Keynes wrote that an asset is more liquid than another "if it is more certainly realizable at short notice without loss". The degree of liquidity of an asset can be measured either by the variability of its price (or return) over time or by the availability of a market that can readily absorb the sale of the asset at the going price if trade is thin and infrequent, one may have to accept a substantial price reduction for the market to absorb a large and sudden sale.Discussions of liquidity often concentrate only on price volatility. In Discussion Paper No. 142, Research Fellow Marco Pagano analyzes instead on the relationship between liquidity and the volume of trading in a market.

Trading volume, he argues, is important not only in markets for heterogeneous goods, housing, but also markets for standardized securities, such as futures markets or stock exchanges.

This suggests that the degree of liquidity of a market can have quite dramatic effects on the number of traders that are attracted to the market. If these traders expect that the market will possess too low a degree of liquidity, they may look for alternative trading opportunities outside the market, such as trade on a parallel market or direct search for a trading partner. In this respect the beliefs or "conjectures" of speculators about the degree of absorptive capacity of a market may easily become self-fulfilling if sufficiently widespread. If traders desert the market because of its supposedly low absorptive capacity, they will reduce its volume of trade and thus may end up validating the initial belief, by leading the market into a low-trade, low-liquidity trap.

This paper Pagano illustrates how such self fulfilling expectations can lead to multiple equilibria can arise in a two- period model of a stock market economy with two types of potential transactors - 'large' and 'small' traders. Large transactors can trade either on the official stock market outside it, either through an informal parallel market among large transactors or by searching for a trading partner.

Pagano assumes that large traders can choose whether to trade on the official market or outside it, but once they have chosen they are locked in.

How does a large trader choose between the two markets? The decision depends on which of the two options is expected to yield the trader a higher level of welfare, conditional upon the conjecture that he holds about the other traders' behaviour and the implied equilibrium in the official market. Pagano considers two extreme conjectures that a large trader might entertain: the first is that only the small traders will trade on the official market; the second is that the other large traders will as well.

Pagano examines the conditions under which each of these will be a "rational" conjecture, in the sense that it will be fulfilled by the resulting equilibrium. He finds that in general two rational expectations equilibria exist: everyone trades on the same market and the other in which where the two types of traders transact in separate markets. If for example, every large trader conjectures that the others will trade on the unofficial market, this conjecture will be fulfilled: all the large traders will in fact trade on that market. If instead, each large trader held the opposite conjecture, that all traders would enter the official market, then the resulting equilibrium will be one in which the larger number of traders translates into a greater price elasticity of market demand for the asset and thereby increases the "absorptive capacity" of the market: this benefits the large traders in particular.

Pagano also considers a version of the model in which large traders can either trade on the stock market or search (at a fixed cost) directly for a trading partner outside the market. Two rational expectations equilibria can arise in this situation as well: one where all large traders operate on the market, and another where they all turn to search. Again, the beliefs of the large traders are important here: traders' conjectures concerning the behaviour of the other traders, and thus on the depth of the market, determine whether the economy will select the first or the second equilibrium.

How traders form their conjectures is therefore an important issue Pagano concludes. If they are formed by extrapolating from past observations, the equilibrium attained by the market - whether a low-level trap or a high-level equilibrium - will tend to perpetuate itself. This may explain why the intermediation of world capital flows has shown a persistent tendency to be concentrated in a few financial centres and why in some markets the volume of trade and the degree of liquidity may remain below their potential level for extended periods.

Empirical studies have found that prices are more volatile in thin speculative markets than in deep ones. Thin markets are generally characterized by small numbers of transactors during any given period of time. Prices in such markets are more sensitive to the impact of individual traders' demand shocks, whereas in deep markets there are so many transactors that the demands of individual traders (if uncorrelated) tend to offset each other and so leave market prices largely unaffected. Market thinness can therefore create additional risk for investors, adding to that arising from asset fundamentals. This explanation, however, takes market size as given but volatile prices may in turn discourage potential traders from entering the market, and trap the market in an equilibrium characterized by low trading volumes and high price volatility. The perverse interaction of market thinness and price volatility has been suggested as one of the causes of the persistent narrowness of continental European equity markets relative to those in Britain, the United States or Japan.

In Discussion Paper No. 146, Pagano constructs a formal model of this 'vicious circle'. The analysis is overlapping generations model, in which people live for three periods, and can invest in a safe asset (default-free debt) and a risky one (equities). They liquidate asset holdings in the last period of their life by selling them to the next generation of investors. Demands for equities differ across individuals because investors are assumed to began life with different endowments of a non-marketable asset. These individual variations introduce shocks into the aggregate demand for equities, and the variance of these shocks is inversely related to the number of traders. Pagano assumes that equities are issued by competitive firms that are entirely equity-financed. The firm chooses its investment plan so as to maximize the current market value of its shares and finances it by retained earnings; firms also act as a price-takers in issuing new equities on the stock market.

Pagano finds that this model, like that analyzed in Discussion Paper No. 142, can yield multiple equilibria: the stock market may remain trapped in an equilibrium characterized by low trading volumes and high price volatility or it may settle into one in which trading volumes are high and price volatility is low. The expectations of economic agents about which equilibrium will prevail are self-fulfilling, and determine which equilibrium is chosen by the economy. The entry of each additional trader into the market generates a positive externality for other (actual or potential) traders, by decreasing the price volatility and hence the riskiness of the stock; lower risk in turn tends to attract more investors. This in turn causes an increase in demand for the stock and raises stock prices; as a result firms find it attractive to issue additional equity. This creates feedback between price volatility and market size, measured either by the number of traders and by the total real stock of equities. If, however, investors face transaction costs in the stock market, the feedback loop may fail to operate: if everyone expects a small volume of trade on the market, the investors with relatively high transaction costs will abstain from trading. Thus the market will remain trapped in an equilibrium with little trading and volatile prices. If the market had a reputation for being large and stable, even investors facing high transaction costs would be willing to trade, and in equilibrium such expectations would be fulfilled. Thinness or depth may be self- perpetuating.

Pagano notes that on this model firms issue additional equity if stock prices rise and this is essential if multiple equilibria are to arise. The opposite assumption of a fixed supply of equities is customary in finance theory: if the supply of equities is given, entry by additional investors permanently increases the price of the stock and the basic feedback mechanism of the model will be inoperative and a unique equilibrium will result.

Pagano also notes that equilibria with high trading volumes yield higher levels of water than do those with little trade: there may therefore be a case for government intervention. Simple incentive schemes such as temporary subsidy to stock market investors, can shift the economy to a high trade equilibrium by inducing new traders to enter the stock market. The increased number of transactors stabilizes the market price and the resulting reduction in risk in turn raises demand for the asset, leading to higher stock prices and to the issue of additional shares. The subsidy need not be permanent, however, since after a certain number of periods higher trading volumes and reduced price volatility become self-sustaining and the economy moves towards the new equilibrium even if the government subsidy is removed.

One of the main functions of the stock market is to "pool" the private information available to market participants: the information thus contained in asset prices allows each trader to sharpen his forecast of future dividends. In Discussion Paper No. 144, Pagano shows that thinness may limit the market's ability to perform this task: in a thin market the individual errors contained in the private information of traders will not cancel out in the aggregate, due to the paucity of investors. As a result, the informational content of the price will be clouded by more noise than would be the case in a deeper market. The conditional variance of stock returns will be correspondingly higher and the current share price will be a less reliable predictor of future dividends.

Market thinness can therefore make investment riskier. Discussion Paper No. 146, analyses a similar link between thinness and risk, generated by differences in individual demands for assets. The focus of the two models is quite different otherwise: in Discussion Paper No. 144. Pagano takes asset supplies as exogenously given, as is customary in finance theory, and analyse the effects of thin trading on the stochastic behaviour of stock prices within a conventional asset pricing model of the Sharpe-Lintner variety. Unlike earlier work in this area, however, Pagano does not assume market thinness but focuses instead on the testable implications of thin trading, based on explicit and consistent modelling of market equilibrium.

In the model Pagano assumes that each trader possesses some "noisy" private information about the future prospects of each firm. The price of a firm's stock summarizes this private information: the larger the number of traders, the larger the amount of information pooled in the stock price, and the more reliable the price becomes as an indicator of the firm's prospects. As the number of investors increases, the conditional variance of the stock return - i.e. the subjective uncertainty of each trade - decreases. The number of traders in each market, is determined endogenously in the model. Trade is assumed to involves fixed costs, which may differ across investors. This assumption is essential, otherwise everyone would trade in all markets and the number of transactors would not differ across markets. The equilibrium number of traders for each stock is positively related to the supply of that stock. As a result, the informational content of stock prices is higher for stocks that are available in larger quantities.

The entry of additional investors into the market reduces the variance of the stock's return, because the larger volume of trade increases the precision of the market forecast of future dividends. Since the supply of the stock is assumed to be fixed, new entrants tend to put upward pressure on the price, and thus to lower the expected return on the stock. A unique equilibrium will exist at the point where the increase in price makes the stock too expensive for additional entrants: the higher the given supply of the stock, the larger the number of traders at which this equilibrium is achieved.

Pagano's model the observed variances of asset returns should be a decreasing function both of the number of investors and of the size of the outstanding asset supply. In addition, the model suggests that investors with comparatively low transaction costs will be more likely to include in their portfolios the stocks of smaller firms and will hold a more diversified portfolio. Their portfolios may nevertheless yield a more variable rate of return because they will also include stocks of the smaller corporations that are characterized by more volatile returns.

Pagano tests these predictions using monthly data on prices and turnover from the Milan Stock Exchange between 1976 and 1984. The theory suggests that there is a negative relationship between the unsystematic component of the variance of returns and the number of market participants or the size of asset supplies. The number of market participants is not observable but can be proxied by average turnover, while the total capitalization of the corresponding firms can serve as a proxy for the size of asset supplies. The data indicate a significant negative relationship between unsystematic risk and both the volume of trade and capitalization at market prices.