In order to lower the estimates of the total project costs, prime contractors often solicit bids from subcontractors which can complete their works with lower costs than they do by themselves, prior to submitting their own bids in procurement auctions. This paper presents a simple model of such two-stage auctions and shows some observations in a laboratory experiment conducted to examine theoretical predictions. Our main observations are as follows. (1) Subcontractors bid more aggressively (i.e., they lower their bids) in the first-price subcontract auctions, as compared to the case where there is no second-stage competition among prime contractors. (2) Second-price subcontract auctions render higher profits to prime contractors than first-price auctions. (3) First-price subcontract auctions more likely achieve \textit{ex post} efficient allocations of a subcontract work than second-price auctions. The policy implications are also discussed.