When such a contract shall have been entered into the prosecution of the work of improvement and the expenditure of funds thereof shall be determined upon, controlled and provided for by joint action of the boards of county commissioners of the two counties. So acting jointly, they shall have power to employ subordinates, purchase material or equipment in open market or by contract, let contracts for work, or cause work to be done by day labor, and to reject any and all bids received for work or material. All vouchers, pay rolls, reports, contracts and bonds on contracts shall be in duplicate, one copy to be filed in the office of the county auditor of each county: PROVIDED, HOWEVER, That the expenditure of said funds must be made in such manner so that the fund from each county is drawn on or expended alternately and such alternate expenditure shall be in proportion to the amount contributed by each county as nearly as may be practicable.