Mich. Comp. Laws § 397.33

Sec. 3. In any county where there is no public library, or in any county within the boundaries of which there are municipalities which have not established public library service or which do not maintain public libraries, the county board of supervisors shall appoint a county library board to receive the per capita portion of penal fine moneys to be allocated for such areas. The county library board shall consist of 5 members appointed by the county board of supervisors for terms of 5 years each, except that the first members shall be appointed for 1, 2, 3, 4 and 5 years respectively. The board may contract with a qualified public library, within or without the county, to provide public library service for all residents of the county without legal access to a public library.