Procurement auctions with losses.

We use a fixed point gradient flow algorithm to compute the equilibria of first-price procurement auctions in the presence of losses and Bayesian priors. We use this efficient algorithm to compare optimal, first-price and VCG auctions. This allows us to numerically estimate the social cost of sub-optimality of the nodal pricing mechanism in wholesale electricity markets. We also derive a closed form expression of the optimal mechanism procurement cost when the types are uniformly distributed. [ABSTRACT FROM AUTHOR]