O PTIMAL R ISK S HARING

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立 政 治 大 學

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N a tio na

l C h engchi U ni ve rs it y

ence is twice continuously differentiable, increasing, and strictly concave; therefore it is given by

U(𝑐₁, 𝑐₂) = { 𝑢(𝑐₁) with probability 𝜔 𝑢(𝑐₂) with probability 1 − 𝜔,

where 𝑐_𝑡 denotes consumption at date 𝑡 = 1, 2, and 𝜔 denotes the probability of being an impatient consumer who withdraws the deposit early. Suppose that 𝜔 has two possible values including a high and a low level, denoted by 𝜔_𝐻 and 𝜔_𝐿, re-spectively. Therefore, the condition, 0 < 𝜔_𝐿 < 𝜔_𝐻< 1, exists.

3.2 Optimal Risk Sharing

3.2.1 The first-best allocation without Deposit Insurance

The optimal risk sharing serves as the solution to a planning problem, that is, the bank makes the decisions of investment portfolio and the demand deposit contract for de-positor’s consumption to maximize the consumer’s expected utility. Let 𝑥 and 𝑦 represent the long asset and the short asset in which the bank could invest, and the average fraction of early consumers be denoted by 𝜋 = (𝜔_𝐻+ 𝜔_𝐿)/2.

We start with planner’s problem without deposit insurance, and extend the origi-nal model to the situation with the deposit insurance mechanism. For a representative bank, the bank’s investment portfolio is derived from consumer’s endowment, which is one unit of consumption good; therefore, the portfolio (𝑥, 𝑦) ≥ 0 must be subject to the feasibility constraint

1

x   y

. (3.1)

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Because the total amount of consumption provided in each period is a constant, the optimal allocation between consumptions and the investment portfolio is that the short asset provides for early consumption at date 1, and the long asset provides for late consumption at date 2. Hence, the feasibility constraint at date 1 is

c

1

y

 

, (3.2)

and the feasibility constraint at date 2 is

 1    c

2

 Rx

_{. (3.3)}

At date 0, each consumer becomes an early or a late consumer with an equal probabil-ity, so the objective function, a consumer’s expected utilprobabil-ity, is given by

  U  u c   

1

1     u c

2

   

. (3.4)

The first-best allocation serves as the unique solution by maximize the objective func-tion (3.4), subject to the constraints (3.1), (3.2) and (3.3).

Let 𝑢(𝑐_𝑖) = ln 𝑐_𝑖, 𝑖 = 1,2, which satisfies characteristics of twice continuously differentiable, increasing, and strictly concave. The equation (3.4) can be rewritten by

  U  ln c

1

 1   ln c

2

   

, (3.5)

and the process of searching optimal allocation is set by maxc_1,c₂ E(𝑈) = 𝜋 ln 𝑐₁+ (1 − 𝜋) ln 𝑐₂

s.t. 𝑥 + 𝑦 ≤ 1 𝜋𝑐₁ ≤ 𝑦

(1 − 𝜋)𝑐₂ ≤ 𝑅𝑥.

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The first-best allocation is achieved as (𝑐₁^∗, 𝑐₂^∗, 𝑥^∗, 𝑦^∗) = *1, 𝑅, (1 − 𝜋), 𝜋+.

The process of the proof is the same as the proof in Appendix A. The first-best alloca-tion satisfies the first-order condialloca-tion 𝑢′(𝑐₁) ≥ 𝑢′(𝑐₂); otherwise, satisfies the incen-tive constraint 𝑐₁ ≤ 𝑐₂, which indicates the late consumers find it weakly optimal to reveal their true type rather than pretend to be early consumers.

3.2.2 The first-best allocation with Deposit Insurance

By analyzing the effectiveness of the deposit insurance, we need to introduce the government-provided deposit insurance into our analysis. In Diamond and Dybving (1983) model, they assumed that how much tax must be raised by a government as the deposit insurance depends on how many deposits are withdrawn at date 1. Therefore, the government is able to levy a proportionate tax on depositors in terms of the amount they have withdrawn at date 1.

Unlike the setting of the deposit insurance mechanism in Diamond and Dybving (1983) model, we simplify the deposit insurance tax rate from a proportionate tax to a lump-sum tax without loss of generality. Then we also expand the Allen and Gale’s (2000) model by assuming that a government can impose a lump-sum tax as the de-posit insurance on each dede-positor regardless of an early or a late consumer. Suppose that the government levies the lump-sum tax with 𝜏 which denotes by percentage, where 0 < 𝜏 < 1, on each consumer at date 0. As a result, consumer’s endowment is reduced by 𝜏%; in other words, the available amount of investment for a bank is re-stricted to 1 ∙ (1 − 𝜏). The process of searching optimal allocation is reset by the fol-lowing equation.

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maxc_1,c₂ E(𝑈) = 𝜋 ln 𝑐₁+ (1 − 𝜋) ln 𝑐₂

s.t. 𝑥 + 𝑦 ≤ 1 ∙ (1 − 𝜏) 𝜋𝑐₁ ≤ 𝑦

(1 − 𝜋)𝑐₂ ≤ 𝑅𝑥.

The first-best allocation became

(𝑐̃₁, 𝑐̃₂, 𝑥̃, 𝑦̃) = *(1 − 𝜏), 𝑅(1 − 𝜏), (1 − 𝜋)(1 − 𝜏), 𝜋(1 − 𝜏)+.

Proof: See the Appendix A for a detailed resolution of the process.

As a consequence, it is easy to find that the new solutions (𝑐̃₁, 𝑐̃₂, 𝑥̃, 𝑦̃) are all less than the originals (𝑐₁^∗, 𝑐₂^∗, 𝑥^∗, 𝑦^∗). Even though the result seems disappointing, depos-itors can be prevented against a bankruptcy by the deposit insurance. Hence, in Sec-tion 6, we will find out an optimal deposit insurance tax rate for maximizing consum-er’s utility, and analyze the advantage and disadvantage of the deposit insurance, and finally discuss the effectiveness of the deposit insurance in enhancing social welfare.

4 Decentralization

Allen and Gale (2000) established various interbank market structures for their analy-sis. In this section, we follow their assumption and sequentially analyze the first-best allocation in different interbank deposit market structures. The economy is assumed to be divided into four identical regions, labeled A, B, C and D, and each region contains identical banks and consumers. In interbank deposit market, except absorbing con-sumers’ deposits for investing in the short and the long assets, banks are also allowed to hold other banks’ deposits from other regions.

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The analysis is based on the assumption that different regions suffer different level of liquidity shocks. The liquidity shock means that an unexpected high liquidity demand takes place in a region at date 1, that is, many consumers rather choose to withdraw their deposits at date 1 than do at date 2. The assumption is precisely illus-trated in Table 1.

Table 1 Regional Liquidity Shocks

State A B C D

S

1 𝜔_𝐻 𝜔_𝐿 𝜔_𝐻 𝜔_𝐿

S

₂ 𝜔_𝐿 𝜔_𝐻 𝜔_𝐿 𝜔_𝐻

Suppose that overall shortage of liquidity demand does not exist, but unbalanced distribution. Hence, interbank deposit market can offer a unique method to overcome the maldistribution of the liquidity. In regard to interbank market structures, here we only focus on the unidirectional incomplete and the complete interbank market struc-tures illustrated in Figure 1 and Figure 2, respectively.

Figure 1 – Unidirectional Incomplete Interbank Market Structure

A B

D C

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