Contributions published at Financial Economics (Felix Kübler)

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We present a gravitational wave (GW) analysis of an extensive series of three-dimensional magnetohydrodynamical core-collapse simulations. Our 25 models are based on a 15 progenitor stemming from (i) stellar evolution calculations; (ii) a spherically symmetric effective general relativistic potential, either the Lattimer-Swesty (with three possible compressibilities) or the Shen equation of state for hot, dense matter; and (iii) a neutrino parametrisation scheme that is accurate until about 5 ms postbounce. For three representative models, we also included long-term neutrino physics by means of a leakage scheme, which is based on partial implementation of the isotropic diffusion source approximation (IDSA). We systematically investigated the effects of the equation of state, the initial rotation rate, and both the toroidal and the poloidal magnetic fields on the GW signature. We stress the importance of including of postbounce neutrino physics, since it quantitatively alters the GW signature. Slowly rotating models, or those that do not rotate at all, show GW emission caused by prompt and proto-neutron star (PNS) convection. Moreover, the signal stemming from prompt convection allows for the distinction between the two different nuclear equations of state indirectly by different properties of the fluid instabilities. For simulations with moderate or even fast rotation rates, we only find the axisymmetric type I wave signature at core bounce. In line with recent results, we could confirm that the maximum GW amplitude scales roughly linearly with the ratio of rotational to gravitational energy at core bounce below a threshold value of about 10%. We point out that models set up with an initial central angular velocity of 2π rad s-1 or faster show nonaxisymmetric narrow-band GW radiation during the postbounce phase. This emission process is caused by a low T/|W| dynamical instability. Apart from these two points, we show that it is generally very difficult to discern the effects of the individual features of the input physics in a GW signal from a rotating core-collapse supernova that can be attributed unambiguously to a specific model. Weak magnetic fields do not notably influence the dynamical evolution of the core and thus the GW emission. However, for strong initial poloidal magnetic fields (≳1012 G), the combined action of flux-freezing and field winding leads to conditions where the ratio of magnetic field pressure to matter pressure reaches about unity which leads to the onset of a jet-like supernova explosion. The collimated bipolar out-stream of matter is then reflected in the emission of a type IV GW signal. In contradiction to axisymmetric simulations, we find evidence that nonaxisymmetric fluid modes can counteract or even suppress jet formation for models with strong initial toroidal magnetic fields. The results of models with continued neutrino emission show that including of the deleptonisation during the postbounce phase is an indispensable issue for the quantitative prediction of GWs from core-collapse supernovae, because it can alter the GW amplitude up to a factor of 10 compared to a pure hydrodynamical treatment. Our collapse simulations indicate that corresponding events in our Galaxy would be detectable either by LIGO, if the source is rotating, or at least by the advanced LIGO detector, if it is not or only slowly rotating.

Recently, simulations of the collapse of massive stars showed that selected models of the QCD phase transitions to deconfined quarks during the early postbounce phase can trigger the supernova explosion that has been searched for over many years in spherically symmetric supernova models. Using sophisticated general relativistic Boltzmann neutrino transport, it was found that a characteristic neutrino signature is emitted that permits to falsify or identify this scenario in the next Galactic supernova event. On the other hand, more refined observations of past supernovae and progressing theoretical research in different supernova groups demonstrated that the effects of multidimensional fluid instabilities cannot be neglected in global models of the explosions of massive stars. We point to different efforts where neutrino transport and general relativistic effects are combined with multidimensional fluid instabilities in supernovae. With those, it will be possible to explore the gravitational wave emission as a potential second characteristic observable of the presence of quark matter in new-born neutron stars.

We have performed a set of 11 three-dimensional magnetohydrodynamical (MHD) core-collapse supernova simulations in order to investigate the dependences of the gravitational wave signal on the progenitor's initial conditions. We study the effects of the initial central angular velocity and different variants of neutrino transport. Our models are started up from a 15Modot progenitor and incorporate an effective general relativistic gravitational potential and a finite temperature nuclear equation of state. Furthermore, the electron flavour neutrino transport is tracked by efficient algorithms for the radiative transfer of massless fermions. We find that non- and slowly rotating models show gravitational wave emission due to prompt- and lepton driven convection that reveals details about the hydrodynamical state of the fluid inside the protoneutron stars. Furthermore we show that protoneutron stars can become dynamically unstable to rotational instabilities at T/|W| values as low as ~2% at core bounce. We point out that the inclusion of deleptonization during the postbounce phase is very important for the quantitative gravitational wave (GW) prediction, as it enhances the absolute values of the gravitational wave trains up to a factor of ten with respect to a lepton-conserving treatment.

In this paper, we examine non-parametric restrictions on counterfactual analysis in a dynamic stochastic general equilibrium model. Under the assumption of time-separable expected utility and complete markets all equilibria in this model are stationary. The Arrow-Debreu prices uniquely reveal the probabilities and discount factor. The equilibrium correspondence, defined as the map from endowments to stationary (probability-free) state prices, is identical to the equilibrium correspondence in a standard Arrow-Debreu exchange economy with additively separable utility. We examine possible restriction on this correspondence and give necessary as well as sufficient conditions on profiles of individual endowments that ensure that associated equilibrium prices cannot be arbitrary. Although restrictions on possible price changes often exist, we show that results from a representative-agent economy usually do not carry over to a setting with heterogeneous agents.

This paper develops a method to compute the equilibrium correspondence for exchange economies with semi-algebraic preferences. Given a class of semi-algebraic exchange economies parameterized by individual endowments and possibly other exogenous variables such as preference parameters or asset payoffs, there exists a semi-algebraic correspondence that maps parameters to positive numbers such that for generic parameters each competitive equilibrium can be associated with an element of the correspondence and each endogenous variable (i.e. prices and consumptions) is a rational function of that value of the correspondence and the parameters. This correspondence can be characterized as zeros of a univariate polynomial equation that satisfy additional polynomial inequalities. This polynomial as well as the rational functions that determine equilibrium can be computed using versions of Buchberger's algorithm which is part of most computer algebra systems. The computation is exact whenever the input data (i.e. preference parameters etc.) are rational. Therefore, the result provides theoretical foundations for a systematic analysis of multiplicity in applied general equilibrium.

In this paper I employ a dynamic general equilibrium model to study macroeconomic effects and welfare implications of alternative reforms to the U.S. health insurance system. In particular, I focus on expanding Medicare to the entire population, extending Medicaid, and having an individual mandate as well as other related medical reforms. All these reforms can be financed in several ways. I consider a stochastic OLG framework with heterogeneous agents facing uncertain health shocks. Individuals make optimal decisions on labor supply, health insurance, and medical services. As the amount of optimal medical consumption and hours worked are endogenous, this environment captures general equilibrium effects. The model is calibrated to the U.S. data. Numerical simulations indicate that reforming the health insurance system has several important macroeconomic effects on health expenditures, hours worked, and welfare.

In this paper we present a recursive method for the computation of dynamic competitive equilibria in models with heterogeneous agents and market frictions. This method is based on a convergent operator over an expanded set of state variables. The fixed point of this operator defines the set of all Markovian equilibria. We study approximation properties of the operator as well as the convergence of the moments of simulated sample paths. We apply our numerical algorithm to two growth models, an overlapping generations economy with money, and an asset pricing model with financial frictions.

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In the interior of supernovae, temperatures and densities exceed the range that is easily accessible by terrestrial experiments. With the improving sensitivities of neutrino and gravitational wave detectors, the chance of obtaining observations providing a deep view into the heart of a close-by supernova explosion is steadily increasing. Based on computational models we investigate the imprint of the nuclear equation of state on the emission of neutrinos and gravitational waves. If a QCD phase transition to quark matter occurs during the immediate postbounce accretion phase, a strong second shock front is formed at a radius of order 10 km. Neutronised hadronic outer layers of the protoneutron star fall into it, are shock-heated and lead to a rapid acceleration of the second shock wave. As soon as this shock reduces the electron degeneracy at the neutrinospheres, a sharp second neutrino burst is emitted, dominated by electron antineutrinos. Together with the abruptly increasing mean energies of μ- and τ-neutrinos it may serve as a clear signature of the phase transition of the protoneutron star core to a more compact state.

In this paper I derive a sucient condition for a numeri-cally computed -equilibrium of a dynamic stochastic economy with heterogeneous agents to be close to an exact equilibrium. If the economic fundamentals are semi-algebraic, one can verify computationally whether this condition holds. The condition can be interpreted economically as a robustness requirement on the set of -equilibria that form a neighborhood of the computed approximation. I use this method of 'self-validating computation' to prove that in realistically calibrated stochastic overlapping generation models, competitive equilibria can often be extremely well approximated by cubic functions, mapping the current shock and the beginning-of-period wealth-distribution to current endogenous variables.

We describe a sparse grid collocation algorithm to compute recursive solutions of dynamic economies with a sizable number of state variables. We show how powerful this method may be in applications by computing the nonlinear recursive solution of an international real business cycle model with a substantial number of countries, complete insurance markets and frictions that impede frictionless international capital flows. In this economy the aggregate state vector includes the distribution of world capital across different countries as well as the exogenous country-specific technology shocks. We use the algorithm to efficiently solve models with 2, 4, and 6 countries (i.e., up to 12 continuous state variables).

Core-collapse and the launch of a supernova explosion form a very short episode of few seconds in the evolution of a massive star, during which an enormous gravitational energy of several times 1053 erg is transformed into observable neutrino-, kinetic-, and electromagnetic radiation energy. We emphasize the wide range of matter conditions that prevail in a supernova event and sort the conditions into distinct regimes in the density and entropy phase diagram to briefly discuss their different impact on the neutrino signal, gravitational wave emission, and ejecta.

We present the gravitational wave analyses from rotating (model s15g) and nearly non-rotating (model s15h) 3D MHD core collapse supernova simulations at bounce and during the first couple of ten milliseconds afterwards. The simulations are launched from 15 progenitor models stemming from stellar-evolution calculations. Gravity is implemented by a spherically symmetric effective general relativistic potential. The input physics uses the Lattimer-Swesty equation of state for hot, dense matter and a neutrino parametrisation scheme that is accurate until the first few ms after bounce. The 3D simulations allow us to study features already known from 2D simulations, as well as nonaxisymmetric effects. In agreement with recent results, we find only type I gravitational wave signals at core bounce. In the later stage of the simulations, one of our models (s15g) shows nonaxisymmetric gravitational wave emission caused by a low T/|W| dynamical instability, while the other model radiates gravitational waves due to a convective instability in the protoneutron star. The total energy released in gravitational waves within the considered time intervals is (s15g) and (s15h). Both core collapse simulations indicate that corresponding events in our Galaxy would be detectable either by the LIGO or Advanced LIGO detector.

This paper develops theoretical foundations for an error analysis of approximate equilibria in dynamic stochastic general equilibrium models with heterogeneous agents and incomplete financial markets. While there are several algorithms that compute prices and allocations for which agents’ first-order conditions are approximately satisfied (“approximate equilibria”), there are few results on how to interpret the errors in these candidate solutions and how to relate the computed allocations and prices to exact equilibrium allocations and prices. We give a simple example to illustrate that approximate equilibria might be very far from exact equilibria. We then interpret approximate equilibria as equilibria for close-by economies; that is, for economies with close-by individual endowments and preferences. We present an error analysis for two models that are commonly used in applications, an overlapping generations (OLG) model with stochastic production and an asset pricing model with infinitely lived agents.We provide sufficient conditions that ensure that approximate equilibria are close to exact equilibria of close-by economies. Numerical examples illustrate the analysis.

We review the recent literature in macroeconomics that analyses Markov equilibria in dynamic general equilibrium model. After defining the Markov equilibrium concept we first summarize what is known about the existence and uniqueness of such equilibria in models where sequential equilibria can be obtained by solving a suitable social planner problem. We then discuss the existence problems of Markov equilibria in models where equivalence of equilibrium allocations and solutions to social planner problems cannot be established and review techniques the literature has developed to deal with the existence problem, as well as recent applications of these techniques in macroeconomics.

This monograph presents a general equilibrium methodology for microeconomic policy analysis. It is intended to serve as an alternative to the now classical, axiomatic general equilibrium theory as exposited in Debreu`s Theory of Value (1959) or Arrow and Hahn`s General Competitive Analysis (1971). The methodology proposed in this monograph does not presume the existence of market equilibrium, accepts the inherent indeterminancy of nonparametric general equlibrium models, and offers effective algorithms for computing counterfactual equilibria in these models. It consists of several essays written over the last decade, some with colleagues or former graduate students, and an appendix by Charles Steinhorn on the elements of O-minimal structures, the mathematical framework for our analysis.

In this article, I review two recent developments in the theory of computation of general equilibria. First, following Brown, DeMarzo and Eaves (1996) several papers have developed globally convergent algorithms for the computation of general equilibria in models with incomplete asset markets. I review some of the developments in that area. Second, new developments in computational algebraic geometry lead to algorithms to compute effectively all equilibria of systems of polynomial equations. I point out some applications of these algorithms to general equilibrium theory.

In this paper I demonstrate how one can generalize finitely many examples to statements about (infinite) classes of economic models. If there exist upper bounds on the number of connected components of one-dimensional linear subsets of the set of parameters for which a conjecture is true, one can conclude that it is correct for all parameter values in the class considered, except for a small residual set, once one has verified the conjecture for a predetermined finite set of points. I show how to apply this insight to computational experiments and spell out assumptions on the economic fundamentals that ensure that the necessary bounds on the number of connected components exist. I argue that these methods can be fruitfully utilized in applied general equilibrium analysis. I provide general assumptions on preferences and production sets that ensure that economic conjectures define sets with a bounded number of connected components. Using the theoretical results, I give an example of how one can explore qualitative and quantitative implications of general equilibrium models using computational experiments. Finally, I show how random algorithms can be used for generalizing examples in high-dimensional problems.