The characteristic polynomial and determinant are not ad hoc constructions

The typical definition of the characteristic polynomial seems totally ad hoc to me. This note gives a canonical construction of the characteristic polynomial as the minimal polynomial of a "generic" matrix. This approach works not just for matrices but also for a very broad class of algebras including the quaternions, all central simple algebras, and Jordan algebras. The main idea of this paper dates back to the late 1800s. (In particular, it is not due to the author.) This note is intended for a broad audience; the only background required is one year of graduate algebra.Comment: v2 is heavily revised and somewhat expanded. The product formula for the determinant on an algebra is prove

Unramified cohomology of classifying varieties for exceptional simply connected groups

Let BG be a classifying variety for an exceptional simple simply connected algebraic group G. We compute the degree 3 unramified Galois cohomology of BG with values in Q/Z(2) over an arbitrary field F. Combined with a paper by Merkurjev, this completes the computation of these cohomology groups for G semisimple simply connected over all fields. These computations provide another example of a simple simply connected group G such that BG is not stably rational

Financial constraints in search equilibrium: mortensenand Pissarides Meet Holmstron and Tirole

The Great Recession has indicated that firms' leverage and access to finance are important for hiring and firing decisions. It is now empirically established that bank lending is correlated to employment losses when credit conditions deteriorate. We provide further evidence of this drawing on a new dataset that we assembled on employment adjustment and financial positions of European firms. Yet, in the Diamond Mortensen Pissarides (DMP) model there is no role for finance. All projects that display positive net present values are realized and financial markets are assumed to be perfect. What if financial markets are not perfect? Does a different access to finance influence the firm's hiring and firing decisions? The paper uses the concept of limited pledgeability proposed by Holmstrom and Tirole to integrate financial imperfections and labor market imperfections. A negative shock wipes out the firm's physical capital and leads to job destruction unless internal liquidity was accumulated by firms. If firms hold liquid assets they may thus protect their search capital, defined as the cost of attracting and hiring workers. The paper explores the trade-off between size and precautionary liquidity holdings in both partial and general equilibrium. We find that if labor market frictions disappear, so does the motive for firms to hold liquidity. This suggests a fundamental complementarity between labor market frictions and holding of liquid assets by firms

In medio stat victus: Labor Demand Effects of an Increase in the Retirement Age

After falling for four decades, statutory retirement ages are increasing in most OECD countries. The labor market adjustment to these reforms has not yet been thoroughly investigated by the literature. We draw on a major pension reform that took place in Italy in December 2011 that increased the retirement age by up to six years for some categories of workers. We have access to a unique dataset validated by the Italian social security administration (INPS), which identifies in each private firm, based on an administrative exam of eligibility conditions, how many workers were locked in by the sudden increase in the retirement age, and for how long. We find that firms mostly affected by the lock in are those that were downsizing even before the policy shock. The increase in the retirement age seems to displace more middle-aged workers than young workers. Furthermore, there is not a one-to-one increase in the number of older workers in the firms where some workers were locked in by the reform. We provide tentative explanations for these results, based on the interaction between retirement, employment protection legislation and liquidity constraints of firms