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Commodity money arises endogenously in a general equilibrium model with separate budget constraints for each transaction. Transaction costs imply differing bid and ask (selling and buying) prices. The most liquid good—with the smallest proportionate bid/ask spread—becomes commodity money. General equilibrium may not be Pareto efficient. If zero-transaction-cost money is available then the equilibrium allocation is Pareto efficient. Fiat money is an intrinsically worthless instrument. Its positive price comes from acceptability in paying taxes, and its use as a medium of exchange is based on low transaction cost.

Two generations ago, Prof. Gerard Debreu suggested that the research agenda for mathematical general equilibrium theory should include a theory of money. Money is used to move purchasing power between markets and transactions, precisely the interaction between markets that general equilibrium emphasizes. Thus, general equilibrium modeling is an appropriate setting for microeconomic foundations of money. Nevertheless, an Arrow–Debreu model cannot successfully provide a role for money. The single budget constraint facing transactors in that model precludes a carrier of value between transactions. In order endogenously to derive the transactions role of money, multiple transactions—each with a budget constraint—and a motive for carrying value between them needs to be introduced. Though there has been progress on this head, e.g. Howitt (2000), Jones (1976), Kiyotaki and Wright (1989), Wallace (1980), full integration of money as a medium of exchange in the general equilibrium model has been incompletely successful.1The general equilibrium foundations of monetary theory should include parsimonious elementary economic conditions that allow commodity or fiat money to be sustained in an individually rational market equilibrium. This paper’s treatment seeks to provide—in a general equilibrium model with complete markets and complete information—weak sufficient conditions to derive a market equilibrium with the elementary properties of actual monetary economies. Under the conditions posited, trade is monetary in equilibrium; one side of almost all transactions is the economy’s medium of exchange (Theorem 3). The key to the formalization is in Hahn (1971) and Foley (1970). Those papers remind us that transaction costs create a bid/ask spread between buying and selling prices. Menger (1892) recognized this price spread as a measure of liquidity and argued that the most liquid assets become endogenous commodity money, “goods [are]. . . more or less saleable, according to the. . . facility with which they can be disposed of. . . at current purchasing prices. . . with less or more diminution. . . men. . . exchange goods. . . for other goods. . . more saleable. . . [which] become generally acceptable media of exchange.” A good is liquid if its bid and ask prices are close together. Thus, price theory implies a theory of liquidity. The most liquid good becomes ‘money.’ That is the outcome of the model below. Fiat money enters when government provides it (backed by the government’s undertaking to accept fiat money in payment of taxes—a notion going back to Adam Smith). Then ‘money’ is government-issued fiat money, trading at a positive value though it conveys directly no utility or production (Theorem 4). This essay proposes a parsimonious model of an economy where existence of a medium of exchange is an equilibrium result of the optimizing behavior of individual firms and households. The monetary character of trade, use of a medium of exchange, is shown to be an outcome of general equilibrium with transaction costs. Markets are assumed to be segmented;2 there is a separate budget constraint at each transaction creating demand for a carrier of value between transactions. Commodity money arises endogenously as the most liquid (lowest transaction cost) asset. Government-issued fiat money sustains its function as a medium of exchange through low transaction cost. This essay presents a full information general equilibrium model with realistic modification of the Arrow–Debreu specification sufficient to derive this monetary structure as an outcome.3The price system itself designates ‘money’ and guides transactors to trade using ‘money.’ It is useful to distinguish search/random matching models of money, e.g. Kiyotaki and Wright (1989), Trejos and Wright (1995), from general equilibrium models with transaction cost, e.g. Foley (1970), Hahn (1971), Starrett (1973), Ostroy and Starr (1974), Iwai (1996), Howitt (2000) and this essay. Search models emphasize very imperfect uncertain markets with limited ability of traders to locate desirable trades and with limited price flexibility. That approach is consistent with Smith (1776). General equilibrium with transaction cost models typically portray complete markets and a fully articulated price system. Using the complete markets approach allows us to pursue a parsimonious theory: What is a minimal set of market imperfections so that money arises endogenously? Starr (2003) and Starr (in press) provide elementary examples in a trading post model of the equilibria investigated here. The random matching/search formalization of the friction in trade has a very classical implication: in the rare case where two agents have a double coincidence of wants and meet to trade, they will trade their goods or services directly for one another, Kiyotaki and Wright (1989), Trejos and Wright (1995). This is a distinctive feature, distinguishing the random matching/search models from complete market general equilibrium with transaction cost models. In actual monetary economies, in those comparatively rare instances where double coincidence ofwants occurs, it is seldom resolved by barter exchange (a supermarket checkout clerk pays for groceries in money and an autoworker pays money to acquire a car). This essay’s model is distinct from the overlapping generations model, Samuelson (1958), Wallace (1980, 2001), etc., emphasizing complete markets and including a transactions demand for money at a point in time, not only over time. In the overlapping generations model, demand for money cannot be sustained in the presence of other intertemporal assets carrying a positive rate of return. In the present model (with time dated goods), there may be a demand for money as the low transaction cost instrument even in the presence of assets whose yield dominates money’s. The present model posits fully informed trade in many separate markets, with a separate budget constraint in each segmented market and transaction costs. The notion of multiple budget constraints is merely the formalization of the observation that budgets balance in each of many transactions separately, Hahn (1971), Ostroy (1973). A typical household will make many distinct transactions, with retailers, service providers, an employer, and so forth. In each of these transactions a budget constraint prevails. At prices prevailing in each transaction, budgets must balance; each party delivers value to the other equal to that he receives. Since there is a multiplicity of separate budget constraints, the market is said to be segmented. In addition, there are transaction costs in each market creating differing bid and ask prices. The notion of transactions as a resource using activity is embodied in market-making firms, Foley (1970), with a production technology transforming the ownership of goods between sellers and buyers.Multiple budget constraints create demand for media of exchange. Liquidity is priced: its price is the bid/ask spread. The most liquid asset, the instrument that provides liquidity at lowest cost, will be chosen as the medium of exchange. Thus, the choice of commodity money is the outcome of optimizing behavior of economic agents in market equilibrium. Fiat money—issued by government—derives its positive value from acceptability in payment of taxes, and it becomes the medium of exchange from its low transaction cost. To prove existence of a general equilibrium in a segmented market with transaction cost this paper combines two available treatments. Foley (1970) provides a demonstration of existence of general equilibrium with bid and ask prices and transaction costs in a single unified market. Arrow and Hahn (1971, Chapter 6) demonstrates the existence of general equilibrium with externalities. The composite household model belowthen expands the commodity space and the population of households. Each commodity is treated as distinct depending on which market segment it trades in. Each household is treated as being many distinct counterparts depending on which market segment it trades in. The counterparts are then combined by formalizing an external effect (in the form of a common consumption and common maximand) among them. The general equilibrium of the composite household model with externalities is then a general equilibrium of the original segmented market economy.

An Arrow–Debreu general equilibrium model modified to include transaction costs and multiple budget constraints implies monetary trade as a consequence of the equilibrium. Commodity (and fiat) money flows are endogenously determined as part of the equilibrium actions of firms and households. Liquidity is priced in the bid/ask spread; prices provide a direct incentive to concentrate the medium of exchange function in goods with the narrowest bid/ask spread. Fiat money’s positive value is supported by acceptability in payment of taxes and it becomes the common medium of exchange because of low transaction cost. Commodity or fiat money with a zero-transaction-cost leads to Pareto efficient equilibrium allocation.