Stabilization Theory

[Ottavia Delamar]

Indynamical systems, an orbit is called Lyapunov stable if the forward orbit of any point is in a small enough neighborhood or it stays in a small (but perhaps, larger) neighborhood. Various criteria have been developed to prove stability or instability of an orbit. Under favorable circumstances, the question may be reduced to a well-studied problem involving eigenvalues of matrices. A more general method involves Lyapunov functions. In practice, any one of a number of different stability criteria are applied.

Stability means that the trajectories do not change too much under small perturbations. The opposite situation, where a nearby orbit is getting repelled from the given orbit, is also of interest. In general, perturbing the initial state in some directions results in the trajectory asymptotically approaching the given one and in other directions to the trajectory getting away from it. There may also be directions for which the behavior of the perturbed orbit is more complicated (neither converging nor escaping completely), and then stability theory does not give sufficient information about the dynamics.

The simplest kind of an orbit is a fixed point, or an equilibrium. If a mechanical system is in a stable equilibrium state then a small push will result in a localized motion, for example, small oscillations as in the case of a pendulum. In a system with damping, a stable equilibrium state is moreover asymptotically stable. On the other hand, for an unstable equilibrium, such as a ball resting on a top of a hill, certain small pushes will result in a motion with a large amplitude that may or may not converge to the original state.

Stabilizing selection (not to be confused with negative or purifying selection[1][2]) is a type of natural selection in which the population mean stabilizes on a particular non-extreme trait value. This is thought to be the most common mechanism of action for natural selection because most traits do not appear to change drastically over time.[3] Stabilizing selection commonly uses negative selection (a.k.a. purifying selection) to select against extreme values of the character. Stabilizing selection is the opposite of disruptive selection. Instead of favoring individuals with extreme phenotypes, it favors the intermediate variants. Stabilizing selection tends to remove the more severe phenotypes, resulting in the reproductive success of the norm or average phenotypes.[4] This means that most common phenotype in the population is selected for and continues to dominate in future generations.

The Russian evolutionary biologist Ivan Schmalhausen founded the theory of stabilizing selection, publishing a paper in Russian titled "Stabilizing selection and its place among factors of evolution" in 1941 and a monograph "Factors of Evolution: The Theory of Stabilizing Selection" in 1945.[5][6]

Stabilizing selection causes the narrowing of the phenotypes seen in a population. This is because the extreme phenotypes are selected against, causing reduced survival in organisms with those traits. This results in a population consisting of fewer phenotypes, with most traits representing the mean value of the population. This narrowing of phenotypes causes a reduction in genetic diversity in a population.[7] Maintaining genetic variation is essential for the survival of a population because it is what allows them to evolve over time. In order for a population to adapt to changing environmental conditions they must have enough genetic diversity to select for new traits as they become favorable.[8]

There are four primary types of data used to quantify stabilizing selection in a population. The first type of data is an estimation of fitness of different phenotypes within a single generation. Quantifying fitness in a single generation creates predictions for the expected fate of selection. The second type of data is changes in allelic frequencies or phenotypes across different generations. This allows quantification of change in prevalence of a certain phenotype, indicating the type of selection. The third type of data is differences in allelic frequencies across space. This compares selection occurring in different populations and environmental conditions. The fourth type of data is DNA sequences from the genes contributing to observes phenotypic differences. The combination of these four types of data allow population studies that can identify the type of selection occurring and quantify the extent of selection.[9]

However, a meta-analysis of studies that measured selection in the wild failed to find an overall trend for stabilizing selection.[10] The reason can be that methods for detecting stabilizing selection are complex. They can involve studying the changes that causes natural selection in the mean and variance of the trait, or measuring fitness for a range of different phenotypes under natural conditions and examining the relationship between these fitness measurements and the trait value, but analysis and interpretation of the results is not straightforward.[11]

The most common form of stabilizing selection is based on phenotypes of a population. In phenotype based stabilizing selection, the mean value of a phenotype is selected for, resulting a decrease in the phenotypic variation found in a population.[12]

Stabilizing selection is the most common form of nonlinear selection (non-directional) in humans.[13] There are few examples of genes with direct evidence of stabilizing selection in humans. However, most quantitative traits (height, birthweight, schizophrenia) are thought to be under stabilizing selection, due to their polygenicity and the distribution of the phenotypes throughout human populations.[14]

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For the purposes of computing resonance parameters an alternative version of the stabilization method is used in a simpler way than previously reported [V. A. Mandelshtam, T. R. Ravuri, and H. S. Taylor, Phys. Rev. Lett. 70, 1932 (1993)]. The method is based on the calculation of the eigenphase sum, which, as is shown, can be extracted easily from a stabilization diagram.

We develop a novel, risk-based theory of the effects of exchange rate stabilization. In our model, the choice of exchange rate regime allows policymakers to make their currency, and by extension, the firms in their country, a safer investment for international investors. Policies that induce a country's currency to appreciate when the marginal utility of inter- national investors is high lower the required rate of return on the country's currency and increase the world-market value of domestic firms. Applying this logic to exchange rate stabilizations, we find a small economy stabilizing its bilateral exchange rate relative to a larger economy can increase domestic capital accumulation, domestic wages, and even its share in world wealth. In the absence of policy coordination, small countries optimally choose to stabilize their exchange rates relative to the currency of the largest economy in the world, which endogenously emerges as the world's \anchor currency." Larger economies instead optimally choose to float their exchange rates. The model therefore predicts an equilibrium pattern of exchange rate arrangements that is remarkably similar to the one in the data.

As we shall see, the relative merits of these two hypotheses can in fact be evaluated by considering which of the two is more consistent with available data. But first, let's look at some examples of stabilization processes.

This paper studies the role of stabilization policy in a model where firm entry responds to shocks and uncertainty. We evaluate stabilization policy in the context of a simple analytically solvable sticky price model, where firms have to prepay a fixed cost of entry. The presence of endogenous entry can alter the dynamic response to shocks, leading to greater persistence in the effects of monetary and real shocks. Entry affects welfare, depending on the love of variety in consumption and investment, as well as its implications for market competitiveness. In this context, monetary policy has an additional role in regulating the optimal number of entrants, as well as the optimal level of production at each firm. We find that the same monetary policy rule optimal for regulating the scale of production in familiar sticky price models without entry, also generates the amount of (endogenous) entry corresponding to a flex-price equilibrium.

Adam Hayes, Ph.D., CFA, is a financial writer with 15+ years Wall Street experience as a derivatives trader. Besides his extensive derivative trading expertise, Adam is an expert in economics and behavioral finance. Adam received his master's in economics from The New School for Social Research and his Ph.D. from the University of Wisconsin-Madison in sociology. He is a CFA charterholder as well as holding FINRA Series 7, 55 & 63 licenses. He currently researches and teaches economic sociology and the social studies of finance at the Hebrew University in Jerusalem.

There are many reasons why a government would seek to implement stabilization policies. For one, such policies may help to even out erratic economic swings like recessions, which can lead to unemployment, price volatility, and reduced output. Stabilization policy can also cool an overheated market.

During the early months of the COVID-19 public health emergency, governments and central banks around the world adopted stabilization policies in an effort to mitigate the risk of severe recession. Examples of stabilization policy during this period included grants and low-interest loans for affected business sectors, stimulus checks for individuals and families, and the lowering of interest rates to keep money circulating in the economy.

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