The local authorities shall pay monthly to the Department of Public Welfare,¹ as such local authorities payment for assistance, the amount expended by the department during the preceding month as assistance on behalf of patients receiving public nursing home care in a medical institution of their county institution district, and on behalf of children in foster family homes and child-caring institutions, plus the cost of administering such assistance, minus the amount of Federal funds properly received or to be received by the Department of Public Welfare on account of such expenditures, increased or reduced, as the case may be, by any amount by which the sum paid for any previous month differed from the amount which should have been paid for such previous month and by the proportionate share of refunds for such assistance. As provided in the “Public Assistance Law,”² the Department of Public Welfare shall certify to the local authorities the amount to be paid by them to the department.