Economic Dynamics Newsletter

Ricardo Lagos on Liquidity and the Search Theory of Money

1. Introduction

There is no ambiguity among economists regarding what it means for an asset to be risky, or for markets to be complete. There seems to be, however, no definition of “liquidity” that is generally agreed upon. The notion of liquidity is sometimes used as an attribute of an asset, e.g., U.S. Treasuries are considered to be more liquid than equity shares. The notion of liquidity is often also used to describe the state or nature of a market, e.g., the market for municipal bonds is commonly regarded as illiquid. Sometimes researchers describe economic agents as being “liquidity constrained,” to mean that the agents face binding borrowing constraints. In what follows, I will focus on the first two notions of liquidity.In financial economics, the liquidity of assets and markets is associated with the cost of trading. An asset is considered to be liquid if trading it entails relatively low transaction costs, e.g., transaction taxes, brokerage fees, or bid-ask spreads. A market is regarded as liquid if individual traders can find a counterpart for trade relatively quickly, and if the out-of-pocket costs involved in consummating trades are relatively small. There is a vast literature in finance that uses liquidity differences measured by differences in transaction costs to rationalize various asset-pricing puzzles, i.e., asset-return differentials that seem too large to be accounted for by the differences in the stochastic properties of the flows of payments that the assets represent. (For an extensive list of references, see Amihud et al., 2005.)

In monetary economics, “liquidity” refers to the degree to which an asset is useful as a medium of exchange. An asset is illiquid if it cannot be used as means of payment in quid pro quo trades. In particular, the Search Theory of Money–by which I mean the class of models that use search theory to provide micro foundations for monetary exchange that was pioneered by Kiyotaki and Wright (1989)–is built around the premise that assets can be valued not only for the intrinsic value of the real payoffs they represent, but also for their usefulness in the mechanism of exchange. This approach has deepened our understanding of the nature of monetary exchange by making explicit the frictions–e.g., specialization patterns, the configuration of the decentralized exchange mechanism, information structure, and so on–that make monetary exchange an equilibrium. In other words, this approach has proven useful for explaining the grandfather of all asset pricing puzzles: the existence of fiat money; an asset that is a formal claim to nothing yet sells at a positive price. Despite the evident conceptual connection and the potential for cross-fertilization, the Search Theory of Money and the mainstream asset-pricing literature have managed to stay disconnected for many years.

In the sections that follow, I will review some of my recent research and an ongoing research agenda, parts of which I have pursued, or am currently pursuing, with Guillaume Rocheteau, Pierre-Olivier Weill, and Randy Wright. In Sections 2-5 I will survey several papers that represent an effort to develop models in which, much as in the Search Theory of Money, financial assets are valued not only as claims to streams of consumption goods but also for their usefulness in the mechanism of exchange–i.e., for their moneyness, or exchange liquidity. In Section 6 I focus on the notion of transaction liquidity associated with transaction costs that is used in the financial literature, and review two papers that represent an effort to trace this notion of liquidity back to the microstructure of the market in which the assets are traded.

2. From indivisibility and upper bounds to Euler equations

The approach to monetary economics that started with Kiyotaki and Wright (1989) consists of using search theory to provide an explicit model of the type of decentralized trading activity that can give rise to a meaningful role for a medium of exchange. This literature made significant progress along several dimensions, but for a number of years, the depth of the analysis and the breadth of its fundamental insights were limited by stark modeling restrictions on asset holdings. Specifically, in order to keep the endogenous distribution of money holdings manageable, agents were typically restricted to hold either 0 units or 1 unit of an indivisible asset. These restrictions meant that agents never faced a standard portfolio problem leading to standard Euler equations, namely, the basic building block of any modern theory that–much like search-theoretic models–seeks to explain asset prices and asset demand patterns.In “A Unified Framework for Monetary Theory and Policy Analysis” (hereafter LW), Randy Wright and I propose a search-based framework that combines some features of search models, such as an explicit mathematical formulation of the decentralized trading and the price-setting mechanisms, with some features of competitive models. Specifically, agents sometimes trade in a decentralized manner as in search theory–bilaterally, at prices determined by bargaining–and sometimes as in competitive theory–with the market, at market clearing prices. There are at least two important benefits from taking this step from pure search theory toward competitive theory. First, it becomes relatively easy to integrate search-based micro foundations for money demand with the rest of macroeconomic theory. For example, some real-world markets may be better approximated by competitive markets, so it may be convenient in applications to be able to incorporate a segment of competitive trades into our search-based theories. Second, it turns out that under certain conditions on preferences (quasi-linearity in one of the goods that is traded competitively), allowing agents periodic access to competitive markets implies that the model remains tractable–even if we work with divisible money and allow agents to carry any nonnegative amount. (Quasi-linearity eliminates wealth effects in the demand for money, which simplifies the dynamics of the endogenous distribution of money holdings. In the simplest version, all agents who trade in the competitive market choose to carry the same money balances into the next round of decentralized trade, therefore periodically, the endogenous distribution of money holdings becomes degenerate. This allows for sharp analytic results, and makes the framework easy to work with. An alternative approach was proposed by Shi, 1997, who develops a tractable search model with divisible money that relies on a large-household construct to avoid distributional issues.)

Just as incorporating a competitive segment brings the search theory of money closer to mainstream macroeconomic theory, relaxing the stark restrictions on money holdings implies that the search-theoretic models can now deliver standard Euler equations. This brings the field within a step of mainstream asset-pricing theory. The missing step is a meaningful portfolio choice problem; something that requires introducing financial assets that agents can hold alongside money balances.

3. A first look at competing assets: money and capital

Initially, the new search-theoretic literature with divisible money and unrestricted holdings tended to shy away from formulations in which the role of money as a medium of exchange could be challenged by other assets. This issue was circumvented either by assuming that money is the only asset (e.g., Lagos and Wright, 2003, Lagos and Rocheteau, 2005), or by assuming that assets other than money simply cannot be used in exchange (e.g., Aruoba and Wright, 2003). The implicit presumption being, perhaps, that in the face of competition from other assets, money would be driven out of circulation, or rendered “inessential” (a term used by monetary theorists to mean that the set of Pareto optimal allocations implementable by a monetary equilibrium is no larger than the set of Pareto optimal allocations implementable by a nonmonetary equilibrium).In “Money and Capital as Competing Media of Exchange,” Guillaume Rocheteau and I build on LW and consider a model where real assets (reproducible capital goods) can compete with fiat money as a medium of exchange. Our theory allows agents to choose which assets to exchange in decentralized trades and imposes no restrictions on their portfolios. We establish that a monetary equilibrium exists if and only if money is essential, and offer a condition on fundamentals under which this is the case. The condition states that money is essential when the capital stock that a social planner would choose to accumulate is smaller than the stock of assets that agents need to conduct transactions. In the nonmonetary equilibrium, this liquidity shortage manifests itself as a premium on the rate of return on the assets that can be used as a medium of exchange, and this premium induces agents to over-accumulate capital. Capital plays two roles in this economy: it has a productive role and it serves as a medium of exchange in decentralized trades. The introduction of fiat money helps to disentangle the productive use of capital from its liquidity use, and induces agents to reduce the inefficiently high stock of capital goods. In the monetary equilibrium, money has the same rate of return as capital since both assets can be used in decentralized trade and agents can exploit arbitrage opportunities in the centralized market. We find that when a monetary equilibrium exists, the policy of contracting the money supply at the rate of time preference is optimal.

4. Exchange liquidity and the asset-pricing puzzles

In “Asset Prices and Liquidity in an Exchange Economy,” I develop an asset-pricing model in which financial assets are valued for their exchange liquidity–the extent to which they are useful in facilitating exchange–as well as for being claims to streams of consumption goods. I use the theory to study the implications of this liquidity channel for average asset returns, the equity-premium puzzle and the risk-free rate puzzle. I consider a real LW economy with two assets: a one-period government-issued risk-free real bill (a “bond”), and an equity share that represents the bearer’s ownership of a “Lucas tree” and confers him the right to collect a stochastic stream of real dividends. The model can be thought of as a version of Mehra and Prescott (1985), but extended to incorporate some trades that take place in a decentralized manner, away from competitive markets. As in the prototypical search model of money, agents face quid pro quo constraints in decentralized trades.In the basic formulation, assets differ only in the stochastic properties of their payoffs, and agents are free to choose which assets to use as means of payment in decentralized trades. The theory unambiguously predicts that someone testing an agent’s Euler equation for the risk-free bill using its measured rate of return would find that, at the margin, this agent can gain from transferring consumption from the future to the present. That is, there would appear to be a risk-free rate puzzle. I also find that, at least qualitatively, the theory may also be consistent with an equity premium in excess of the risk premium that compensates equity holders for bearing undiversifiable aggregate risk. I calibrate the model economies and study the extent to which they are able to generate average equity returns and risk-free rates that are in line with U.S. data. Mehra and Prescott’s test of their theory essentially consisted of experimenting with different values of the curvature of the agent’s utility function (call it sigma) to find the values for which the average risk-free rate and equity premium in the model matched those in the U.S. economy. I carry out a similar exercise.

First, I consider a baseline economy in which agents are free to use bonds and equity shares in all decentralized trades. I find that for relatively low values of sigma, the liquidity mechanism is inactive and the equilibrium is the one in Mehra-Prescott. For larger values of sigma (equal to or greater than 8) equity shares and bills are valuable in decentralized exchange. This additional motive for holding the assets lowers the return on equity and the risk-free rate from what they would be in the Mehra-Prescott economy, and brings them closer to the data. There is a precise sense in which even if shares are just as useful as bonds for exchange purposes, quantitatively, the model performs better than the Mehra-Prescott frictionless benchmark. For example, with standard constant relative risk aversion preferences, it takes a value of sigma of about 10 for the model to be consistent with asset return data in the Hansen-Jagannathan sense (as opposed to a value of sigma above 20 in Mehra-Prescott). Despite equity shares and bonds being equally acceptable in decentralized exchange, the model is capable of increasing the size of the equity premium somewhat relative to the Mehra-Prescott benchmark. The reason is that equity is a worse hedge against the risk of facing binding trading constraints in the decentralized market than bonds (trading constraints tend to bind more in states where the equity return is low).

Equity and bonds are equally liquid (equally useful to finance decentralize trades) in the baseline model. In order to assess the extent to which liquidity differences (differences in acceptability) can magnify the return differential between equity and bonds, I also analyze specifications in which exogenous restrictions, which can be interpreted as arising from institutional or legal arrangements, give bonds an advantage over equity as a medium of exchange. Specifically, I consider the case in which equity shares cannot be used in a fraction theta of decentralized exchanges (theta =0 is the case with no exogenous liquidity differences analyzed previously). For a calibrated version of this model, I formulate the following question: For a given value of sigma, how large does theta (the relative illiquidity of equity) have to be for the model to generate an average yearly risk-free rate of 1% and an equity premium that matches the long-term average for the U.S. economy? The answer is, quite small: If one allows for the fact that bonds may be slightly better suited than equity shares to play the medium-of-exchange role, then the model is able to match the historical average return to equity and the risk-free rate for the United States with values of sigma between 3 and 5.

I take these results as an indication that prying deeper into the microeconomics of the decentralized exchange process can add to our understanding of how asset prices and returns are determined in actual economies. The quantitative results also highlight the importance of a fundamental question: why is asset X more generally accepted or better suited to function as a medium of exchange than asset Y? The finding that small differences in the acceptability of an asset can generate relatively large return differentials underscores the importance of tackling this question. Some research along these lines is already underway. For example, Kim and Lee (2008), Lester et al. (2008), and Rocheteau (2008) are exploring the possibility that differences in acceptability may arise due to a moral hazard problem in an environment where some assets are more susceptible to counterfeiting than others. Lagos and Rocheteau (2008) are focusing on the differences in acceptability that arise from an adverse selection problem due to the fact that some agents may be better informed about the return characteristics of certain assets than other agents. Nosal and Rocheteau (2008) are exploring trading mechanisms that may give some assets an advantage in exchange as in Zhu and Wallace (2007).

5. Asset prices, exchange liquidity, and monetary policy

The work described in the previous section studies the implications of exchange liquidity for real asset returns in economies with no money. Since much of the existing work on asset pricing abstracts from monetary considerations, this seemed like a natural starting point. However, given the emphasis on exchange liquidity, it would be odd to neglect fiat money–the quintessential medium of exchange–indefinitely. For one thing, the primary role of fiat money is precisely to provide exchange liquidity, so to the extent that the valuations of other financial assets have a liquidity component, the interactions between these other assets and money through the agents’ portfolio choices should not be ignored. Also, by setting monetary policy, governments can affect real money balances and in this way supply the economy with the liquidity it needs to lubricate the mechanism of decentralized exchange. So again, to the extent that the valuations of financial assets have a liquidity component, monetary policy will be a key determinant of their (real) measured returns.In “Some Results on the Optimality and Implementation of the Friedman Rule in the Search Theory of Money,” I consider a physical environment similar to the one described in the previous section, but replace the one-period government-issued risk-free real bill with government-issued fiat money. In this context, I show that the quantities in a monetary equilibrium are Pareto optimal if and only if the nominal interest rate, i.e., the marginal (indirect) utility of holding an additional dollar, is constant and equal to zero. Thus, a monetary authority that wishes to maximize welfare ought to follow Milton Friedman’s prescription that monetary policy should be conducted with the objective of inducing a zero nominal interest rate. I characterize a large family of deterministic monetary policies that implement Milton Friedman’s prescription, in the sense that these policies are consistent with the existence of a monetary equilibrium with zero nominal interest rates. This family of optimal policies is defined by two properties: (i) the money supply must be arbitrarily close to zero for an infinite number of dates, and (ii) asymptotically, on average, the growth rate of the money supply must be at least as large as the rate of time preference. Interestingly, even though the agents’ liquidity needs are stochastic in this environment (because equity, whose price is stochastic, can be used alongside money in decentralized trades) this is the same class of monetary policies that implements the Friedman rule in the context of deterministic cash-in-advance economies, as shown by Cole and Kocherlakota (1998) and Wilson (1979).

Under an optimal policy, the quantity of real money balances is large enough so that agents’ liquidity needs are satiated, so the real equity price and return are independent of monetary and liquidity considerations. The monetary equilibrium under an optimal monetary policy also exhibits some peculiarities: the price level is indeterminate, and the inflation rate can be independent of the path of the money supply, as was emphasized by Cole and Kocherlakota (1998) in the context of their deterministic cash-in-advance economy. Some may find the failure of the quantity theory and the ensuing price-level indeterminacy unappealing if the model is to be used for applied research. One way to eschew these issues, is to consider a perturbation of a certain class of optimal policies that implements a constant but positive nominal interest rate. A policy that targets a constant nonzero nominal rate in this stochastic environment, however, will typically involve a stochastic monetary policy rule. This is the direction I pursue in “Asset Prices, Liquidity, and Monetary Policy in an Exchange Economy” to study the positive implications of monetary policy for asset prices and returns. The analysis provides insights on how monetary policy must be conducted in order to support a recursive monetary equilibrium with a constant nominal interest rate (with the Pareto optimal equilibrium in which the nominal rate is zero as a limiting case): The growth rate of the money supply must be relatively low in states in which the real value of the equilibrium equity holdings is below average. Something similar happens with the implied inflation rate: it is relatively low between state x and a next-period state x’, if the realized real value of the equilibrium equity holdings in state x’ is below its state-x conditional expectation.

I also find that on average, liquidity considerations can introduce a negative relationship between the nominal interest rate (and the inflation rate) and real equity returns. Intuitively, since agents are free to use any combination of assets for exchange purposes, even if the equity yields a real and exogenous dividend stream, part of the equity return will be linked to its liquidity return, and this liquidity return in turn depends on the quantity of real money balances–which is a function of the inflation rate. On average, if the rate of inflation is higher, real money balances are lower, and the liquidity return on equity rises, which causes its price to rise and its real measured rate of return (dividend yield plus capital gains) to fall. This type of logic could perhaps help to rationalize the fact that historically, real stock returns and inflation have been negatively correlated–an observation long considered anomalous in the finance literature (e.g., Fama and Schwert, 1977).

The model has a number of implications for the time-paths of output, inflation, interest rates, equity prices, and equity returns, and it would be interesting to explore these implications further. For example, even though variations in aggregate output are effectively exogenous under the types of monetary policies considered, for some parametrizations the theory produces a negative correlation between the inflation rate and the growth rate of output–a short-run “Phillips curve” –but one that is entirely generated by a monetary policy designed to provide liquidity in an economy with stochastic liquidity needs.

6. Transaction liquidity and the structure of financial markets

In the introductory section I mentioned that in finance, the liquidity of assets and markets is associated to the cost of trading: this transaction liquidity is, loosely speaking, the ability to trade cheaply. Central to this literature is the notion that actual financial trade is not costless and seamless as in competitive theory. To capture the idea that trade is not costless, much of the work in this area maintains the competitive market structure, but incorporates transaction costs of various forms (e.g., fixed costs of trading, or costs that are proportional to the size of the trade). In many of these models, transaction costs affect asset prices because in equilibrium, investors must be compensated for bearing these costs. Hence for any given dividend flow that the asset generates, transaction costs lower the asset price and increase the asset return, i.e., higher transaction costs reduce the transaction liquidity of the asset.A recent related strand of work in theoretical finance started by Duffie et al. (2005) subscribes to the notion of market liquidity as trading costs, but goes deeper into the nature of these trading costs by building explicit models of the mechanism of exchange in financial markets. The starting point is the observation that many financial securities (e.g., unlisted stocks, currencies, derivatives, corporate bonds, municipal bonds, federal funds) are traded in over-the-counter (OTC) markets. The defining feature of OTC markets is that they have no formal organization: unlike organized securities exchanges, OTC markets are completely decentralized and do not operate in a particular location at set times. An agent wishing to trade a security in an OTC market is confronted with two fundamental trading frictions: he must first search for a counterpart, and once a potential counterpart has been found, the two parties will typically negotiate the terms of the trade to share the gains.

Duffie et al. (2005) show that a search-based model of an OTC market can parsimoniously rationalize standard measures of transaction liquidity discussed in the finance literature, such as trade volume, bid-ask spreads, and trading delays. A virtue of their formulation is that it is analytically tractable, so all these mechanisms can be well understood. The literature spurred by Duffie et al. (2005), however, keeps the framework tractable by imposing a stark restriction on asset holdings: agents can only hold either 0 units or 1 unit of the asset.

In “Liquidity in Asset Markets with Search Frictions,” Guillaume Rocheteau and I develop a search-based model of trade in an OTC market with no restrictions on investors’ asset holdings. The model is close in structure and spirit to Duffie et al. (2005): there are two types of agents, investors and dealers. The asset is only held by investors (they can hold any nonnegative position), and their idiosyncratic willingness to hold the asset changes over time, which creates a motive for trade among investors. Trades are intermediated by dealers, and investors contact dealers at random times, with the period of time preceding a trade interpreted as an execution delay. We find that as a result of the restrictions they impose on asset holdings, existing search-based theories of trade in OTC markets neglect a critical feature of illiquid markets, namely, that market participants can mitigate trading frictions by adjusting their asset positions to reduce their trading needs.

The key theoretical observation is that an investor’s asset demand in an OTC market depends not only on his valuation for the asset at the time of the trade, but also on his expected valuation over the holding period until his next opportunity to trade. A reduction in trading frictions (e.g., a reduction in the average time it takes for an investor to contact a dealer) makes investors less likely to remain locked into an undesirable asset position and therefore induces them to put more weight on their current valuation. As a result, a reduction in trading frictions induces an investor to demand a larger asset position if his current valuation is relatively high, and a smaller position if it is relatively low, which tends to increase the spread of the distribution of asset holdings. We find that this effect is a key channel through which trading frictions determine trade volume, bid-ask spreads, and trading delays–the dimensions of transaction liquidity that search-based theories of financial intermediation are designed to explain.

In “Search in Asset Markets: Market Structure, Liquidity, and Welfare,” Guillaume Rocheteau and I consider a version of this model with free entry of dealers as a way to endogenize trading delays. We show that when interacted with investors’ unrestricted asset holding decisions, the dealers’ incentives to make markets generate a liquidity externality that can give rise to multiple steady states. This finding suggests that all the symptoms of an illiquid market–large spreads, small trade volume, and long trading delays–can simultaneously arise as a self-fulfilling phenomenon in asset markets with an OTC structure.

In many financial markets, the search-for-a-counterpart problem is alleviated by dealers who trade assets from their own inventories. During market crashes, for instance, it can take a long time for an investor to find a counterpart for trade, either because of the technological limitations of order-handling systems or, as in OTC markets, due to the decentralized nature of the trading process. In these circumstances, liquidity provision by dealers (in the sense of their alleviating the investors’ search problem by becoming themselves a counterpart for trade) can become critical, as shown by Weill (2007). In “Crashes and Recoveries in Illiquid Markets,” Guillaume Rocheteau, Pierre-Olivier Weill and I extend the model in “Liquidity in Asset Markets with Search Frictions” to allow dealers to accumulate asset inventories. We use the theory to study the equilibrium and the socially optimal inventory policies of dealers during a market crash, which we model as a temporary negative shock to investors’ willingness to hold the asset, followed by a (possibly stochastic) recovery path.

Our model can rationalize why dealers intervene in some crises and withdraw in others. We derive conditions under which dealers will find it in their interest to provide liquidity in the aftermath of a crash, as well as conditions under which their incentives to provide liquidity are consistent with market efficiency. We relate the liquidity provision by dealers to the details of the market structure, e.g., dealers’ degree of market power or the extent of the search frictions (the average length of time it takes for an investor to contact a dealer), and the characteristics of the crash, such as the severity and persistence of the shock to investors’ demands.

We find that the amount of liquidity provided by dealers following a crash varies nonmonotonically with the magnitude of search frictions. When search frictions are small, investors with higher-than-average utility for assets become more willing to hold larger-than-average positions and absorb more of the selling pressure coming from investors whose demands for the asset are lower than normal. In some cases, the former are so willing to buy large quantities of the asset from the other investors, i.e., so willing to serve as a counterpart for trade, that dealers don’t find it profitable to step in. If, on the contrary, search frictions are large enough, dealers do not accumulate inventories either, but for a different reason: Trading frictions reduce the investors’ need for liquidity provision by dealers. Indeed, in order to reduce their exposure to the search frictions, investors choose to take less extreme asset positions, so the problem of finding a counterpart for trade becomes less relevant. In fact, it is possible that the investors reduce their trading needs so much that dealers don’t find it profitable to accumulate inventories following a crash. Thus, if one considers a spectrum of asset markets ranging from those with very small search frictions, such as the New York Stock Exchange (NYSE), to those with severe search frictions, such as the corporate bond market, one would expect to see dealers accumulate more asset inventories during a crash in markets which are in the intermediate range of the spectrum.

7. Concluding remarks

A single underlying theme runs through the work I have reviewed in the preceding sections: in many instances, the demand for an asset (and consequently the asset price and return) depends not only on the demand for the fundamentals that the asset represents, but also on the nature of the mechanism of exchange through which the asset is traded. Financial economists associate the “mechanism of exchange” to the microstructure of the market where the asset trades, and call the asset “liquid” if it can be traded cheaply in terms of time, and out-of-pocket costs. Monetary theorists associate the “mechanism of exchange” to the explicit microeconomic process through which goods and the assets used to pay for those goods flow between buyers and sellers, and call an asset “liquid” if it is generally accepted and used in this exchange process.On the one hand, by conceptualizing financial assets in the way monetary theorists think of fiat money, we can develop theories of liquidity that afford new insights into the fundamental features of assets and markets that can make some assets useful in the mechanism of exchange, and hence more prone to exhibit returns that exceed their fundamental valuations. A better understanding of the deeper determinants of the liquidity component of asset valuations can shed new light on a number of asset-pricing anomalies, as well as on the role that monetary policy plays, and the role it ought to play, in the determination of asset prices. On the other hand, the finance microstructure perspective reviewed in Section 6, formalizes some of the key micro elements of the exchange process in actual financial markets, which include trading delays, bilateral decentralized exchange, and other distinctively non-Walrasian elements, much like those which characterize the stylized “decentralized market” of the macro models in Sections 2-5. It seems to me that incorporating these microstructure considerations into those macro models would be a fertile avenue for future research. The hope I hold for this research agenda, is that by continuing to develop the association between the monetary and the financial perspectives on liquidity, we may be able to make progress on some deeper questions, such as which assets are better suited to satisfy the quid pro quo constraints in the Search Theory of Money, and why certain assets seem more prone to be traded in frictional market structures than others.

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