Kiyotaki–Wright model of money

The Kiyotaki–Wright model of money, also called the Kiyotaki–Wright search model, is a search-theoretic model in monetary economics developed by Nobuhiro Kiyotaki and Randall Wright. Introduced in 1989 and elaborated in later papers, it studies how a medium of exchange can arise in decentralized trade when barter is limited by the double coincidence of wants problem.^[1][2] The framework has influenced later work in New Monetarist economics.^[3][4]

The original 1989 paper emphasized money's medium-of-exchange role rather than treating money only as a store of value or imposing its use through a cash-in-advance constraint. In the model, agents specialize in consumption and production, are matched randomly over time, and can trade only bilaterally and quid pro quo. Under those frictions, some commodities can become commodity money, and intrinsically worthless fiat money can also circulate in equilibrium when supported by beliefs.^[1]

Kiyotaki and Wright's original 1989 paper modeled an economy in which agents specialize in production and consumption, meet randomly over time, and trade bilaterally. In that setting, certain commodities can emerge endogenously as media of exchange, depending on their intrinsic properties and on agents' beliefs; the model also admits equilibria with valued fiat currency.^[1]

The 1989 article also distinguished between fundamental equilibria, in which agents rank assets according to storage costs except when they encounter their own consumption good, and speculative equilibria, in which some agents accept a higher-storage-cost object because it is more marketable in future matches.^[1] That distinction became important in later search-theoretic monetary models because it showed that equilibrium monetary roles can depend on both fundamentals and self-fulfilling expectations.^[1][4]

In A Contribution to the Pure Theory of Money (1991), Kiyotaki and Wright analyzed a more general environment with search frictions and differentiated commodities, proving the existence of equilibria with valued fiat money and showing that those equilibria are robust to several changes in the environment.^[5]

Their 1993 paper presented a more tractable version focused on fiat money and used it to analyze welfare, specialization, and the possibility of multiple currencies.^[2] Early extensions included results on mixed strategies and steady states by S. Rao Aiyagari and Neil Wallace,^[6][7] and by Timothy J. Kehoe, Kiyotaki and Wright.^[8]

In the canonical Kiyotaki–Wright environment, agents are infinite-lived and meet randomly in pairs. They are specialized: each agent can consume only a subset of goods and does not consume his own output. Trade is decentralized and bilateral rather than coordinated through a centralized market.^[1][2]

In the 1989 formulation, time is discrete and there are three indivisible commodities. There is a continuum of agents split equally among three types. A type ${\displaystyle i}$ agent consumes only good ${\displaystyle i}$, produces only one other good ${\displaystyle i^{*}}$, can store only one unit at a time, and pays good- and type-specific storage costs ${\displaystyle c_{ij}}$, ordered so that good 1 is most storable and good 3 least storable: ${\displaystyle c_{i3}>c_{i2}>c_{i1}>0}$ for all ${\displaystyle i}$.^[1]

Because exchange is quid pro quo, barter succeeds only when each trader simultaneously wants what the other is offering. This is the double-coincidence-of-wants problem. Agents therefore choose trading strategies partly on the basis of an object's future marketability: they may accept an item that they do not wish to consume if doing so improves their chances of trading later.^[2]

The 1989 paper defines a steady-state Nash equilibrium as a set of trading rules and an invariant distribution of inventories such that each agent's strategy is optimal given the strategies of others and the steady-state distribution, and the distribution is itself generated by those strategies.^[1] If ${\displaystyle \tau _{i}(j,k)=1}$ denotes that type ${\displaystyle i}$ wants to trade inventory good ${\displaystyle j}$ for good ${\displaystyle k}$, then trade occurs only if both traders agree. An agent accepts good ${\displaystyle k}$ over good ${\displaystyle j}$ when the continuation value of leaving the match with ${\displaystyle k}$ exceeds that of leaving with ${\displaystyle j}$.^[1]

In the 1989 version, this mechanism can make a relatively marketable commodity circulate as commodity money.^[1] In the 1991 and 1993 variants, fiat money can also be valued and circulate as a medium of exchange.^[5][2] The tractable 1993 model assumes indivisible goods and money, with a fraction of agents initially endowed with fiat money and the rest holding one unit of a real commodity.^[2]

The original paper studies two benchmark production patterns, often called model A and model B.^[1]

In model A, types produce in the cycle ${\displaystyle 1^{*}=2}$, ${\displaystyle 2^{*}=3}$, ${\displaystyle 3^{*}=1}$. Depending on parameter values, there is either a unique fundamental equilibrium, in which good 1 is the unique commodity money, or a unique speculative equilibrium, in which type 1 agents accept good 3 because it is more marketable and both goods 1 and 3 serve as commodity monies.^[1]

In model B, the production pattern is ${\displaystyle 1^{*}=3}$, ${\displaystyle 2^{*}=1}$, ${\displaystyle 3^{*}=2}$. A fundamental equilibrium always exists, with goods 1 and 2 functioning as media of exchange. For some parameter values, a speculative equilibrium also exists in which types 2 and 3 speculate and goods 2 and 3 become commodity monies, so multiple equilibria coexist for the same fundamentals.^[1]

The model showed how money can have an essential medium-of-exchange role in equilibrium without relying on a Walrasian auctioneer or a centralized trading mechanism.^[1][2] It also implied that monetary exchange can improve welfare relative to barter by reducing search frictions and, in some versions, by supporting greater specialization and productivity.^[1][2][7]

A central result of the 1989 paper is that whether an object becomes money depends on both intrinsic properties and extrinsic beliefs. Lower storage costs make a commodity a natural candidate for monetary use, but a more costly object can still circulate if agents expect it to be easier to pass on in future trade. The paper therefore gives a search-theoretic account of marketability in the sense associated with Carl Menger.^[1]

The model also generated rate-of-return dominance. In the speculative equilibrium of model A, a type 1 agent may prefer to hold good 3 rather than good 2 even though good 3 has a higher storage cost, because good 3 is more marketable and therefore yields a higher continuation value in decentralized exchange.^[1] This result was important because earlier monetary models often required auxiliary assumptions to explain why a dominated asset could circulate.^[1][4]

At the same time, the Kiyotaki–Wright framework typically admits multiple equilibria, and its monetary outcomes are not generally Pareto efficient.^[1][6][7] In the 1989 paper, equilibria in model B are not generally Pareto-comparable across types, and even when a commodity-money equilibrium exists, decentralized agents may reject trades that would make everyone better off ex ante.^[1]

In On Money as a Medium of Exchange, Kiyotaki and Wright also introduced a fiat object, denoted good 0, that yields no direct utility and is not used in production. Agents can hold either one real object or fiat money, but not both simultaneously. The paper shows that fiat money is tenuous: if nobody expects others to accept it later, nobody accepts it now; but if everyone expects universal acceptance, fiat money can circulate as the general medium of exchange.^[1]

In the fiat-money equilibrium analyzed in the 1989 paper, fiat money has zero storage cost and is accepted by all types, while at least one commodity may continue to circulate as a more specialized medium of exchange alongside it.^[1] The paper studies the quantity of real balances ${\displaystyle S=M/P}$, where ${\displaystyle M}$ is the nominal stock of fiat money and ${\displaystyle P}$ is the money price of one unit of a real good, and shows that welfare depends on real balances rather than on nominal balances alone.^[1]

The 1989 paper also studied several measures of moneyness: the stock of each asset, the number of times it is traded, its velocity, its acceptability, and its liquidity.^[1] It found that velocity is not, by itself, a reliable indicator of whether an object is money, because an asset can have high velocity simply because its equilibrium stock is small.^[1] By contrast, acceptability—the probability that an offered object is accepted in trade—tracks monetary status more closely; fiat money has acceptability 1 in the fiat-money equilibrium because it is the general medium of exchange.^[1]

Liquidity in the model is defined by the expected time it takes an agent starting with a given object to obtain his consumption good. This depends on the object's marketability and on the stock of real balances in circulation. In the fiat-money equilibrium, larger real balances reduce the stock of real goods available for trade and lengthen waiting times to consumption, illustrating the possibility of "too much money chasing too few goods" in the model's search environment.^[1]

In Search, Bargaining, Money, and Prices (1995), Alberto Trejos and Wright added bilateral bargaining to endogenize the price level, showing that monetary equilibria are generally inefficient and that non-stationary inflationary equilibria can exist.^[9]

Later work relaxed some of the original indivisibility assumptions. Shouyong Shi's divisible-money model severed the link between the money supply and the number of money holders in the original framework, allowing the model to be used more directly for questions about neutrality, superneutrality and optimal money growth.^[10]

The Kiyotaki–Wright model became the benchmark starting point for the search-theoretic approach to money.^[3][4] It has been used to study commodity money, fiat money, multiple currencies, bargaining, the distribution of money holdings, and the microfoundations of liquidity.^[4]

The 1989 paper was especially influential because it connected classic themes in monetary theory—specialization, double coincidence of wants, marketability, and the emergence of a general medium of exchange—to an explicit non-cooperative equilibrium framework.^[1][3] Its distinction between fundamental and speculative monetary equilibria also helped motivate later work on multiplicity, self-fulfilling beliefs, and the welfare properties of decentralized trade.^[1][4]

A later development was the Lagos–Wright model, which preserved decentralized trade frictions while making the framework more tractable for quantitative and policy analysis.^[11]

• Double coincidence of wants
• Lagos–Wright model
• Medium of exchange
• Monetary economics
• Search theory (economics)
• Trejos–Wright model

• Kiyotaki, Nobuhiro; Wright, Randall (1989). "On Money as a Medium of Exchange". Journal of Political Economy. 97 (4): 927–954. doi:10.1086/261634. JSTOR 1832197.
• Rupert, Peter; Schindler, Martin; Shevchenko, Andrei; Wright, Randall (2000). "The Search-Theoretic Approach to Monetary Economics: A Primer". Economic Review (Q IV). Federal Reserve Bank of Cleveland: 10–28.
• Lagos, Ricardo; Wright, Randall (2005). "A Unified Framework for Monetary Theory and Policy Analysis". Journal of Political Economy. 113 (3): 463–484. doi:10.1086/429804.