Stochastic stability in monotone economies

Kamihigashi, Takashi; Stachurski, John

doi:10.3982/te1367

Cited by 35 publications

(26 citation statements)

References 65 publications

(89 reference statements)

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“…Therefore, we do not give here all these assumptions and refer the reader to the above-mentioned works. Further results on invariant distributions are also widely reported in Meyn and Tweedie (2009), Bhattacharya and Majumdar (2007) and in the recent papers of Kamihigashi and Stachurski (2014), Zhang (2007).…”

Section: Stationary Distributionssupporting

confidence: 55%

Stochastic optimal growth model with risk sensitive preferences

Bäuerle

Jaśkiewicz

2018

Journal of Economic Theory

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This paper studies a one-sector optimal growth model with i.i.d. productivity shocks that are allowed to be unbounded. The utility function is assumed to be non-negative and unbounded from above. The novel feature in our framework is that the agent has risk sensitive preferences in sense of Hansen and Sargent (1995). Under mild assumptions imposed on the productivity and utility functions we prove that the maximal discounted non-expected utility in the infinite time horizon satisfies the optimality equation and the agent possesses a stationary optimal policy. A new point used in our analysis is an inequality for the so-called associated random variables. We also establish the Euler equation that incorporates the solution to the optimality equation.Keywords. stochastic growth model, entropic risk measure, unbounded utility, unbounded shocks.onward. Here, E t stands for the expectation operator with respect to period t information. The parameter γ affects consumer's attitude towards risk in future utility. The form of preferences in (1) is due to Hansen and Sargent (1995), who used them to deal with a linear quadratic Gaussian control model. The preferences defined in (1) has several advantages. First of all, they are not time-additive in future utility. Time-additivity, however, requires an agent to be risk neutral in future utility. Risk sensitive preferences, on the other hand, allow the agent to be risk averse in future utility in addition to being risk averse in future consumption. This fact results in partial separation between risk aversion and elasticity of intertemporal substitution. 1 Moreover, as argued by ? risk sensitive preferences are also attractive, because they can be used to model preferences for robustness. It is worth emphasising that risk sensitive preferences of form (1) have found several applications, for instance, in the problems dealing with Pareto optimal allocations (see Anderson (2005)) or small noise expansions (see Anderson et al. (2012)).Our main results are two-fold. First we establish the optimality equation for the nonexpected utility in the infinite time horizon, when the agent has risk sensitive preferences (1). The proof as in the standard expected utility case is based on the Banach contraction principle, see Stokey et al. (1989). However, in oder to show that the dynamic programming operator maps a space of certain functions into itself, we have to confine our consideration to concave, non-decreasing and non-negative functions that are bounded in the weighted supremum norm. A novel feature in this analysis is an application of some inequality for the so-called associated random variables. This inequality also plays a crucial role in proving that the a fixed point of dynamic programming operator is indeed the value function. Moreover, it naturally fits into our model, in which the production and utility functions satisfy some mild conditions such as monotonicity and concavity. Here, we would like to emphasise that similarly as in Kamihigashi (2007), we do not assume the Inada...

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Section: Stationary Distributionssupporting

confidence: 55%

Stochastic optimal growth model with risk sensitive preferences

Bäuerle

Jaśkiewicz

2018

Journal of Economic Theory

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“…The area of fixed point theory received much attention from mathematicians due to its vast scope of applications, see [1,2]. More on applications of fixed point theory can be found in [3][4][5][6][7][8][9]. Furthermore, spaces and mappings are very important when studying fixed points; metric spaces, partial metric spaces, fuzzy spaces, smooth spaces, contractive mappings, monotone mappings and so on, see [1,2,4,[10][11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, another significant area of fixed point theory was brought into light, by the advent of fixed point theorems in ordered spaces, especially the Knaster-Tarski fixed point theorem [9,21]. Later, many important results were established as improvement or generalization of the Knaster-Tarski fixed point theorem, see [3,5,6,[22][23][24]. In this direction, Heikkila [4] proved some fixed point results of increasing operators in a partially ordered set, and presented some applications in partially ordered Polish spaces.…”

Section: Introductionmentioning

confidence: 99%

Fixed points of terminating mappings in partial metric spaces

Batsari

Dhompongsa

2019

J. Fixed Point Theory Appl.

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In this paper, we introduce a mapping called a terminating mapping. The existence of a unique, and globally stable fixed point of terminating mappings were established in partial metric spaces. Also, some application theorems in the space of probability density functions and example on quantum operations were presented. The results obtained in this paper, extend, and improve some existing results in the literature.

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“…driving sequence) and establish convergence and uniqueness for monotone economies. Kamihigashi and Stachurski [25] introduce a strong reversing condition that requires that if there are two mutually independent trajectories X (y0) t and X (x0) t for the pair of initial conditions x 0 < y 0 , then, there is an N ≥ 1 and δ > 0 such that P(X Szeidl [40] considers a model with an ordered state space that has no maximal and/or minimal element. The author suggested a reasonable "replacement", say, for a top point (if one does not exist) by a random "top" point.…”

Section: Discussionmentioning

confidence: 99%

“…It follows that lim T →∞ḡ (T ) (g(k ′′ , z 0 )) = k ′ . 25 Therefore, fixing somek ∈ (k ′ , k ′′ ), there exists a finite sequence of productivity shocks (z t ) T t=1 with z t ∈ arg min z∈Z {g(ḡ (t−1) (g(k ′′ , z 0 )), z)} such that the occurrence of (z t ) T −1 t=1 implies k T ≤k. Moreover, the sequence (z t ) T t=1 , with z T = z 0 , has positive probability since all the transition probabilities are positive.…”

Section: One-sector Stochastic Optimal Growth Modelmentioning

confidence: 99%

Stochastic stability of monotone economies in regenerative environments

Foss

Shneer

Thomas

et al. 2018

Journal of Economic Theory

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We introduce and analyze a new class of monotone stochastic recursions in a regenerative environment which is essentially broader than that of Markov chains. We prove stability theorems and apply our results to three canonical models in recursive economics, generalizing some known stability results to the cases when driving sequences are not independent and identically distributed. 2 We follow the terminology of Meyn and Tweedie [33] and use the term Markov Chain to refer to any discrete-time Markov process (DTMP) whether the state space is finite, countable or continuous.3 An i.i.d. process is one that is regenerative at every date. 4 We are not the first to consider regenerative processes in the economics literature. For example, Kamihigashi and Stachurski [26] consider perfect simulation of a stochastic recursion of the form X t+1 = f (Xt, ξt)1{Xt ≥ x}+ǫt1{Xt < x} where X = [a, b], x ∈ (a, b), f is increasing in X and {ξt} and {ǫt} are i.i.d. The process regenerates for values Xt < x. This process arises in models of industry dynamics with entry and exit [see 20]. It is a Markov process, but it is not monotone unless the distribution of f (x, ξ) stochastically dominates the distribution of ǫ.5 Stokey et al. [39] use the Feller property, which is a continuity requirement, together with monotonicity and a mixing condition to derive the results. Hopenhayn and Prescott [21] develop an existence result using monotonicity alone, and combined with a mixing condition, establish that uniqueness and stability follow.

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Stochastic stability in monotone economies

Cited by 35 publications

References 65 publications

Stochastic optimal growth model with risk sensitive preferences

Stochastic optimal growth model with risk sensitive preferences

Fixed points of terminating mappings in partial metric spaces

Stochastic stability of monotone economies in regenerative environments

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