Under what conditions do the poor hold cash in pairwise-trade models?

Most existing pairwise-trade models of money predict that the poor may not hold any money in the presence of interest-bearing liquid assets. We resolve this seemingly implausible prediction by incorporating cash and interestbearing checking account into a standard search-theoretic model. If the transaction cost of a debit card is neither too small nor too large, there is a coexistence equilibrium in which the poor hold cash for their small transactions, whereas the rich hold interest-bearing deposits and use a debit card for their large transactions.