An economic project(abstract base)

Financial Regulation Lecture 9

Based on notes by Marlena Eley

March 22, 2019

1

1 Concept Review

We have been studying Cooper and Ross’s paper, which demonstrates how deposit insurance causes moral hazard. Their work uses the Diamond and Dybvig model and adds:

1. Risky Technology

2. Monitoring by Households (HH)

The introduction of deposit insurance reduces HH’s incentives to monitor. HHs would have to pay a fixed cost Γ, which is measured in utils (an effort cost rather than a resource cost), to monitor the bank. When HHs do monitor, they are able to FORCE the bank to invest in the safe technology. We know that HHs always prefer the safe long term technology investment to the risky technology investment because they are risk averse (u′′ < 0).

1.1 LEMMA

Let q denote the probability of a run - this q is exogenously given. A Bank that maximizes HH’s utility solves the following problem:

if q ≤ q∗ the deposit contract offered allows runs if q > q∗ the deposit contract offered is a run preventing contract

When q ≤ q∗, banks are providing liquidity insurance, cE > 1, which means that a run is a potential equilibria. When q > q∗, banks provide contracts similar to autarky allocations, cE = 1,cL = R.

Intuitively, if runs are not very likely, then the bank prefers to offer a contract that allows for runs but offers some liquidity insurance, which is valuable to households. We’re studying when q ≤ q∗ because we want deposit insurance to be offered so that moral hazard is an issue.

1.2 Risky Technology

Risky technology is defined as:

(−1, 0,

{ λR with probability ν

0 with probability 1 −ν

Where:

1. λ > 1

2. νλ ≤ 1

2

Sidenote: when νλ = 1, we say this technology has a mean preserving spread.

Moral hazard is introduced here and this implies that rather than strictly maximizing HH’s utility, Banks are now maximizing profit. The “managers” of the Banks get whatever is leftover after feeding the consumers.

Banks then maximize profit as follows:

max i∈[0,1]

ν[iλR + (1 − i−πcE)R− (1 −π)cL]

+ (1 −ν)max(i∗ 0 + (1 − i−πcE)R− (1 −π)cL, 0)

We see that the objective function is linear in i which tells us that there may be corner solutions where i∗ = 0 (investment is only made in the safe technology) or i∗ = 1 (investment is only made in the risky technology).

1.3 The Threshold ī

We study this decision by establishing a threshold of indifference, ī - when they are indifferent between investing everything in the risky technology and investing in the safe technology.

We can establish ī by looking at the investment decision in the low state, (1 −ν) when the risky technology fails.

Conditional on being in the low state (1 −ν):

ī = {i ∈ [0, 1] : (1 − i−πcE)R− (1 −π)cL = 0}

∀i > ī, (1 − i−πcE)R− (1 −π)cL < 0, max = 0 therefore invests in risky. ∀i < ī, (1 − i−πcE)R− (1 −π)cL > 0, max = (1 − i−πcE)R− (1 −π)cL therefore invests in safe.

Our next step is to plug these values back into the objective function for each case.

If i ≥ ī, hence the max = 0, then the objective function becomes: ν[iλR + (1 − i−πcE)R− (1 −π)cL] which is increasing in i

⇒ i∗ = 1

If i < ī, max = (1− i−πcE)R−(1−π)cL and the objective function becomes: ν[iλR + (1 − i−πcE)R− (1 −π)cL] + (1 −ν)[(1 − i−πcE)R− (1 −π)cL] νiλR + (1 − i−πcE)R− (1 −π)cL which is decreasing in i ⇒ i∗ = 0

Now, we need to know which investment strategy results in the higher profit and therefore which investment strategy will be picked?

3

1.3.1 i∗ = 0

Well when i∗ = 0, the profit becomes: (1 −πcE) − (1 −π)cL. We have been solving the Bank’s problem under the assumption that they would be offering the First Best contracts of c

sp E and c

sp L . We know at the First

Best the resource constraint faced by the Social Planner resembles very closely (exactly) the profit of the bank. The resource constraint being: (1 −π)cL = (1 −πcE)R. This then tells us the the Bank’s profit will be 0 when i∗ = 0!

1.3.2 i∗ = 1

Now we have to evaluate the profit when i∗ = 1, which is νλR−πcER− (1 −π)cL. Again, part of this profit, πcER− (1 −π)cL resembles very closely, actually exactly, the Resource Constraint faced by the Social Planner at the First Best solution, where (1 −π)cL = (1 −πcE)R. This then tells us the profit when i∗ = 1 is νλR, which we know is greater than 0!

Comparing these two profits, we know that the Bank will invest in the risky technology for the larger profit! This is the risky technology, so i∗ = 1.

2 Monitoring

Given what we know:

1. δ = (cE,cL) is the first best allocation and

2. I(cE) and I(cL) are the deposit insurances offered if the bank fails in t=1 and t=2, respectively

If the bank is investing in the safe technology, HH won’t monitor. Why would they pay the fixed cost Γ if they know they will get the safe payout?

Given we know that the Bank will invest in the risky asset (which we proved in part 1), we want to find the Γ for which a HH will choose to monitor. Remember, by monitoring, HH FORCE the Bank to invest in the safe technology. We can find the Γ for which HH will monitor by finding the following:

EUtility without monitoring < E Utility with monitoring

2.1 E (Utility without monitoring)

The Expected utility without monitoring is as below:

π{(1 −q)[U(cL) + qU(I(cE))}+ (1 −π){[νU(cL) + (1 −ν)U(I(cL))] + qU(I(cE))

Broken down we have:

4

1. π{(1 −q)U(cE) + qU(I(cE))}, which is the expected utility of impatient agents. This multiplies π, the fraction of impatient agents by (1 −q), the probability of no run, times the utility they would get in this equilibria, U(cE), plus q, the probability of a run times the utility they would get in a run U(I(cE)). 2.(1 −π){1 −q)[νU(cL) + (1 −ν)U(I(cL))] + qU(I(cE)), the expected utility of patient agents. This multiplies (1 −π), the fraction of patient agents, by (1 −q), the probability of no run, by the payoff received in no run which is νU(cL) + (1 −ν)U(I(cL)), which itself is an expectation of the payoff based on the success or failure of the risky technology, and q, the probability of a run in t = 1, multiplied by U(I(cE)).

2.2 E(Utility with monitoring)

The expected utility with monitoring is as below:

−Γ + [π((1 −q)U(cE) + q(I(cE)))] + (1 −π)[(1 −q)U(cL) + qU(I(cE))]

Remember, by monitoring and incurring the effort cost Γ, HH force the Bank to invest in safe technology so we don’t even consider payoffs of risky technologies!

Broken down we have:

1. π((1 −q)U(cE) + q(I(cE))), which is the utility of impatient agents and 2. (1 −π)[(1 −q)U(cL) + qU(I(cE)), the utility of patient agents.

2.3 E(Utility without monitoring) < E (Utility with monitoring)

This inequality is as below: π{(1 −q)[U(cE) + qU(I(cE))} + (1 −π){(1 −q)[νU(cL) + (1 −ν)U(I(cL))] + qU(I(cE))}≤−Γ+[π((1−q)U(cE)+q(I(cE)))]+(1−π)[(1−q)U(cL)+qU(I(cE))]

This equation greatly simplifies, namely terms for early consumers cancel out, which makes sense as they withdraw at t = 1 anyway so they don’t care about whether the bank invests in the risk technology (they are not around at t = 2 anyway as they consume before and their preferences are not defined over consumption at t = 2). 1 In other words, if the bank invests in risky technologies, they are not the ones who face an insolvent bank.

This then simplifies to: (1 −π){(1 −q)[νU(cL) + (1 −ν)U(I(cL))]}≤−Γ{(1 −π)[(1 −q)U(cL)]}

⇒ Γ ≤ (1 −π)(1 −q)(1 −ν)[U(cL) −U(I(cL))]

Here the tradeoff between deposit insurance and monitoring is greatly apparent, look at the term U(cL) −U(I(cL))!

1This is equivalent to saying they exit the economy, for our purposes.

5

So far, we have been considering deposit insurance to be complete, that is that I(cL) = cL. If this is the case then,

Γ < 0 for HH to monitor the banks!

Therefore monitor will never take place under complete deposit insurance for any value of Γ > 0.

2.4 Finding Γ

Well, lets try to figure out if there is a way to have HH monitor AND still implement the First Best allocation. Let’s consider the assumptions that we’re making:

1. cL = c sp L

2. If there was no insurance at all monitoring would take place if we make the following assumption:

Γ ≤ (1 −π)(1 −q)(1 −ν)U(cspL )

Now we have to consider whether we can implement the first best allocation. Well, if we do, there is a potential for a run equilibrium and HH will monitor. But by monitoring, HH incur the cost Γ and there is a potential for a run.

Well, what happens with complete deposit insurance? HH don’t monitor and Banks end up investing in risky technology. HH are paid even in the event of a run or a Bank failure. Can the First Best be implemented in this situation? The solution offered by Cooper and Ross is Capital Requirements. They suggest that if Banks are solely gambling with depositors money what would happen if Banks were forced to invest some of their own resources?

3 Capital Requirements

In the case of Capital Requirements we have to make an additional assumption.

1. Banks’ shareholders have ”deep pockets”

This just says that Banks get some money from outside the model. These shareholders have a large endowment of the good.

Cooper and Ross then introduce Capital Requirements which require banks to invest at least K units of their own endowment per unit of deposit. With the introduction of Capital Requirements the Banks problem becomes:

max i∈[0,1+K]

{ν[iλR + (1 + K − i−πcE)R− (1 −π)cL]

+ (1 −ν)max((1 + K − i−πcE)R− (1 −π)cL, 0)}

6

We solve this the exact same way that we solved the previous problem!

∃ī ∈ [0, 1 + K] such that the payoff in the low state is the same regardless whether the bank is solvent or not.

(1 + K − ī−πcE)R− (1 −π)cL = 0 ∴ (1 + K − ī−πcE)R = (1 −πcL)

For i ≥ ī: max (1 + K − ī−πcE)R− (1 −π)cL, 0) = 0 ⇒ i∗ = 1 + K (invest in the risky technology)

For i ≤ ī ⇒ i∗ = 0 (invest only in the safe technology)

The question then becomes, is there a value of K such that the First Best allocation can be implemented as an equilibrium outcome?

3.1 Finding K

There are three steps to finding this K:

1. Use the First Best allocation

2. Evaluate the Bank’s profit from investing all in safe or all in risky technology

3. Solve for the smallest K such that i = 0 (investment in all safe technology) yields the highest profit.

Solving for K in this way is solving for the minimum capital requirement necessary for the bank to be indifferent between investing in risky technologies and safe technologies. We know how to solve this: we evaluate the investment strategies at the two corners!

Step 1 - First Best Allocation c sp E > 1 ⇒ c

sp L < R

From the resource constraint at the First Best, (1 −π)cspL = R(1 −πc sp E )

Step 2 - The Bank’s Payoffs

max i∈[0,1+K]

ν[iλR + (1 + K − i−πcE)R− (1 −π)cL]

+ (1 −ν)max((1 + K − i−πcE)R− (1 −π)cL, 0)

Plugging in the Resource Constraint for (1 −π)cL = R(1 −πcE), this simplifies to:

max i∈[0,1+K]

{ν[iλR + (K − i)R] + (1 −ν)max((K − i)R, 0)}

The payoff at i = 0 is then KR. The payoff at i = 1 + K is then ν[(1 + K)λR−R].

7

Step 3 - Finding the Smallest K This then sets us up with the inequality KR ≥ ν[(1 + K)λR−R], from which we can solve for a K.

KR = ν[(1 + K)λR−R] KR = ν(1 + K)λR−νR K = νλ + νKλ−ν

K −νλK = νλ−ν K(1 −νλ) = ν(λ− 1)

∴ K = ν(λ− 1) (1 −νλ)

For K = ν(λ−1) (1−νλ) , there is no monitoring AND we are implementing the First

Best allocation! We are disciplining the moral hazard by implementing these Capital Requirements. Because we know what λ and ν are, we can talk about how K changes when the technology becomes riskier.

3.2 How K Changes with λ and ν

If we assume that the risky technology has a mean preserving spread, i.e. λ̂ν̂ = λν and λ̂ > λ (the technology has higher returns) and ν̂ < ν (the technology has a lower probability of success), then:

K̂ = ν̂λ̂−ν̂ 1−ν̂λ̂

= λν−ν̂ 1−λν ⇒ K̂ > K

This tells us that the riskier the technology, the higher the capital requirement needed to discipline the moral hazard and implement the First Best allocation.

8