Uniform-price auctions with adjustable supply

In the uniform-price auction with adjustable supply, the seller decides how much to sell after receiving the bids so as to maximize its ex post profit. Given N bidders and adjustable supply, all equilibria of the uniform-price auction lead to price on order 1/N3 below the Walrasian price. By contrast, given the usual market-clearing rule it is well-known that the uniform-price auction can lead to equilibrium prices on order 1/N below the Walrasian price