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In this paper we extend the work by Ryuzo Sato devoted to the development of economic growth models within the framework of the Lie group theory. We propose a new growth model based on the assumption of logistic growth in factors. It is employed to derive new production functions and introduce a new notion of wage share. In the process it is shown that the new functions compare reasonably well against relevant economic data. The corresponding problem of maximization of profit under conditions of perfect competition is solved with the aid of one of these functions. In addition, it is explained in reasonably rigorous mathematical terms why Bowleys law no longer holds true in post-1960 data.
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We deal with an infinite horizon, infinite dimensional stochastic optimal control problem arising in the study of economic growth in time-space. Such problem has been the object of various papers in deterministic cases when the possible presence of stochastic disturbances is ignored. Here we propose and solve a stochastic generalization of such models where the stochastic term, in line with the standard stochastic economic growth models, is a multiplicative one, driven by a cylindrical Wiener process. The problem is studied using the Dynamic Programming approach. We find an explicit solution of the associated HJB equation and, using a verification type result, we prove that such solution is the value function and we find the optimal feedback strategies. Finally we use this result to study the asymptotic behavior of the optimal trajectories.
We give a new proof of Gromovs theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. Unlike the original proof, it does not rely on the Montgomery-Zippin-Yamabe structure theory of locally compact groups.
We exhibit a regular language of geodesics for a large set of elements of $BS(1,n)$ and show that the growth rate of this language is the growth rate of the group. This provides a straightforward calculation of the growth rate of $BS(1,n)$, which was initially computed by Collins, Edjvet and Gill in [5]. Our methods are based on those we develop in [8] to show that $BS(1,n)$ has a positive density of elements of positive, negative and zero conjugation curvature, as introduced by Bar-Natan, Duchin and Kropholler in [1].
The agent-based Yard-Sale model of wealth inequality is generalized to incorporate exponential economic growth and its distribution. The distribution of economic growth is nonuniform and is determined by the wealth of each agent and a parameter $lambda$. Our numerical results indicate that the model has a critical point at $lambda=1$ between a phase for $lambda < 1$ with economic mobility and exponentially growing wealth of all agents and a non-stationary phase for $lambda geq 1$ with wealth condensation and no mobility. We define the energy of the system and show that the system can be considered to be in thermodynamic equilibrium for $lambda < 1$. Our estimates of various critical exponents are consistent with a mean-field theory (see following paper). The exponents do not obey the usual scaling laws unless a combination of parameters that we refer to as the Ginzburg parameter is held fixed as the transition is approached. The model illustrates that both poorer and richer agents benefit from economic growth if its distribution does not favor the richer agents too strongly. This work and the accompanying theory paper contribute to understanding whether the methods of equilibrium statistical mechanics can be applied to economic systems.
We describe how orbital graphs can be used to improve the practical performance of many algorithms for permutation groups, including intersection and stabilizer problems. First we explain how orbital graphs can be integrated in partition backtracking, the current state of the art algorithm for many permutation group problems. We then show how our algorithms perform in practice, demonstrating improvements of several orders of magnitude for some problems.
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