We consider a first-price sealed-bid procurement auction between two capacity constrained bidders. Either bidder, due to the capacity constraint, is unable to fulfill the auctioned project from a buyer. We consider two formats of capacity cooperation between the two competing bidders: ex ante subcontracting and ex post subcontracting. In the former case, two bidders first bargain over the capacity cooperation contract and then compete with each other by bidding for the buyer’s project. In the latter case, two bidders first submit bids to the buyer and then bargain over the capacity cooperation contract. With ex ante or ex post subcontracting, two bidders have the so-called co-opetition relationship: they compete to win the auction but cooperate through subcontracting. We find that bidders’ subcontract timing has a fundamental impact on their competitive bidding behavior and profits. Both bidders will benefit from the ex ante subcontracting, and submit higher equilibrium bid prices, compared to the case without subcontracting. The ex post subcontracting, however, will lower the equilibrium bid price and may hurt the bidder with capacity advantage when bidders’ relative bargaining power is not consistent with the relative capacity advantage.