DEVELOPING STOCK MARKET NANOTECHNOLOGY SYSTEMS AND GEOMETRIC FRACTAL REPRESENTATIONS OF DATA USING ECONOPHYSICS HAAG’S THEOREM Soumitra K. Mallick1 IISWBM & Calcutta University, Management House, College Square West, Kolkata 700 073, India
[email protected] April 17, 2015 ABSTRACT: This paper develops a stock market nanotechnology system by considering the quantum algorithmic nanosteps of Integrating Learning Systems and Asset Markets over time and current and information flow modelled by time flow and space flow , which gives the interaction picture in this Genetic Meanfield, by developing and proving that Learning and Stock Trading in a General Equilibrium Market model results in a D-Branes String and creates algorithmic nanotechnology steps in the case of Indian Stock Markets Nanotechnology systems which are systems integrable. KEY WORDS : M-Branes String Integration of Stock Markets, Econophysics Haag’s Theorem. PACS Classification Nos. 3.67 Ac, 89.65 Gh; JEL Classification nos. C60, G10 1
This paper has benefitted from comments on related papers presented at NYU, at
Statistics-Economics conferences held at Kalyani University, at the Indian Statistical Institute, Calcutta & New Delhi, a winter school of the Econometric Society, S & SE Asia, IIMC, IIEST, IISWBM, Calcutta University, Far Eastern Meetings of the Econometric Society, Beijing, R.Radner, J.Benhabib, Y.Nyarko, F.Hahn, S.Pudney, P.Dutta, I. Mallick, P.Siconolfi, G.Tian, T.Duncan, A. Sen, P. Sehgal, S. Raychaudhury, A. Kaban, P. Higgs and three earlier referees. The author remains responsible for all remaining errors.
1. THE NATURE OF THE STOCK MARKET ENVIRONMENT
In Stock Markets the business environment which is comprised of an Economic System and its Social Environment the Physical Social interaction which results in sustainable dynamic equilibrium in stock markets over time integrating the two systems through the utility and budget constraints in the Lagrangean of the market actions, conditions on the Learning Behaviour which involves complexity and learning needs to be engineered. Such approach to the Econophysics of Stock Markets which result in social engineering have been found (Mallick (1993)) to give rise to sustainable information fields (Mallick (2014)) which convert energy into money and consumption of goods, implicitly, through competitive goods markets. This ensures the quantum integrability using the Haag’s theorem with some modifications involving the setting of a special relativistic setting of Planck’s constant at zero in the limit of time where time is integrated over discrete nanosteps and information flows continuously. This formulation of the Lagrangean problem it has been proved elsewhere (Mallick (2014)) ensures the setting up of the algorithmic quantum system of markets with learning bounded rational real human subjects (genetic meanfield (Mallick(2014)) and in the presence of discrete nanosteps in the time space which are ”small” but finite so that infinities are cancelled out from the algorithmic steps by taking fractal ratios, as is ensured by the Arrow-Debreu model of stock markets quantum integrated with its business environment (Mallick (2014a)) over string theoretic time and industrial networks. This gives rise to a linear nanotechnology which with the logarithmic utility discovery through the learning algorithm satistifes the systems integration theorem (Mallick (2014b)) which provides the actions necessary for engineering the market with learning and laws of trading usually
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specified for stock market systems in all tradeable goods indutries and over time, like the Bombay Stock Exchange. The learning algorithms developed in Radner & Rothschild (1975) and adapted to economic market nanotechnology systems particularly the Indian Stock Market nanotechnology systems like the Bombay Stock Exchange (with open trading in foreign exchange and tick size of 1 paisa i.e. minimum size of price changes allowed per second). This paper discusses the Engineering of this social system with the Stock Markets acting on trades of the human subjects. We use two controlled second order stochastic processes (characterization of historical learning behavior) on asset returns with a discrete and bounded state space, finite number of agents and finite number of stochastic assets, whose transition probabilities are themselves functions of a random assignment of weights. The assignment of weights follows a control based on the past realization of values by the stochastic assets. Such controlled second order stochastic processes which have been described by Radner & Rothschild (1975)(modified in Mallick (1993)), have problems of existence and /or uniqueness of integrating probability measures characterizing history, future and the present. This satisfies the ”Systems Integration theorem” which further satisfies a constant exogenous nanostep ”Classification Theorem” developed and proved in Mallick (2014a).
These are the Systems integrability properties corresponding to the
Haag’s Theorem with a zero Planck’s Constant for Economics based monetary Systems with money-energy implicit conversion hence no specific energy requirements for the Equilibrium actions. Coincidentally, description of risk in models of economics usually involve proba-
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bility distributions associated with Von Neumann Morgenstern utilitarian choices. In such situations if one proceeds without the decomposition of such probability distributions and hence the associated measures of risk aversion (like the Arrow Pratt measure) which arise out of such probability distributions and the associated state contingent utility functions, one could be faced with the problem of nonuniqueness as two different probability distributions and utility functions could give rise to the same measurement of risk aversion.Hence, the same set of fundamentals could give rise to different temporal outcomes. As an example let’s discuss the following two alternatives : V (c1 , c2 ) = π1 u(c1 ) + π2 u(c2 ), u0 > 0, u00 < 0, π1 + π2 = 1 u u(c2 ) V˜ (c1 , c2 ) = aπ1 (c1 ) + (π2 + (1 − a)π1 ) π +(1−a)π 2 1 a π2
Here, V (c1 , c2 ) = V˜ (c1 , c2 ), however the probability distributions & utility functions (in their cardinal sense) are different.This has alternative effects on asset markets which result. It has been observed in Mallick, Sarkar & Roy (2006), that in 2000 at the end of the 90s decade, companies in India which were highly efficient in terms of market price of equity shares being ten times or more of their face value, i.e. the price at which the equity shares were initially issued, did not have significant weightage on dividend distributed in their asset pricing equation at the close of the year, but the market as a whole had significant weightage on dividends, which included significant number of companies which had market prices much less than ten times their initial prices. This suggests that ”subadditivity” does not work always in terms of asset pricing in the aggregate when one considers efficient companies visa-vis all types of companies pooled together. This requires alternative models for
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asset pricing behavior of alternative types of companies, especially in a dynamic sense, based on the corporate fundamentals. Various approaches can be taken to decompose such ”fundamental” measures with Markov Chains (history) and Riemann Integrals (future) and temporal risks in expected utility analysis may be rid of the existence or nonuniqueness problems of measures (present). This paper discusses one such approach. The importance of there being a bound on the variance of the ”evolutionary” Markov process is highlighted as also the properties of the resultant aggregate demand and the state contingent excess demand which satisfy the Integrability properties - which are the necessary and sufficient properties on excess demand functions, a la Sonnenschein’s famous result, which ensure continuous decentralization. The continuous time generalised financial market experiment and the discrete nanostep time asset market experiment are solved by Special Relativity formation of nanosteps which are fractal architecture of markets (as in Mallick (2014b)). Section 2 sets up the model, sections 3 contains the results, section 4 concludes. 2.1 THE NATURE OF THE PHYSICAL ENVIRONMENT (This model and some of the mathematical results have been taken from Mallick (1993, 2011b))
The nature of the physical environment which leads to learning and discovery of the fundamentals of the Financial Markets viz. utility, prices, stock of goods, stock of money, market closing and opening times on subsequent dates in this state preference theoretic financial market system is as follows. According to the beliefs agents are born with at each date t there are a finite number -S- and ordered possible states of the world. I denote ω = {1, 2, . . . , S} as this set. Each economy E in this world is associated with a subset (proper or not) of the set ω which are
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likely to occur infinitely often in this economy and hence are ”relevant” for it. Agents are not born with the knowledge of such a subset but can learn about the possibilities, as we shall describe later on. Each state s at date t is characterized by a random state variable Xs (t) which can take on values over [0, ∞). The Xs (t)’s for any date t are independent but they are correlated over t’s. A value of 0 for any Xs (t) signifies the non-occurrence of ”irrelevance” of state s are date t. The return on assets specific to each state (to be discussed in detail shortly) is determined by the realization of the state variable corresponding to that state. Agents do not observe the realization of the state variables but observe the realization of their share of asset returns by allocation of some effort (to be described in detail later). Learning in this model is thus a ”filtering” problem (see for e.g. Lipster & Shiryayev (1984)) of learning about the true states of the world from the asset returns2 Learning leads to formation of beliefs about the probabilities of the various states of the world. This Econophysics model of learning, relating the Statistical properties of the Market System to the Market Structure and separating the effect of genetic behaviour, was first developed in Mallick (1993). To denote the information structure in this economy I will regard all random variables as measurable functions on a common probability space, which is endowed with a sigma field F of measurable subsets. Corresponding to each date τ is a sigma field Fτ of subsets representing all events that can be observed upto and including date τ . Each Fτ contains the preceding one, and all Fτ are contained in 2
However, instead of taking an optimal filtering approach like Lipster & Shiryayev do
we will be taking a stopped random walk approach along the lines of for e.g. Gut (1988), to learning.
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F (Mallick(1993)). In other words to simplify discussion I shall put structure on the random return vectors directly instead of the state variables X. Thus the learning process and the updated probabilities will be defined directly in terms of the returns. The underlying description of uncertainty as I just now described is important for motivating the problem (Mallick (1993,2005) solves some of this). This gives rise to a D-Branes String with the Dirichlet fundamentals discovered through the learning technology, the direction of change given by the sign of the financial asset created out of the Lagrangean consumption action and the magnitude given by the Lagrangean also by the Kuhn-Tucker-Kharush Theorem. The String of actions which arises out of the Market Algorithm integrates the actions over time is described below. The continuous time general financial market specification and the discrete time asset market specification are special relativity specifications over trading fractals or the fractal geometric architecture of discrete nanosteps (Mallick (2014a)).
2.2 STEP I : THE CONSUMPTION ACTION MODEL
Each agent i in economy E starts period 0 (the initial period) with a prior probability distribution over the various states of the world for each date in the future, a known endowment stream in the form of shares in assets , and ”nontradeable rules” for allocating effort to the assets when optimization is not possible (to be discussed at length later). Each agent i is infinitely lived. However, his preferences are defined only over Si states of the world where Si < S.us (ci (s, t)) represents the state contingent utility function with agent i has for s ≤ Si , us (ci (s, t)) has
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the further property that u0s (.) > 0 and u00s (.) < 0. Since i’s preferences are not complete he cannot make optimizing actions without learning. Before participating in trade at period 1, agents come to know about the distribution of the return on each asset (indexed by s) for each date in the future and also how the distribution would be altered (in a stochastic sense) over time by the allocation of effort to any particular asset or assets. The return on the assets signals the occurrence of the corresponding state infinitely often in the future. Using this expectation of the future distribution of assets returns subject to the behavioral rule followed the agent updates his probability beliefs about the future states of the world. Then with this updated probability beliefs he enters the market in period one to trade in assets. Thus the consumer’s action problem involves two steps: (a) The plan of allocating effort in the future following some behavioral rule and the corresponding learning and updating of probability beliefs, (b) the optimizing problem with respect to asset trades given these updated probability beliefs and the knowledge over asset returns. It must be emphasized at this point that all the trading that takes place will be at date 1. Hence the agents have rational expectations about the future returns on assets. Trades occur with respect to this perceived process of state contingent returns (which depend on the effort allocation behavior to be followed in the future).
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2.3 STEP II : THE LEARNING ACTION MODEL
There are stocks in the economy, 3 each of which produce output only in one state of the world, and by observing each stock one can gain information about that particular state of the world occurring in the future4 . There are S states in the economy each indexed by a stock to which it pertains. Stocks are ”potentially” infinitely lived. Stocks are divisible and are tradeable. The returns on the stocks can be altered by allocation of effort to the stocks. This allocation of effort alters the return on the stock owned by the agent (no externalities are present between the shares of different agents) in a stochastic way. This alteration of the return process has two effects on the agent’s consumption action - (a) it changes the future income stream of the agent and (b) since evolving performance of the stocks signal the possibility of the corresponding state occurring in the long run, changes the agent’s probability beliefs over the future states of the world. Thus, learning here is ”learning by doing”. Let R(t) = (R(s, t))Ss=1 represent the return vector on stocks of S firms at time t. At each date t = 0, 1, 2, . . . and for each stock s, the individual i perceives a return Ri (s, t, ) on his share in stock s. Ri (s, t) can range all over R ∪ {−∞, +∞}. 3
Day (1967) has suggested a learning model for producers with an unknown profit
function, who learn about their optimizing objective from their past output actions and the resultant profit levels. The learning rules in this paper form a system of nontradeable fundamentals of this economy which can give rise to alternative market possibilities.Thus post trade payoffs can be fundamentally divergent in this economy as opposed to the fundamentally convergent results discussed in Day. 4 ”Stocks” represent behind their ”corporate veil” firms in the economy which I do not discuss explicitly in this pure exchange model.This note has been motivated by an anonymous referee of Mallick (2005)
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As a function of the history of the vector Ri (n), n = 0, . . . , ` agent i allocates to each stock s some fraction ais of his effort, during the coming period of time. The vector ai = (ais ) of fractions is nonnegative, and it is assumed that it sums to unity. Agent i’s probability beliefs about the various states of the world follow the following rule: πi (s) = π(s) lim f (E(Ri (s, t))
(1)
t→∞
where, πi (s) represents the updated probability belief about state s, π(s) the initial probability beliefs about state s, E is the expectations operator based on his perception of the future stream of returns taken with respect to date 0 (information set (F0 ), f (.) the learning function and has the following properties: (a)f ∈ C1 {(−∞, ∞) :→ [0, ∞)},
(1a)
(b)f (E(Ri (s, t)) = 0, ifE(Ri (s, t)) ≤ 0,
(1b)
(c)f (.) is non − decreasing in E(Ri (s, t)) > 0,
(1c)
(d) and since πi (.) is a probability measure has the further property that P
πi (s) = 1, at any date t (1d)
An example of such a function is: f (E(Ri (s, t)) = (π(s))−1 max[0, (ERi (s, t))/
X
max(0, ERi (s, t))]
ω
My modelling objective is as follows. Since, each agent i’s preferences are defined over Si < S he cannot optimize using the rule max
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P
π(s)us (ci (s, t))
where us (ci (s, t)) represents his utility from consumption in state s. If, however it so happens the πi (s) = 0 for (S − Si ) of the states then the following problem max max
P
ω
P
ωi ={1,...,Si } πi (s)us (ci (s, t))
is defined and has the same solution as
πi (s)us (ci (s, t)), whatever us (ci (s, t) might be for s 6∈ ωi , since the weights
on those numbers will be zero. 2.4 STEP III : THE RETURN RESULTANT MONEY TO GOODS CONVERSION PROCESS & THE INFORMATION (LEARNING) TO MARKET CONVERSION PROCESS OF GOODS DISCOVERED: Given past history up through date t, the conditional distribution of the next vector, Ri (t + 1), of returns to any agent i depends upon the vector ai (t) of allocations at date t I define Zi (t + 1) = Ri (t + 1) − Ri (t)
(2)
The sequence of vectors Zi (t) is the sequence of successive increments in the vectors of returns Ri (t). I shall make the following assumption about the conditional distribution of Zi (t + 1), given the sequence Ri (0), . . . , Ri (t) and the allocation ai (t): 3a)The distribution of Zi (t + 1) depends only on ai (t) 3b)EZi (s, t + 1) = ai (s, t)ηs −[1−ai (s, t)]νs , where ηs and νs are given positive parameters 3c)V arZi (s, t+ 1) = σs2 (ai (s, t)), where σs is a given strictly positive continuous function. 3d)The coordinates of Zi (t + 1) are mutually independent. 3e)I also assume that the coordinates of Zi (t + 1) are integer-valued and uniformly bounded.
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3f)Ri (s, 0) is uniformly bounded. 3g)Also, for any s ≤ u, the random vectors Ri (s), Zi (s) are measurable with respect to Ft (see previous section). A behavior for any agent i is a sequence of random allocation vectors ai (s) such that each ai (s) is measurable with respect to Ft for all s ≤ u. Average return is defined in such a way that expected increments in average returns do not depend on the agent’s behavior. Define a vector w = (ws ) of ”weights” by X
ws = (1/(ηs + νs ))(
1/(ηi + νj ))−1
(4)
j
These weights are positive and their sum is unity. Define the corresponding weighted averages of performance and performance increments by Rim (t) =
X
ws Ri (s, t), Zim (t) =
s
X
ws Zi (s, t)
(5)
s
Notice that Zim (t + 1) = Rim (t + 1) − Rim (t)
(6)
It can be verified from (3b) that the conditional expected value of Zim (t), given the past history of the system up through date t − 1, is ζ m = (1 −
X
X
(ηs /ηs + νs ))(
s
(1/ηi + νi ))−1
(7)
i
I shall focus attention on what Radner & Rothschild call ”putting out fires” and ”staying with the winner” behaviors. ”Putting out fires” behavior involves at every date allocating effort to the stocks which have the lowest return, while
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”staying with the winner” involves allocating effort to the stocks which have the highest return. (a) Putting out fires DEFINITION 1: The formal definition is as follows. Let, Mi (t) = min Ri (s, t) s
(8)
then ”putting out fires” behavior is defined as follows: (a) if Ri (s, t) > Mi (t) then ai (s, t) = 0. (b) If Ri (s, t) = Mi (t) and ai (s, t − 1) = 1, then ai (s, t) = 1. (c) If neither (a) nor (b) holds, then ai (s, t) = 1 for s =smallest j such that Ri (j, t)= Mi (t).
(9)
The following results characterize the return process under this kind of behavior. PROPOSITION 1: Suppose that K is any proper subset of the state space ω = {1, . . . , S} that K 0 is the complement of K in ωi and that putting out fires is practiced by i on the activities in K while no effort is allocated to those in K 0 . Then given 0 < θ < ∞ there is a nonrandom Ti (θ) (which is finite) for any i such that E min Ri (k, t) ≥ min Ri (k, 0) + θforallt ≥ Ti (θ) k∈K
k∈K
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E max Ri (k, t) ≥ max Ri (k, 0) + θforallt ≥ Ti (θ) k∈K
k∈K
Ti (θ) can be chosen so that the above results hold for any K properly contained in ω. Also, Ti (θ) is nonotonically increasing in θ. Proof: The proof follows lemma 1 in Radner & Rothschild (1975).Q.E.D. PROPOSITION 2: If ”putting out fires” is practiced with respect to all assets, if ζ m > 0 and if P {Zi (s, t + 1) = 0|a, (s, t)} > 0, P {Zi (s, t + 1) = 1|a, (s, t) = 1} > 0, P {Zi (s, t + 1) = −1|a, (s, t) = 0} > 0, then, P {Ri (s, t) > 0; 1 ≤ s ≤ S, 0 < t ≤ ∞} > 0.
Proof: This is a restatement of theorem 3 of Radner & Rothschild (1975).Q.E.D. (b) Staying with the winner DEFINITION 2: The formal definition is as follows. Let, Mi∗ (t) = max Ri (s, t) s
(10)
Then, ”staying with the winner” is defined as follows: ai (s, t) = 0 for all s such that Ri (s, t) < Mi∗ (t).
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(11)
DEFINITION 3: An asset s can compete if, for any date t any allocation of effort at the date, and any other asset j, P {Zi (s, t) − Zi (j, t) > 0} > 0.
(12)
The following result characterizes the return process under this kind of behaviour. This gives the fractal geometric market architecture for both the long and short trading specifications. PROPOSITION 3: If for any i every asset can compete (see definition (1)) and is aperiodic (refer to Kemeny et.al. (1966) for definition), then the following statements are true with probability one: (a) ∃s∗i (random) and a finite Ti (random) such that ∀t > Ti , ai (s, t) > 0 only for s = s∗i (b) ∀s, P {s∗i = s} = pi > 0, (c) P {Ri (s, t) > 0∀t|s = s∗i } > 0, (d) P {Ri (s, t) > 0i.o|s 6= s∗i } = 1, (e) P {Mi∗ (t) > 0∀t} > 0. Proof: This is a restatement of theorem 5 of Radner & Rothschild (1975)Q.E.D..
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2.5 STEP IV : HUMAN INTEGRATION BY LEARNING-BY-DOING ACTIONS: Given the return processes corresponding to the two behavioral rules given in the above three propositions each agent i follows the following reasoning to reduce the state space under his consideration: The limit of the expectation of the returns of each stock signals the likelihood of the corresponding state occurring infinitely often in the future. This means that if the return on the stock with respect to a certain state is to become nonpositive beyond certain finite date then the possibility of the corresponding state occurring more than rarely (infinitely often) in the future is negligible. This has been subsequently integrated into the ECONOPHYSICS HAAG’S THEOREM.
2.6 STEP V : THE LAGRANGEAN MODEL :
If the allocation of effort by agent i in the manner just described and the consequent learning process updates his probability belief in such a way that he can now focus attention only on ωi = {1, . . . , Si } of the states for the future then he uses maximizing actions with respect to trades in contingent consumption. Thus agent i’s problem at date 1 is :
max
{ci (s,t)≥0}
s.t.
X
XX t
δ t πis us (ci (s, t))
ωi
p(s, t)(Ri (s, t) − ci (s, t)) = 0t = 1, . . . , ∞
(13)
ωi
where, 0 < δ < 1 is the discount factor, (and represents the rate of time preference) . πis is the updated probability belief already described with respect to the learning
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behavior ∧, ci (s, t) refers to the consumption of the only good in the economy by agent i in state s at date t, p(s, t) is the futures price of the good in state s at date t, Ri∧ (s, t) refers as already stated to the physical units of goods received by agent i by virtue of ownership of shares in asset s at date t state s. This model is equivalent to: max
{ci (s,t)≥0}
s.t.
X
XX t
δ t πis us (ci (s, t))
ω
p(s, t)(Ri (s, t) − ci (s, t)) = 0t = 1, . . . , ∞
(130 )
ω
as for Si < s ≤ S , πi (s) = 0 . With this description of the optimizing behavior of each agent i I now go on to the dynamic general equilibrium asset pricing model.
2.7 STEP VI : THE LEARNING SOCIAL ENGINEERING RESULTANT DYNAMIC GENERAL EQUILIBRIUM ASSET PRICING MODEL
There are N > 0 agents in the competitive economy, each differing from the other with respect to the state space over which preferences are defined and with respect to the initial holdings of assets. Each agent is born in period 0 and is infinitely lived. Agents are endowed with shares of the S different stocks at birth, one unit of time in each period which they can allocate to learning as described before, and a set of probability beliefs (π(s))s∈ω over the possible states of the world. Each agent i is characterized by the number of states Si over which his preferences are defined. Each agent has the same learning function as described in (1). Formally, this economy E is characterized at date 0 by E = {(us (ci (s, t))s∈ωi ,i=1,...,N, (ψi (s))s∈ωi ,i=1,...,n , (f (.)), ∧, (R(t))},
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where us (ci (s, t)) refers to the time stationary utility function of any agent i in state s defined only for the range of states over which his preferences are defined for any date t in the future ψi (s) refers to the share of stock s that the agent i receives at the start of date 0, the function f (.) refers to the learning function of every agent as discussed in (1), ∧ refers to the effort allocation behavior followed, and (R(t)) is the exogenously specified return vector on the trees as specified before. Thus, the agents differ in terms of the state space over which their preferences are defined and with respect to the initial endowment of stocks. The only income which the agents earn is the return on their stocks. There is only one good in this economy which is produced by the stocks. The perceived return vector is with respect to the physical units (not value) of these stocks. Their values will be determined at equilibrium prices (if they exist). The state contingent commodity is perishable and therefore cannot be stored beyond the period in which it is produced by the stock. There are two stages in the action process in this economy for each agent as has already been described. First, he has to allocate his time to learning about a set of states or in other words about the stocks appropriate to those states. At this stage since he cannot form optimal actions in terms of contingent consumption, he chooses either one of the nontradeable work rules detailed already: (a) putting out fires or (b) staying with the winner He then uses updated probability beliefs to maximize expected utility subject to the budget constraints. I first make the assumption that there is no variable cost attached to learning, for e.g. with respect to the size of the state space with respect to which learning is practiced or the length of the history which is taken into consideration for each
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of the work rules. Hence, learning can be with respect to any size of the statespace or in other words with respect to any number of assets without changing the resource cost of learning. I also assume that allocation of effort by any agent does not alter the return process and hence expectations of other agents. DEFINITION 4: A ”partially non-tradeable” Arrow-Debreu equilibrium at date 0 is defined as: A learning behavior ∧ , a list of contingent commodity prices (p(s, t)), s = 1, . . . , Simax , t = 1, 2, . . . , (where Simax = max{Si }i=1,...,N ), and contingent consumption allocation vector (ci (s, t)), s = 1, . . . , Si , t = 1, 2, . . . , for each i = 1, . . . , N such that Given the learning behavior ∧ (a) Each agent maxci (s,t)≥0 s.t.
X
P
tδ
t π ∧ ()u (c∗ (s, t)) s i i
p(s, t)(Ri∧ (s, t) − ci (s, t) = 0, f ort = 1, . . . , ∞, 0 < δ < 1
ω
(b) For each i, πi∧ (s) = 0 for s > Si (c)
∧ N (Ri , t)
P
− ci (s, t)) = 0, ∀s, t
where, πi∧ (s) = π(s) limt=∞ f (E(Ri∧ (s, t)) and Ri∧ (s, t) = ψi (s)R∧ (s, t) R∧ (s, t) = R(s, t) where ∧ is the non tradeable learning rule (14)
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DEFINITION 5:
I shall define existence of equilibrium for this economy in two
stages. First, I define an epsilon equilibrium for this economy as follows: An ”epsilon equilibrium” for E is {∧, (p (s, t)), (ci (s, t)), s ≤ Simax , t ≥ 1, i ≤ N, p (s, t) ∈ (0, 1) X
p (s, t) = 1}
ω max
such that : Given the learning behavior ∧ and any > 0, (a) Each agent i, max{ci (s,t)≥0} s.t.
X
P
iδ
tP ωi
π(s)f (E(Ri∧ (s, t))us (ci (s, t))
p (s, t)(Ri∧ (s, t) − ci (s, t)) = 0 f or t = 1, . . . , ∞
ωi
(b) ∃ a finite T () s.t. π(s)f (E(Ri∧ (s, t)) ≤ , ∀s ≥ Si , ∀t ≥ T (), i ≤ N.
(c) X
(Ri∧ (s, t) − ci (s, t)) = 0, ∀s ≤ max{Si , i ≤ N }, ∀t
(15)
i
Then, I shall define the partially non-tradeable Arrow-Debreu equilibrium in the economy E as the limit of the sequence of epsilon equilibria as → 0 from above.
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2.8. STEP VII : ALGORITHMIC NANOSTEPS WHICH CHARACTERIZE THE LEARNING (QUANTUM) LONG MARKET EQUILIBRIUM OVER TIME :
The following theorems characterize the existence of equilibrium in this interpretation of partially non-tradeable Arrow-Debreu economy : THEOREM 1: With learning following ”putting out fires” behavior practised by each i over ωi∧ only, an Arrow-Debreu equilibrium (henceforth referred to as such which is of the type being studied here)exists for E. Proof: I shall first prove existence of an ”epsilon equilibria” then take limits to get the Arrow-Debreu equilibrium. I shall prove existence of an ”equilibrium” in the following steps: NanoStep 1: I show that the demand function for contingent consumption with respect to any finite data-state pair for any agent, within his defined class of states, is continuous and single-valued in the contingent price. NanoStep 2: I construct an ”extended demand correspondence” of each agent over maxi ωi for each date within a finite set such that the extended demand correspondence of each agent is single-valued and continuous. NanoStep 3: I show that the aggregate extended demand correspondence is single-valued and continuous over maxi ωi and that Walras’ Law holds. NanoStep 4: I use the Gale-Debreu-Nikado lemma to show existence of an equilibrium with respect to any positive epsilon. NanoStep 5: I show that this existence result holds in the limit as epsilon
21
goes to zero from above. This action operates on the Lagrangean because of the ECONOPHYSICS HAAG’S THEOREM limit stated in the proof. It proves the problem with the interaction picture between the two actions in the Long Run and Short Run Nanosteps analysed in this paper and the time (Vectorial Arrow of Time (see Mallick (2014b)) in the Genetic Stock Market Meanfield. Each Short Run Financial Stream leading to another short run savings (positive charge) and short run investment (negative charge) and the continuity of the field flow given by the Expectation model and the information field constructed before (Earman & Fraser (2006), Mallick (2014b) for the interaction picture in this Genetic Meanfield). This has been established in Mallick (2014a) as the D-Branes String for Econophysics Field. Notice that the criticality of the A and A(ψ) matrices establish the Boolean fundamentals of this field for Econophysics convergence. At Equilibrium the A(ψ) matrix can be solved by inverting the equilibrium. This establishes the graph in less than 23 dimensions (Mallick (2014a)). However, the behavioural criticality of the ”Putting out Fires” behaviour embeds this model in National Econophysics systems. Since time is always the dimension of convergence in Step 5 therefore the Vectorial three dimensional resolution along resultant dimensions can always be resolved Lagrangeally hence ensuring current and information flow (Mallick (2014a, 2014b)). NanoStep 1: By virtue of assumption (1a) -(1d) on f , for any 0 < < ∞ there exists some 0 < γ < ∞ such that if E(Ri (s, t)) ≤ γ then π(s)f (ERi (s, t)) ≤ . Now, I define Θ = maxk∈(ω−ωi ) Ri (k, 0) − γ. Since maxk Ri (k, 0) is bounded by the assumption (3f), Θ is bounded and hence by Proposition 1 ∃ finite Ti (Θ) such that: E
max Ri (k, t) ≤ γ∀ ≥ Ti (Θ)
k∈(ω−ωi )
22
This implies that f (E maxk∈(ω−ωi ) Ri (k, t) ≤ ∀t ≥ Ti (Θ) For any agent i since (us (ci (s, t)) is well defined for s ∈ ωi , continuous and strictly concave, hence the agent’s maximizing problem will give a single valued, continuous demand function p(s, t) for each s ∈ ωi and each t ≤ maxi Ti (Θ). Let us call this demand function di (p(s, t)) and the vector of demand functions Di = (di (p(s, t))s∈ωi t≤maxi Ti (Θ) Di is continuous and single-valued for each p(s, t) maxi Ti (Θ)
ωi over R+ XR+
.
NanoStep 2: Let’s consider the following extension of the demand correspondence Dt just derived (henceforth referred to as the ”extended demand correspondence” and labeled Di∗ ) as follows: Di∗ = Dt if, s ∈ ωi = Ri (s, t) if , s ∈ (max(ωi − ωi )) , ∀ p(s, t) such that p(s, t) ∈ ∆max Si X∆max Ti (Θ) i
(13) This demand correspondence is single valued and continuous because each component is single valued and continuous. NanoStep 3: Now, consider the ”aggregate extended excess demand corresponmax Ti (Θ)
max Si dence” from (13) (D∗ : ∆max Si X∆max Ti (Θ) :→ R+ XR+
D∗ =
X
as follows:
{Di∗ (p(s, t)) − Ri (s, t))s∈ωi ,t≤max Ti (Θ)
i
+(Ri (s, t) − Ri (s, t))s∈(max ωi −ωi ),t≤max Ti (Θ) } D∗ is single-valued and continuous over its range, since each of the elements are single- valued and continuous from step 2.
23
NanoStep 4: From the construction of the extended demand correspondence in step 2 it is clear that for any agent i the following holds: X
p(s, t)(Di∗ (p(s, t)) − Ri (s, t)) +
s∈ωi
X
p(s, t)(Ri (s, t) − Ri (s, t))
s∈(max ωi −ωi )
= 0(from the budget constraints in the individual maximizing problems). This implies, denoting p = (p(s, t))s∈max ωi ,t≤max Ti (Θ) p.D∗ = 0, which further implies that Walras’ Law holds. Thus, all requirements of the Gale-Debreu-Nikaido lemma and satisfied with respect to D∗ and therefore an epsilon equilibrium (15) exists for any > 0 (see for e.g. Nikaido (1956)). NanoStep 5: Since, the arguments in steps 1-4 hold with respect to any > 0 it holds in the limit as ↓ 0, and for all t ≤ lim↓0 supi (Θ)), the r.h.s. of which is finite for any positive . Hence, a ”partially nontradeable” Arrow-Debreu equilibrium (14) exists for E.Q.E.D. THEOREM 2: If ”putting out fires” is practised with respect to all assets by any agent i, if ζ m > 0 and if:
P {Zi (s, t + 1) = 0|ai (s, t)} > 0,
P {Zi (s, t + 1) = 1|ai (s, t) = 1} > 0, P {Zi (s, t + 1) = −1|ai (s, t) = 0} > 0,
24
then, P {a ”partially nontradeable” Arrow − Debreu equilibrium exists for E} < 1.
Proof: All I need to show here is that the argument of step 1 in Theorem might fail to hold true for i with positive probability. Since, the conditions of Proposition 2 are satisfied hence, P {Ri (s, t) > 0, ∀ s ∈ (ω − ωi ), ∀ t} > 0.
Hence, ∃γ > 0 such that,
P {Ri (s, t) ≤ γ, foranys ∈ (ω − ωi ), foranyt} > 0. This implies that for s ∈ (ω − ωi ), ∃ > 0 such that P {π(s)f (E(Ri (s, t)) ≤ foranyt} > 0.
Hence, condition (b) of the definition of ”epsilon equilibrium” (15) fails with positive probability, thus showing that an ”epsilon equilibrium” may fail to exist. This implies that the probability of a ”partially nontradeable” Arrow-Debreu equilibrium (14) existing for the economy E is less than 1. Q.E.D. THEOREM 3: If for any agent i every asset can compete (see definition (1)) and is aperiodic and the agent practises ”stating with the winner” behavior with respect
25
to ωi assets only, then, P {a ”partially nontradeable” Arrow − Debreu equilibrium exists for the economyE} < 1.
Proof: Since, the conditions of Proposition 3 are satisfied hence, P {Ri (s, t) > 0, ∀s ∈ (ω − ωi ), ∀t} X
=
P {(Ri (j, t) > 0∀t&j = s}
s∈(ω−ωi )
=
X
P {(Ri (j, t) > 0∀|j = s}P {j = s}
s∈(ω−ωi )
> 0.(Proposition3)
This implies that for s ∈ (ω − ωi ), ∃ > 0such that P {π(s)f (E(Ri (s, t)) ≤ for any t} > 0.
This implies that condition (b) in the definition of ”epsilon equilibrium” (15) may fail to hold with positive probability. This implies, that the probability of a ”partially nontradeable” Arrow-Debreu equilibrium (14) existing for E is less than 1.Q.E.D.
26
3. AN ALTERNATIVE LEARNING (QUANTUM) SHORT MARKET APPROACH TO DYNAMIC EQUILIBRIUM USING THE ECONOPHYSICS HAAG’S THEOREM : An alternative approach to the dynamic equilibrium of the Stock Market Econophysics Social Engineering Model is given by the model of Incomplete Asset Markets as Fractal Geometrically Imperfect Financial Markets structurally as developed in Mallick (2003) which establishes a correspondence between the Hilbert Spaces of Economically valued dated Commodity Markets and dated Asset Markets the distinction being established by the price vectors (p(s)) and (q(s)) respectively. PROPOSITION 4: The Hilbert Spaces of [p(s).(ci (s) − wi (s))]i and q.A are related by the noncommutative relation lim t [δ t [p(s).(ci (s)−wi (s))]i |s∗ =[I.q(s).θi .A.I]i a.e. given At = A. Proof: The Hilbert Space is vectorial by the Span of the S* matrix and covectorial by the dimension [S-S*] since the basis of the S* and [S-S*] matrices in this paper are the matrix of 1s and 0s, by referring to the diagram of the A matrix below. Therefore, also setting the Dynamic Limit in Theorem 1 to zero of the Planck’s constant -ψ- multiplied by the rate of discount δ t in the limit, which would imply δ t xψ lim t = 0 a.e. would satisfy the Econophysics HAAG’S Theorem. The context makes it clear as to which matrix is being referred to in the two parts of the proof. Q.E.D. 4.THE MODEL (Mallick (2011c)): 4.1 Uncertainty :
27
ξ ≡< ((usi , πi (s))s≤si )i≤N , ((wis )s≤S i≤N ) > There are N > 1 agents in the economy. Agents differ possibly in terms of their Von-Neumann-Morgenstern utilities (usi , πi (s))s≤si and endowments (wis )s≤S . There are two dates 0 and 1. Each participant i is characterized by a discrete subset ωi to which he attaches positive probability
ωi = {1, 2, . . . , si } (ωi )c = {si + 1, si + 2, . . . , S} s.t.P {some s ωi occurs at t = 1/i} = 1
(2)
Further, (ωi )c 6= φ for all i, ωi ⊂ ωj for j 0 ∀ sωi
(4)
πi (s) = P { State sωi occurs at
4.2 : Choices : There is one commodity in the economy. consumption by participant i at t = 0 is denoted by c0i and contingent consumption in state s at t = 1 is denoted by csi . Any particular realization of participant i’s endowment at t = 0 is wi0 and (S+1)
in state s at t = 1 wis , (wis )R+
& is µ measurable, where µ is the Lebesque
measure over the Borel subsets of the positive (Real) orthant.
28
The price of the commodity at t = 0 is denoted p(0) ≥ 0 while its statecontingent price in state s at t =1 is p(s) ≥ 0. Participant i’s utility function with respect to state s is u(.).u(.) is bounded, increasing and strictly concave. His rate of time discount is δ : 0 < δ < 1. Participant i’s choices at t = 0 is described by :
max
(c0i ,(csi ))≥0
s.t. p(0)c0i +
u(c0i ) +
si X
si X
δπi (s)u(csi )
1
p(s)(csi − wis ) − p(0)wi0 ≤ 0
1
csi
=
wis for
s > si if (ωi )c 6= φ
(5)
(s = 0 corresponds to t = 0, φ denotes the null set ) . 4.3 Imperfect Markets : Primary security n is a contract which pays off 1 unit of the commodity in state n and 0 units in all other states, at t = 1 Let, cni ((p(s))) denote the demand for contingent consumption by participant i for state n at t = 1 given the price vector (p(s)) in terms of (5). The contingent commodity market is ’imperfect at t = 0 at prices (p(s))’ iff ∃ i,n :
cni ((p(s))) − win 6= 0
29
but, cnj ((p(s))) − wjn = 0 for all j 6= i. (6)
The contingent commodity market is ’imperfect at t = 0’ iff it is imperfect in the limit for all convergent price sequences in the positive orthant. 5
4.4 Incomplete markets : Suppose ∃ at t = 0 a pre-available vector of primary securities which pay off at t = 1, with row rank ≤ maxi si . (Iff column rank < maxi si , the security market is incomplete). The budget constraint is rewritten in (5) as
p(0)c0i + q.θi ≤ p(0)wi0 p(s)csi ≤ p(s)wis + p(s)θi (s)r(s), s ≤ si where, θi = (θi (s))s is the demand for securities by i and q is the security S×S price vector. θi (s)R, q(s)R+ ∀s, (r(s))R+ is the security payoff matrix, and
the no-arbitrage condition holds for security prices. q.θi =
X
6
Also,
q(s)θi (s)
s
. In this paper the pre-available payoff vector can be decomposed into four matri5
The author thanks an anonymous referee for improving the exposition of this section
6
The author thanks an anonymous referee for highlighting this definition.
30
ces, one non-singular and positive and the others zeros,
r(1)
0
0
0
0
0
0
0
r(2)
0
0
0
0
0
0
0
r(3)
0
0
0
0
0
0
r(4) . . .
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
r(s∗) . . . 0
0 S × S 0 0
0
0
∞ > r(.) > 0. This is henceforth referred to as the ’A’ matrix. Also, s∗ = maxi si . 5. Results : Lemma 1 : Span A ≡ Span of Arrow payoff matrix with dimension s∗. Proof : Let r(i), i = 1, . . . , s∗ represent the column vectors of A, for the first s∗ columns. The rest of the columns and rows are zeros, so w.l.o.g. the same arguments would hold if we considered S instead of s∗ as being the dimensions of the matrix generated by r(i)s.
7
Let A = [r(1), r(2), . . . , r(s∗)] 7
The author thanks an anonymous referee for improving this exposition.
31
s∗ . Since, r(i) s are independent we shall show that Span A ≡ R+ s∗ = s∗, the set [r(1), . . . , r(s∗), y] is dependant for every yRs∗ .A is Dim.R+ + s∗ (Rudin independent, it follows that y is in the span of A Hence, A Spans R+
(1976)) Dim. ~0 = 0. Hence Span A ≡ Span A. Nothing in this proof fails if in particular r(i) = 1∀i ≤ s∗ . Hence, Span A ≡ Span of Arrow payoff matrix with dim. s∗ . s∗×s∗ s∗ , q Rs∗×s∗ q .A is Lemma 2 : Consider the mapping qn .A : R+ → R+ n n +
continuous. Besides Span qn .A ≡
Span qn .I, I is the identity matrix spanning
s∗×s∗ R+ . s∗×s∗ Proof : If qn intR+ , qn .A is of full rank
Hence, if qn → q, qn .A → q.A Also, qn .I → q.I since Span A ≡ Span I (Lemma -1)
rkqn .A = rkqn .I,
s∗×s∗ qn R++
(because of no arbitrage condition) suppose span qn .A ⊂ spanqn .I
⇒ rkA < rkI a contradiction similarly the converse.
32
Hence, span qn .A ≡ spanqn .I. Hence, the above Lemma. Proposition 1 : Iff the contingent commodity market is imperfect then the corresponding asset market is incomplete. Proof : The budget constraint with asset structure ’A’ for participant i and any bounded price sequence (pn (s)) of spot prices and (qn (s)) of security prices satisfying, the no-arbitrage condition can be written as:
n 0 pn (0)c0,n i + qn θi ≤ pn (0)wi
≤ pn (s)wis + pn (s)θin (s)r(s)f ors ≤ si pn (s)cs,n i where, qn θin =
X
qn (s)θin (s)
s
This is equivalent to the budget constraint in (5) by Lemma 2. Using the Lagrangian multipliers λi (0) at t=0 and λi (s) at t=1 state s and aggregating over the budget constraints one obtains :
λi (0)(pn (0)(c0,n i
−
wi0 )
+
si X
qn (s))θin (s)
+
1
si X
λi (s)pn (s).
1
n s (cs,n i − θi (s)r(s) − wi ) ≤ 0 si +1 f or(λi )R+
33
(ii)
Hence the combined budget constraint can be written as :
0 λi (0)(pn (0)(c0,n i − wi ) +
si X
qn (s)θin (s)) +
si X
n s λi (s)pn (s)(cs,n i − θi (s)r(s) − wi ) ≤ 0
si +1 f or(λi )R+ ((pn (s), qn (s)), R+ )
(iii)
From Lemma 2 of Florenzano and Gourdel (1994) the security market is incomplete iff λi (s) 6= λj (s) for some i,j for any s ≤ max si i
where (pn (s)) and (qn (s)) satisfy the no arbitrage condition i.e.
λi (0)qn (s) = λi (s)pn (s)r(s)∀i, s,
(iv)
when agents are heterogeneous securities market can either be complete or incomplete (exhaustive) Hence, from (iii) and (iv)
0 λi (0)pn (0)(c0,n i − wi ) +
si X
s λi (s)(cs,n i − wi ) ≤ 0
(v)
From non-satiation (5) and perfect divisibility the budget constraints are satisfied with equality at equilibrium, for s ≤ si ∀i. If the Arrow-Debreu market is imperfect at t = 0 by (7) then for some i : s,n s s maxj sj = si , (cs,n i − wi ) 6= 0f ors = si but (cj − wj ) = 0 for j 6= i from (5).
34
s This applies in the limit since cs,n i ((pn (s))) − wi is continuous in (pn (s)) by the
Maximum Theorem. Hence, since λi (s)p(s) > 0 (see Florenzano et.al (1994) Lemma 1) at equilibrium for s ≤ maxi si λi (s) can be any number in (v) for s = si for j 6= i and satisfy (iv) but not so for i for which it is a particular real number at equilibrium (Florenzano et. al. (1994) Lemma 2) Hence, the uniqueness of λi (s)∀i is violated. Hence, (7) ⇒ incomplete markets with respect to A. To prove the converse. Suppose w.l.o.g. only participant j is different from the rest. Let, λi (s) 6= λj (s) for s = s˜, for any particular i 6= j The utility functions are strictly increasing (5) holds and the commodity is perfectly divisible. ∗
λi (0)p∗ (0)(c0i − wi0 ) +
si X
s λi (s)p∗ (s)(cs∗ i − wi ) = 0
(a)
s λj (s)p∗ (s)(cs∗ j − wj ) = 0
(b)
1 0 λj (0)p∗ (0)(c0∗ j − wj ) +
sj X 1
at equilibrium. (* indicates equilibrium values) Let, w.l.o.g. sj > si and s˜ : sj ≥ s˜ > si Then, (a) and (b) ⇒
35
0∗ 0 0 λi (0)p∗ (0)(c0∗ i + cj − wi − wj ) +
si X
s∗ s s λi (s)p∗ (s)(cs∗ i − wi + cj − wj )+
1 si X
s∗ λi (s)p∗ (s)(cs∗ s)p∗ (˜ s)(csj˜∗ − ws˜) = 0 j − wj ) + λj (˜
s=si +1
At equilibrium summing over all i s
(λj (˜ s) − λi (˜ s))p∗ (˜ s)(csj˜∗ − wis˜) = o λj (s)p∗ (s) > 0f ors ≤ max sj (λj (˜ s) − λi (˜ s)p∗ (˜ s) 6= 0 ⇒ (csj˜∗ − wjs˜) = 0. Hence, markets cannot be imperfect, since p∗ (s)) is the limit of some convergent price sequence in the positive orthant. Hence the Proposition. Proved Theorem : If the contingent commodity market is imperfect an equilibrium does not exist, generically. Proof : Consider the saddle function
Φi ((csi ), (wis ), (p(s))) ≡ u(c0i ) +
si X
δπi (s)u(csi ) + λi [p(0)(wi0 − c0i ) +
1
si X 1
36
p(s)(wis − csi )], λi R++
Define the extended saddle function Z=
X
S X
(Φi +
s
s
λi (wis − csi )) ∀ λi R++ ∀ such s.
s=si +1
i
Let, the objective be max Z. It can be easily verified that iff the interior solution set of max Z is of measure 0 with respect to any non-atomistic measure over the domain of definition of ((ci ), ((λi ), (λi )), (p(s))i then Arrow-Debreu market at t = 0 is imperfect a.s. (Note 1). W.l.o.g. choose s
any λi > 0 f or all s > si ∀ i. By the Maximum Theorem, ((ci )i ) : ((usi , πi (s))sωi × (wis )sω ) → RN (S+1)+N S By the Kuhn-Tucker theorem (Rockafeller (1970) theorem 28.4) the solution set of the above objective is given by the set of equations DZ = 0, where DZ refers to the Jacobian of Z with respect to the above variables. The utility maps are strictly concave, bounded and monotonically increasing (S+1)
si +1 in int R+ and (wis )intR+
∀i, the mapping Z is differentiable in N S ×RS+1 ) N (S+1)×R+ +
int(R+
N (S+1)
with rank less than (2N S +S +N +1) due to assumption 3.Z : ⇒ (R++
NS × ×R++
S+1 R++ ) - a manifold (Auslander and Mackenzie (1963) pg.33). Hence by Sard’s
Theorem (Auslander et.al.pg. 108) the solution set of DZ=0 has measure zero on the space of ξ (indexed by (λi )i ’s) (i.e. every economy of class ξ is characterized by at most a column vector NS (λsi )R++
37
(see Florenzano et.al (1994) Lemmas (1 and 2)).
Note 1 : This is the antithesis to the application of the ”Cass trick” in proving existence of incomplete markets equilibrium as discussed in Geanokoplos (1990) sec 4.2, see also Duffie and Shafer (1985). Let i∗ : max si = si ∗ and s∗ = max si /i∗, si ∗ > s∗, then rk(Dp Z) = (s∗ + 1) a.s. and (s∗ + 1) < (S + 1). Also, the above result is invariant to any extension to R(si∗ +1) × R(S−si∗ −1) . Hence the theorem. Proved. 6.ECONOPHYSICS INTREPRETATION OF HAAG’S THEOREM (Earman & Fraser (2006)): Corollary 1: In the Econophysics limit of δ t ψ in the Hilbert Spaces of [(pt (s)).((ci s) − wi (s))] and [(qt (s)).A] is zero where ψ is the eigenvalue of the matrix A(ψ) given by :
38
ψ
0
0
0
0
0 0
0
ψ
0
0
0
0 0
0 S×S 0 0 0
0
0
ψ
0
0
0
0
0
0
ψ. . .
0
0
0
0
0
0
ψ at s∗ . . .
0
0
0
0
0
0
0
0
0
0
0
0
0 0
ψ is the Planck’s constant. This is henceforth referred to as the ’A(ψ)’ matrix. Also, s∗ = maxi si . Proof: The result of the Theorem derives from the facts that neither is rk(DP Z) upper semicontinuous in p nor is DP Z of full rank exceptin measure zero cases, which together lead to failure of upper semicontinuity of the demand for contingent consumption a.e.. This result is similar to the HAAG THEOREM OF QUANTUM FIELD THEORY in that multiplication of the A Matrix by the Identity Matrix of dimension of I does not give a square matrix which implies that the Incompleteness due to S < S∗ dimension does not lead to an equilibrium in the Hilbert Space of Asset Payoffs given by [(pt (s)).((ci s) − wi (s))] 39
and [(qt (s)).A] matrix even if in the limit the value of δ t ψ = 0. Corollary 2 : If the securities market is incomplete an equilibrium does not exist for ξ, a.s. Proof : Follows immediately from the Proposition and the Theorem. Proved 7.APPLICATION TO TRADING ”PLATFORMS” (ALGORITHMS WITH INTEGRATED SYSTEMS) INTEGRATING ASSET MANAGEMENT AND FINANCIAL MANAGEMENT NANOSTEPS : COROLLARY 3: The Systems Algorithms described in NanoSteps 1-6 defines a M-Branes Trading platform with Financial & Asset Markets integration. PROOF: The ECONOPHYSICS HAAG’S THEOREM proves the existence of MBranes String Integration of the Asset Markets (discrete periodic) over the continuous time interval generic financial markets with the integral over time 40
, over time, being achieved by the Steps 1-6 systems integration nanosteps using the Quantum Asset Matrices A(ψ) and δ t A(ψ) trading actions respectively over the short actions and long actions respectively. This will give rise to the Nanotechnology Algorithm of Systems Trading => M-Branes Trading Platform, created in NanoSteps 1-6...Q.E.D. The existence of such optimising systems integrated trading actions have been experimentally verified in Mallick (2011a) and Mallick, Sarkar & Roy (2006), establishing the existence of δ t A(ψ) action in Indian Industry. 8. FURTHER APPLICATIONS:
I have in this paper discussed the existence of equilibrium in an ArrowDebreu type economy where agents have Bounded Rationality in the sense that their preference ordering are not complete ex-ante over all the possible states of the world. I have seen that this provides a natural synthesis of optimizing and rule following behavior. Thus, I have provided a microfoundation for the existence of rule following behavior where the rules are nontradeable in the sense that they form a fundamental property of the economy and stock trades are conditional on these fixed rules (which span the entire set of agents). An intrepretation of Econophysics Haag’s Theorem shows why because of minimum energy requirements for running companies which implies minimum energy requirements for keeping every share of stock surviving in the long run it is not possible to have completely egalitarian social engineering, some have to be left out of financial markets. In the positive intrepretation δ t .ψ is the minimum interest cost to be invested in the 41
production of energy for every share of stock valued at Re. 1 which is true in the case of India after globalisation. This is independant of the form of the comunication network (Mallick (2014a)). This algorithmic description provides an inegrated nanotechnology system where the nanotechnology has been algorithmically constructed for the actions of Dynamic Asset Market equilibrium systems integrated with learning behaviors of human subjects. The Simple String theory developed in Arachane, Iyotama, Kunitomo, Tokuro (1997) and Sheppherd, Terrell & Henkelman (2008), has been algorithmically developed into D-Branes upgraded string of asset trading and bits flow (Mallick (2014a,b)) establishing existence by construction and applied to engineer a nanotechnology stock exchange using systems integration properties of nanotechnology stock exchanges, over time and space calibrated Networked National Stock Exchanges Nanosteps with Genetic Survival and Uncertain Genetic Growth Lagrangean Actions in Meanfields, developed elsewhere (Mallick (1993, 2014a,b)).
42
REFERENCES
ARAKANE, Y., H. ITOYAMA, H. KUNITOMO, A. TOKURA, Infinity Cancellation, Type l’ compactification and String S-Matrix Functional, Nucl. Phys., B486 (1997), 149-163. ARROW, K.J. & G.DEBREU, Existence of an equilibrium for a competitive economy, Econometrica, 22: 265-290, 1954. BERMAN, S. & A.SCHOTTER, Supergames & Diffusion Processes, Discussion Paper no. 79-01, C.V. Starr Center for Applied Economics, New York University, 1979. BHATTACHARYA,R. & M.MAJUMDAR, Random Exchange Economies, Journal of Economic Theory, 6: 37-67, 1973. BILLINGSLEY,P., Probability & Measure, John Wiley & Sons, New York, 2nd. edition, 1986. CYERT, R.M. & J.G.MARCH, A behavioral theory of the firm, Prentice Hall, Englewood Cliffs, NJ, 1963. DAY, R.H., Profits, Learning and the Convergence of Satisficing to Marginalism, Quarterly Journal of Economics, 81: 302-311, 1967. EARMAN, J. & D. FRASER, Haag’s Theorem and Its Implications for the Foundations of Quantum Field Theory, Erkenntnis 64: 305, 2006, online at philsci-archive
43
KABAN, A., L. NOLAN & S. RAYCHAUDHURI (2006), Factorisation of positive valued functions, ICA Research Network International Workshop (ICArn06), Liverpool. KEMENY, G. & J.L.SNELL, Denumerable Markov Chains, Springler Verlag, New York, 1966. KRICHEL, T., (2007), Information retrieval performance measures for a current awareness report composition aid, Journal of Information Processing & Management, 43(4), 1030-1043. KRUSSEL, P. & A.A.SMITH, (2012), Asymmetric Information and Signal Extraction among Entrepreneurs in a Competitive Economy, IIES working paper, Stockholm. LIPSTER, R.S. & A.N.SHIRYAYEV, Statistics of Random Process I : General Theory, Springer Verlag, New York, 1984. MAJUMDAR, M. & R.RADNER., (1991), Linear models of economic survival under production uncertainty, Economic Theory, 1: 13-20, 1991. MALLICK, S.K., Bounded rationality and Arrow-Debreu economies, Ph.D. dissertation, Dept. of Economics, New York University, 1993. −−−−, A. SARKAR & K.K.ROY , Financial Fragility in Emerging Markets in India : A cross-sectional study, Indian Economic Review, submitted, 2006. −−−− , Implementing Pareto Optimal Asset Prices, Quantitative Finance, Archived, 2011a. 44
− − −− , Integrating Markov Processes and Expected Utility Analysis, Mathematical Finance , Archived, 2011b. − − −− , Observability and existence of equilibrium under bi-annual planning with incomplete asset market, Mathematical Finance, Archived, 2011c. −−−− , The String Theory of Indian Stock Market Nanostructure System, Journal of Physics A, Archived, 2014a. − − −− , Deriving genetic algorithm for stock market systems, Journal of Physics D, Archived, 2014b. MAS-COLLEL, A.,An equilibrium existence theorem without complete or transitive preferences, Journal of Mathematical Economics, 1: 237246,1974. NIKAIDO, H., On the classical multilateral exchange problem, Metroeconomica 8: 135-145, 1956. − − −−, Convex Structures and Economic Theory, Academic Press, New York,1968. ROTHSCHILD, M.,A two-armed bandit theory of market pricing, Journal of Economic Theory, 9:185-202,1974. RADNER, R., Equilibrium under uncertainty, in Handbook of Mathematical Economics, Arrow, K.J. & Intriligator, M. eds., Vol II, North Holland, 1982. 45
− − −−, & M.ROTHSCHILD, On the allocation of effort, Journal of Economic Theory, 10: 358-376, 1975. SCHOTTER, A., The economic theory of social institutions, Cambridge University Press,New York, 1981. − − −−, Free market economics: a critical appraisal, St. Martin’s Press, New York, 1985. − − −−, K.WEIGELT & C.WILSON, A laboratory investigation of multiperson rationality and presentation effects, Discussion Paper no. 94-24, C.V. Starr Center for Applied Economics, New York University, 1990. SEN, A. (2013), S-duality improved superstring perturbation theory , arXiv:1304.0458. SHEPHERD, D., R. TERRELL & G. HENKELMAN : Optimization methods for finding minimum energy paths, Journal of Chemical Physics, 128, 134106 (2008). SIMON, H.A., A behavioral model of rational choice, Quarterly Journal of Economics, 69:99-118, 1955. − − −−, Rational Choice and the structure of the environment, Psychological Review, 63: 129-138, 1956. − − −−, Theories of Bounded Rationality, ch. 8 in C.B. McGuire and R. Radner eds. Decision and Organization, North Holland, Amsterdam, 1972.
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