# New Monetarist Economics

- Chao Gu, Chao GuDepartment of Economics, University of Missouri
- Han HanHan HanSchool of Economics, Peking University
- and Randall WrightRandall WrightDepartment of Economics, University of Wisconsin–Madison

### Summary

This article provides an introduction to New Monetarist Economics. This branch of macro and monetary theory emphasizes imperfect commitment, information problems, and sometimes spatial (endogenously) separation as key frictions in the economy to derive endogenously institutions like monetary exchange or financial intermediation. We present three generations of models in development of New Monetarism. The first model studies an environment in which agents meet bilaterally and lack commitment, which allows money to be valued endogenously as means of payment. In this setup both goods and money are indivisible to keep things tractable. Second-generation models relax the assumption of indivisible goods and use bargaining theory (or related mechanisms) to endogenize prices. Variations of these models are applied to financial asset markets and intermediation. Assets and goods are both divisible in third-generation models, which makes them better suited to policy analysis and empirical work. This framework can also be used to help understand financial markets and liquidity.

### Introduction

New Monetarist Economics is a branch of macro and monetary theory and policy analysis that developed over the last three decades.^{1} The New Monetarist label suggests a comparison to Old Monetarism and New or Old Keynesianism. Some contributors to the New Monetarist literature find many ideas in the Old Monetarism of Friedman and his followers. However, New Monetarist Economics puts greater emphasis on microfoundations than Old Monetarism or Old Keynesianism and differs from New Keynesianism in its focus on particular underlying frictions in the economy. While Keynesians rely almost exclusively on nominal price rigidity as the critical distortion relevant for all macro theory, empirical work, and policy analysis, New Monetarists focus on imperfect commitment and information problems as well as difficulties in coordinating trade that are often captured using search theory (for more on these issues and, in particular, on differences across the various macro camps, see Williamson & Wright, 2010a).

This article provides a summary of research on New Monetarist Economics, providing a rigorous theoretical approach to identify the key frictions and to endogenize institutions like money (as well as banking and various alternative credit arrangements) by explicitly developing models in which liquidity is endogenously valued. While initially concerned with theory, the field has become increasingly policy oriented as the theories have matured, and recent events have put monetary matters front and center, including those related to interest rates, credit conditions, financial markets, and liquidity.

While the literature is diverse, papers do share a set of principles and methods. Common themes include the following: What exactly is liquidity and why does it matter? What role do liquid assets, including money, play in improving the allocation of resources and welfare? Which objects serve this capacity, or should serve this capacity, in equilibrium and in optimal arrangements? How can intrinsically worthless currency be valued—or, more generally, how can asset prices differ from their fundamental values? How does credit work without commitment, and what are the roles of collateral and reputation? Under what conditions can money and credit coexist and how do they interact? What are the functions of intermediation, including banking but also other financial institutions? What can monetary policy achieve and how should it be conducted?

The research concerns much more than just pricing currency, although that was once a major goal and indeed was one of the classic issues in economics. On “the old conundrum” of fiat money, for example, Hahn (1987) says:

At a common-sense level almost everyone has an answer to this, and old-fashioned textbooks used to embroider on some of the banalities at great length. But common sense is, of course, no substitute for thought and certainly not for theory. In particular, most of the models of an economy which we have, and I am thinking here of many besides those of Arrow and Debreu, have no formal account for the exchange process. (p. 23)

Similarly, Clower (1970) says:

conventional value theory is essentially a device for logical analysis of virtual trades in a world where individual economic activities are costlessly coordinated by a central market authority. It has nothing whatever to say about delivery and payment arrangements, about the timing or frequency of market transactions, about search, bargaining, information and other trading costs, or about countless other commonplace features of real-world trading processes. (p. 427)

^{2}

Modern monetary theory rectifies this situation, and it is a major accomplishment that the “old conundrum” of currency is now a solved problem, even if research continues to fine-tune our knowledge. Yet more generally, this research seeks to understand the process of exchange in the presence of frictions and how this might be facilitated by various monetary or credit arrangements. It emphasizes the process by modeling agents that trade with each other, which is *not* the case in much of economics. In general equilibrium(GE) theory, agents merely slide along budget lines: they are endowed with a vector $\overline{x}$, and choose another $x$ subject only to $px\le p\overline{x}$, with $p$ taken as given; and as the previous paragraph indicates, getting from $\overline{x}$ to $x$ is not up for discussion.^{3} The research discussed here studies transactions between individuals in the presence of explicit frictions that hinder their interactions, and then analyzes how institutions ameliorate these frictions.

While borrowing from GE theory, the approach relies heavily on search theory, which is all about agents trading with each other. It also uses game theory, naturally, since the issues are inherently strategic—for example, what one accepts as a medium of exchange or as collateral should generally depend on what others accept. Other common tools include mechanism design and bargaining theory, which are standard tools in microeconomics, if less so in macro. Key frictions that make the models interesting include spatial and temporal separation, which affect who meets or communicates with whom. At least as important are limited commitment and information, which hinder credit and thus can make arrangements involving money socially beneficial. To be sure, this is not the case in models with cash-in-advance (CIA) assumptions, where, similar to sticky-price models, money considerations lower welfare. Similar comments apply to money-in-utility-function (MUF) models. If it were really such a hindrance, how did money survive all these centuries?

New Monetarist research tries to get phenomena like monetary or credit arrangements to arise endogenously as beneficial institutions. Earlier work that is similar in spirit includes research on overlapping-generations (OLG) economies, with seminal contributions by Samuelson (1958), Lucas (1972), and Wallace (1980). Since Kiyotaki and Wright (1989), many monetary economists have employed search theory. However, it has been clarified over the years that, as emphasized by Kocherlakota (1998), we also need commitment and information frictions to make money *essential* in the sense made more precise below. On banking, past work similar in spirit includes Diamond and Dybvig (1983), Diamond (1984), and Williamson (1986, 1987). On intermediation more generally, we mention Rubinstein and Wolinsky (1987) and Duffie, Gârleanu, and Pederson (2005), while on secured and unsecured credit, we mention Kiyotaki and Moore (1997) and Kehoe and Levine (1993).^{4}

The preference for modeling monetary, credit, and other arrangements explicitly is related to the Lucas (1976) critique, and the general approach follows ideas espoused in Townsend (1987, 1988). This involves writing down an environment (i.e., an explicit description of agents, their preferences, technologies, etc.) that combine to constitute an artificial economy that we can study the way other scientists proceed in their laboratories. We ask what the agents do in our artificial economies and compare that to what happens in reality, at least in a stylized sense. When simple artificial economies do not resemble the institutions we want to understand, we add complications, or frictions. For instance, it is natural that agents want to engage in risk sharing, which corresponds to insurance in the real world. However, in simple artificial economies agents get more insurance than we seem to see in the data. To understand that, we might consider information or commitment frictions to reconcile theory and observation. But in using this approach to help us understand the world, one should not assume full insurance is impossible without explicitly modeling why.

More generally, market structure ought to emerge endogenously from a description of an environment with frictions. Many economists seem to agree that a pressing goal for mainstream macro is to incorporate more interesting financial considerations. We concur, but insist that financial considerations must be incorporated with solid micro foundations. A related point is that one should explain and not assume the kinds of contracts that agents use in our models. It is flawed to not let agents in our artificial economies enter into agreements that are obviously better than those we impose. Imposing sticky nominal contracts may make a model look more realistic and policy relevant, but one really has to ask why prices are sticky: If agents in our artificial economy want indexed contracts, why cannot we provide them? Or at least we need to say exactly why they cannot use real contracts.

This relates to the Wallace (1998) dictum: “*Money* should not be a primitive in monetary theory—in the same way that *firm* should not be a primitive in industrial organization theory or *bond* a primitive in finance theory” (p. 20). Gu, Mattesini, Monnet, and Wright (2013a, 2013b) further suggest that a bank should not be a primitive in intermediation theory. Further along these lines, Wright (2017) promotes the idea that monetary, credit, and other arrangements should be outcomes of our theories, not inputs to our theories. In other words, we want models of these institutions, not just models that include them. For many issues, it does not suffice to assert that households cannot lend to firms, but they can lend to banks, and banks can lend to firms. That would be a model *with* banks, but not *of* banks, because it does not make explicit the frictions generating these arrangements endogenously. If households and firms want to trade intertemporally the model should let them, or preclude that by laying out explicit details about commitment, information, or other frictions.^{5}

This article is organized as follows: Section “First-Generation Models” uses the simple first-generation model of money in Kiyotaki and Wright (1993) to illustrate some very basic ideas and methods, and to show that it is not hard to formalize liquidity considerations. Such models are rudimentary in that they have indivisible goods and assets, plus restrictions on inventories, that make every trade a one-for-one swap, which is crude but allows us to illustrate several key ideas with minimum complication. Section “Second-Generation Models” sketches the second-generation model in Shi (1995) or Trejos and Wright (1995), where prices are introduced by making goods divisible but keeping assets indivisible to maintain tractability. These models are interesting in their own right and provide a stepping stone to third-generation models, where everything is divisible.

Section “Third-Generation Models” takes up third-generation models, discussing computational work following Molico (2006) and some more tractable analytic work following Shi (1997) or Lagos and Wright (2005). In particular, we go into detail showing how to solve the Lagos-Wright model, which has become a workhorse in the field. This model exemplifies principles behind the New Monetarist method—even though it is just one of many specifications that might be useful in various applications—and we want to illustrate that it is very tractable and flexible. Section “Applications and Extensions” follows up on this by discussing some of applications from the literature. Section “Banking” concentrates on banks and Section “Conclusion” provides a brief conclusion.

Before embarking, let us mention that much of what follows frames the discussion in terms of households trading consumption goods, but one can alternatively proceed in terms of firms and production inputs, financial institutions and assets, or other scenarios. In part, this follows early papers in the New Monetarist literature, which themselves follow classics like Jevons or Menger. Depending on the substantive issues at hand, it can be better to use firms and inputs, banks and reserves, or other options. Let us also mention that given our perceived mandate for this article, we have aimed to summarize research at or near the frontier but still discuss some earlier work in the area. Indeed, we go further by trying to put it all into historical perspective by referencing the classics where appropriate.

### First-Generation Models

Here is a simple model based on Kiyotaki and Wright (1993), which is itself a simplification of Kiyotaki and Wright (1989, 1991), and can also be considered an extension of early equilibrium search models along the lines of Diamond (1982). The object of all these models is to derive equilibrium transactions patterns endogenously and ask if they resemble trading arrangements in actual economies, if only in a stylized sense, since the framework is fairly abstract. To this end, consider an environment with a $\left[0,1\right]$ continuum of infinitely lived agents that meet bilaterally, at random, in continuous time. Denote by $\alpha $ the (Poisson) arrival rate of potential trading partners. There is a set of goods that is nonstorable and hence must be produced on the spot for immediate consumption at cost $c>0$.^{6}

To introduce gains from trade, suppose agents have specialized tastes and technologies. A simple way to capture this is to assume the following happens when a random pair $i$ and $j$ meet: $\delta $ is the probability of a double coincidence, where $i$ likes what $j$ produces and $j$ likes what $i$ produces; and $\sigma $ is the probability of a single coincidence, where $i$ likes what $j$ produces but $j$ does not like what $i$ produces. The concepts of single- and double-coincidence meetings are used to good effect by Jevons (1875), but the idea goes back much further.^{7} Also, while it is interesting to relax this, for the sake of simplicity we impose symmetry: In a single-coincidence meeting each agent has an equal probability of liking a partner’s output, and any good the agent likes gives the same utility $u>\mathrm{}c$ while all other goods give utility $0$.

Agents trade and consume in double-coincidence meetings; what is to be determined is the outcome in single-coincidence meetings. To discuss monetary matters later, we now introduce a perfectly storable asset in addition to the nonstorable consumption goods. While no one consumes the asset, it yields a flow utility $\rho $ to anyone holding a unit. One can interpret $\rho >0$ as a dividend, like the proverbial fruit from a Lucas (1978) tree in standard asset-pricing theory; one can interpret $\rho <0$ as a storage cost, as in theories of commodity money like Kiyotaki and Wright (1989); and if $\rho =0$ the asset is by definition fiat currency, as discussed in Wallace (1980). Let $A\in \left[0,1\right]$ be the fixed supply, and for now assume the asset is indivisible, with each individual restricted to hold $a\in \left\{0,1\right\}$. This is a strong assumption that allows us to make the points relatively easily (and in any case, the restriction is relaxed below).

Let us begin with $A=0$, to see what happens without assets. The simplest reasonable arrangement is pure barter, where agents trade only in double-coincidence meetings. The flow payoff to this is $r{V}^{B}=\alpha \delta \left(u-c\right)$, where $r$ is the rate of time preference—in other words, the payoff to living in a barter economy ${V}^{B}$ is the present value of meeting people at rate $\alpha $, trading with probability $\delta $, and enjoying a surplus $u-c$ in each exchange. Given $\delta >0$, this beats autarky, ${V}^{B}>\mathrm{}{V}^{A}=0$, but, given $\sigma >0$, it does not do all that well becayse there are meetings where $i$ wants to trade but $j$ does not oblige, and that is bad for everyone in the long run.

Therefore, suppose we try to institute the following idealized credit system: let agent $i$ produce for agent $j$ whenever $j$ likes $i$’s output.^{8} The flow payoff to this arrangement satisfies

Now, in addition to barter, in single-coincidence meetings, with equal probability an agent produces without consuming and consumes without producing. If $\sigma >0$ then ${V}^{C}>\mathrm{}{V}^{B}$. Ergo, if agents could commit they would commit to abide by this arrangement, and that maximizes ex ante utility conditional on the search and matching process. However, if they cannot commit, we must check if this is dynamically incentive feasible.

The potentially binding incentive condition arises when an agent is supposed to produce in a single-coincidence meeting. If ${V}^{D}$ is the deviation payoff, this is given by

where $\mu $ is the probability that deviators are caught and punished. This says the payoff to producing and continuing on the equilibrium path must be at least as high as not producing and taking one’s chances with the consequences, where $\mu <1$ captures imperfect monitoring, record keeping, or communication, so that deviations are only detected and punished probabilistically.

As usual, in terms of dissuading opportunistic behavior, the best punishment is the harshest. We take that to be no worse than banishment to autarky. While in principle one can consider harsher punishments, like flogging, this discussion assumes that such punishment is beyond the pale. In fact, even autarky may or may not be a credible threat depending on details, and the harshest punishment may instead be continuation with no more credit, only barter (one can show that is always credible). Let us pursue the case ${V}^{D}={V}^{B}$, and leave ${V}^{D}={V}^{A}$ as an exercise, since the results are similar. Then (1) can be reduced to

which says that $r$ must be low, or agents must be patient, for credit to work, as is standard when we try to support any cooperative behavior without commitment. Equivalently, instead of saying $r$ must be low, we can say $\mu $ must be high, naturally, since credit requires monitoring and record keeping.

If credit is not viable because $\mu $ is too low, what can we do? Reintroducing the asset in supply $A\in \left(0,1\right)$, we now show how that can help, even if $\rho =0$. Let ${V}_{a}$ be the value function for agents with $a\in \left\{0,1\right\}$, where those with $a=1$ are called buyers and those with $a=0$ are called sellers, anticipating that the former can use the asset to pay the latter in single-coincidence meetings. Then payoffs are

where $\tau $ is the probability a seller produces to get the asset and we include $\rho $ for future reference, even if the main case interest is $\rho =0$ for now. After trade using the asset, notice how the buyer becomes a seller and vice versa.

Letting $\Delta ={V}_{1}-{V}_{0}$, we write the seller’s best-response condition as

(in principle, for trade, the buyer must also agree to trade, which requires $u+{V}_{0}\ge {V}_{1}$, but that is not binding as long as $\rho $ is not too big). Then a stationary equilibrium is defined as a list $\u3008{V}_{0},{V}_{1},\tau \u3009$ satisfying (3)–(5). Given $\rho =0$, the best-response condition for $\tau =1$ reduces to

We conclude that exchange using fiat currency is consistent with incentives—in other words, that a monetary equilibrium exists—if and only if $r\le {\widehat{r}}_{M}$.

Intuitively, in a monetary economy people are willing to work to get currency since they (correctly) believe that others will work for them, to get currency from them, in the future. No one is forced to use currency—they are free to exclusively barter—but they choose to use it because it is advantageous to trade goods for money and then money for goods rather than only goods for goods. In other words, people value money because it conveys liquidity. This is not a surprise—the idea has been around for centuries—but there is still value in rigorous, precise formalization.

Money is said to be *essential* when we can achieve better incentive- and resource-feasible allocations with money than without it. When is this the case? It is easy to check that welfare is higher in monetary equilibrium than under pure barter but lower than under credit.^{9} Moreover, notice that ${\widehat{r}}_{C}>\mathrm{}{\widehat{r}}_{M}$ when $\mu =1$. This means that money is not essential when the monitoring probability is high, since then credit is viable and delivers higher payoffs. What if $\mu $ is small? Then we cannot support credit, but potentially can support monetary exchange, which does not depend on $\mu $. This exemplifies a general result in Kocherlakota (1998): When the monitoring/record-keeping technology—what he calls memory—is perfect in the sense of $\mu =1$. Money cannot be essential; and when $\mu $ is small money can be essential. This shows why and when monetary exchange can be a socially beneficial institution.

Money is also tenuous, or fragile. There is always an equilibrium where fiat currency is not accepted. There are also sunspot equilibria, where its acceptability fluctuates randomly over time as a self-fulfilling prophecy even when all parameters are constant, so it can be said to be prone to excess volatility. Yet in another sense money is robust: equilibria with $\tau =1$ exist for $\rho <0$, as long as $|\rho |$ is not too big, because agents are willing to tolerate a bad return on an asset that conveys liquidity, as long as the return is not too bad. A general message is that a liquid asset can be valued above its *fundamental value*, defined here as the present value of its dividend stream $\rho /r$ (this is obvious for $\rho \le 0$, but also true for $\rho >0$). Another message is that whether or not an asset circulates as payment, an instrument need not be pinned down by primitives. For some $\rho $ there coexist equilibria where it circulates and equilibria where it does not. So liquidity is at least partially a self-referential concept.

Based on these results and others in the early New Monetarist literature, we suggest that this setup delivers results that ring true, and are easy enough to teach at any level, on topics such as the tradeoff between acceptability and return, the tenuousness/robustness of currency, the self-referential nature of liquidity, and necessary conditions for essentiality. However, this model is too primitive to provide a general framework for thinking about financial issues, mainly because everything is indivisible. Hence, rather than cataloguing additional applications, the next step is to relax that restriction.

### Second-Generation Models

Shi (1995) and Trejos and Wright (1995) generalize the framework by incorporating divisible goods and letting agents negotiate the terms of trade. This is natural since they are trading bilaterally, which makes Walrasian price taking somewhat unpalatable. Having said that, there are alternatives: Julien et al. (2008) use auctions in a version of the environment where multiple buyers might encounter the same seller; Burdett, Trejos, and Wright (2017) consider price posting with noisy search, as in Burdett and Judd (1983), where an individual buyer may encounter many sellers; Zhu and Wallace (2007) use pure mechanism design; and so on. It speaks to the flexibility of the approach that it accommodates all these options, rather than only Walrasian pricing.

In any case, these models maintain $a\in \left\{0,1\right\}$, but let goods be divisible and say consumption of $q$ units gives utility $u\left(q\right)$ for any good the consumer likes, while production gives disutility $c\left(q\right)$. The functions $u\left(q\right)$ and $c\left(q\right)$ satisfy the usual monotonicity and curvature conditions, plus the normalization $u\left(0\right)=c\left(0\right)=0$. Also, there is a $\overline{q}>0$ with $u\left(\overline{q}\right)=c\left(\overline{q}\right)$, so that $u\left(q\right)>\mathrm{}c\left(q\right)$ if and only if $q\in \left(\mathrm{0,}\overline{q}\right)$. To reduce notation, let us set $\delta =0$ for now.^{10} Then the analogs to (3) and (4) are

where again $\Delta ={V}_{1}-{V}_{0}$, and we set $\tau =1$ to focus on equilibrium where agents trade and study how much they trade.

To begin, we need to discuss what is feasible. Recall that with indivisible goods, credit is viable iff $r\le {\widehat{r}}_{C}$ and fiat money is viable iff $r\le {\widehat{r}}_{M}$, where ${\widehat{r}}_{C}$ and ${\widehat{r}}_{M}$ are given by (2) and (6). The analogue results are that now we can support credit exchange, where agents always produce $q$ for anyone that likes their output, for any $q\le {\widehat{q}}_{C}$, and we can support exchange where agents trade $q$ for fiat money, for any $q\le {\widehat{q}}_{M}$, where ${\widehat{q}}_{C}$ and ${\widehat{q}}_{M}$ solve

The applicable version of Kocherlakota (1998) is: high $\mu $ implies ${\widehat{q}}_{C}>\mathrm{}{\widehat{q}}_{M}$, so money is inessential; and low $\mu $ implies money is essential, because for low $\mu $ we can support some $q$ with money that we cannot support with credit.

To determine $q$ in equilibrium we need another condition, and, as mentioned previously, there are alternatives. An easy option is Kalai’s (1977) proportional bargaining solution, where the buyer has bargaining power $\theta $.^{11} In this context, where the buyer’s and seller’s trade surpluses are $u(q)-\Delta $ and $\Delta -c\left(q\right)$, respectively, we can find the Kalai solution by giving the buyer a share $\theta $ of the total surplus, which yields

In fact, many other mechanisms for determining $q$ can be accommodated simply by changing the function $v(\cdot )$ defined in (9) that maps $q$ into the value of the transfer from buyer to seller. Hence, sometimes in what follows we use a generic $v(\cdot )$, imposing only natural properties, like $v\left(0\right)=0$ and ${v}^{\prime}\left(q\right)>0$. Also, note that the price level in this formulation is simply $1/q$.

A stationary equilibrium is a list $\u3008{V}_{0},{V}_{1},q\u3009$ satisfying (7)–(9) with $q\in \mathrm{0,}\overline{q}]$, since $q\in \mathrm{0,}\overline{q}]$ means $u\left(q\right)\ge \Delta \ge c\left(q\right)$, and that means voluntary trade. One can characterize the set of equilibria rather completely. As in Section “First-Generation Models,” the asset can be valued for more than its fundamental $\rho /r$ because agents have the *option* to trade it for $q$. Depending on parameters, there may be multiple stationary equilibria, again because the value of liquidity is partly self-referential: if you think others give a low $q$ for the asset then you only give a little; but if you think they give more for it then you give more. There are also dynamic equilibria where $q$ increases or decreases deterministically over time, sunspot equilibria in which it fluctuates randomly, and sometimes continuous-time cycles revolving around steady state. Again, due to liquidity considerations, the value of the asset can be above its fundamental, can display multiplicity, and is subject to excess volatility.

Compared to first-generation models there is more to say about efficiency. Clearly, $q={q}^{\ast}$ is optimal, where ${u}^{\prime}\left({q}^{\ast}\right)={c}^{\prime}\left({q}^{\ast}\right)$. If $\rho =0,$ for example, when $r$ is not too big we can construct ${\theta}^{\ast}\le 1$ such that we get $q={q}^{\ast}$ in equilibrium, while when $r$ is too big $\theta =1$ gets us as close as possible (this is related to general discussions of efficiency in search theory by Mortensen, 1982, and Hosios, 1990). One can also take $\theta $ as given and maximize welfare with respect to $A$. With $q$ exogenous, as in Section “First-Generation Models,” the solution is ${A}^{\ast}=1/2$, but with $q$ endogenous, if $q<\mathrm{}{q}^{\ast}$ at $A=1/2$, then ${A}^{\ast}<1/2$. This captures in a stylized way the notion that monetary policy should balance liquidity provision and control of the price level. Obviously, in a sense, this is an artifact of $a\in \left\{0,1\right\}$, but it also illustrates a serious point: It is not the nominal amount of money that matters, but the distribution, or, as Francis Bacon put it, “Money is like muck, not good except it be spread.”^{12}

In general, there are several forces that impinge on efficiency. For the sake of illustration consider Nash bargaining, which in this application is easier than Kalai. First, there is bargaining power, which can make $q$ too high or too low depending on $\theta $; to neutralize that, let us set $\theta =1/2$. Second, there is market power coming from the threat points in the buyer’s and seller’s surpluses; to neutralize that, set $A=1/2$. Then one can show $q<\mathrm{}{q}^{\ast}$ for $r>0$ but $q\to {q}^{\ast}$ as $r\to 0$. The intuition is compelling: In frictionless economies agents work to acquire purchasing power that can be turned into immediate consumption, and hence work until ${u}^{\prime}\left({q}^{\ast}\right)={c}^{\prime}\left({q}^{\ast}\right)$; in monetary economies, however, they work for assets that provide consumption only in the future when they meet an appropriate counterparty, so they produce less than ${q}^{\ast}$ as long as they discount the future.

Papers using this framework study interactions between money and other assets, the relationship between money and output, currency shortages, maturity and denomination structures, issues in monetary history, and international monetary matters. One can look to the literature for details, but let us highlight some work on banking by Cavalcanti and Wallace (1999a, 1999b). They have a fraction of agents, called bankers, that are monitored in all transactions, while the rest, called nonbankers, are never monitored. All agents can issue notes, which are pieces of paper with their names on them having no coupon, $\rho =0$, but potentially having value in exchange. Notes issued by nonbanks are never accepted in payment, but those issued by banks can be accepted. They then use the setup to compare regimes with inside and outside money (notes and fiat currency). This approach has the virtue of allowing one to concretely discuss the merits of different transaction arrangements.^{13}

Related to second-generation monetary theory are search-based models of middlemen. This is natural since middlemen, like currency, are institutions that facilitate trade in the presence of frictions. Rubinstein and Wolinsky (1987) analyze goods markets with intermediaries that hold inventories restricted, for simplicity, to $a\in \left\{0,1\right\}$. While there are many extensions and variations of this framework, from our perspective Nosal et al. (2017) is worth a nod. In a generalization of the standard Rubinstein-Wolinsky environment, they show that goods markets with intermediaries have a unique equilibrium unless one adds well-understood devices like increasing returns and asset markets with intermediaries can have multiple equilibria and complicated dynamics without such devices. This supports the venerable notion that financial markets are especially susceptible to fragility or volatility.

Another contribution is the search-and-bargaining model of over-the-counter financial markets by Duffie et al. (2005). They have investors that hold assets positions restricted to $a\in \left\{0,1\right\}$—again, a strong assumption, but the model still delivers interesting insights. Their investors experience idiosyncratic shocks to the valuations of assets, so there are gains from trade between low-valuation agents with an asset and high-valuation agents without one. Agents meet bilaterally and bargain over the terms of trade, as in monetary theory, although there are also interesting differences (Trejos & Wright, 2016). Duffie et al. (2005) also take a step further by introducing dealers that buy assets from low-valuation agents and sell them to high-valuation agents. Due to space considerations we cannot dedicate more to this work, which is a vibrant area of current research that illustrates how the general framework has applications well beyond explaining fiat currency.

### Third-Generation Models

If $a\in \mathcal{A}$, for some set $\mathcal{A}$ less restrictive than $\left\{0,1\right\}$, the main complication is that we have to somehow handle the endogenous distribution of assets across agents, $F(a)$. One option is to let $\mathcal{A}=\{0,\mathrm{1...}\widehat{a}\}$, where $\widehat{a}$ may be finite or infinite, and proceed with a combination of analytic and computational methods.^{14} Molico (2006) considers $\mathcal{A}=[\mathrm{0,}\infty )$, and studies the case of fiat currency, where the supply evolves in discrete time according to ${A}_{+1}=\left(1+\pi \right)A$, with subscript $+1$ indicating next period. Here $\pi >0$ is the rate of monetary expansion generated by a lump-sum transfer $T$ each period.

As above, Molico’s agents can be buyers or sellers, depending on who they meet, but they are not constrained by $a\in \left\{0,1\right\}$. Maintaining the commitment and information assumptions precluding credit, plus $\delta =0$, let $q\left({a}_{B},{a}_{S}\right)$ be the quantity of goods and $d\left({a}_{B},{a}_{S}\right)$ the dollars traded when a buyer has ${a}_{B}$ and a seller has ${a}_{S}$, assuming $\left({a}_{B},{a}_{S}\right)$ is observed by both, for simplicity. The value function is

where $\beta =1/\left(1+r\right)$. The first term on the Right hand side (RHS) is the expected value of not trading, the second is the value of buying from a random seller, and the third is the value of selling to a random buyer.

As in second-generation models, there are various options for determining the terms of trade, but Molico uses bargaining with $\theta =1$, subject of course to feasibility, $d\left({a}_{B},a\right)\le {a}_{B}$. Different from those models, there is now an endogenous law of motion for $F\left(a\right)$ induced by agents meeting and trading. A stationary equilibrium is a list $\u3008V,q,d,F\u3009$ satisfying (10), the bargaining equations, and a stationarity condition for $F\left(a\right)$, which is similar in spirit to earlier models but more mathematically and economically complicated. Molico analyzes his model numerically to study the relationship between inflation and price dispersion, to ask what happens when frictions are reduced, and to quantify the welfare effects of inflation. In terms of inflation, the transfer $T>0$ compresses the distribution of real balances, since it raises the price level, and when the value of a dollar falls it hurts those with high $a$ more than those with low $a$. Since those with low $a$ do not buy very much, and those with high $a$ do not sell very much, in equilibrium this compression stimulates economic activity by spreading liquidity around. At the same time, inflation detrimentally reduces real balances, and policy must balance these effects.

Chiu and Molico (2010, 2011) and Molico and Zhang (2006) extend this computational approach in various ways. Chiu and Molico (2014) and Jin and Zhu (2017) go beyond stationary equilibrium to study dynamics after various types of monetary interventions and show how the redistributional impact of policy can have rather interesting effects on output and prices. In Jin and Zhu’s formulation, for example, for some parameters output in a match $q\left({a}_{B},{a}_{S}\right)$ is decreasing and convex in ${a}_{S}$. Hence, a policy that increases dispersion in real balances increases average output across trades, $\mathbb{E}q$. Now, there are other effects, and this is a numerical result about the net effect, but that does not diminish the importance of their findings. One fascinating result is that a monetary injection can increase dispersion in real balances, and hence $\mathbb{E}q$, and that leads to slow increases over time in the average price $\mathbb{E}p$ after the injection, where in any trade $p\left({a}_{B},{a}_{S}\right)=d\left({a}_{B},{a}_{S}\right)/q\left({a}_{B},{a}_{S}\right)$.

The reason is *not* that prices are sticky, since $q$ and $d$ are determined endogenously by bargaining in each and every trade, but rather that the increase in $q$ keeps $p$ from rising too quickly during the transition. In the long run, as the economy returns to stationary equilibrium after an increase in the money supply, $\mathbb{E}p$ increases in proportion to the injection, but in the short run it is mainly output that adjusts. This shows that nominal rigidities are *not* needed to capture the time-series observation that monetary injections affect mainly output in the short run and prices in the long run. That message is important, because those observations are a major component of Keynesians’ defense of their assumptions and policy prescriptions. Thus, it provides a clear example where microfoundations provide a voice of caution about the legitimacy of those assumptions and prescriptions.

A different approach when $\mathcal{A}=[\mathrm{0,}\infty )$ is to harness the distribution $F(a)$. One method, due to Shi (1997), assumes a continuum of households, each with a continuum of members, to get $F\left(a\right)$ degenerate.^{15} This approach has many applications, as discussed in Shi (2006). Another method from Menzio et al. (2013) uses directed search with free entry. Here we focus on the approach in Lagos and Wright (2005), which delivers tractability by combining search-based models like those presented previously with frictionless models. While one might imagine different ways to proceed, the basic setup divides each period into two subperiods: In the first, agents interact in a decentralized market, or DM, like the markets discussed earlier; in the second, they interact in a frictionless centralized market, or CM. The Lagos-Wright framework is widely used in the literature.

The DM good is still denoted $q$, plus there is a different good $x$ that serves as CM numeraire. Also, although this is easily relaxed, to reduce notation we assume $x$ is produced one-for-one using labor $\mathcal{l}$ in the CM, so the real wage is $1$. In the DM, agents can again be buyers or sellers depending on who they meet. In the former case period utility is $U(x)-\mathcal{l}+u\left(q\right)$, and in the latter case it is $U(x)-\mathcal{l}-c\left(q\right)$. The DM value function is like (10), except wherever $\beta {V}_{+1}(\cdot )$ appears on the RHS it is replaced by $W(\cdot )$, since before going to the next DM agents now visit the CM, where $W(\cdot )$ is the value function. It satisfies

where $a$ and $\widehat{a}$ are asset holdings when the CM opens and closes, $\varphi $ is the price of $a$ in terms of $x$, $\rho $ is a dividend, and $T$ is a transfer.

As usual, a special case is fiat currency, where $\rho =0$ and $T<0$ can be used to contract or $T>0$ can be used to expand the money supply. However, the theory prices assets, in general, and not only currency. In any case, assuming interior solutions, after eliminating $\mathcal{l}$, we have

From this the following results are immediate: $x={x}^{\ast}$ where ${U}^{\prime}\left({x}^{\ast}\right)=1$; $W(a)$ is linear with slope $\varphi +\rho $; and $\widehat{a}$ is independent of $a$, implying $F\left(\widehat{a}\right)$ is degenerate at the start of DM trade.^{16}

We now move to the DM, where whether $a$ is a real asset or fiat money makes a slight difference. The reason is that in equilibrium the constraint $d\le {a}_{B}$, which again is simply resource-feasibility in bilateral trade, always binds with $\rho =0$ (this takes a proof, but, intuitively, there is no sense bringing more money to the DM than you will ever spend because fiat currency is not a good savings vehicle). By contrast, with $\rho >0$, this constraint turns out to bind if and only if $\rho A$ is low, meaning there is a scarcity of liquid assets (delivering the result for fiat money as a special case). If liquid assets are not scarce (in a sense made precise later) then $d\le {a}_{B}$ is slack and assets are priced fundamentally: $\varphi ={\varphi}^{\ast}\equiv \beta \rho /\left(1-\beta \right)$. So, let us consider the more interesting case, where liquidity is scare.

For the DM terms of trade we use a generic $v(\cdot )$ function.^{17} Then $v\left({q}_{+1}\right)=({\varphi}_{+1}+\rho )\widehat{a}$, where as the timing indicate, the $\widehat{a}$ taken out of the CM in one period is used in the DM in the next period. This expression is very simple because of the result that $W(\cdot )$ is linear. Moreover,

where ${q}_{+1}$ solves $v\left({q}_{+1}\right)=({\varphi}_{+1}+\rho )\widehat{a}$ and hence depends on the asset position of this agent when he is a buyer, while ${Q}_{+1}$ solves $v\left({Q}_{+1}\right)=\left({\varphi}_{+1}+\rho \right)A$ and depends on the asset position of the buyers to whom this agent sells (as usual, although $\widehat{a}=A$ in equilibrium, we have to let them differ to find equilibrium). Notice (12) is similar in terms of economics to (10) from Molico’s model, but because it is much simpler it yields more analytic results. In particular, into the FOC for $\widehat{a}$, $\varphi =\beta {V}^{\prime}\left(\widehat{a}\right)$, we can insert the derivative of (12) to get^{18}

Given a time path for $\varphi $, this Euler equation describes a path for the demand for $\widehat{a}$. To reduce notation, let $\lambda \left(q\right)\equiv {u}^{\prime}\left(q\right)/{v}^{\prime}\left(q\right)-1$ denote what is usually called the liquidity premium (in fact, it is the Lagrangian multiplier on the constraint $d\le \widehat{a}$) and write

One can now eliminate ${q}_{+1}$ using $v\left({q}_{+1}\right)=({\varphi}_{+1}+\rho )\widehat{a}$ and $\widehat{a}={A}_{+1}$ to get a difference equation in $\varphi $. Equivalently, one can eliminate $\varphi $ to get a difference equation in $q$,

then recover asset prices from $\varphi =v\left(q\right)/A-\rho $.

An equilibrium is given by paths for $\u3008q,\varphi \u3009$ satisfying these conditions. For the sake of illustration, let us fix the supply $A$ and consider a stationary equilibrium, which is a steady state of the system, or a solution to

Stationary equilibrium is unique (Gu & Wright, 2016). It involves $q<\mathrm{}{q}^{\ast}$ and $\varphi >\mathrm{}{\varphi}^{\ast}\equiv \rho /r$ as long as $\alpha \sigma \lambda \left(q\right)>0$, which means liquidity considerations are operative, and obtains when $\rho A<\mathrm{}\beta v({q}^{\ast})$. So when assets are not too abundant they are priced above their fundamental value. Moving beyond steady state, even with parameters fixed there are dynamic equilibria where $q$ and $\varphi $ vary over time—both cyclic equilibria where they oscillate deterministically and sunspot equilibria where they fluctuate stochastically (Ferraris & Watanabe, 2011; Rocheteau & Wright, 2013). As in the other models presented earlier, to some extent liquidity is self-referential, potentially leading to excess volatility as a self-fulfilling prophecy. This is an important message, we think, from the New Monetarist approach, as it transcends details of the specification.

With fiat money it is natural to let $A$ grow over time at rate $\pi >0$ or $\pi <0$, since policymakers can obviously issue or retire currency as part of conventional monetary policy, while it is less conventional for them to issue or retire Lucas trees—although one could study that, too. When ${A}_{+1}=\left(1+\pi \right)A$ with $\pi $ time-invariant, it is standard to denote real balances by $z=\varphi A$ and rewrite (13) as

where ${q}_{+1}={v}^{-1}\left({z}_{+1}\right)$. This difference equation has a unique monetary steady state $q>0$ as well as a nonmonetary steady state $q=0$. And as with $\rho >0$, even with parameters fixed there are equilibria where $q$ and $z$ vary over time as self-fulfilling prophecies.

In steady state, $\pi $ is the inflation rate because, for $z$ to stay constant, $\varphi $ must fall as $A$ rises (a version of the Quantity Equation). Now suppose we ask agents how much they would require in terms of $x$ in the next CM to give up $1$ unit of $x$ in this CM. They would answer $1+r=1/\beta $, thus defining the real interest rate on an illiquid asset. Similarly, suppose we ask how much money they would require in the next CM to give up a dollar in this CM. They would answer $1+\iota $, thus defining the nominal interest rate on an illiquid asset. It is easy to see that $1+\iota =\left(1+r\right)\left(1+\pi \right)$ (a no-arbitrage condition known as the Fisher Equation). Hence (14) reduces to

an elegant condition determining $q$, and hence $z=v\left(q\right)$, and the price level $1/\varphi =M/v\left(q\right)$ (another version of the Quantity Equation). Note how these depend on the nominal rate $\iota $, the search and matching parameters $\alpha $ and $\sigma $, and the liquidity premium $\lambda \left(q\right)={u}^{\prime}\left(q\right)/{v}^{\prime}\left(q\right)-1$, which itself depends on the mechanism $v\left(q\right)$. All of these matter qualitatively and quantitatively.

As we did in Section “Second-Generation Models,” consider efficiency under Nash bargaining where $\theta $ is buyers’ bargaining power. Then $q={q}^{\ast}$ under two conditions. As in most models, one is the Friedman Rule $\iota =0$, which is necessary to eliminate the wedge in money demand on the Left hand side (LHS) of (15).^{19} The other is $\theta =1$, as otherwise agents bring too little cash to the DM, because when they try to spend it sellers capture part of the surplus. For any $\theta \in (\mathrm{0,}1)$ we have $q<\mathrm{}{q}^{\ast}$ for all $\iota \ge 0$, and one can show $\partial q/\partial \iota <0$. This may suggest we should set $\iota <0$, but as is apparent from (15) there is no equilibrium with $\iota <0$, a New Monetarist version of the zero-lower-bound problem. However, this result depends on the mechanism—with Kalai bargaining, e.g., $\iota =0$ implies $q={q}^{\ast}$ for any $\theta $, while Hu, Kennan, and Wallace (2009) design a mechanism that delivers $q={q}^{\ast}$ even at $\iota >0$ as long as it is not too big. Moreover, given any $v(\cdot )$, the distortion induced by $\iota >0$ is greater when the search and matching frictions are more severe ( $\alpha $ and $\sigma $ are lower). These details of the micro market structure matter.

To see how they matter quantitatively, Lagos and Wright (2005) calibrate the model to match the relationship between $z$ and $\iota $ in the data, and use it to calculate the welfare cost of inflation. A great many papers previously performed similar exercises in reduced-form (CIA or MUF) models, and while there are differences across studies, as a rough average, the finding was that households would be willing to give up about $0.5\%$ of consumption to eliminate $10\%$ annual inflation. In the model of Lagos and Wright (2005), the finding is closer to $5.0\%$, an order of magnitude higher. The exact number depends on details and honing measurement along these lines is an ongoing research program, but this illustrates sharply how taking microfoundations more seriously can make a big difference for quantitative policy issues and not only for theory.

### Applications and Extensions

There are many uses for the benchmark model just presented. In addition to fiat currency, we have discussed real assets in fixed supply as first analyzed in the New Monetarist framework by Geromichalos, Licari, and Suárez-Lledó (2007). One can alternatively consider neoclassical capital, which is similar, except (speaking heuristically) the supply curve is horizontal instead of vertical, so liquidity considerations are manifested not by $\varphi >\mathrm{}{\varphi}^{\ast}$ but by $k>\mathrm{}{k}^{\ast}$ (Lagos & Rocheteau, 2008). Housing is similar when it conveys liquidity via home-equity lending, except supply need not be horizontal or vertical (He, Wright, & Zhu, 2015). Bonds that convey liquidity are also similar (Rocheteau, Wright, & Xiao, 2018; Williamson 2012). We can also incorporate multiple assets in various combinations, and importantly, there is still a role for fiat currency even with liquid real assets, as long as they are not too abundant.^{20}

In Lagos and Rocheteau (2005) and Rocheteau and Wright (2005), agents know if they will be buyers or sellers in the next DM before leaving the current CM, which is obviously useful information when they decide how much money to take. This can be due to permanently heterogeneous agents—some are always buyers in the DM while others are always sellers—or due to homogeneous agents getting shocks to preferences and technologies each period in the CM making them buyers or sellers in the next DM. This provides a natural setting in which to introduce price posting and directed search, instead of bargaining and random search. Moreover, having dedicated buyers and sellers in the DM makes it a two-sided market, which allows one to incorporate a general meeting technology that depends on market tightness (the buyer/seller ratio), and to add entry by one side to endogenize tightness, similar to Pissarides’s (2000) generalization of Diamond’s (1982) one-sided market.

Several papers consider situations where capital is a factor of production but does not compete with other assets as a medium of exchange. Applications in growth theory and in real business cycle theory include Aruoba, Waller, and Wright (2011) and Aruoba (2011), respectively. Another application studies optimal monetary and fiscal policy from a public finance perspective, as in Aruoba and Chugh (2008), which is especially interesting because is overturns conventional wisdom from the reduced-form literature. In particular, under certain conditions, CIA or MUF models predict that $\iota =0$ is optimal even when other taxes are distortionary, but Aruoba and Chugh show this is not necessarily true even under similar conditions in their model. Moreover, the results are quantitatively relevant. This again shows how microfoundations can make a qualitative and quantitative difference.

Other work abstracts from capital and focuses on employment, including versions with frictional labor markets, such as Berentsen, Menzio, and Wright (2011) and several follow-up papers that analyze the Phillips curve (the relationship between unemployment and inflation). Among other findings, at anything other than very high frequencies, the work shows the Phillips curve in U.S. data slopes upward, opposite to what Keynesians contend. One version of this that should be more appreciated is shown in Figure 1, based on empirical work discussed in Haug and King (2014). This shows inflation and unemployment data filtered to eliminate higher frequencies, with a phase shift of 13 quarters, as dictated by fit. In terms of the Old Monetarist notion of long and variable lags between monetary policy and outcomes, the implication of this evidence is that the lag between inflation and unemployment may be long, but in fact it is not variable. In any case, the inflation-unemployement relationship clearly looks positive, casting reasonable doubt on the Keynesian Phillips curve and policy prescription based on it.

The framework can also explain—not only assume—sticky prices, defined as some firms leaving their nominal prices the same when the aggregate price level rises. Head, Liu, Menzio, and Wright (2012) model the DM after Burdett and Judd (1983), where first prices are posted, then each buyer sees a random number of them and of course picks the lowest price he sees. Burdett-Judd markets display price dispersion: there is density $f\left(p\right)$ on a nondegenerate interval $[\underset{\_}{p},\overline{p}]$ such that profit is the same for all $p\in \underset{\_}{p},\overline{p}]$. Intuitively, low- $p$ sellers earn less per unit, but make it up on the volume, given a buyer seeing multiple prices picks the lowest. When there is inflation, many sellers can stick to their nominal $p$’s for a while, even though it means their real prices fall, because their volume increases, and the equilibrium $f\left(p\right)$ is such that these effects offset.^{21} This provides a reasonable microfoundation for sticky nominal prices, which are too often imposed in models where they imply potentially huge gains from trade left sitting on the table. Furthermore, it is worth emphasizing that stickiness emerges endogenously, the policy prescriptions are not at all the same as those coming from models with exogenous stickiness.

Aruoba and Schorfheide (2011) estimate a model integrating New Monetarist and Keynesian features. They compare the importance of exogenous sticky price distortions, which imply $\pi =0$ is optimal, and the effects emphasized here, which imply $\iota =0$ is optimal. Estimating the model under various specifications, $\iota =0$, or at least $\pi <0$, turns out to be optimal. Craig and Rocheteau (2008) reach similar conclusions in a menu-cost version of our benchmark model. On that, some have argued that a little inflation is good because it chips away at the market power of sellers when there are menu costs, but Craig and Rocheteau show that for reasonable parameters this effect is dwarfed by the wedge making $\iota >0$ suboptimal. On net, deflation, not inflation, is best in their setup. So a little inflation may be good, for some reasons, but according to these findings not for menu cost reasons.^{22}

There is also work using versions of the above model to study interactions between money and credit, such as Lotz and Zhang (2013) or Gu et al. (2016). To pick up on just one theme, many papers following Kiyotaki and Moore (1997) interpret intertemporal trade as *collateralized credit*, which in this context means that in the DM a buyer can pledge to pay a seller in numeraire in the next CM any amount $D$ up to a fraction $\chi $ of the value of his portfolio. The idea is that, even with limited commitment, an agent may choose to honor his debts since otherwise he will have part of his asset portfolio seized, although in general he may be able to abscond with a fraction $1-\chi $. Clearly, he prefers to honor his debt obligation if and only if $D\le \overline{D}\equiv \chi \left({\varphi}_{+1}+\rho \right)\widehat{a}$, where again $\widehat{a}$ is the asset position chosen in the CM this period and brought into the DM next period. Hence, $\overline{D}$ defines an endogenous DM debt limit.

When $\chi =1$, a buyer can promise the full value $\left(\varphi +\rho \right)\widehat{a}$. But the buyer may as well just hand over $\widehat{a}$ in the DM, replacing deferred with immediate settlement. As discussed by Lagos (2011), this logic says that secured credit à la Kiyotaki-Moore is equivalent to assets serving as media of exchange à la Kiyotaki-Wright—they provide different narratives, but lead to the same equations. What about $\chi <1$? It’s still the same if one merely deletes the story about renegers running away with part of their portfolio and replaces it with a story about agents only able to bring part of their portfolio to the DM. This gives a different narrative but the same equations. One can also be more sophisticated. According to Li et al. (2012), when assets can be counterfeited at a cost, in equilibrium sellers that cannot detect low-quality versions accept only a fraction $\chi $ of a buyer’s holdings. Heuristically, consider fiat money, and suppose it costs $10$ dollars to counterfeit $20$ dollars; a buyer with $20$ can then credibly convince sellers to accept up to $10$ since, after all, who would pay $10$ to pass less than $10$ in bad bills? So, in principle cash can involve $\chi <1$.^{23}

Different assets can have different pledgeability (or acceptability) if they are more or less costly to counterfeit. Also, when pledgeability (or acceptability) is endogenous, variables like $\chi $ (or $\tau $) are not invariant to policy changes—an application of the Lucas critique that is as far-reaching as it is ignored by reduced-form practitioners. Further, endogenous pledgeability (or acceptability) can generate multiple equilibria for natural reasons, and it turns out that the answers to policy questions can depend delicately on the equilibrium one selects. Lester et al. (2012) indicate that where sellers can choose to become informed about the quality of assets at a cost and more sellers pay the cost, the DM acceptability of an asset increases. This raises its CM price, which means more sellers want to invest in information. For this and other reasons, information-based models are potentially quite important in this literature.^{24}

### Banking

Similar to asking what money is and does, one can ask questions about banks. We first discuss Gu et al. (2013a), where banking emerges endogenously as an institution to mitigate commitment problems. This model has two types of infinitely lived agents, $j=1,2$, where the number of each type is the same, but it does not matter at this point if that number is large or small. Each period in discrete time these agents want to trade with probability $\gamma $, while with probability $1-\gamma $ there are no gains from trade and hence no interaction that period. When there are gains from trade, type 1 wants to consume good $x$ and can produce good $y$, while type $2$ wants to consume $y$ and can produce $x$. This proceeds in two stages. Production must take place in stage $1$. Type $1$ also consumes at stage 1, but type $2$ does not consume until stage $2$. Type $1$ has a technology to store or otherwise invest good $y$ across stages, for a gross return $\rho $, while type $2$ does not (what matters is only that type $2$ has an inferior storage or investment opportunity). For simplicity, suppose goods are only storable across stages and not across periods.

To get type $2$ to give $x$, type $1$ can produce $y$, invest it, and promise to deliver $\rho y$ in the second subperiod. Type $1$’s period payoff from this arrangement is ${U}^{1}\left(x,y\right)$, while type $2$’s is ${U}^{2}\left(\rho y,x\right)$, where both are increasing (decreasing) in the first (second) argument, plus satisfy the usual curvature assumptions and the normalization ${U}^{j}\left(\mathrm{0,0}\right)=0$. Agents discount across periods by $\beta \in \left(0,1\right)$. As type $1$ consumes before delivering payment to type $2$ any trade must use credit. The way the environment is designed, one can interpret $y$ as collateral: type $2$ sees type $1$’s output, and hence is willing to give him $x$, because type $1$ can credibly pledge to deliver $\rho y$ next subperiod, even without commitment. This is because his production cost is sunk at stage $2$, and he has no opportunity cost, since he does not consume his own output.

We do not take a stand here on implementing a particular allocation, although it is easy enough to adopt a mechanism (e.g. bargaining), and solve for the equilibrium outcome; instead let us here simply ask what is feasible. The answer is, any $\left(x,y\right)$ as long as ${U}^{1}\left(x,y\right)\ge 0$ and ${U}^{2}\left(\rho y,x\right)\ge 0$. But now, to give the limited commitment friction some bite, let us assume that at stage $2$ type 1 can renege on delivery of $\rho y$ to obtain a payoff $\lambda $ per unit, for a total period payoff ${U}^{1}\left(x,y\right)+\lambda \rho y$ (similar to the “cash-diversion” models of Biais, Mariotti, Plantin, & Rochet, 2007, or DeMarzo & Fishman, 2007). If $\lambda =0$ type $1$’s output/investment works well as collateral; if $\lambda >0$ it works less well and the allocation in general is constrained. Assume ${U}^{1}\left(x,y\right)+\lambda \rho y\le {U}^{1}\left(x\mathrm{,0}\right)$ for all $x$ and $y$, so that ex ante it is never efficient for type $1$ agents to produce and invest for themselves, but still they might consider acting opportunistically ex post. If a type $1$ agent does so, he gets caught with probability $\mu $, the same random monitoring technology used in the monetary models, and if he is caught the punishment going forward is permanent autarky.

Now, in addition to ${U}^{1}\left(x,y\right)\ge 0$ and ${U}^{2}\left(\rho y,x\right)\ge 0$, we have to be sure type $1$ does not want to opportunistically default. Let us restrict attention to stationary allocations for simplicity (see Gu et al., 2013b for the general case). Then type $1$’s lifetime payoff is ${V}^{1}=\gamma {U}^{1}\left(x,y\right)/\left(1-\beta \right)$, since trade takes place with probability $\gamma $ each period, in which case he gets ${U}^{1}\left(x,y\right)$. Thus, he will not renege if $\left(x,y\right)$ satisfies the *repayment constraint*,

Letting $\psi =\left(1-\beta \right)\lambda \rho /\beta \gamma \mu $, we reduce this to

Notice (16) makes ${U}^{1}\left(x,y\right)\ge 0$ redundant. Hence, (stationary) feasible allocations are given by pairs $\left(x,y\right)$ satisfying ${U}^{2}\left(\rho y,x\right)\ge 0$ and (16).

This is shown in Figure 2, where in the left (right) panel the constraint is fairly loose (tight). It is obvious from (16) that the constraint is less tight when type $1$ is relatively patient (high $\beta $), more visible (high $\mu $), more connected to the system (high $\gamma $), or less tempted to divert resources (low $\lambda $). It is less obvious but still true that (16) is less tight when he has better investment opportunities (high $\rho $). All these characteristics combine to make type $1$ more trustworthy, which implies he has a bigger debt limit, in the sense that he can credibly promise a bigger payment at stage $2$, and that allows him to get more $x$ at stage $1$. While this does not yet provide the foundation for a theory of banking, it is still interesting—basically, it is a version of Kehoe and Levine’s (1993) model of credit with limited commitment, differing only in details concerning timing and the monitoring technology.^{25}

To think about banks, suppose now that there are two groups, called $a$ and $b$, each qualitatively like the baseline model, but potentially different in parameters $\left(\beta ,\gamma ,\mu ,\lambda ,\rho \right)$. To make a stark point, suppose agents do not like to consume the $x$ or $y$ produced by the other, so there is no motive for trade absent strategic considerations. However, type $1$ from either group can invest the output $y$ from both groups, and can divert the return $\rho y$ for an opportunistic payoff $\lambda \rho y$. Also, suppose the combination of parameters is such that the constraint (16) is tight in group $a$ and loose in group $b$, making type $1$ in group $b$ more trustworthy than those in group $a$. Then, even without working through the equations, it should be clear that the following arrangement is useful: type $1$ in group $a$ produces and deposits his output with type $1$ in group $b$; then type $1$ in group $b$ invests it and promises to deliver the proceeds to type $2$ in group $a$ at stage $2$, in addition to promising to deliver something to type $2$ in his own group $b$; and based on this promise type $2$ in group $a$ produces for type $1$ in group $a$.

This arrangement is useful because it slackens the constraint in group $a$, and while of course it tightens the constraint in group $b$, the latter was less tight. Also note that in general type $1$ in group $a$ may just give some, not all, output to type $1$ in group $b$. The bigger point is that this arrangement resembles banking, with type $1$ in group $b$ serving as bankers, in several senses: they accept deposits; they are delegated to make investments on behalf of depositors; and most significantly, their liabilities—claims on said deposits—facilitate transactions between the depositors and third parties (i.e., between types $1$ and $2$ in group $a$). This works because type $1$ in group $a$ does not have to rely on his own reputation as a trustworthy trading partner, but can bank on the good name of type $1$ in group $b$.

Moreover, a natural way to implement the scheme is this: type $1$ in group $a$ gives deposits to his banker, type $1$ in group $b$, in exchange for a note; type $1$ in group $a$ then trades the note to type $2$ in group $a$; and type $2$ in group $a$ carries the note to stage $2$ when he redeems it at the bank for consumption. Thus, bank notes serve as inside money, circulating in trade between types $1$ and $2$ in group $a$ (and presumably could circulate longer if we added more stages and types). It is the liabilities of the more patient, visible, or connected, and less tempted, that tend to serve in this capacity, as they are the more trustworthy, and hence their notes convey liquidity. It also helps if they have a high return $\rho $, but notice that other parameters can compensate for a low $\rho $—hence, in this model, it can be wise to sacrifice return for liquidity.

Huang (2016) extends the framework by making the probability $\mu $ endogenous. In her setup there are economies of scale in monitoring, so even with ex ante homogeneous agents it is efficient to choose a fraction of them to act as bankers. However, other things equal, fewer bankers means more deposits per bank, and that implies a greater incentive for them to act opportunistically. Efficiency implies finding the right trade-off between fewer large banks and more small banks. Huang shows how the optimal number and size depends on parameters and uses this to interpret trends in U.S. banking data. Also, the theory makes clear why bankers need big rewards—what keeps them from acting opportunistically is the risk of losing these rewards. She also points to a tension between free entry and the positive profits required for incentives, and shows equilibrium is only efficient if we limit entry by taxation or a quota on bank charters. This analysis provides a fresh perspective on topical issues. See Donaldson, Piacentino, and Thakor (2016) for a related banking model with different applications.

In a different strand of research, Berentsen, Camera, and Waller (2007) integrate banking into a standard New Monetarist alternating-market model of money. Buyers in their model receive preference shocks after the CM has closed, independent across agents, indicating whether or not they will be active (i.e., want to consume) in the next DM. At that point they cannot alter their labor supply $\mathcal{l}$ to adjust their money balances, but there is a financial market, call it the FM, where they can potentially trade. In the original specification the FM has banks where active buyers can borrow to top up their cash and inactive buyers can deposit money they do not need. It is legitimate to think of this as banking, and not a pure credit market, because buyers lack commitment and are to some degree anonymous, so they cannot borrow and lend among themselves, but can lend to (deposit with) bankers because banks can be trusted to honor deposit contracts, and can borrow from bankers because banks have a technology to enforce repayment.

Is this any better than just saying borrowers cannot borrow directly from lenders, but have to use banks, something we criticized above? We think so, because the lack of commitment and enforcement is not something “tacked on” in an ad hoc fashion—it is an integral part of the environment, given that it is in fact the reason money is essential in the first place. Here banks are essential in intermediating between those with too much liquidity and those with too little. The ability to earn interest on deposits when they are not needed increases the demand for money in the CM, and banks can pay interest on deposits because active buyers are willing to pay interest on loans. This is related to the liquidity insurance in Diamond and Dybvig (1983), except here, more realistically, banks deal in money and not real goods. Further, we can reinterpret buyers as not carrying any money out of the CM—all the cash can be in banks, with inactive buyers leaving it there and active buyers taking it out after the shocks are realized, making this arrangement look even more like liquidity insurance.^{26}

In fact, Berentsen et al. (2007) do not only assume agents will honor their debts, they also make this self-enforcing, as in Kehoe-Levine models. Still they assume the FM is perfectly competitive, so the deposit rate equals the loan rate, but that can be generalized in many ways. In Chiu and Meh (2011) banking entails a transaction cost, and agents have different needs for liquidity depending on who they meet, where now one should think of the FM and DM operating simultaneously so that agents can use banking while actively trading. In this setting, only buyers with particularly good (bad) trading opportunities borrow from (deposit in) the bank, and there is a spread between the deposit and loan rates reflecting the transaction cost. In a further extension, Chiu, Meh, and Wright (2017) show how banking can be useful for strategic reasons: While bargaining with a seller, a buyer has an outside option to walk away and take his money to the next CM, but that is not much of a threat because, again, currency is not a good savings vehicle. If banks are open during the negotiations, however, the buyer has a better option: depositing the money at interest. This gets him a better deal with the seller, reducing the holdup problem, which gives agents the incentive to bring more real balances to the DM and enhances efficiency.^{27}

### Conclusion

The approach taken in this article can be summarized as an attempt to take microfoundations seriously. Alternatively, it may be described as an attempt to analyze various notions of liquidity in one more-or-less unified framework. This sample of the literature will hopefully stimulate readers to look in more depth at the ongoing research in the area.

### Acknowledgments

For comments we thank David Andolfatto, Chao He, Alberto Trejos and Liang Wang. Wright acknowledges research support from the Ray Zemon Chair in Liquid Assets at the Wisconsin School of Business for research support. The usual disclaimers apply.

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### Notes

1. Other recent surveys of this work include Rocheteau and Nosal (2017) and Lagos, Rocheteau, and Wright (2017). We borrow heavily from these sources, but also add new material. For other related surveys or discussions, see Shi (2006), Lagos (2008), Wallace (2001, 2010), and Williamson and Wright (2010a, 2010b).

2. Quotations to this effect can be found going back much further, but it is hard to top Menger (1892): “that every economic unit in a nation should be ready to exchange his good for little metal disks apparently useless as such, or for documents representing the latter, is a procedure so opposed to the ordinary course of things . . . [that it is] downright ‘mysterious’.”

3. Earlier research surveyed by Ostroy and Starr (1990) tried to analyze some of the issues that concern New Monetarists using classical GE theory augmented in various ways. The attempt was admirable but the problems were far from solved.

4. We go into detail on specific models in what follows; at this point the goal is to just provide a flavor of what we do by mentioning a few well-known papers.

5. By analogy, the GE theory in Debreu (1959) is a model

*with*households and firms and not*of*households and firms—they are primitive entities in his GE approach. Obviously that is fine for some purposes, but perhaps not for others, like studying family formation or industrial organization.6. These modeling ingredients all have a purpose. An infinite horizon is crucial for understanding currency, liquid assets more generally, and some types of credit. A large number of agents makes it harder to sustain credit, as emphasized by Araujo (2004). Having nonstorable consumption precludes commodity money, which is studied in other papers, to focus here on other issues. Random matching, while a simplistic view of the world, is a convenient way to capture the realistic notion that finding trading partners is not always easy, although it is worth noting that Corbae, Temzilides, and Wright (2003) and others have redone these models with directed instead of random search and the basic results survive.

7. An example of specialization used by, e.g., Aiyagari and Wallace (1991) is this: There are $N$ types of agents and $N$ goods, where type $j$ consumes good $j$ but produces good $j+1$ modulo $N$. Thus, $N=2$ implies $\delta =1/2$ and $\sigma =0$, while $N>2$ implies $\delta =0$ and $\sigma =1/N$. The easiest case that makes money useful is $N=3$ (in Wicksell 1898/1965, there are Swedes, Norwegians, and Danes trading wheat, timber, and fish). Also, we reiterate that we usually call the agents consumers but similar considerations apply to firms, and instead of saying individual $j$ consumes good $j$, we can say firm $i$ wants input $i$; this generates the same motives for, and difficulties with, trade.

8. This is understood as credit because agents sometimes produce while receiving nothing by way of quid pro quo, with the understanding, or maybe the promise, that people will do the same for them in the future. So while it is not bilateral credit, we still call it credit.

9. Money is inferior to credit because: (1) potential sellers may have $a=1$; (2) potential buyers may have $a=0$; and in either case they cannot trade. Now (1) may seem an artifact of the restriction $a\in \left\{0,1\right\}$ , but even without that, it can seller with (lots of) money may still not produce (very much). It is easier to see that (2) is robust, since agents can always run out of cash. In some specifications with directed instead of random search, this is less of a problem, and money can be just as good as credit, but it can never be better.

10. This eases the presentation because $\delta >0$ implies we have to determine $q$ in both barter and monetary trade, and with $\delta =0$ we can focus on the latter. But it is important to say the results go through with to deflect the critique that we are forcing agents to use cash here—in fact, as in first-generation models, what makes monetary exchange useful is $\sigma >0$, not $\delta =0$.

11. While early work on this model used Nash bargaining, the analysis with Kalai’s solution is similar and simpler, as in some other models with liquidity considerations (Aruoba, Rocheteau, & Waller, 2007). The simplest case is actually $\theta =1$, which makes Kalai and Nash the same, but we think the implications of different $\theta $ are too interesting to ignore.

12. Saying that the nominal amount of money does not matter simply means that the number written on fiat notes is irrelevant. This is a stark version of classical neutrality that is only violated in silly models, e.g., ones that assume sellers must produce $q$ in proportion to the number written on a buyer’s note because prices are sticky.

13. Related studies of banking in second-generation models include Cavalcanti and Wallace (1999a, 1999b) and He, Huang, and Wright (2005). In fact He et al. (2005) also derive insights about banking in a first-generation model, as does Lester (2009).

14. In this vein there is a body of technical work spawned by Green and Zhou (1998). Examples of different but related approaches include Camera and Corbae (1999), Berentsen (2002), Zhu (2003) and Deviatov and Wallace (2009).

15. His decision-making units are families whose members search randomly, but at the end of each trading round they return home and share the proceeds. Hence, each family starts the next trading round with the same $a$.

16. While these results follow from quasilinear utility here, Wong (2012) shows they go through for any CM utility function over $\left(x,1-\mathcal{l}\right)$ that is homogeneous of degree $1$ (see also Rocheteau, Rupert, Shell, & Wright, 2008, for a different way to generalize these results).

17. Again this encompasses a variety of bargaining solutions, including Nash and Kalai. One can also use Walrasian price taking by letting $v\left(q\right)=Pq$, where $P$ is the price of DM goods in terms of numeraire, which individuals take as given, but it satisfies $P={c}^{\prime}\left(q\right)$ in equilibrium. To motivate price taking, Rocheteau and Wright (2005) describe DM meetings in terms of large groups, as opposed to bilateral trade—intuitively, it may help to think about the Lucas-Prescott (1974) job search model, as opposed to Mortensen-Pissarides (1994).

18. In case it is not obvious, the second line uses ${{W}^{\prime}}_{+1}\left(\widehat{a}\right)=\left({\varphi}_{+1}+\rho \right)$ and $\partial {q}_{+1}/\partial \widehat{a}=\left({\varphi}_{+1}+\rho \right)/{v}^{\prime}\left({q}_{+1}\right)$, where the latter comes from $v\left({q}_{+1}\right)=({\varphi}_{+1}+\rho )\widehat{a}$.

19. In steady state, the money growth or inflation rate that implements the Friedman Rule is $\pi =\beta -1<0$. Andolfatto (2010, 2013) discusses how this raises issues if agents can avoid paying the taxes necessary to contract the money supply, and how paying interest on currency can achieve the same results as $\pi =\beta -1$.

20. While some of these papers concentrate on steady state, in principle they can all display complicated dynamic equilibria as a self-fulfilling prophecy. The models are also applied to various issues in financial economics, including the credit card debt puzzle, on-the-run phenomena, the equity premium and risk free rate puzzles, home bias, the term structure and housing bubbles.

21. To be clear, it is an endogenous outcome that profits are constant for all $p\in \underset{\_}{p},\overline{p}]$; it is not an assumption.

22. Another reason inflation may be beneficial involves standard second-best reasoning—e.g., when capital taxation makes $k$ too low, inflation can mitigate this if money and capital are substitutes in the payment process (Venkateswaran & Wright, 2013). Yet another is that inflation can improve the mix of sellers in some models (Bethune et al. 2018).

23. In fact Li et al. (2013) use a fixed, not proportional, cost of counterfeiting, but see Rocheteau et al. (2018). A related approach in Lester et al. (2012) assumes that counterfeiting or otherwise misrepresenting the value of one’s assets has no cost; this means uninformed sellers will not accept any of them, thereby endogenizing acceptability, similar to the way Li et al. endogenize pledgeability.

24. Also, while counterfeiting is a simple story, the general messages apply to many situations with information frictions, and the recognizability of assets has been recognized as a crucial property at least since Law (1705). As Alchian (1977) put it “ It is not the absence of a double coincidence of wants, nor of the costs of searching out the market of potential buyers and sellers of various goods, nor of record keeping, but the costliness of information about the attributes of goods [or assets] available for exchange that induces the use of money.” That’s probably going to far, as various frictions may all be relevant, but it speaks to the potential importance of information.

25. Gu et al. (2013b) pursue this model of credit by studying dynamic equilibria, and show how credit markets can be prone to instability in the form of cyclic and sunspot equilibria. See Bethune, Choi, & Wright (2018), Bethune, Hu, and Rocheteau (2018), and Carpella and Williamson (2015) for additional extensions and applications.

26. This depends on there being no search frictions in the FM, so it does not matter who holds the money, buyers or bankers. If there are search frictions in the FM, buyers choose to hold money because they may not be able to find a bank when they need liquidity. Papers following up on this idea included Geromichalos and Herrenbrueck (2016), Mattesini and Nosal (2016), Lagos and Zhang (2018), and Han (2014), where agents try to rebalance their portfolio in OTC markets. In some of these models, agents do not lend to and borrow from banks but trade assets with each other—e.g., active buyers try to swap fewer liquid assets for cash while inactive buyers try to do the opposite. This is in the spirit of Duffie, Gârleanu, and Pedersen (2005), but we think the approach is more detailed and more realistic in terms of liquidity needs.

27. A different model of banks in the alternating-market monetary model is contained in He, Huang, and Wright (2008). There, cash is risky—it can be lost or stolen—while bank deposits are relatively safe. Hence, buyers may prefer to use deposit claims as payment instruments in the DM, even if deposits bear negative interest (i.e., even if there is a fee for using them, like American Express Travelers’ Checks; see also Sanches & Williamson, 2010). Other related applications include Li (2011), Li and Li (2013), and He et al. (2015).