Auctions with almost homogeneous bidders

We prove that, around the symmetric case, where the values are identically distributed, the equilibrium of the first price auction is jointly differentiable with respect to general bidder-specific parameters of the value distributions. We show that the revenue equivalence between the first-price and the second-price auctions to the first-order in the size of the parameters is an immediate consequence of this differentiability and the Revenue Equivalence Theorem; thereby formally establishing the first-order equivalence Fibich et al. G. Fibich, A. Gavious, A. Sela, Revenue equivalence in asymmetric auctions, J. Econ. Theory 115 (2004) 309-321] noticed for their particular perturbation.