Unique Stable Equilibrium Research Articles

Evolutionary stability of bargaining and price posting: implications for formal and informal activities

In this paper we study the co-existence of two well known trading protocols, bargaining and price-posting. To do so we consider a frictional environment where buyers and sellers play price-posting and bargaining games infinitely many times. Sellers switch from one market to the other at a rate that is proportional to their payoff differentials. Given the different informational requirements associated with these two trading mechanisms, we examine their possible co-existence in the context of informal and formal markets. Other than having different trading protocols, we also consider other distinguishing features. We find a unique stable equilibrium where price-posting (formal markets) and bargaining (informal markets) co-exist. In a richer environment where both sellers and buyers can move across markets, we show that there exists a unique stable dynamic equilibrium where formal and informal activities also co-exist whenever sellers’ and buyers’ net costs of trading in the formal market have opposite signs.

Uniqueness, stability and comparative statics for two-person Bayesian games with strategic substitutes

This paper considers a class of two-player symmetric games of incomplete information with strategic substitutes. First, we provide sufficient conditions under which there is either a unique equilibrium which is stable (in the sense of best-reply dynamics) and symmetric or a unique (up to permutations) asymmetric equilibrium that is stable (together with an unstable symmetric equilibrium). Thus, (i) there is always a unique stable equilibrium, (ii) it is either symmetric or asymmetric, and hence, (iii) a very simple local condition—stability of the symmetric equilibrium (i.e., the slope of the best-response function at the symmetric equilibrium)—identifies which case applies. Using this, we provide a very simple sufficient condition on primitives for when the unique stable equilibrium is asymmetric (and similarly for when it is symmetric). Finally, we show that the conditions guaranteeing the uniqueness described above also yield novel comparative statics results for this class of games.

Metastability for small random perturbations of a PDE with blow-up

We study random perturbations of a reaction–diffusion equation with a unique stable equilibrium and solutions that blow-up in finite time. If the strength of the perturbation ε>0 is small and the initial data is in the domain of attraction of the stable equilibrium, the system exhibits metastable behavior: its time averages remain stable around this equilibrium until an abrupt and unpredictable transition occurs which leads to explosion in a finite time (but exponentially large in ε−2). Moreover, for initial data in the domain of explosion we show that the explosion times converge to the one of the deterministic solution.

Flow-Inducing Networks

We consider flow-inducing networks, a class of models that are well-suited to describe important biochemical systems, including the MAPK pathway and the interactions at the trans-Golgi network. A flow-inducing network is given by the interconnection of subsystems (modules), each associated with a stochastic state matrix whose entries depend on the state variables of other modules. This results in an overall nonlinear system; when the interactions are modeled as mass action kinetics, the overall system is bilinear. We provide preliminary results concerning the existence of single or multiple equilibria and their positivity. We also show that instability phenomena are possible and that entropy is not a suitable Lyapunov function. The simplest non-trivial module is the duet, a second order system whose variables represent the concentrations of a species in its activated and inhibited state: under mass action kinetics assumptions, we prove that: 1) a negative loop of duets has a unique equilibrium that is unconditionally stable and 2) a positive loop of duets has either a unique stable equilibrium on the boundary or two equilibria, of which one is unstable on the boundary and one is strictly positive and stable; both properties 1) and 2) hold regardless of the number of duets in the loop.

Compatibility in tax reporting

Purpose – to describe a compliance-monitoring equilibrium in presence of compatibility costs in a setting when managers and other parties have different attitude towards compliance.
 Design/Method/Approach. Classical game theory – Nash equilibrium.
 Findings. If compatibility costs are small, there exist a unique stable Nash equilibrium of the game between the tax authority and a population of heterogeneous firms. In this equilibrium, the relation between compatibility costs and compliance is non-monotonic and depends on the curvature of auditing function. However, compatibility costs reduce non-compliance in low cheating regimes and may enhance it when many firms are cheating.
 Limitations. The model is at high level of abstraction and neglects many important detail that characterize each field where it could be potentially applied.
 Theoretical implications. The results provide one rationale for developing countries to be cautious with employing refined auditing schemes and for developed countries to promote complicated accounting procedures.
 Originality/value. Compatibility costs are not previously considered in economic analysis of compliance.
 Paper type – conceptual.

The Ganikhodjaev Model of ABO Blood Groups

In 2010, N. Ganikhodjaev proposed the models of ABO and Rh blood groups of Malaysian people. Based on some numerical simulations, it was showed that the evolution of ABO blood groups of Malaysian people has a unique stable equilibrium. In this paper, we analytically prove that the Ganikhodjaev model of ABO blood groups has a unique fixed point.

Stability and bifurcation analysis in the coupled HR neurons with delayed synaptic connection

This paper presents stability and bifurcations of two synaptically coupled identical Hindmarsh–Rose neurons with one time delay. The parameters we choose contribute to the single neuron exhibiting excitable behavior with a unique stable equilibrium. In the absence of time delay, we find coupling-induced oscillations, i.e., the excitable neurons can fire with different types of regular or irregular periodic spiking/bursting behaviors due to the synaptic coupling. With the help of stability and bifurcation theory, the asymptotic stability of equilibrium, fold and Hopf bifurcation are studied from the corresponding characteristic equation. In case of time delay, a detailed Hopf bifurcation analysis is given. And an explicit formula about the coupling strength and time delay for the occurrence of Hopf bifurcation is derived, based on which a series of periodic orbits generate when time delay passes through the critical value. Finally, numerical simulations are carried out for supporting our theoretical results; meanwhile, branches of Hopf bifurcation curves are plotted in the two-parameter plane.

Diffusive heat transport in Budyko's energy balance climate model with a dynamic ice line

M. Budyko and W. Sellers independently introduced seminal energy balance climate models in 1969, each with a goal of investigating the role played by positive ice albedo feedback in climate dynamics. In this paper we replace the relaxation to the mean horizontal heat transport mechanism used in the models of Budyko and Sellers with diffusive heat transport. We couple the resulting surface temperature equation with an equation for movement of the edge of the ice sheet (called the ice line), recently introduced by E. Widiasih. We apply the spectral method to the temperature-ice line system and consider finite approximations. We prove there exists a stable equilibrium solution with a small ice cap, and an unstable equilibrium solution with a large ice cap, for a range of parameter values. If the diffusive transport is too efficient, however, the small ice cap disappears and an ice free Earth becomes a limiting state. In addition, we analyze a variant of the coupled diffusion equations appropriate as a model for extensive glacial episodes in the Neoproterozoic Era. Although the model equations are no longer smooth due to the existence of a switching boundary, we prove there exists a unique stable equilibrium solution with the ice line in tropical latitudes, a climate event known as a Jormungand or Waterbelt state. As the systems introduced here contain variables with differing time scales, the main tool used in the analysis is geometric singular perturbation theory.

Discrimination and Inequality in an Integrated Walrasian-General-Equilibrium and Neoclassical-Growth Theory

The purpose of this study is to examine dynamic interdependence between economic development, wealth and income distributions, and discrimination in an integrated Walrasian-general-equilibrium and neoclassical-growth theory. We build a dynamic economy with one consumer goods sector, one capital goods sector, and heterogeneous households. We build a model in which wealth accumulation, income and wealth distribution, time distribution and division of labor interact with each other under a fixed pattern of discrimination. For illustration, we simulate the motion of the economic system with three groups, called the discriminator, the neutral group (neither discriminating nor being discriminated), and the discriminated group. We identify the existence of a unique stable equilibrium point. Our comparative dynamic analyses with regard to different discrimination rates provide some insights. For instance, we show that when the discriminator strengthens its discrimination against the discriminated group and the discriminated group “positively” reacts the strengthened discrimination, we have the following effects: the national output, the national wealth, the total labor supply, and the output levels and the input factors of the two sectors are increased; the lump sum transfer from the discriminated group to the discriminator is increased; the discriminated group’s wage rate is augmented and the other two groups’ wage rates are slightly affected; the discriminator’s work time is reduced, the discriminated group’s work time is increased, and the neutral group’s work time is slightly affected; the discriminated group’s and the discriminator’s consumption and wealth levels are increased, and the neutral group’s consumption and wealth levels are slightly affected.

Global Determinacy Under Monetary and Fiscal Policy Switchings

This paper examines equilibrium determinacy in the context of a structural regime switching macro model with monetary and fiscal policy switching. We first show theoretically and in practice that the long-run Taylor principle proposed by Davig and Leeper (2007) is necessary but not sufficient for determinacy in mean-square stability sense. We then show that stochastic switching in policy stances substantially changes the determinacy region. On the one hand, a unique stable equilibrium under regime switching may exist when monetary and fiscal policies are both recurrently passive or active. On the other hand, a switch between two alternative policy combinations which imply determinacy under two fixed-regimes often requires a large swing in fiscal policy in order to achieve global determinacy, ruling out some traditionally determinate equilibria. Finally, we show that money growth management can be an alternative to active fiscal policy in a zero lower-bound scenario in order to lead the economy to global determinacy.

Phase-Coupled Oscillators with Plastic Coupling: Synchronization and Stability

In this article we study synchronization of systems of homogeneous phase-coupled oscillators with plastic coupling strengths and arbitrary underlying topology. The dynamics of the coupling strength between two oscillators is governed by the phase difference between these oscillators. We show that, under mild assumptions, such systems are gradient systems, and always achieve frequency synchronization. Furthermore, we provide sufficient stability and instability conditions that are based on results from algebraic graph theory. For a special case when underlying topology is a tree, we formulate a criterion (necessary and sufficient condition) of stability of equilibria. For both, tree and arbitrary topologies, we provide sufficient conditions for phase-locking, i.e., convergence to a stable equilibrium almost surely. We additionally find conditions when the system possesses a unique stable equilibrium, and thus, almost global stability follows. Several examples are used to demonstrate variety of equilibria the system has, their dependence on system's parameters, and to illustrate differences in behavior of systems with constant and plastic coupling strengths.

On Bistable Region of Slotted-ALOHA Systems with Delay Constraints

Bistable behavior of slotted ALOHA systems has been extensively investigated. It is proved that bistable region can be eliminated by limiting the number of allowable transmission attempts per packet to eight or less. In this paper, we investigate bistable behavior of slotted ALOHA systems with delay constraints in terms of the catastrophe theory. A lifetime of a packet is confined within D slots, that is, a packet will be discarded after timer of D time-slots expired, unless packet transmission succeeds. Letting r be the packet transmission probability in a slot, we indicate that there exists bistable region, if the product Dr is greater than approximately 8.30, and that the system can operate with the unique stable equilibrium point, if Dr is less than approximately 8.30.

Cyclic Feedback Systems with Quorum Sensing Coupling.

Synchronization and desynchronization is of great interest in the study of circadian rhythms, metabolic oscillations and time-dependent cell aggregate behaviors. Several recent studies examine synchronization and other dynamics in models of repressilators coupled by a quorum sensing mechanism that uses a diffusive signal. Their numerical simulations have shown the complexity of the collective behavior depends sensitively on which protein upregulates diffusive signal. In this paper, we rigorously prove that the collective dynamics indeed strongly depends on how the signaling network integrates into the repressilator network. In fact we prove a general result for a class of negative cyclic feedback systems with signaling of which the repressilator is but one example. We show that if the feedback along the signaling loop is also negative, the resulting negative feedback, negative signaling (Nf-Ns) system admits either unique stable equilibrium, or a stable oscillation. When a positive signaling feedback is included, the system is no longer (Nf-Ns) and numerically exhibits multistable dynamics (Ullner etal. in Phys Rev Lett 99:148103, 2007; Phys Rev E 78:031904, 2008). We demonstrate that this multistability emerges through saddle node bifurcations of a sole cubic curve-as in generic bistable models.

CAN RESIDENTIAL INTEGRATION BE A STABLE EQUILIBRIUM?

Shelling’s model and its different versions predict that segregation is the unique stable equilibrium. However, cities around the world show different levels of intermingle, and it is very uncommon to observe fully segregated urban patterns even when segregation is high. We argue that individuals do not take into account only their neighbours characteristics for making a location decision. In particular, if residential segregation generates ghettos of low-skilled labourers, the societal level of production would diminish and, consequently, individuals’ level of consumption too, affecting all society members’ well-being. When agents care about the impact of segregation on their own consumptions, we prove that the set of system’s equilibria can be one containing states of residential integration. Besides, using simulations, we compare the aggregated utility related to these equilibria with the maximum level of aggregated utility.

Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system

Based on Rabinovich system, a 4D Rabinovich system is generalized to study hidden attractors, multiple limit cycles and boundedness of motion. In the sense of coexisting attractors, the remarkable finding is that the proposed system has hidden hyperchaotic attractors around a unique stable equilibrium. To understand the complex dynamics of the system, some basic properties, such as Lyapunov exponents, and the way of producing hidden hyperchaos are analyzed with numerical simulation. Moreover, it is proved that there exist four small-amplitude limit cycles bifurcating from the unique equilibrium via Hopf bifurcation. Finally, boundedness of motion of the hyperchaotic attractors is rigorously proved.

Quality improvement and process innovation in monopoly: A dynamic analysis

We investigate the R&D portfolio of a monopolist investing in cost-reducing and quality enhancing R&D. Incentives along the two directions are inversely related to the size of market demand, and independent of each other. The stability analysis shows the existence of a unique stable steady state equilibrium, which is a saddle point. Finally, we show that the monopolist undersupplies product quality as compared to the social optimum, while its investment in the abatement of marginal cost is socially efficient.

Values of Land and Renewable Resources in a Three-Sector Economic Growth Model

Abstract This paper studies dynamic interdependence of capital, land and resource values in a three sector growth model with endogenous wealth and renewable resources. The model is based on the neoclassical growth theory, Ricardian theory and growth theory with renewable resources. The household’s decision is modeled with an alternative approach proposed by Zhang two decades ago. The economic system consists of the households, industrial, agricultural, and resource sectors. The model describes a dynamic interdependence between wealth accumulation, resource change, and division of labor under perfect competition. We simulate the model to demonstrate the existence of a unique stable equilibrium point and plot the motion of the dynamic system. The study conducts comparative dynamic analysis with regard to changes in the propensity to consume resources, the propensity to consume housing, the propensity to consume agricultural goods, the propensity to consume industrial goods, the propensity to save, the population, and the output elasticity of capital of the resource sector.

Endogenous formation of free trade agreements in vertically related markets

We study the endogenous formation of free trade agreements (FTAs) in a symmetric three-country model with vertically related markets characterized by Cournot competition. We analyze the coalition-proof Nash equilibria in an FTA formation game in which each country forms an FTA with one, both, or none of its trading partners. We show that multilateral free trade is the unique stable equilibrium of the FTA formation game. In other words, FTAs act as building blocks for multilateral trade liberalization in the presence of vertical trade structures.

On the Interplay between Cannibalism and Harvest in Stage-Structured Population Models

By use of a nonlinear stage-structured population model the role of cannibalism and the combined role of cannibalism and harvest have been explored. Regarding the model, we prove that in most parts of parameter space it is permanent. We also show that the transfer from stability to nonstationary dynamics always occurs when the unique stable equilibrium undergoes a supercritical Neimark-Sacker (Hopf) bifurcation. Moreover, the dynamic consequences of catch depend not only on which part of the population (immature or mature) is exposed to increased harvest pressure but also on which part of the immature population (newborns, older immature individuals) suffers from cannibalism. Indeed, if only newborns are exposed to cannibalism an enlargement of harvest pressure on the mature part of the population may act in a stabilizing fashion. On the other hand, whenever the whole immature population is exposed to cannibalism there are parts in parameter space where increased harvest on the mature population acts in a destabilizing fashion.

Development, Preference Change, and Inequality in an Integrated Walrasian-General-Equilibrium and Neoclassical-Growth Theory

This study builds a model to analyze dynamic interdependence between economic growth, structural change, inequality in income and wealth, habit formation and preference change in a heterogeneous-households economy. The analytical framework is based on the Walrasian general equilibrium theory and the neoclassical growth theory. We show that the motion of the economy is described by a set of nonlinear differential equations. Analytically, we focus on transitional processes as well as economic equilibrium. For illustration, we simulate the motion of the economic system with three groups. We identify the existence of a unique stable equilibrium point and carry out comparative dynamic analysis with regard to some parameters. We show how, for instance, habit formation of a group affects the dynamic paths of all the variables of the economic system.