Asymptotic Exit Location Distributions in the Stochastic Exit Problem

Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point $S$. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength $\epsilon$, the system state will eventually leave the domain of attraction $\Omega$ of $S$. We analyse the case when, as $\epsilon\to0$, the exit location on the boundary $\partial\Omega$ is increasingly concentrated near a saddle point $H$ of the deterministic dynamics. We show that the asymptotic form of the exit location distribution on $\partial\Omega$ is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter $\mu$, equal to the ratio $|\lambda_s(H)|/\lambda_u(H)$ of the stable and unstable eigenvalues of the linearized deterministic flow at $H$. If $\mu<1$ then the exit location distribution is generically asymptotic as $\epsilon\to0$ to a Weibull distribution with shape parameter $2/\mu$, on the $O(\epsilon^{\mu/2})$ length scale near $H$. If $\mu>1$ it is generically asymptotic to a distribution on the $O(\epsilon^{1/2})$ length scale, whose moments we compute. The asymmetry of the asymptotic exit location distribution is attributable to the generic presence of a `classically forbidden' region: a wedge-shaped subset of $\Omega$ with $H$ as vertex, which is reached from $S$, in the $\epsilon\to0$ limit, only via `bent' (non-smooth) fluctuational paths that first pass through the vicinity of $H$. We deduce from the presence of this forbidden region that the classical Eyring formula for the small-$\epsilon$ exponential asymptotics of the mean first exit time is generically inapplicable.Comment: This is a 72-page Postscript file, about 600K in length. Hardcopy requests to [email protected] or [email protected]

Branching of the Falkner-Skan solutions for λ < 0

The Falkner-Skan equation f'" + ff" + λ(1 - f'^2) = 0, f(0) = f'(0) = 0, is discussed for λ < 0. Two types of problems, one with f'(∞) = 1 and another with f'(∞) = -1, are considered. For λ = 0- a close relation between these two types is found. For λ < -1 both types of problem allow multiple solutions which may be distinguished by an integer N denoting the number of zeros of f' - 1. The numerical results indicate that the solution branches with f'(∞) = 1 and those with f'(∞) = -1 tend towards a common limit curve as N increases indefinitely. Finally a periodic solution, existing for λ < -1, is presented.

Heterogeneity and the dynamics of technology adoption

We estimate the demand for a videocalling technology in the presence of both network effects and heterogeneity. Using a unique dataset from a large multinational firm, we pose and estimate a fully dynamic model of technology adoption. We propose a novel identification strategy based on post-adoption technology usage to disentangle equilibrium beliefs concerning the evolution of the network from observed and unobserved heterogeneity in technology adoption costs and use benefits. We find that employees have significant heterogeneity in both adoption costs and network benefits, and have preferences for diverse networks. Using our estimates, we evaluate a number of counterfactual adoption policies, and find that a policy of strategically targeting the right subtype for initial adoption can lead to a faster-growing and larger network than a policy of uncoordinated or diffuse adoption

European Electricity Market Reforms: Any Signs of Efficiency Improvements?

This paper investigates whether European electricity market refirms have induced any changes infirm efficiency either through productive, allocative or dynamic efficiency improve- ments. In particular, this ex-post analysis looks closely at productivity e ects of changing industry structure, ownership structure and regulation with respect to barriers to entry and access to wholesale and retail markets. Based on the Europeanfirm-level data for the period 1996-2007, the results indicate sluggish productivity improvements of European electricityfirms due to refirms implemented in the last decade. In particular, productivity gains are associated with high-productivityfirms close to the technology frontier, while no signi cant impact is found for the laggards. Looking from a dynamic perspective, it seems that the clos- est are thefirms to the frontier the more they are able to improve productivity in response to liberalization e orts stimulating competition

Productivity in Electricity Retail after Market Liberalisation: Analysing the Effects of Ownership and Firm's Governance Structure

This paper, which is one of the first to estimate productivity in retail electricity for a European country after liberalisation, analyses the effect of ownership and governance structure by using a unique dataset of German electricity retailers from 2003 to 2012. An innovative service production function for the retail sector is derived with labour and external services as the main inputs. A structural model is used with a proxy function for productivity to overcome the endogeneity of input choice. Ownership is controlled for in the law of motion for productivity. The results of the dataset used to validate the model show that firm-level productivity did not increase after 2008 and that ownership had no effect on productivity. The results provide useful insights into the link between ownership and productivity in modern public enterprises after liberalisation

R&D and Non-Linear Productivity Growth of Heterogeneous Firms

The present paper studies the relationship between R&D investment and firm productivity growth by explicitly accounting for non-linearities in the R&D-productivity relationship and inter-sectoral firm heterogeneity. In order to address these issues, we employ a two step estimation approach, and match two firm-level panel data sets for the OECD countries, which allows us to relax both the linearity and homogeneity assumptions of the canonical Griliches (1979) knowledge capital model. Our results suggest that: (i) R&D investment increases firm productivity with an average elasticity of 0.15; (ii) the impact of R&D investment on firm productivity is differential at different levels of R&D intensity - the productivity elasticity ranges from -0.02 for low levels of R&D intensity to 0.33 for high levels of R&D intensity; (iii) the relationship between R&D expenditures and productivity growth is non-linear, and only after a certain critical mass of R&D is reached, the productivity growth is significantly positive;(iv) there are important inter-sectoral differences with respect to R&D investment and firm productivity - high-tech sectors' firms not only invest more in R&D, but also achieve more in tfirms of productivity gains connected with research activities