Market behavior and formation of value
under absence of information on prices
The intent of this paper is to stress on the three following items:
(a) the general character of the methodology which is expected to be appropriate for the development of the theory of value;
(b) the process of formation of the economic value under the most general assumptions on making economic decisions;
(c) the problem of a measurement for the economic value.
The most of the results presented here come from the article which is expected to be published in Russian in The Proceedings of the Independent Agroeconomic Society of the Russian Federation, issue 2, Moscow, Izdatelstvo MSHA, 1998.
In the majority of economic models (in particular of the Walrasian type) the concept of value is external in relation to models. It does not result from their character. Value variables are comprised in them with no regard to what should be an object of the model for value is inherent in its commodities. The reason to enter the value in the model is just existence of the value in the reality. Such models are suitable for research of properties, but not of essence of the value.
Leonid V. Kantorovich used alternative approach to research the value. He applied this approach to economic interpretation of a problem of linear programming and afterwards expanded it on the general optimization problem. In these problems, value is a result of the nature of the object. Being an inevitable property of the optimal solution it appears when solving for optimum rather than at a stage of the formal description.
The model described below follows this approach in research of decentralized economic systems. The purpose of modeling is a study of formation of value in conditions, when the economic agents have no information on the prices and on the intents of other agents. Therefore behavior of the agents and, in particular, the supply and demand depend neither on the prices, nor on the fact of their existence in the market. The preferences of systems are assumed given, transitive and complete. No other assumptions on preferences needed. The agents have opportunities of free interchange.
The economy is represented in the form of the economic system H = {Hk | k Î K}. Hk is an elementary system (in economic interpretation it corresponds to an agent). Each the elementary system follows the rules
Under these constrains the elementary system Hk selects the preferred state given the preferences which orders the field of its possible states hk = (ak, xk) . Here and in the rest of the paper K means the set of indices of elementary systems; xk , xkt — an array of outputs and inputs (positive and negative components respectively) of the system Hk; ak , akt — non-negative array of commodities (stock) under the ownership of Hk ; ekt — an array of net result of interchange operations of Hk ; t — time index; Xk — the production set of Hk. Each Hk given any instant of t aims to achieve the most preferable state hkt+1 by means of both commodities in ownership and exchange. The #I´ #K-matrix Et consists of the columns ekt such that and represent some interchange between some subset of elementary systems.
The preferences set relations between two given states h'k and h''k of the system Hk. One can easily see that this concept is an equivalent to ordering the arrays of stock: under the constraints of their stocks the elementary systems ever select the state which is the best from their points of view. So, it is reasonable to let the notation have the same meaning as , where stands for the best (the most preferable) state of Hk given the commodities ak under its ownership. Speaking about some array ek which is complementary to ak , its desirability in comparison with other arrays complementary to the same ak depends on both the nature of and the current stock ak . This kind of desirability of any change in the stock (I suggest the term "marginal preferences") can be formally described in the following way:
Here denotes the relation of marginal preferences of Hk. To simplify notation it is supposed that Hk is currently in the state hk and, therefore, owns ak. So the base to compare different changes in the stock is the stock ak. As far as is transitive, is also transitive. The corresponding relations of the strict marginal preferences and the marginal indifference » k are also transitive.
Assume the set I of all the commodities in the economy H consists of two subsets, namely In and Ir, so that In I Ir = Æ , In U Ir = I. The former embodies the commodities that can be measured with non-negative integers, the latter — with non-negative real numbers. Now I'll introduce the set I$ Í Ir of the commodities having the following attributes despite of k:
a) given the complementary stock yk (which, generally, can have negative components assuming ak + yk belongs to the set of the possible states of Hk) one can nominate a real number c so that cii » k yk ;
б)
c > 0 purports and c < 0 purportsGenerally I$ can be empty.
Interchange E is considered to be expedient for Hk, when the following is true (assuming ek is a column of E):
(2)
Hk blocks the interchange E, if the following conditions are false:
(3)
Interchange E is possible, if no one of Hk blocks it. Possible interchange should be qualified as expedient if it is expedient for at least one Hk.
Notation Ak means the set of such the arrays of stock, that the preferences of Hk are defined for every two arrays of this set. No assumptions made of the topology of the set. Let be the set of the arrays ak0 of commodities under the ownership of system Hk , for which where ak denotes some array from Ak. Ek means the set of all the net interchanges meeting the condition Hence, Hk do not have enough commodities to participate in any interchange but in that belonging to Ek. If there is such an interchange that all the elementary systems have enough commodities to take part in it then this interchange is a member of the set of all the interchanges which meet the conditions for any k Î K.
Considering the aim of this study it is reasonable to make an assumption according to which the interchange E to be actualized in the economy H is a random member of E. The fact that we make no additional assumptions on the distribution of probabilities of different interchanges of E may mean that E is really selected by chance; or that we do not know how it is selected so it seems to us that this is a random process; at last, that we do know the rule of selection but ignore it in our study. Probably taking this rule into consideration is a way to achieve additional significant results.
In the context of economy H the following statements are true.
1. If exists some chain of interchanges due to which the economy H shifts from the state h1 to the state h2 then in the state h1 a possible interchange exists which shifts H directly to the state h2. 2. Suppose that some elementary system Hk have shifted from hk1 to hk2 due to some interchange. Then the set of states, which Hk can reach due to another interchange starting from the state hk2, must be the subset of the set of states which Hk can reach due to an interchange starting from the state hk1. If the interchange moving Hk from hk1 to hk2 is expedient then the former and the latter sets of reachable states must not be the same.The two statements directly follow from the assumption of transitive character of preferences.
The relation is a condition under which Hk does not block the interchange E. This condition leads to the following definition: the set of all the arrays of net results of interchanges which are not blocked by Hk in the state hk is
3. Assume the following: (a) in each set regardless of k there is an array the most preferable for Hk; (b) in the economy H there are the possible interchanges. Then among these possible interchanges at least one interchange exists which shifts to the Pareto snare where there are no expedient interchanges. This conclusion follows out from statements 1 and 2.In particular the most preferable array in ak + Ek exists when: (a) preferences are continuous and all the sets Ek are compact or (b) when I = In. If the preferences are continuous and the stock exists which is more preferable than any array in any ak + Ek but Ek are not sure compact then the Pareto optimum can not exist. In this case among the variety of interchanges one can find such interchange that shifts the economy to the state in which all the components of any expedient interchange are less than a given very small number.
The net interchange result ek may be represented in a form where means the quantities of commodities sold, bought. It is natural that If the economy persists in the state of the Pareto optimum then, if there are possible interchanges, it must be Let's study the interchange between two participants, namely Hk and Hl (where k, l Î K) in the state of Pareto snare. To simplify notation, I shall denote the state of Hk before exchange E as and of Hl — as Here it must be and Then one can easily achieve the following result.
4. Two statements are not jointly feasible in the state of the Pareto snare regardless of k:(a) there exist semi-positive и such that
(b) the relation is true.
In the Pareto optimum it is obvious that Furthermore the sentence (a) implies hence it is true that Sentence (b) implies that each one of the arrays and represents the better suit for Hl than Therefore Hl is interested in changing for each of them, or else the three corresponding interchanges should be qualified as expedient for this elementary system. Among them at least change for is not blocked by another side, namely Hk. The conclusion is that there is at least one expedient interchange in the economy H. This sentence is in contradiction with the assumption that H persists in the Pareto optimum. This proves the statement 4.
Assume I$ ¹ Æ . This immediately implies the cardinal character of all the preferences of elementary systems in the economy. If cii ~ k ak where i Î I$ then c perfectly characterizes the rate of desirability of the stores ak. Note that the assumption of non-empty I$ requires, in turn, a variety of formal conditions which are the subject of separate study. In particular, this assumption is not contradictory if the conditions of Debreu theorem on the cardinal measurement of desirability take place in H.
Now let again Hk persist in a state and Hl in a state
5. If one can qualify some i Î I$ and non-negative ck, cl such that then ck = cl.This statement is directly follows from the statement 4.
As a consequence, if in the Pareto snare one can qualify some stock y 0 such that:
(a) the given Hk has the property y + ak , ak 0;
(b) some other elementary systems Hl , l Î K' Í K, have the properties y + al , al 0;
(c) y » k cii , i Î I$;
(d) cii (y + al) " l Î K';
then it can be easily proved that for each Hl the relation y » l cii is true. The way to prove that is a study of a specific possible interchange, namely changing y for y.
The properties of the formal economy H can be transferred to the real economy in the case of:
(a) absence of appreciable externalities;
(b) the conditions under which the preferences can be treated as typically transitive.
To achieve wider generality, an additional study needed, which is expected not to be very complex. In the framework of real equivalent of H, in this study it is shown that assuming other factors are the constants every interchange reduces the further interchange opportunities (this results from statement 2).
In the reality, it is reasonable to consider the terms of statement 3 acceptable. As a result among the variety of possible interchanges (if possible interchanges exist) there is at least one interchange (maybe between more than two agents) which shifts the economy in the Pareto optimum. In this particular state, if reached, in accordance to statement 4, two sets of commodities are marginally equivalent if each of these sets is indifferent relative to the sets supplied by the two agents taking part in two-side interchange.
If in the formal economy persisting in the Pareto optimum the following statements take place:
(a) there exists a commodity i which due to its nature is able to be a measure of value (that means i Î I$);
(b) there is a possible interchange between exactly two elementary systems Hk and Hl ;
(c) both Hk and Hl have the commodity i in amount exceeding some minimum;
then both and are the equivalents to the same amount of the commodity i from the point of view of both Hk , and Hl . This consequence from the statement 5 is a subject to be treated as the possibility of formation of the value having the following properties:
(a) it is formed as a result of random expedient interchanges;
(b) it requires a commodity (at least one) which is potentially able to play a role of measure of the value;
(c) given the store of commodities its value is common for all the agents in the market.
Some agent who does not have enough commodity i to buy some store of commodities worth the amount c of commodity i is able to consider it more preferable than the amount c of commodity i (unlike every agent having enough i). Nevertheless the only real opportunity selling this store is to gain the only c of i or equivalent. Thus the "effective" value, in other words the value which control prices, is the equivalent of the opportunity cost and is the same for each participant of a possible interchange.
If more than two sides participate in an interchange then and may differ (sure k ¹ l). But nevertheless and must be equivalent if the Pareto optimum persists. This easily implies expanding properties of value on the case of interchanges with more than two participants.
For the sake of interpretation, it is natural to treat the statement 5 in terms of value. The statement 4 has, generally, the same meaning as the statement 5 though one meets difficulties trying to directly interpret it. If one introduces a value as some reality which does not necessary require quantification but is able to be reflected by means of stores of commodities then she can treat the statement 4 as the possibility of formation of the same value for all the economic agents (and hence for the economy as a whole) due to any chain of expedient interchanges (suppose there is enough time to complete all the possible interchanges) under the conditions accepted here with no regard to the existence of a potential measure of value.
Mathematical notations used in this paper
an array resulting from x my means of replacing negative components with zeroes.
an array resulting from x my means of replacing positive components with zeroes.
#X — the number of elements of a countable set X.
ik — the k-th column of the unit matrix I.