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Equally central to economics, however, is the idea of the evolution of an economic system over time. To avoid an inappropriate bipolarity of these two key concepts in economics, however, a synthesis of both seems to be desirable. Fortunately, economic theory has proposed ways to tie the two strings together. A particularly unsatisfactory feature of this conceptualization of an equilibrium, however, is the fact that the intertemporal optimizing approach completely predetermines the whole future of the economic system.

This is the point at which the new branch of evolutionary economics has made its entrance. To be sure, there have been attempts to reconceptualize the notion of equilibrium from the evolutionary viewpoint. However, these proposals also appear, in one way or another, to hinge on the ideas of rest. This particularly applies to the branch of nonlinear dynamics and deterministic chaotic motion. To further the latter idea we will put forth here a new attempt to synthesize the two ideas of economic equilibrium and evolution. This means, we do not think of any connotation of progress, or directed development in any sense anagenesis when speaking of an evolving economic system.

In any of these two conceptualizations an evolution consists of a succession of states of the economic system under consideration. As a general remark we would like to emphasize that throughout our whole study geometrical imagination is always a good guide for intuition. In Part I, the concept of an evolution of economies is formalized analytically. They have been partly adopted from the literature, partly they are new. This way of analytically formalizing evolutions is not only intuitive, but it also appears to be the only reasonable one for our purposes.

Moreover, continuously one-parametrized economies have also been analyzed for instance by A. Mas-Colell in his comprehensive monograph from Chapters 5 and 8. Indeed, both approaches have originated in complete independence of each other. The reader should note, however, that the study by Mas-Colell only provides an analytical treatment of one-parametrized economies, but gives no further economic applications. All our constructions and results are solely based on assumptions of continuity.

In Chapter 4 evolutions in three basic models from the Walrasian exchange framework, one of which is a model of large exchange economies will be formalized. In fact, this model is similar to the one used by Mas-Colell as a basis for oneparametrizations , Section 5. These models are inspired by a former model by N. Schulz the purpose of which has been to model a subsystem of the system of all conceivable markets in an economy. Nevertheless, the relaxation of these two standard assumptions will prove to be of great help later in our study when a new formalization of an economy evolving in historical time is developed Section In Chapter 6, evolutions are formalized in two models with production, tax, and subsidy schemes originally developed by T.

Kehoe b. Finally, in Chapter 7 evolutions are formalized in two models from the quantity constrained equilibrium framework. Furthermore, a new model is designed with many productive sectors on a medium level of aggregation. Any proofs in Part II which employ advanced mathematical results are relegated to the appendices at the end of this monograph. The central analytical results of this study are given in Chapter This result is certainly not at all clear from the outset since even for simple examples a total indeterminacy of the equilibrium set of one-parametrizations can be observed.

The intuitive geometrical meaning of a near-equilibrium path is that of a polygonal path, which lies in the graph of the Walras correspondence of the given evolution of economies. Another related result for a basic model of a large exchange economy has been provided by Mas-Colell , 5. A mathematical criterion is provided for checking which points lie on nearequilibrium paths.

In Chapter 11, it is shown how any evolution can be approximated so that there even exists a geometrically, nicely behaved equilibrium path, i. To our knowledge so far there is no precursor in the literature of our class of well-behaved paths and our approximating evolutions. We will come back to this below. To achieve our aims, we have to accomplish three tasks.

Second, we must provide a general construction of approximating evolutions for each of our basic models, and third we have to verify that our approximating evolutions always possess an equilibrium path from the designed 1 General Introduction 5 well-behaved class. In Section Both methods have advantages. Balasko and others see Balasko , for surveys, see also Balasko, Lang and Bonnisseau, Cayupi At this point it is also natural to examine the relationship of our results in this monograph to the results of the so-called law of demand Hildenbrand , , , a, b.

Actually, the validity of the law of demand would ensure uniqueness of the equilibrium set of any single state economy of an evolution. Then the existence of geometrically well-behaved near -equilibrium paths of our type would directly follow from the continuity of evolutions. Chapter 12 provides further natural interpretations and extensions of the general concept of an economic evolution developed here. Obviously our conceptualization of economic evolution by one-parametrizations gives room for two economic interpretations.

On the one hand, one may emphasize the aspect that a one-parametrization connects its initial state with its terminal state. On the other hand, one may understand an evolution in this context as starting from its initial state and openly evolving in some continuous way. A particularly interesting question is whether for each of the basic models there is always a connection evolution for any two given economies. This will be our theme in Section The structure results of the existence of near -equilibrium paths from Chapters 10 and 11 raise the following natural question: Is this the only structural property of the equilibrium set of evolutions that generally holds?

As a notable corollary of our result, any two nonempty compact subsets of the price domain can be realized as the equilibrium sets of two arbitrarily close exchange economies.

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Our results are also closely related to the results on the local surjectiveness of the graph of the Walras correspondence by B. Allen As we will see in Chapter 13 our results and those by B. Allen neither extend, nor contain each other, but are complementary in their characterization of the graph of the Walras correspondence. Together with the above mentioned global results by Y. Balasko and others, these results provide a fairly detailed understanding of the shape of the graph of the Walras correspondence.

In Chapter 14, we present a detailed comparison of our results with related results in the literature. All of our conceptualizations and results are based solely on continuity. Section In a nutshell our conclusion is that the static regular theory produces stronger results than ours in the local sense, but if one leaves a connected component of the subspace of regular economies, these strong results break down.

There is certainly a close relationship between the theory of regular oneparametrizations and the approximation results in Section However, there are advantages of our approach: Our method of achieving approximating evolutions by well-behaved equilibrium paths is constructive, whereas the theory of regular one-parametrizations merely provides an abstract existence result. Of course, equilibrium paths for an evolution are just selections from its equilibrium set. Moreover, we can show here that the compartmentalization of the space of exchange economies as well as of the space of exchange one-parametrizations by the subspace of critical economies and critical one-parametrizations, respectively, is fairly complicated.

We emphasize that this weakens the structure results of the regular theory considerably. Actually, slightly perturbing a critical economy, or one-parametrization, leads to a regular economy with probability one. But the complex structure of the subspace of critical economies makes it almost impossible to predict the properties of the obtained regular economy, or one-parametrization,respectively. To be precise, this implicitly presumes a uniform probability distribution on the space of economies one-parametrizations.

However, so far no consistent underpinning has been provided by economic theory which would justify the assumption of negligibility. In Part III of the study the economic content of the preceding conceptualizations and results are explored. More precisely, it is shown that our results, in a certain sense, achieve an extension of the well-known path following computational method of equilibria of regular exchange economies see e.

Mas-Colell, , Section 5. This is, however, at the cost of loss of algorithmic comfort. In our second atemporal application the notorious paralysis of comparative statics caused by multiplicity of equilibria is dealt with. In fact, it is our conviction that the multiplicity phenomenon is intrinsically linked to the present-day way of economic thinking. Our conclusion from this is that a way should be sought to give comparative statics a meaning, also in the multiplicity case. In Chapter 18, the methodological viewpoint 8 1 General Introduction and the scope of the analysis are explained at some length.

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Kinetics does not inquire into the causal explanation of the individual evolution of the real economic system in historical time, but searches for general regularity, or structural, properties of the dependent evolutions of the endogenous key variables. In this sense our temporal applications can be seen as being complementary to dynamics, and especially to evolutionary economics. Essentially our applications in this context are based on the atemporal applications given in Chapter The main body of our temporal applications, however, are applications in historical time Sections Its main idea is to describe the evolution of a market over time by varying time intervals between two successive demand, or supply, events.

What are the economic achievements of the application of the analytical work from Part II to these conceptualizations of evolving economies in historical continuous time? In a nutshell, it provides the opportunity to tune equilibria, at least piecewise, continuously to their changing values when the economy undergoes an evolution. In particular, our understanding of the notion of equilibrium is not that of a description of the real state of an economy.

Indeed, this is made impossible by the multiplic- 1 General Introduction 9 ity of equilibria. Instead, we understand the equilibria of a given momentary state of an evolving economy here solely as momentary, or transitory, coordination solutions to this state. We will come back to this issue shortly. At this point, however, we would like to mention a direct application of this result to the issue of time consuming equilibrium adjustment processes.

It has been known for a long time that, in general, a time consuming equilibrium adjustment process faces a moving target e. Kloek , for a comprehensive survey see e. Fisher This has already been illustrated by V. The adjustment of a moving equilibrium is symbolized by him as a running hare being tracked by a hound. In fact, three of the nine basic models are, from their economic conceptualization, suitable for this.

These are the two models from the framework with production, taxes, and subsidies Chapter 6 and the multisectoral quantity constrained model Chapter 7. All these models have in common that they contain explicit parameters that are, in principle, accessible to an external control by some economic policy institution. These are prices and wages in the case of the quantity constrained multi-sectoral model from Chapter 7, and tax and subsidy rates in the case of the two models from Chapter 6. In order to ensure a perfect homeostatic equilibrium, i. We also want to emphasize again that our result only provides the general opportunity to realize a perfect homeostatic equilibrium during an evolution to an external policy institution, but does not endogenously model policy institutions, or their actions.

Hatta , Hettich Fellner et al. Moreover, gradual control makes at least partial foresight possible for the economic agents. In economics this is generally considered as favorable for a stabilized evolution of the economy. However, this does not mean that we take a one-sided position favouring a strict gradualism in economic policy making. For both positions of a gradualistic policy and a shock therapy there are striking metaphors: How would it be possible on one hand to change moving forward to moving backward other than gradually?

The shock therapy position, on the other hand is favoured by the metaphor of changing from driving on the left to driving to the right in a state. We are well aware of the disadvantages of the gradualistic principle. What we want to say is that it seems to be worthwhile investigating the conditions and opportunities for enacting a gradual, shock-free policy. A thorough assessment to decide whether a gradual adjustment, or a shock therapy policy adjustment is preferable can only be made on a case-by-case basis.

I have now reached the point where I would like to take the opportunity to thank all who have helped me with their comments and suggestions. In fact, there is a number of people who have contributed to the evolution of 1 General Introduction 11 my personal ideas and views on my subject over the years and who have helped me to make them precise and comprehensible. Our exposition will be self-contained as regards our subsequent analysis.

The reader who still misses further background informations is referred to the relevant textbook literature. At the heart of our formalizations stands the notion of a continuous one-parametrization. For instance, any continuous movement process is an example of a continuous one-parametrization cf. Figure 2. Evolutions of economic systems which we will employ in our study will always be formally representable by one-parametrizations.

Rather, we will introduce evolutions of economic systems in the general notion of any continuous changes governed by a scalar parameter s. In Part III of our study we will study evolutions of economies in both interpretations of the evolution parameter s: in the technical atemporal interpretation, and in the interpretation as elapsing historical time. A contractible topological space X is homotopic to the single point space, i. Contractible spaces are special examples of acyclic spaces. The Lefschetz number of a space is an algebraic topological characteristic.

Clearly, a star-shaped space is contractible. A connected component Z of some topological space X cannot be separated into two disjoint open subsets, i. A path connected component is maximal with this property. One has to distinguish carefully between the notion of a path w and of its arc, i. This is also our intuition in this study. In other words they do not display any features of impassableness. We will come back to this later in our study Section In this terminology, the vertices are precisely the 0-faces, and the 1-faces are called the edges of the simplex.

The intersection of any two simplices from X is either empty or is a common face. Generally, a Euclidean neighborhood retract A in Rn is a subspace which is a retract of some of its neighborhoods, i. For the purpose of our present study, i. The reason for this is that the continuity of a homotopy is a fairly weak property still allowing for some pathologies of the one-parametrized family of state mappings if the domain is not a compact space.

More formally, the continuity of a homotopy F is equivalent to C 0 -uniform convergence of the state mappings Fs on compacta, i. This is possible since the critical arguments run to the boundary of the non-closed domain. Clearly such a behavior strongly contradicts the intuition underlying the notion of a continuous evolution of economic behavior functions. That means that neighboring state functions of an evolution should have similar values on their whole domain — not only on compacta. Thus, throughout our whole study we will employ the stronger concept of overall C 0 -uniform convergence for one-parametrizations instead of mere continuity, i.

R[x1 ,. Thus, R[x1 ,. For our purposes the zero sets of polynomials are crucial. Evidently, the semi-algebraic subsets of Rn form a rich class. As our last geometrical concept we introduce the central projection mapping in Rn together with its inverse. For further details see Hildenbrand , Section I. D, or Mas-Colell , Chapter 1. E, and the references given there. Mas-Colell, , p. The function f has the usual properties: it is continuous and bounded from below, i. Naturally the notion of a continuous one-parametrization of exchange economies with l agents is central for our analysis.

Here sco we will employ the following intuitive notion which is in the lines of Debreu cf. Hildenbrand , Notes 1. Some of our basic set—ups are adopted from the literature Chapters 4 and 6 , whereas the others are new Chapters 5 and 7. Naturally, these are also inspired by existing frameworks. The notion of an evolution of an economic system usually has the connotation of historical time. This can be compared with a cinematic study of a movement process in sports. On the one hand, one can use it for recording and representing the movement process in real time. The metaphor of a movie recording naturally suggests how to analytically formalize an evolution.

In fact, this is most naturally achieved by a one— parametrization, or say a one-parameter family, of single shot states of the economic system considered. This will be our approach in this study. In order to stress our generalized usage of the notion of an evolution with respect to the aspect of historical time, we generally use the symbol s, and not t, for the scalar evolution or say variation, or deformation parameter throughout 28 3 Introduction to Part I the book. In his book, Mas-Colell , Section 5. Nevertheless, he attributes the greatest importance to the one-parametrized case ibid.

In fact, it is this property of an evolution of economies as we formalize it that will turn out to be the essential prerequisite the results in Part II. There are two main reasons for us to do so. Or, as Balasko , p.

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Besides the intrinsic interest in such an undertaking, the insights gained from understanding the mathematical structure of the simpler pure exchange model can be invaluable when dealing with more general models. Indeed, several times in our study we easily can transfer an analysis which we have carried out for the Walrasian set—up to other basic set—ups.

Our main concern is with the result by Mas-Colell who has shown that the feature of indeterminateness also pertains to the equilibrium set itself. More precisely, this means that any compact set of the price simplex can be realized as the equilibrium set of some reasonable exchange economy described by individual preferences and endowments. In Part II of our study we will extend the examination of indeterminateness from the static level to the one-parametrized level.

Chapter 4 presents three well—known models of a pure exchange economy and puts them into the forms needed for our later analysis. It will later turn out to be considerably well—suited for generalization to other basic set—ups. In his analysis of one-parametrized economies A. Mas-Colell , Section 5. There are n markets as it will be the case throughout the whole study. Every individual becomes satiated with respect to any particular good. Note that this particularly means that excess supply on some, or even on all markets, is not inconsistent with equilibrium.

This means that free disposal is implicitly assumed for all commodities. Clearly this assumption does not mean a severe restriction from the economic viewpoint. Consistency with the previous assumptions is evident. Now we come to the crucial notion of an evolution of economies in our present set—up. First, it is weaker than the C 0 —uniform continuity assumption on the oneparametrization, and second it will later turn out to be most useful for the formalization of an evolution of large exchange economies see Section 4.

Figure 4. Convention When addressing an evolution of economies henceforth we will call the underlying static model the basic model, or the basic set—up. It is easy to get a geometric intuition of the presented analytical formalization of an evolution of economies. Moreover, if we endow Eex with the stronger topology of overall C 0 —uniform convergence then any exchange—I—evolution generates a continuous path, and vice versa. We will postpone the proof of Proposition 4. There it will turn out to be a Corollary of the more general result of Proposition 5.

We will call such a mapping an equilibrium equivalent self—mapping in the sequel. To this end let us recall the well-known equilibrium equivalent self— mapping provided by Arrow and Hahn , Section 2. Particularly, this is true for equilibrium price vectors. More precisely, the 4. Nevertheless, as we have argued before from the economic viewpoint this case might be well considered as the most plausible one.

Actually, his construction will prove to be most useful for generalizations later in our analysis. This will become important later. Actually, a model of a large exchange economy has also been used 38 4 Evolutions in the Traditional Walrasian Framework by Mas-Colell , Chapter 5, especially Sections 5. While Mas-Colell, however, uses individual preferences and endowments as primitives, here we basically will follow the lines of Dierker , Chapter 12 who directly builds on demand functions see also Hildenbrand In order to ensure later that evolutions satisfy a uniformized desirability condition we still impose an additional boundary assumption on any individual demand function.

Actually, this is achieved by the topology of weak convergence of probability measures. Moreover, MT becomes a compact, separable, and complete metric space cf Dierker, , p. Trying to formalize the notion of an evolution of economies in this basic set—up shows that the model of a large exchange economy is considerably more abstract than the ordinary exchange model.

In fact, we cannot any longer use the geometrically intuitive notion of perturbed behavior functions. Instead, we have to switch to the concept of a continuous path in the topological space of large economies see Mathematical Preliminaries. Actually, all necessary preparations have already been done in the preceding Subsection. The latter is due to the additional boundary assumption on the individual demand functions of D0. However, to make it comparable to our analysis here we report the general continuous case.

Schulz, , introductory remarks. Relaxing the budget identity will furthermore turn out to be essential for our analysis of evolving economies in Chapter However, there is still a second type of motive for studying such a general framework. This raises the natural question how deeply the consistency of the model under consideration depends on these assumptions. Nevertheless, our second model Section 5. Section 4. But this is precisely what we are not allowed to do.

We have to cope with a given system of n markets, no more and no less. Clearly, J pm is always non—empty. Possibly, the 5. Actually, the zero—vector has only been included here in order to lower notational expense and to simplify the formal analysis in the Subsections ii and iii below.

We will see that it also can be excluded in the more general setting of the following Section 5. The shaded area in the following illustrating Figure 5. Note that by Fig. Admittedly, the bound 1,1 10n seems to be somewhat arbitrary. Apparently, the class of basic economies in this set—up is considerably large. Figure 5. In the introductory remarks to this Fig. Now we have to ensure that a basic economy in the present context is not essentially the same as a basic economy from the traditional Walrasian exchange framework.

To show this we will proceed in the following way. Afterwards we will see that his proposal does not work: 5. After this digression we come to the precise notion of an evolution of economies in the present context. Correspondingly to the uniformized desirability condition for Walrasian exchange evolutions it prevents the equilibria of exchange—III—evolutions from running to the boundary of the homotopy price space.

However, from the economic viewpoint it is clearly not a restrictive assumption. In Section 4. Let Enwh denote the space of economies in the present context endowed with the product topology of uniform convergence on compacta. Thus the continuity of the path z is proven. We introduce the state index s already here in order to make the constructions in Subsection iii below more intuitive.

This contradiction proves assertion 1. Choose a continuous separating function. Consider the component sequence y m. Consider now the sequence y mk , smk. The main question is clearly whether it will be still possible to provide an equilibrium equivalent self—mapping if the basic model from Section 5. Actually, we will be able to show that at the cost of 5. Let us take these points one at a time. Furthermore, for economic reasons the origin 0 shall be contained in every set Di.

Actually, it generates a set of spaces of economies parametrized with n—tuples D1 ,. However, we will not pursue this further since we will not need this topology in our study. Analogously to the preceding basic model from Section 5. Nevertheless, this is immediate. Particularly, this applies to the introduction of the more intuitive index subset I pm. However, there is one point where we have to be cautious. Now let us examine which adaptions of the constructions from Section 5. The following Figure 5. It still Fig.

An appropriate criterion for this is the following standard result from general topology. Proposition 5. Going through the constructions from Section 5. Actually, the proof of the preparatory result 1 is even shorter. But this is completely straightforward from the preceding considerations. We employ the formulation by Kehoe b. He ensures overall consistency of the model by providing a proof of existence of equilibrium that is both simpler and more general than those given previously by, for example, Shoven and Whalley and Todd Kehoe considers this framework particularly intended for researchers who employ empirical general equilibrium models for policy analysis cf.

Kehoe, b, p. Actually, for the purposes of our study this framework has still a further remarkable advantage: it contains parameters which in principle can be controlled by an economic policy institution. This will become important for our later applications in Chapter An equilibrium in these models means a state of the endogenous variables which simultaneously makes all interdependent plans by the government and the individuals consistent.

Thus, from the equilibrium theoretical viewpoint there is no circularity problem with the mutually dependent tax redisbursals and tax payments. Later we will see that this implies that the equilibrium set in fact is contained in a compactum which is essential for the construction of an equilibrium equivalent self-mapping see Kehoe, b, p.

However, the following additional requirement to 6. Note that this aggregate budget constraint still does not indicate how the total tax revenues r is actually generated. Generation of r will be become clear below when also the production sphere will be introduced. Accordingly, the utility maximization problem of consumer j is the following: max uj xj1 ,.

This condition means that, anything else being equal, 63 64 6 Evolutions in a General Equilibrium Framework if tax revenue becomes arbitrarily large, then the income of at least one consumer the government for instance becomes arbitrarily large, which in turn implies that excess demand for some good becomes arbitrarily large cf. For a further example which also allows for tax rates and revenue shares varying with income the reader is referred to Kehoe b, p. Now let us come back again to the general model. Formally, this means that the last n columns of A form the negative n-dimensional unit-matrix.

From E. Actually, it is this equivalent formulation of equilibrium condition E. Together with E. This is formalized by the second equivalence of the following chain of equivalences. Kehoe b , p. Kehoe proposes the following construction for an equilibrium equivalent self-mapping b, pp. We have to verify four issues: 1 The constraint set is non-empty. This follows directly from assumption 2 on the production sphere. This follows from the assumption that there are no taxes on free disposal activities. This follows from the facts that for any pair of arguments p, r the constraint set obviously is closed and convex and varies continuously as a point-to-set mapping, and the objective function of the program is strictly convex.

This is shown in the proof of Theorem 1 by Kehoe b, p. We need to be able to guarantee, however, that the government can pay these subsidies out of its tax revenues. Again, it will be this reformulated form which will be crucial for our later constructions.

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An evolution of economies with production, taxes, and subsidies is in complete analogy to the preceding model in Section 3. The analogue to item 4 , i. We will present two new models which are suitable for our purposes. The second model has not been adapted from the literature. The model combines in an intuitive way the principles of the well-known quantity constrained macromodel with the idea of several interdependent sectors industries. From the macromodel it inherits the opportunity of a geometrical representation in two dimensions. What it makes particularly appealing to the economist is that the prerequisitory problem of existence of equilibrium reduces to a remarkably simple mathematical situation which furthermore is formalizable by an intuitive function.

For the microversion presented in Section 7. As usual, demand is given positive sign, and supply negative. Final allocation in case of a non-market-clearing price system p is achieved through a deterministic rationing scheme. We furthermore assume that all perceived rationing bounds z ai and z ai , and all realized demand and supply transactions Fia.

Clearly, this is not a restrictive assumption. In other words, in an equilibrium no plan revisions are necessary since the supposed bounds for all individual plans turn out to be correct. We have to verify several things. This follows directly from the assumption that realized transactions must be voluntary, i. The following Figure 7. The thick line is the graph Fig.

Accordingly, the closure of the shaded areas shows the range of Fia. To show this let us start with the lia -components of g. Figures 7. The prices for the m commodities y1 ,. To simplify notation the vector p will be subsequently suppressed. This means that there are 2m positive real numbers l1max ,. Evidently, this is not a restrictive assumption. The inner-sectoral spillovers are modelled in complete analogy to the wellknown one-sectoral macromodel.

This corresponds to the well-behavedness of the quantity constrained macromodel usually adopted in the literature cf. The possibility to hold inventories leads to the wedge. On the other hand, in each sector households will reduce maintain their labor supply by substituting leisure for labor if the commodity which is produced in the sector is is not a consumer good and demand is rationed on the commodity market. If it is is not a consumer good, households will reduce maintain their demand for the commodity of the sector if they get rationed on the sectoral labor market.

The wedge is due to the possibility of saving. Particularly, all supply intentions yis li and lis yi must be technologically and physically feasible for the economy. For the derivation from optimization programs cf. Note that the example drawn in Figure 7. But the reader should note that this is not inconsistent with our model, since constant reaction functions are admitted. Moreover, all inter-sectoral constraint spillovers will be incorporated in the next step.

The usual regime terminology shall also be maintained — here now characterizing, however, the state in each sector. Unfortunately, so far the model obviously is still unsatisfactory since each sector is isolated from any possible quantity rationing signals from the other sectors. Our proposal to take the inter-sectoral spillovers into account is to continuously parametrizing the 4 reaction functions lis yi , lid yi , yis li , yid li of each sector i with constraint signals lj and y j from [0, ljmax ] and [0, yjmax ] transmitted from the other sectors. Thus, one obtains 4m continuous functions lis l1 ,.

We will assume that for each i and any parametrizing constraint tuple l1 ,. Equivalently, we can formally represent the system 7. Generally spoken these will be the l, y -coordinates of the states of the other sectors. However, whether these states are the sectoral equilibria or are any other states we will not decide in this study. We can do so since this question has no relevance for the equilibrium analysis we are purposed to do here. In particular, the geometrical representation in two dimensions familiar from the macromodel can be maintained see below.

Clearly, like in the unparametrized case above all demand and supply reactions of the fully parametrized model 7. As it is usual for the macromodel see e. The total interdependence of the reaction functions of an economy 7. Thus, in each of the m two-dimensional sectoral boxdiagrams the reaction functions are shifted when the constraint signals from the other sectors change. Geometrically this results in a right-downwards shifting of the Hir -wedge cf. Figure 7. This will result in a left-downwards shifting of the Fir -wedge.

It interlinks internal planning of the economic agents in each sector which only takes account of inner-sectoral constraint signals with externally determined planning depending on the signals from the other sectors. More precisely, there is a hierarchy of these two principles of planning: the sector-internal planning is completely carried out under every perceived tuple of sector-external signals. Or to say it in other words, in each sector the agents plan as regards the constraint signals from the other sectors. In general, in each sector i any shiftings of the two sectoral reaction wedges will also lead to a displacement of the sectoral quantity constrained temporary equilibrium Gi , i.

Let us now come to the notion of an equilibrium of an economy 7. Thus, in an equilibrium the m sectoral quantity 7. Or, in other words, all sectoral equilibrium signals are just mutually reproduced by the reactions of the agents when the reaction functions in each sector become reparametrized by the sectoral equilibrium values of the other sectors. It is noteworthy that in contrast to the other basic models here any evolution can be geometrically visualized.

Dropping the parametrizing constraint tuple for the moment in order to simplify notation the sectoral reaction functions yis li , lid yi , lis yi , and yid li can be extended beyond the vertices Fi and Hi by constant functions up to the upper bounds limax and yimax respectively. This is indicated in the box-diagram of Figure 7. Clearly, the properties of the mapping G are essentially determined by the properties of the economy. Finally, some remarks are in order.

Actually, this generally does not mean that re-parametrizing the reaction functions in each sector diagram by the coordinates of the sectoral equilibria of the other sectors necessarily leaves the reaction functions unchanged. What it means is just that no changes of any reaction functions are possible which also change the position of any sectoral equilibrium.

However, a further re-re-parametrization does not change the reaction functions anymore. Clearly, G Ce is a self-mapping of C. Particularly, it does not require that also the vertices of the reaction wedges move continuously. On the whole, continuous movements of the sectoral reaction functions do not appear to be more restrictive than the continuity of the reaction functions themselves in the traditional macromodel.

This means that also our multi-sectoral model has the familiar feature of indeterminacy as regards the equilibrium set. Now draw for each yi sector a box-diagram cf. Extension of this type of example to produce continua of equilibria is obvious. Actually it is not hard to think of inner- and intersectoral interdependencies i.

Some of them have been adopted from the literature Chapters 4 and 6 , while the others are new Chapters 5 and 7. Thus the basic models have not been introduced as an end in themselves, but as basic set—ups for constructing the main subject of investigation, i. All nine types of evolutions of economies have been constructed on the same formal principle as continuous one-parametrizations, or say one—parameter families, of economies. It has been emphasized that in the study any successions of states that satisfy our assumptions, not only evolutions of an economic system ongoing over historical time, have been admitted.

In fact, this has become one of the major concerns of theoretical economics since the fundamental indeterminateness of the basic exchange framework was discovered in the early s. Actually, the results in the following Part II will be fairly surprising to the reader who is familiar with the examples illustrating regular equilibrium theory, which strongly suggest that there is a maximal arbitrariness of the dependently evolving equilibrium set. All results are economically motivated and discussed. Thus, the reader who is mainly interested in the analytical results of Part II can start here.

In fact, it is this property of an evolution of economies as we have formalized it that will turn out to be the essential prerequisite for the results in Part II. The central analytical result is given in Chapter It ensures that the equilibrium set of any evolution considered in Part I has a certain structural property. Loosely speaking, this means that above any evolution of economies, there is a path of equilibria in the graph of the equilibrium correspondence which is either itself geometrically well-behaved, or which can be approxi- 92 9 Introduction to Part II mated by a well-behaved path.

This result has also been given independently for the basic model of a large exchange economy by Mas-Colell , Section 5. How large is the subclass of those evolutions that already have well-behaved equilibrium paths in each basic set-up? Or, to formulate it more stringently, can any evolution be approximated by one that has a well-behaved equilibrium path? For the exchange framework a closely related result has been shown by A.

Nevertheless, here we provide constructive methods to achieve nice approximating evolutions, whereas Mas-Colell just gives an abstract existence result. From our constructions, we derive a further new result that shows that the equilibrium correspondence of each of the nine basic models introduced in Part I is extremely regularly connected. This result considerably extends the result of the manifold property of the graph of the Walras correspondence by Y.

Our conceptualization of an evolution of economies admits two obvious economic interpretations, which will be analyzed in Chapter On the one hand, one may stress the aspect of course. Then an evolution describes an economy, which from its initial state economy, evolves somewhere in the space of economies — the only restriction being that it has to obey the weak conditions indicated in Part I. On the other hand, one may stress the aspect that an evolution connects its initial state economy to its terminal state economy. This interpretation immediately raises the question of whether there is always an evolution connecting any two given economies.

When considering evolutions as evolutions in historical time, it seems to be most natural to also include new commodities entering the economy during its evolution and old commodities leaving it. The formal extensions necessary to achieve this in all basic set-ups are provided in Section However, the natural question remains whether there are possibly other structural properties of the equilibrium set of evolutions that also generally hold.

Furthermore our result is to be seen as complementary to the related results on the local surjectiveness of the graph of the Walras correspondence by B. Actually, our constructions in subsections ii and iii of the Chapters 4 to 7 put us in a position to show for any presented type of evolution that there is a certain general structure property of its equilibrium set.

The existence of such a joining connected equilibrium component in turn ensures the existence of a nicely behaved joining path which approximates the equilibrium set of the evolution arbitrarily closely. Actually, it is the existence of near-equilibrium paths which proves to be crucial for our further applications in Part III of the monograph.

Browder, however, presented his result in a fairly general setting unfortunately obscuring his achievements somewhat. Applied to our situation his result, however, does not allow for boundary equilibria.

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  • Fortunately, there is an extended result by Dieter Puppe , Corollary 5. Having achieved the general structure property of the existence of nearequilibrium paths for evolutions it is most natural to ask whether possibly there are still further general structure properties of the equilibrium sets of evolutions. We will settle this question in Chapter 13 for the exchange framework. Actually, we will be able to demonstrate that the existence of a joining connected equilibrium component is the only structure property which is generally valid. Theorem Moreover, the method which we will provide is particularly powerful for detecting initial points of joining connected equilibrium components.

    Figure Nevertheless, we will Fig. Now we are ready to state the following result which will be central for our study. The following proof of Theorem This is also the reason why we will not relegate it to the appendix. Fortunately, the major part of work has already been done in Part I. Actually, from the subsections iii of the Sections 4. Let us now look at the properties of K. K is compact, and, particularly, it is a Euclidean neighborhood retract.

    Since K is furthermore contractible, it is also acyclic. Let K be a compact subset of Rn and a neighbourhood retract. We will call a connected component of the equilibrium set of an evolution of economies which meets bottom and top of the homotopy space a joining equilibrium component of the evolution. The Proposition has been proven by Puppe , Corollary 5. However, such a connected joining equilibrium component may still display some bad geometrical features.

    A closely related result has been proven by F. Browder , Theorem 2, p. However, Browder uses a more restrictive boundary assumption. Actually, this result would not work for those of our basic models which admit boundary equilibria, i. Now, let us do the last step of our proof of Theorem We now proceed by pointing out the reasons why it is generally necessary to approximate a joining equilibrium component by near-equilibrium paths in order to get a nicely behaved path in the homotopy space.

    In Chapter 11 we will give an analytically precise and comprehensive characterization of the class of nicely behaved paths and of the class of evolutions which produce them. Let us now look at the possible geometrically bad behavior of joining equilibrium components. First and foremost a joining equilibrium component need not be path connected. We will give an example of an evolution producing this below. Unfortunately, any of these complications actually can occur in the equilibrium set of an evolution.

    They even cannot be removed by additional differentiability conditions on the evolution. The following example makes this clear. For the exchange framework we begin with a generalization of the wellknown and fairly weak approximation criterion using the supremum norm. Note that this notion still allows for shifts in the state-parameter in order to achieve a true equilibrium. This, however, is excluded by the following last notion of approximating equilibria of evolutions.

    Also the converse directions do not generally hold. Apparently, this amounts to the general question of how one can discern for any given evolution whether a certain point in the homotopy space lies on one of its joining equilibrium components or not. Moreover, we will see that this result also provides a partial answer to our general question.

    We will not give the details of the proof of Theorem The following Figure Naturally this requires explicit knowledge of the selfopen neighborhoods W mapping g0 to some extent. However, since we will not make here further use of the result of Theorem We are still left to demonstrate that Theorem Nevertheless, Fig. Just apply Theorem When both times the criterion of Theorem This means, the dotted case in Figure What we will do now is to show precisely that intuition actually led us the right way.

    Unfortunately, however, this program would not only mean a hard piece of technical work. Thus, we will adopt another way — starting from the opposite direction. This especially means that none of the geometrically bad features discussed in Section Moreover, we will also be able to show that it is even possible to approximate any evolution arbitrarily closely by such a well-behaved evolution.

    The second one is based on an approximation method which is even simpler than polynomial approximation and which has the further advantage of being entirely constructive. In the last Section Let us conclude this Introduction by the following remark. There is also a close relationship of our results given in this Section to a result by Mas-Colell given in his book , Section 8. Particularly, regular one-parametrizations have rather well-behaved joining equilibrium components. It does not provide any constructive method how to achieve them.

    In this Chapter we will go beyond this abstract existence result in that we explicitly provide methods to construct approximating evolutions with nicely one-parametrizable joining equilibrium components. We do so since this model notably well suits as a reference model for later generalizations. We are going to present two intuitive general methods for approximating any given Walrasian exchange evolution by a well — behaved and economically intuitive evolution of this type having, moreover, nicely behaved joining equilibrium paths.

    To prove the well — behavedness of the equilibrium set, however, requires an advanced result from algebraic geometry. Even though, this method is far more constructive than the addressed abstract existence result by Mas-Colell. The reader should This reservation does not apply to our second method, which, however, is a little bit more laborious. It achieves approximating evolutions in a completely constructive manner using as main analytical tool piecewise linear functions.

    This construction has the further advantage that it makes the well — behavedness of the equilibrium set still more intuitive. Now we have a twofold purpose. Nevertheless, from the discussion in Chapter 10 it is intuitively clear what we expect from a joining equilibrium component deserving this attribute.

    In a word it should be well passable in the intuitive geometrical sense from bottom to top of the homotopy price prism. But how can we formalize this in a precise analytical way? Fortunately, we can present a class which serves all of our purposes. From our construction the reader will immediately see that it is built on a geometrically intuitive principle.

    This means that the path w represents the one-parametrization of the arc. In fact, analytical paths are geometrically remarkably well—behaved curves. This will be made lucid by the characterizations provided by Proposition Proposition It makes essential use of what we call a normalized tangent direction.

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    Evidently the tangent direction is only determined up to a multiple. Actually, the normalized tangent direction gives us an intuitive and exact instrument to study the behaviour of an analytical path in dependence of the parameter t. This is realized by Proposition However, the following example exhibits that we are still not quite done. Consider the subspace of R2 formed by the two coordinate axes. Clearly, there exists an analytical path joining 1, 0 and 0, 0 , and one joining 0, 0 and 0, 1. But, there is no analytical path joining 1, 0 and 0, 1!

    This follows immediately from the following Proposition Nevertheless, there is a simple way to cure our approach from this unwelcome oversharpness. Now we are prepared to tackle our main task of this Section. CBPR rests on principles of empowerment. The researchers are responsible for the scientific quality and that ethical standards are met. Evaluation objectives are to compare outcome between children in intervention and control areas, conduct health economic assessments HEA and evaluate the processes of the project.

    HEI is a repeated cross-sectional and longitudinal study. Primary contact groups are children in disadvantaged communities. Core efforts are to confirm and convey knowledge, elucidate and facilitate on-going health work and support implementation of continuous health work. Socioeconomic status is assessed on area level by the parameters yearly average income, degree of employment, tertiary education and percent of inhabitants born in countries where violent conflicts recently have taken place or were ongoing.

    Anthropometry, food patterns, physical activity and belief in ability to affect health; together with learning, memory and attention assessment will be assessed in children born Examinations will be repeated after two years, forming the basis of a health economic analysis. Inviting, awaiting and including local perspectives create mutual confidence and collaboration. Enhanced self-efficacy and access to relevant knowledge has potential to enable individuals and communities to choose alternatives that are relevant for their health and well-being in a long perspective.

    The economic of this study may contribute in decision- making processes regarding appropriate public health interventions. Deviation from normal weight i. Obesity, a medical condition in which excess body fat has accumulated to the extent that it may have an adverse effect on health, has strong associations with other conditions such as non-insulin dependent diabetes and high blood pressure [ 5 ]. In high income countries, obesity is more common than underweight. Obesity is difficult to treat, at all ages and obese children often become obese adults.

    Prevalence of childhood obesity is inversely associated with parental education and income. It is also associated with parental occupation and migrant status [ 6 ]. Mechanisms behind these associations are complex and involve several determinants i. For public health interventions to be successful in narrowing health inequalities, theories and approaches drawing on empowerment are needed [ 7 ]. The focus of the intervention is on promoting healthy weight in children and narrowing the health gap.

    Primary outcomes will be weight and body composition in children. A major challenge in research today is low response rate [ 8 ]. In epidemiological studies, participation is normally lower in groups with less than optimal health outcomes [ 9 ]. Resting on principles of participation, influence and empowerment, CBPR has potential to narrow the health gap and to raise participation rates, especially among hard-to-reach groups [ 10 ].

    Health of an individual is formed depending on which challenges we meet, meaning that health is related to learning, judgment of risks, impulse control and coping strategies. This development period shapes the cognitive abilities of the child [ 12 ] and from around the age of 7 many of these abilities are present [ 13 ]. Some children may perform exceptionally well and already have abilities equal to an adult but others may have troubles [ 14 , 15 ].

    Tests of planning, executive function and attention are good markers of a normal development for a child [ 16 ]. Nutrition itself is also a critical factor in normal cognitive development among children [ 20 ] so having a method to monitor learning skills in combination with BMI measures is very relevant. A recent study found that consumption of a Western diet increased the risk for developing ADHD in adolescents [ 21 ].

    However, positive cognitive effects of adding more physical activity to the school curriculum of children could be measured [ 22 ], again suggesting the value of measuring these parameters in addition to those related to anthropometry. Considering the fact that there is now a widespread awareness of the problem of childhood obesity many governments are seeking to invest in prevention and management programs.

    If unlimited resources were available for obesity prevention activities, program planners could simply implement any effective program without regard to expense. However, because public health resources are limited, prevention interventions must not only be effective but also be cost-effective [ 23 , 24 ]. One of the objectives of this study is to conduct a health economic assessment HEA , hoping that it could contribute in the decision making process for public health intervention planners.

    Evaluation objectives are 1 to compare the outcome between the children in the intervention areas with a comparison sample and 2 to scrutinize the program to see whether the processes are helpful in order to fulfill the aims and whether they are implemented. In the areas where the current study will be performed, HEI is funded for three years and this includes an evaluation. In a dialogue with the regional and municipal health planners, the outcome focus of the intervention was chosen to be children 6—12 years in disadvantaged communities.

    Primary contact groups are children including schools, families and their networks. The wider community, including youth recreation centers and other arenas for siblings and parents is also included in the intervention plan. HEI prioritizes groups with elevated risk of adverse health outcomes; supporting them to improve health from their own perspective. HEI aims to broaden the spectra of participants that take active part in societal matters and makes efforts to involve people that are not already spokespersons in other contexts.

    Core efforts are to confirm and convey knowledge, elucidate and facilitate on-going health work and to support implementation of continuous health work. In dialogue with municipal health planners, plans emerged in terms of which schools and other stakeholders should be approached. Via this method, theoretically-informed and locally-anchored interventions emerge following discussions with schools and other organizations, parents and children. Thus, the intervention to be performed will be designed continuously so it is not possible to precisely describe beforehand what the program output will be.

    Program logic model. Through dialogues and meetings with actors on arenas in the local community, program output is formed and activities are planned. Logic of change: selected examples of performance and change objectives. Performance objectives are chosen to support the overarching aim of the intervention. Determinants for change objectives are affected through specific strategies, based on Social Cognitive Theory.

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    Numbers refer to specific determinants listed below the table. Performance objectives are theoretically informed and developed in discussions with residents and local professionals. The lists will develop during the interventions. This is a repeated cross-sectional and longitudinal study. Outcomes will be compared between children in the intervention areas and control areas. The design will be explorative in the first step. In the second step it will be quasi-experimental with the hypothesis that the intervention yields a difference in body mass index BMI -development between intervention and control schools, leading to a long-term favorable health economic impact for healthcare and the wider society.

    A sample size of children in each group, i. Taking dropouts into account the aim is to recruit participants from control schools and from intervention schools. Control schools are purposively selected from areas where HEI has not been active. They are chosen from local government statistical data to match the interventions schools by area level index of socioeconomic status [ 27 ].

    Health promoting activities of varying quality and sustainability are initiated and carried out continuously in Swedish schools and municipalities. It is therefore necessary to separate potential effects of HEI from those of other health promoting activities for intervention schools and areas , and to account for interventions that may affect the children in the control schools and areas.

    Activities and interventions will be listed, described and compared to HEI program output, by communication with school staff and municipal health planners and by reviews of relevant documents. Thus, children in their first year in primary school, most of them born in , are invited to participate. They are invited for 1 anthropometric examination weight, height, body composition and waist , 2 short interviews about food pattern, physical activity and belief in their ability to affect health.

    The examinations will be repeated after two years and form the basis of a health economic analysis. Parents are invited to participate. Much effort is put into organizing the study so as to cause the least possible burden on the schools. Parents are welcome to be in the room when the child is interviewed and measured. After the examination the family is asked if they have any comments on the school as a place for health promotion. After the measurements are finished at each school the classes with participating children are offered health lessons to give children feedback on their participation, and with a focus on self-efficacy.

    For ethical reasons, health lessons are offered also in control schools. The interviews are conducted with the children, at each school. If the parents express any worries concerning any of the outcomes the research team will support them and be sure that adequate medical or nursing care is received. Schools were approached through headmasters or school nurses. Throughout, the measurements were presented and regarded as part of the health intervention in the intervention schools.

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    Parents are informed in parental meetings. Letters with information and forms for consent and acceptance are sent to all parents. Those who do not send the forms back are contacted by telephone at one occasion. Families are offered different options concerning time points for the measurements. If the parent does not participate, the HEI staff picks up the child from the school class and then follows them back afterwards.

    All HEI interviewers have degrees in health professions clinical nutrition, nursing and public health and have been trained together. Interviews, anthropometric measurements and computer exercises are standardized and calibrated before the measurements and after two months. Children will be weighed in light indoor clothes. Typical indoor clothes for children of this age were weighed and the result, grams, will be subtracted from the total.

    The child will be encouraged to state its age. Total body weight and weight of muscles will be reported to the child and parent, when present , in a non-problematic way. Checking is done that the floor under the scale is flat and solid, that the child is standing up straight, that their heels touch the ground and that shoulders are level and not raised. The child should look straight ahead and breathe normally. Waist circumference will be measured using a SECA measuring tape, with a standardized procedure. The child is then asked to stand straight with the abdomen relaxed, the arms at the sides and the feet pointing forwards and together.

    After finding the correct points together, the researcher marks the spots and uses the measuring tape to find the point in the middle of the distance between them. The child is encouraged to exhale normally and the measurement is taken around the trunk at the end of such expiration, without the tape compressing the skin. The measurement will be taken in centimetres.

    In case of uncertainty the measurement will be repeated and the mean of the measurements will be used. Children will be asked whether they can do something to affect their health or wellbeing. This question has been used in a school survey with children of age 11—12 where it on a group level was associated with healthier food habits and BMI within normal ranges, both cross-sectionally and longitudinally [ 28 ].

    Consumption of breakfast, bread, milk, fruit, vegetables, fish, legumes, sweets and snacks and sweet drinks is included, together with items of favorite food, and drink for dinner at home. It draws on the Eating Choice Index, which is a validated tool [ 29 ] and a questionnaire from the Swedish Food and Nutrition Board [ 30 ]. These instruments, originally constructed for adults, were adapted for Swedish children and complemented with a question that examines their knowledge of the widely-used Swedish green keyhole symbol [ 31 ].

    Pictures of different kinds of bread, legumes and milk packages [ 32 ] were used to facilitate understanding and improve the validity. After consulting physiotherapists and fellow researchers we decided to use a questionnaire earlier used by Bonnevier in a study of children in third grade approximately aged A shortened version of this instrument is used [ 33 ].

    They are also asked to characterize themselves as active, sedentary or in between. These include the following:. This test teaches the child how to press the touchscreen carefully when an X appears on the screen. It can also be an index of reaction time and motor ability. In this test children are instructed to watch the centre of the screen where random numbers appear. They are supposed to look for when the sequence 1, 2, and 3 appears and then press a button only when the 3 appears, not before or after.

    This measures the ability of the child to stay focused on the task, to pay attention to the patterns and also measures their premature responses. This is one of the most commonly used tests of impulsivity and can relate to ADHD. This test measures so-called executive function and is a marker of normal development of the brain. The children are asked to find blue boxes hidden under an increasing number of other boxes. This is a test of searching but also of developing a strategy to systematically search through the boxes one-by-one. At the age of 7 children often have no strategy and just search randomly while by the age of 9 they should search in a more orderly fashion similar to teens and adults.

    When possible scores from these tests will be compared groupwise with the reference population built into the CANTAB system in order to see whether the children in this study differ in general. The Control and Intervention groups will also be compared. Providing that results from effectiveness evaluation will show a significant decrease in the BMI, a simple cost-effectiveness analysis CEA will be conducted. Changes in BMI will be derived at the baseline and follow up among the children in the intervention and the control schools.

    In the comparison two types of costs will be considered:. Comparisons between intervention and control schools will be made using quantitative analysis of the cross-sectional impact and outcome data using descriptive statistics and regression analysis, as appropriate. A process evaluation procedure will monitor the processes in terms of reaching the intended outcome. Summative uses include assessing whether the chosen strategies and underlying methods were appropriate to reach the goals. The specific purpose of the process evaluation in HEI is to develop knowledge about participatory research on a community level.

    Socioeconomic status is assessed on the area level, based on local governmental statistics. Parameters used are yearly average income, degree of employment, tertiary education, and percent of inhabitants born in countries where violent conflicts recently had taken place or were ongoing. Public health interventions often have insufficient funding, with respect both to time frame for the project and to evaluation and this can often lead to failure or a lack of sustainability long-term. Currently HEI is active in more than half of Gothenburg.

    The outcome evaluation presented above is funded for approximately one third of this domain, while process evaluation is a continuous part of the intervention. Inviting, awaiting and including local perspectives and discussions on the program are time-consuming i. If the only result of this effort would be to enable researchers to collect a certain amount of data, it might be difficult to justify this expense. But in these processes mutual confidence emerges together with awareness of possibilities for collaboration.

    This builds prerequisites for complex, supportive structures that constitute the basis for implementation of sustainable health promoting programs within the communities. The programs will not be a locally adapted version of a general model but rather internalized in the communities. However, if the participants do not experience continuous benefits from the project like influence on prerequisites for health, on publicity and research , it may not continue.

    The concept of empowerment has had a strong impact on health promotion theory and practice. It is used in various ways, some of which diverge from the original concept, developed by Paulo Freire [ 35 ]. One example is when corporations use it as a label to denote their efforts to make their employees feel more valued and creative [ 36 ].

    In public health work the autonomy of the participants may be violated, namely when public health workers rather than the participants themselves define lifestyle changes that individuals or groups need to carry through, for example to start having breakfast or decrease sedentary time.

    Even if the methods used to support such lifestyle changes build on empowerment theories it is not self-evident that the term can be used. There is a need to reflect on whether there is some paternalism embedded. However, measures taken within the project need to be compatible with the intent of the funding agency. Here is a potential conflict, i. For HEI, the goals are broad and include possibilities for a wide range of interventions, none of which are defined from the beginning.