Liquidity needs and pricing of liquid assets:-

Consider the liquidity-needs model with a fixed investment and two possible liquidity shocks. The borrower has cash A and wants to finance a fixed-size investment I>A at date 0. At date 1, a cash infusion equal to ρ is needed in order for the project to continue. If ρ is not invested at date 1, the project stops and yields nothing. If ρ is invested, the borrower chooses between working (no private benefit, probability of success pH) and shirking (private benefit B, probability of success pL = pH - ?p). The project then yields, at date 2, R in the case of success and 0 in the case of failure. The liquidity shock is equal to ρL with probability (1 - λ) and to ρH with probability λ, where

There is a single liquid asset, Treasury bonds. A Treasury bond yields 1 unit of income for certain at date 1 (and none at dates 0 and 2). It is sold at date 0 at price q ≥  1. (The investors' rate of time preference is equal to 0.)

(i) Suppose that the firm has the choice between buying enough Treasury bonds to withstand the high liquidity shock and buying none. Show that it chooses to hoard liquidity if

(ii) Suppose that the economy is composed of a continuum, with mass 1, of identical firms with characteristics as described above. The liquidity shocks of the firms are perfectly correlated. There are T Treasury bonds in the economy, with Show that when λ is small, the liquidity premium (q - 1) commanded by Treasury bonds is proportional to the probability of a high liquidity shock. (Hint: show that either (2) or (3) must be binding, and use (1) to conclude that (3) is binding.)

(iii) Suppose that, in the economy considered in the previous subquestion, the government issues at date 0 not only the T Treasury bonds, but also a security that yields at date 1 a payoff equal to 1 in the good state (the firms experience liquidity shock ρL) and 0 in the bad state (the firms experience liquidity shock ρH). What is the equilibrium date-0 price q of this new asset? (Prices of the Treasury bonds and of this new asset are market clearing prices.)