Equilibrium Deposit Contract

The model assumes that the bank offers deposit contracts to the consumers. These deposit contracts allow consumers to withdraw c₁ units of goods from the bank in period 1 if they wish, or they can leave their deposits in the bank until period 2 and receive c₂ units of goods.

Then, the model further assumes free entry into banking, implying that the bank earns zero profits in the equilibrium. The bank makes each depositor as well off as possible while earning zero profits in periods 1 and 2. Because if this does not happen, some other bank could enter the market offering alternative deposit contracts and attract all consumers away from the first bank.

Since all consumers deposit in the bank in period 0, the bank has N units of goods to invest in the technology in period 0. In period 1, the bank must choose the fraction x of the investment to interrupt so that it can pay c1 to each depositor who wishes to withdraw at that time.

If only early consumers show up at the bank to withdraw in period 1, we must have

Ntc1 = xN (Eq-1)

Implying that the total quantity of withdrawals equals the quantity of production interrupted. Then, in period 2, the quantity of uninterrupted production matures; this quantity is used to make payments to those consumers who chose to wait, who are supposed to be only late consumers.

N (1-t)c1 = (1-x) N (1+r) Eq(2)

Implying that the total payout to the late consumers is equal to the total return on uninterrupted production. From Eq (1) and Eq (2) we derive

Eq-3

The above equation is like lifetime budget constraint for the bank that governs how deposit contracts (c₁, c₂) can be set. The bank’s lifetime budget constraint in slope-intercept form can be expressed as

Eq-4

The following figure depicts the lifetime budget constraint. Points A, B and D lie on the constraint, which has a vertical intercept of 1+r/1-t; this is the maximum payout to a late Consumer if the bank does not interrupt any of its production. The horizontal intercept 1/t is the maximum amount that could be withdrawn by the early consumers when all the production is interrupted by the bank.

Point A lies on the tangency between the bank’s lifetime budget constraint and the consumer’s indifference curve, which is the equilibrium deposit contract. Point B would have equal consumption for early and late consumers, and point D is what the consumer could achieve in the absence of the bank.

The equilibrium deposit contract at point A lies north-west of point B, which is the point on the bank’s lifetime budget constraint where the bank’s payouts to early and late consumers are the same.

The equilibrium deposit contract at point A lies in the south-east of point D, which is what the consumer would choose in the absence of the bank.

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