Sciences, Department of Economics.
    Chaos in duopoly pricing2004In: Complexity in Economics, Edward Elgar Publishing , 2004Chapter in book (Other academic)
  • 34.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics.
    Chaos in duopoly pricing1991In: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 1, no 6, p. 573-581Article in journal (Refereed)
    Abstract [en]

    The dynamics of two competing firms in a market is studied in terms of Cournot's duopoly theory. Assuming iso-elastic demand and constant unit production costs the iterative mapping

    for the outputs of the two firms ensues. The two constants are the unit production costs. The fixed point, the Cournot equilibrium, has earlier been assumed to be the only interesting feature of this model. It is, however, shown that the model can produce persistent motion, periodic or chaotic. 

  • 35.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics.
    Complex dynamics in economic and social systems: Introduction1996In: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 7, no 12, p. R7-R7Article in journal (Other academic)
  • 36.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics.
    Complex dynamics with three oligopolists1996In: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 7, no 12, p. 2075-2081Article in journal (Refereed)
    Abstract [en]

    The adjustment process by three Cournot oligopolists is studied. An iso-elastic demand function and constant marginal costs are assumed. The system can easily result in chaotic behaviour, and a much richer variety of bifurcations will occur than in the case of duopoly with two agents, discussed before under the same assumptions.

  • 37.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM). Umeå University, Faculty of Social Sciences, Department of Economics.
    Complex oligopoly dynamics2005In: Nonlinear Dynamical Systems in Economics, Springer , 2005Chapter in book (Other academic)
  • 38.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Continuous economic space modelling: Draft of a survey2009In: The annals of regional science, ISSN 0570-1864, E-ISSN 1432-0592, Vol. 43, no 1, p. 5-25Article in journal (Refereed)
    Abstract [en]

    This paper attempts a survey of continuous modelling in spatial economics. The classics of spatial economics, from von Thünen on, like geographers always did, considered phenomena in the two dimensional plane, though later development was in favour of modelling discrete location point sets connected by communication arcs. The models discussed here are strongly focused around Beckmann’s continuous space market model from the early 1950s.

  • 39.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Cournot Oligopoly2011In: Oligopoly: Old Ends - New Means / [ed] Tönu Puu, Berlin: Springer Berlin/Heidelberg, 2011, p. 17-41Chapter in book (Other academic)
    Abstract [en]

    As mentioned in Chap. 1, Cournot’s oligopoly model was one of the first mathematical models proposed in the field of economics. It addresses the functioning of a market with numerous atomistic demanders versus few relatively large suppliers. This implies that all the suppliers influence market price appreciably, and hence, like monopolists, take account of the demand function of the consumers on the market in order to calculate their best moves. As a rule, demand is a decreasing function of price. In equilibrium demand equals supply, and one can also speak of the inverse demand function which states how market price depends on supply.

  • 40.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Disequilibrium trade and the dynamics of stock markets2014In: Complexity in economics: cutting edge research / [ed] Marisa Faggini, Anna Parziale, Springer-Verlag New York, 2014, p. 225-245Chapter in book (Refereed)
    Abstract [en]

    The present work considers pricing and trade dynamics for stock commodity markets, which, unlike flow commodity markets have been little studied, if at all. Concepts and tools in economics are shaped to deal with flow markets, where commodities disappear in each period and then reemerge. This allows one to define unique demand and supply functions and their equilibria. A durable commodity, a stock, in contrast, remains on the market to the next period and may just change owner through exchange. This, however, changes demand and supply functions, and hence the equilibrium state to which a dynamic process may be heading. Dynamic processes are provided with memory of the actual exchange history. We also need to state how disequilibrium trade in stock markets takes place. This is another neglected issue, though a fact of reality. Using a case with only two traders of two stock commodities, and focusing pure trade, it is possible to specify the exact conditions for disequilibrium trade in each step of the dynamic process. In the end any of an infinity of equilibria can be reached, or trade can stick in some disequilibrium point while complex, even chaotic, price dynamics goes on.

  • 41.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Duopoly and Complex Dynamics2011In: Oligopoly: Old Ends - New Means / [ed] Tönu Puu, Berlin: Springer Berlin/Heidelberg, 2011, p. 43-67Chapter in book (Refereed)
    Abstract [en]

    As mentioned in Chap. 1, Rand (1978) conjectured that with suitable shapes of the reaction functions the outcome of dynamic duopoly would be chaotic. His purely mathematical treatment does not, however, include any substantial economic assumptions under which this becomes true. In what follows a very simple duopoly model, already encountered in Chap. 2, based on traditional microeconomis, will be discussed.

  • 42.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics.
    Economic Development in the Arts, Crafts, and Sciences2001In: Evolutionary Controversy in Economics, Tokyo: Springer-Verlag , 2001, p. 59-72Chapter in book (Other academic)
  • 43.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics.
    From catastrophe to chaos- a general theory of discontinuities- J B Rosser1994In: Environment and Planning, B: Planning and Design, ISSN 0265-8135, E-ISSN 1472-3417, Vol. 21, no 3, p. 379-380Article, book review (Other academic)
  • 44.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics.
    Hotelling's "ice cream dealers" with elastic demand2002In: The annals of regional science, ISSN 0570-1864, E-ISSN 1432-0592, Vol. 36, no 1, p. 1-17Article in journal (Refereed)
    Abstract [en]

    This article reconsiders the Hotelling duopoly model of 1929, but under elastic demand, more precisely a linear demand function. The equilibrium state for identical firms is fully described, and the intervals of different regimes: independent monopolies, genuine duopoly competition, and price cutting wars, are specified in terms of one single compound parameter (maximum price, minus marginal production cost, divided by transportation cost).

  • 45.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics.
    Hotelling's migration model revisited1991In: Environment and planning A, ISSN 0308-518X, E-ISSN 1472-3409, Vol. 23, no 8, p. 1209-1216Article in journal (Refereed)
    Abstract [en]

    In the present article Hotelling's model of population growth and migration of 1921 is 'revisited'. After a discussion of the stationary solutions and their stability the main point is made. The model itself is structurally unstable, but can be easily stabilized by adding a simple autonomous migration component. By this, the solution curves, in the shape of constant amplitude population waves over space for the original model, either become damped in one direction and explosive in another or are replaced by just one single spatial limit cycle.

  • 46.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Introduction2011In: Oligopoly: Old Ends - New Means / [ed] Tönu Puu, Berlin: Springer Berlin/Heidelberg, 2011, p. 1-16Chapter in book (Other academic)
    Abstract [en]

    According to Joseph Schumpeter’s “History of Economic Analysis” from 1954 - the still unsurpassed source book for economics, mathematical modelling in the field had two roots. The first was Johann Heinrich von Thünen’s theory of land use dating from 1827, implicitly containing the most general theory of the emergence of specialization and trade ever presented up to now, as it was based on the mathematical principle of transversality alone; much more general than the Ricardian comparative advantages or the later Heckscher–Ohlin theory which also was based on immobile resources. As a historical irony the theory has not to this day been recognized as anything more than a theory in agricultural economics.

  • 47.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Introduction to Mathematical Economics2007In: Mathematical Models in Economics: UNESCO Encyclopedia of Life Support Systems, Oxford: EOLSS Publishers , 2007Chapter in book (Other academic)
  • 48.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Layout of a New Industry: From Oligopoly to Competition2005In: Pure Mathematics and Applications, ISSN 1218-4586, Vol. 16, no 4, p. 475-492Article in journal (Refereed)
    Abstract [en]

    Oligopoly models with constant marginal costs for the competitors tend to produce destabilization of the Cournot equilibrium point when the number of competitors increases. It therefore becomes difficult to explain how an oligopoly can evolve into perfect competition through an increase in the number of competitors. In the present study it is explored how cost functions with built in capacity limits can eliminate this problem. It is also shown how such cost functions can be derived from CES functions when the input of capital is fixed through an act of investment. It turns out that the durability of capital equipment is what is needed to stabilize the system, and so make it possible for an oligopoly to seamlessly transform into a competitive equilibrium.

  • 49.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Mathematical Location and Land Use Theory: An Introduction2003 (ed. 2)Book (Other academic)
  • 50.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Multistability2011In: Oligopoly: Old Ends - New Means / [ed] Tönu Puu, Berlin: Springer Berlin/Heidelberg, 2011, p. 127-150Chapter in book (Other academic)
    Abstract [en]

    In a short printed abstract of contributions to a Cowles Commission conference at Colorado Springs in 1936, and an extensive follow up, in Swedish, dating from 1939, Tord Palander focused some interesting dynamics problems in Cournot duopoly when the demand curve was kinked linear, of the type suggested by Joan Robinson in 1933, where the marginal revenue curve jumped up, producing two different local profit maximising intersections with the curve of marginal cost. Palander took it constant or even zero.

123 1 - 50 of 106
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