Pursuant to an agreement entered into under division (A)(10) of section 5153.16 of the Revised Code respecting the operation, acquisition, or maintenance of a children’s home, training school, or other institution for the care of children maintained by a municipal corporation or other political subdivision, the public children services agency may acquire, operate, and maintain such an institution. The agency may enter into an agreement with a municipal corporation, a board of education, and the board of county commissioners, or with any one of them, to provide for the maintenance and operation of children’s training schools. The agreement may provide for the contribution of funds by the municipal corporation, board of education, or board of county commissioners, in such proportions and amounts as the agreement states. The agreement also may provide for the operation and supervision of the training school by any one of them, or by the joint action of two or more of them, provided that municipal corporations, boards of education, and boards of county commissioners may expend moneys from their general funds for maintaining and operating the joint children’s training school.