Sensitivities in Models with Backward Dynamics

In this paper, we study some properties of economic model with backward dynamics. We mainly introduce the concept of Psensitivity andP-Li-Yorke sensitivity especially for themultivalued forward dynamics of suchmodels and characterizeP-sensitivity andP-Li-Yorke sensitivity of themultivalued forward dynamics in terms of sensitivity and Li-Yorke sensitivity of an induced singlevalued 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 suchmodels in terms of the corresponding sensitivities of an induced single-valued dynamics.


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-inadvance (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.

Preliminaries
. . 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 ( +1 , ) = 0. There are more than one solution for the difference equation when was given, but given +1 there is only one solution for ( +1 , ) = 0; i.e., the implicitly-defined difference equation can be simplified as = ( +1 ). 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,[5][6][7]).
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 Discrete Dynamics in Nature and Society and it was clearly represented by a utility function of the form in which the discount factor 0 < < 1, 1 , 2 represent cash goods and credit goods at time , respectively. We take the following form as utility function as Kennedy did in [4]: with > 0 and > 0. The household needs cash carried from −1 to pay for the cash good 1 at time . The credit good 2 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 1 + 2 = . 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 taking as given 0 and { , +1 } ∞ =0 . The government supply money { } at a constant growth path +1 = (1 + ) , where is the growth rate and 0 > 0 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.
In [8] the authors use this relationship to get a difference equation in alone that characterizes equilibria in the model: or where One can always solve for the backward map = −1 ∘ ; when the function is invertible, let = ( +1 ). The dynamic going forward are multivalued when (⋅) is not invertible. In [8], the authors set = 0.98, = 0.5, = 4, = 2, and equal to 0, 0.5, 1.0. Then the function is not invertible and there exists an invariant set [ , ℎ ], such that the backward map exhibits sensitivity. The backward map with = 0 is shown in Figure 1 reproduced from Kennedy et al. [4]. In this case, the CIA model has the property with backward dynamics. . . 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,[9][10][11]).
Suppose {( , )} ∞ =1 is a sequence of compact metric spaces and : is defined as the following: If : → is continuous and satisfies that ∘ = ∘ +1 for each ∈ N, then { } ∞ =1 induces the map ∞ on the inverse limit space lim ← ( , ) ∞ =1 in the following way: In particular, if = , = and = = for each ∈ N, then the space is called the ℎ 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 { 1 , ⋅ ⋅ ⋅ , } be a finite collection of open sets in . Define Discrete Dynamics in Nature and Society 3 The collection {⟨ 1 , 2 , ⋅ ⋅ ⋅ , ⟩ : { 1 , 2 , ⋅ ⋅ ⋅ , } 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 metric̃on ∞ .
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 lim ← ( , ) ⊂ ∞ 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.
. 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 by We 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 −1 (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 .
. . 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 . 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 > 0 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.
we say that is Li-Yorke sensitive.
It was pointed out by Akin and Kolyada [14] that Li-Yorke sensitivity is strictly stronger than sensitivity.
. . 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 . Let ( , ) be a metric space, : → is a map, and ⊂ ∞ . For a positive we consider pairs of points (x, y) ∈ × 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 −1 and is -sensitive, then we say −1 is -sensitive on .
Remark . Let ( , ) be an inverse system. The direct limit space of ( , ) was defined as . 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 . Let ( , ) be a metric space and : That is if is sensitive or Li-Yorke sensitive then issensitive or -Li-Yorke sensitive.

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 ( , −1 ) 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.
The following result characterizes -sensitivity of a multivalued dynamical system ( , −1 ) in terms of sensitivity of the induced single-valued dynamical system ( , − ).
The result follows from the definitions of sensitivity andsensitivity immediately. ⇐ : Let = max , ∈ ( , ). Since − is sensitive, there exists > 0 such that for any > 0 and every x ∈ there exists y ∈ (x, ) such that Our aim is to show that there exists a subsequence of { ( (x), (y))} ∈N converging to something strictly positive. Since is compact. This subsequence will have a convergent sub-subsequence strictly bounded way from 0, Discrete Dynamics in Nature and Society 5 i.e., lim sup →∞ ( (x), (y)) = * > 0. Choose 1 = 1 such that By Lemma 8, then there exists an 1 ≥ 1 such that Then we can choose > 1 , calling it 2 . Then by the same reasoning and repeating this process, we get an sequence < +1 and → ∞ with the property that Since is compact then Since The following result characterizes -Li-Yorke sensitivity of a multivalued dynamical system ( , −1 ) in terms of Li-Yorke sensitivity of the induced single-valued dynamical system ( , − ).

(44)
Since this is true for any > 0, we have So − is Li-Yorke sensitive. ⇐ : Since − is Li-Yorke sensitive, there exists > 0 such that for any > 0 and every x ∈ there exists y ∈ (x, ) such that Note that by definition of̃and the fact that is a metric, we havẽ( The same argument from Theorem 10 gives the need condition for lim sup →∞ ( (x), (y)) > . So −1 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 ( , ).
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 ( , ).

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