Capital dependent population growth induces cycles

Cobb-Douglas type production functions and time-delay are not sufficient for the economy to behave cyclic. However, capital dependent population dynamics can enforce Hopf bifurcation.     

A microeconomic model of business growth,decay and cycles

Summary This paper presents a dynamic theory of price, output, working capital and interest for the firm and industry. It yields many of the standard properties of the firm and industry at equilibrium, and in addition, a new microeconomic theory of growth, decay and business cycles. This is achieved by adding a working capital or budget constraint to the conventional atomistic single product firm and assuming a one period input-output lag. The theory leads to a simple but instructive example of a new mathematical system called recursive programming that represents ex ante optimizing while at the same time exhibiting ex post nonoptimality under certain conditions.

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Zusammenfassung Die Arbeit bringt eine dynamische Theorie von Preis, Ausstoß, Betriebskapital und Zins für die Unternehmung und Branche. Es werden allgemeine Eigenschaften einer Unternehmung und der Branche im Gleichgewicht und darüber hinaus Grundzüge einer mikroökonomischen Theorie von Wachstum, Schrumpfung und Konjunkturzyklen dargelegt. Dies wird erreicht, indem bei der konventionellen, sich in atomistischer Konkurrenz befindlichen Einproduktunternehmung eine Betriebskapital- oder Budgetbeschränkung eingeführt wird und ein einperiodischer input-output lag angenommen wird. Die Theorie führt zu einem einfachen aber instruktiven Beispiel einer neuen, Rekursive Programmierung genannten Methode der ex ante Optimierung, wobei aber unter bestimmten Bedingungen ex post Nicht-Optimilität auftreten kann.

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Vorgel. v.:W. Wittmann.     

A bivariate model for total fertility rate and mean age of childbearing.

Annual estimates of total fertility rate and mean age of childbearing from the years 1921-80 in the US were fitted to a bivariate autoregression model incorporating regression adjustments for the effects of World War II on fertility. The time series model reduces to a transfer function model in which the total fertility rate is a function of its own past and noise, while mean age at childbearing is a function of its own past, the past of the total fertility rate, and noise. Although contemporaneous changes in mean age of childbearing tend to be negatively correlated with changes in total fertility rate, there is a tendency for adjustments in mean age to be in the same direction as changes in total fertility in the long run. The dynamic tendency in the relationship between these 2 variables is not obvious in simple graphs of these quantities. Forecasts beyond the year 1944 were found to include high standard errors and thus are considered useless as guides to the future behavior of these variables. However, other time series could be added to the model, and the judgements of demographers, economists, and sociologists could be blended with the model-based extrapolations to provide useful data.     

Endogenous cycles in discrete symmetric multisector optimal growth models

This paper discusses the asymptotic stability of the steady state in discrete symmetric multisector optimal growth models. Using a variational method, we provide a new proposition which gives some conditions ensuring the local saddle point property. A characterization of the bound above which the steady state is locally unstable is also proposed in terms of the indirect utility function concavity properties. On this basis, some sufficient conditions for the existence of competitive cycles are stated. We thus prove the existence of a Flip bifurcation.The authors would like to thank J. M. Grandmont, P. Michel, and L. Montrucchio for helpful discussions which greatly improved the exposition of the paper. The present work also benefited from comments received during a presentation at the Meetings of the Society for Economic Dynamics and Control, UCLA, June 30–July 2, 1994.     

Limit cycles in a chemostat with general variable yields and growth rates

A three-dimensional chemostat model with general variable yields and general growth rates is studied. The stability of the equilibrium points, the existence of limit cycles, and the Hopf bifurcation are investigated. Examples are used to illustrate that our theorems are easily applied in the practical models.     

Economic intermittency in a two-country model of business cycles coupled by investment

Intermittent behavior of economic dynamics is investigated by a two-country model of Keynes-Goodwin type business cycles. Numerical simulations show that after an economic system evolves from weak chaos to strong chaos the system keeps its memory before the transition and its time series alternates episodically between periods of weakly and strongly chaotic fluctuations. In addition, we examine the intermittent phenomena from the view point of business cycle patterns near the crisis point.     

Volume growth and escape rate of symmetric diffusion processes

We give an upper rate function, in terms of the volume growth of the underlying state space, for the symmetric diffusion process associated with a symmetric, strongly local regular Dirichlet form. It extends the main result of Hsu and Qin [Ann. Probab. 38(4) 2010], where an upper rate function was given for Brownian motion on Riemannian manifold.     

The growth rate of symplectic Floer homology

The main theme of this paper is to study for a symplectomorphism of a compact surface, the asymptotic invariant which is defined to be the growth rate of the sequence of the total dimensions of symplectic Floer homologies of the iterates of the symplectomorphism. We prove that the asymptotic invariant coincides with asymptotic Nielsen number and with asymptotic absolute Lefschetz number. We also show that the asymptotic invariant coincides with the largest dilatation of the pseudo-Anosov components of the symplectomorphism and its logarithm coincides with the topological entropy. This implies that symplectic zeta function has a positive radius of convergence. This also establishes a connection between Floer homology and geometry of 3-manifolds.     

A theory of average growth rate indices

This paper develops an axiomatic theory of an economic variable average growth rate (average rate of change) measurement. The obtained structures generalize the conventional measures for average rate of growth (such as the difference quotient, and the continuously compounded growth rate) to an arbitrary domain of the underlying variable and comprise various models of growth. These structures can be described with the help of intertemporal choice theory by means of parametric families of time preference relations on the “prize-time” space with a parameter representing the subjective discount rate.     

Growth rate,growth curve and growth prediction of tumour in the competitive model

ABSTRACT

The growth of cancer is still the focus of many research works in the scientific community. So far, various models have been introduced to analyse the behaviour of cancers, including the mathematical growth models such as Logistic, Gompertz and Bertalany. Despite the advances in the analysis of the cancer behaviour, the lack of definitive treatment of cancer disease indicates the need for new perspectives which are supported by more biological background. Recently, a model has been proposed, in which, the tumour growth is interpreted as the outcome of the competition of healthy and cancer cells over the available oxygen, nutrients and space. We have modified this model in order to provide the necessary preparations for wider use of the model in growth rate, growth curve and growth prediction of tumours. Meanwhile, the model is performed on some experimental data to show its capabilities.     

Generating all cycles, chordless cycles, and Hamiltonian cycles with the principle of exclusion

An efficient method to generate all edge sets XE of a graph G=(V,E), which are vertex-disjoint unions of cycles, is presented. It can be tweaked to generate (i) all cycles, (ii) all cycles of cardinality 5, (iii) all chordless cycles, (iv) all Hamiltonian cycles.     

Hamiltonian cycles and 2-dominating induced cycles in claw-free graphs

Let G = (V, E) be a connected graph. For a vertex subset , G[S] is the subgraph of G induced by S. A cycle C (a path, respectively) is said to be an induced cycle (path, respectively) if G[V(C)] = C (G[V(P)] = P, respectively). The distance between a vertex x and a subgraph H of G is denoted by , where d(x, y) is the distance between x and y. A subgraph H of G is called 2-dominating if d(x, H) ≤ 2 for all . An induced path P of G is said to be maximal if there is no induced path P′ satisfying and . In this paper, we assume that G is a connected claw-free graph satisfying the following condition: for every maximal induced path P of length p ≥ 2 with end vertices u, v it holds:

Hamiltonian cycles and dominating cycles passing through a linear forest

Let G be an (m+2)-graph on n vertices, and F be a linear forest in G with |E(F)|=m and ω₁(F)=s, where ω₁(F) is the number of components of order one in F. We denote by σ₃(G) the minimum value of the degree sum of three vertices which are pairwise non-adjacent. In this paper, we give several σ₃ conditions for a dominating cycle or a hamiltonian cycle passing through a linear forest. We first prove that if σ₃(G)≥n+2m+2+max{s−3,0}, then every longest cycle passing through F is dominating. Using this result, we prove that if σ₃(G)≥n+κ(G)+2m−1 then G contains a hamiltonian cycle passing through F. As a corollary, we obtain a result that if G is a 3-connected graph and σ₃(G)≥n+κ(G)+2, then G is hamiltonian-connected.     

Endogenous cycles with small discounting in multisector optimal growth models: Continuous-time case

This paper discusses the role of the trade-off between the level of discounting and the curvature of preferences and technology in establishing the existence of endogenous cycles. Using the equivalence between the Euler-Lagrange equations and a modified Hamiltonian dynamic system, in a general continuous-time multisector optimal growth model, it is proved that, if the indirect utility function is weakly concave (i.e., concave-, with >0 arbitrarily close to 0), then the discount rate values compatible with endogenous fluctuations are arbitrarily low. We show by numerical simulations that our result explains and generalizes to the multisector case the recent contribution of Benhabib and Rustichini in this field.The author would like to thank P. Cartigny, J. M. Grandmont, P. Michel, and L. Montrucchio for helpful discussions which greatly improved the exposition of the paper. He is also indebted to G. Sorger for a fruitful discussion which clarified some recent contributions. The paper also benefited from comments received during a presentation at the Meeting of the Society for Economic Dynamics and Control, University of California, Los Angeles, California, 1994.     

The growth rate of the harmonious chromatic number

In this paper we investigate the growth rate of the harmonious chromatic number of a family of regular graphs, as a function of the number of vertices, when the valency or the diameter is fixed. We also look at families of graphs where the harmonious coloring is minimal.