Differentiable comparative statics with payoff functions not differentiable everywhere

In this note we consider Cournot oligopoly. Due to the consideration of several consumers and/ or technologies, the profit function is not differentiable everywhere. We show that Cournot equilibrium never occurs in outputs where profit functions are not-differentiable. This result validates the procedure of differentiating profit functions to perform comparative statics in models where profit functions are not differentiable everywhere.Publicad

Classification of Complex Wishart Matrices with a Diffusion-Reaction System guided by Stochastic Distances

We propose a new method for PolSAR (Polarimetric Synthetic Aperture Radar) imagery classification based on stochastic distances in the space of random matrices obeying complex Wishart distributions. Given a collection of prototypes $\{Z_m\}_{m=1}^M$ and a stochastic distance $d(.,.)$, we classify any random matrix $X$ using two criteria in an iterative setup. Firstly, we associate $X$ to the class which minimizes the weighted stochastic distance $w_md(X,Z_m)$, where the positive weights $w_m$ are computed to maximize the class discrimination power. Secondly, we improve the result by embedding the classification problem into a diffusion-reaction partial differential system where the diffusion term smooths the patches within the image, and the reaction term tends to move the pixel values towards the closest class prototype. In particular, the method inherits the benefits of speckle reduction by diffusion-like methods. Results on synthetic and real PolSAR data show the performance of the method.Comment: Accepted for publication in Philosophical Transactions

Welfare losses under cournot competition

We find that in a market for a homogeneous good where firms are identical, compete in quantities and produce with constant returns, the percentage of wel-fare losses (PWL) is small with as few as five competitors for a class of demand functions which includes linear and isoelastic cases. However with fixed costs and asymmetric firms PWL can be large. We provide exact formulae of PWL and robust constructions of markets were PWL is close to one in these two cases. We show that the market structure that maximizes PWL is either monopoly or dominant firm, depending on demand. Finally we prove that PWL is minimized when all firms are identical, a clear indication that the assumption of identical firms biases the estimation of PWL downwards

The Long-Run Keynesian Multiplier

We study the impact of investment on employment. In the short−run an increase in investment stimulates employment (this is the standard Keynesian multiplier). However increases of investment translate into increases in the capital stock. If labor and capital are substitutes (resp. complements), an increase in investment today decreases (resp. increases) employment tomorrow. We provide a formula to measure the overall effect of an increase in investment on emplyment, assuming that certain regularities hold.Publicad

Comparative Statics for Market Games: The Strong Concavity Case

In this paper we study the effects of a change in sorne exogenous variable (the number of players or a parameter in the payoff functions) on the strategies played and payoffs obtained in a Nash Equilibrium in the framework of a Market Game (a generalization of the Cournot model)o We assume a strong concavity condition which implies that the best reply function of any player is decreasing on sum of the strategies of the remaining players (Le. strategic substitution). Our results generalize and unify those known in the Cournot model