On the Smallest Eigenvalue of Grounded Laplacian Matrices

We provide upper and lower bounds on the smallest eigenvalue of grounded Laplacian matrices (which are matrices obtained by removing certain rows and columns of the Laplacian matrix of a given graph). The gap between the upper and lower bounds depends on the ratio of the smallest and largest components of the eigenvector corresponding to the smallest eigenvalue of the grounded Laplacian. We provide a graph-theoretic bound on this ratio, and subsequently obtain a tight characterization of the smallest eigenvalue for certain classes of graphs. Specifically, for Erdos-Renyi random graphs, we show that when a (sufficiently small) set $S$ of rows and columns is removed from the Laplacian, and the probability $p$ of adding an edge is sufficiently large, the smallest eigenvalue of the grounded Laplacian asymptotically almost surely approaches $|S|p$. We also show that for random $d$-regular graphs with a single row and column removed, the smallest eigenvalue is $\Theta(\frac{d}{n})$. Our bounds have applications to the study of the convergence rate in continuous-time and discrete-time consensus dynamics with stubborn or leader nodes

Microeconomic Inventory Behavior and Aggregate Inventory Dynamics

The slow adjustment of inventory stocks to changes in sales has been a puzzle for the inventory literature since at least Auerbach and Feldstein (1976). Recent evidence suggests that estimated firm-level adjustment speeds of inventory stocks are significantly higher than estimates based on aggregate data. This paper investigates the circumstances under which such bias occurs using an industry equilibrium model where, consistently with empirical evidence, some firms smooth production while others bunch it. The model can account for the significant downward bias documented empirically when a subset of firms displays countercyclical mark-up movements.

Capital Ownership Under Incomplete Markets: Does it Matter?

In the macroeconomic literature, the implications of a context with household heterogeneity and incomplete financial markets have been mostly studied under the assumption that households own the physical capital and undertake the intertemporal investment decision. Further, firms rent capital and labor from the households to maximize period profits. The present paper provides the conditions under which this assumption is still irrelevant when markets are incomplete. It shows that, if firms own the physical capital and undertake the investment decision to maximize their asset value, in the sense that they discount future cash flows with positive state price processes that are consistent with security prices, the equilibrium allocations are the same as in the standard setting with static firms. On the other hand, the firm valuation of future cash flows only coincides with the valuation of the unconstrained shareholders. Given this, value maximization may still lead to shareholder disagreement in the presence of effectively binding portfolio restrictions.

Moderazione del vivere cittadino: il "Testamentum" di Pietro da Fermo (1292)

Il saggio prende in esame un manoscritto inedito e fino ad ora sconosciuto, conservato presso la Biblioteca Civica \uabRomolo Spezioli\ubb di Fermo: si tratta di un testamento \u2018morale\u2019 scritto alla fine del XIII secolo da un funzionario podestarile, Pietro da Fermo, mentre si trovava al seguito del podest\ue0 Offreduccio di Falerone nella citt\ue0 di Rieti e indirizzato ai suoi figli. Il testo si articola in ottantatre rubriche, che raccolgono prevalentemente sententiae e proverbi derivanti dalla tradizione classica e medievale. Il dettato del testo non ambisce ad alcuna pretesa di originalit\ue0 n\ue9 di letterariet\ue0, ma risulta nondimeno significativo, dal punto di vista euristico, in quanto denota la circolazione, all\u2019interno di una cultura media come quella interpretata da Pietro da Fermo, delle idee variamente sviluppate nella coeva letteratura didattica. Quella che l\u2019autore veicola nel suo testo \ue8 un\u2019educazione di tipo pragmatico, rivolta al cittadino consapevole degli assetti istituzionali e delle pratiche sociali della citt\ue0 comunale, nei delicati equilibri della quale si svolge concretamente l\u2019esperienza di vita dell\u2019autore e dei suoi destinatari

Le \u2018origini\u2019 del comuni rurali nelle Marche: un tema storiografico nella medievistica del primo Novecento

This essay investigates a notable period of historiography on medieval Italy, in the early 20th Century, in relation to a peripheral area: the \u2018Marche\u2019. At that time the interest was focused on the origin of \u2018comune rurale\u2019 and in Tuscany this topic was investigated by leading scholars, such as Gioacchino Volpe and Romolo Caggese. Also in the Marche a broad discussion expanded on this issue, launched by scholars who had carried out thorough research in archives of the smaller towns, mainly Gino Luzzatto, Francesco Filippini, Andrea Menchetti. Through essays, reviews and replies, the issue of \u2018origins\u2019 and social composition of \u2018comune rurale\u2019 was further discussed and analysed from different points of view. It was a comparison far from a provincial one, because it was based on the achievements of recent studies, wisely combined with the territorial area investigated

Immigrants, schooling and background. Cross-country evidence from PISA 2006

Using data from PISA 2006, we examine the performance of immigrant students in different international educational environments. Our results show smaller immigrant gaps – differences in scores with respect to natives - where educational systems are more flexible and students’ mobility between courses and school programs is higher. Unlike previous studies, our analysis reveals no direct relation between these gaps and education models, be they comprehensive or tracking, adopted by countriesInternational migration; educational systems; PISA;