Cities, depending on their transportation connections, will have access to other cities as potential markets or suppliers. At the beginning, cities will indeed be mainly bartering with whatever goods they have. But because food has only a limited shelf life, and wood/stone require too much storage accommodation, at some point wealth will be more conveniently accumulated in the form of valuable objects that can serve as money (e.g. conch shells or gold coins). Example of the transition:
City A, which is running out of storage space/capacity for its wealth, may request that City B exchange gold for wheat, rather than wood for wheat. Or it may be marble for wheat, just whatever can serve as a convenient store of value. Then eventually City A may just ask for a piece of paper that says "City B will redeem this paper for X amount." For convenience, a bank or the government may rather issue pieces of paper "Anyone can redeem this paper for X amount." And the final stage is the government issuing pieces of paper "This paper is worth X amount, because Government Y says so."
Perhaps that's more than answering the question, but I like the UET to be generally applicable to all periods of history.