Price versus Quality Competition: In Search for Schumpeterian Evolution Mechanisms

Transcription

1 Price versus Quality Competition: In Search for Schumpeterian Evolution Mechanisms Authors Agnieszka Lipieta Department of Mathematics Cracow University of Economics Rakowicka 27, Cracow, Poland Andrzej Malawski Department of Mathematics Cracow University of Economics Rakowicka 27, Cracow, Poland Abstract The paper extends the research program of modeling the Schumpeterian vision of innovative development in the Arrow Debreu theory of general equilibrium. The core of this set up is based on modeling the two fundamental forms of economic life distinguished by Schumpeter, namely the circular flow and economic development by specific extensions of the production system being part of the Debreu private ownership economy, so that the analysis includes static as well as dynamic forms. In this context the paper aims at studying the mechanisms of Schumpeterian evolution in the conceptual apparatus of the modern theory of economic mechanisms. To this purpose two kinds of mechanisms of Schumpeterian innovative evolution are studied. First, price mechanisms corresponding to the circular flow, and second, qualitative mechanisms based on innovations.

2 Key Words: Schumpeterian evolution, mechanism, designing mechanism, Debreu economy 1. Introduction The social sciences aim, in contrast to natural ones, at explanation of phenomena and processes under study not by subsuming them under a causal law, but by discovering the rules or mechanisms which guide them, render meaningful, and deliver a deeper insight into their intrinsic logic. In this context, Schumpeter (1912), (1939), (1950) studied the mechanisms of economic evolution based on the processes of innovations and governed by the principle of creative destruction. It is presently 100 years since Schumpeter (1912) published his book Die Theorie der wirtschaftlichen Entwicklung which came out late in This book presented his theory of economic evolution as driven by innovative entrepreneurs. Thus, it is justified in considering Schumpeter as the initiator of an evolutionary economic program which has recently enjoyed a remarkable revival not only in economic theorizing but also in real economic life worldwide. Indeed the Innovative Economy Operational Programme , approved by the decision of the European Commission of 1 st October 2007 and by the resolution of the Council of Ministers of 30 th October 2007 can serve as an example. This program aims at supporting innovativeness in its broad sense and is directed mostly to all entrepreneurs who want to implement innovative projects connected with research and development, modern technologies, and investments of high importance for the economy. Let us recall that Schumpeter distinguished two fundamental forms of economic life, namely the circular flow and economic development. Specifically, within the circular flow economic life goes smoothly along the beaten tracks keeping the governing rules unchanged year after year. Indeed, Schumpeter wrote (1961, p. 108): In the circular flow, from which we always start, the same products are produced every year in the same way. For every supply there waits somewhere in the economic system a corresponding demand, for every demand the corresponding supply. All goods are dealt in at determined prices with only insignificant oscillations, so that every unit of money may be considered as going the same way in every period.... There is no market for the bearers of the original productive services themselves,..., and there is also no price for them within the normal circular flow. Consequently, the process reminds us (cf. ibid., p. 61) of the blood circulation in a living organism and can be interpreted as the stagnation of economic life. On the other hand, economic life goes beyond these narrow patterns of continuous evolution taking development as described by Schumpeter (ibid., p. 64): Development in our sense is a

3 distinct phenomenon, entirely foreign to what may be observed in the circular flow or in the tendency towards equilibrium. It is spontaneous and discontinuous change in the channels of the flow, disturbance of equilibrium, which forever alters and displaces the equilibrium state previously existing. The driving force and the key to the understanding of the phenomenon of economic development is the creative destruction principle. This principle can be interpreted as the synthesis of two opposing tendencies present in economic development: creative innovations and the elimination of existing, products and organizational structures and their replacement by new ones. Indeed, Schumpeter (1950, pp. 82 3) wrote: The fundamental impulse that acts and keeps the capitalistic engine in motion comes from the new consumers goods, the new methods of production, the new forms of industrial organization that capitalist enterprise creates.... The opening up of new markets, foreign or domestic, and the organizational development... illustrate the same process of industrial mutation if I may use that biological term that revolutionizes the economic structure from within, incessantly destroying the old one, incessantly creating a new one. Economic development so understood is dynamic and discontinuous, it continuously destroys the existing old economic structures, substituting for them new structures. It has to be remembered, though, that while the fundamental force explaining the rules of circular flow are consumers needs and preferences, economic development is initiated by producers who become innovators and entrepreneurs (cf. Schumpeter 1961, p.65). However, the relationship between these two forms of economic life has not been explained by Schumpeter in a satisfactory way. This transition from the circular flow to economic development can be characterized as follows (Hanusch and Pyka, 2007, p. 23): It is no longer price competition only, but following his ideas of development, quality competition, driven by innovations and imitations of economic actors, takes over the leading role. In other words, profit opportunities are signaled not exclusively by market prices but also by creativity and daringness of entrepreneurial actors which change the relative scarcity in an economic system. Moreover, studying the relationship between the circular flow and economic development Day (2007) emphasizes that the coordination mechanisms of transactions between agents, within the forms of economic life studied above, are dramatically different. In the former it is based on a Bidding Negotiation Bargaining (BNB) process (ibid, p. 748), which results in competitive equilibrium prices. But, in the latter, this mechanism is replaced by (ibid) a Inventory Order Backlog Price Adjustment (IOPA) procedures to assume bounded rationality of economic agents instead of perfect rationality under the BNB terms. Actually, it is the role of

4 entrepreneurs innovators to drive the development of human creative intelligence, which Day distinguishes from rationality. The more detailed analysis of evolutionary mechanism in Schumpeter s writings has been elaborated by Andersen (2009). He emphasizes that Schumpeter used the expression the capitalist engine as a rhetorical device and pedagogical metaphor to influence the reader s imagination. However, Andersen suggests that it seems to be a strange engine of evolution. Indeed, he writes (ibid, p. 147): The fuel of this internal-combustion engine is provided by Schumpeterian entrepreneurs. Their innovations locate the ignition of the process (Cycles, 102; Cycles Abr., 76), and this process includes responses in the form of imitation, adaptation, and destruction. The result of the working of the capitalist engine is economic evolution. In other words, Andersen (ibid, p.149) identifies two opposing evolutionary mechanisms. First, the mechanism of innovations which moves the economic system from a stationary state to its maximally disequilibrated state. Second, the mechanism of adaptation to move the system back to a new stationary state, where previous innovations have been absorbed in an equilibrated system of economic routines. However, the structure mechanism of innovations turns out to be more complex, and at least two versions of the capitalist engine can be distinguished, labeled after Freeman (1982) as Mark I and Mark II. The first is based on establishment of new firms by innovative entrepreneurs. In this linear model exogenous innovations are then diffused into economic life. Contrary to this, Mark II assumes the innovative activity of incumbent firms in their oligopolistic competition that focuses on new products, new technologies and new organizations, which leads exogenously to increased market concentration. A more rigorous analysis of the problem of Schumpeterian evolution mechanisms has been elaborated within the theory of endogenous economic growth (see: Aghion and Howitt, 1992, 1998). However, this approach, as Foster (2011) suggests, is, at best, tenuously connected with Schumpeter himself, and the neo Schumpeterian paradigm originated in Nelson and Winter (1982). This analysis of Schumpeterian evolution mechanisms, presented above, suggests that the mechanisms under study can be classified into two categories: price mechanisms corresponding to circular flow and qualitative mechanisms based on innovations. However, our general, more rigorous setting implies that it is difficult to preserve this dichotomy and a variety of economic mechanisms should be taken into account. The main premise in this context is based on our viewpoint that the consequences of innovative changes contribute to more diversified results. Specifically, this variety of qualitative mechanisms concern the agents whose position get better

5 off. Thus we can distinguish qualitative mechanism with respect to the given set of agents, for example the set of innovators or all producers. Now, it is worth including the research trend presented above in the formal modeling of the Schumpeterian vision of innovative development in the Arrow Debreu theory of general equilibrium (cf. Malawski 2004, 2005, 2008; Malawski, Woerter 2006; Ciałowicz, Malawski 2010, 2011) so that it would embrace studies on mechanisms of economic evolution. To justify this setting it should be recalled that Schumpeter s work was strongly inspired by Walrasian thinking (cf. Hodgson 1993, Anderson 2008), which has been neglected in the current literature. Needless to say, Walrasian equilibria are now being studied within the mathematical framework of the Arrow Debreu general equilibrium theory and its extensions (cf. Magill, Quinzii 2002, Duffie 1988). In this approach a Debreu (1959) private ownership economy has been presented in the form of a relational system which consists of interrelated subsystems of production and consumption operating on a commodity price space of. In our setting producers activity is still independent of choices made by consumers, despite their shares in producers profits. Consequently, the behavior of producers can be reduced to the selection of a production plan maximizing profit under a given price vector, which means that issues corresponding with firms management do not go beyond profit calculating. The core of this set up is based on modeling of the two fundamental forms of economic life distinguished by Schumpeter (1912), namely the circular flow and economic development by specific extensions of the production system being part of the Debreu private ownership economy, so that the analysis considers static as well as dynamic forms. Specifically, cumulative and innovative extensions correspond to the circular flow and economic development, respectively. This research program will be now extended in a new direction heretofore neglected. Namely, the paper aims at studying the mechanisms of Schumpeterian evolution in the conceptual apparatus of the modern theory of economic mechanisms designing (cf. Hurwicz, Reiter 2006), where the central category of economic mechanism is understood as a mathematical structure that models institutions through which economic activity is guided and coordinated (ibid). More formally, the mechanism is represented by a message space, containing messages available to communications within agents, a message correspondence, which associates with each element of the given environment a set of messages and an outcome function (see def. 3.1 below). Specifically two kinds of mechanisms will be distinguished. First, the mechanism in which prices of commodities are elements (not necessarily unique) of the message space and they influence the set of outcomes, called the price mechanism. Second, if in the mechanism at least one agent from the given set will be better off due to a given criterion, without making the rest of agents from this

6 set worse off, then the mechanism will be called the qualitative mechanism with respect to the given set. In this context some mechanisms are constructed, which correspond to the production system as well as to cumulative and innovative extensions of the production system being the part of Debreu economy. The criteria in question can be assumed as profit maximization. It allows us to discuss, in some particular cases, the way of creating the innovative and non innovative activity in production sphere. Summing up, the current neo Schumpeterian studies suggest that the Schumpeterian innovative evolution is governed by two different kinds of mechanisms. First, quantitative, price mechanism to determine the prices and quantities of goods in the state of Walrasian equilibrium corresponding to the circular flow, and, second, qualitative mechanisms typical for economic development explaining innovations. In this context the paper is aimed at verifying this hypothesis within the rigorous analysis of the Schumpeterian evolution based on a dynamical version of the Arrow Debreu model as well as Hurwicz's theory of economic mechanisms designing. In connection with the above, the paper is organized as follows. In the next section the basic model of the production system in the form of a relation system is presented. The third section includes the analysis of mechanisms corresponding to cumulative and innovative extensions of a production system. 2. The Model The Debreu private ownership economy (Debreu 1959) can be studied in the form of a multi range relational system (see: Malawski 2005; Malawski, Woerter 2006; Lipieta 2010; Ciałowicz, Malawski 2011) that consists of the combination of the production and the consumption systems. In this paper our study will be reduced to the production system as a component of the Debreu economy. The linear space with the standard scalar product, is interpreted as the dimensional space of commodities and prices. Let. Suppose that two groups of agents: producers and consumers operate in. The producers try to maximize their profits and the consumers tend to maximize their utilities on the budget sets. If there exists a price vector, such that both of them manage to realize their tasks at the price system and

7 the total supply equals the total demand, then, consequently, such vector p is called the equilibrium price vector. Now we precisely define production system. Let be a finite set of producers, be the correspondence of production sets, which to every producer assigns a nonempty production set be a price vector. of the producer s feasible production plans, Now, the model of a production sphere of an economy will be constructed in the two steps. First, a quasi production system will be defined, where the aims of producers are not specified. Thus the quasi type model can serve as the tool of modeling supply side of an economy to function under both perfect and bounded rationality paradigms. Second, a quasi production system will be expended to production one by adding some characteristics which define the goals of producers. Specifically, in our set up they will be coherent with the perfect rationality framework. Definition The two range relational system, is called the quasi production system. The set of all quasi production systems will be denoted by. Definition 2.2. If for the given price vector, then is called the correspondence of supply, which to every producer assigns the set of production plans maximizing their profit at the price system, R is the maximal profit function, where for every where. the two range relational system, is called the production system. The set is also called the set of optimal plans of producer.

8 Let us notice, that every production system is also the quasi production system (which will be denoted by ). The production systems are more complex structures than quasi production ones. In production systems, the producers aim at maximizing their profits at a given price system and technologies, what is not defined in a quasi production system. Now let us recall some definitions which will be of use later. Let. We say that if and only if for every. For, we define the natural projection on space according to the following. Now we can define two kinds of extensions for production system which correspond to the two forms of economic life studied by Schumpeter (1912), namely the circular flow and economic development. Let two production systems, be given. Definition 2.3. (cf. Ciałowicz, Malawski 2011, p. 46) A production system called a cumulative extension of a production system, shortly, if is 1., 2., 3. and for every a. b. c.. According to the definition, in a cumulative extension of a production system, new firms or commodities may appear. However, in particular, when,, and consequently, the respective projections are identity mappings, the above is ruled out and, what is more, neither new firms nor commodities are eliminated from the production process. All the old technologies are still used (Condition 3.a) and, with non decreasing prices (Condition 2), the firms optimal production plans remain optimum (Condition 3.b) and result in not less profit (Condition 3.c). All this means that the old core of the production sector is preserved implying that the idea of a cumulative extension can be interpreted as one modeling circular flow of the production sphere, as it follows the rules ascribed by Schumpeter. In other words the intended interpretation of a cumulative extension of a

9 production system suggests that the production systems P and should be the same, otherwise only small, quantitative changes are allowed. It is also evident that the producers play a purely passive, managerial role adapting actions to the given prices and technologies. In the sequel, in modeling such situation, it is assumed that, in a cumulative extension of a production system. We also consider a strong version of a cumulative extension of a production system with respect to its given component. Definition 2.4. A cumulative extension of a production system is said to be a strong cumulative extension with respect to: 1. the number of commodities if and only if, 2. the price system if and only if, 3. the producers if and only if, 4. the production set if and only if for some, 5. the supply correspondence if and only if for some, 6. the maximal profit if and only if for some. The concept of a strong cumulative extension goes beyond the idea of the circular flow, because of the possibility of opening new markets and firms (conditions 1, 3), introducing new technologies and profit maximizers (conditions 4, 5), and additional profit gained by producers (condition 6). Thus, it is a link to study innovations in the production sphere. What is more, in terms of economic mechanisms discussed in introduction, condition 1 corresponds to Mark I mechanism and conditions 1, 4, 5 to Mark II. However, these cases do not cover the whole complexity of innovative changes in the production sphere of an economy. Let us now consider a different form of a production system extension called an innovative extension (cf. (Ciałowicz, Malawski 2011), p. 48). Definition 2.5. A production system production system, shortly, if is called an innovative extension of 1., 2., 3.

10 a. b. c.. The producers satisfying condition 3 are called the innovators. In the further part of the paper, the set of innovators will be denoted by ( ). According to the definition, there may appear in a production system at least one new product or commodity. At the same time, the definition does not say whether new products are put out by brand new firms or by the ones already existing but modernized. All that is demanded here is that in a production system, there is at least one producer whose technological abilities go beyond the abilities of all producers acting within a production system P. Hence, the optimal (i.e. maximizing the profit) production plans of the producer cannot be reduced to the analogous plans being realized by the producers in a production system P. Moreover, although the prices of old products do not change, meaning that incumbent firms are not able to react at once to innovations, the fixed producer s maximum profit is greater than the one any of the producers in system can make. Thus, according to the Schumpeterian terminology, the producer can be treated as an innovator and entrepreneur who makes the profit resulting from his initiative and who should be given a leading role in a production system. It is easy to see, that, in particular, when, definition 2.5 covers at least four cases of five internal changes characterized by Schumpeter as development (Schumpeter 1961, p. 66), i.e. (1) the introduction of a new good condition 1, (2) the introduction of a new method of production condition 3.a, (3) the opening of a new market condition 1, (4) the carrying out of the new organization of any industry condition 3 as a whole. The above allows us to say that the innovation in the production system takes place if a new commodity or technology are introduced by innovator. Let us also notice, that in the spirit of the creative destruction principle the set can be entirely different from the set. 3. Mechanisms and mechanisms designing First, we will elaborate the structure of economic mechanism introduced by Hurwicz (1960). We start with the description all the components of this structure.

11 It is assumed that there is a finite number of agents in the analyzed economic process. The number of agents is denoted by, and set denotes the set of traders. All the elements that constrain the situation form so called environment denoted by ( ). The set consists of technological possibilities, preferences, endowments, etc. The desired outcomes of the process under study form the outcome space. It is assumed that outcome space is not empty ). Moreover, it is assumed that there is possibility to communicate within the agents. The definitions presented below are borrowed from Hurwicz, Reiter (2006). Definition 3.1. The triple, (3.1) where the set, called the message space, contains the messages available to communications within agents, the correspondence, (3.2) called the message correspondence, associates with each element of environment, the set of messages the function, (3.3) called the outcome function, to every message assigns the outcome, is called the economic mechanism. In some cases, the specific environment of every agent is considered. Denote by environment of agent. Then, environment is of the form the (3.4) Analogously, the message correspondence of every agent is analyzed. Let (3.5) be the message correspondence of agent. It associates with each element of environment, the set of messages. In this context a privacy preserving mechanism is defined:

12 Definition 3.2. The mechanism is the privacy preserving mechanism if for every, of the form of (3.4),. (3.6) If is a privacy preserving mechanism then correspondence (see (3.5)), depends only on the characteristics of agent. It is the consequence of nonexistence the arrangements within agents. It means that every agent knows only his own environment and on the basis of the above he sends his messages. Let s recall that a mechanism, in which prices of commodities are elements of the message space will be called the price mechanism. However, the price mechanism may have many consequences, like the improvement in the agents position. If in the given mechanism at least one agent from the given set will be better off due to a given criterion, without making the rest of agents (from this set) worse off, then this mechanism will be called the qualitative mechanism with respect to the given set. The criterion in question can be assumed as profit maximization. We say that a producer is better off in a production system than in production system P if and only if, The idea of economic mechanisms leads to the next challenge, namely designing economic mechanism (see Hurwicz, Reiter 2006). To define the problem we firstly recall the following definition: Definition 3.3. It is said that mechanism correspondence) if realizes the goal function (or. Now, the problem of designing economic mechanism is as follows: for a given (defined) goal function (correspondence), which to every elements of the environment assigns the desired outcomes, find a mechanism, or choose from the set of possible mechanisms, that realizes this goal function (correspondence). An example of the price mechanism, where the set of outcomes consists of producers and consumers optimal plans, is a production system. Some specific forms of the below presented mechanism the reader can find in Hurwicz, Reiter (2006).

13 Theorem 3.1. The production system is a privacy preserving price mechanism realizing the goal correspondence which to producers, represented by their production set assigns the optimal producers plans. Proof. Let a production system producer (see def. 2.2) is given, then we define the environment of every,, the environment, the message space and the message correspondence (see (3.2)): :, where (nonempty by def. 2.2). Putting set of outcomes : }, nonempty by definition 2.1, and outcome function (see (3.3)) we get that production system is the privacy preserving price mechanism, where message correspondence of every producer is of the form,. This mechanism realizes goal correspondence. (3.7)

14 Consequently, the cumulative or innovative extensions of a production system, which are also the production systems, are the privacy preserving price mechanisms realizing the goal function of the form (3.7). Now we define the mapping which will be essential for analyzing the mechanisms of cumulative and innovative changes in production sphere. The transition from a quasi production system to a quasi production system is a mapping (see definition 2.1), where and. The transition from the quasi production system to quasi production system can be identified with the pair. Hence, below we will denote the transition from a quasi production system to a quasi production system by. Analogically, the concept of transition can be applied to a production system. We show that the transition from a production system to its cumulative extension is the price mechanism. Let and be the production systems where and. It means that we consider a simplified case of a cumulative extension of production system, which will be generalized later. Theorem 3.2. If is a cumulative extension of (, then, the transition from production system to its cumulative extension, is the price mechanism. Proof. For, we define (3.8) and vector of the form. (3.9) Hence. (3.10) Put. Defining message space, (3.11) message correspondence (see (3.2))

15 , (3.12) set of outcomes, (3.13) and outcome function (see (3.3)), (3.14) we get, by (3.8) (3.14), that the transition from the production system to its cumulative extension is the price mechanism. It is worth emphasizing that in contrast to the proof of theorem 3.1 the price system p has been included as a component of the environment. Remark 3.1. The price cumulative mechanism described in theorem 3.2 realizes the goal correspondence, (3.15) where the set is defined by (3.11) and the set by (3.13). It means that correspondence assigns to production system its cumulative extension satisfying conditions (1) (3) by definition 2.3. In the further part of the paper, the price mechanism describing the transition from production system to its cumulative extension will be called the price cumulative mechanism. Informally speaking, in the structure called the price cumulative mechanism, the initial production system is identified with environment and its cumulative extension with the set of outcomes. The concept of a strong cumulative extension of a production system (see definition 2.4) shows that under additional assumptions the price cumulative mechanism might be a qualitative mechanism. Indeed the following theorem is true. Theorem 3.3. If and and is a strong cumulative extension of with respect to profit maximization, then the transition from production

16 system to this extension is price cumulative mechanism which is also the qualitative mechanism with respect to the set of all producers. Proof. The proof of the theorem goes on the same way of the proof of theorem 3.2, except definition of the message correspondence by condition (3.12). Here the condition (3.12) should be replaced by the formula. Now we focus on a more general case of price cumulative mechanisms. To elaborate mechanisms with more complex structure, firstly we introduce a production system with inactive producers. Let and be a production system (see def. 2.2). If production system satisfies,,, (3.16) then. Let us notice that a production system satisfying condition (3.16), further denoted by is the strong cumulative extension of production system with respect to producers. Production system differs from production system mainly in the existence of some producers (from set ) who are irrelevant for the action structure, because their only feasible production plan is. Producers from set will be called the inactive producers (in system ), contrary to producers from set, who will be called the active producers (in system. Keeping this terminology, production system will be called the production system with inactive producers based on production system. The existence of the inactive producers in a given production system, does not influence the optimal production plans and the maximal profits of the active producers. The inactive producers implement nothing important in a production system, so every production system might be interpreted as the production system with dormant producers (based on itself). Hence, in the further part of the paper, the components (namely the correspondence of production sets, the supply correspondence and the maximal profit functions) of production system will be denoted in the same way as the proper components of system. Hence, if is a production system, is the set of agents containing set, where agents from set are the inactive producers, then the production system with inactive

17 producers based on with set as the set of producers, will be denoted by. Theorem 3.4. If is a strong cumulative extension of production system with respect to producers, then the transition from production system to strong cumulative extension of production system, is the price cumulative mechanism. Proof. For producer, we define by (3.8), for producer (3.17) and vector. (3.18) Hence. (3.19) Put. Define message space, (3.20) message correspondence, (see (3.2)), (3.21) set of outcomes (3.22) and outcome function (see (3.3))

18 . (3.23) By (3.17) (3.23) the transition from production system (exactly from production system with inactive producers based on ) to its cumulative extension is the price mechanism. Remark 3.2. The price cumulative mechanism described in theorem 3.3 realizes the goal correspondence (3.15) where the set is defined by (3.20) and the set by (3.22). Like above, correspondence to production system assigns its cumulative extensions satisfying conditions (1) (3) by definition 2.3. The reasoning, analogous to the proofs of theorems , gives by definition 2.4, that the following is satisfied. Theorem 3.5. If is a strong cumulative extension of production system with respect to profit maximization, then the transition from production system (if, and from (if ), to extension is the price cumulative mechanism which is the qualitative mechanism with respect to the set of all producers. Now we present a different kind of mechanism whose set of outcomes includes innovations in a production system. The price mechanism describing the transition from production system to its innovative extension will be called the price innovative mechanism. In price innovative mechanism, the initial production system is identified with environment and the innovative extension of the initial production system is the outcome. Let and be the production systems (we do not assume conditions ). Then we have Theorem 3.6. If, are production systems and, then, the transition from production system to production system, is the price innovative mechanism which is the qualitative mechanism with respect to the set of innovators. Proof. Let be an innovative extension of production system, be the production system with inactive producers based on production system and let be the production system with inactive producers based on production system. If, then the environment of agent is defined by, (3.24)

19 if (if ) then. (3.25) Let. Then Hence, the environment of the transition from production system to the innovative extension of system, is defined by, (3.26) Notice, that set, by condition 2 of definition 2.5, is not empty. Let. Put message space (see definition 3.1) of the form:, (3.27) message correspondence, (see (3.2)), (3.28) set of outcomes (3.29) (nonempty, by definitions 2.1 and 2.5), and outcome function (see (3.3)). (3.30) By (3.24) (3.30), the transition from production system to the production system with inactive producers based on the innovative extension of production system has its performance as the mechanism. By definition of the set of outcomes (see (3.29)) we get that this mechanism is qualitative with respect to the set of innovators. Let us notice that, if in the definition of the message correspondence (3.28) there is a producer for whom, then the producer will become the inactive producer.

20 Hence in the spirit of the creative destruction principle, the elimination of existing products and organizational structures take place and they are replaced by the new ones. The qualitative character of a strong cumulative extension with respect to profit maximization of a production system should be explained. To this aim we use a specific version of innovative extension of production system such that it preserves a cumulative extension in the set non innovators (cf. Ciałowicz, Malawski 2011). Let production system and its innovative extension are non empty. be given. Let s assume additionally that sets Definition 3.4. An innovative extension of production system production system preserves a cumulative extension in the set of non innovators, if is a cumulative extension of production system. It means (by definition 3.4) that the production system preserves a cumulative extension in the set of non innovators, if production system restricted to non innovators is the cumulative extension of the production system restricted to set. This way two mechanisms, i.e. price cumulative and price innovative studied up to now separately can be combined as price cumulative/innovative mechanism. Now it is easy to see (by definition 3.4) that for the innovative extension preserving a cumulative extension the following theorem is true. Theorem 3.7. The price innovative mechanism which moves a production system to the innovative extension of production system preserving a cumulative extension in the set of non innovators, is the qualitative mechanism with respect to the set of all producers. Let us notice that the above qualitative mechanism with respect to the set of all producers does not eliminate any producer from the market. 4. Conclusion

21 In modeling the transition from a production system to its cumulative (theorems ) or innovative extension (theorems ) we deal practically with composition of mechanisms. Indeed, the environment and the set of outcomes of that mechanism form production systems which also have the structure of the mechanism (theorem 3.1) in the meaning of definition 3.1. Moreover, the rigorous analysis of the Schumpeterian evolution in the Arrow Debreu set up suggests that the structure of mechanisms of this innovative evolution is more complex and diversified than in exploring them in the current neoschumpeterian literature. Searching for the best or at least the better than mechanisms presented above with innovations as the set of outcomes (generally or in the particular cases) is interesting in the light of the role of innovations in the Schumpeterian economic development, and it still remains within our research plans. Acknowledgment This work was supported by Polish Ministry of Science and Higher Education - Grant NN References Aghion P, Howitt P (1992) A Model of Growth through Creative Destruction, Econometrica 60, p Aghion P, Howitt P (1998) Endogenous Growth Theory, MIT Press, Cambridge, Mass. and London Andersen E.S (2009) Schumpeter s Evolutionary Economics, Anthem Press, London Ciałowicz B, Malawski A (2010) Demand driven Schumpeterian innovative evolution, the paper presented at the 13th International Schumpeter Society Conference, Aalborg, June, Denmark Ciałowicz B, Malawski A (2011) The role of banks in the Schumpeterian innovative evolution an axiomatic set up, in: Catching Up, Spillovers and Innovations Networks in a Schumpeterian Perspective, A. Pyka, F. Derengowski, M. da Graca (eds.), Springer, Heidelberg, Dordrecht, London, New York Day R.H (2007) The mechanisms of economic evolution: completing Schumpeter s theory, in: Elgar Companion to Neo-Schumpeterian Economics (2007) H. Hanusch, A. Pyka, Edward Elgar, Cheltenham, UK, Northampton, MA, USA, pp eds., Debreu G (1959) Theory of value. New York, Viley Duffie D (1988) Security Markets, Stochastic Models; Academic Press

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