Stability Properties Of Model Research Articles

Characterizing the Degree of Stability of Non-linear Dynamic Models

The purpose of this paper is to show how the stability properties of non-linear dynamic models may be characterized and studied, where the degree of stability is defined by the effects of exogenous shocks on the evolution of the observed stochastic system. This type of stability concept is frequently of interest in economics, e.g., in real business cycle theory.We argue that smooth Lyapunov exponents can be used to measure the degree of stability of a stochastic dynamic model. It is emphasized that the stability properties of the model should be considered when the volatility of the variable modelled is of interest. When a parametric model is fitted to observed data, an estimator of the largest smooth Lyapunov exponent is presented which is consistent and asymptotically normal. The small sample properties of this estimator are examined in a Monte Carlo study. Finally, we illustrate how the presented framework can be used to study the degree of stability and the volatility of an exchange rate.

Continuous-time estimation of an endogenous growth model of an open economy

Most recent theoretical research on endogenous economic growth has been conducted with continuous-time models that embody the assumption of an intertemporally optimizing representative agent. Yet virtually none of these models has been confronted with empirical data. In this paper we make use of a model of an open economy recently developed by Turnovsky to demonstrate one approach to the estimation and analysis of such models. We present the results of the estimation, analyze the estimated model's stability properties around its steady-state solution, and explore the model's behavior in response to alternative tax policies.

QUANTITATIVE EVALUATIONS OF EFFICIENT TAX POLICIES FOR LUCAS' SUPPLY SIDE MODELS

Robert Lucas's recent paper on supply-side economics (1990) finds a large welfare loss from taxation of interest income in the U.S. economy. The present work extends the analysis of steady-state equilibria to cover the transition paths that accompany tax reform. Calculations then show a 25 percent reduction in taxes on capital's income, instituted through a carefully designed sequence of steps, yielding a 1-1.5 percent welfare gain. The analysis also considers government debt and an investment tax credit. In addition, it reveals a potential problem with one aspect of the model's stability properties, when growth is purely endogenous. Copyright 1995 by Royal Economic Society.

Atmospheric Transports, the Thermohaline Circulation, and Flux Adjustments in a Simple Coupled Model

A theoretical analysis of the interactions between atmospheric meridional transports and the thermohaline circulation is presented, using a four-box ocean-atmosphere model in one hemisphere. The model is a simplified version of that developed by Nakamura Stone, and Marotzke and is amenable to analytical solutions. The ocean model is Stommel's; the atmospheric model gives the surface heat and freshwater fluxes as residuals of the atmospheric energy and moisture budgets, assumed in balance. Radiation at the top of the atmosphere depends linearly on surface temperature; atmospheric meridional heat and moisture transports are proportional to the meridional temperature gradient. A Newtonian cooling law is derived for differential surface heat flux. The restoring coefficient is proportional to the efficiency of atmospheric transports and inversely proportional to the relative ocean area compared to total surface area. Surface freshwater flux increases with increasing temperature gradient and is inversely proportional to the ratio of ocean area to catchment area. The range of stable solutions with high-latitude sinking is smaller than in related, uncoupled box models due to the dependence of freshwater flux on the temperature gradient, which leads to a positive feedback with the thermohaline circulation. A strong control of the temperature gradient by atmospheric transports enhances the positive feedback between the salinity gradient and thermohaline Circulation simultaneously, it weakens the positive feedback between atmospheric moisture transport and the thermohaline circulation. Overestimating the atmospheric moisture transport and underestimating oceanic mass transport both artificially destabilize the high-latitude sinking state. Overestimating the atmospheric heat transport and hence the Newtonian restoring coefficient can be artificially stabilizing or destabilizing. These erroneous sensitivities ate not alleviated if flux adjustments are added to obtain the correct mean climate, and then held fixed in climate change experiments. We derive alternate flux adjustment schemes, which do preserve the model's stability properties for particular sources of error.

Size-Selective Sex-Allocation and Host Feeding in a Parasitoid--Host Model

1. Hymenopterous parasitoids frequently exhibit (a) size-selective sex-allocation, laying predominantly male eggs in smaller host individuals and female eggs in larger hosts, and (b) size-selective host-feeding, i.e. feeding on and killing, but not parasitizing, smaller host individuals. We abbreviate size-selective sex-allocation and host-feeding by SSH. We analyse a parasitoid--host model incorporating SSH that recognizes the following: overlapping generations, an invulnerable adult host stage, a young immature stage, an old immature host stage, and only the female parasitoid. We assume that young immature hosts are attacked by the parasitoids, die as a consequence, but do not contribute to the juvenile female parasitoid population; each attack on an old immature host produces a juvenile female parasitoid. 2. SSH leads to delayed pseudo-density-dependence in recruitment to the parasitoid population because the current attack rate on young immatures, which is a function of parasitoid density, influences the future number of old immatures and hence the future per head rate of recruitment of searching parasitoids. SSH has two effects on the model's stability properties. (a) It is potentially stabilizing because it tends to suppress the inherent long-period host-parasitoid cycles. This stabilizing propensity is enhanced when the adult host stage lasts about as long as or langer than the entire immature stage, the immature parasitoid stage is short-lived, and the young immature host stage is not too long-lived or the attack rate on it is not too high. (b) SSH can also destabilize the model by creating a region of instability in which cycles occur with a period related to the total length of the host immature stages. These shorter-period cycles are promoted by the same features that tend to suppress the underlying parasitoid--host cycles. 3. Small perturbations from equilibrium in the locally stable region of parameter space are followed by damped oscillations. However, because of the presence of a multiple attractor, in much of the locally stable region large perturbations may be followed by limit cycles, commonly of the type seen in the new unstable region.

Are pelagic systems inherently unstable? a model study

Recent model work on patchiness of biological factors in the open sea (Steele, 1974a; Dubois, 1975) neglect the recycling of nutrients, which is necessary for continuous primary production. Steele shows that an inherently unstable primary producer—herbivore system together with diffusive processes can cause patchiness in these populations. The purpose of the present study is to investigate the stability properties of phytoplankton-zooplankton models, when recycling is taken into account, and how these properties are influenced by the use of Michaelis-Menten expressions for grazing and nutrient uptake. Also the effects of two different so called threshold functions in the grazing are investigated (i.e. functions, which give low or zero zooplankton filtration rate at low food concentrations). An analytical investigation of a two-component model shows that the incorporation of recycling pathways radically changes the stability properties of the model. Simulations of a four-component model with nutrient recycling confirm this conclusion and furthermore given the following results: 1. (1) If one or two of the three recycling pathways in the model are omitted, the model remains stable, but exhibits a greater tendency to damped oscillations if perturbed. 2. (2) When Michaelis-Menten functions are used for both the nutrient dependence of phytoplankton production and the dependence of zooplankton grazing on food concentration or only for the latter, asymptotically stable as well as nonconvergent oscillating cases can be obtained depending on the parameter values. With a Michaelis-Menten function only in the primary production, exclusively asymptotically stable cases were obtained, but some of these exhibited so weakly damped oscillations, that they in practice Could be considered non-convergent. 3. (3) Only one of the two tested threshold responses in zooplankton grazing have a marked stabilizing effect on the model It is pointed out, that during the highly productive seasons the recycling of matter is a vital feature of the pelagic phytoplankton-zooplankton system, and that only in special cases can an omission of the recycling processes in a functional model of this system be justified.

Baroclinic instability of stratified shear flows in the ocean and atmosphere

As an extension of Eady's model, two new baroclinic models are studied in this paper: Model 1 consists of a layer with constant shear surmounting a quiescent layer, and model 2 consists of a more statically stable uniform layer of constant zonal flow surmounting a layer with constant shear. In model 1 two special cases are considered, one in which the upper shear layer is more stably stratified than the lower quiescent layer (1a) and another in which the upper layer is less stably stratified than the lower layer (1b). The stability properties of model 16 and model 2 turn out to be very similar, since for the same shear, depth ratio, and static stability ratio the growth rates are the same and the wave structure of model 1b is the symmetric image of that of model 2. The wavelengths of the stability cutoff wave and the most unstable wave and the growth rate of the most unstable wave are considerably influenced by the presence of a quiescent layer in model 1a but are only moderately influenced by this quiescent layer in model 1b or by the uniform layer in model 2. The unstable wave moves eastward with a velocity slower than that of the vertically averaged mean current in the shear layer in model 1 but faster than the mean current in the shear layer in model 2. The deviation of the phase velocity from this mean current becomes larger the longer the wave. Typical parameters for an oceanic current system, such as the Gulf Stream, are applied to model 1a, and typical parameters for an atmospheric mid-latitude zonal flow are applied to model 2. The implications regarding long waves are that just as the long atmospheric waves penetrate the tropopause into the stratosphere, so the long oceanic waves penetrate the base of the thermocline to great depths, although the horizontal scales for the two systems are considerably different.