A general and intuitive envelope theorem

We present an envelope theorem for establishing first-order conditions in decision problems involving continuous and discrete choices. Our theorem accommodates general dynamic programming problems, even with unbounded marginal utilities. And, unlike classical envelope theorems that focus only on differentiating value functions, we accommodate other endogenous functions such as default probabilities and interest rates. Our main technical ingredient is how we establish the differentiability of a function at a point: we sandwich the function between two differentiable functions from above and below. Our theory is widely applicable. In unsecured credit models, neither interest rates nor continuation values are globally differentiable. Nevertheless, we establish an Euler equation involving marginal prices and values. In adjustment cost models, we show that first-order conditions apply universally, even if optimal policies are not (S,s). Finally, we incorporate indivisible choices into a classic dynamic insurance analysis

Envelope Theorems for Non-Smooth and Non-Concave Optimization

We study general dynamic programming problems with continuous and discrete choices and general constraints. The value functions may have kinks arising (1) at indifference points between discrete choices and (2) at constraint boundaries. Nevertheless, we establish a general envelope theorem: first-order conditions are necessary at interior optimal choices. We only assume differentiability of the utility function with respect to the continuous choices. The continuous choice may be from any Banach space and the discrete choice from any non-empty set

Money cycles

Classical models of money are typically based on a competitive market without capital or credit. They then impose exogenous timing structures, market participation constraints, or cash-in-advance constraints to make money essential. We present a simple model without credit where money arises from a fixed cost of production. This leads to a rich equilibrium structure. Agents avoid the fixed cost by taking vacations and the trade between workers and vacationers is supported by money. We show that agents acquire and spend money in cycles of finite length. Throughout such a “money cycle,” agents decrease their consumption which we interpret as the hot potato effect of inflation. We give an example where money holdings do not decrease monotonically throughout the money cycle. Optimal monetary policy is given by the Friedman rule, which supports efficient equilibria. Thus, monetary policy provides an alternative to lotteries for smoothing out non-convexities.

Money Cycles

Classical models of money are typically based on a competitive market without capital or credit. They then impose exogenous timing structures, market participation constraints, or cash-in-advance constraints to make money essential. We present a simple model without credit where money arises from a fixed cost of production. This leads to a rich equilibrium structure. Agents avoid the fixed cost by taking vacations and the trade between workers and vacationers is supported by money. We show that agents acquire and spend money in cycles of finite length. Throughout such a "money cycle," agents decrease their consumption which we interpret as the hot potato effect of inflation. We give an example where money holdings do not decrease monotonically throughout the money cycle. Optimal monetary policy is given by the Friedman rule, which supports efficient equilibria. Thus, monetary policy provides an alternative to lotteries for smoothing out non-convexities.

Hybrid Collaborative Filtering with Autoencoders

Collaborative Filtering aims at exploiting the feedback of users to provide personalised recommendations. Such algorithms look for latent variables in a large sparse matrix of ratings. They can be enhanced by adding side information to tackle the well-known cold start problem. While Neu-ral Networks have tremendous success in image and speech recognition, they have received less attention in Collaborative Filtering. This is all the more surprising that Neural Networks are able to discover latent variables in large and heterogeneous datasets. In this paper, we introduce a Collaborative Filtering Neural network architecture aka CFN which computes a non-linear Matrix Factorization from sparse rating inputs and side information. We show experimentally on the MovieLens and Douban dataset that CFN outper-forms the state of the art and benefits from side information. We provide an implementation of the algorithm as a reusable plugin for Torch, a popular Neural Network framework

16 years of Ulysses Interstellar Dust Measurements in the Solar System: II. Fluctuations in the Dust Flow from the Data

The Ulysses spacecraft provided the first opportunity to identify and study Interstellar Dust (ISD) in-situ in the Solar System between 1992 and 2007. Here we present the first comprehensive analysis of the ISD component in the entire Ulysses dust data set. We analysed several parameters of the ISD flow in a time-resolved fashion: flux, flow direction, mass index, and flow width. The general picture is in agreement with a time-dependent focussing/defocussing of the charged dust particles due to long-term variations of the solar magnetic field throughout a solar magnetic cycle of 22 years. In addition, we confirm a shift in dust direction of $50^{\circ} \pm 7^{\circ}$ in 2005, along with a steep, size-dependent increase in flux by a factor of 4 within 8 months. To date, this is difficult to interpret and has to be examined in more detail by new dynamical simulations. This work is part of a series of three papers. This paper concentrates on the time-dependent flux and direction of the ISD. In a companion paper (Kr\"uger et al., 2015) we analyse the overall mass distribution of the ISD measured by Ulysses, and a third paper discusses the results of modelling the flow of the ISD as seen by Ulysses (Sterken et al., 2015).Comment: 41 pages, 10 figures, 5 table

A stoichiometric reaction scheme for Saccharothrix algeriensis growth and thiolutin production

A new bacterial species, Saccharothrix algeriensis NRRL B-24137, was isolated in 1992 in the Sahara desert. This filamentous bacterium is able to produce dithiolopyrrolones, molecules presenting antibacterial, antifungal, and anticancer properties. In this study, a “reaction engineering” approach was adopted to gain more knowledge on the growth of Sa. algeriensis and its dithiolopyrrolone production on a semi-synthetic liquid medium. The objective is to establish a reaction scheme of the bacterium metabolism from extracellular experimental information, relatively easy to obtain. The approach enabled us to show that Sa. algeriensis could grow using several substrates that were sequentially consumed and that substrate limitation may induce a secondary metabolism in antibiotic production. From these qualitative data, a general reaction scheme was extracted consisting of four reactions: growth via amino acids, glucose consumption for maintenance, growth using glucose, and thiolutin production. The stoichiometric coefficients and the reaction extends were identified using a factorial analysis based on the bilinear structure of the component mass balances in a batch reactor. The analysis of the reaction stoichiometry enabled us to draw some conclusions concerning the substrate consumption pathway