Problem with defining liquidity I have heard multiple times that a consistent definition of liquidity does not exist. The two wikipedia articles describing liquidity (Market liquidity and accounting liquidity) lack a discussion of this apparent problem within economics. Could someone perhaps provide a reference to a discussion of the problem of defining liquidity and maybe provide a brief summary of the issue?
Etymology and Introduction

As a concept to measure the interchangeability of assets and money, liquidity is a new word. It first appears in 1923 in a use by Hawtrey (The Oxford English Dictionary (1989)). The underlying idea however is much older. Menger (1892) calls a good more or less saleable according to the facility to which it can be disposed of at current purchasing prices with less or more diminution. He is talking about the origin of money, and not the disposition of financial assets, but this concept of saleability is very much like the modern concept of liquidity. This sense of saleability goes back at least as far as Jehan Palsgrave's usage in 1530 (The Oxford English Dictionary (1989)). Although the absence of liquidity is now commonly referred to as "illiquidity", Marschak (1938) offers "frozen" as an alternative that did not catch on.

Hicks (1962) says that the use of the word liquidity in a financial sense was popularized by Keynes and the Macmillan Report in the 1930's (Macmillan Committee of HMSO (1931)). In The General Theory, Keynes says that liquidity justifies money trading at a premium over bills or bonds and causes the existence of an interest rate. Hicks teases from Keynes' Treaty on Money (Keynes (1930)) that one calls an asset more liquid than another if it is "more certainly realizable at short notice without loss." 

This quote is more ambiguous than it may seem at first glance. Hicks offers several interpretations. The first, which he flatly rejects, define liquidity as the difference between the price the owner carries on his books for an asset and the price they could sell it for on the market. The second, is an interpretation of marketability. Hicks defines a security as marketable if it is sold just as well after negotiation, search and advertising as it is without it. That is, we can compare the liquidity of two assets by the relative sacrifice one makes from a rapid sale. He claims this interpretation is "more appealing" but still not what Keynes meant. He understands Keynes's definition of liquidity to require perfect marketability, but even perfectly marketable assets can be more or less liquid. The difference here is in the moments of the asset's price. Among marketable financial assets, we can regard them as more or less liquid by using a utility function to manage the trade-offs between maximizing the desirable odd moments (e.g., positive mean and skew) and minimizing undesirable even moments (e.g., variance and leptokurtosis) of asset returns.

Definitions of Liquidity

A treatment of market liquidity is critical if we are to realistically model the behavior of markets of financial assets that are traded with transaction costs. The liquidity literature primarily follows three meanings of liquidity. The first and oldest class of measures of liquidity relate the size of loss to the amount of notice. That is, what is the fraction of the best possible price that a seller can net as a function of time allotted to conduct a loss minimizing sale? Keynes's definition is a specific example where he is interested only in short notice. For example, compare trying to sell a round lot (100 shares) of IBM stock today with doing the same with a home. The IBM shares will sell without diminution. The home will sell at an enormous discount because of product heterogeneity, heterogeneous buyers, and the skipping of time consuming risk avoidance techniques (e.g., title search and property inspection). A classic statement of this sense of liquidity comes from Hirshleifer (1968). He calls liquidity "as asset's capability over time of being realized in the form of funds available for immediate consumption or reinvestment -- proximately in the form of money." Admati and Pfleiderer (1988) also care about liquidity in this first sense. They see exogenous liquidity events (say a change in margin requirements) causing a "demand for immediacy" (a term also found in Grossman and Miller (1988)), that is, the willingness to sell rather than wait when doing so costs the seller. According to Greenbaum (1971), who called this the L1 definition of liquidity, the earliest work on this sense of liquidity was in Tobin's unpublished manuscript. Pierce (1966) elaborates this notion and explores this measure in the context of commercial bank portfolio management.

The second meaning understands liquidity as the expected time to sale without diminution. Returning again to the example of selling the lot of IBM stock and a home, the expected time to getting the best price on your shares of IBM is almost zero (at least during business hours and with access to a computer or phone) whereas, on average, the home would take a few months to sell optimally. Lippman and McCall (1986) explore this sense of liquidity. They take an agent as choosing a stopping rule $\tau^* \in T$ of all possible stopping rules to maximize expected net receipts (under that stopping rule) of $E [R(\tau^*)]$. They define liquidity as $E [\tau^*]$, the expected time to sale under the optimal stopping rule. Krainer and LeRoy (2002) also suggest this expected time to sale under an optimal selling rule a measure of liquidity.

The third definition of liquidity involves the uncertainty of an asset's value. As discussed above, Hicks (1962) sees this as the critical attribute of liquid assets. Proponents of this definition argue that it is of little importance if you can sell an asset on short notice and with small loss if the asset itself is worth little when you need it. That is reasonable as long as investors are risk averse. Tobin (1958) introduces this sense of liquidity in a framework of risk averse investors and uncertainty of future interest rates. Tobin's paper resolved the paradox (to economists) that consoles (perpetual bonds) have higher expected rates of return than cash investments. Lagos (2008) is able to explain much of the risk-free-rate and equity-premium puzzles in an economy with riskless, liquid assets and risky, illiquid ones when agents hold assets for the liquidity services.

Deaton (1991) equates a model's liquidity constraints with limitations on borrowing future income. In that sense, liquid assets are more effective in moving income though time. Holmstrom and Tirole (1998) also explore this meaning of liquidity as a way of storing wealth between periods in search of recommendations for efficient market making of financial assets and optimal provision of liquidity services by the government. Hovakimian, Opler, and Titman (2001) continue to explore this inter-temporal liquidity concept in economies with human capital that creditors cannot seize or make claims upon. He experiments with two an extra features, either no short selling of physical capital or limited borrowing by agents.

The concept of liquidity is a mixture of the attributes of liquidity discussed above. Households without liquid assets want to borrow with their illiquid assets as collateral. They cannot do so. Creditors are unwilling to lend because in the event of default, lenders will have to sell collateral with serious discount from the true value, with long waiting times for finding a good selling price, or with unfavorable co-movement (e.g., covariance or other forms of joint distribution) with other asset holdings. This idea of liquidity as having wealth when you need it is similar to the CAPM insight of weighting future returns by the marginal utility. Krainer and LeRoy (2002) argue that the CAPM holds for illiquid assets as well as liquid ones, but that observing the relevant shadow prices is difficult. However, the ambiguity of asset values in the presence of illiquidity (explored later) ensures that these shadow prices will not be unique. The general idea is that illiquidity can drive a a wedge between the discounted expected risk adjusted returns and the price for which you can sell an asset. As such, the traditional CAPM model is not entirely appropriate. An area for future investigation is expanding the CAPM to make this distinction. Holmstrom and Tirole (2001) was one attempt to so, but focused on firm decisions and not investor valuations.

Can the Alternative Definitions of Liquidity be Reconciled?

Here is an example of the wedge between expected returns and price that also highlights that an asset can be liquid in one sense and yet is illiquid in another. Imagine an investor with a US Treasury bond with a 14% coupon in a market where newly issued bonds have a coupon of 7%. Selling these highly appreciated bonds (which traded far above par) would require paying significant capital gains taxes, while holding them involves minimal credit risk and the interest would be state and local tax exempt.  Therefore, to sell would be expensive for the owner and therefore illiquid in sense 1. Nevertheless, these securities could surely be sold rapidly though the secondary US Treasury Bond market and there would be little benefit to waiting and therefore highly liquid in sense 2. It is certainly the case that assets that are liquid in one sense are often liquid in the others too. But as this example shows, that need not be so.

Therefore we cannot collapse these three definitions into a single numerical measure of liquidity capturing all the desired attributes. Even if we could, Hicks (1962) and Admati and Pfleiderer (1988) have concluded that liquidity is an ordinal property. Marschak (1938) suggests measuring the distinct properties of liquidity separately. Pierce (1966) disagrees that such an ordering is possible, at least with respect to sense 1 of liquidity. He says that "apart from those assets that are perfectly liquid and those for which sale prior to maturity is impossible, assets cannot be uniquely ranked by degrees of liquidity." His example is an asset A that is easy to  sell well with short searches (but no better after long ones) and an asset B that sells more poorly than A for short searches and better than A for long ones. Pierce also notes (in describing the weakness of his notion of liquidity) that "the price per unit often depends on the number of units sold." This allows for another example of the crossing behavior, where one asset may be more liquid at small quantities but less liquid for large ones. 

Admati and Pfleiderer (1988) argue that liquidity of the third type actually encourages liquidity of the first type. They argue that commodities with successful futures markets have demand for immediacy because price volatility and risks of delaying sales are large (illiquid in sense 3). Such markets also help spread the fixed costs of market making (waiting around for buyers and sellers to want to trade, as well as infrastructure) across a large number of market participants.  In contrast, home sellers are less concerned
with short-term price volatility and instead prefer an extended search for potential buyers.

Alternative Definitions

Krainer and LeRoy (2002) argue that liquidity is a feature of markets and not of assets. They offer as a first example that a Ford automotive factory is illiquid but Ford stock is liquid. A second example is a pool of mortgages which is more illiquid than than the underlying mortgages. However, perhaps it suffices to refer to an asset's liquidity as its liquidity in the most liquid market an agent can trade it in. Tobin calls reversibility "the value of the asset to its holder expressed as a percentage of is contemporaneous cost to the buyer" (Tobin and Golub (1998)). Hahn (1990) sees asset liquidity as being closely related "to the cost reversing a decision taken earlier." His example is that for economic agents, the cost of selling an asset in period 2 will factor into the decision to invest in it in period 1. This meaning of reversibility is a mixture of sense 1 and sense 2 of liquidity. If waiting for suitable trading opportunities is the expensive part of reversing a trade then reversibility is primarily the second sense of liquidity. However, if transaction costs are the expensive part of reversal then the first sense of liquidity will be the relevant one. Marschak (1938) develops a concept of plasticity that is essentially this concept of reversibility. He considers plasticity as a
more general concept than saleability that includes flexibility. Greenbaum (1971) is an early reference that notes the interrelatedness of reversibility and liquidity.

Jones and Ostroy (1984) see an essential attribute of liquidity in the related measure of flexibility. Financial investments that leave agents with a larger set of intermediate and final choices are more flexible. Lippman and McCall (1986) are similarly interested in liquidity as flexibility. They show in a simple search model that if the investor opportunity set changes over time, even risk neutral investors will demand a more
liquid asset because it improves the expected returns of their portfolio. Hahn (1990) also sees liquidity as tied to the "speed of response to new information," but it is hard to see if this is a cause of liquidity or caused by it. Intuition suggests the former. Few market participants, high transaction costs and other direct causes of illiquid markets all make it more difficult to trade on new information. Further, the trees model
of Lucas (1978) shows that prices can move even in the absence of trade. Frequent trade is not synonymous with liquidity. 

The frequency of order arrival is a determining factor of liquidity in markets with market makers. This implies that markets need not be liquid to have fast adjusting prices. Flexibility can be about more than investment opportunities. If agents are borrowing constrained and goods can be purchased only with liquid assets, then agents must hold some liquid assets to purchase goods for immediate consumption, even though doing so does not maximize their portfolio return. If consumption opportunities vary over time then agents will hold relatively more ready assets to take advantage of fleeting consumption opportunities. If agents were not so constrained, then they could simply borrow against their illiquid assets (repaying debts as the returns arrived or when prudent sale was possible) to consume as they like. This is also true for investment flexibility. If investors could borrow against their illiquid assets then they would not need liquid holdings to take advantage of future investment opportunities. The necessity of holding liquid assets for consumption purposes is one motivation for the money-in-the-
utility function (MIU) literature. The MIU framework puts money, an asset without consumption value nor any investment return, into the utility function as a reduced form representation of all the ways that money (as the most liquid asset) makes consumption easier and more efficient.