Our interest here concerns liquidity supply as a distinctive feature of the bank-borrower relationship. Any agent facing an opportunity or a commitment may find him/herself unexpectedly illiquid, and hence he/she may find it profitable to borrow "on call" if this costs less than missing the opportunity or defaulting on the commitment, or costs less than using non-money goods as means of payment. This is the essence of what Hicks called "the overdraft economy". Accordingly, we call a debt contract inclusive of the liquidity service an "overdraft debt contract", and we investigate its efficiency properties in a continuous time stochastic model of a repeated bank-borrower relationship where the key problem is the credibility of their mutual commitment between the two parties. Our main finding is that efficient, i.e. cost-minimising, overdraft debt contracts emerge in the absence of perfect commitment and enforceability as the borrower and the bank can exert mutual threat of termination.