The county commission of all such counties shall have the power and authority to provide for the payment of the cost and expenses of such inventories and to prorate such cost and expenses between the county, the state, each municipality, the county board of education, and each city board of education in the county, such proration to be approximately in the proportion that the revenue received by the state, the county, such municipalities, and such boards of education, respectively, bears to the total amount of ad valorem taxes collected in the county. When the board of equalization or like boards or agencies, with final authority to fix the value of property for the purposes of taxation of any such county shall provide for the taking of any such inventory and appraisal, the Comptroller, subject to the approval of the Governor, shall have the authority to pay the state’s pro rata thereof, and each municipality and each board of education in such county shall have authority to pay its pro rata of said cost and expenses, and the county governing body shall have authority to provide for the payment of the county’s pro rata thereof; provided, that the failure of either the state or any of said municipalities or boards of education to provide for the payment of their pro rata portion of said cost and expenses shall not prevent the county, or any of the other governmental bodies or agencies mentioned herein, continuing such inventory and appraisal and paying the entire cost thereof, upon such terms as may be agreed upon between the county and such other governmental bodies or agencies.