No contract for any of the improvements provided by sections 14-384 to 14-3,127 shall be let unless first the city shall have made a detailed estimate of the costs of the contemplated improvement, nor shall any such contract be let until after the city has advertised for and received bids for the performance of such work. If no bid is received within the estimate, no award shall be made upon any bids received until after fifteen days after the time for receiving bids under such advertisement shall have expired. Within such time anyone desiring to do so may file a bid within the estimate and award may be made on such bid in like manner as if such bid had been received in pursuance to the advertisement calling for bids. All improvements authorized by sections 14-384 to 14-3,127 shall be done under contract with the lowest responsible bidder, except that when bids are called for by advertisement for grading in a street or alley and no bid is received within the estimate, the city may enter into a contract to do such grading without further advertisement for bids if the contract price be within the estimate and the contract be entered into within thirty days after the time for receiving bids under the advertisement calling for bids.