Mich. Comp. Laws § 404.3
The cost of establishment and construction of said county medical care facility shall be borne by each county in proportion to the assessed valuation of said county. The board of supervisors of each county becoming a party to the erection of a joint county medical care facility under the provisions of this act, may raise in any 1 year for construction or maintenance purposes a sum not exceeding .1 mill on each dollar of assessed valuation of said county. Such tax shall be regarded as a special tax, and moneys received therefrom shall be transmitted by the treasurer of the county in which it is collected to the treasurer of the county in which said joint county medical care facility is to be located and constructed. All such moneys shall be and remain in a special fund and shall be used solely for the purposes for which the tax is spread: Provided, however, That the money so received for construction purposes and not needed therefor, may be expended by the board of trustees for the maintenance and operation of said joint county medical care facility. Money expended for the construction, equipment and installation of equipment of any joint county medical care facility shall be paid out by the county treasurer having such fund in charge, on the order of the board of trustees of such joint county medical care facility. The maintenance and operation of such joint county medical care facility shall be borne by each county in proportion to the number of persons kept by each county in said medical care facility.
MCL 404.3