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This holds only when the agents' valuations are statistically independent; when the valuations are dependent, we have a common value auction, and in this case, the revenue in a second-price auction is usually higher than in a first-price auction. The item for sale may not be sold if the final bid is not high enough to satisfy the seller, that ...
Without a tax, the equilibrium price will be at Pe and the equilibrium quantity will be at Qe. After a tax is imposed, the price consumers pay will shift to Pc and the price producers receive will shift to Pp. The consumers' price will be equal to the producers' price plus the cost of the tax.