Economics has entered the stage of deep transformation of its bases. The traditional method of constructing a scientific theory is first to synthesize and investigate mathematical framework; this traditional approach was taken as a principle of our research. Finally the mathematical theory of the general dynamic market model has recently been constructed, the main elements of which are given in this paper. The next step is to build models of specific real markets based on the general theory. The C# application Model was created especially to support the synthesis of concrete models based on the general theory. The most important goal of this paper is to propose cooperation in such research. Results of the research: the crucial factors, which ensure the market stability, are the market coherence and the market intention to adaptive expectations. If no any firm uses naive expectations in the market then with sufficiently small incoherence there is unique Nash equilibrium, which is stable for all acceptable values of parameters. The increase of naive expectations leads to stability loss, to flip bifurcations and finally to chaos. The increase of number of firms also as a rule leads to stability loss and finally to chaos. At sufficiently small changes in production per step, systems of general dynamic market model turns into systems of neoclassical microeconomics.
Introduction
Increasingly, processes and systems are researched or developed through computer
simulations and this trend is likely to continue [
The real economic processes make a clear demonstration that neoclassical "rational
man" is not their subject. In real economy "optimal imperfect decisions" are taken by
simple and non-expensive calculations, well adapted to frequent repetitions, to
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evolution [
Now institutional school of economics analyzes economic systems as a result of
evolutionary process of participants’ interaction [
Modern development of dynamic paradigm in economics is a wide stream of
researches. However it is a stream of examples which are not developing in the
general theory; their relations with real markets are often problematic [
Indeed, the idea of this research arose in the course of computational experiments
with the two-dimensional market model [
The next step is to build models of specific real markets based on the general
theory. In this direction, we have just started: we explored the Australian retailer
market, based on the general market model. For this purpose Data Republic
technologies were used, which provided comprehensive initial data for research, as
well as new methods of machine learning [
Generally speaking, computational modeling derives from two steps: (i) modeling,
i.e. finding a model description of a real system, and (ii) solving the resulting model
equations using computational methods. In the natural sciences it is often not so
difficult to find a suitable model, however the resulting equations tend to be very
difficult to solve, and can in most cases not be solved analytically at all [
The paper goal is to introduce the general dynamic market model; to introduce the specialized C# application Model and to demonstrate the use of the Model in building the general market model; the most important goal is to propose cooperation in future research.
The paper is organized as follows: in part 2 we introduce the general dynamic market model; in part 3 we demonstrate desktop application Model; part 4 concludes.
We had to omit proofs of propositions in order to fit the paper format. 2 2.1
First of all let’s introduce the basic concepts of the model. We consider a market of homogeneous product, where exogenous parameter n(t) indicates how many firms operate at time t . Each firm produces output xi (t) , where i 1,..., n(t) . Thus the n industry output of the market is Q(t) xi (t) at time t . Product price P(t) is i1 given by isoelastic demand function P(t) b(t) / Q(t) ( b(t) 0 ).
Formally the firm is defined by its objective function. Firm maximizes both its own profit X (P v) x fc , where v v(t) is the firm’s competitive marginal cost per unit in the market, fc fci (t) is fixed cost, and consumer surplus CS CSi (t) is a difference between maximum price which consumer can pay and Q real price. Here CS P(q)dq P Q , where parameter is the minimal technologically possible product quantity, i (t) specifies the segment of the market, which the firm believes its own and optimizes; usually i (t) i (t) , where n(t) i (t) is the given model parameter. Then
Q CS (b ln( ) b Q
Q Q Q) b (ln( ) 1) b ln where e (specific choice of does not affect the model dynamics and so further we suppose 1). Then general profit function П Пi (t) ( i 1,..., n(t) ) of firm is: П CS ((P v)x fc) b ln Q , where i (t) is share of short-run own profit i (t) in the objective function, i (t) 1 is share of consumer surplus CS , fc fci (t) is a fixed cost. As a matter of fact П is the weighted average of short-run profit and expected stable long-run profit.
The methods used by firms for planning are extremely diverse and hardly a general uniform description of them is possible in principle. Here, in the General dynamic market model, we use only one obvious and universal consideration. If firms i and j are identical at moment t and in particular they have the same planning at moment t , then it is natural for firm when planning to suppose that their production quantities will be equal at next moment t 1 too.
In most general case for mixed naïve and adaptive expectations xe (t 1) ij (t)xi (t 1) ij (t)x j (t) ( 0 ij (t) rij (t) , ij (t) (1 pij ) (rij (t) ij (t)) 0 ). (1) ij
Here ii (t) 1 , ii (t) 0 for all i and t . Thus according to (1) prospective industry output of a market expected by a firm i during next time period t 1 equal to is n(t) n(t) n(t) Qie (t 1) xiej (t 1) ij (t) xi (t 1) ij (t) x j (t)
j1 j1 j1
Then under planning of their quantity xi (t 1) in next period t 1 each firm i
(2)
maximizes objective function П ie i 1,..., n(t) . Thus we obtain the equations of
dynamics of the general market model [
In this paper we consider all actions, expectations and strategies of firms in shortrun time period , therefore the equations parameters n , ci , di , ij , ij etc. are assumed further as constants which are independent of time. 2.2
Let’s define the key notion of market coherence in the general market model. Let xiek (t 1) is quantity of output of firm k expected by a firm i , n Qie (t 1) xiek (t 1) is prospective industry output of a market expected by a k 1 firm i during next time period t 1 from (2). Then for firms i and j we put ij max t
Qie (t 1) Qej (t 1) , where Q(t) is industry output of the market in period
Q(t) t , is the time period considered. Value ij characterizes incoherence in expectations of firms i and j . Therefore value max ij we will call the level of i, j n xk (t) x j (t) k 1
n and all t Q(t) ( t ). Then for sufficiently small , for all i 1,..., n
n incoherence of market expectations or simply the market incoherence. Finally, the value 1 we will call the level of coherence of market expectations or simply the market coherence.
Let
P min P(t) , t maix i , B 4 v n ,
P
D 2n , max i , i 1 i n ci ik , Cij ci , ij (t) xi (t) , where i, j 1,..., n .
k 1 c j x j (t)
Proposition 1. (Main lemma) Let 2Q(t) Qie (t) for all i 1,..., n , t and j be the number of a firm with a production volume x j (t) above the average: ij (t) Cij (1 D ) B .
Let’s define the second key notion in the general market model. According to (1), the higher intention of a firm to plan with adaptive expectations the closer ij to rij . So in the notations (1) the value
n ij x j i, j 1(i j )
( 0 1) rij x j i, j 1(i j ) we will call the intention to plan with adaptive expectations in the market or simply the market intention to adaptive expectations.
Proposition 2. At 1with all sufficiently small and there is the unique Nash equilibrium for the dynamic system (3) and this equilibrium point is stable for all admissible values of the parameters.
Proposition 3. At 0 with all sufficiently small and there is the unique Nash equilibrium and this equilibrium point is unstable and hyperbolic for all admissible values of the parameters of the almost any dynamic system (3).
But what is the behavior of system (3) for 0 < 1 ?
Proposition 4. For dynamic system (3) with all sufficiently small and with decrease in from 1 to 0 flip bifurcations (cycle doubling bifurcations) occur throughout the Sharkovskii’s order and finally the state of dynamic chaos arises.
This proposition, unlike the previous ones, even formally uses the results of computational studies of dynamics of the two-dimensional market models. For such application computations through Model were carried out.
Proposition 5. For dynamic system (3) there is such a segment [ a; b ] ( 0 a b 1), that for all [a; b] , with all sufficiently small and , with increasing n flip bifurcations (cycle doubling bifurcations) occur throughout the Sharkovskii’s order and finally the state of dynamic chaos arises.
Proposition 6. For sufficiently small xi (t 1) – xi (t) and respectively small i ( i 1,..., n , t ) the dynamics of system (3) has the unique stable Nash equilibrium point with an equilibrium price in the market.
Proposition 6 means fulfilling the premises of classical economic theory at sufficiently small changes in production per step. Speaking informally, at sufficiently small changes in production per step, systems of general dynamic market model turns into systems of neoclassical microeconomics. 3.
C# application Model created to support the modeling process. Its main goal is to maximize research support, to provide the best service for a regular cycle: hypothesis experiment hypothesis. The basic requirements implemented in the specialized software application are i) obtain a model in the subject language without codes; ii) immediately obtain all the necessary research tools already configured for this model; and most importantly, iii) it is easy to modify the model depending on the results of computer experiments. The main requirement is that new ideas should immediately be put into experiments for verification. In natural experiments it is impossible to immediately implement a new idea, immediately creating a new device. But in Model with computer simulations we can do this using the program window with the appropriate tools: new experiment results create new ideas that we test immediately by creating appropriate new windows.
The following figure shows the main program window, which automatically appears when you open the Model.
But the main tool to support the synthesis of concrete models using the Model is a simple modification of the current model. After pressing the second menu button (the “Edit” button) we get the window of fig. 8.
The editing window on the screen is located above the current model window, which allows both windows to be used simultaneously. Left-click on the model equation in The dynamical system field to go to the Equation field, where you can change it. After clicking the Add button the modified equation is written back to The dynamical system field. Similarly, such a procedure can be performed with parameters in the Add Parameter field. 3
Economics has entered the stage of deep transformation of its bases. The traditional method of constructing a scientific theory is first to synthesize and investigate mathematical framework – the general market model according to the new dynamic paradigm of economics. And then we can study complex real systems which are grounded on this basis. This traditional approach was taken as a principle of our research.
The mathematical theory of the general dynamic market model has recently been constructed, the main elements of which are given in this paper. We show that the definition of our model really consists only of simple and obvious constructions; that there is nothing in it that would not inevitably enter into the model of any market of homogeneous products.
Dynamics of the general market model with sufficiently small incoherence in the market is stratified on dynamics of two-dimensional markets. The main lemma of theory allows us to generalize properties of simple two-dimensional market models by means of formal reduction to the general market model of this paper.
The crucial factors which ensure the market stability are the market coherence and the market intention to adaptive expectations. If no any firm uses naive expectations in the market then with sufficiently small incoherence there is unique Nash equilibrium which is stable for all acceptable values of parameters. The increase of naive expectations leads to stability loss, to flip bifurcations and finally to chaos.
The increase of number of firms as a rule also leads to stability loss, to bifurcations and finally to chaos in the market. Thus behavior of general view markets is sharply different from their usual behavior in neoclassical microeconomic theory. However at sufficiently small changes in production per step, systems of general dynamic market model turns into systems of neoclassical microeconomics. This means that general dynamic theory does not contradict logically the neoclassical static theory, despite their striking unlikeness.
The next step is to build models of specific real markets based on the general theory. The C# application Model was created especially to support the synthesis of concrete models based on the general theory. To date, we are not aware of applications created for this purpose. The most important goal of this paper is to propose cooperation in such researches.