Stability of nonlinear homogeneous difference equations

A class of nonlinear homogeneous difference equations is considered in which each coefficient is a homogeneous function of variables. A sufficient condition of relative stability is presented using a nonlinear generalization of the Perron-Frobenius theorem due to Morishima (“Equilibrium, Stability and Growth,” Oxford Univ. Press, Oxford, 1964).     

Optimal control problems from second-order difference equations

We examine solutions x?l_∞(Rⁿ) of the equations $\text{F}{}_{\mathrm{t}}\text{(}\text{x}{}_{\mathrm{t?1}}\text{,}\text{x}{}_{\mathrm{t}}\text{,}\text{x}{}_{\mathrm{t+1}}\text{) = 0}$ and derive conditions for the existence of objective functions G_t so that $\text{x}$ solves max_x?l∞ΣG_t(x_t, x_t+1). We specialize the conditions to time autonomous equations and apply them to a competitive industry in temporary equilibrium. The objective function in the example is of the cost-benefit form: consumer surplus minus opportunity cost of labor minus industry-wide cost of investment.     

Nonlinear dynamics in CEE stock markets indices

We investigate the existence of nonlinearities in the dynamics of the returns of stock markets indices from CEE economies. We use several types of nonlinear tests. We discuss the implications of the results with respect to the efficient market hypothesis.     

The use of linear difference equations in manpower planning: A criticism

Systems of linear difference equations have been used extensively to project educational and manpower needs. The dynamic properties of such models appear never to have been considered. A number of these models were examined and, without exception, were highly unstable. Sensitivity analyses indicated that the source of model instability is not in parameter estimation error. The consistently unsatisfactory results suggest that linear difference equations do not adequately model manpower needs. The more general point made is that validation of any dynamic model should always include an investigation of the dynamic characteristics.     

Nonlinear dynamics and covered interest rate parity

This paper examines the dynamics of deviations from covered interest parity using daily data on the UK/US spot, forward exchange rates and interest rates over the period January 1974 to September 1993. Like other studies we find a substantial number of instances during the sample in which the covered interest parity condition exceeds the transaction costs band, implying arbitrage profit opportunities. While most of these implied profit opportunities are relatively small, there is also evidence of some very large deviations from covered interest parity in the sample. In order to examine the persistence of these deviations, we estimated a threshold autoregression in which the dynamics behavior of deviations from covered interest parity is different outside the transaction costs band than inside them. We find that while the impulse response functions when inside the transaction costs band are nearly symmetric, those for the outside the bands are asymmetric-suggesting less persistence outside of the transaction costs band than inside the band.The authors would like to thank Jeanette Diamond, Michael O'Hara, Baldev Raj, and two anonymous referees for helpful suggestions on an earlier draft. The views expressed herein are solely those of the authors and should not be construed to be those of the Federal Reserve Bank of Dallas or the Federal Reserve System.     

Housing prices and population dynamics in urban China

In this study, a dynamic stochastic general equilibrium (DSGE) model is used to investigate the influence of population on housing price dynamics in urban China. The contribution of population growth to the housing price trend and the contribution of population shocks to housing price fluctuations are quantitatively measured. The empirical results indicate that the ongoing policy to control the population size in large cities in China can decrease the growth rate of housing prices by 0.49% every year, which accounts for 1/10 of the total housing price trend. Furthermore, this policy cannot stabilize housing prices because population shocks have negligible effects on the housing price cycle. Housing technology shocks and housing preference shocks are responsible for most of the housing price fluctuations.     

Nonlinear exchange rate dynamics under stochastic official intervention

Many studies employ non-linear models to explain or forecast the exchange rate and find their superiority. This article builds an exchange rate model of managed float under conditional official intervention. In the model, the government minimizes social loss through a trade-off between targeting the exchange rate and lowering intervention costs. We obtain an endogenous threshold model and derive an analytical solution of the exchange rate stochastic interventions. The implication of a managed float causing a lower volatility of the exchange rate has been found by past empirical studies. Our model provides not only a justification for the central banks' conditional interventions but also a rationale for the use of regime-switching models of two states (intervention vs. non-intervention) in the empirical studies of exchange rates.     

Nonlinear dynamics: Evidence for a small stock exchange

This paper models the main stock index of the Vienna Stock Exchange with daily data from 1986 to 1992. We find that returns are nonnormal and show linear and nonliner dependence. On that basis we compare the fit of alternative specifications of Generalized Autoregressive Conditional Heteroscedasticity (GARCH) to the Markov-Switching approach. The models are evaluated with diagnostic tests on the standardized residuals. We consider evidence for deterministic structures and for infinite variance. Our main result is that a parsimonious model from the GARCH – class can generate the statistical properties of daily returns. The behavior of the two types of models with respect to temporal aggregation is found to differ significantly. First version received: January 1996/Final version received: December 1997     

Nonlinear behavior of the socio-economic dynamics for lake eutrophication control

To succeed in combating lake eutrophication, cooperation of local inhabitants, small factories, and farmers in reducing phosphorus discharge is very important. But the willingness of each player to cooperate would depend on the cooperation of other players and on the level of environmental concern of the society in general. Here we study the integrated dynamics of people's choice of behavior and the magnitude of eutrophication. Assumptions are: there are a number of players who choose between alternative options: a cooperative and environment-oriented option is more costly than the other. The decision of each player is affected by “social pressure” as well as by economical cost of the options. The lake pollution increases with the total phosphorus released, and a high pollution level in the lake would enhance the social pressure. The model includes a positive and a negative feedback loops which create diverse dynamical behavior. The model often shows bistability — having an equilibrium with a high level of cooperation among people and clean water, and the other equilibrium with low cooperation and polluted water, which are simultaneously stable. The model also shows fluctuation between a high and a low levels of cooperation in alternating years, cycle with a longer periodicity, or chaotic fluctuation. Conservatism of people stabilizes the system and sometimes helps maintaining cooperation. The system may show unexpected parameter dependence — the improved phosphorus removing efficiency might make water more polluted if it causes the decline in the environmental concern and cooperation among people.     

Convergence for difference equations with vanishing time-dependence, with applications to adaptive learning

Summary. We provide conditions for local stability and instability of an equilibrium point in certain systems of nonautonomous nonstochastic difference equations. In the systems under study the influence of time is present through a positive scalar “gain” parameter which converges in the limit to zero. These systems have recently been used to study the dynamics of adaptive learning in economic models, and we provide two economic illustrations of the formal results. Received: October 7, 1997; revised version: February 8, 1999     

The systems dynamics of endogenous population growth in a renewable resource-based growth model

This paper evaluates the dynamic effects of adding an endogenous process for human population growth into a renewable resource-based economic growth model. Endogenizing human population growth in a static, constant technology form of the model gives rise to a dynamically complex system, with the possibility of multiple steady states of several types, and unusual comparative static responses to changes in the system's parameters. Adding technological progress to the model gives rise to the possibility of multiple sustainable paths for the variables in the system. These results reinforce concerns raised by ecological economists about systems stability and sustainability, since exogenous shocks to the system could move the economy from higher welfare to lower welfare equilibria or, in the model with technological progress, from higher welfare to lower welfare sustainable growth paths. Moreover, this kind of dynamic complexity adds to the management challenge faced by policy makers, who could confront the necessity of maneuvering the economy among different equilibria or sustainable growth paths.     

The employment dynamics of different population groups over the business cycle

We examine differences in employment dynamics across population groups using Bayesian vector autoregressions. We document that groups who are particularly strongly affected by business-cycle fluctuations (males, young people, non-whites, the less educated, and workers in blue-collar occupations) also tend to be affected early in the build-up of a boom or bust. We further identify the drivers of the different cyclicalities across population groups. Supply shocks seem to be most important for the heterogeneous employment fluctuations and particularly for the early effects of recessions and booms on the most affected groups. Dynamics in sectoral activity and in hiring rates can help to understand our findings.     

A note on the compatibility of a system of difference equations and a time-independent linear equation

Solutions to systems of difference equations subject to linear constraints are studied. Restrictions on the coefficients necessary to maintain a compatible system are developed. Applications to the dynamics of expenditure shares and to lagged adjustment models with adding up constraints are given.     

Nonlinear dynamics in exchange rate deviations from the monetary fundamentals: An empirical study

This paper reexamines empirical performance of the monetary exchange rate model with nonlinear dynamics of exchange rate deviation from the monetary fundamentals. First, we apply unit root test of Park and Shintani (2005) to post-Bretton Woods exchange rate data and able to reject the null of unit root deviation from monetary fundamentals against alternative hypothesis of nonlinear stationary process for deutschemark, pound, and Swiss franc. Our empirical results find that exchange rates show high degree of mean-reversion with larger deviation and long periods of overvaluation and undervaluation of dollar. We also find empirical evidence of predictability of the monetary fundamentals at longer horizons.     

Random perturbations of deterministic equilibria

This paper examines whether a small random deviation from a non-random policy process will destabilize the equilibrium exchange rate. The results focus on exchange rate dynamics when the government maintains a target zone or makes unanticipated deviations from a policy rule. With rational expectations small random shocks do not create large deviations from the policy rule exchange rate. If agents are approximately rational large deviations between exchange rates are possible.     

Nonlinear dynamics in arbitrage of the S&P 500 index and futures: A threshold error-correction model

Using a three-regime threshold error-correction model, we investigate the nonlinear dynamics of the S&P 500 index and futures. First, using the SupLM statistic, we report estimates of two thresholds for the three-regime model to explain the nonlinear dynamics in arbitrage of the S&P 500 index and futures. This provides empirical evidence of the no-arbitrage band predicted by the cost-of-carry model. Second, using quasi-maximum likelihood estimation, we demonstrate that those indexes that are located outside the no-arbitrage band are a nonlinear stationary process of mean-reversion to the no-arbitrage band. However, index and futures that are located within the no-arbitrage band are non-stationary. Third, we confirm an earlier finding that futures price leads the nonlinear mean-reverting behavior of the index but not vice versa. Impulse response function analysis and forecasting performance of three-regime error-correction model reinforce our findings and our estimation results are robust with different specifications of pricing error terms and endogenous variables.     

Renewable resource market obeying difference equations: Stable points,stable cycles,and Chaos (1991)

In this paper, the basic principles of the Chaos Theory are applied to dynamic systems involving renewable resource and harvest. The results of simulations are presented regarding harvested resource stock time path. It is shown that stationary steady-state is only one of many different possible outcomes of bioeconomic equilibrium. The varying impacts that different management policies have on an unstable dynamic system are also studied. It is shown that in unstable resource systems, the effects of some policy measures will be very different from the ones obtained through traditional resource analysis. It also shows that policies which induce a stabilizing effect in stock time series will generally provide greater welfare gains. The results are illustrated by two market examples.     

Non-linear dynamics of population and natural resources: The emergence of different patterns of development

This paper studies the long-term dynamic interaction between the exploitation of natural resources and population growth. Brander and Taylor [Brander, J.A. and Taylor, M.S. (1998) “The Simple, Economics of Easter Island: a Ricardo–Malthus Model of Renewable Resource Use”. American Economic Review, 88 (1) 119–138.] started a sequence of papers which sought to deduce from historical and archaeological studies some stylized links between ecological and economic systems. In this strand of literature all the services of the natural environment are aggregated in an ecological complex which is characterized by a simple logistic dynamics. Given such assumptions, these models show a unique long-term steady state. The aim of this paper is to obtain a more general framework that could account for the heterogeneity of environmental development paths followed by past societies. Two new assumptions are introduced: i) the disaggregation of the ecological complex into two different resources; ii) irreversibility — namely, an inexorable tendency to exhaustion when the renewable resource stock is below a certain threshold. Analysis of the dynamic properties of the system shows a multiplicity of steady states which makes it possible to consider the effects of technical progress, cultural and climate changes on the resilience of the existing development path.