Darboux-Egorov system, bi-flat FF-manifolds and Painlev\'e VI

This is a generalization of the procedure presented in [3] to construct semisimple bi-flat $F$-manifolds $(M,\nabla^{(1)},\nabla^{(2)},\circ,*,e,E)$ starting from homogeneous solutions of degree -1 of Darboux-Egorov-system. The Lam\'e coefficients $H_i$ involved in the construction are still homogeneous functions of a certain degree $d_i$ but we consider the general case $d_i\ne d_j$. As a consequence the rotation coefficients $\beta_{ij}$ are homogeneous functions of degree $d_i-d_j-1$. It turns out that any semisimple bi-flat $F$ manifold satisfying a natural additional assumption can be obtained in this way. Finally we show that three dimensional semisimple bi-flat $F$-manifolds are parametrized by solutions of the full family of Painlev\'e VI.Comment: 20 page

A bi-Hamiltonian approach to the sine-Gordon and Liouville hierarchies

In this paper we study the sine-Gordon and the Liouville hierarchies in laboratory coordinates from a bi-Hamiltonian point of view. Besides the well-known local structure these hierarchies possess a second compatible non-local Poisson structure.Comment: 17 pages. v2: Additional reference

From Darboux-Egorov system to bi-flat FF-manifolds

Motivated by the theory of integrable PDEs of hydrodynamic type and by the generalization of Dubrovin's duality in the framework of $F$-manifolds due to Manin [22], we consider a special class of $F$-manifolds, called bi-flat $F$-manifolds. A bi-flat $F$-manifold is given by the following data $(M, \nabla_1,\nabla_2,\circ,*,e,E)$, where $(M, \circ)$ is an $F$-manifold, $e$ is the identity of the product $\circ$, $\nabla_1$ is a flat connection compatible with $\circ$ and satisfying $\nabla_1 e=0$, while $E$ is an eventual identity giving rise to the dual product *, and $\nabla_2$ is a flat connection compatible with * and satisfying $\nabla_2 E=0$. Moreover, the two connections $\nabla_1$ and $\nabla_2$ are required to be hydrodynamically almost equivalent in the sense specified in [2]. First we show that, similarly to the way in which Frobenius manifolds are constructed starting from Darboux-Egorov systems, also bi-flat $F$-manifolds can be built from solutions of suitably augmented Darboux-Egorov systems, essentially dropping the requirement that the rotation coefficients are symmetric. Although any Frobenius manifold possesses automatically the structure of a bi-flat $F$-manifold, we show that the latter is a strictly larger class. In particular we study in some detail bi-flat $F$-manifolds in dimensions n=2, 3. For instance, we show that in dimension 3 bi-flat $F$-manifolds are parametrized by solutions of a two parameters Painlev\'e VI equation, admitting among its solutions hypergeometric functions. Finally we comment on some open problems of wide scope related to bi-flat $F$-manifolds.Comment: 32 pages, eliminated a remark at the end of proof of Theorem 6.

Recent Developments in Business Cycle Theory: News, Expectations and Demand Shocks

This paper surveys some recent work in the theory of business cycles, which emphasizes the role of public news and consumer expectations as driving forces behind short-run aggregate fluctuations. The paper uses a simple two period model to introduce and discuss three issues regarding this class of models: (1) what is the role of nominal rigidities, (2) what are the testable implications of these models, (3) what are their implications for monetary policy.

Inefficient Credit Booms

This paper studies the welfare properties of competitive equilibria in an economy with financial frictions hit by aggregate shocks. In particular, it shows that competitive financial contracts can result in excessive borrowing ex ante and excessive volatility ex post. Even though, from a first-best perspective the equilibrium always displays under-borrowing, from a second-best point of view excessive borrowing can arise. The inefficiency is due to the combination of limited commitment in financial contracts and the fact that asset prices are determined in a spot market. This generates a pecuniary externality that is not internalized in private contracts. The model provides a framework to evaluate preventive policies which can be used during a credit boom to reduce the expected costs of a financial crisis.

A Theory of Demand Shocks

This paper presents a model of business cycles driven by shocks to consumer expectations regarding aggregate productivity. Agents are hit by heterogeneous productivity shocks, they observe their own productivity and a noisy public signal regarding aggregate productivity. The shock to this public signal, or "news shock," has the features of an aggregate demand shock: it increases output, employment and inflation in the short run and has no effects in the long run. The dynamics of the economy following an aggregate productivity shock are also affected by the presence of imperfect information: after a productivity shock output adjusts gradually to its higher long-run level, and there is a temporary negative effect on inflation and employment. A calibrated version of the model is able to generate realistic amounts of short-run volatility due to demand shocks, in line with existing time-series evidence. The paper also develops a simple method to solve forward-looking models with dispersed information.

News Shocks and Optimal Monetary Policy

This paper studies monetary policy in a model where output fluctuations are caused by shocks to public beliefs on the economy's fundamentals. I ask whether monetary policy can offset the effect of these shocks and whether this offsetting is socially desirable. I consider an environment with dispersed information and two aggregate shocks: a productivity shock and a "news shock" which affects aggregate beliefs. Neither the central bank nor individual agents can distinguish the two shocks when they hit the economy. The main results are: (1) despite the lack of superior information an appropriate monetary policy rule can change the economy's response to the two shocks; (2) monetary policy can achieve full aggregate stabilization, that is, it can induce a path for aggregate output that is identical to that which would arise under full information; (3) however, full aggregate stabilization is typically not optimal. The fact that monetary policy can tackle the two shocks separately is due to two crucial ingredients. First, agents are forward looking. Second, current fundamental shocks will become public information in the future and the central bank will be able to respond to them at that time. By announcing its response to future information, the central bank can influence the expected real interest rate faced by agents with different beliefs and, thus, induce an optimal use of the information dispersed in the economy.