Networks versus vertical integration

We construct a theory to compare vertically integrated firms to networks of manufacturers and suppliers. Vertically integrated firms make their own specialized inputs. In networks, manufacturers procure specialized inputs from suppliers that, in turn, sell to several manufacturers. The analysis shows that networks can yield greater social welfare when manufacturers experience large idiosyncratic demand shocks. Individual firms may also have the incentive to form networks, despite the lack of long-term contracts. The analysis is supported by existing evidence and provides predictions as to the shape of different industries.

Spin-state splittings, highest-occupied-molecular-orbital and lowest-unoccupied-molecular-orbital energies, and chemical hardness.

It is known that the exact density functional must give ground-state energies that are piecewise linear as a function of electron number. In this work we prove that this is also true for the lowest-energy excited states of different spin or spatial symmetry. This has three important consequences for chemical applications: the ground state of a molecule must correspond to the state with the maximum highest-occupied-molecular-orbital energy, minimum lowest-unoccupied-molecular-orbital energy, and maximum chemical hardness. The beryllium, carbon, and vanadium atoms, as well as the CH(2) and C(3)H(3) molecules are considered as illustrative examples. Our result also directly and rigorously connects the ionization potential and electron affinity to the stability of spin states.