Abstract

In this communication, the generalized continuous economic model for random markets is revisited. In this model for random markets, agents trade by pairs and exchange their money in a random and conservative way. They display the exponential wealth distribution as asymptotic equilibrium, independently of the effectiveness of the transactions and of the limitation of the total wealth. In the current work, entropy of mentioned model is defined and then some theorems on entropy growth of this evolutionary problem are given. Furthermore, the entropy increasing by simulation on some numerical examples is verified.

1. Introduction

In the last years, it has been reported [1, 2] that, in western societies, around 95% of the population, the middle and lower economic classes of society, arrange their incomes in an exponential wealth distribution. The incomes of the rest of the population, around 5% of individuals, fit a power law distribution [3]. A kind of models considering the randomness associated to markets is the gas-like models [4]. These random models interpret economic exchanges of money between agents similarly to collisions in a gas where particles share their energy [5]. In this communication, the increasing of the entropy when these systems evolve toward the asymptotic equilibrium is checked. This is associated with the existence of an H-theorem for all these economic models [6, 7]. Exponential distribution is ubiquitous in the framework of multiagent systems. Usually, it appears as an equilibrium state in the asymptotic time evolution of statistical systems. It has been explained from very different perspectives. In statistical physics, it is obtained from the principle of maximum entropy [8]. In the same context, it can also be derived without any consideration about information theory, only from geometrical arguments under the hypothesis of equiprobability in phase space [9]. Also, several multiagent economic models based on mappings, with random, deterministic, or chaotic interactions, can give rise to the asymptotic appearance of the exponential wealth distribution [10–13]. Concretely, the continuous economic model for random markets is given byIt is found that the exponential distribution is a stable fixed point of this type of system. In this work, first entropy of mentioned model is defined and then we give some theorems on entropy growth of this evolutionary problem in the next section. Moreover, we illustrate the entropy increasing by simulation on some numerical examples in Section 3. Finally, some concluding remarks are given in Section 4.

2. Entropy Growth

Definition 1. We define space as follows:

Definition 2. For each , we define as entropy of.

Definition 3. Assume ; we introduce a sequence of real numbers aswhere .

Lemma 4. Suppose that is decreasing function; then there exists , so that .

Proof. It is clearly a consequence of this fact that for each there exists so that . Now from this point that is decreasing, function tends to zero. Therefore, there exists surely a large number as so that we have in the interval .

Theorem 5. Entropy function is bounded. In other words, there exist so that .

Proof. Suppose that ; then it is clear that there exists so that is decreasing function in the interval , because in the case of the fact that function is not decreasing we have which is contradiction by definition of the set . Now, it is obvious from Lemma 4 for decreasing function to say that there exist and and so that and ; thereforeso the proof is completed.

Corollary 6. The sequence of real numbers , , is bounded sequence.

Proof. It is obviously the consequence of Theorem 5.

Theorem 7. Suppose that , exists, and then .

Proof. We know that . Now, since the function is continuous in the interval , then we have

Theorem 8. Suppose that , , is increasing sequence; then the entropy of is the supremum among entropies of all distributions in the set .

Proof. Assume that is increasing sequence; on the other hand by Corollary 6 we know that is bounded. Therefore there is a limit for sequence ; from Theorem 5 this limit should be ; that is,Then the entropy of is the supremum.

Theorem 9. Suppose that , , exists and the entropy of is the supremum among the entropies of all distributions in the set ; then , , is increasing sequence for enough large .

Proof. Existence of implies that ; then for each there exist such that for all we have . On the other hand we know that is supremum; therefore for all we haveHence, which means is increasing sequence.

3. Numerical Simulations

Example 1. Consider; then and so and for some primary have been calculated as follows:

Example 2. Consider ; then and so and for some primary have been calculated as follows:

Example 3. Consider ; then and so and for some primary have been calculated as follows:

Example 4. In this example we have provided Table 1 of other types of distributions in and calculated some primary and corresponding for each function as well.

4. Conclusion

Different versions of a continuous economic model that takes into account idealistic characteristics of the markets have been reconsidered. In these models, the agents interact by pairs and exchange their money in a random way. The asymptotic steady state of these models is the exponential wealth distribution. The system decays to this final distribution with a monotonic increasing of the entropy taking its maximum value just on the equilibrium.

Competing Interests

The authors declare that they have no competing interests.