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conIInuaIIon on .,.... 271
Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and W. Krelle
249
Spatial Price Equilibrium:
Advances in Theory,
Computation and Application
Papers Presented at the Thirty-First North American
Regional Science Association Meeting
Held at Denver, Colorado, USA
November 1984
Edited by Patrick T. Harker
Springer-Verlag
Berlin Heidelberg New York Tokyo
Editorial Board
H. Albach M. Beckmann (Managing Editor) P. Ohrymes
G. Fandel J. Green W. Hildenbrand W. Krelle (Managing Editor) H.P. KOnzi
G.L Nemhauser K. Ritter R. Sato U. Schittko P. Schonfeld R. Selten
Managing Editors
Prof. Dr. M. Beckmann
Brown University
Providence, RI 02912, USA
Prof. Dr. W. Krelle
Institut fOr Gesellschafts-und Wirtschaftswissenschaften
der Universitat Bonn
Adenauerallee 24-42, 0-5300 Bonn, FRG
editor
Dr. Patrick T. Harker
The Stephen M. Peck Assistant Professor of Decision Sciences
Department of Decision Sciences, Wharton School, University of Pennsylvania
Philadelphia, PA 19104, USA
ISBN-13: 978-3-540-15681-9 e-ISBN-13: 978-3-642-46548-2
DOl: 10.1007/978-3-642-46548-2
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© by Springer-Verlag Berlin Heidelberg 1985
2142/3140-543210
TABLE OF CONTENTS
Editor's Introduction ••••••••••••••••••••••••••••••••••••••••••• V
R.L. TOBIN and T.L. FRIESZ: A New Look at Spatially Competitive
Facility Location Models •••••••••••••••••••••••••••••••••••••••• 1
H. HASHIMOTO: A Spatial Nash Equilibrium Model ••••••••••••••••• 20
P.T. HARKER: Investigating the Use of the Core as a Solution
Concept in Spatial Price Equilibrium Games ••••••••••••••••••••• 41
J.E. FALK and G.P. McCORMICK: Computational Aspects of the
International Coal Trade Model ••••••••••••••••••••••••••••••••• 73
P.C. JONES, R. SAl GAL and M. SCHNEIDER: Demand Homotopies for
Computing Nonlinear and Multi-Commodity Spatial Equilibria •••• 118
J.S. PANG and Y.Y. LIN: A Dual Conjugate Gradient Method for
the Single-Commodity Spatial Price Equilibrium Problem •••••••• 136
R.L. TOBIN: General Spatial Price Equilibria: Sensitivity
Analysis for Variational Inequality and Nonlinear Complementarity
Formulations .................................................. 158
J.E. SOHL: An Application of Quadratic Programming to the
Deregulation of Natural Gas ••••••••••••••••••••••••••••••••••• 196
B.F. HOBBS and R.E. SCHULER: Evaluation of Electric PowerDeregula
tion Using Network Models of Oligopolistic Spatial Markets •••• 208
E. ERICKSON and R. HOUSE: Multiple Objective Analysis for a
Spatial Market System: A Case Study of U.S.AgriculturalPolicy.255
EDITOR'S INTRODUCTION
The problem of predicting interregional commodity movements
and the regional prices of these commodities has intrigued
economists, geographers and operations researchers for years. In
1838, A.A. Cournot (1838) discussed the equilibrium of trade
between New York and Paris and noted how the equilibrium prices
depended upon the transport costs. Enke (1951) recognized that
this problem of predicting interregional flows and regional
prices could be formulated as a network problem, and in 1952,
.Paul Samuelson (1952) used the then recent advances in mathe
matical programming to formalize the spatial price equilibrium
problem as a nonlinear optimization problem. From this formula
tion, Takayama and Judge (1964) derived their quadratic program
ming representation of the spatial price equilibrium problem,
which they and other scholars then applied to a wide variety of
problem contexts. Since these early beginnings, the spatial
price equilibrium problem has been widely studied, extended and
applied; the paper by Harker (1985) reviews many of these
results.
In recent years, there has been a growing interest in this
problem, as evidenced by the numerous publications listed in
Harker (1985). The reasons for this renewed interest are many.
First, new applications of this concept have arisen which
challenge the theoretical underpinnings of this model. The
spatial price equilibrium concept is founded on the assumption of
perfect or pure competition. The applications to energy markets,
steel markets, etc. have led scholars to rethink the basic
structure of this model. Second, the new mathematical theories
and algorithms in the areas of variational inequalities/comple
mentarity problems and non convex optimization have provided the
tools which are necessary to make meaningful and useful exten
sions of the spatial price equilibrium concept. Finally, new
policy debates concerning natural gas deregulation, electric
generation deregulation, and farm subsidies have provided the
incentive for these theoretical and algorithmic advances.
VI
This volume presents ten papers which constitute a sig
nificant advance in the theory, computation and application of
the spatial price equilibrium concept. The majority of these
papers were presented at the Thirty-First North American Regional
Science Association Meetings held at Denver, Colorado in November
of 1984. Three sessions at this meeting were devoted to the
spatial price equilibrium problem and thanks must be given to the
Regional Science Association for providing the necessary forum
for the exchange of our research on this topic.
The first three papers in this volume present theoretical
extensions of the spatial price equilibrium concept. The
original Samuelson-Takayama-Judge formulation is static in the
sense that producers and consumers are fixed in location. Tobin
and Friesz explore the addition of locational decisions to the
spatial price equilibrium model in the first paper. The original
formulation also assumes that each producer acts as a price
taker, thereby ignoring any spatial advantages which he or she
may possess (see Sheppard and Cury 1982 for a discussion of this
issue). Hashimoto and Harker both overcome this restriction by
stating and analyzing noncooperative and cooperative game
theoretic models of spatial competition, respectively.
The next four papers address several computational issues in
spatial equilibrium modelling. Falk and McCormick discuss the
computation of a solution to the International Coal Trade Model,
which is a model of spatial competition based on Nash's bargain
ing solution and used extensively by the United States' Depart
ment of Energy. Two new algorithms for the standard spatial
equilibrium problem are presented by Jones et ale and Pang and
Yu. Finally, Tobin presents an application of his recent
extension of nonlinear programming sensitivity analysis to the
problem of computing the sensitivity of a spatial equilibrium as
a function of changes in the problem data.
The last three papers present innovative uses of models of
spatial competition to analyze current public policy issues.
Sohl uses the standard spatial equilibrium model to analyze the
effects of various deregulation proposals on the natural gas
market in the United States. Hobbs and Schuler present network
VII
models of oligopolistic spatial markets and then use these models
to analyze the impacts of deregulating the electric power
industry in New York State. Finally, Erickson and House imbed a
spatial equilibrium model in a multi-objective programming
problem in order to analyze the various pricing policies under
consideration in the 1985 United States' Farm Bill.
In summary, this volume presents a very good overview of the
current state-of-the-art in the modelling of spatially separated
markets. I would like to thank all of the authors for their
cooperation in putting this volume together and hope that several
new research topics can be spawned by the ideas developed within
this volume.
REFERENCES
Cournot, A.A. (1838). Mathematical Principles of the Theory of
Wealth, translated by N.T. Bacon, Kelley, New York, 1960.
Enke, S. (1951). Equilibrium Among Spatially Separated Markets:
Solution by Electric Analogue. Econometrica 19, 40-47.
Harker, P.T. (1985). The State of the Art in the Predictive
Analysis of Freight Transport Systems. Transport Reviews 5,
143-164.
Samuelson, P.A. (1952). Spatial Price Equilibrium and Linear
Programming. American Economic Review 42, 283-303.
Sheppard, E. and L. Curry (1982). Spatial Price Equilibria.
Geographical Analysis 14, 279-304.
Takayama, T. and G.G. Judge (1964). Equilibrium Among Spatially
Separated Markets: A Reformulation. Econometrica 32, 510-524.
P.T. Harker
Philadelphia, Pennsylvania
July, 1985
A NEW LOOK AT SPATIALLY COMPETITIVE FACILITY LOCATION MODELS
Roger L. Tobin
Environmental Research Division
Argonne National Laboratory
Argonne, IL 60439
Terry L. Friesz
Department of Civil Engineering
university of Pennsylvania
Philadelphia, PA 19104
ABSTRACT
This paper presents some new formulations of models for locating a
firm's production facilities while simultaneously determining
production levels at these facilities so as to maximize the firm's
profit. Existing firms, as well as the new entrant, are assumed to
act in accordance with an appropriate model of spatial equilibrium. A
heuristic algorithm is proposed.
2
1.0 Introduction and Notation
In this paper we are concerned with locating a firm's production
facilities and determining production levels at these facilities so as
to maximize the firm's profits taking into account the effect the
firm's production will have on market prices. We assume that competi
tion exists among all firms and, in particular, between the locating
firm and those already in place. Furthermore, we restrict the possible
locations to a subset of nodes of a graph representing the transporta
tion network and require that movements of the commodity produced and
resulting prices correspond to a competitive equilibrium.
Most formulations for optimally locating production facilities assume
a fixed demand at the markets to be served and that the prices at
these markets will not be changed by the introduction of the new
production. Exceptions to this are formulations presented in Hansen
and Thisse (1977) and in Erlenkotter (1977). In both of these formula
tions, although the market price is related to the locating firms
production, there is no interaction among firms - since these authors
consider a spatial monopolist. The models presented here take into
account the changes in prices at each of the spatially separated
markets that would result from the increase in supply provided by the
new facilities and also from the response of the competing firms.
The following notation will be used throughout the paper:
a denotes an arc of the network
p denotes a path of the network
Jl ,m, i denotes nodes of the network
T (Jl ) is the set of arcs leaving node Jl
H( Jl ) is the set of arcs entering node Jl
p is the full set of paths
is the set of paths connecting origin-destination pair (Jl,m)
°
=
is an element of the arc - path incidence matrix; 1 if
arc a belongs to path p, 0 otherwise ap
is the flow (of a single commodity) on path p
= ( ... , hp' ... )
is the flow (of a single commodity) on arc a; note fa
p
0ap hp
3
t!
is the flow on arc a originating at node £
f = ( ••• , fa' .•• )
ca(f) is the unit cost of transportation on arc a as a function
of flow
= ( ...
c(f) ,ca(f), .•. )
Cp(h) is the unit cost of tra&ns portation on path p as a function
=
of flow; note cp(h) cap ca
D£ is the demand (for the single commodity) at node £
D~ is the demand at node £ for commodity originating at node k
D ( ... , OQ' ... )
6Q(0) is the inverse demand function at node Q
= ( ...
6(0) ,6Q(0), ..• )
S£ is the supply (of the single commodity) at node £
S = ( ... , SQ' ... )
~£(S) is the inverse supply function at node £
= ( ...
~(S) ,~£(S), ... )
is the (single commodity) price at £
is a discrete location decision variable; y£ = I if a
production facility is located at node £, 0 otherwise
N denotes the set of all nodes of the network
NOCN denotes those nodes at which the firm of interest may locate
NOCN denotes those nodes at which there is a demand
NsCN denotes those nodes at which there is a supply
F£ is the fixed cost of establishing a production facility at £
Q is the maximum amount of new production which may be
established
is the production level of the firm of interest at
node £ e No
is the production capacity of the facility located at £ e No
is a large number, greater than any nodal supply level
anticipated
V£(QQ) is the total variable cost of producing Q£ at node £
L(k) is the set of nodes at which firm k has production
facilities
A\B is the set of elements of A which are not contained in B
