Each county in a district formed under this chapter shall pay from its general fund its proportionate share to the board of directors of such amount as the board may designate to constitute a cash revolving fund to carry on the work and expense of maintaining such joint county jail. Each month a statement of the expense of the joint county jail shall be sent to the board of supervisors of each county in the district, together with a claim for its proportionate share of expenses. Amounts when received shall be paid into the cash revolving fund.