Abstract. Spectral properties of the Schrödinger operator on a finite compact metric graph with delta-type vertex conditions are discussed. Explicit estimates for the lowest eigenvalue (ground state) are obtained using two different methods: Eulerian cycle and symmetrization techniques. In the case of positive interactions even estimates for higher eigenvalues are derived.

This paper examines hedging against a large market‐wide shock in a model with heterogeneous firms and sunk costs of entry. If hedging is voluntary only the most efficient firms hedge against this shock, a finding in line with empirical evidence but at odds with standard motivations for risk management. Hedging affects the critical level of the marginal cost needed to operate in the market. A setting with mandatory hedging is associated with stronger competition than when hedging is voluntary which, in turn, is associated with stronger competition than when hedging is unavailable.