具有三次代数曲线解的五次系统的极限环

本文利用Maple软件强大的符号计算能力,得到了以 y=ax~3+bx~2+cx+d 为解的平面五次多项式系统的特征,由此证明了该系统在奇点的邻域内可以存在极限环。     

Pre‐procreative ages in population stability and cyclicity

Age‐specific models of population renewal (with and without feedback) which imply convergence to a stable state for some levels of fertility or feedback may imply the presence of long‐term cycling around a constant or exponentially changing equilibrium for other levels of fertility or feedback. The switch from one regime to the other is a “bifurcation.”; The conditions for bifurcation involve the roots of an analogue of Lotka's Equation.

Typically bifurcation is induced by raising the strength of feedback or the level of fertility. It has been known since the early 1980s, however, that this is sometimes impossible. It is sometimes impossible even with the linear renewal equation itself and with the most basic of non‐linear models, Lee's cohort feedback model.

Here it is proved that this typical route to bifurcation does not fail for these basic models in the presence of a condition which always holds for realistic applications with higher organisms: the existence of a span of ages before the onset of fertility.

Specifically, a strictly positive lower bound on ages of procreation is proved to be sufficient to guarantee the existence of a rescaling of Lotka's Equation for which the real part of some complex root vanishes. This result holds for absolutely Lebesgue‐integrable (signed) net maternity functions on the positive real line and for absolutely summable (signed) net maternities on the positive integers.

It follows that Coale's rescaling device for the analysis of approach to stability in stable population theory can be implemented for all realistic human net maternity schedules. It also follows that the many special cases of the cohort feedback model throughout population biology will all generate persistent cycling instead of stability if feedback is sufficiently strong.     

论经济周期波动的随机统计规律

从经济周期波动的特征出发，分析经济周期现象中考虑随机性的合理性和必要性，可看出经济周期运动的内在机制中蕴含着大量的不确定随机因素及对经济周期波动的影响，从随机周期现象入手，针对周期波动的研究历史和现状，就可得到新的启示和评价．     

Inference on Filtered and Smoothed Probabilities in Markov-Switching Autoregressive Models

ABSTRACT

We derive a statistical theory that provides useful asymptotic approximations to the distributions of the single inferences of filtered and smoothed probabilities, derived from time series characterized by Markov-switching dynamics. We show that the uncertainty in these probabilities diminishes when the states are separated, the variance of the shocks is low, and the time series or the regimes are persistent. As empirical illustrations of our approach, we analyze the U.S. GDP growth rates and the U.S. real interest rates. For both models, we illustrate the usefulness of the confidence intervals when identifying the business cycle phases and the interest rate regimes.     

Sentiments

This paper develops a new theory of fluctuations—one that helps accommodate the notions of “animal spirits” and “market sentiment” in unique‐equilibrium, rational‐expectations, macroeconomic models. To this goal, we limit the communication that is embedded in a neoclassical economy by allowing trading to be random and decentralized. We then show that the business cycle may be driven by a certain type of extrinsic shocks which we call sentiments. These shocks formalize shifts in expectations of economic activity without shifts in the underlying preferences and technologies; they are akin to sunspots, but operate in unique‐equilibrium models. We further show how communication may help propagate these shocks in a way that resembles the spread of fads and rumors and that gives rise to boom‐and‐bust phenomena. We finally illustrate the quantitative potential of our insights within a variant of the RBC model.     

Do Political Institutions Shape Economic Policy?

Do political institutions shape economic policy? I argue that this question should naturally appeal to economists. Moreover, the answer is in the affirmative, both in theory and in practice. In particular, recent theoretical work predicts systematic effects of electoral rules and political regimes on the size and composition of government spending. Results from ongoing empirical work indicate that such effects are indeed present in the data. Some empirical results are consistent with theoretical predictions: presidential regimes have smaller governments and countries with majoritarian elections have smaller welfare‐state programs and less corruption. Other results present puzzles for future research: the adjustment to economic events appears highly institution‐dependent, as does the timing and nature of the electoral cycle.     

Sticky Price Models of the Business Cycle: Can the Contract Multiplier Solve the Persistence Problem?

We construct a quantitative equilibrium model with firms setting prices in a staggered fashion and use it to ask whether monetary shocks can generate business cycle fluctuations. These fluctuations include persistent movements in output along with the other defining features of business cycles, like volatile investment and smooth consumption. We assume that prices are exogenously sticky for a short time. Persistent output fluctuations require endogenous price stickiness in the sense that firms choose not to change prices much when they can do so. We find that for a wide range of parameter values, the amount of endogenous stickiness is small. Thus, we find that in a standard quantitative model, staggered price‐setting, alone, does not generate business cycle fluctuations.     

A Dynamic Model of Investment in the U.S. Beef-Cattle Industry

Reproduction constraints lead to some a priori information that may prove valuable in econo- metrically modeling inventory dynamics within the U.S. cattle herd. The distributed-lag models estimated for animal inventories suggest an implicit incorporation of the age distribution of the herd that is crucial to understanding the retention/culling (investment) decisions. The estimated equations are validated using post-sample observations that were withheld prior to estimation. The final-form dynamic equations have complex roots and protracted price and investment effects. The parameter estimates are used to calculate intermediate and long-run elasticities at the means of relevant variables.     

Codependent Cycles of Chinese Inflation

A test and decomposition of the codependent cycles of the consumer price index (CPI), agricultural producer price index (APPI) and M2 reveals that the sharp cyclical fluctuations in agricultural producer prices largely stem from the grain price fluctuations induced by factors such as land enclosure and the impact on APP of the abrupt changes in pork prices and other factors. Therefore, changing the current policy of grain purchasing price subsidies to one of subsidies to grain production and sales will reduce the impact of grain prices and other factors on APP and inflation. The weak cyclic codependence among APP, CPI and M2, along with the strong cyclic codependence between M2 and CPI, has given rise to a “shuttle-shaped” intersection of cyclic elements. This indicates that China’s monetary policy goal has been alternating between curbing inflation and stimulating growth. When the APP cycle is basically stable, China can implement a moderately easing monetary policy to stimulate economic growth, but this must be based on a 1.5 or so margin of increase in the codependent cycle components of M2. When APP rises sharply or is in the rising phase of cycle elements, China should emphasize APP control, and should delay the moderately easing monetary policy till the APP cycle becomes stable or is in the downlink phase.     

The Secular and Cyclical Behavior of Real GDP in 19 OECD Countries, 1957–1983

Log per capita real gross domestic product is modeled as a third-order autoregression with a pair of complex roots whose amplitude is smaller than the amplitude of the real root. The behavior of this terms series is interpreted in terms of these two amplitudes, the periodicity of the complex roots, and the standard deviation of the disturbance. Restrictions are evaluated and inference is conducted using the likelihood principle, applying Monte Carlo integration with importance sampling. These Bayesian procedures efficiently cope with restrictions that are awkward taking a classical approach. We find very little difference in the amplitudes of real roots between countries and of complex roots relative to within-country uncertainty. There are some substantial differences in the periodicities of complex roots, and the greatest differences between countries are found in the standard deviation of the disturbance.