## ASSET PRICING MODEL: Liquidity premium 2

The optimal contract between a representative entrepreneur and the investors maximizes the social surplus generated by the entrepreneurial activity,

so firms are liquidity constrained in low-income states.

Let us turn to the date-0 choice of liquidity. From our previous characterization, L is chosen to maximize

The liquidity premium is equal to the expected marginal value of the liquidity service. In states of liquidity-shortage (x < y* — L), an extra unit of liquidity allows the firm to increase its reinvestment by 1 and the private benefit by b — (x + L). This marginal private benefit, expressed in monetary terms, is equal to [b — (x + L)]/ц. The increase in reinvestment has monetary cost 1. This yields the expression for the liquidity premium (6). We have looked for an equilibrium {q, ц} in which Treasury bonds command a liquidity premium. Such an equilibrium satisfies the break-even constraint (2) (with equality) and the asset pricing equation (5), for L = L (because q > 1, investors do not hold Treasury bonds), and for the optimal reinvestment policy y(-) defined by (3) and (4). If the solution to this system yields q < 1, then Treasury bonds command no liquidity premium (q = 1); that is, even if liquidity is free, firms may not hold all available liquid claims.

It is easy to show that i) there exists an L* such that q > 1 if and only if L < L*; and ii) the value of the marginal liquidity service m(-) and the price of liquid claims q are monotonically decreasing in L. These results are illustrated in figure 2, where m^(-) denotes the marginal liquidity service for bond supply L. When L = L3 > L*, the economy has surplus liquidity, and there is no liquidity premium.10 Hence, bond prices are low. The cases I. I.\ or L-> depict the interesting case of scarce liquidity.

Remark: Note that the value of liquidity m(x) is linear in x in this example. Linearity requires that the date-2 payoff be nonverifiable, that is, the payoff is purely a private benefit. Alternatively, the date-2 payoff could be verifiable but not fully pledgeable as in Holmstrom-Tirole (1998), for instance, because moral hazard problems require the entrepreneur to keep a stake in the final payoff. Under mild conditions, the value of liquidity m(x) would then differ from that depicted in figure 2 only in that the decreasing part would be convex rather than linear.