An Interaction Model

Methodologically, the IIASA research program on Regional Development reflects the general attitude of the majority of regional scientists. Among other things, this means that the models developed deal with discrete sets of regions or locations. For planning purposes, this approach is extremely efficient, due to computational advantages. On the other hand, systematic information about regional structures, of the geometric flavor associated with classical location theory, is hard to obtain if one discretizes space from the outset. To complement this main stream of regional analysis, two scientists currently trying to revive continuous modeling of the space economy, Martin J. Beckmann and Tonu Puu, were invited to IIASA in September 1979. They started writing a comprehensive monograph intended to present the state-of-the-art in the field of continuous regional modeling. The completion of such an extensive work was not possible in the brief period of three weeks. The authors are currently continuing work on the project. This paper by Tonu Puu constitutes one chapter of the forth-coming monograph. It was completed during his visit to IIASA in August 1982 and follows the chapter circulated as CP-82-11. Whereas the preceding chapters dealt with commodity trade models with unique patterns of flow, the present one describes a simple interaction model cast in a continuous format. Given a specified need for communication and accommodation, optimal land use (balancing traffic congestion and population crowding) is discussed. In addition to the problems of optimal communication routes, the paper focuses on equilibrium population distributions such that communication and housing costs are in balance

On the stability of Cournot equilibrium when the number of competitors increases

This article reconsiders whether the Cournot equilibrium really becomes a perfect competition equilibrium when the number of competitors goes to infinity. It has been questioned whether the equilibrium remains stable with an increasing number of firms. Contraindications were given for linear and for isoelastic demand functions. However, marginal costs were then taken as constant, which means adding more potentially infinite-sized firms. As we want to compare cases with few large firms to cases with many small firms, the model is tuned so as to incorporate capacity limits, decreasing with an increasing number of firms. Then destabilization is avoided

New properties of the Cournot duopoly with isoelastic demand and constant unit costs.

The object of the work is to perform the global analysis of the Cournot duopoly model with isoelastic demand function and unit costs, presented in Puu (1991). The bifurcation of the unique Cournot fixed point is established, which is a resonant case of the Neimark-Shacker bifurcation. New properties associated with the introduction of horizontal branches are evidenced. These properties di¤er significantly when the constant value is zero or positive and small. The good behavior of the case with positive constant is proved, leading always to positive trajectories. Also when the Cournot fixed point is unstable, stable cycles of any period may exist.Cournot duopoly, isoelastic demand function, multistability, border-collision bifurcations.

An Attempt at Restoring von Thuenen - A Topological Model

This paper uses a topological model together with a structural stability principle as a means to identify and characterize long run solutions as regards regional specialization, direction of trade, and spatial organization. Within this setting not too restrictive assumptions are used to deduce results which shed new light on von Thunen's theory of location and spatial interaction

Long-Run Planning for Capital and Labour Allocation in Space

Methodologically, the IIASA research program on Regional Development reflects the general attitude of the majority of regional scientists. Among other things, this means that the models developed deal with discrete sets of regions or locations. For specific planning purposes, this approach is extremely efficient, due to computational advantages. On the other hand, systematic information about regional structures, of the geometric flavor associated with classical location theory, is hard to obtain if one discretizes space from the outset. To complement this main stream of regional analysis, two scientists currently trying to revive continuous modeling of the space economy, Martin J. Beckmann and Tonu Puu, were invited to IIASA in September 1979. They started writing a comprehensive monograph intended to present the state-of-the-art in the field of continuous regional modeling. The completion of such an extensive work was not possible in the brief period of three weeks. The authors currently continued to work on the project. The present paper by Tonu Puu is one chapter of the forthcoming monograph, and it was finished during his renewed visit to IIASA in April 1982. It deals with planning models for the allocation of available labor and capital resources within a continuous two-dimensional space economy. The main results of the paper concern the advantages of specialization and trade, in the absence of even comparative advantages or localized input supplies. So, the usual conditions for trade, as developed in general (spaceless) economic theory, are not needed, as specialization and trade seem to develop from the nature of bounded two-dimensional space itself. Moreover, the close parallel between the planning and competitive equilibrium solutions is brought out

Mathematical Properties of a Combined Cournot-Stackelberg model.

The object of this work is to perform the global analysis of a new duopoly model which couples the two points of view of Cournot and Stackelberg. The Cournot model is assumed with isoelastic demand function and unit costs. The coupling leads to discontinuous reaction functions, whose bifurcations, mainly border collision bifurcations, are investigates as well as the global structure of the basins of attraction. In particular, new properties are shown, associated with the introduction of horizontal branches, which di¤er significantly when the constant value is zero or positive and small. The good behavior of the model with positive constant is proved, leading to stable cycles of any period.Cournot-Stackelberg duopoly, Isoelastic demand function, Discontinuous reaction functions, Multistability, Border-collision bifurcations.

Continuous Flow Modeling in Regional Science

Although discontinuous methods of regional problem analysis are now widely used, in some cases continuous models might be useful. Professor T. Puu's work is an introduction to the fundamental tools of continuous-flow modeling. Numerous possible implementations of this approach are analyzed in the paper

The dynamics of a triopoly Cournot game when the competitors operate under capacity constraints

Oligopoly theory, i.e., the economic theory for competition among the few, goes back to 1838 and Augustin Cournot [7]. See also [11]. Quite early it was suspected to lead to complex dynamic behaviour and chaos. See Rand 1978 [13]. The probably simplest case under which this happens with reasonable economics assumptions was suggested by one of the present authors in 1991, see [9]. It assumes an isoelastic demand function, which always arises when the consumers maximize utility functions of the Cobb-Douglas type, combined with constant marginal costs. The particular layout was a duopoly, the case of only two competitors. The model was shown to produce a period doubling sequence of ip bifurcations ending in chaos for the outputs of each of the two competitors. Later the triopoly case under these assumptions was studied. See [2], [3], and [4] for examples. An interesting fact is that with three competitors the main frame becomes the Neimark-Hopf bifurcation, which provides new and di erent scenarios. The main reason for economists to study increasing numbers of competitors is to nd out whether it is the number of competitors that uniquely decides a road from monopoly over duopoly, oligopoly, and polypoly, to perfect competition, a state where each rm is so small that its actions cannot in uence the market at all. To nd out about this it is of primary interest to know whether the number of competitors stabilizes or destabilizes the equilibrium state. Some authors have questioned the assumption, to which most economists adhered, that increasing numbers of competitors bring stabilization. However, we must be clear about what we compare. If we study increasing numbers of competitors with constant unit production costs, we are not reducing the size of the rms when their number increases. Constant marginal cost means that potentially each rm has in nite capacity, and adding such rms is not what we want for comparison. It is therefore interesting to combine an increased number of rms with decreasing size of each rm, but in order to do so we have to introduce capacity limits. Already Edgeworth [8] insisted on the importance of capacity limits. It is not so easy to nd non-constant marginal cost functions which allow us to solve for the reaction functions for the rms in explicit form, but one of the present authors, see [12], found one type of function, which models the capacity limit by letting marginal cost go to in nity at a nite output. That paper discussed the competition between two duopolists. The objective of the present paper is to nd out the facts when there are three competitors, and we still keep the assumption of capacity limit

Structurally Stable Transport Flows and Patterns of Location

This report describes developments of the continuous model of trade and equilibrium in two-dimensional space introduced by Martin J. Beckman in the early 1950s. The original model is extended by treating several interrelated commodity flows and an explicit production activity, transforming the contents of one flow (labor services) into another (finished goods). A residential-industrial agglomeration pattern arises that corresponds to the two flows. This general model, which is capable of representing very diverse spatial organizations, but at the same time contains very little information, is specified by using the generic theory of differential equations. Therefore, if structural stability of the flows of commodities is assumed, it is possible to obtain a rather precise topological characterization of the stable flow and of the corresponding spatial organization. The main conclusion reached is that extreme care should be taken when deriving the results of classical market area from nonlinear models. The classical theory is linear and, therefore, always structurally stable. Without linearity (i.e. homogeneous space) stability is no longer guaranteed, but must be expressly assumed. The conclusions about basic spatial organization then become very different

Bertrand oligopoly revisited

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