Auctions purchases represent an effective mechanism for public authorities, designed to provide the buyer (public authority), products and services with a convenient time to complete the transaction and the option to set a minimum price. In this paper we will study the problem of the optimal public expenditure rules using Bayes-Nash equilibrium in an symmetrical auction with knowledge of independent value, meaning each bidder knows only his own information. After setting the function for optimum balance to profit for the bidders by minimizing this function (the derivation of I order) and maximize it (the derivation of II order), it has to identify the optimal range where the offer of a bidder for products and services will stand
