McMaster University February, 2012
Liquidity preferences An asset is illiquid if its liquidation value at an earlier time is less than the present value of its future payoff. For example, an asset can pay 1 r 1 r 2 at dates t = 0, 1, 2. The lower the ratio r 1 /r 2 the less liquid is the asset. At time t = 0, consumers don t know in which future date they will consume. The consumer s expected utility is ωu(r 1 ) + (1 ω)u(r 2 ), where ω is the probability of being an early consumer (type 1). Sufficiently risk-averse consumers prefer the liquid asset.
Example: Diamond (2007) Let A = (r 1 = 1, r 2 = 2) represent an illiquid asset and B = (r 1 = 1.28, r 2 = 1.813) a liquid one. Assume investors with power utility u(c) = 1 c 1 and ω = 1/4. The expected utility from holding the illiquid asset is E[u(c)] = 1 4 u(1) + 3 u(2) = 0.375 4 By comparison, the expected utility from holding the liquid asset is E[u(c)] = 1 4 u(1.28) + 3 u(1.813) = 0.391 4 Observe, however, that risk-neutral investors would prefer the illiquid asset, since: E[A] = 1.75 > 1.68 = E[B]
Liquidity risk sharing Consider an economy with dates t = 0, 1, 2, a liquid asset (numeraire) (1, 1) and an illiquid asset (technology) (r, R), with r 1 and R > 1. Suppose that consumer s preferences are given by { u(c1 ) with probability ω U(c 1, c 2 ) = (1) u(c 1 + c 2 ) with probability 1 ω Denoting by ck i the consumption of agents of type i at time k, the optimal risk sharing for publicly observed preferences is c 2 1 = c 1 2 = 0 (2) u (c 1 1 ) = Ru (c 2 2 ) (3) ωc 1 1 + (1 ω) c2 2 R = 1 (4)
Private information and incentives However, liquidity preferences are private unverifiable information! Fortunately, it follows from R > 1 that the optimal risk sharing solution satisfy the self-selection condition 1 < c 1 1 < c 2 2 < R. (5) This implies that there exists a contract that achieves (2) (4) as a Nash equilibrium. The key insight of Diamond (1983) is that such a contract can take the form of a demand deposit offered by a bank.
A model for banks Suppose now that a bank offers a fixed claim r 1 per unit deposited at time 0. Assume that withdrawers are served sequentially in random order until bank runs out of assets. Denoting by f j the fraction of withdrawers before j and by f their total fraction, the payoffs per unit deposited are V 1 (f j, r 1 ) = r 1 1 {fj <r 1 1 } V 2 (f, r 1 ) = [R(1 r 1 f )/(1 f )] + Setting r 1 = c1 1, a good equilibrium corresponds to f = ω, since this leads to V 2 = c2 2 > c1 1 = V 1. However, it is clear that f = 1 (run) is also an equilibrium leading to V 1 c 1 1 and V 2 = 0 < c 2 2.
Example revisited: Diamond (2007) Let the illiquid asset be A = (1, 2), u(c) = 1 c 1 and ω = 1/4 Then the marginal utility condition becomes c 2 2 = Rc 1 1. Substituting into the budget constraint (4) gives R c1 1 = 1 ω + ω R = 1.28, c2 2 = 1.813. Suppose the bank offers the liquid asset B = (1.28, 1.813) to 100 depositors each with $1 at 0 and invests in A. If f = 1/4, the bank needs to pay 25 1.28 = 32 at t = 1. At t = 2 the remaining depositors receive 68 2 75 = 1.813. Therefore a forecast ˆf = 1/4 is a Nash equilibrium. However, the forecast ˆf = 1 is another Nash equilibrium.
A model for interbank loans Consider an economy with 4 banks (regions) A, B, C, D. There is a continuum of agents with unit endowment at time 0 and liquidity preferences given according to (1). The probability ω of being an early consumer varies from one region to another conditional on two states S 1 and S 2 with equal probabilities: Table: Regional Liquidity Shocks A B C D S 1 ω H ω L ω H ω L S 2 ω L ω H ω L ω H Each bank can invest in a liquid asset (1, 1) and an illiquid asset (r < 1, R > 1) and promises consumption (c 1, c 2 ).
The central planner solution The central planner solution consists of the best allocation (x, y) of per capita amounts invested in the illiquid and liquid assets maximizing the consumer s expected utility. This is easily seen to be given by γc 1 = y, (1 γ)c 2 = Rx, where γ = ω H + ω L is the fraction of early consumers. 2 Once liquidity is revealed, the central planner moves resources around. For example, in state S 1, A and C have excess demand (ω H γ)c 1 at t = 1, which equals the excess supply (γ ω L )c 1 from B and D. At t = 2 the flow is reversed, since the excess supply (ω H γ)c 2 from A and C equals the excess demand (γ ω L )c 2 from B and D.
Optimal interbank loans In the absence of a central planner, interbank loans can overcome the maldistribution of liquidity. Suppose that the network is completely connected (i.e links between all banks). To achieve the optimal allocation, it is enough for banks to exchange deposits z i = (ω H γ)/2 at time t = 0. At t = 1, a bank with high liquidity demand satisfies [ ω H + ω H γ 2 which reduces to γc 1 = y. At t = 2, the same bank satisfies ] c 1 = y + 3(ω H γ)c 1, 2 [(1 ω H ) + (ω H γ)]c 2 = Rx, which reduces to (1 γ)c 2 = Rx.
Shocks and stability Allen and Gale then analyze the effects of small shocks to interbank markets with of the form: They show that the complete network absorbs shocks better than the incomplete one. Their analytic model is difficult to generalize to arbitrary (asymmetric).
Financial Accelerator Firms need external financing to engage in profitable investment opportunities. Their ability to borrow depends on the market value of their net worth. If an initial shock induces a fall in asset prices, it deteriorates the balance sheets of the firms and their ability to borrow declines. Tightening financing conditions limit investment, which in turn reduces output. Decreased economic activity further cuts the asset prices down, which leads to deteriorating balance sheets, tightening financing conditions and declining economic activity.
A simple theoretical model Consider a firm with cash holdings C and illiquid assets A. To produce output Y the firm uses inputs X financed with borrowed funds B. Suppose that the interest rate is zero and that A can be sold with a price of P per unit after the production, and the price of X is normalized to 1. Thus X = C + B. Suppose now that it is costly for the lender to seize firms output Y in case of default, but ownership of A can be transferred. Then B PA, which implies X C + PA. Thus, an initial decline in the asset prices limits borrowing and leads to decreased economic activity, which feeds back to a fall in asset demand and further fall is asset prices further, causing a vicious cycle.