Whenever a village or city or cities and a county or township shall desire to act under the provisions of this law, the relationship established between such municipalities and governmental agencies shall be fixed by contract and such contracts may be made by cities and counties under the provisions of this act in the manner and to the extent that natural persons might make contracts for like purposes. Such contracts before becoming operative shall be approved by a vote of a majority of all the members-elect of the legislative bodies of the counties and cities operating under the provisions of this act.