The Reconstruction of the Theory of Production and Preference

Abstract

This article was originally designed to comment an unprecedented bibliometric study of evolutionary economics by Hodgson and Lamberg (2016). However, Hodgson only reported the historical trajectory and the current situations of evolutionary economics initiated by since Nelson RR Winter SG Nelson and Winter (1982). He gave an insightful consideration about the raison d’être of evolutionary economics. According to his opinion, evolutionary economics is still failing to equip the core theories. We naturally agree with his remarks on the current situations around evolutionary economics. We take this opportunity to squarely address the subject about how to insert the core theories into evolutionary economics. We argue that the alternative candidates to be replaced with the main stream core theories are in the following ordering:(1) the theory of production to invalidate myopic optimization, (2) the theory of preference to invalidate myopic optimization, (3) the SMD Theorem to invalidate invisible hand, and (4) the market mechanism to invalidate the efficient market hypothesis. Needless to say, the alternative theories shown in these arguments imply the reconstruction of economics. Finally, we address the current analysis of bibliometrics.

1 Toward a Reconstruction Plan of Economics

The work of Hodgson and Lamberg (2016) is an unprecedented work on bibliometric analysis of evolutionary economics in its range and scope. First, this analysis retrieved 8,474 articles. Second, it separated the development of the time profiles in the subsequent five year periods: 1986–1990, 1991–1995, 1996–2000, 2001–2005, and 2006–2010. Thus this study was able to refer to the present trend and the fashions around the different and cross disciplines. In this sense, the observed facts that Hodgson and Lamberg discovered seem to be almost all reasonable from our institutional overview. Furthermore, in connection with the author’s great intelligence, this bibliometric analysis has finally given an insightfully useful perspective to reconstruct economics for us.

In spite of wider penetrations/success over diverse fields except for economics over years, as Hodgson and Lamberg (2016) stated, it must be true that almost no one was successful in replacing the core engine with the main stream’s one. His suggestions fortunately coincide with our original plans appeared in Aruka (2017a) to revive economics ontologically. This is the reason why I decided to comment on this article. As Hodgson and Lamberg recommended, the raison d’être of evolutionary economics must be the innovation of the core theories of the traditional economics. In other words, the belief of the main stream economics is that everything is at best. On the contrary, our belief is that everything is evolving. We continue to comprehend that “evolutionary economics” as “economics to be evolved”. In this point of view, first of all, the reconstruction of political economy mattered. In this sense, Japan Association for Evolutionary Economics (JAFEE) adopted another purpose against the standard definitions of evolutionary economics at the very outset of the foundation. She was sustaining her effort to construct the core theories, such as the theory of value, as an alternative to the main stream. Based on that, Shiozawa (2007) and (2015) used to sub-tropical algebra to provide a general proof of international trade theory of value in the case of the transaction of three commodities including intermediate goods among three countries. This is the great innovation since Ricardian theory of value. Due to Shiozawa’s contributions, the existing theory of comparative cost and its marginal version has lost their justification.

Once upon a time at Cambridge UK, Kaldor (1955) purposed constructing “alternative theories” of distribution in order to replace the neoclassical marginal productivity theory. He and his followers’ attempts have never been successful at rejecting the existing core theory of marginal productivity with their doctrine. The Sraffian theory of value survived longer in being an alternative candidate in value and distribution. However, the classical principles of political economy has been throughly replaced with equilibrium economics. Equilibrium economics always depends on the belief that everything is at best, even in serious depressions. It is interesting to note that the Leibzian idea of everything is at best was criticized by Candide in [?].

We regard evolution as an idea of the antithesis against the everything-at-its-best proposition. The real world of evolution is then irrelevant to the grand design of God. Biological evolution does not contain an original intention to optimize the whole system to coordinate other organs consistently, at least at the outset of mutation. In this context, evolution is rather motivated by creative coincidence in Mainzer (2007) as a consecutive source of innovation. The same mechanism that causes the skin’s blood to condense causes blood condensation in the brain. In general, evolution does not give any normative optimality at all. Thus the use of the idea of evolution becomes a strong support to the construction of the core theories of production and preference without the idea of optimization and human rationality.¹

The main stream economics updated herself to survive. Despite of Hirofumi Uzawa’s (1928–2014) expectation, the rational expectational hypothesis internationally spread in the economics departments. As demonstrated in Aruka (2015a), it is noted that the idea of bounded rationality is considered the raison d’être of rationality. That is to say, bounded rationality is not any modification of rational economics at all but a refined form of it. Reality is always skewed and polluted because the secular world is not the world of God if we are faithful to the tradition of Thomas Aquinas.

2 The Preliminaries to Replace the Core Theories of Economics

We first mention the recent prospects to innovate the core theories of economics. Hildenbrand (1981, 1994) has already given us a set of great hints to innovate the theories in both production and consumer preference. Although my exposition is not exhaustive, in the following, we will refer to how several eminent scholars prepared for the alternative theories. In my opinion, the present time is overdue to innovate on the traditional ideas. Now, we mention the main points and illustrate them very shortly.

• 1. Theory of production to invalidate myopic optimization

  • a. Hildenbrand (1981) and the zonotope set of production

  • b. A fallacy of the specification on a particular technique and the zonotope set of the international trade

• 2. Theory of preference to invalidate myopic optimization

  • a. A sufficient condition to confirm the positive income effect in Hildenbrand (1994)

  • b. The utility theory of imperfect identification in Luce (1959) and Saari (2005)

• 3. The SMD Theorem to invalidate the invisible hand

  • a. A generalization on SMD theorem in Saari (1992)

  • b. Aggregation and choice paradoxes

• 4. The market mechanism to invalidate the efficient market hypothesis

  • a. The U-Mart system as an ontological construction of the exchange system by AI

  • b. Class 4 of Fully Random Iterated Automata (FRIA)

2.1 Theory of Production to Invalidate Myopic Optimization

2.1.1 Hildenbrand (1981) and Zonotope Set of Production

There were few to understand the importance (Hidldenbrand, 1981). Dosi (2016b) and Scuola Superiore Sant’Anna noticed its importance. Hildenbrandt proved that the short-run production function was empirically measured did not fulfil the contours of the neoclassical production function but, rather, the form of zonotopic set of the discrete basic activities of production in a Minkowski space. This set geometrically forms a shape like a rugby ball. Most importantly, the use of zonotopic production set permits us to analyze the recursive property of activities/techniques. Either the neoclassical school or Leontief’s input-output analysis were not able to implement a recursive process of production in general.

In short, “[l]eaving the special space restricted by the traditional production function, we can observe various possible combinations to accept institutional effects. Broadening the space, from where technology can be chosen, implies to introduce all possible combinations of activities” (Aruka, 2017a, 410). Dosi (2016a) successfully applied this kind of production set to technical progress by calculating the volume of this set. This must be a great progress in the theory of production to invalidate the myopic optimization in production.

2.1.2 A Fallacy of Specialization on a Particular Technique and the Zontopic Set of the International Trade

A good example of the benefit of specialization in a particular techniques is the merger or an integration of two basic activities/plants. However, as Aruka (2017b) demonstrated, this does not depict the effect of specialization in the international trade between two countries. So far, no one was successful to formulate a general model of international trade of intermediate commodities. This was due to the different rates of distribution among both countries are complicatedly intermingled with the country-wise profile space and the commodity-wise profile space. This interrupted us in confirming equilibrium prices. Fortunately, Shiozawa (2007 and 2015) gave historically great contributions to find equilibrium in this case. By this proof, it was verified that we never required a particular specification of commodity production for the world efficient production.

This theorem permits us to model the international trade between the same industries, even in equilibrium. Thus we have also been released also from the neoclassical bindings in the international trade theory. It is also noted that the world production system of three countries and three commodities are exactly the zonotopic set, as shown in Aruka (2017b).

According to sub-tropical algebra, let denote the max-times semi-ring a set of all positive numbers by $a \oplus _{\max } b = \max \{a,b\}$ and $a \odot b = a \cdot b$

On the other hand, we describe Ricardian economy  of M countries and N commodities by

$$\begin{aligned} \mathcal {E} =\{A=[q_{ij}], q=(q_i) \end{aligned}$$

(8.1)

Let the wage-price system be

$$\begin{aligned} (w, p) =(w_1, \ldots , w_M; p_1, \ldots , p_N) \end{aligned}$$

(8.2)

where $w_i$ is the wage of Country i, $p_j$ is the price of commodity j. It must then holds in economic point of view:

$$\begin{aligned} w_i a_{ij} \ge p_j, \forall i \in [M] \; \text{ and } j \in [N]. \end{aligned}$$

(8.3)

Expressing this in terms of subtropical matrix form, it follows:

$$\begin{aligned} w \ge p \otimes _{\max } B. \end{aligned}$$

(8.4)

Here B is a transposed matrix whose (j, i) component is the reciprocal of $1/a_{ij}$.²

Shiozawa (2015, 188) called the frame simplex  such as Fig. 8.1.

Vertices of price simplex represent commodities. The hyperplane for max-times semi-ring separates the simplex into three domains of different types. Correspondingly to this illustration, we define the wage simplex, i.e., subtropical hyperplanes with the min-times semi-ring  as detailed later.

Fig. 8.1

Max-times hyperplane of exotic algebra *The figure was designed by the author by referring to Shiozawa (2015, 188)

2.2 Theory of Preference to Invalidate Myopic Optimization

2.2.1 A Sufficient Condition to Confirm the Positive Income effect in Hildenbrand (1994)

The great success of Pareto-Slutsky equation (Slutsky, 1915) in analyzing consumer demand was given as one of the raison d’être of the main stream economics. But this analysis never was completed to derive the law of consumer demand, which says that the demand decreases as the price increases. Despite being able to confirm the direction of income effect, surprisingly, many insisted to call Pareto-Slutsky equation the law of demand with a studied nonchalance. This attitude was considerably frivolous. The usual practice to justify Slutsky equation is to impose the assumption of gross substitutability (GS). This assumption is equivalent with respect to the income effect being negligible, despite of the discovery of Giffen effect. It is quite easy to find Giffen effect in the simplest exchange economy of two commodities if the budget is presented in terms of the real endowments of the economy and not using income in terms of monetary units, as shown in Aruka (2015b, 49). That is to say, the Giffen effect is not an exceptional case. The followers of the General Equilibrium Model were too idle to look for a sufficient condition to establish the direction of sings of the income effect.

Hildenbrand (1994) has smartly given several sufficient conditions to establish the positiveness of income effect. One of these conditions is that as income x increases, the spread of demand f, which is the sum of variance and square mean of demand f, increase. In other words, the consumption patterns of the richer class become diverse. Here it is useless to apply a monotheistic idea of myopic optimization. This suggests a great conversion from the neoclassical method of homogeneous agents to the new method of heterogeneous gents. The idea of mathematical closeness both in the commodity space and in terms of personal satisfaction must be a fallacy if we envisage the way in which patterns of consumption will be formed in reality. Outside sophisticated mathematical modeling, we need a different idea of consumption such as shown in Aruka (2015b, 51–62).

2.2.2 The Utility Theory of Imperfect Identification in Saari (2005)

The alternative theory of choice must be the probabilistic choice theory of utility given by Luce (1959). However, the generic usage of this theory is almost for a practical one to predict the consumer demand of transportation or sightseeing, including mode choices. This method became well known by McFaden’s Nobel Prize in the academic world. When Luce’s utility is algebraically considered, this is currently dealt with as the multi-logit model, which can illustrate either successive choices or mode choices. Alternatively, it may be possible to derive the multi-logit model type utility from the neoclassical assumptions.³ A restrictive assumption like independence from irrelevant alternatives is often imposed. However, almost all of such an attempts depend on the assumption that the set/subset of preference are universal.

However, the original version of Luce (1959) does not need to assume universal preference set. In a serial paper of topological geometrical investigation, Saari (2004) showed that imperfect identification of alternative issues could generate paradoxes that are sensitive to a small change of circumstances. Given the preference set/subsets, it turns out that there are trivially formed probability profiles and ranking profiles. Either in voting or commodity exchanges, the ranking result of candidates depends on a relative position of the preference subset in relation to the probability profile.

It is noted that the ranking results of candidates may be changed because agent’s preferences changes. His/her position never changes but a preference subset to which he/she belongs changes either by forming a coalition of several agents or by the new introduction of an alternative issue, for instance. In short, his/her relative distance from the candidates changes. Thus such a change of a relative position can be imperfectly identified to give rise any counter-intuitive paradox, thus contradicting myopic optimization.

3 Durlauf’s Random Preferences Influenced by Social Interaction

Luce=Saari’s studies will be detailed in the next chapter. In this subsection, we illustrate how the main stream economics deal with a random choice but without leaving methodological individualism.

3.1 Individualistic Utility

Durlauf (1997) and (2000) in principle never part with the methodological individualism in orthodox economics but depends essentially on the above formulation of interactive network of agents when he seriously is concerned with the effects of heterogeneous subgroups on individualistic choice. His case may be regarded as an application of the above minimization problem in view of social costs. In particular, he utilized the logistic distribution as a specific form of random preferences to derive a probability distribution of choice as resulting some similar idea of Gibbs distribution.

In Durlauf’s idea the utility V of agent has been construed as a union of three different types of utility for individual $i \in I$: private utility $\omega$ as private and deterministic utility; social interaction S as social and deterministic utility; random shock $\epsilon$ as private and stochastic utility.

$$\begin{aligned} V_i = \omega _i + S + \epsilon _i. \end{aligned}$$

(8.7)

3.2 Private Utility

Binary Choice.:

We indicate the outcome of binary choice whether an individual $i \in I$ can be a painter or a musician by $\omega _i$, and its support by $\Omega _i = \{ -1, 1\}$. $\omega _i =-1$ means painter, $\omega _i =1$ a musician, for example.

Heterogeneous Agent.:

Agents playing in the system are heterogeneous, whose property is expressed as $X_i$.

Random Shocks.:

We assume independent stochastic disturbances among heterogeneous agents. In the case of $\omega _i=1$ which suggests to be a musician, if utility of random shock $\epsilon (1)$ is relevant, utility of choice for i to be a musician will increase. $\epsilon (1)$ exhibits unobservable musical talent, while $\epsilon (-1)$ unobservable artistic talent.

There may be several possibilities for each individual. A particular one will be realized by a probabilistic distribution of random shocks. We for convenience assume a logistic distribution of the probability which the difference of the random shocks on a binary choice $\epsilon _i (-1) - \epsilon _i (1)$ can be within a particular z.⁴

$$\begin{aligned} \text{ Prob }(\epsilon (-1) - \epsilon (1) \le z) = \frac{1}{1 + e^{-\beta _i z}}. \end{aligned}$$

(8.8)

The density function of logistic distribution is symmetrically bell-shaped. $\beta _i$ indexes unobservable heterogeneity. The larger $\beta$ is, the lower a probability which the difference of utility falls within a certain value of z may be.

3.3 Social Utility

Agent i has his own belief on choices of the other members of the group I based on a certain information: $\mu _i^e (\omega _{-i} | F_i)$.⁵ Individuals are inclined to conform to or deviate from others’ choices of the group. When all the members have the same choice, social utility for individual is of zero. When each deviates from each other, social utility for agent i is negative.

$$\begin{aligned} S(\omega _i, X_i, \mu _i^e) = - E_i \Bigg (\sum _{j \ne i} \frac{J_{i,j}}{2} (\omega _i - \omega _j)^2\Bigg ). \end{aligned}$$

(8.9)

Weight on interaction$J_{j} (X_i)$ simply is denoted by $J_{i,j}$, as relating the choice for i to one for j. Thus the square $(\omega _i - \omega _j)^2$ will give conformity effects. Hence social utility S is just a subjectively expected value as based on agent i’s belief of the distribution of social interaction $[J_{j}(X_i)]$.

3.4 Derivation of the Probability on Individual Choice

The decision process of individual i is as follows:

$$\begin{aligned} \omega _i = \arg \max _{\omega \in \{-1,1\}} V(\omega , X_i, \mu _i^e, \epsilon _i). \end{aligned}$$

(8.10)

Linearization of Utility Function

A linearized function of utility is given where a slope is $h_i$, an intercept $k_i$⁶:

$$\begin{aligned} u(\omega _i, X_i) = h_i \omega _i + k_i. \end{aligned}$$

(8.11)

Under a certain set of information, a choice $\omega _i$ is generated only if the gain of the choice is greater than the opposite. A conditional choice $\mu (\omega _i)$ is the probability when the gain of $\omega _i$ is greater than the gain of $-\omega _i$.

$$\begin{aligned} \mu (\omega _i|X_i, \mu _i^e) = \mu (V(\omega _i, \epsilon _i(\omega _i))>V(-\omega _i, \epsilon _i(-\omega _i)). \end{aligned}$$

(8.12)

Taking account of the above (8.7) and (8.9) into (8.12), it follows:

$$\begin{aligned} {} & {} \mu (h_i \omega _i - E_i \sum _{j \ne i} \frac{J_{i,j}}{2} (\omega _i - \omega _j)^2 + \epsilon _i (\omega _i) \nonumber \\ {} & {} > - h_i \omega _i - E_i \sum _{j \ne i} \frac{J_{i,j}}{2} (-\omega _i - \omega _j)^2 + \epsilon _i (-\omega _i)) \nonumber \\ = & {} \mu (h_i \omega _i + \sum _{j \ne i} J_{i,j} \omega _i E_i(\omega _i) + \epsilon (\omega _i) > - h_i \omega _i - \sum _{j \ne i} J_{i,j} \omega _i E_i(\omega _j) + \epsilon _i (-\omega _i)) \nonumber \\ = & {} \mu (\epsilon _i (\omega _i) - \epsilon _i (-\omega _i) > -2h_i \omega _i - \sum _{j \ne i} 2J_{i,j} \omega _i E_i(\omega _j)). \end{aligned}$$

(8.13)

Since the random shocks are assumed to be subject to the logistic distribution (8.12), the probability distribution on the choice of individual i, i.e., $\omega _i$ can be solved:

$$\begin{aligned} \mu (\omega _i | X_i, \mu _i^e) \propto e^{\beta _i h_i \omega _i + \sum _{j \ne i} \beta _i J_{i,j} \omega _i E_i(\omega _j)}. \end{aligned}$$

(8.14)

The joint probability distribution of the whole population on the choice of $\omega$ then is:

$$\begin{aligned} \mu (\omega | X_1, \ldots , X_n, \mu _1^e, \ldots , \mu _n^e) = \Pi _i \mu (\omega _i | X_i, \mu _i^e) \propto \Pi _i e^{\beta _i h_i \omega _i + \sum _{j \ne i} \beta _i J_{i,j} \omega _i E_i(\omega _j)} . \end{aligned}$$

(8.15)

3.5 A Generalization of the SMD Theorem to Invalidate Invisible Hand

3.5.1 A Generalization on SMD Theorem in Saari (1992)

A serial theorem argued by Sonnenschein (1972, 1973), Mantel (1974), and Debreu (1974) is called the SMD Theorem. SMD states that the aggregate excess demand function can always be constructed only by the classical assertions about continuity, homogeneity and Walras’ Law. There is no need any other general properties. Hence the SMD theorem, at first look, seems the most robust theorem to guarantee an equilibrium in any circumstance to justify the invisible hand. Furthermore, many are convinced that the Walrasian market mechanism are universal almost in any circumstance. Due to this conviction, many are willing to generalize this result in the financial market. This enthusiasm still supports the belief of efficient market hypothesis (EMH) in financial markets.

On the other hand, Saari (1992), in the context of the original Luce theory, carefully inspected this theorem to generalize this theorem and provide an alternative proof to show that to prove a general equilibrium of the market is essentially equivalent to prove the winning process of voting.

[Saari’s generalized theorem (Saari, 1992)] asserts that almost anything imaginable can happen! Examples can be created where Smith’s invisible hand grasps nothing. Using only diagrams of the form used in a freshman economics course, examples can be created that support Adam Smith’s story for all pairs of commodities, but, with the same agents, the price dynamic is chaotic for all triplets, then all sets of four commodities have several equilibria where some are attractors and other are not, and then... (Saari, 2015, 134)

3.5.2 Aggregation and Choice Paradoxes

Roughly speaking, any variation of a purchased commodity changes a next optimal position of the individual. The purchase of “IPad, for instance, can radically change customer demand structure (Saari, 2015, 134)”. Suppose that a subset where this change occurs is separated from a larger subset of commodities. We do not need to know the aggregated excess demand of any larger subset. In short, the aggregated demand function never specifies how a new equilibrium will be generated by what kind of adjustment process. The same reasoning will also be valid to theory of vote, or social choice. The latter theory is often faced with a dead end like voting paradoxes.

Additionally, here, the method of aggregation matters. It is quite interesting to see that an intervention of aggregation rather causes to reflect the peculiarities of the particular rule but not the real data (Saari, 2015, 125).

“Positional elections” (or “positional election rules”)⁷is the name attached to election rules where ballots are tallied by assigning points to candidates according to how they are positioned (i.e., ranked) on the ballot. The widely used “plurality vote,” where each voter votes for one candidate and the candidate with the largest number of votes wins, assigns one point for a top-ranked candidate and zero for all others. (Saari, 2004, 232)

To be sure, we are used to use an extreme evaluation like plurality vote. Therefore, a small positional change of candidates will give rise chaos.

3.6 The Market Mechanism to Invalidate the Efficient Market Hypothesis

3.6.1 The U-Mart System as an Ontological Construction of the Exchange System by AI

In the above subsections, we referred to the invalidation of myopic optimization in theoretical point of view. Economics depicted the market as the integration of production and preference. General equilibrium theory is simply one of the pictures to sketch the market economy. Thus, it must be indispensable for us to argue how the market mechanism really is working, without introducing myopic optimizing agents.

The AI market is no longer a virtual market. This is actually a real market. It is not realistic to argue about the stock exchange without referring to the high frequency transaction(HFT). The computerization of the market does not guarantee a purity of competitiveness of the market. In April 2018, Japan Exchange decided to adopt the policy of the advanced registration of the HFT agent. This may not be a natural outcome for the general equilibrium and/or the invisible hand’s supporters.

The AI market based on algorithm rationality is evidence that the market can accept another principle different from human rationality. Constructing the AI market system is an attempt to construct the market transaction without resorting to the neoclassical principles. In the real market, dominant agents are the so-called technical agents rather than the myopic optimizing and/or rational expectation agent. On the contrary, we can employ the AI market simulator like the U-Mart System ⁸ to run our experiments without damaging the functional terms structured by existing entities of the real market.

3.6.2 The Role of Randomness/No-intelligence in the Market

In an actual market, participating agents are sending orders either randomly or non-intelligently, even though they depend on their own unambiguous strategies. It is also noted that purely random orders often bring the best performance in the market. Thus the market system may be a system with many redundancies.⁹

In this century, it turned out that the automaton of the Wolfram rule 110 fulfilled the criteria of Turing completeness. This is among major problems that continued to interest Stephen Wolfram.¹⁰ Now, the rule 110 and the similar rules are being explored. The cellular automaton has a simple structure, but it is potent in terms of generating complicated behavior like Class 4:

Class 4:

 Nearly all initial patterns evolve into structures that interact in complex and interesting ways, with the formation of local structures that are able to survive for long periods of time.¹¹

Now, we accommodate the idea of Class 4 to the market system. In the market, at first, the participants of various types are locally formed, and then they mutually interact in complex and interesting ways, with the formation of local structures that are able to survive for long periods of time. The Wolfram ICA simulations examined a kind of attractor formation. Conversely, in the market experiment of the U-Mart system, we often observe a certain correspondence generated in a relationship between an initial strategy configuration and its final performance configuration. A final performance is represented by some special form, like attractor. We can examine whether a final performance configuration is wired to its initial strategy configuration or not. A similar set of strategies, in spite of different environments and experimental modes of the market, may fall into a similar result. Irrespective of the rigid assumption of myopic optimization or rational expectation, we can run the analysis of the market.

4 Micro-based Theories for Evolutionary Economics

4.1 Methodological Biases

Also In the financial society established since the 1990s, economists has strangely learned to generalize the “efficienct market hypothesis” to financial markets.

Mysterious situations have thus been dominant as follows:

• 1. First of all, the expected utility theory is often used in financial theory. But the basis of this theory is given by Daniel Bernoulli. Its basis does not have any intersection with the classical utility theory at all. In the first place, the domain for both theories is different. The argument of expected utility is money or income. On the other hand, the argument of the classical utility theory is usually defined as physical quantities or services enjoyed. Moreover, in addition to utility theory and expected utility theory, mode selection based on Luce’s discrete selection theory in Luce (1959) provides a sequential utility theory. However, as we will see later, an alternative geometrical analysis such as found in Saari’s works proves that traditional utility functions do neither contribute to selection theory, nor can Luce’s selection theory contribute to unique decisions. The so-called social choice theory such as Amartya Sen is also under pressure to change significantly (Lin & Saari, 2008; Saari, 2006).

• 2. In financial markets, especially in the artificial market for securities or stock trading, we know that so-called zero intelligence is the agent that is not defeated. We know a reason why this agent is a winner, but random agents are not intellectually rational in terms of traditional economics. Interestingly, however, it is not defeated by an intellectually complex agent. This is what economics, which insists on intellectual optimization, should explore properly.

• 3. In addition, Mirowski (2007) has already pointed out that when the market is stacked on top of the market, the original market meaning is lost. Financial markets are typical of hierarchical markets. In such a hierarchical network, the working of the individual market will have to be examined by computer simulation. However, there is little sign that economics will solve the problem of hierarchical interaction. Now financial markets themselves are more essential to technical efficiency calculations than economic efficiency calculations. With the advent of HFT, microsecond payments are now made, which means that physical distance determines the settlement. Originally, the Zaraba rule was a time-first principle, but the Zaraba rule itself was custom to stabilize the transaction. But under HFT, the economic meaning of the rule has changed, and even the traditional stock market rule of membership has virtually changed with the lending of high-performance servers on stock exchanges.

4.2 Encounters with New Science and Innovation in Economics

As Arthur (2009) pointed out, the aircraft structure of modern state-of-the-art fighters does not incorporate stability for fail-safe. It increases the room to become uneasy structurally of the aircraft, and enables agile action. This is because ICT technology can restore instability instantaneously.

Relaxed Static Stability

This observation may call us “the relaxed static stability”. Depending on the idea of “fail safe”, structural stability has been important in designing planes and ships up to now. On the contrary, a modern stealth fighter like Lockheed F35 is designed by the idea of relaxing “static stability”. It is well known that bicycle behaves much more quickly than tricycle, and tries to restore its instability by resort to its counter motion. The counter motion will be controlled by computational powers. Thus we expect that some nonlinear effect is implemented behind the flash crashes, always accompanying a countervailing power (Fig. 8.2).

Fig. 8.2

Lockheed Martin F-35 Lightning II, Wikipedia

As symbolized by Germany’s Industrial Policy Industry 4, it goes without saying that now that ICT is essential to infrastructure, the scope of economic policy applications needs to be reconsidered. However, there is only a great emotional backlash against Arthur’s theory of technology, and few people, like Arthur, seriously consider the true situation of modern society. It was not actually realized from the theory of financial and economics the reason why the option market  has publicly become opened again at the end of the last century. In order to reopen it, technological advances that can accurately calculate high risk by polynomial calculation, that is, a dramatic increase in computer computing power, were indispensable. That’s Arthur’s point, but in a recent example, the same is true of the emergence of Bitcoin in 2008.

Bitcoin assumes a Peer-to-Peer payment network based on an open source protocol. The task of booking transaction settlements for this kind of other person is not possible without the establishment of Git-hub technology. The Git-hub system has made it possible to create a new system without server management, but bitcoin is only feasible with such a technical foundation. Of course, Bitcoin also has many properties in common with other regional currencies and recently emerged digital coins, but it should not be forgotten that they are inherently different. Bitcoin can be an important entity of the society’s economic system, and so it will also carry the risk of system destruction. Bitcoin is a prime example, but for survival, the “computability problem” always comes with it, which essentially leads to the “Turing problem”. It should be noted that it is not a problem of economics by any way. Economics often emphasizes economic decisions, but it will be necessary to explore how much economic decisions contribute to “economic problem” decisions.

Wolfram’s New Science

Mathematica’s Stefan Wolfram’s New Kind of Science (Wolfram, 2002) is not very famous in the world of economics. John Conway simplified the Turing problem with life games and contributed to the development of cellular automata (see Gardner (1970)). Cellular automata promoted the rise of computer science in the 1990s with the development of John Holland’s complex adaptive systems and genetic algorithms. With the spread of agent-based modeling ABM, computational electronics is now one of the standards. The important point here is that these developments follow Turing traditions properly. ABM can be criticized by economics for its lack of a theoretical basis. However, ABM itself has a mathematical basis in the Turing sense if the algorithm can be proposed neatly like the bucket relay algorithm of the gene. It is equally important that interactions containing nonlinear interactions that cannot be solved by mathematical analysis have no choice but to simulate in ABM.

In the 1930s, the theoretical basis of genetic research was almost established. The problem was that it was computationally possible, as was the optional risk calculation, so it would take more than 50 years to demonstrate the theory. It will be condesteming in the reader’s understanding of the theory of light of Wright (1932) and Turing (1952) about the morphological model (Aruka, 2017a).

In the era of classical political economy, the method was ontological centric. But, in the last century, the method was replaced with the belief of so-called hypothesis testing, in particular, efficient-market hypothesis testing. The idea of hypothesis testing generally presumes that an object to be tested is really unknown. This is reminiscent of agnosticism. In statistical test, while there is some unknown process f in itself, there are supposed to be alternative statistical models to represent an imprecise but a supposed nearly true process. A best practical way to detect f may be then a comparison of alternative statistical models $\{g_1, g_2\}$ by the famous Akaike Information Criterion (AIC), for example.

• 1. This kind of statistical customs, however, is categorized within null-hypothesis orientation.

• 2. The acceptance or rejection does not mean any visualization of the unknown process under test.

The statistical test of efficiency can simply judge whether there exists market whose attribute is more efficient or not. Even if a model with statistically higher efficiency were chosen, the test would never state how such an efficient-market was constructed. The test only states that an unknown market might be efficient. Thus, even if any stochasticity should be incorporated into some general equilibrium model, the stochastic parts would make the system neither realistic nor empirically verified, so that a dynamic stochastic general equilibrium model will not be bestowed reality. An unknown market is not resolved at all. In ontology, on the contrary, how to exist matters. An entity specific to a particular domain is constructed by ontological blocks, which are either virtual or physical existing in the domain.

5 Ontological Method of Production Theory

Ontology in information science, a fortiori, makes us build some ontological construction, whether virtual or physical, using artificial intelligence. Due to a drastic change of the social system, the efficient-market hypothesis has lost its validity. It seems to us unavoidable to restore ontology to economics. However, this does not mean to return to the approach of classical political economy again. We should learn a new ontology defined in information science. It is noted that entity is already embodied as operator/function in Wolfram’s Mathematica. Entity can be of functional form in information science.

As artificial intelligence is employed to run entities, it is easier to argue their classes, instances, attributes, their relationships, and functional terms among complex interactions which are mathematically not identified in advance. Thus, an ontological reasoning on the market domain easily brought an idea to apply agent-based modelling (ABM) to emulate an actual entity such as actual market, macro economy, and so on. Among them, a market simulator such as U-Mart system  can emulate an actual exchange market, potentially in some cases become a miniature of an actual market itself. That is to say, the artificial market is an ontological experiment in the market domain to reproduce a market. To employ ABM, without restricting excessively a domain under research and/or damaging functional terms structured by existing entities, we will be able to reproduce some complicated interactions.

To part with the efficiency hypothesis in the domain of production, some ontological consideration may be required. A custom sanctified by long practice in production was to presuppose optimizers and their environments fitted to them. As Hildenbrand (1981) noticed, the traditional production function is described as the projection of Y on the input space for some X of the input space is V:

$$\begin{aligned} \text{ Maximize } \; X \; \text{ for } \,(V, X) \;\text{ in } \; Y \end{aligned}$$

As an optimizing rule/agent is imposed in advance, the domain of technology set is then limited to a convex set of original single activities.

Hildenbrand (1981, 1097) mentioned:

[T]he economic relevance of the efficient production function is questionable. Other concepts of production functions for an industry are indeed conceivable; one might take into account certain institutional barriers to factor mobility in aggregating the individual production sets.

Leaving the special space restricted by the traditional production function, we can observe various possible combinations to accept institutional effects. Broadening the space, from where technology can be chosen, implies to introduce all possible combinations of activities. By nature, on the other hand, a zonotope of production permits to represent all possible combinations of activities. It then turns out that traditional economics is entirely irrelevant to an ontological thinking about how the production function exists. Given four basic activities, a zonotope of production is described in Fig. 7.2 of the previous chapter.

The same consideration as a zonotope of production applies to gene analysis, as Wright (1932) described a diversity of huge size, which allelomorphs of genes can generate, as Fig. 8.3. Figures 7.2 and 8.3 are of formally similar forms. Furthermore, we can immediately grasp a common idea between both figures.¹²

Interestingly, Alan Turing exhibited a model of the Chemical Basis of Morphogenesis in Turing (1952). Here, he suggested that concentrations and asymmetries are keys to understanding evolution starting from the embryo. Evolving dynamics is modeled as a single activator a single inhibitor model to generate various singularities. There exists an asymmetry between the inhibitor and the activator. The inhibitor’s diffusion from one cell to another is much bigger than the activator’s one. More interestingly, Turing in 1952 properly predicted that artificial intelligence would reproduce this theory. In 2006, Stefanie Sick et al. by computer simulation on the growth of hair has shown that two types of molecules that play an important role in the growth of hair have all the characteristics of Turing’s morphogens.¹³

Fig. 8.3

The 16 homozygous combinations: a; b; c; d are allemorphs. *Cited from Wright (1932, 357)

It is noted that the article of Rosser and Rosser (2017) shares with the spirit common to Wright (1932) and Turing (1952). By resorting to Thorstein Veblen, this article examined the basis for increasing returns, multiple equilibria, and bifurcations in the evolution of institutions.

6 Production Theory and Zonotope Production Set

6.1 Traditional Production Functions

With the birth of economic physics, the problems of so-called “aggregate production functions” have theoretically come to be recognized at a deep level. The theory of production functions starting with Knut Wicksell has completely lost its original meaning in the last century. Originally, the variables of the production function were land and labor, both as the original production element, but in the 20th century, land was replaced with capital artificially produced. Naturally, economists intentionally abandoned this problem. Speaking precisely, capital must be manufactured by the production function as long as it is an artifact.

We discuss the problem of the short-term production function. First, we describe a production function of thee factors of production. It is noted that the third factor of production $x_3$ is measured by the z on the cube. Namely, the figures here are drawn so that the output level increases in a stepped direction toward the upper right corner of the box. In the following diagrams of Figs. 8.4, 8.5, all coordinates of composed by the $x_1$,$x_2$, $x_3$, show the input amounts.

Fig. 8.4

A production function of increasing return

Fig. 8.5

A production function of decreasing return

Figure 8.4 shows a production function of increasing return to scale such as the type:

$$\begin{aligned} y = x_1^3 + x_2^3 + x_3^3. \end{aligned}$$

(8.16)

On the other hand, Fig. 8.5 shows a production function for diminishing returns to scale such as the type:

$$\begin{aligned} y = x_1^{0.33} + x_2^{0.33} + x_3^{0.33} \end{aligned}$$

(8.17)

6.2 Recursive Properties of Production Technology

Wassily Leontief is well known for his formulation of input-output table, but it is not well known that he examined the recursive nature of technology in his paper “The Internal Structure of Functional Relationships” (Leontief, 1947). Recursing, i.e., self-reference, is when describing something, a reference to itself appears in the description. $x_1$ and $x_2$ need each other for their own production. Hence The function $f^1$ represents the recursive relationship.

Now we assume that the production function $F(x_1,x_2,x_3)$ is continuous and can be second-order differential. It then hols:

Proposition 8.1

$$\begin{aligned} \frac{\partial (\frac{\partial F}{\partial x_1}/\frac{\partial F}{\partial x_2})}{\partial x_3} \equiv 0, \end{aligned}$$

(8.18)

$$\begin{aligned} \text{ i.e., } \frac{\partial R_{12}}{\partial x_3} \equiv 0 \end{aligned}$$

(8.19)

If and only if the condition is fulfilled, it exists the two functions $f^1 (x_1, x_2)$ and $f^0(f^2 (x_1, x_2), x_3)$ satisfying

$$\begin{aligned} F(x_1, x_2, x_3) \equiv f^0 (f(x_1, x_2), x_3) \end{aligned}$$

(8.20)

Here, $R_{12}$ is the so-called marginal substitution rate.

It is noted that this proposition can be generalized for over 3 arguments $x_i$.

Theorem 8.1

Let S be a partial set of $\overline{X}$. Also be S a complement set of S. S, a partial set of X is then separable if and only if

$$\begin{aligned} \frac{\partial R_{s_i s_j}}{\partial \overline{s}} \equiv 0 \; \text{ for } \text{ all } \; \overline{s} \in \overline{S} \end{aligned}$$

(8.21)

Primitive Inseparable Relationships

The following function is called a primitive inseparable function:

$f^1 (x_1, x_2 )=f^1 (f^2 (x_1,x_2 ), f^3 (x_1,x_2 ))$

6.3 Zonotope Production Set

In this subsection, we would like to consider the recursive relationship between production technologies through the Zonotope production set.

Well, I want to start from the introduction to theory of production function by Hildenbrand who is a giant head of mathematical economics. It was published in the journal Econometrica (Hildenbrand, 1981), but this work was not reconsidered at all until (Dosi et al., 2016b) enlighted on it in his own new reformulation.

Hildenbrand’s Short-Run Production Function

As is well known, the law of profit maximization is assumed in advance to derive traditional production functions. However, Hildenbrand (1981) drew a short-term production function using the Norwegian tanker industry (377 vessels, load capacity of 15,000 tons or more) in 1967 as an numerical example. This figure was produced as Fig. 3 of Hildenbrand (1981). The shape is of zonotope, i.e., a ruggby ball. This remebers Fig. 7.2. In this figure, the types of tankers are 57 turbine-driven, 320 motor-driven, and the date of manufacture is varied from 1950 to 1966. The tanker industry produces tons of transport miles per day. In keeping with economics tradition, the inputs are assumed to be only two elements: fuel and labor. Inputs are valued by the price and wages of a given base year. Hildenbrand then cited the work of Johansen and Eide to reproduce the production function of the Norwegian tanker industry.

After all, traditional production functions are not trying to see the relationship between input and output only in areas limited by operator max. It is different from the actual production functions that Hildenbland tested in the Norwegian tanker industry. On the other hand, the actual production set was a zonotope production set.

Fig. 8.6

The activity analysis of zonotope-basis

Fig. 8.7

The zonohedron, given the original 4 activities $\{a_1,a_2,a_3,a_4\}$: $\sum _{i=1}^4 [0,a_i] \subset R^3$

Figure 8.6 is a productivity set consisting of a possible combination of basic activities. Combinations are not limited to efficient combinations like traditional production functions. They contain all possible combinations, institutional, practical and policy combinations. Figure 8.7 depicts changes in the input surface when the input scale is increased by $1x, 2x, \text{ and }\; 3x$ at the same time. The scale of the input is increasing toward the upper right side of the box, but the surface of the isoquant of equal input varies according to the input scale. The bottom isoquant faces down toward the bottom, the middle one is relatively closer to the plane, and the upper level is depicted convex toward the top. As seen from Figs. 8.4 and 8.5, the upper isoquants exhibit increasing returns, while the lower isoquants exhibit decreasing returns. Thus, a zonotope production set, in general, will not establish some unique returns to scale.

6.4 Production Set in the Minkowski Space

We describe the ex post technology of a production unit as follows:

$$\begin{aligned} (a_1,...,a_l,a_{l+1}) \in \Re \end{aligned}$$

We then observe the production set Y. Given a family $\{a_n\}_{n \in N}$, the short-run total production set is written in the following manner:

$$\begin{aligned} Y = \sum _{n \in N}[0, a_n] \end{aligned}$$

In economics, there were few to discuss production in context of zonotope. Zonotope is one of the Minkowski space family. A convex polygon is a zonotope if and only if all its 2-dimensional faces have a center of symmetry. By the use of this idea, we can observe all possible combinations for production without resort to any restriction of efficiency.

Hildenbrand in (1981) defines the short-run total production set associated to them as the zonotope. The short run production possibilities of an industry with N units at a given time is a finite family of vectors $\{a_i\}_{1\le i \le N}$ of production activities.

$$\begin{aligned} Y = \{y \in \Re _{+}^{l+1} | y = \sum _{i=1}^N \phi _i a_i, 0<\phi _i<1 \} \end{aligned}$$

According to Hildenbrand, this idea differs from the traditional production function. We define the projection of Y on the input space $\Re _{+}^{l}$:

$$\begin{aligned} D= \{V \in \Re _{+}^{l} | (V, X) \in Y \, \texttt {for some} \, X \in \Re _{+} \} \end{aligned}$$

It then hods the traditional production function:

$$\begin{aligned} F(V) = \max \{ X \in \Re _{+} | (V, X) \in Y \} \end{aligned}$$

The operator $\max$ in the above has excluded the possibilities of “certain institutional barriers to factor mobility in aggregating the individual production sets” (Hildenbrand, 1981, 1097).

In general, the ex post technology of a production unit is a vector is a production activity a that produces, during the current period, $a_{l+1}$ units of output by means of $(a_1,...,a_l)$ units of input. The size of the firm is the length of vector a, i.e., a multi-dimensional extension of the usual measure of firm size.

Given $Y=(a_1,...,a_l)$, a set of generators for n, the zonotope Y is the convex hull of all vectors of the form a; that is (Y) is the Minkowski sum of all segments $[0, a_i]$, where $a \in Y$, i.e., $\sum _{a_i \in Y} a_i$. Also Z(Y) is the shadow of the r-dimensional cube $[0, 1]^{r}$ via the projection.¹⁴ It is noted that the application of znotpoe to production set was recently achieved by Dosi et al. in (2016a). They employed the volume of zonotope Y in $\Re$:

$$\begin{aligned} Vol (Y) =: \sum _{1\le i_1 \le i_2 \cdots \le i_l \le N} |\Delta i_1, ...,i_l| \end{aligned}$$

Here $|\Delta i_1, ...,i_l|$ is the module of the determinant. This kind of discussion will suggest a new growth/innovation theory of production. This approach may be classified into the complex adaptive system. This matter may be referred by the author elsewhere.

Exercises

8.1

Discuss the ontological background when it can be concluded that Leibzian idea of everything is at best will be justified

8.2

According to the main stream random choice theory, formulate descriptively the private utility, the social utility, and its interaction. Then examine whether this kind of simplification could be joustified in case that the set/subset of preference are not universal.

8.3

Explain the differences between this theory and Ruth’s random choice theory. In particular, discuss what difficulties exist with many multi-logit utility models.

8.4

Try to sort out the difference between the mainstream random choice theory and the original Luce-Saari theory of random choice.

8.5

Give a numerical example of discrete basic production activities of 3 sectors in Minkowski space, and make out the diagram. Then calculate the volume of this zonotope set.

8.6

In the text above, Class 4 property of cellular automaton was interpreted by exemplifying the AI market transactions. Detect another economic instance of Class 4 property.

8.7

Explain descriptively why the efficient market hypothesis is indispensable for the SMD theorems in the barter transactions but the efficient market hypothesis will be invalidated if it were incorporated into the finanical market.

8.8

Brian Arthur employed the term “the relaxed static stability” to characterize one of new features of the new innovation. Find another instance other than the F-35 to discuss the new direction of the current innovation.

8.9

Create a numerical example of a production function with the primitive inseparable relationships and verify Theorem 8.1.

8.10

We have a traditionally analytical tool such as indifference curve either of the production function or the utility function. Ensure that the idea of Zenohedron is also available for utility functions, and then try to make out such a figure in the 3-dimensional space.