Dynamic Network Performancewith an Application to Japanese
Cooperative Shinkin Banks
Hirofumi Fukuyama1* and William L. Weber2
1. Faculty of Commerce, Fukuoka University, Japan2. Department of Economics and Finance, Southeast Missouri State University, U.S.A.
• Efficiency Measures-Distance Functions
• Farrell (JRSS-1957), Shephard (1970)
• Data Envelopment Analysis-Charnes, Cooper, Rhodes (EJOR-1978)
• Färe, Grosskopf, and Lovell (Production Frontiers-1994)
• Directional Distance Functions-Chambers, Chung, and Färe (JET-1996, JOTA-1998), Färe and Grosskopf (2004)
Production With Undesirable Outputs
• Färe, Grosskopf, and Weber (Ecol. Ec.-2006)-Agriculture
• Färe, Grosskopf, Noh, and Weber (J.Econometrics-2005)- Färe, Grosskopf, Pasurka, and Weber (App. Ec- 2011)-Electric Utilities
• Fukuyama and Weber (2008, 2009, 2010, 2011)-Financial Institutions
• Rogers and Weber (2011)-Transportation
Standard Black Box Model
x=(x1,…xN) inputs
P(x)=the output possibility set={(y,b): x can produce (y,b)}
y=(y1,…,yM) desirable outputs
b=(b1,…,bJ) undesirable outputs
Directional Distance Function
( , , ; ) max{ : ( , ) ( )}y bD x y b g y g b g P x
y
b
(b,y)
y+βgy
b-βgb
gy
gb
P(x)
y1
y2
P(xd, xu)
0
P(xd’,xu’)
1 1
desirable inputs undesirable inputs
( ,..., ) ( ,..., )d d u d Nx x x x x x
' , 'd d u ux x x x
y
b
P(xd,xu)
0
P(xd’,xu’)
' , 'd d u ux x x x
DEA (CRS) Production Technology
1
1
1
( , , ) : , 1,..., ,
, 1,..., ,
, 1,..., ,
0, 1,...,
Jt t t
nj j nj
Jt tmj j m
j
Jt tlj j l
j
tj
T x y b x x n N
y y m M
b b l L
j J
1
1
1
( , , ; ) max{ : , 1,..., ,
, 1,..., ,
, 1,..., ,
0, 1,..., }
Jt t
o o o nj j no xj
Jt tmj j mo y
j
Jt tlj j lo b
j
tj
D x y b g x x g n N
y y g m M
b b g l L
j J
y=loans, securities investments
xd=desirable inputs=labor, physical capital, net assets (equity capital) b=non-performing (bad) loans
xu=undesirable input=bt-1
Are deposits an input (x) or an output (y)? Both?
Sealey and Lindley (J. of Finance -1977)-intermediation approach
Hancock (JPE-1985)-User cost approachCore deposits=inputTransaction deposits=output
Berger and Humphrey (NBER-1992, EJOR-1997)
Barnett and Hahm (J. Bus. Ec. Stat.-1994)-Banks produce the money supply
Fukuyama and Weber (2010)-Deposits are an input to one stage of production and an output at another stage of production.
Network Production Models
• Färe and Grosskopf (Ec.Letters-1996, SEPS-2000)
• Färe and Whitaker (1996) (Dynamic and Network)
• Kao and Hwang (EJOR-2008)
• Tone and Tsutsui (EJOR-2009)
• Fukuyama and Weber (Omega-2010)
• Färe, Fukuyama, and Weber (IJISSC-2011)
• Akther, Fukuyama, and Weber (Omega-2012))
A Two Stage Network Model
Stage 1P1(x,b)={z that can be produced by (x,b)}
Stage 2P2(z)={(y,b) that can be produced by z}
xt =(xt1,…xt
N), bt-1=(bt-11,…bt-1
J)
y t=(yt1,…,yt
M) bt =(bt1,…,bt
J)
zt =intermediate output=deposits
1
1
{ , , , , such that
, , 1 and , , 2 }.
t t t t t t
t t t t t t t t
N b x z b y
b x z P z b y P
1
11
1 11
1
11
21
21
21
( , , , , ) :
Stage 1:
, 1,..., ,
, 1,..., ,
, 1,..., ,
Stage 2:
, 1,..., ,
, 1,..., ,
t t t t t t
Jt t tn nj j
j
Jt t tl lj j
j
Jt tq qj j
j
Jt tq qj j
j
Jt t tm mj j
j
Jt tl lj j
j
T x b z y b
x x n N
b b l L
z z q Q
z z q Q
y y m M
b b
1 2
, 1,..., ,
0, 0, 1,..., ,
t
t tj j
l L
j J
The Network Technology
11
, 1,..., ,J
t tq qj j
j
z z q Q
21
, 1,..., ,J
t tq qj j
j
z z q Q
1 21
0, 1,..., ,J
t t tqj j j
j
z q Q
The two constraints
First Stage
Second Stage
Can be rewritten as
• Dynamic Models
• Färe and Grosskopf (1996, 1997)
• Bogetoft, Färe, Grosskopf, Hayes, and Taylor (JORSJ-2009)
• Färe, Grosskopf, Margaritis, and Weber (JPA-2011)
Dynamic ModelProduction in period t-1 affects the technology in period t
Intermediate output produced in the second stage of production= ct
ct affects stage 2 production in period t+1
ct = carryover assets= Assets – Required Reserves – physical capital – loans - securities
Bad loans produced in period t-1, bt-1, become an undesirable inputin stage 1 production in period t
Total output consists of final outputs and carryover assets
t t tm m my fy c
Dynamic Network Model (y=fy+c)
P1(xt,bt-1) P1(xt+1,bt) P1(xt+2,bt+1)
P2(zt P2(zt+1, ct) P2(zt+2, ct+1)
xt,bt-1 xt+1 xt+2,
ztzt+1 zt+2
(yt, bt) (yt+1,bt+1) (yt+2,bt+2)
bt
ctct+1
ct-1
bt+1
, ct-1)
bt+2
ct+2
Dynamic Network DEA Technology
1 1 1 1 1 11 2
1 1
0 0 11
at 1,
Stage 1 Stage 2
, 1,..., , 1,..., J J
n nj j q qj jj j
l lj j
DN
t
x x n N z z q Q
b b
1 1 12
1 1
1 1 1 1 1 1 11 2
1 1
11
, 1,..., , 1,...,
, 1,..., , 1,..., ,
0, 1,...,
J J
l lj jj j
J J
q qj j m m mj jj j
j
l L b b l L
z z q Q fy c y m M
j J
0 0 12
1
12
, 1,...,
0, 1,...,
J
m mj jj
j
c c m M
j J
1 1
1 1 1 1 1 1
1 1
{ , , , , such that , , , , , ,
, , , , , , and
, , , , , }.
t t t t t t t t
t t t t t t t t
T T T T T T T T
DN b x z c fy b x z c b fy c N
b x z c b fy c N
b x z c b fy c N
1 21 1
1 11
1
Stage 1 Stage 2
, 1,..., , 1,...,
, 1,...,
J Jt t t t t tn nj j q qj j
j j
Jt t tl lj j
j
x x n N z z q Q
b b l L
21
1 21 1
1
, 1,...,
, 1,..., , 1,...,
0, 1,..., , 2,..., 1
Jt t tl lj j
j
J Jt t t t t t tq qj j m m mj j
j j
tj
b b l L
z z q Q fy c y m M
j J t T
1 12
1
2
, 1,...,
0, 1,..., , 2,..., 1
Jt t tm mj j
j
tj
c c m M
j J t T
In the intermediate periods, t=2,…,T-1
1 21 1
1 11
1
Stage 1 Stage 2
, 1,..., , 1,..., J J
T T T T T Tn nj j q qj j
j j
T T Tl lj j
j
x x n N z z q Q
b b
21
1 21 1
1
, 1,..., , 1,...,
, 1,..., , 1,..., ,
0, 1,...,
J JT T Tl lj j
j
J JT T T T T T Tq qj j m m mj j
j j
Tj
l L b b l L
z z q Q fy c y m M
j J
1 12
1
2
, 1,...,
0, 1,...,
JT T Tm mj j
j
Tj
c c m M
j J
And in the final period, T,
1 2
1 1 11 1
1
( , , ; ) max{ ... ... ) :
1,
Stage 1 Stage 2
, 1,...,
k k k t T
Jx
nk n nj jj
D x y b g
t
x g x n N z
1 1 12
1
0 0 1 1 1 11 1 2
1 1
1 1 1 11 1
1
, 1,...,
, 1,..., , 1,...,
, 1,...,
J
q qj jj
J J
lk lj j lk b lj jj j
J
q qj j mk yj
z q Q
b b l L b g b l L
z z q Q fy g
1 1 12
1
0 0 12
1
, 1,..., ,
, 1,...,
J
m mj jj
J
mk mj jj
c y m M
c c m M
1 1 11 2 1
Choice variables in t=1 are
and , 1,..., , , 1,..., ,j j mj J c m M
2 1 21 1
1 11 1
Stage 1 Stage 2
, 1,..., , 1,...,J J
t t t t t tnk x nj j q qj j
j j
t t tlk t b lj j
j
x g x n N z z q Q
b g b
21 1
1 21 1
, 1,..., , 1,...,
, 1,..., , 1,...,
J Jt t tlk t b lj j
j
J Jt t t t t t tq qj j mk t y m mj j
j j
l L b g b l L
z z q Q fy g c y m M
1 12
1
, 1,..., J
t t tm mj j
j
c c m M
In the intermediate periods, t=2,…,T-1
1 2
Choice variables in t=2,...,T-1 are
and , 1,..., , , 1,..., ,t t tj j m tj J c m M
1 21 1
11
Stage 1 Stage 2
, 1,..., , 1,...,J J
T T T T T Tnk T x nj j q qj j
j j
Tlk T b
x g x n N z z q Q
b g
11 2
1 1
1 21
, 1,..., , 1,...,
, 1,..., , 1
J JT T T T Tlj j lk T b lj j
j j
JT T T T T T Tq qj j mk T y mk mj j
j
b l L b g b l L
z z q Q fy g c y m
1
1 12
1
,..., ,
, 1,..., .
J
j
JT T Tm mj j
j
M
c c m M
And in the final period, T,
1 2
Choice variables in t=T are
and , 1,..., ,T Tj j Tj J
• Network Links:
• in t,
• In t+1,
• In t+2,
• Etc.
1 21
0, 1,...,J
t t tqj j j
j
z q Q
1 1 11 2
1
0, 1,...,J
t t tqj j j
j
z q Q
2 2 21 2
1
0, 1,...,J
t t tqj j j
j
z q Q
• Dynamic links:
• Between t and t+1, Undesirable output at stage 2 in t becomes and input to stage 1 in t+1
Carryover assets from period t become an input to stage 2 in period t+1
• Similar dynamic links between t+1 and t=2, etc.
12 1
1 1
J J
t t t t t tj j j j
j j
b b b b
12 2
1 1
and J J
t t t t t t tj j j j
j j
fy c y c c
269 Japanese Shinkin Banks, 2002-2009
• Shinkin Banks are cooperative
• Accept deposits from members, make loans (real estate and commercial) to member firms within a given prefecture.
• Decline in Shinkin banks from 401 to 271 during 1998-2011 and shrank in size relative to for profit Regional Banks and City Banks
• Research by Nishikawa (1973) , Miyamura (1992) , Miyakoshi (1993) , and Hirota and Tsutsui (1992) has generally found some scale economies, not many scope economies.
• Fukuyama (1996) - large banks more technically efficient than small banks: better managerial oversight dominates any scale economies.
• Färe, Fukuyama, and Weber (2010)-ex ante merger gains: for infra-prefecture mergers biggest gains in Fukuoka and Saga, for inter-prefecture mergers, biggest gains between banks in Miyazaki and Nagasaki.
• Fukuyama and Weber (2008)-For profit regional banks were more efficient, had greater technical progress, but a higher shadow cost of reducing bad loans than cooperative Shinkin banks.
Mean Std. dev. Min. Max.y1=loans 246.2 321.7 18.6 2409.3y2=securities 118.8 139.7 2.0 1119.1c1+c2=carryover assets 90.9 111.2 5.4 1023.2x1=labor 412 408 35 2651x2=physical capital 7.2 9.7 0.2 69.3x3=net assets (equity) 23.8 27.7 0.9 204.6z=deposits 431.0 523.4 33.1 4263.6b=non-performing loans 19.5 24.5 0.8 211.9
Except labor, all variables in billions of Japanese yen deflated by the Japanese GDP deflator
Descriptive Statistics (Pooled data 269 banks x 8 years, 2002-2009
Directional Vector( , , ) ( , , )x y bg g g g x y b
Model uses a three period window: t, t+1, t+2Need 4 years of data, t-1, t, t+1, t+2
1
1
2
is fixed
and are endogenous
is fixed
tk
t t
tk
c
c c
c
100%t Is the percent of mean inputs and undesirable outputsthat can be contracted and percent of mean desirable outputsthat can be simultaneously expanded.
mean Std. dev.
Min. Max. # on frontier
2003-20050.045 0.039 0 0.238 10
0.045 0.038 0 0.225 9
0.047 0.042 0 0.257 9
0.137 0.115 0 0.674 6
1̂2̂3̂
1 2 3ˆ ˆ ˆ
Estimates for 2003-2005
Estimates of Dynamic Inefficiency
2003-2005 2004-2006 2005-2007 2006-2008 2007-2009Karatsu Shinkin Bank xKanonji Shinkin Bank x xThe Kyoto Shinkin Bank x x xYamanashi Shinkin Bank xSapporo Shinkin Bank xJohnan Shinkin Bank x xChoshi Shinkin Bank xSawayaka Shinkin Bank xOsaka Higashi Shinkin Bank
x x x x
Himawari Shinkin Bank x x xKochi Shinkin Bank x x x x x
Frontier Banks
Actual Optimalt-value
(prob>t)
Actual Optimalt-value
(prob>t)
2003-2005 83.4(104.7)
53.9(78.6)
10.81(.01)
87.8(107.9)
71.5(100.2)
6.82(.01)
2004-2006 87.8(107.9)
70.2(94.6)
7.28(.01)
86.7(108.7)
64.1(87.1)
8.54(.01)
2005-2007 86.7(108.7)
63.4(88.1)
8.5(.01)
89.8(106.0)
57.9(75.2)
9.93(.01)
2006-2008 89.8(106.0)
56.4(74.6)
10.53(.01)
97.5(117.8)
52.2(80.1)
11.33(.01)
2007-2009 97.5(117.8)
48.7(66.6)
11.67(.01)
97.2(119.4)
58.5(94.9)
10.43(.01)
tc ˆtc 1tc 1ˆtc
Optimal and Actual Values of Carryover Assets
Calculating optimal deposits from the intensity variables two
max 11
min 21
ˆmaximum value:
ˆminimum value :
Jt t t
j jj
Jt t t
j jj
z z
z z
2 11 1
ˆ ˆJ J
t t t t tj j j j
j j
z z z
Mean(s) Min. Max.
Mean(s) Min. Max.
Mean(s) Min. Max.
2003-2005 0.869(.097)
0.612
1.313
0.868(.095)
0.516
1.190
0.893(.082)
0.545
1.132
2004-2006 0.863(.100)
0.511
1.362
0.870(.095)
0.500
1.118
0.895(.080)
0.544
1.114
2005-2007 0.868(.099)
0.499
1.178
0.862(.097)
0.479
1.115
0.903(.077)
0.574
1.175
2006-2008 0.859(.106)
0.473
1.268
0.856(.097)
0.488
1.203
0.921(.072)
0.646
1.253
2008-2009 0.855(.099)
0.480
1.122
0.874(.097)
0.546
1.328
0.922(.070)
0.658
1.252
ˆt
t
z
z
1
1
ˆt
t
z
z
2
2
ˆt
t
z
z
Ratios of Optimal Deposits to Actual Deposits
• Extension
• Dynamic Luenberger Productivity Growth
• Policy Implication-”Easy to fix” versus “Hard to Break”
1
1 1 1 1 1 1 1
1[ ( , , ; ) ( , , ; )
2
( , , ; ) ( , , ; )]
t t t t t t t t
t t t t t t t t
DL x y b g x y b g
x y b g x y b g
