ASSET PRICING MODEL: Liquidity premium

There are three periods, t = 0,1, 2, one good, and a continuum (of mass 1) of identical entrepreneurs, each with one project. Entrepreneurs are risk neutral and do not discount the future. They have no endowment at date 0 and must turn to investors in order to defray the fixed date-0 set up cost, / . of their project. At date 1, the project generates a random verifiable income x. The realization of x is the same for all entrepreneurs and so there is aggregate uncertainty. The distribution G(x) of x is continuous on [0, сю), with density g(x) and a mean greater than I.

At date 1, the firm reinvests a monetary amount у > 0, which at date 2 generates a private benefit by — \ for the entrepreneur, with b > 1. This private benefit cannot be recouped by the investors; it is nonpledgeable.

The only noncorporate financial asset is a Treasury bond. At date 0 the government issues L bonds that mature at date 1 and yield one unit of the good in every state. The date-0 price of a bond is q. The repayment of the Treasury bond is financed through taxes on consumers.

Consumers (investors) are also risk neutral and do not discount the future. They value consumption stream (со, ci, C2) at C0+C1+C2. A key assumption is that consumers cannot individually commit to provide entrepreneurs with funds at date 1, because they cannot borrow against their future income.8 The consumers’ preferences imply that they will hold only assets whose expected rate of return is nonnegative. This implies in particular that q > 1. If q > 1, consumers hold no Treasury bonds and we will refer to q — 1 as the liquidity premium. The liquidity premium can be strictly positive, because consumers cannot commit to date-1 payments and hence cannot short-sell assets.

A contract between a representative entrepreneur and the investors specifies a quantity L of Treasury bonds to be held by the firm and an income contingent reinvestment policy y(x). Because consumers cannot commit to provide more funds at date 1, the reinvestment policy must satisfy
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For notational simplicity, we will assume that investors receive x + L — y. It is easy to show that this is indeed the case if the expected date-1 income is not too large.9 Investors expect a nonnegative rate of return, and so
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where is a date-0 expectation, and expectations are taken with respect to the random variable x. A competitive capital market guarantees that (2) is satisfied with equality. Because investors break even, entrepreneurs receive the social surplus associated with their activity. Figure 1 summarizes the timing.

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