Existence Of Multiple Equilibria Research Articles

Asymmetric interaction and diverse forms in public goods production in volunteer dilemma game

Theoretical prediction of classical evolutionary games shows that public goods is contributed either by strong players in asymmetric games or by all of the involved players in symmetric games. However, it is inconsistency with experimental observations that weak players could also contribute to public goods in real cooperation systems. To reconciling these conflicts, we here study an evolutionary volunteer's dilemma game by assuming different roles can be interchangeable. In this model, not only the existence of multiple equilibria is possible but also the evolutionary result of the system depends on initial conditions, which can be interpreted as the production modes of public goods under different circumstances. Specifically, we found that the survival of strong individuals with mixed strategies is associated with two different outcomes: one is similar to Selten's model, that is, public goods are produced by strong players if the defectors are weak players and strong defectors are scarce in the initial condition; and the other is that the weak individuals with mixed strategies produce public goods if the defectors are strong individuals, where the strong cooperators are absent in the initial condition. Moreover, the weak individuals with mixed strategies produce public goods while strong individuals mainly display defection if the cooperation of strong players is absent in the initial condition. Our main results may help to explain the emergence of diverse forms of cooperation in asymmetric evolutionary games.

The many faces of the taylor rule for advanced undergraduate macroeconomics

The Taylor Rule and Fisher Relation can be represented on a graph that allows for discussion of the zero lower bound on interest rates, the existence of multiple equilibria, secular stagnation and Japan’s lost decade, among other issues. The Taylor Rule and Fisher Relation can also be included in a small macroeconomic model that can be used to study the stability under various interest rate rules under adaptive expectations or solved under rational expectations. There are a number of related empirical exercises using both descriptive statistics and regressions for undergraduates with intermediate macro level knowledge.

Existence of multiple equilibria in a short-term market with persistent liquidity trading

There is an extensive body of theory on information heterogeneity, the impact of higher order beliefs and short-termism on asset pricing, yet empirical studies testing their implications are scarce. The objective of this paper is to provide empirical evidence on the existence of multiple equilibria in a marked short-term stock market, Borsa Istanbul. Results indicate existence of two extremal equilibria with distinct levels of volatility, liquidity and informativeness under persistent liquidity trading.

Detecting multiple equilibria for continuous dependent variables

This article considers structural equations where continuous dependent variables are related to independent variables and unobservables through a nonparametric function. Multiple equilibria may arise when the structural equations admit multiple solutions. This article proposes a detecting criterion for the existence of multiple equilibria. The main finding is that multiple equilibria would reveal itself in the form of jump(s) in the density function of the dependent variables. When there is a unique equilibrium, the density function of dependent variables will be continuous, whereas when there are multiple equilibria, the density will have jump(s) under reasonable conditions.

Dynamic analysis of the role of innate immunity in SEIS epidemic model.

Consideration of every important aspect while modeling a disease makes the model more precise and the disease eradication strategy more powerful. In the present paper, we analyze the importance of innate immunity on SEIS modeling. We propose an SEIS model with Holling type II and type III functions representing innate immunity. We find the existence and stability conditions for the equilibria. When innate immunity is in the form of Holling type II function, the disease-free equilibrium exists for reproduction number less than unity and is locally asymptotically stable, and supercritical transcritical (forward) as well as subcritical transcritical (backward) bifurcation may occur where the contact rate beta =beta ^* acts as the bifurcation parameter. Hence, disease-free equilibrium need not be globally stable. For reproduction number greater than unity unique endemic equilibrium exists which is locally asymptotically stable. The global stability conditions for the same are deduced with the help of Lozinskibreve{i} measure. When innate immunity is considered a Holling type III function, the disease-free equilibrium point exists for reproduction number less than unity and is locally as well as globally stable. The existence of either unique or multiple endemic equilibria is found when reproduction number is greater than unity, and there exists at least one locally asymptotically stable equilibrium point and bistability can also be encountered. The conditions for the existence of Andronov–Hopf bifurcation are deduced for both cases. Moreover, we observe that ignoring innate immunity annihilates the possibility of Andronov–Hopf bifurcation. Numerical simulation is performed to validate the mathematical findings. Comparing the obtained results to the case when innate immunity is ignored, it is deduced that ignoring it ends the possibility of backward bifurcation, Andronov–Hopf bifurcation as well as the existence of multiple equilibria, and it also leads to the prediction of higher infection than the actual which may deflect the accuracy of the model to a high extent. This would further lead to false predictions and inefficient disease control strategies which in turn would make disease eradication a difficult and more expensive task.

A discrete-time second-best dynamic road pricing scheme considering the existence of multiple equilibria

When multiple equilibria exist, the desired traffic equilibrium, may not be reached through a day-to-day dynamic adjustment process. Han et al. [“Discrete-Time Day-to-Day Dynamic Congestion Pricing Scheme Considering Multiple Equilibria.” Transportation Research Part B: Methodological 104: 1–16.] proposed a day-to-day dynamic pricing scheme charged on all links to ensure the desired traffic equilibrium can be achieved even when multiple equilibria exist. However, due to many practical constraints, the second-best pricing scheme has more practical interests. This study first discussed the issues of existing congestion pricing schemes for second-best case. Then, this study developed a second-best dynamic road pricing scheme that is implemented on a dynamic subset of the road links in the network. Our rigorous theoretical proof and numerical tests prove that the dynamic pricing scheme can drive the traffic state evolution towards the desired second-best traffic user equilibrium state from any initial traffic state.

A behavioral model of evolutionary dynamics and optimal regulation of tax evasion

The paper studies the dynamics of compliance in a population of agents that decide whether to engage in tax evasion depending on an evolutionary adaptation process, when payoffs are assumed to have the realistic features of Prospect Theory utilities. The paper also considers the optimal control problem of a tax authority that targets the maximization of the expected stream of tax revenues choosing auditing effort. The analysis produces novel and rich results, including conditions for the convergence to an asymptotically stable interior equilibrium, the existence of multiple equilibria and discontinuities in the optimal control.

Public Debt and Low Interest Rates

This lecture focuses on the costs of public debt when safe interest rates are low. I develop four arguments.First, I show that the current U.S. situation in which safe interest rates are expected to remain below growth rates for a long time, is more the historical norm than the exception. If the future is like the past, this implies that debt rollovers, that is the issuance of debt without a later increase in taxes may well be feasible. Put bluntly, public debt may have no fiscal cost.Second, even in the absence of fiscal costs, public debt reduces capital accumulation, and may therefore have welfare costs. I show that welfare costs may be smaller than typically assumed. The reason is that the safe rate is the risk-adjusted rate of return to capital. If it is lower than the growth rate, it indicates that the risk-adjusted rate of return to capital is in fact low. The average risky rate however also plays a role. I show how both the average risky rate and the average safe rate determine welfare outcomes.Third, I look at the evidence on the average risky rate, i.e. the average marginal product of capital. While the measured rate of earnings has been and is still quite high, the evidence from asset markets suggests that the marginal product of capital may be lower, with the difference reflecting either mismeasurement of capital or rents. This matters for debt: The lower the marginal product, the lower the welfare cost of debt.Fourth, I discuss a number of arguments against high public debt, and in particular the existence of multiple equilibria where investors believe debt to be risky and, by requiring a risk premium, increase the fiscal burden and make debt effectively more risky. This is a very relevant argument, but it does not have straightforward implications for the appropriate level of debt.My purpose in the lecture is not to argue for more public debt. It is to have a richer discussion of the costs of debt and of fiscal policy than is currently the case. Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

Public Debt and Low Interest Rates

This lecture focuses on the costs of public debt when safe interest rates are low. I develop four main arguments. First, I show that the current US situation, in which safe interest rates are expected to remain below growth rates for a long time, is more the historical norm than the exception. If the future is like the past, this implies that debt rollovers, that is the issuance of debt without a later increase in taxes, may well be feasible. Put bluntly, public debt may have no fiscal cost. Second, even in the absence of fiscal costs, public debt reduces capital accumulation, and may therefore have welfare costs. I show that welfare costs may be smaller than typically assumed. The reason is that the safe rate is the risk-adjusted rate of return to capital. If it is lower than the growth rate, it indicates that the risk-adjusted rate of return to capital is in fact low. The average risky rate however also plays a role. I show how both the average risky rate and the average safe rate determine welfare outcomes. Third, I look at the evidence on the average risky rate, i.e., the average marginal product of capital. While the measured rate of earnings has been and is still quite high, the evidence from asset markets suggests that the marginal product of capital may be lower, with the difference reflecting either mismeasurement of capital or rents. This matters for debt: the lower the marginal product, the lower the welfare cost of debt. Fourth, I discuss a number of arguments against high public debt, and in particular the existence of multiple equilibria where investors believe debt to be risky and, by requiring a risk premium, increase the fiscal burden and make debt effectively more risky. This is a very relevant argument, but it does not have straightforward implications for the appropriate level of debt. My purpose in the lecture is not to argue for more public debt, especially in the current political environment. It is to have a richer discussion of the costs of debt and of fiscal policy than is currently the case. (JEL E22, E23, E43, E62, H63)

Finite-Time Boundedness and H∞ Control for Affine Switched Systems

For affine switched systems, the existence of multiple equilibria is related to subsystems owing to the affine terms, which makes asymptotic and finite-time stability analysis nontrivial. In this paper, the problems of finite-time boundedness (FTB) analysis and stabilization are addressed for affine switched systems, and several definitions and sufficient conditions are proposed to study FTB and H∞ performance. At first, the definition of FTB for affine switched systems is improved concerning the affine terms and multiple equilibria. Based on the FTB definition, sufficient conditions ensuring finite-time boundedness for affine switched systems under a prespecified state boundary are given. Then the results are extended to solve H∞ finite-time boundedness problem, in which the H∞ controllers are designed to guarantee the finite-time boundedness of affine switched system with H∞ performance. In our investigation, average dwell-time approach is employed to study the time-dependent constrained switching case. Finally, several numerical examples are given to illustrate the effectiveness of the proposed results.

Modelling and stability analysis of switching impulsive power systems with multiple equilibria

ABSTRACTThis paper tries to model power systems accompanied with a series of faults in the form of switched impulsive Hamiltonian systems (SIHSs) with multiple equilibria (ME) and unstable subsystems (US), and then analyze long-term stability issues of the power systems from the viewpoint of mathematics. According to the complex phenomena of switching actions of stages and generators, impulses of state, and existence of multiple equilibria, this paper first introduces an SIHS with ME and US to formulate a switching impulsive power system composed of an active generator, a standby generator, and an infinite load. Then, based on special system structures, a unique compact region containing all ME is determined, and novel stability concepts of region stability (RS), asymptotic region stability (ARS), and exponential region stability (ERS) are defined for such SIHS with respect to the region. Third, based on the introduced stability concepts, this paper proposes a necessary and sufficient condition of RS and ARS and a sufficient condition of ERS for the power system with respect to the region via the maximum energy function method. Finally, numerical simulations are carried out for a power system to show the effectiveness and practicality of the obained novel results.

MODELING THE INFECTION DYNAMICS OF ONCHOCERCIASIS AND ITS TREATMENT

Onchocerciasis is one of the neglected tropical diseases caused by Onchocerca volvulus. Ivermectin is known to be effective in the treatment of onchocerciasis because it suppresses the production of microfilariae by the adult female worms for a few months following treatment thus reducing transmission. In this study, a deterministic model is developed to assess the effect of mass treatment of onchocerciasis with ivermectin. The basic reproduction number, [Formula: see text], of the model system is determined and it is observed that the model exhibits backward bifurcation for some parameters implying the existence of multiple endemic equilibria when [Formula: see text]. The existence of multiple equilibria emphasizes the fact that [Formula: see text] is not sufficient to eradicate the disease and the need is to lower [Formula: see text] much below one to make the disease-free equilibrium globally stable. Numerical simulations are done and conclusions drawn with respect to the known treatment protocols in endemic areas. The study results suggest that the mass treatment of the disease with ivermectin should cover a higher proportion of the population to control the disease and eventually eliminate it from the population.

A Simple Dynamical Model of an Industrial Laboratory Incorporating Knowledge Stock

A model of an industrial laboratory is presented and analyzed. It is a highly simplified abstraction, and consists of a network with two stages in series, Research (R) and Development (D). Ideas and prototypes are incubated in the R stage, possibly patented and documented, and brought to a sufficiently stable level for transfer to teams of professional developers in the D stage. Revenue is generated from sale and licensing of patents at both stages, and the sale of products/solutions that are outputs of the D stage. The model is dynamic, evolving with time, which is discrete. In each time period the parent corporation gives the managers of the industrial laboratory a fixed, constant amount of money to invest in the two stages. The investments determine the capacities of the stages based on linear unit costs. A novel feature of the model is for the stages, which represent the accumulated know-how from practicing research and development activities. Obsolescence and replenishment are responsible for their dynamic evolution, with replenishments owing to investments. The impact of the knowledge stocks is on the unit costs, which decrease by factors that are polynomials of the stocks. The memory in knowledge stocks make current investment decisions have long term impact on costs and profits. Three modes of profit maximizing decision-making are investigated, namely, myopic, long term average and an optimal control-based strategy that combines short and long term profits via discounting. The time evolutions are studied as nonlinear dynamical systems. For instance, we find, in the case of myopic profit maximizations, the existence of multiple equilibria and the phenomenon of entrapment in suboptimal regimes. Numerical results illustrate the main features of the model.

Heterogeneity of expectations and financial crises: a stochastic dynamic approach

This study provides a heterogeneous agent-based microfoundation to the concept of “liquidity trap” with a binary choice model, in which an economic agent stochastically changes her decisions. The transition rates from one state to the other vary, depending on the degree of diversity in expectations. Applying this model to the money/bond choice, this study seeks to derive the money demand function proposed by Keynes and analyze how the heterogeneity of expectations affects it. The model demonstrates that money holding becomes relatively advantageous as the proportion of money holders increases and that such a situation could bring about multiple equilibria. Through comparative statics, this study finds that the heterogeneity of expectations plays a crucial role for existence of multiple equilibria. Demonstrating that a financial crisis is a leap from one equilibrium to the other, the model helps to explain the recent crisis and offers practical implications for monetary policies. In particular, in analyzing the influences of heterogeneous expectations on the economy, this study uncovered an interesting fact that unconventional monetary policies work better than conventional ones in fighting against crises induced by a flight to liquidity.

Effects of Transmission Congestion on Different Incentive Policies for Renewable Energy

AbstractThis paper analyzes the effects of relieving power transmission congestion to encourage the development of renewable energy (RE) under different regulatory policies. The present model is based on Cournot competition and incorporates uncertainty and the variability of both renewable resources and demand. Transmission congestion particularly affects the development of RE under subsidy policies (feed-in tariff and premium payment). By investing in flexible AC transmission system (FACTS) and/or other devices that increase transmission capacity, all policies have positive incentives from the social perspective. From a private perspective, the tax policy does not provide sufficient incentives. Higher levels of transmission investment are obtained from a social perspective than from a private perspective. Finally, the existence of multiple equilibria affects the resulting social welfare and market outcomes. In particular, CO2 emissions and social welfare may significantly vary from equilibrium to equilib...

The role of radiation in organizing convection in weak temperature gradient simulations

AbstractUsing a cloud system resolving model with the large scale parameterized by the weak temperature gradient approximation, we investigated the influence of interactive versus noninteractive radiation on the characteristics of convection and convective organization. The characteristics of convecting environments are insensitive to whether radiation is interactive compared to when it is not. This is not the case for nonconvecting environments; interactive radiative cooling profiles show strong cooling at the top of the boundary layer which induces a boundary layer circulation that ultimately exports moist entropy (or analogously moist static energy) from dry domains. This upgradient transport is associated with a negative gross moist stability, and it is analogous to boundary layer circulations in radiative convective equilibrium simulations of convective self‐aggregation. This only occurs when radiation cools interactively. Whether radiation is static or interactive also affects the existence of multiple equilibria‐steady states which either support precipitating convection or which remain completely dry depending on the initial moisture profile. Interactive radiation drastically increases the range of parameters which permit multiple equilibria compared to static radiation; this is consistent with the observation that self‐aggregation in radiative‐convective equilibrium simulations is more readily attained with interactive radiation. However, the existence of multiple equilibria in absence of interactive radiation suggests that other mechanisms may result in organization.

Market structure and the value of overselling under stochastic demands

In the operations management literature, traditional revenue management focused on pricing and capacity allocation strategies in a two-period model with stochastic demand. Inspired by travel and lodging industries, we examine a two-period model in which each seller may also adopt the overselling strategy to customers whose valuations are differentiated by timing of arrivals. Widely seen as a popular hedge against consumers’ skipping reservations, we extend the stylized approaches of Biyalogorsky, Carmon, Fruchter, and Gerstner (1999) and Lim (2009) to understand the value of overselling under various market structures. We find that contrary to existing literature, the impact of period-two pricing competition from overselling spills over to period-one such that overselling may not always be a (weakly) dominant strategy once unlimited early demand ceases to hold in a duopoly regime. We provide some numerical studies on the existence of multiple equilibria at the capacity allocation level which actually lead to different selling strategies at the equilibrium despite identical market conditions and firm characteristics.

Forecast combination, non-linear dynamics, and the macroeconomy

This paper introduces the concept of a Forecast Combination Equilibrium to model boundedly rational agents who combine a menu of different forecasts in a way that mimics the behavior of actual forecasters. The equilibrium concept is consistent with rational expectations under certain conditions, while also permitting multiple, distinct, self-fulfilling equilibria, many of which are stable under least-squares learning. The equilibrium concept is applied to a Lucas-type monetary model and to a Fisherian monetary model with a Taylor rule. The existence of multiple equilibria is shown to depend on the aggressiveness of monetary policy in both models. In the latter, a more aggressive response to inflation is required in the Taylor rule than is typically found in this class of model to ensure a unique and learnable equilibrium. Real-time learning simulations with a constant gain illustrate some appealing properties of this approach including time-varying volatility and sharp movements in inflation, similar to actual data, while assuming only i.i.d. random shocks.

Intercomparison of methods of coupling between convection and large-scale circulation: 1. Comparison over uniform surface conditions.

As part of an international intercomparison project, a set of single‐column models (SCMs) and cloud‐resolving models (CRMs) are run under the weak‐temperature gradient (WTG) method and the damped gravity wave (DGW) method. For each model, the implementation of the WTG or DGW method involves a simulated column which is coupled to a reference state defined with profiles obtained from the same model in radiative‐convective equilibrium. The simulated column has the same surface conditions as the reference state and is initialized with profiles from the reference state. We performed systematic comparison of the behavior of different models under a consistent implementation of the WTG method and the DGW method and systematic comparison of the WTG and DGW methods in models with different physics and numerics. CRMs and SCMs produce a variety of behaviors under both WTG and DGW methods. Some of the models reproduce the reference state while others sustain a large‐scale circulation which results in either substantially lower or higher precipitation compared to the value of the reference state. CRMs show a fairly linear relationship between precipitation and circulation strength. SCMs display a wider range of behaviors than CRMs. Some SCMs under the WTG method produce zero precipitation. Within an individual SCM, a DGW simulation and a corresponding WTG simulation can produce different signed circulation. When initialized with a dry troposphere, DGW simulations always result in a precipitating equilibrium state. The greatest sensitivities to the initial moisture conditions occur for multiple stable equilibria in some WTG simulations, corresponding to either a dry equilibrium state when initialized as dry or a precipitating equilibrium state when initialized as moist. Multiple equilibria are seen in more WTG simulations for higher SST. In some models, the existence of multiple equilibria is sensitive to some parameters in the WTG calculations.

Determinacy and Sunspots in a Nonlinear Monetary Model

In this paper we analyze a basic sticky price model with monopolistic competition and price stickiness à la Calvo. Starting by the relations describing a general economic equilibrium model (see Woodford in Interest and Prices, Foundations of a Theory of Monetary Policy, The MIT Press, 2003), as it results from the optimizing behavior of the private agents, we provide a nonlinear model for the monetary policy analysis. This kind of model is a candidate for the existence of multiple equilibria, with a dependence of exogenous sunspots. We explore the stability of such a model combined with interest rate rules in order to investigate the determinacy of the model and we find, for some policy and elasticity parameters, the conditions under which it is possible.