are both satisfied. In addition, defineη∗∗by q2(1−θ)
ζ¯2−η∗∗+ η∗∗
0
F(z)dz (1−τ)(1−θ−ρ) =η∗∗ . Then the solution to (17) and (21) satisfies (15) iff
η∗∗/(1−θ) (24)
>[θ(1−θ)/(1−θ−ρ)[(1−τ)(1−θ−ρ) +θρ]]
+{(1−θ)/[(1−τ)(1−θ−ρ) +θρ]}ζ[η∗∗, q1;γ]. We then have the following claim.
Proposition 2. A steady state equilibrium in which banks follow strategy 2 at each date exists iff (22), (23) and (24) hold.
As we have noted, there is at most one steady state in which banks follow strategy 1 (2) for all time. However, there still remains the possibility that there are multiple steady state equilibria. In particular, we can assert the following.
Proposition 3. Suppose that (19), (20), (22), (23), and (24) are all satisfied. Then there are two steady state equilibria. In one, banks follow strategy 1 at each date and the reserve requirement binds. In the other, banks follow strategy 2 at each date and the reserve requirement binds.
This proposition is of interest because it describes conditions under which multiple steady states may arise. If there are two steady states, and if the economy ends up in a steady state where strategy 2 is followed in each period, then there is a strong sense in which the bank failure rate,F(η), is higher than it needs to be. Therefore, financial markets may operate “poorly” for purely endogenous reasons.
Having described conditions under which one or more steady states exist, we now turn our attention to the issue of how steady state equilibria depend on various possible policy choices that the government may make.
η RHS (25)
LHS (25) A
B
τ↑ r′
Figure 6.A steady state in which banks follow strategy 1 at each date, and the effects of an increase in the lump sum tax
Indeed, in this context, a narrow banking proposal amounts to nothing more than a 100% reserve requirement.14Finally, it is often viewed as socially “irresponsible” to monetize the losses associated with “bailing out” banks. Together, these assertions amount to the statements thatρ,θ, andτ should all be set at relatively high values.
We now examine the validity of these assertions in the context of our model.
In order to derive comparative statics results about how the choices ofρ,θ, and τaffect an equilibrium, we proceed as follows. We provisionally assume that there is a steady state in which banks follow strategy 1 in each period. We then show how changes in policy parameters affect the candidate equilibrium value ofη. Since the value ofη¯is independent of these parameters, we can then not only draw an inference regarding howηchanges for steady states in which strategy 1 is followed.
We can also infer how changes in policy variables may impact on the existence of such an equilibrium. Finally, we do not formally derive comparative statics results for economies in which there is a steady state (or in which there is also a steady state) where banks follow strategy 2. Such results are qualitatively similar to those for steady states in which strategy 1 is followed.
To begin, it will be useful to eliminaterfrom (16) and (18). Doing so yields the condition that determines the steady state value ofηwhen banks follow strategy 1:
q1 π¯1−η+ η
0
G(z)dz
={θ(1−θ)(1−τ)/[(1−τ)(1−θ−ρ) +θρ]}
+{(1−θ)(1−τ)(1−θ−ρ)/[(1−τ)(1−θ−ρ) +θρ]}π(η, q1;γ). (25) The determination ofηis depicted in Figure 6. We can now depict the consequences of changes in policy parameters diagrammatically.
14100% reserve requirements were also advocated by Friedman (1960).
7.1 A change inτ
The effects of an increase in the volume of deposit insurance costs that are financed out of general revenue is depicted in Figure 6. Evidently, the left-hand-side of (25) is not affected by a change inτ. The derivative of the right-hand-side of (25) with respect toτis given by the expression
−{θρ(1−θ)/[(1−τ)(1−θ−ρ) +θρ]2}[θ+ (1−θ−ρ)π(η, q1;γ)]≤0. Thus increases inτshift the right-hand-side of (25) downwards in Figure 6,unlessρ (the deposit insurance premium) is zero.As a consequence, the candidate equilibrium value ofηrises ifρ >0holds. There are now two possible equilibrium outcomes.
One is that a steady state equilibrium in which banks follow strategy 1 exists, both before and after the increase inτ. In this event, sinceG(η)is the probability of bank failure, an increase inτ leads to a higher rate of bank failures. Thus, an increased reliance on general revenue to fund the costs of deposit insurance provision will generally have adverse consequences for the health of the banking system.
A second possibility is that there exists a steady state equilibrium in which banks follow strategy 1 before the increase inτ. However, the increase inτraisesηabove
¯
η(which is independent ofτ). In this situation the increase inτ implies that there is no longer a steady state in which banks follow strategy 1. As a result, either there is a steady state in which banks follow strategy 2 or there is no steady state. In the former case, the increase inτcontinues to have the effect of raising the rate of bank failure. Thus, funding a deposit insurance program largely or entirely out of general revenue will very generally have a negative impact on bank failure rates, so long as ρ >0.
When ρ = 0 an interesting possibility arises. In this case, an increase in τ has no effect onη, or onr. Therefore the rate of bank failure is independent of the government’s financing scheme. Moreover, since neitherη norr changes asτ varies, the level ofτhas no consequences for the rate of inflation. Thus monetizing losses associated with bank bailouts is not inflationary when no deposit insurance premium is levied. Intuitively, as the level of general taxation rises, the implied government deficit associated with operating a deposit insurance system falls. By itself, this decline in the operating deficit would act to reduce the rate of inflation.
However, the increase in taxation also reduces deposits, and hence the demand for reserves. Other things equal, this would tend to increase the rate of inflation. When ρ= 0holds, these two effects exactly offset each other. Moreover, whenρis small, these effects should “nearly” offset each other. Thus the inflationary consequences of monetizing bank bailouts will be small when deposit insurance premia are small, as they typically are in practice.
Parenthetically, one interpretation of ρ = 0is that there is no formal deposit insurance system in place. However, the government is committed to prevent de- positor losses, perhaps because banks are “too big to fail.” Under this situation, it is economically irrelevant whether bank bail-outs are financed with general revenue, or with income from the inflation tax. Of course whenρ >0holds, we reiterate that it
will generally matter how the government finances the insurance of depositors, and that a greater reliance on general revenue will, in this case, lead to more frequent bank failures.
7.2 A change in the deposit insurance premium
The effects of changes in the deposit insurance premium and in the reserve require- ment are generally ambiguous. To see this it suffices to focus on a simple special case.
We therefore consider the following example:τ = 0holds (all losses from deposit insurance provision are monetized). Whenτ= 0holds, equation (25) reduces to
q1 π¯1−η+ η
0
G(z)dz
= [θ+ (1−θ−ρ)π(η, q1;γ)]/(1−ρ). (26) Evidently, the left-hand side of (26) is unaffected by changes inρ. The derivative of the right-hand-side of (26) with respect toρis given byθ(1−π(η, q1;γ)]/(1−ρ)2. It follows that an increase inρreducesη if 1 > π(η, q1;γ)and that it increases η otherwise. Since the probability of bank failure is G(η), the effect of a higher deposit insurance premium on bank failure rates depends critically on the magnitude ofπ(η, q1;γ).
In order to give some economic content to this quantity, it is useful to ask how a small increase in the deposit insurance premium affects the economy whenρ is initially set equal to zero. In this case,θ+ (1−θ)π(η, q1;γ) = r(1−τ)is satisfied initially. Thus ifr(1−τ)<1,1 > π(η, q1;γ)will hold as well. Thus in an economy where real deposit rates are low and/or the reliance on general revenue for the financing of deposit insurance programs is high, it will be the case that an increase in the deposit insurance premium will lead to a reduction in bank failure rates. It will lead to an increase in bank failure rates when the real deposit rate is high, and/or the reliance on general revenue for the support of deposit insurance is low. Parenthetically, a change in the deposit insurance premium can also affect the number of steady state equilibria.
7.3 Changes in reserve requirements
To see the effects of changes inθ, it will be useful to consider the special case:ρ= 0 holds (so no explicit deposit insurance premia are charged). Now (25) reduces to
q1 ¯π1−η+ η
0
G(z)dz
=θ+ (1−θ)π(η, q1;γ). (27) Clearly the left-hand-side of (27) is independent ofθ, while the derivative of the right- hand-side of (27) with respect toθis given by the expression1−π(η, q1;γ). Thus if1> π(η, q1;γ)holds, a higher reserve requirement leads to a lower rate of bank failures, while if this condition fails the converse is true. Economically speaking, it continues to be the case in this example thatθ+ (1−θ)π(η, q1;γ) =r(1−τ)
obtains. Thus for economies with low (high) real deposit rates and/or heavy (limited) reliance on general revenue to fund deposit insurance, it will be the case that higher reserve requirements lead to lower (higher) rates of bank failure. Again the number and type of steady state equilibria that exist can also depend in a complicated way on the choice ofθ.