Abstract
In this paper, we study some properties of economic model with backward dynamics. We mainly introduce the concept of -sensitivity and -Li-Yorke sensitivity especially for the multivalued forward dynamics of such models and characterize -sensitivity and -Li-Yorke sensitivity of the multivalued forward dynamics in terms of sensitivity and Li-Yorke sensitivity of an induced single-valued dynamics by using the theory of inverse limits. Similarly, we also characterize the sensitivity and Li-Yorke sensitivity of the single-valued backward dynamics of such models in terms of the corresponding sensitivities of an induced single-valued dynamics.
1. Introduction
In applications, trajectories generated by a dynamic system are often used to characterize the equilibrium of a dynamic economic model. In economics, there are lots of nonlinear dynamical systems with the property of single-valued moving backward, but multivalued going forward in time. In this paper, we call them models with backward dynamics the same as Medio and Raines called. Among models with backward dynamics, some of them are very popular, such as cash-in-advance (CIA) model, overlapping generations (OG) model, model of credit with limited commitment, and so forth. Just as Stockman characterized in [1] those models were often investigated by a local analysis method or by analyzing the models using their well-defined backward maps. But local analysis method may ignore some potentially interesting equilibria. In [2] Medio notes that the backward map solution is not entirely satisfactory because trajectories for the backward map and the equilibria for the models run in the opposite direction. In [1], the author also characterized that certain properties of the models with backward dynamics can be obtained by using the backward maps of the systems and introduced the concept of chaos for a multivalued dynamical systems where a concept of sensitivity of the multivalued map was raised as well. In [3] it was proved that Li-Yorke sensitivity does not imply Li-Yorke chaos. Inspired by the work of the authors mentioned above, in this paper our main purpose is to consummate the theoretical framework for analysis models with backward dynamics.
2. Preliminaries
2.1. Model with Backward Dynamics
In this section, we introduce one model with backward dynamics mainly cited from [1, 4].
Mathematically, as the references mentioned an equilibrium in model with backward dynamics can be characterized by an implicitly-defined difference equation . There are more than one solution for the difference equation when was given, but given there is only one solution for ; i.e., the implicitly-defined difference equation can be simplified as . In our paper we just take CIA model as an example for models with backward dynamics. Much more models with backward dynamics can be found in ([1, 5ā7]).
After Lucas and Stokey introduced CIA model in [6], it was studied by many economists such as Michener and Ravikumar [8], David R. Stockman [1], Kennedy et al. [4], and others. The same as the references characterized CIA model, there is a representative agent and a government, but government do nothing except setting monetary policy using a money growth rule. It is an endowment economy with credit goods and cash goods, the preferences of household were represented by choosing a sequence of , and it was clearly represented by a utility function of the formin which the discount factor , represent cash goods and credit goods at time , respectively. We take the following form as utility function as Kennedy did in [4]:with and . The household needs cash carried from to pay for the cash good at time . The credit good can be bought on credit. In each period the household has an endowment which can be transformed into the cash and credit goods such that . Technology allows the credit good to be substituted for the cash good one-for-one. Credit good, cash good, and endowment (per unit) sell at the same price in equilibrium.
The aim of household is to maximize (1) by choosing sequence constrained by ,taking as given and . The government supply money at a constant growth path , where is the growth rate and is given. In each period the government transfer cash to the household in the amount . A perfect foresight equilibrium is defined in the usual manner.
The same as the references [1, 4] did, let denote the level of real money balance, and let be the unique solution to . When the cash-in-advance constraint (3) binds, let ; otherwise let . In [8] the authors use this relationship to get a difference equation in alone that characterizes equilibria in the model:orwhereOne can always solve for the backward map ; when the function is invertible, let . The dynamic going forward are multivalued when is not invertible. In [8], the authors set , , , , and equal to 0, 0.5, 1.0. Then the function is not invertible and there exists an invariant set , such that the backward map exhibits sensitivity. The backward map with is shown in Figure 1 reproduced from Kennedy et al. [4]. In this case, the CIA model has the property with backward dynamics.
2.2. Inverse Limits
In this section, we give a brief introduction of the concept of inverse limit and the method on how to use inverse limits to study systems with backward dynamics (for more details see [1, 9ā11]).
Suppose is a sequence of compact metric spaces and is a continuous map. Let ; then is called the inverse limit space of the maps and are called bonding maps of the space. Note that is a compact metric subspace of the product space , where a metric on is defined as the following:
If is continuous and satisfies that for each , then induces the map on the inverse limit space in the following way: for each . We say original maps. In particular, if , and for each , then the spaceis called the inverse limit space of . Let ; the map defined by is called the projection map. Let be the infinite product of endowed with usual product topology. Recall that the product topology is generated by the following basic open sets. Let be a finite collection of open sets in . DefineThe collection is a finite collection of open sets in X is the collection of basic open sets of with product topology. Since is a compact metric space, is a compact metric space and the product topology on is same as the topology generated by the metric on .
In the context of the economic model with backward dynamics, denote the state space and backward map, respectively. The pair is called inverse system. It is obvious that and points in the inverse limit space have special structure which forms backward solutions to the dynamical system . For is the backward map of model, the points in the limit space also correspond to forward solutions of the implicit difference equation characterizing equilibrium in the model; i.e., the set of equilibrium in the model is an inverse limit space.
Let . Now we induce a natural map on the inverse limit space by the bonding map as the following: for each ,It means that the induced map is a homeomorphism from onto , the inverse of is defined byWe call the shift homeomorphism of . Thus dynamical system was induced by the inverse system , which is single-valued going both forward and backward in time. The approach is new to use inverse limits to analyze models with backward dynamics. In [9, 10] Medio and Raines use it to analyze the long-run behavior of an OG model; they show that typical long-run behavior of equilibria in the model corresponds to an attractor of the shift map on the inverse limit space. Kennedy et al. [11] discuss the topological structure for the inverse limit space associated with the CIA model. Recently, Stockman discusses the chaos properties for models with backward dynamics by using inverse limits approach in [1]. The complexity of inverse limit space and the complexity of the dynamical system are closely related. The main advantage of the inverse limit approach is to study models with backward dynamics. Treating an equilibrium in the model as a single point in a larger space can help us to study the sensitivity property of the dynamics of (which is multivalued) on by studying the sensitivity property of the dynamics of on . Similarly, it also helps us to study the sensitivity property of the dynamics of on by studying the sensitivity property of the dynamics of on .
2.3. Sensitivity of Dynamical Systems
The concept of sensitivity was first introduced by Auslander and Yorke [12] and popularized by Devaney [13]. The following is its definition.
Definition 1. Let be metric space; is a map. Given a positive , we consider pairs of points whose orbits are frequently at least apart, that is,The system or is sensitive on if for some positive the set of pairs which satisfy this condition is dense in .
Recently many mathematicians generalized the definition of sensitivity. Here we mainly focus on the one generalized by Akin and Kolyada [14]; a system is called Li-Yorke sensitive if there exists such that every is a limit of points such that the pair is proximal but whose orbits are frequently at least apart. See the following.
Definition 2. Suppose is a metric space and is a map. If there exists such that for any and every there exists such thatwe say that is Li-Yorke sensitive.
It was pointed out by Akin and Kolyada [14] that Li-Yorke sensitivity is strictly stronger than sensitivity.
2.4. Sensitivity of Backward Dynamics
In this section, we extend the definition of sensitivities to models with backward dynamics. What does it mean that the model with backward dynamics is sensitive? In the following definitions, we give our ways to describe the sensitivities of such models by treating the trajectories as points in a larger metric space.
Definition 3. Let be a metric space, is a map, and . We say that generates if for each we have for . We say that generates if for each we have for .
Definition 4. Let be a metric space, is a map, and . For a positive we consider pairs of points whose components are frequently at least apart, that is,We say the projection map is sensitive on or is -sensitive if for some positive the set of pairs which satisfy the above condition is dense in . If is generated by and is -sensitive, then we say is -sensitive on .
Definition 5. Let be a metric space, is a map, and . We say that is -Li-Yorke sensitive if there exists such that for any and every there exists such thatIf is generated by and is -Li-Yorke sensitive, then we say is - Li-Yorke sensitive on .
Remark 6. Let be an inverse system. The direct limit space of was defined asLet , then is generated by and is generated by . These are different subsets of (and when sensitivity occurs, they are very different spaces topologically). The examples to illustrate the relationship between these sets could be found in Stockman [1]. If is -sensitive or -Li-Yorke sensitive, then we say the inverse system is -sensitive or -Li-Yorke sensitive.
Remark 7. Let be a metric space and a map. is the infinite product metric space with , and , where is generated by . Then for any pair of , and . We have . So if , then . SinceandThat is if is sensitive or Li-Yorke sensitive then is -sensitive or -Li-Yorke sensitive.
3. The Main Results
In this section, we use the theory of inverse limit to characterize the -sensitivity and -Li-Yorke sensitivity of a multivalued backward dynamical system in terms of the sensitivity and Li-Yorke sensitivity of the induced dynamical system . We also characterize the sensitivity and Li-Yorke sensitivity of the single-valued dynamical system in terms of sensitivity and Li-Yorke sensitivity of the induced dynamical system . In order to show our results, the following lemmas (details of proof: see ([1]) are needed.
Lemma 8. Let be the metric on and let be the metric on induced by . Then for and , if for some , then there exists such that .
Lemma 9. Let be the metric on such that is a compact space and let be the metric on induced by . Suppose is continuous and onto, and . Let . Then for , and given , if for some with , then there exists such that .
The following result characterizes -sensitivity of a multivalued dynamical system in terms of sensitivity of the induced single-valued dynamical system .
Theorem 10. Suppose that is a compact metric space, is continuous, and is the infinite product space with the metricLet , be the shift homeomorphism. Then is -sensitive if and only if is sensitive.
Proof. ā: Since for any and any , we haveThe result follows from the definitions of sensitivity and -sensitivity immediately.
ā: Let . Since is sensitive, there exists such that for any and every there exists such thatLet It means there exists a subsequence such that For , there exists an such that for all Our aim is to show that there exists a subsequence of converging to something strictly positive. Since is compact. This subsequence will have a convergent sub-subsequence strictly bounded way from 0, i.e., . Choose such that By Lemma 8, then there exists an such that Then we can choose , calling it . Then by the same reasoning and repeating this process, we get an sequence and with the property that Since is compact then Since is compact, there exists a subsequence of converging to a point ; i.e., there exists a subsequence with hence
The following result characterizes -Li-Yorke sensitivity of a multivalued dynamical system in terms of Li-Yorke sensitivity of the induced single-valued dynamical system .
Theorem 11. Suppose that is a compact metric space, is continuous, and is the infinite product space with the metricLet and be the shift homeomorphism. Then is -Li-Yorke sensitive if and only if is Li-Yorke sensitive.
Proof. ā: Since is -Li-Yorke sensitive, there exists such that for any and every there exists such thatSince for any , we have and Soand it follows that Then we need to show that To this end, it is sufficient to show that for any , there exists a subsequence , with such that Let . ThenPick large enough such that . Then pick such thatSince is continuous and is compact, is uniformly continuous on for any . For the given , there exists such that if , ,. Let . Since , there exists such that , and . Let . We have Since , we have for . Then Since this is true for any , we have So is Li-Yorke sensitive.
ā: Since is Li-Yorke sensitive, there exists such that for any and every there exists such thatNote that by definition of and the fact that is a metric, we haveThis implies that Hence we haveThe same argument from Theorem 10 gives the need condition for . So is -Li-Yorke sensitive.
The following result characterizes Li-Yorke sensitivity of a single-valued dynamical system in terms of Li-Yorke sensitivity of the induced single-valued dynamical system .
Theorem 12. Suppose that is a compact metric space, is continuous, and is the infinite product space with the metricLet and be the induced homeomorphism. Then is Li-Yorke sensitive if and only if is Li-Yorke sensitive.
Proof. ā For is Li-Yorke sensitive, let and ; there exists such that for any there exists such thatLet . Sinceit follows that Then we need to prove that That is, for any , there exists a positive integer sequence with such that Let . ThenPick large enough so that . Then pick such that Since is continuous and is compact, is uniformly continuous on for any . Given , there exists such that if , ,. Let . Since , there exists such that , and . Let . ThenSince , we have for . For any , the following inequality holds That is,For any , we choose large enough such that , with . Since is uniformly continuous on , there exists such that whenever . For any , let such that ; then . By the above arguments, we have and Since hence . So is Li-Yorke sensitive.
(ā) Suppose that is Li-Yorke sensitive. Then there exists , for any and for any , there exists such thatBy the definition of , we have that . And similarly, for any we have which implies thatSoLet This implies the existence of a subsequence with property For , there exists an such that for all we have Let such that . Without loss of generality assume that and . Let , and for . Note for each , we have . Since by Lemma 9, there exists with with This implies that Hence is Li-Yorke sensitive.
Similar to the proof of Theorem 12, we have the following theorem, which characterizes sensitivity of single-valued dynamical system in terms of sensitivity of the induced single-valued dynamical system .
Theorem 13. Suppose that is a compact metric space, is continuous, and is the infinite product space with the metricLet and be the induced homeomorphism. Then is sensitive if and only if is sensitive.
4. Conclusion
In some economic models, the dynamical system characterizing equilibria in the model has multivalued forward dynamics but single-valued backward dynamics, i.e., models with backward dynamics. In this paper, we offered a definition of -sensitive and -Li-Yorke sensitive for such multivalued forward dynamical systems, and by using of theory of inverse limits, we characterize -sensitive and -Li-Yorke sensitive of a multivalued dynamics on in terms of sensitivity and Li-Yorke sensitivity of a single-valued forward dynamics on a subspace of .
In the future research, we will continue to investigate how does -sensitivity and -Li-Yorke sensitivity about the backward orbits of correspond to equilibria in the model. We are also interested in the tracing property for such system with backward dynamics which are motivated by the work of Gu and Sheng [15] and Wu and Chen ([16ā18]).
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.