Discussion Papers In Economics
And Business

Tax-Financed Public Funds Injection into Banksand its Welfare Implications
Ishikawa Daisuke

Discussion Paper 03-02

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANTax-Financed Public Funds Injection into Banksand its Welfare Implications
Ishikawa Daisuke

Discussion Paper 03-02
January 2003

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANTax-Financed Public Funds Injection into Banksand its Welfare Implications
Ishikawa Daisuke∗
Graduate School of Economics, Osaka University

Abstract
I construct a model of a dynamic economy in which the government
collects taxes and injects them into banks’ capitals for stabilization offinancial system. In theoretical part, I derive loan demand and supply
functions from dynamic optimizing problems of households and banks.
Carrying out a simulation, I show that the injection improves welfare
in some first periods, although it rather aggravates in the long term. I
also show that the injection induces efficient investments in some first
periods, while, it induces inefficient investments in the long term.
Keywords: Public funds injection; Stabilization of financial system;
Welfare analysis
JEL classification number : E17; E59; E65∗

Correspondence: Machikaneyamacho 30-4-206, Toyonaka, Osaka 560-0043, Japan; Email: ishikawadaisuke@srv.econ.osaka-u.ac.jp1

1

Introduction

The purpose of this paper is to evaluate the effects of public funds injection
into private banks. In Japan, to mitigate a systemic risk, huge amounts ofpublic funds were injected in 1998 and 1999, and currently there is a heateddispute over whether to inject them or not. The discussion of public funds
injection so far often focuses on moral hazard problem of a bank manager(see Osano, 2001).
Another problem of the injection, which has not been focused on so far,is who bears its cost. This paper considers this unheeded cost and assumes
it is financed by taxes. The actual injection in Japan was carried out byborrowing 7.5 trillion yen (63 billion dollar) from the Bank of Japan andthe private financial institutions, and investing in the preferred stocks andsubordinated debentures which the recipient banks newly issued1 . Seemingly, people do not have any burdens. However, this is not true because
the government fully guarantees its debts. If the preferred stocks or subordinated debentures become valueless by the failures of the recipient banks,
those losses will be compensated by taxes.
The injection may also increase bank lendings and lower a lending rateof interest. This happens because the injected public funds can be used
for the lendings and a blip of the bank’s capital ratio, the ratio of capitals
to assets, lowers its credit cost under the Basel Accord. Figure 1 implies
this is actually the case: when the public funds were injected in 1999, thelending attitude of financial institutions in Japan were greatly improved foraccommodative.
To analyze how these effects influence the macroeconomy through time, I
construct a model of a dynamic economy in which the government collects
taxes and injects them into banks’ capitals as a financial system stabiliza1

The breakdown of the borrowed money was as follows: about 1.2 trillion yen (10billion dollar) from the Bank of Japan, and about 6.3 trillion yen (53 billion dollar) fromthe private financial institutions.

2tion measure. Since it is too difficult to solve analytically the influencesof the public funds injection on the economy, I carry out a simulation byusing the theoretical model and the empirically-known parameters. From
this simulation, it is shown that the injection improves welfare in some first
periods; however, it rather aggravates in the long term. It is also shownthat the injection induces efficient investments in some first periods, while,
it induces inefficient investments in the long term.
To the best of our knowledge, there are few studies on public funds injection
into banks. Kobayashi(2002) makes a survey of financial system stabilization measures, including public funds injection. Osano(2001) studies the
optimal scheme of injection to prevent a bank from taking moral hazard
action.
The rest of this paper is organized as follows. Section 2 derives theborrowing-demand function and the bank’s loan-supply function. Section
3 analyzes the effects of the tax-financed public funds injections into banks.
Section 4 carries out the simulation. Section 5 concludes.

2

The model

2.1

The household production

Consider a discrete-time economy in which agents borrow from a bank andcarry out investments, productions, and consumptions. Hereafter, we callthose agents household productions. They are homogeneous with population
size one. Each household production is risk-averse, and has the following
CRRA type expected utility function at date t:
Et [

∞
X

c1−σ − 1],
β s t+s
1−σ
s=0

(1)

where ct+s is his or her consumption at date t + s, and Et denotes expectations formed at date t. The subjective discount factor β takes the value
between zero and one strictly. σ expresses the degree of relative risk aver3

sion. These parameters are common to all agents.
Each household production has the following Cobb-Douglas type production function:αyt = f (kt−1 ) = kt−1,(2)where kt−1 is the capital used for production at date t − 1, and yt is theoutput (GDP) at date t. α is a capital share relative to the output and lies
between zero and one.
Each household production can borrow from a bank. The amount of theloan made at date t − 1 and the gross rate of interest at date t − 1 aredenoted by bt−1 and Rt−1 , respectively. At date t, he or she consumes ct ,repays Rt−1 bt−1 , and invests kt −kt−1 . Since these expenditures are financedby the output yt and the new loan bt , each household production faces thefollowing budget constraint:f (kt−1 ) + bt ≥ ct + Rt−1 bt−1 + (kt − kt−1 ) .

(3)All these variables are measured by the consumption goods.
Each household production maximizes expected discounted utility (1) subject to intertemporal budget constraint (3) and the perfect foresight assumption Et [xt+s ] = xt+s :max : Et [{c,b,k}s.t.

∞X

β s u(ct+s )] =

s=0

∞
X

c1−σ − 1
β s t+s1−σs=0

f (kt−1 ) + bt ≥ ct + Rt−1 bt−1 + (kt − kt−1 ) .

By appealing to the Bellman’s principle of optimality, the solutions are obtained as follows2 :1

u0 (ct−1 ) = βRt−1 u0 (ct )or ct = (βRt−1 ) σ ct−1 ,1
Rt − 1 α−1
f 0 (kt ) = Rt − 1 or kt = (
)
,
α

2

Refer to Appendix A for the detailed derivation.

4

(4)
(5)

1
Rt − 1 α−1)α1αRt−1 − 1 α−1
Rt−1 − 1 α−1
−(
)
)
−(
.
α
α
1

bt = (βRt−1 ) σ ct−1 + Rt−1 bt−1 + ((6)

Equation (4) claims that he or she makes an optimal consumption choice.
Equation (5) claims that he or she makes an optimal investment choice.
Equation (6) is interpreted as the borrowing-demand function. Equation (6)is derived by substituting optimal paths (4) and (5) for budget constraint(3) with equality.
From equations (2) and (5), the optimal output yt can be obtained asαyt = f (kt−1 ) = kt−1=(

α
Rt−1 − 1 α−1
)
.
α

(7)

To ensure long-term rationality of household productions, we rule out exploding bubbles in the gross rate of interest:
lim Et [β −s Rt+s ] = 0 .

s→∞

(8)

Before we turn to dynamics, it is useful to look at the steady-state equilibrium. The steady-state variables should satisfy
R∗ = 1/β ,1
1 − β α−1k∗ = (
)
,
αβ
α
β−1 ∗
β−1 ∗
1 − β α−1c∗ = ()b + k ∗α = ()b + (),ββ
αβ
α
1 − β α−1y ∗ = k ∗α = ().
αβ
Now I consider the dynamics. Since optimal paths (4)-(6) are non-linear,
it is difficult to derive an analytical solution. Then, I analyze this system
by linearizing around the steady state. Denoting the percentage deviationb t ≡ (Xt − X ∗ )/X ∗ , thefrom the steady state of an optimal path {Xt } as X

linear approximations of equations (4)-(6) are
cbt =

1b
Rt−1 + cbt−1 ,
σ
5

(9)

bt ,
kbt = −a1 R
b
bt

ybt

(10)

1b
b t + a4 R
b t−1 ,
bt−1 − a1 a3 Rβb t−1 ,
= −αa1 R= a2 cbt−1 +

(11)
(12)

where
1
k∗
1
c∗
1
(> 0) , a2 ≡ ( ∗ ) , a3 ≡ ( ∗ ) , a4 ≡ a2 +(1 − α)(1 − β)
b
b
σ
β
1
a1 a3 −a3 .
(α − 1)β

a1 ≡
+

2.2

The representative bank

The balance sheet identity of a bank at date t is given as
CASHt + Lt = Dt + CAPt ,(13)

where CASHt is the amount of cash (riskless assets)3 at date t, Lt is theamount of loans (risky assets) at date t, Dt is the amount of debts at datet, and CAPt is the amount of capitals at date t. We assume that the bankraises funds of debts from the outside perfect financial market. Among thesevariables, CASHt and Lt are endogenous variables, and Dt and CAPt areexogenous variables4 .
At date t, the bank aims to maximize its next date profit Πt+1 underbalance sheet identity (13)5 :max : Πt+1 = (Rt − 1)Lt − (RtD − 1)Dt − C(F Lt , BISt ) ,
Lt

(14)

where Rt is the gross rate of interest on the loans from date t to datet + 1, which is assumed to to be the same as that on the borrowings of3

Actually, banks often hold the government bonds as riskless assets. Here, for simplicity, I assume banks to hold cash as riskless assets.
4
In order to focus on the bank lendings, I assume that the amount of debts is exogenous.
5
Readers may wonder why the bank does not aim to maximize its profit over the”infinite” horizons. We have two reasons to adopt the simplified optimization. First, thedynamic paths become too complicated if we adopt the infinite problem. Second, eachapproach yields the similar results.

6the household productions, RtD is the gross rate of interest on the debts
from date t to date t + 1, Ct is the cost function of the bank at date t,
F Lt ≡ Lt − Lt−1 is a net increase of the loans from date t − 1 to date t,
and BISt ≡ CAPt /Lt is the bank’s capital ratio at date t. Among thesevariables, Rt is an endogenous variable, and RtD is an exogenous variable.
The cost function of the bank at date t, Ct , has two elements: adjustment
costs and credit costs (BIS costs). The bank bears adjustment costs when it
increases new loans (see Ogawa and Kitasaka, 2000, and E.Elyasiani et al.,1995). The cost function Ct satisfies the following conditions:





∂Ct
∂F Lt
∂ 2 Ct
∂F L2t>0,≥0.

The bank also incurs a penalty from the financial market when the bank’s
capital ratio becomes smaller. Hereafter, we call this penalty credit costs(BIS costs) (see Ito and Sasaki, 2000). The cost function Ct also satisfies
the following conditions:





∂Ct
∂BISt
∂ 2 Ct∂BISt2<0,≥0.

I specify the cost function Ct , which satisfies the above conditions, as
C(F Lt , BISt ) = γ0 F Lt +γ1
γ2 1
F L2t +
2
2 BISt2(γ0 , γ1 , γ2 > 0) .

(15)

The first order condition of this profit maximization problem is(Rt − 1) =∂Ct ∂F Lt∂Ct ∂BISt+.
∂F Lt ∂Lt
∂BISt ∂Lt

(16)

Equation (16) claims that the marginal revenue and marginal cost are equal6 .
The optimal path is obtained from equation (16) as
Lt =

γ1
1
γ0 + 1,γ2 Lt−1 +γ2 Rt −γ1 +γ1 +γ1 + γ2 2
2
2
CAPt

6

The second order condition,

CAPt∂ 2 Πt+1∂L2
t≤ 0, is also satisfied.

7

CAPt

(17)

which is interpreted as the loan-supply function.
Now, let us consider the dynamics. The linear approximation of the loansupply function and the steady-state value of Lt are calculated as follows
respectivley:db t + δ3 CAP
b t−1 + δ2 R
b t = δ1 L
L
t ,

L∗ =

CAP
γ2

∗2

(

(18)

1
− 1 − γ0 ) ,
β

where
γ1

(< 1) , δ2 ≡

1

δ1 ≡

γ1 + γ2 ∗2
CAP

2.3

The market equilibriumβL∗ (γ1 +2γ2, δ3 ≡
.
γ2
∗2
∗2 )
CAP
γ
+
γ
1
2
CAP

The market equilibrium condition of the bank loans and borrowings is Lt =bt , which determins Rt . The linear approximation of the above equation isbt = bbt:Lbt , which determins R
bt =
R

1
1
b t−1 } ,{a2 cbt−1 + ( − δ1 )bbt−1 + a4 R
δ4
β

(19)

where
δ4 ≡ δ2 + a1 a3 (> 0) .
Equation (19) shows that a contemporaneous lending rate of interest depends on the various ”lagged” variables.

3

Tax-financed public funds injection into the bankIn this section, I evaluate the effects of tax-financed public funds injection
on the economy. To examin, I adopt the technique employed in Kiyotaki
and Moor (1997): the injection and taxation are considered as unanticipatedshocks to the steady-state equilibrium, and the response to that impulse isconsidered.
8Suppose that the economy is in the steady state in which a bank’s capitalratio is relatively low. Then, in order to improve the bank’s fragility andstabilize the financial system, the government collects taxes from the household productions and injects them into the bank’s capital at date t = T .
The taxes are collected by a lump-sum manner.
1. At date t = T
Each household production pays a lump sum tax τ to the government,
and the government injects them to the bank. Moreover, I assume that the
proportion λ(0 ≤ λ ≤ 1) of the injected public funds are used for the loans,and the rest 1 − λ of those are left as cash. Denoting the ratio of the tax tob T −1 = 0 holds, equations
the output (GDP) as τb(≡ τ /y ∗ ) and noting that X(9) − (12) and (18) at date t = T become
cbT

= 0,

kbT

bT ,
= −a1 RbbT∗
b T + ( y )τb ,= −a1 a3 R
∗
b

bT
L

d
b T + δ3 CAP= δ2 RT + λ(

d
CAP
T

= (

y∗
b.
∗ )τ
CAP

y∗
)τb ,
L∗

bT ,
The percentage deviation of the gross rate of interest at date t = T , R

is calculated by the market equilibrium condition b̂T = L̂T . The result isbT =Ry∗ 1 − λδ3b.
{( ∗ ) − (
∗ )}τ
δ4
b
CAP

(20)

The first term (1 − λ)/b∗ expresses the taxation effect: the lending rate of∗interest rises at date t = T . The second term (δ3 /CAP ) is the injection
effect: the lending rate of interest falls at date t = T . The total effect on
the lending rate of interest at date t = T , which is the sum of the first andsecond effects, depends on parameters.

9

2. At date t ≥ T + 1
Equations (19) − (12) directly apply at date t ≥ T + 1. Equation (18) at
date t ≥ T + 1 becomesb t + δ3 (b t−1 + δ2 R
b t = δ1 L
L

y∗
b.
∗ )τ
CAP

The percentage deviation of the gross rate of interest at date t ≥ T + 1,
b t , is calculated by the market equilibrium condition b̂t = L̂t . The result is
R
bt =
R

4

∗
1
1
b t−1 − δ3 ( yb} .
{a2 cbt−1 + ( − δ1 )bbt−1 + a4 R
∗ )τ
δ4
β
CAP

(21)

Simulation
It is too difficult to solve analytically the influences of the public funds

injection on the economy. Thus, we carry out a simulation to show numerically how the economy developes through time7 . In order to carry out thesimulation, we need to determine values of parameters in a model. As thevalues of the parameters, I adopt the values compatible with the real economy and those based on empirical studies.
Six baseline (deep) parameters appear in this model: α, β, γ0 , γ1 , γ2 , λand σ. For α(the capital share relative to the output) and β(the subjective
discount rate), Cooley and Prescott (1995) reported α = 0.40 and β = 0.98,
where one period is taken as three months. In this simulation, I adopt thesevalues and the interval.
For the parameters in the cost function of the bank, the values of γ0 = γ1 =∗∗1.4 × 10−2 and γ2 = 3.6 × 10−3 imply that ((RD − 1)D )/C(F L∗ , BIS ∗ ) =(costs of paying the interest on debts)/(adjustment costs + credit costs)=98 ,
∗

∗

∗

∗

BIS ∗ (capital ratio) ≡ CAP /L∗ = 0.08, RD = 1.00115 9 and D +CAP =7

For the general instructions of the simulation methods, refer to Kawasaki(1999).
This calibration reflects the fact that about 90 % of the total costs are used for paying
interest on debts in the Japanese banks.
∗
9
We take RD as the average deposit rates from 1995 to 2001 in Japan. As for theamount and term, I adopt ”no less than 10 million yen” and ”3 months less than 6 months”.
810

L∗ . We assume λ = 0.5 and σ = 1. Table 1 collects the values of these baseline (deep) parameters.
When the values of the baseline (deep) parameters in Table 1 are known,
the steady-state values of R∗ , k ∗ and y ∗ can be determined. The results are∗R∗ = 1.018, k ∗ = 1.7 × 102 and y ∗ = 7.8 . The values of CAP =9.4 and∗b∗ = L∗ = 1.2 × 102 imply that BIS ∗ (capital ratio) ≡ CAP /L∗ =0.08 andb∗ /k ∗ =0.7 . And, when these values are known, we can calculate the amountof the consumption in the steady state c∗ . The result is c∗ = 5.6. Finally,
we assume that the tax rate in the steady state, τb∗ = τ ∗ /y ∗ , is 0.02. Table
2 collects the steady-state values of these variables. The values of a1 ∼ a4
and δ1 ∼ δ4 can be calculated from the values in Tables 1 and 2. Table 3
presents the results.
We are now in a position to investigate stability of the dynamics around thesteady state. Using the results of Tables 1 and 3, it can be shown that one
eigenvalue is larger than 1, which corresponds to an explosive path; and thatthe other two eigenvalues lie strictly between -1 and 1, which corresponds tostable paths10 . Therefore, this dynamics has a saddle-stable path. We takethe gross rate of interest to be a jump variable so that the dynamic pathα
0.40

Table 1: Values of the baseline (deep) parameters
β
γ0
γ1
γ2
λ σ
0.98 1.3 × 10−2 1.3 × 10−2 3.6 × 10−3 0.5 1

R∗
1.018

Table 2: Steady-state values of the variables∗
τb∗k∗y ∗ b∗ (= L∗ ) c∗ CAP
BIS ∗1.7 × 102 7.8 1.2 × 102 5.6
9.4
0.08 0.02

a1
91.7

a2
4.8 × 10−2

10Table 3: Values of the coefficients
a3
a4
δ1
δ2
δ3
1.4 1.3 × 102 0.997 0.63 5.9 × 10−3

Refer to Appendix B for the derivation.

11

δ4
1.3 × 102

lies on a two-dimensional stable manifold:bT =Ry∗ 1 − λδ3b jump ,b+R{( ∗ ) − (
∗ )}τ
δ4
b
CAP

b jump is a jump component of Rb T , which is chosen so as to satisfy
where Rthe long-term rationality condition (8). Actually, we can find the value ofb jump such that R
b T +s converges to some finite value for large s.
R

Figure 2 shows how the lending rate of interest developes through time.
At the initial date, it jumps upward greatly, which can be understood as
an overshooting phenomenon of the taxation effect. After the initial date,it gradually falls toward the new steady-state equilibrium, which is about
0.002 % less than the original steady state. One possible explanation ofthis gradual decline is that the injection into the bank’s capitals reduces
the credit cost permanently. Fourth term in the R.H.S. of equation (21),
−δ3 (

y∗
b, captures this effect.
∗ )τ
CAP

Figure 3 shows how the borrowings develope through time. At the initialdate, they jump downward greatly, which is triggerd by the upward jump of
the lending rate of interest. After the initial date, they gradually increase
toward the new steady-state equilibrium, which exceeds the original steadystate by about 1.0 %.
Figure 4 shows how the outputs (GDP) develope through time. At thesecond quarters, they jump downward greatly, which is triggerd by the upward jump of the lending rate of interest. After the second quarters, they
gradually increase toward the new steady-state equilibrium, which exceeds
the original steady state by about 0.8 %.
Figure 5 shows how the capital stocks develope through time. At the initialdate, they jump downward greatly, which is triggerd by the upward jump of
the lending rate of interest. After the initial date, they gradually increase
toward the new steady-state equilibrium, which exceeds the original steadystate by about 0.2 %.
Finally, Figure 6 shows how the consumption developes through time. From
12

the initial date to the 40th quarters, the consumption exceeds the original
steady state: the injection improves welfare in some first periods. However,after the 40th quarters, it gradually decreases toward the new steady-state
equilibrium, which is about 0.25 % less than the original steady state: theinjection rather aggravates welfare in the long term.
The intuition for why this is the case is given by considering equation (9).
b t−1 in the R.H.S. of equation (9) captures the substitution
The first term σ1 R

effect by interest rate fluctuations: when the interest rate at date t − 1,
b t−1 , rises, an agent finds it profitable to consume more at the next date t
Rbecause the gain from tilting its consumption from date t − 1 to date t also
increases. The second term cbt−1 in the R.H.S. of equation (9) captures theeffect of diminishing maginal utility: when an agent consumes more at datet, she finds it profitable to consume more at next date t because the utility
gain from consuming more at date t decreases. Therefore, the consumption
fluctuates parallel to the interest rate with some lags.
Putting Figure 5 and 6 together yields another important implications.
From 14th quarters to 40th, the increase of the capital stocks results in
the increase of the consumption: the injection induces efficient investments.
However, after the 40th quarters, the increase of the capital stocks never results in the increase of the consumption: it induces inefficient investments.

5

Conclusion

I have analyzed how tax-financed public funds injection into banks influence a macroeconomy. First, I carried out theoretical analyses to derive loan
demand and supply functions from dynamic optimizing problems of households and banks. Then, I carried out a simulation by using the theoreticalmodel and the empirically-known parameters. From the simulation, it isshown that the injection improves welfare in some first periods; however, it

13rather aggravates in the long term. It is also shown that the injection induces efficient investments in some first periods, while, it induces inefficientinvestments in the long term. These results have important policy implications: public funds injection itself can mitigate a systemic risk, however, it
can aggravate welfare and induce inefficient investments in the long term.
Therefore, the government should take these things into consideration when
it carries out public funds injection.
Finally, in the future research, it would be very interesting to discuss towhat extent the public funds injection can mitigate a systemic risk. This
discussion is important when we consider the total quantitative evaluationof the public funds injection.

14

Appendix A: derivation of the optimal paths of the household production
Under the perfect foresight assumption Et [xt+s ] = xt+s , the utility maximization problem of the household production over infinite terms is
max : Et [{c,b,k}s.t.

∞X

β s u(ct+s )] =

s=0

∞
X

c1−σ − 1
β s t+s1−σs=0

f (kt−1 ) + bt ≥ ct + Rt−1 bt−1 + (kt − kt−1 ) .

This maximization problem over infinite terms can be solved by the Bellman’s principle of optimality. First, the variables decided at date t − 1 andknown at date t can be proxied by a state variable ωt at date t. In thisproblem, ωt is chosen as
ωt ≡ f (kt−1 ) + kt−1 − Rt−1 bt−1 .
The discounted present value of lifetime utility from date t,

(22)
∞
P
s=0

β s u(ct+s ),is determined by ωt , and let V (ωt ) denote the value function of this lifetime
utility. Then, the above maximization problem over infinite terms can be reduced to a two-period problem with the state variable ωt and value function
V (ωt ) (Bellman equation):
V (ωt ) = max{ct ,bt ,kt }s.t.

[u(ct ) + βV (ωt+1 )]ωt ≡ f (kt−1 ) + kt−1 − Rt−1 bt−1 = ct + kt − bt .

(23)
(24)

When ωt+1 in equation (23) is written in terms of ct , bt and kt by using
equation (24) at date t + 1, the above maximization problem becomes
V (ωt ) = max{ct ,bt ,kt }[u(ct ) + βV (f (kt ) + kt − Rt bt )] s.t. Eq.(44) .

(25)

Although equation (25) is the maximization problem with respect to threevariables ct , bt and kt , this problem can be reduced to that of two variables
ct and kt by eliminating bt :V (ωt ) = max [u(ct ) + βV (f (kt ) + kt − Rt (ct + kt − ωt ))] .
{ct ,kt }

15

(26)

Hereafter, we use this simplified equation (26).
The first order conditions with respect to ct and kt are obtained as follows
respectively:
∂ωt+1
∂ct
∂ωt+1
βVω (ωt+1 )∂kt

u0 (ct ) + βVω (ωt+1 )= u0 (ct ) + βVω (ωt+1 )(−Rt ) = 0 ,(27)= βVω (ωt+1 )(f 0 (kt ) + 1 − Rt ) = 0 . (28)And, the following equation is obtained by differentiating both sides of equation (26) with respect to ωt (envelope theorem):Vω (ωt ) = βVω (ωt+1 )∂ωt+1= βVω (ωt+1 )Rt .
∂ωt

(29)

Substituting Vω (ωt+1 ) in equation (29) for equation (27) yields u0 (ct−1 ) =βRt−1 u0 (ct ), which is the same as equation (4). And equation (28) is equivalent to f 0 (kt ) = Rt − 1, which is the same as equation (5).

16Appendix B: calculating the eigenvalues of the dynamic
systemThe independent variables are only ct , Rt and bt . From equations (15),b t , the transition equations are surmarized(27) and (30) together with bbt = L

into the following matrix form:










cbt−1
cbt
1
1/σ
0

 b
 b  1/δ4 (1/β − δ1 )a4 /δ4   Rt−1  .
 Rt  =  a2 /δ4bb(a2 δ2 )/δ4 δ1 + δ2 /δ4 (1/β − δ1 ) (a4 δ2 )/δ4
bt−1
bt
Using the results of Tables 1 and 3, we can calculate the eigenvalues of thistransition matrix. The results are 1.3628, 0.9994 and -0.7321.

Acknowledgements
I would like to thank Yasuo Maeda, Kazuo Ogawa and Shinichi Kitasaka
for their thoughtful comments and help. Particularly, I would like to thankYoshiro Tsutsui for his extensive instruction on an earlier version of thispaper. I am also indebted to participants in the Monetary Economics Workshop (MEW) held at Osaka University. An earlier version of this paper waspresented at the 2002 Annual Meeting of the Japan Society of Monetary
Economics in Osaka. Of course, I am solely responsible for any remaining
errors.

17

References[1] Cooley, T.F. and E.Prescott, 1995, Economic growth and business cycles,
in: Cooley, T.F (Ed.) Frontiers of Bussiness Cycle Research, Princeton
University Press, Princeton NJ.
[2] Elyasiani, E., K.Kopecky and D.Hoose, 1995, Costs of adjustment, portfolio separation, and the dynamic behavior of bank loans and deposits,
Journal of Money, Credit and Banking, 27, 955-974.
[3] Ito, T. and Y.Nagataki-Sasaki, 1998, Impacts of the Basel Capital Standard on Japanese bank’s behavior, NBER Working paper No.6730.
[4] Kawasaki, K., 1999, Ooyou ippan kinko moderu no kiso to ooyou: kouzou
kaikaku no simyureshon (Foundation and Application of General Equilibrium Model: Simulation Analysis on Economic Structure Reform),
Nihon Hyoronsha, Tokyo.
[5] Kiyotaki, N. and J.Moor, 1997, Credit Cycles, Journal of Political Economy, 105, 211-248.
[6] Kobayashi, K., 2002, Ginko shisutemu kiki eno seisaku taiou (Policy
measures on banking crisis), RIETI Discussion Paper Series 02-J-0016,
The Research Institute of Economy, Trade and Industry (RIETI), Tokyo.
[7] Ogawa, K. and S.Kitasaka, 2000, Bank lending in Japan: its determinants and macroeconomic implications, in: Hoshi, T. and H.Patrick(eds.) Crisis and Change in the Japanese Financial System, Kluwer Academic Publishers.
[8] Osano, H., 2001, Injection of public funds into banks under deposit insurance and bank regulation, in: Osano, H. and T.Tachibanaki (eds.)
Banking, Capital Markets and Corporate Governance, Palgrave 51-84.

18

[9] Walsh, C.E., 2000, Monetary theory and policy, The MIT Press, Cambridge MA.

19

Diffusion Index of "Accommodative"
minus "Severe"

40
30
20
10
0
-10
-20

19

93

-O

c
Ap t
r-9
Oc 4
t-9
Ap 4
r-9
Oc 5
t-9
5
Ap
r-9
Oc 6
t-9
Ap 6
r-9
Oc 7
t-9
Ap 7
r-9
Oc 8
t-9
Ap 8
r-9
Oc 9
t-9
9
Ap
r-0
Oc 0
t-0
Ap 0
r-0
Oc 1
t-0
1

-30

Large Firms

Quarters Firms
Medium-Sized

Small Firms

Fig.1. Lending Attitude of Financial Institutions in Japan(Source: TANKAN (Short-term Economic Survey of All Enterprises in Japan) 2001, the Bank of Japan )0.012

Percentage Deviation from the SteadyState (%)

0.01
0.008
0.006
0.004
0.002
0

-0.002-0.0041

7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115
Quarters
Lending Rate of Interest
Fig.2. Dynamics of the Lending Rate of Interest

1.5

Percentage Deviation from the SteadyState (%)

1
0.5
0

-0.5
-1

-1.5
1

8

15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134
Quarters
Borrowings

Fig.3. Dynamics of the Borrowings
0.1

Percentage Deviation from the SteadyState (%)

0
-0.1
-0.2
-0.3
-0.4
-0.5
1

7

13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115
Quarters
Outputs(GDP)Fig.4. Dynamics of the Outputs(GDP)

0.4
0.2
Percentage Deviation from the SteadyState (%)

0
-0.2
-0.4
-0.6
-0.8
-1

-1.2
1

7

13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115
Quarters
Capital Stocks

Fig.5. Dynamics of the Capital Stocks
0.05

Percentage Deviation from the SteadyState (%)

0
-0.05
-0.1

-0.15
-0.2

-0.25
-0.3
1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201 211
Quarters
Consumption

Fig.6. Dynamics of the Consumption