Research goals and objectives: to study of the simplest possible mathematical model of microeconomic system with different social responsibilities of agents in accordance with agent based computational economics paradigm using a desktop application. Object of research: microeconomics system with heterogeneous agents. Subject of research: mathematical model of microeconomic system with different social responsibilities, equilibrium and disequilibrium states of the systems using desktop application. Research methods are: optimization methods, bifurcation analysis, stability analysis, simulation methods, game theoretic approach. Results of the research: dynamic models of microeconomic system with different social responsibilities (reciprocator and selfish types) were created using specially developed desktop application. Based on software module the conditions of stability, bifurcation and analysis were obtained. As a result of numerical investigation we have found that flip bifurcations occur with increasing of firms' number in the market. If two-thirds of firms use naive expectation, then there appears the state of dynamic chaos.
Information technology in the economy made it possible to model artificial societies
and study economic models through the computer simulation. This new school in
science is called agent based computational economics (ACE) and creates absolutely
new possibilities in economic research of microeconomic systems [
Evolution appeared due to variation and selection process [
Microeconomics has entered the stage of deep transformation of its bases. In recent
years, researchers have abandoned the traditional main assumption - the perfect
rationality as the basis of unconditional behavior of the economy. Neoclassical
"rational man" does not exist in reality, as individuals act according to established
rules, do not have full information and do not always maximize benefits [
Unlike traditional simultaneous, instantaneous achieving equilibrium by perfectly
rational firm in the real economy, "best imperfect decisions" taken by the simple and
non-consumable calculations, are well adapted to frequent repetitions in the evolution
process. If the system has multiple equilibria, the repetitive interactions, evolutionary
dynamics of selection mechanism is a better equilibrium [
New paradigm of microeconomics is a combination of the dynamical systems’
nonlinear theory and mathematical programming, including game theory and optimal
control theory [
Example of this new paradigm is an evolutionary model of oligopoly competition
where agents can select between different behavioral rules to make decisions on
quantity or price settings [
The traditional method of constructing a scientific theory is first to synthesize and investigate the example of the simplest possible mathematical model of microeconomic system.
These new approaches make a clear explanation for some events in economics
rather than traditional mainstream. The evolutionary approach and analysis of the
dynamics allow to explain why one type of firm ousts another from the market, why
sometimes the economic system is stable, but in other cases is unstable [
Microeconomic system consists of two types of agents with heterogeneous responsibilities, such as selfish and reciprocator firms. Firms’ social responsibility implies that they have not only selfish goal of increasing their own profits immediately, but are also willing to sacrifice a part of their short-run profits and to save consumer surplus in return for stable nonmaximal long-run profit. In other words reason of reciprocator firms’ appearance is their desire to obtain stable profit in longrun period instead of short-run maximal profit.
The purpose of the paper is a study of the dynamic microeconomic system through synthesis of the simplest possible mathematical model according to agent based computational economics paradigm using our specially developed software module.
This paper is a direct continuation of research [
The paper is organized as follows: part 2 describes the simplest mathematical model of microeconomic system according to new paradigm; part 3 demonstrates desktop C#-application for numerical experiments; part 4 includes numerical investigations of microeconomic system using this application; part 5 concludes. 2
First of all we show that our mathematical model is the simplest one in the ACE paradigm. In general, almost any microeconomic market model is constructed as follows: 1) n firms operate in the market (to simplify the notation suppose n = 2 ); 2) these firms produce homogeneous products in quantities x1 (t) and x2 (t) in time period t ; 3) they use adaptive approach, i.e. they try to predict the quantity of their competitor in the next time period t + 1 where xej (t +1) be expected quantity of rival j by a firm i in period t . Then under planning of its quantity xi (t +1) in the next period the firms decide the following optimization problem:
MaxП1(x1(t + 1); x2e (t +1)) , MaxП2 (x1e (t +1); x2 (t + 1)) , where Пi , i = 1, 2 is a profit function of firm i. The assumption about unchangeable quantity of the competitor (i.e. firm i will use x j (t) instead of x ej (t +1) when it solves the optimization problem) is an example of imperfect, bounded rationality in firm’s strategies; it is called naive expectations. As a rule these two approaches (adaptive and naive) coexist in the market with a certain probability. Our model is based on this assumption.
b
Q
We consider a market of homogeneous product, where n firms operate, among them are k identical reciprocator firms with the same output x and n - k identical selfish firms with the same output y . Thus the industry output of the two types of firms is Q = k ⋅ x + (n - k⋅) y . Product price P is given by the inverse market demand function P = P(Q) =
( b > 0 ). This is simplest demand function leads to a nonlinear dynamics. Alternative demand function is linear P = P(Q) = b - c⋅ Q ( b, c > 0 , wherefrom Q = k ⋅ x + (n - k⋅) y≤ b ) is used to test the general model’s properties.
c This model is uniquely defined by objective functions of firms and types of their expectations. It does not use any additional assumptions or restrictions.
The objective function of selfish firm-egoist is a profit π Y = (P - v⋅) y , where v is the firm’s cost per unit in the market. Reciprocator firm maximizes both its own profit π X = (P - v⋅) x and consumer surplus CS of its market segment (loyal consumers) CS = γ Q ∫ P(q)dq - P⋅ Q , where parameter γ specifies the segment of the market, k ε which the reciprocator firmr believes its own and optimizes it ( 0 < γ ≤ k ); ε is the minimal technologically possible product quantity. Then CS = γ b ⋅ ln( Q ) - b⋅ Q = bγ⋅ ln( Q-) 1= b⋅γ ln Q , where εˆ = ε ⋅ e (specific k ε Q k ε k εˆ choice of ε does not affect the model dynamics). Using CS (difference between price which consumer can pay and real price) profit function of reciprocator firm is: ПX = α ⋅ ( P - v⋅) x+ (1- α⋅ ) C=S ⋅α (- P ⋅ v+) x - (1⋅ α ⋅) bγ ln Q , k εˆ where α is share of own profit π X in the objective function. In other words ПX is a weighted average short-run profit π X and expected factor of stable long-run profit CS from loyal consumers.
Now let us consider the dynamics of this model with discrete time t = 0,1, ... . Let xi (t) , y j (t) be the outputs at time t of reciprocator ( i = 1,..., k ) and selfish firms ( j = 1,..., n - k ), respectively. On the basis of these values at time t each firm finds the optimal value for its own quantity setting in the next moment t + 1 , maximizing its objective function.
Each reciprocator firm i ( i = 1,..., k ) is looking for such value of xi (t + 1) at which it maximizes its own profit function, suggesting that all other firms leave their quantities x- i (t) , y j (t) unchanged: xse (t + 1) = xs (t ) , yej (t + 1) = y j (t ) : MaxПi (x1e (t + 1),..., xie- 1 (t + 1), xi (t + 1), x+ie 1(t + 1),..., xke (t + 1); y1e (t + 1),..., yne- k (t + 1)) =
MaxПi (x1 (t ),..., xi - 1 (t ), xi (t + 1), x+i 1 (t ),..., xk (t); y1 (t),..., yn- k (t))
Hence, in view of [
Similarly each selfish firm j ( j = 1,..., n - k ) is looking correspondingly for such value of y j (t + 1) at which it maximizes its profit π j , suggesting that all other firms leave their quantities xi (t) , y- j (t) unchanged:
Maxπ j ( x1e (t + 1),..., xke (t + 1); y1e (t + 1),..., yej- 1 (t + 1), y j (t + 1), y +ej 1(t + 1)..., yn-e k (t + 1)) = Maxπ j ( x1(t),..., xk (t); y1 (t),..., y j- 1 (t ), y j (t + 1), y +j 1 (t ),..., yn- k (t )). (1) (2)
The last equations of this system mean that k reciprocator firms and n - k selfish firms are identical for all t .
Since all selfish firms and all reciprocator firms are assumed as identical and they have the same strategies at moment t so it is natural to suggest that their production quantities will be equal at next moment t + 1 too. In accordance with such expectations each reciprocator firm under quantity setting assumes that xi (t + 1) = x1e (t + 1) = ... = xke (t + 1) . Therefore, to determine its quantity in the next period this firm solves the following optimization problem:
MaxПi (x1e (t + 1),..., xie- 1 (t + 1), xi (t + 1), x+ie 1(t + 1),..., xke (t + 1); y1e (t + 1),..., yne- k (t + 1)) =
MaxПi (xi (t + 1),..., xi (t + 1), xi (t + 1), xi (t + 1),..., xk (t + 1); y1(t),..., yn- k (t)). Similarly each selfish firm j in accordance with common sense believes under y j (t + 1) = y1e (t + 1),..., yne- k (t + 1) . So this firm solves the quantity setting that following optimization task:
As above, the last equations of this system means the identity of all reciprocator and selfish firms.
Maxπ j ( x1e (t + 1),..., xke (t + 1); y1e (t + 1),..., y ej- 1(t + 1), y j (t + 1), y+ej 1 (t + 1),..., yn-e k (t + 1)) =
Maxπ j ( x1(t),..., xk (t); y1 (t + 1),..., y j (t + 1), y j (t + 1), y j (t + 1)..., y j (t + 1)). y1(t) = ... = yn- k (t), j = 1,..., n - k.
It leads to such dynamic system [
In real life both decision making approaches (adaptive and naive) coexist in the market with a certain probability p for adaptive and correspondingly q = 1 - p for naïve expectations. According to such expectations typical (representative) reciprocator firm i suggests that production quantities of its rival j will be equal to xej (t +1) = p ⋅ x j (t) + q ⋅ xi (t + 1) ( j = 1,..., k , j ≠ i ). Typical reciprocator firm i ( i = 1,..., k ) resolves following optimization problem:
MaxПi (x1e (t +1),..., xi (t + 1),..., xke (t + 1); y1e (t +1), ..., yne- k (t + 1)) =
MaxПi ( px1 (t) + qxi (t),..., xi (t +1),..., pxk (t) + qxi (t +1); y1(t),..., yn- k (t)). Similarly typical selfish firm problem:
j ( j = 1,..., n - k ) solves following optimization
Maxπ j (x1e (t +1),..., xke (t + 1); y1e (t +1),..., y j (t +1),..., yne- k (t +1)) =
Maxπ j (x1(t),..., xk (t); py1(t) + qy j (t + 1),..., y j (t + 1),..., pyn- k (t) + qy j (t + 1)).
This hybrid case leads to following dynamics [
y y1 (t) = ... = yn- k (t), j = 1,..., n - k, d , where d = 1 1- α b(1+ p( k- 1)) and wx = q(k - 1)xi (t+) ( n- k) y j (t), 2 α vk wy = kxi (t) + q(n - k- 1) y j (t). (3) 3
For our research we developed desktop application Model for numerical experiments with dynamical systems on two-dimensional phase space. The main purpose of the application is to provide the best service for research cycle: hypothesis → computing experiment → hypothesis. For a given differential equations system and parameters set of the model the application immediately generates a window of this model. It makes it easy to specify and modify the considered model. Window tools allow us to obtain trajectories, phase curves, bifurcation diagrams and its animation after setting of the initial parameters.
Note that C#-application is created on GUI-based C # System.DrawingiSystem.Windows. All calculations concerned with a model, are localized in method Calc of application Model which allows us to modify easily the equations of the model, or switch to other models. To work with continuous models, i.e. with differential equations, the program uses OpenMaple interface access to the Maple computational kernel from various programming languages, such as C#, Java, VisialBasic etc. The program also uses namespace System.Runtime.InteropServices, allowing to make reference to the dynamic assembling of Maple kernel - maplec.dll. The application window for the model of this paper is shown in the following fig.1.
The right side presents 6 kinds of graphs displayed by the application; their examples are set forth in the paper. Selected switch indicates that here the graph of trajectory x(t) is selected. On the left side counters allow us to specify the parameters of the model and the initial values of the trajectory. After their setting automatically and immediately appear iterations of calculating the coordinates of a trajectory below the counters and their image in the center of the chart window, at that the number of iterations can be set on the scroll bar over graph. Software Module displays an animation of a selected path after pressing the button near with scroll bar. After pressing button Modelview we can see on the left and above information about the model, its equations and parameters. The model data and stored and we can go immediately to its window using the name of the model.
4
One of the main assumptions of orthodox neoclassical microeconomics is the idea of automatic stabilization of a market as a result of increasing of independent firms’ number under quantity competition. This is the realization of Adam Smith’s ‘invisible hand’. Let consider the behavior of our model with the growth of parameter n (total number of firms). Let n = 34 ; the number of reciprocator firms k = 32 ; b = 200 ; marginal cost v = 2 ; the share of profit in the objective function of reciprocator is α = 0.9 ; probability of naive expectations is q = 0.65 . The trajectory of dynamical system (3) with such initial parameters and initial output point x0 = 0.1 , y0 = 0.1 is shown in the following fig.2.
Fig. 2. Quantity trajectory of firm-reciprocators under initial conditions ( n = 34 , k = 32 , b = 200 , v = 2 , α = 0.9 , q = 0.605 , x0 = 0.1 , y0 = 0.1 ) Here along horizontal axis are given iteration of system (3) from t = 1 to t = 200 , along ordinate axis are given corresponding quantities of reciprocator firm xi (t) , i = 1,..., k . As you can see from the graph, the path quickly converges to the equilibrium quantity x* = xi* ≈ 1.5 . The graph for quantity path of selfish firms y j (t) , i = 1,..., n - k looks like this one ( y* ≈ 0.5 ) under same conditions.
Let consider the graph of the trajectory for the same parameters except n . Now let n = 36 (fig. 3). In fig. 3 instead of equilibrium point there appeared bifurcation and a stable cycle where x(t) approximates to point x* ≈ 1.9 for even t and to point x* ≈ 0.7 for odd t . After doubling the lag between iterations is either even or odd iterations. Thus either quantity x* ≈ 1.9 or x* ≈ 0.7 respectively will be equilibrium output.
Stable cycle has four points for n = 44 (fig. 4). There was a flip bifurcation. The more firms’ number the more series of doubling bifurcation cycle according to Shvarkovskii’s scale. State of dynamic chaos already exists for n = 100 (fig. 5).
To understand chaos effect which contradicts to orthodox microeconomics during growth in the number of firms, let us consider how the number of reciprocator firms impacts on model dynamic for fixed n. Let k = 1 , n = 100 , all the other parameters are the same as above. The following figure shows graph x(t) of corresponding trajectory (fig.6). This trajectory converges to Nash equilibrium. Now let k = 3 . There exists a flip bifurcation (fig.7). Assume further that k = 10 . We have a new flip bifurcation (fig.8).
If k = 32 or k = 100 - 32= 68 we will get the same chaos as in fig. 5. For
k = 100 - 10= 90 , k = 100 - 3= 97 , k = 100 - 1= 99 we obtain the same dynamics as
in fig. 8, fig. 7 and fig. 6 correspondingly. Thus if different types of firms are
uniformly presented in the market, quantity dynamics can be complex and transform
to chaos after increasing firms’ number. The destabilizing role of agents’ number due
to evolution is well-known for oligopoly games [
The reason of instability market share with increasing firms’ number n is revealed
in the following proposition 1. According to [
Here x* = x1* = … = xk*, y* = y1* = … = yn-k* . Then Proposition 1. For any given b, v> 0 and α (0≤α≤1) Jacobian J of system (1) for Nash equilibrium (4) is proportional to value n–1 for sufficiently large n. Its absolute value increases with growth of n, if k > ε and k - 3> ε for any ε > 0 .
n n 4
Since the volume of the phase space under the influence of the dynamics of (1) at fixed point (4) is proportional to the absolute value of the Jacobian at this point, then proposition 1 means increased instability with the increase of n .
Proof. Here Jacobian J of system (1) at point (4) equals J xx = J yx = b (k - 1) v )→ 0 for n→∝. Similarly we obtain following expression for second b ⋅ ((k - 1⋅) x*+ ( n- ⋅k ) y+* ) v d=2 b 2 k ⋅ - 1 + v n
1 o .
n 2 b ((k - 1)x+t ( n- k )+yt d 2
v possible n, k, b, v> 0 and α (0≤α≤1), Q.E.D.
b 2 v But by the data k - 3 > ε for ε> 0, which guarantees that the following expressions n 4 do not equal zero: b b v v
How we can achieve stability of a competitive market with a large firms’ number?
We found that adaptive behavior is a way of achieving of market steady state [
(5) Here x* = x1* = … = xk*, y* = y1* = … = yn-k*. The trajectories of the system (2) converge to Nash equilibrium (5) for any acceptable initial values.
With the growth of adaptive expectations (i.e. with increase in p) stability enhances, predictability of the market becomes stronger; with the growth of naive expectations (i.e. with increasing q = 1 - q) the market loses stability, chaos increases. The process of loss of stability and transition to chaos of dynamic system (3) is the most visual in the bifurcation diagram (fig.9).
As the above flip bifurcation can be interpreted as splitting of equilibrium state into several directions, one of which is selected by the market in the evolution of firms’ strategies. If two-thirds of firms use naive expectation (q≈0.67), then there appears the state of dynamic chaos in the market. Facilities of Model application allow us to make sure that the above number is a universal constant which does not depend on model parameters and demand function.
If any type of firms increases their profit more quickly than their rivals then these
firms will survive and expand their type of social responsibilities between all firms
[
One more unexpected finding of our research during computational experiment is that in this model λxy(t) is adiabatic invariant (constant) of a dynamical system, i.e. is almost independent for t > 3 for all acceptable values of parameters. As example, consider the phase curves for certain sets of parameter values used in section 4.1. The ration between the outputs of reciprocator and selfish firms remains unchanged both for steady state and dynamics chaos. For example profit ratio (phase curve) for dynamic chaos is presented in fig. 10. This phase curve corresponds to state of dynamic chaos in fig. 5 (fig. 10). ratio between firms’ output with different responsibilities remains almost unchanged. Every conceivable examples and parameters set can be easily viewed through the application Model and gives the same result. x(t) λxy (t)
Finally, consider one more property of model with adaptive expectations (2).
Proposition 3. The total quantity of reciprocator firms exceeds the total quantity of selfish firms in model (2) for sufficiently large t ( t > 3 ) for all values of parameters.
Proof. In accordance with proposition 2 the trajectories of system (2) converge to a x(t) Nash equilibrium (5) for any acceptable initial values. Since value λxy = = v(2bα )2 (α + (1- α )γ ){(α+ (1- α )γ- ) - (α(1 =α )γ )} = v(2bα )2 (α + (1- α )γ )2(1- α )>γ Q
Proposition 4. There is unique Nash equilibrium for dynamic microeconomic system which consists of selfish and reciprocity firms with adaptive expectation and linear demand function: . , (6) b - v where M = , x* = x1* = … = xk*, y* = y1* = … = yn-k*. The trajectories of this c system converge to a fixed point (6) for any acceptable initial values.
Proof. Without loss of generality we assume that α 1 = ... = α k = α we simplify the system: 2α - 1 2α- 1 x + (k - 1⋅) 3α - 1⋅ x+ ( n- ⋅k) 3α- ⋅ 1 =y M , (7) 1 k ⋅ x + y + 1 (n - k- ⋅1) =y M .
2 2 2 where x1 = ... = xk are quantities of reciprocity firms; y1 = ... = yn- k - quantities of selfish firms. The solutions of system (7) are the equilibrium quantities in proposition 4, Q.E.D.
Proposition 5. Reciprocator firm (for k ≥ 2 ): (а) produces more product in the market than selfish one x* > y* if and only if the share of his private interest is within the interval: α ∈ (0;α1 ) ∪ (α 2 ;1) ; (b) produces less product in the market than selfish firm x* < y* if the share of his
The solutions of corresponding equation are α 1 = , α 2 = So in compliance with proposition 3 λxy = = 1 if k = n - k for all α .
However according to proposition 5 λxy = < 1 if α ∈ (α 1;α 2 ) where α 1 < α 2 for k = n - k , k > 1 . Thus the result of proposition 3 is not generalized for linear demand functions.
The following fig.11 shows the graphs of dependance of Nash equilibrium point (6) coordinates’ x* and y* on firms’ number n according to proposition 4 at fixed parameters k = 4 , α = 0.04 and M = 50 . x * y *
1 n + 2 x(t) > n - k y(t) k
k n + k +1 , Q.E.D. y* = 0 there are no selfish firms in the market; for n = 26 , x* = 0 reciprocator firms have been pushed out. As we see the ratio of profit λ xy can vary from zero to infinity, depending on market conditions, in particular on firms’ number.
5
Thus we have synthesized the simplest possible mathematical model of microeconomics in accordance with agent based computational economics paradigm. This is the model of competition between reciprocator and selfish firms which plan their output using adaptive approach with probability p and naïve one with a probability 1 - p .
Desktop C# application Model has been created specially for our research for the computational experiments. As a result of simulation experiment we have found that flip bifurcations occur with an increase firms’ number in the market. Such bifurcations can be interpreted as separation of equilibrium state into several ways, one of which is selected by the market due to the evolution of firms’ strategies. A market moves from stability to chaos with an increase in parameter n and finally has reached dynamic.
The crucial factor which ensures sustainable equilibrium in the market is the adaptive approach. In the market with only adaptive expectations there is unique Nash equilibrium which is stable for all possible values of the parameters. If no less than two-thirds of firms use naive expectations, then it will appear the state of dynamic chaos in the market. During numerical investigations we found that the ration between the outputs of reciprocator and selfish firms remains unchanged both for steady state and dynamics chaos.
The total quantity of reciprocator firms exceeds the total quantity of selfish firms for nonlinear demand function. This property is not generalized for linear demand functions. Reciprocity firms will have more market share than selfish ones if their private interest is either sufficiently high or very low. Selfish firms will have more output than reciprocity ones if their reciprocity share is average.
On this basis we plan to study complex real systems, which, in our opinion, involve the construction of a neural network which simulates the real market based on a very simple model.