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