Numerical Solution of Dynamic Equilibrium Models under Poisson Uncertainty

We propose a simple and powerful numerical algorithm to compute the transition process in continuous-time dynamic equilibrium models with rare events. In this paper we transform the dynamic system of stochastic differential equations into a system of functional differential equations of the retarded type. We apply the Waveform Relaxation algorithm, i.e., we provide a guess of the policy function and solve the resulting system of (deterministic) ordinary differential equations by standard techniques. For parametric restrictions, analytical solutions to the stochastic growth model and a novel solution to Lucas' endogenous growth model under Poisson uncertainty are used to compute the exact numerical error. We show how (potential) catastrophic events such as rare natural disasters substantially affect the economic decisions of households.continuous-time DSGE, Poisson uncertainty, waveform relaxation

Beyond mean-field dynamics of small Bose-Hubbard systems based on the number-conserving phase space approach

The number-conserving quantum phase space description of the Bose-Hubbard model is discussed for the illustrative case of two and three modes, as well as the generalization of the two-mode case to an open quantum system. The phase-space description based on generalized SU(M) coherent states yields a Liouvillian flow in the macroscopic limit, which can be efficiently simulated using Monte Carlo methods even for large systems. We show that this description clearly goes beyond the common mean-field limit. In particular it resolves well-known problems where the common mean-field approach fails, like the description of dynamical instabilities and chaotic dynamics. Moreover, it provides a valuable tool for a semi-classical approximation of many interesting quantities, which depend on higher moments of the quantum state and are therefore not accessible within the common approach. As a prominent example, we analyse the depletion and heating of the condensate. A comparison to methods ignoring the fixed particle number shows that in this case artificial number fluctuations lead to ambiguities and large deviations even for quite simple examples.Comment: Significantly enhanced and revised version (20 pages, 20 figures

Quantifying Optimal Growth Policy

The optimal mix of growth policies is determined within a comprehensive endogenous growth model. The analysis captures important elements of the tax-transfer system and accounts for transitional dynamics. Currently, for calculating corporate taxable income US firms are allowed to deduct approximately all of their capital and R&D costs from sales revenue. Our analysis suggests that this policy leads to severe underinvestment in both R&D and physical capital. We find that firms should be allowed to deduct between 2-2.5 times their R&D costs and about 1.5-1.7 times their capital costs. Implementing the optimal policy mix is likely to entail huge welfare gains.economic growth, endogenous technical change, optimal growth policy, tax-transfer system, transitional dynamics

The Macroeconomics of TANSTAAFL

This paper shows that dynamic inefficiency can occur in dynamic general equilibrium models with fully optimizing, infinitely-lived households even in a situation with underinvestment. We identify necessary conditions for such a possibility and illustrate it in a standard R&D-based growth model. Calibrating the model to the US, we show that a moderate increase in the R&D subsidy indeed leads to an intertemporal free lunch (i.e., an increase in per capita consumption at all times). Hence, Milton Friedman’s conjecture There ain’t no such thing as a free lunch (TANSTAAFL) may not apply.intertemporal free lunch, dynamic inefficiency, R&D-based growth, transitional dynamics

Quantifying Optimal Growth Policy

The optimal mix of growth policies is derived within a comprehensive endogenous growth model. The analysis captures important elements of the tax-transfer system and takes into account transitional dynamics. Currently, for calculating corporate taxable income US firms are allowed to deduct approximately all of their capital and R&D costs from sales revenue. Our analysis suggests that this policy leads to severe underinvestment in both R&D and physical capital. We find that firms should be allowed to deduct between 2-2.5 times their R&D costs and about 1.5-1.7 times their capital costs. Implementing the optimal policy mix is likely to entail huge welfare gains.economic growth, endogenous technical change, optimal growth policy, tax-transfer system, transitional dynamics

Multi-Dimensional Transitional Dynamics: A Simple Numberical Procedure

We propose the relaxation algorithm as a simple and powerful method for simulating the transition process in growth models. This method has a number of important advantages: (1 It can easily deal with a wide range of dynamic systems including stiff differential equations and systems giving rise to a continuum of stationary equilibria. (2) The application of theprocedure is fairly user friendly. The only input required consists of the dynamic system. (3) The variant of the relaxation algorithm we propose exploits in a natural manner the infinite time horizon, which usually underlies optimal control problems in economics. As an illustrative application, we simulate the transition process of the Jones (1995) and the Lucas (1988) model.transitional dynamics, continuous time growth models, saddle-point problems, multi-dimensional stable manifolds

A Macroscopic Portfolio Model: From Rational Agents to Bounded Rationality

We introduce a microscopic model of interacting financial agents, where each agent is characterized by two portfolios; money invested in bonds and money invested in stocks. Furthermore, each agent is faced with an optimization problem in order to determine the optimal asset allocation. The stock price evolution is driven by the aggregated investment decision of all agents. In fact, we are faced with a differential game since all agents aim to invest optimal. Mathematically such a problem is ill posed and we introduce the concept of Nash equilibrium solutions to ensure the existence of a solution. Especially, we denote an agent who solves this Nash equilibrium exactly a rational agent. As next step we use model predictive control to approximate the control problem. This enables us to derive a precise mathematical characterization of the degree of rationality of a financial agent. This is a novel concept in portfolio optimization and can be regarded as a general approach. In a second step we consider the case of a fully myopic agent, where we can solve the optimal investment decision of investors analytically. We select the running cost to be the expected missed revenue of an agent and we assume quadratic transaction costs. More precisely the expected revenues are determined by a combination of a fundamentalist or chartist strategy. Then we derive the mean field limit of the microscopic model in order to obtain a macroscopic portfolio model. The novelty in comparison to existent macroeconomic models in literature is that our model is derived from microeconomic dynamics. The resulting portfolio model is a three dimensional ODE system which enables us to derive analytical results. Simulations reveal, that our model is able to replicate the most prominent features of financial markets, namely booms and crashes.Comment: arXiv admin note: substantial text overlap with arXiv:1711.0329

Una carta inédita de Pascual de Andagoya.

Sin resume

Multi-dimensional transitional dynamics : a simple numerical procedure

We propose the relaxation algorithm as a simple and powerful method for simulating the transition process in growth models. This method has a number of important advantages: (1) It can easily deal with a wide range of dynamic systems including multi-dimensional systems with stable eigenvalues that di.er drastically in magnitude. (2) The application of the procedure is fairly user friendly. The only input required consists of the dynamic system. (3) The variant of the relaxation algorithm we propose exploits in a natural manner the in.nite time horizon, which usually underlies optimal control problems in economics. Overall, it seems that the relaxation procedure can easily cope with a large number of problems which arise frequently in the context of macroeconomic dynamic models. As an illustrative application, we simulate the transition process of the well-known Jones (1995) model.saddlepoint problems, transitional dynamics, economic growth, multidimensional stable manifolds