The provisions of this chapter shall not apply in respect to the use of any sand, gravel, or rock to the extent of the cost of or charges made for labor and services performed in respect to the mining, sorting, crushing, screening, washing, hauling, and stockpiling such sand, gravel, or rock, when such sand, gravel, or rock is taken from a pit or quarry which is owned by or leased to a county or a city, and such sand, gravel, or rock is (1) either stockpiled in said pit or quarry for placement or is placed on the street, road, place, or highway of the county or city by the county or city itself, or (2) sold by the county or city to a county, or a city at actual cost for placement on a publicly owned street, road, place, or highway. The exemption provided for in this section shall not apply to the use of such material to the extent of the cost of or charge made for such labor and services, if the material is used for other than public road purposes or is sold otherwise than as provided for in this section.