Diamond-Dybvig Banking Model

7 months ago

Atiqur Rahman

9 minutes

This model was developed by Douglas Diamond and Philip Dybvig in the early 1980s. The model was published in 1983 by Douglas W. Diamond of the University of Chicago and Philip .H. Dybvig of then Yale University and now of Washington University in St. Louis.

The model primarily shows how an institution with long-maturity assets and short-maturity liabilities can be unstable. It is a simple model that captures an intertemporal aspect of consumption and investment. It further explains the bank runs and related financial crises. The model exhibits how banks’ portfolios of illiquid assets like mortgages or business loans and liquid liabilities like deposits, which can be withdrawn anytime, would lead to a situation of panic among depositors.

The model emphasizes that all business investments often require expenditure in the present in order to obtain results in future. Therefore, investors prefer loans with long maturity, implying low liquidity. The same rule is applied to individuals looking for finance housing or automobiles. While individual cavers, including both households and firms, may have sudden requirements of funds due to unforeseen and unpredictable needs, therefore they require funds which permit immediate access to deposits, implying that they would prefer short-maturity deposit accounts.

Since banks act as intermediaries between savers who prefer liquid assets and borrowers who prefer long-maturity loans, the banks, therefore, play a valuable role by channelling funds from individual deposits into loans for borrowers. Banks help depositors save money on transaction costs they will have to pay while lending to investors directly, while investors will only pay off in the future. Since banks provide valuable services to both, they can charge higher interest rates on loans than they pay on deposits and thus earn profits on margin.

As far as the Central bank is concerned, Diamond and Dybvig consider a role for central bank policy originating from the central bank’s superior ability to make payoffs to depositors contingent on realized aggregate outcome. Keeping this theoretical background in consideration, the model can now be illustrated as follows.

Let us consider the three time periods: 0, 1 and 2. There are N consumers where N is very large, and each consumer is endowed with one unit of a good in period 0, which can serve as input to production. The production technology takes one unit from period 0 and converts it into 1+r units of consumption goods in period 2. If production is interrupted, then nothing is produced in period 2.

A consumer may wish to consume early in period 1 or to consume late in period 2. However, in period 0, individual consumers do not know whether they are early or late consumers; they will get to know this in period 1. In period 0, each consumer knows that he has a probability t of being an early consumer and a probability (1-t) of being a late consumer.

Further, in period 1, t*N consumers learned that they were early consumers, and (1-t) N consumers came to know that they were late consumers. 0 < t <1, the production technology captures utility in a simple way that it reflects investing in long-maturity assets that could be sold with some loss before it mature. Such production technology takes care of the possibility that the consumer might consume early, taking into account the need for liquid assets required for unforeseen circumstances.

A consumer receives a satisfaction or utility given by U(c) where U is the Utility function and c is consumption. The Utility function is assumed to be concave, implying that the marginal utility of consumption declines as consumption increases. The Marginal Utility of consumption is given by MUc. Fig (1)

Under uncertainty, a consumer maximizes the expected utility given by

Expected Utility = tU (c₁) + (1- t) U (c₂) (Eq-1)

Where c₁ is consumption if the consumer consumes in an early period and c₂ is consumption if the consumer is a late consumer. Therefore, expected utility is a weighted average of utilities that occur over time, and weights are probabilities. The consumer’s expected utility preferences can be represented in terms of indifference curves with c₁ as early consumption and c₂ as late consumption, as shown in Figure (2).

These indifference curves are downward-sloping and convex. The marginal rate of substitution of early consumption for late consumption for the consumer is given by:

Where MRS c₁,c₂ is minus the slope of the indifference curve in Fig (2)

When c₁=c₂, it implies that early consumption and late consumption are equal. This further means MUc₁=MUc₂ because if consumption is the same, the marginal utility of consumption is also the same.

MRS c₁,c₂ = t/(1-t) (Eq-3)

The consumer invests one unit of endowment in the technology in period 0.
The consumer in period 1 interrupts the technology and is able to consume c1=1
If the consumer is a late consumer, then technology is not interrupted, and the consumer gets c₂=1+r in period 2 when the investment matures.

The above model can also be represented as follows: there is an agent who has an initial endowment of one unit in period 1 and none in subsequent periods. The agent deposits this endowment with an intermediary in period 0 in return for consumption at periods 1 and 2. The intermediary has access to two investment technologies.

The first corresponding to short-term asset yields one unit of consumption in period 1 for every unit of investment at the beginning of that period. The second corresponds to investment in the long–term asset yields R>1 unit of consumption in period 2 for every unit of investment in period 1. Individuals are also able to store consumption between period 1 and period 2. The cross-section distribution of preference shocks realized at period 1 is the same as the probability distribution of these shocks for an individual agent at period 0.

In exchange for deposits, the intermediary offers each agent a contract which allows him to withdraw either c₁ units of consumption in period 1 or c₂ units in period 2. In order to finance withdrawals, the intermediary liquidates L units per capita in period 1 and hence receives R(1-L) units per capita in period 2.

The intermediary chooses (c₁,c₂, L) to maximize the ex-ante expected utility of individual agents. Then, it solves the problem.

Max tU (c₁) + (1-t) U (c₂) (Eq-4)

Subject to t₁ c₁ = L

The following are the first-order conditions

U’(c₁*) = R U’( c₂*)
C₁* = L*/t and c₂ = (1 – L*) R/(1-t)

Clearly, c₁*>c₂* is implied by the above (1) and (2) and the condition that R>1.

A sufficient condition for c2*/ c1* to be less than R is that U’ ( c ) be decreasing in c. Thus, if U has a coefficient of relative risk aversion greater than unity, then early consumers will share higher returns on illiquid assets.

The first best allocation (c*₁,c*₂, L* ) is obtained by (1) and (2).

There are two Pareto-ranked Nash equilibrium, according to Diamond and Dybvig. When the agents who genuinely have a preference for early consumption choose to make an early withdrawal. While the second best Pareto-dominated equilibrium occurs when agents who actually prefer late consumption but fear withdrawal by others of the same type also choose to withdraw early. The inefficiency of these bank runs arises directly from premature and unnecessary disinvestment of higher-yielding assets.

Diamond-Dybvig Banking Model

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