Any city improvement costing more than twenty-five thousand dollars ($25,000.00) shall be executed by contract except where such improvement is authorized by the City Council to be executed directly by a city department in conformity with detailed plans, specifications, and estimates. All such contracts for more than twenty-five thousand dollars ($25,000.00) shall be awarded to the lowest responsible bidder bidding in conformity with detailed plans and specifications approved by the City Council after such public notice and competition as may be prescribed, provided the City Council shall have the power to reject any or all bids. Except in case of emergency, alterations in any contract not to exceed twenty-five (25) percent of the total contract may be made when authorized by the Council. The contract may be decreased more than twenty-five (25) percent with the consent of the contractor.