, each maximizing its
utility function 
,
Certificate Trade and Solution Concepts
Abstract
emission permits in the
framework of a competitive economic equilibrium. For this purpose we
integrate well established regional models called Markal-Macro
[Manne and Wene1992, Ahn1992] in one multi-regional CGE model. Emphasize
is put on doing as few changes as possible in the existing formulation
of the regional models. Based on two possible integration schemes we
discuss two solving concepts which are in a sense dual to each other.
We conclude by presenting numerical results of three CGE models: a
small one used for testing purposes, a medium model with modified
Markal-Macro agents based on a reduced data set, and finally a model
with modified Markal-Macro agents and the full data sets.
Given is a set of regional agents , each maximizing its
utility function ,
The first integration concept, which could be called `primal', can be
set up as follows. In a first step the regional models are modified by
introducing a budget constraint and by properly taking into account the
exchange of the goods considered. This modified agents will now return
a regional excess 
(exports minus imports) for a given price
signal p. The overall excess is the sum of this regional excesses,
. In a second step an artificial price agent
tries to find an equilibrium price 
, which is defined by
(demand is less or equal than supply), 
(no
negative price component), and 
(for each good with a
positive price, demand equals supply). Given a set of excess
information 
, the question is, which
strategy should be followed by the central price agent, in iteration
k, to determine a new price 
.
In this framework, existence of an equilibrium can be proved by using a homotopy concept following Garcia and Zangwill [Garcia and Zangwil1981].
A `dual' integrating approach is based on a fixed point map proposed by
Negishi [Negishi1972]. Starting from a set of regional agents
(1), denote the set of tradeable goods by w and the regional
trade balance as 
(exports minus imports). In a
closed economy the global trade balance
has to be
zero. Assign each region a so called Negishi weight NW and define
the Negishi problem
Under some conditions it can be shown [Negishi1972, Ginsburgh and Waelbroeck1981]
that all competitive equilibria can be obtained as
solutions of the Negishi problem for an appropriate set of Negishi
weights. In this `dual' setting an abstract agent tries to adjust the
Negishi weights until all regions obey their budget constraints, where
prices are the dual prices of the global trade balance
.
In the primal framework the price agent has only the (regional) excess
available, which is homogeneous of degree zero. Define therefore the
set feasible prices 
. Then the
equilibrium problem can be stated as the following variational
inequality problem (VIP):
From economic theory one can expect that the `law of demand and supply'
holds, which gives evidence that the overall excess is somehow close to
be (pseudo-)monotone [Dafermos1990, Hildenbrand1983]. Recalling
the definitions, e(p) is monotone over D, if
, and
pseudo-monotone over D, if 
implies
. It is well known from VIP
theory [Kinderlehrer1980] that in case of pseudo-monotonicity the
set 
, with 
,
contains all solutions. Based on that, a cutting plane algorithm can be
established which, in iteration k, determines a new `center'
in 
and performs a new cut 
. Based on Markal-Macro
agents, we implemented this concept and found an equilibrium.
In the Negishi framework little is known about updating strategies of
the weights and about convergence. Following Negishi [Negishi1972],
we implemented the following scheme. In iteration
k take the marginals of the global trade balance as price .
Compute for each region the excess of its budget and add a certain
fraction 
to the Negishi weights,
, and normalize
the weighting vector again
.
To use existing models requires either aggregating the regional models into one large optimization model, or the application of appropriate decomposition techniques.
Two approaches, how existing models can be integrated into an overall competitive equilibrium framework, are studied. On a mathematical level, we are interested in finding weak conditions under which the solution strategies described converge to an equilibrium. It is further interesting to compare efficiency of the two approaches.
From an economic point of view, the integration of other (regional) agents or the introduction of more traded goods is of interest.
Finally, from an algorithmical point of view, we try to develop components which can be generally useful, which are efficient and numerically robust. The goal is to support economic researchers as much as possible when integrating existing agents into a competitive equilibrium framework.
