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continuation on page 137
Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and H. P. KUnzi
Mathematical Economics
168
Convex Analysis
and Mathematical Economics
Proceedings of a Symposium, Held at the
University of Tilburg, February 20, 1978
Edited by Jacobus Kriens
Springer-Verlag
Berlin Heidelberg New York 1979
Editorial Board
H. Albach' A V. Balakrishnan' M. Beckmann (Managing Editor)
P. Dhrymes . J. Green . W. Hildenbrand' W. Krelle
H. P. KUnzi (Managing Editor) . K. Ritter' R. Sato . H. Schelbert
P. Schonfeld
Managing Editors
Prof. Dr. M. Beckmann Prof. Dr. H. P. KOnzi
Brown University Universitat ZOrich
Providence, RI 02912/USA 8090 ZUrich/Schweiz
Editor
Prof. J. Kriens
University of Tilburg
Department of Econometrics
Hogeschoollaan 225
5037 GC Tilburg/The Netherlands
AMS Subject Classifications (1970): 90A 15
ISBN-13: 978-3-540-09247-6 e-ISBN-13: 978-3-642-95342-2
DOl: 10.1007/978-3-642-95342-2
This work is subject to copyright. All rights are reserved, whether the whole
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a fee is payable to the publisher, the amount of the fee to be determined by
agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1979
2142/3140-543210
PREFACE
On February 20, 1978, the Department of Econometrics of the University
of Tilburg organized a symposium on Convex Analysis and Mathematical
Economics to commemorate the 50th anniversary of the University.
The general theme of the anniversary celebration was "innovation" and
since an important part of the departments' theoretical work is con
centrated on mathematical economics, the above mentioned theme was
chosen.
The scientific part of the Symposium consisted of four lectures, three
of them are included in an adapted form in this volume, the fourth lec
ture was a mathematical one with the title "On the development of the
application of convexity".
The three papers included concern recent developments in the relations
between convex analysis and mathematical economics.
Dr. P.H.M. Ruys and Dr. H.N. Weddepohl (University of Tilburg) study
in their paper "Economic theory and duality", the relations between
optimality and equilibrium concepts in economic theory and various
duality concepts in convex analysis. The models are introduced with an
individual facing a decision in an optimization problem. Next, an n
person decision problem is analyzed, and the following concepts are
defined: optimum, relative optimum, Nash-equilibrium, and Pareto-optimum.
These concepts are shown to be closely related. Various duality opera
tions are defined and used to give better insight into the structure,
and to be applied in different economic situations. Applications are
given (e.g. the theory of public goods), and generalizations are deve
loped. The results are, finally, used and adapted for decision pro
blems over (discrete and finite) time periods.
The title of Dr. J.J.M. Evers' (University of Technology, Twente) paper
runs "The dynamics of concave input-output processes". Representing
economic activities by a set S c R~xRn of feasible input/output combi
nations, and associating with each pair (x,y) E S a utility value
~(x,y), ~:S+Rl is called an input/output process if a free disposal con
dition on the inputs is satisfied and, in addition, the hypograph is
closed and convex. A specific property of the concept is, that, combi
ning any number of I/O-processes in any sensible way, the logical struc
ture is preserved. Moreover, a duality transformation is introduced,
IV
resulting into dual systems with again the same structure. Special at
tention is given to dynamic I/O-processes, invariant dynamic optimality
and the existence of optimal trajectories.
The third paper is by Prof. Dr. R.T. Rockafellar (University of Washing
ton, Seattle) and has a title "Convex processes and Hamiltonian dynami
cal systems in economics". The state x(t) of an economic model in conti-.
nuous time is constrained by x(t) A(x(t», where A is a set-valued
£
mapping which represents the underlying technology. If the graph of A
is a convex cone, A is called a convex process. The "efficient" state
trajectories x(t) can then be characterized in terms of the system
(-p(t), x(t» £ aH(x(t), pet»~, where H is a concave-convex function
called the Hamiltonian associated with A, and pet) is a price vector.
Much can be learned by studying the behavior of this system around a
relative saddlepoint of H.
Finally, I want to express may sincere thanks to all who contributed to
make the symposium a success and to Mrs. Ella Broks for doing a diffi
cult typing job excellently.
J. Kriens, editor
Tilburg University
The Netherlands
February 1979.
CONTENTS
I. P.H.M. RUYS and H.N. WEDDEPOHL
page
ECONOMIC THEORY AND DUALITY 1
1. Introduction. 1
2. Abstract economies. 2
3. Introduction to duality. 11
4. An economy with public goods (only). 19
5. Optimality and Nash-equilibrium. 23
6. Competitive equilibrium. 27
7. Intertemporaloptimality. 33
A1. Duality operations on sets and correspondences. 49
A2. Dual programs. 60
II. J.J.M. EVERS
THE DYNAMICS OF CONCAVE INPUT/OUTPUT PROCESSES 73
1. Input/Output processes; logic and economic relevance. 73
2. Prices and dual Input/Output processes. 80
3. Dynamic Input/Output processes. 88
4. Invariant dynamic processes and stationary optimal
trajectories. 93
5. Existence of optimal trajectories in dynamic open
horizon Input/Output systems and approximation by
finite horizon systems. 100
6. Continuity and stability of ~-horizon dynamic systems. 114
III. R.T. ROCKAFELLAR
CONVEX PROCESSES AND H~1ILTONIAN DYNAMICAL SYSTEMS 122
I. ECONOMIC THEORY AND DUALITY
P.H.M. Ruys and H.N. Weddepohl
UNIVERSITY OF TILBURG, NETHERLANDS.
1. Introduction.
Convex analysiS plays an irr.portant role in mathematical economics,
particularly in relation to the notions of optimum and equilibrium.
Also, most equilibrium definitions in economics are closely related
to the (game theoretical) concept of Nash-equilibrium (see Debren [ 3] ) .
The definitions of an optimum (including Pareto optimum), and a Nash
equilibrium will be applied to concepts defined in a finite euclidean
space. In this case they can be associated with convex sets separated
by a hyperplane.
The vectors in the dual space which define the separating hyperplanes
may be interpreted as prices in economic theory. It is the theory of
duality which analyses the relation between sets, functions or correspon
dences, and their dual characteristics in terms of prices.
A characterization of a model or a concept in the dual space gives
usually a better insight in the problem and its solution. There are,
however, also direct applications of duality theory.
A classical application of duality to economics is the so called in
direct utility function (see e.g. [5, p. 120]. Shephard [22] has
applied duality in production theory; a survey is given by Diewert [5].
The authors have worked on duality in relation to equilibrium and other
economic concepts for several years ([ 16], [17] , [18] , [19] , [24] , [25] ,
[ 26], [27]). Some of their earlier results are included in this paper and
some new applications are introduced.
A summary of results on duality is given in Appendix A.l. In Appendix
A.2 the relation between this concept of duality and the one in mathe
matical programming, is considered.
In section 2 we introduce four optimality concepts: optimum, relative
optimum, Nash-equilibrium and Pareto-optimum. The claim that these con-
2
cepts are closely related is proven in section 5. These proofs are ba
sed upon the introduction of one or more fictitious agents, and upon
duality operations. Both techniques will appear also in proofs of the
orems in other sections.
Section 3 is devoted to duality operations on economic notions, which
are used in the next sections.
Three models are given which can be considered as applications of the
optimality theory outlined in section 1 or as concrete interpretations
of the abstract economies defined: an economy with public goods (section
4), a competitive economy (section 5) and an intertemporal economy (sec
tion 7).
2. Abstract economies.
An abstract economy is defined by four primitive concepts: a set of
agents H, a set of actions Xi' for each i E H, a preference correspon
C
dence Pi' for each i E H, and a constraint set i or a constraint cor
respondence Ci for each i E H.
2.1. Optimum.
We first consider an abstract economy with a single agent that has a
C,
constraint set and denote it by ~O:
~o:= (X, P:X t X, C).
The set of actions X is a subset of the finite euclidean space Rn.
It can be interpreted as a consumption set if the agent is a consumer
who considers all conceivable consumption bundles. The preference cor
respondence P:X t X represents a strong preference relation on X. The
set P(x) consists of all actions in X that are better than x. For a
strong preference relation it is assumed that x ~ P.(x), i.e. P is
irreflexive. If preferences are given by a weak correspondence R (which
is also reflexive), a strong preference correspondence P can be derived
from it by defining P(x) :=R(X)\R-1(x) which is the set of elements in
R(x) that are not a member of R-1(X). The constraint set ceRn repre
sents a feasibility constraint on actions: in choosing an action from
C
X, the agent is constrained to remain in n X. It may be noticed that
C
in the definition of ~O' may without loss of generality be replaced
3
C
by n X. In the applications below it is however more convenient to
allow for constraints that are not a-priori in X.
Definition 2.1. Given the abstract economy &0' an action x is called
an optimum if it is:
(i) feasible X E X n C;
(ii) maximal P(x) n C = ~.
A simple example of an abstract economy &0 is given by the following
Linear Programming problem:
maximize a.x
subject to: bk.x < ck' k=1, •.• ,m; x ~ O.
Then &0 is defined by:
X:=R~;
P(x) :={y E xla.y > a.x};
x
An optimal action in &0 is evidently a solution of the L.P.-problem.
Necessary conditions for the existence of an optimum in &0 are given
by the following theorem, due to Ky Fan [ 6] :
Theorem 2.2: Let &O:=(X, P:X ~ X, C) be such that:
(i) X n C is compact, convex and nonempty;
(ii) P han an open graph in XxX, and x ~ Conv P(x) ;
x.
then there exists an optimum
If also:
(iii) x E CI P(x), for all x E X n C,
x C.
then E Bnd
Proof:
Define P(x) :=Conv P(x). Then P also has an open graph, and x ~ P(x).
C
Clearly, P(x) n C = ~ implies P(x) n = ~. Suppose that there does not
exist an optimum with respect to P, then F(x) :=P(x) n C is non-empty
C. C,
for all x E X n Since F is lower hemi continuous in X n and convex
valued, there exists a continuous selection f: (X n C)+(X n C) in
4
F: (X n e) t (X n e), according to Micheal [11]. Brouwer's fixed-point
theorem then implies that there exists an action x=f(x) E F(x). This
contradicts x ~ P(x) •
e. e
Suppose further that x E Cl P(x) and x E Int Then P(x) n ~ ~, and
x cannot be an optimum. o
Condition (ii) above requires that the graph of the correspondence P,
i.e. {(x,y) E Xxx\y E P(x)}, is open relative to XxX, and that the
convex hull of P(x) does not contain x. Condition (iii) is called local
non-satiation; it requires that any neighbourhood of x contains a better
element and it implies that the restrictions given by the constraint set
are actually active (or binding) at an optimum x. In this case,
C, C
X E Cl P(x) n (and P(x) n = ~).
Y
From condition (ii) it follows that Conv P(x) n = ~, which is neces
sary and sufficient for the existence of a hyperplane
H(p,a.) := {y E Rn\p.y = a.}
e,
separating the sets P(x) and and containing the optimum x. The theory
of duality (which is further studied in section 3 and the Appendix A.l)
is based on this property of an optimum.
2.2. Relative optimum.
Next we consider the abstract economy with a single agent that faces a
constraint correspondence instead of a constraint set, and denote this
economy by &R:
The constraint correspondence C: X ~ Rn determines the constraint set
C(x) which the agent faces if he is considering the action x E X.
Clearly the considered action x is feasible if and only if x E C(x) n X.
C(x) may depend on x.
Definition 2.3: Given the abstract economy &R' an action x is said to
be a relative optimuml) in &R' if it is:
(i) feasible: x E X n C(x);
(ii) maximal: P(x) n C(x) = ~.
1) See Borglin and Keiding [2] where such an action is called an equi
librium choice.
