A Dynamic Network Production Model for Bangladeshi Banks

Seyd Akther1, 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.

Standard Black Box Model

x=(x1,…xN) inputs

P(x)=the output possibility set

y=(y1,…,yM) desirable outputs

b=(b1,…,bJ) undesirable outputs

y=loans, securities investments

x=labor, physical capital, equity

b=non-performing (bad) loans

bt-1 is an undesirable input that impacts the period t technology

bt is an undesirable output for the period t technology that becomesan undesirable input for the period t+1 technology

Bank Production Model-Asset Approach

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

z=intermediate output=deposits

Are deposits an input or an output?

Both? Core deposits=inputTransaction deposits=output

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),

Final Outputsy t=(yt

1,…,ytM) bt =(bt

1,…,btJ)

zt =intermediate output

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 ModelProduction in period t-1 affects the technology in period t

Intermediate output produced in the second stage of production= iyt

iyt affects stage 2 production in period t+1

iyt = Assets – Required Reserves – physical capital – loans - securities

We will assume that intermediate and final outputs are additive

fyt + iyt

Dynamic Network Model

P1(xt,bt-1) P1(xt+1,bt) P1(xt+2,bt+1)

P2(zt P2(zt+1, iyt) P2(zt+2, iyt+1)

xt,bt-1 xt+1 xt+2,

ztzt+1 zt+2

(fyt,bt) (fyt+1,bt+1) (fyt+2,bt+2)

bt

iyt

iyt+1iyt-1

bt+1

, iyt-1) iyt+2

bt+2

Dynamic Network Technology

1 1 1 1 1 11 2

1 1

0 0 11

1

at 1,

Stage 1 Stage 2

, 1,..., ,

, 1,...

J J

n nj j q qj jj j

J

l lj jj

t

x x n N z z

b b l

1 1 12

1

1 1 1 1 1 1 1 11 2

1 1

, , 1,...,

, ( ) , 1,..., ,

J

l lj jj

J J

q qj j m m mj mj jj j

L b b l L

z z fy iy y iy m M

0 0 12

1

, 1,...,J

m mj jj

iy iy m M

1 21 1

1 11

1

Stage 1 Stage 2

, 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

b b l L

21

1 21 1

, 1,...,

, ( ) , 1,...,

Jt t tl lj j

j

J Jt t t t t t t tq qj j m m mj mj j

j j

b b l L

z z fy iy fy iy m M

1 12

1

, 1,...,J

t t tm mj j

j

iy iy m M

In the intermediate periods, t=2,…,T-1

1 21 1

1 11

1

Stage 1 Stage 2

, 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

b b l

21

1 21 1

, , 1,...,

, ( ) , 1,..., ,

JT T Tl lj j

j

J JT T T T T T T Tq qj j m m mj mj j

j j

L b b l L

z z fy iy fy iy m M

1 12

1

, 1,...,J

T T Tm mj j

j

iy iy m M

And in the final period, T,

1 2

1 1 1 11 1

1

( , , ; ) max{ ... ... ) :

1,

Stage 1 Stage 2

, 1,...,

t T

Jx

n n nj j qj

D x y b g

t

x g x n N z z

1 12

1

0 0 1 1 1 11 1 2

1 1

1 1 1 1 11 1

1

, 1,...,

, 1,..., , 1,...,

, 1,..., (

J

qj jj

J J

l lj j l b lj jj j

J

q qj j m y mj

q Q

b b l L b g b l L

z z q Q fy g iy y

1 1 12

1

0 0 12

1

) , 1,..., ,

, 1,...,

J

mj mj jj

J

m mj jj

iy m M

iy iy m M

1 1 11 1 2Choice variables are: , , , , 1,...,miy m M

2 1 21 1

1 11 1

1

Stage 1 Stage 2

, 1,..., , 1,...,J J

t t t t t tn x nj j q qj j

j j

Jt t tl t b lj j

j

x g x n N z z q Q

b g b

21

1 21 1

, 1,..., , 1,...,

, 1,..., ( ) , 1,...,

Jt t tl t b lj j

j

J Jt t t t t t t tq qj j m t y m mj mj j

j j

l L b g b l L

z z q Q fy g iy fy iy m M

1 12

1

, 1,..., J

t t tm mj j

j

iy iy m M

In the intermediate periods, t=2,…,T-1

2 1 1 2Choice variables are: ,..., , , , 2,..., 1

, 1,..., , 2,..., 1

t tt

tm

t T

iy m M t T

1 21 1

11

Stage 1 Stage 2

, 1,..., , 1,...,J J

T T T T T Tn T x nj j q qj j

j j

Tl T b

x g x n N z z q Q

b g b

11 2

1 1

1 21

, 1,..., , 1,...,

, ( ) , 1,...,

J JT T T T Tlj j l T b lj j

j j

JT T T T T T T Tq qj j m T y m mj mj j

j

l L b g b l L

z z fy g iy fy iy m

1

1 12

1

,

, 1,..., .

J

j

JT T Tm mj j

j

M

iy iy m M

And in the final period, T,

1 2Choice variables are: , , , , 1,...,T T TT miy m M

mean Std. dev. Minimum Maximum

Required reserves 1534 1834 282 8591Unused assets= iy1 5232 8184 98 51652Loans= y1 29510 31516 1413 165043Investments=y2 7139 11303 695 62793

NPL=b 3009 7719 17 40510Capital=x2 757 1195 50 6446Equity=x3 4783 18196 527 144249

Deposits=z 38360 45860 7049 214787Employees=x1 3045 5270 260 24450

assets 47182 58267 9126 301001

iy1y

Table 1. Descriptive Statistics, 20 Banks, 2004 to 2009

( , , ) ( , , )y b xg g g g y b x

The choice of directional vector:

1,..., T will be the percent of the mean

Let T=3.

t=1 in:2005 2006 20070.024

(0.062)0.039

(0.063)0.022

(0.035)0.044

(0.060)0.023

(0.037)0.029

(0.039)0.044

(0.054)0.039

(0.051)0.042

(0.063)0.112

(0.165)0.101

(0.131)0.093

(0.125)# of banks with

8 7 6# of banks with 13 11 10# of banks with

9 9 9# of banks with

8 8 9

1̂

2̂

1 2 3ˆ ˆ ˆ 0

1̂ 0

2ˆ 0

3ˆ 0

3̂

1 2 3ˆ ˆ ˆ

Estimates

t=1 in:2005 2006 2007

2160(1812)

1564(1039)

5309(11309)

1588(1092)

5191(11349)

4986(10299)

874(1903)

605(1128)

768(2801)

2868(2482)

3463(3948)

6768(11773)

3463(3948)

6768(11773)

6446(10935)

6768(11773)

6446(10935)

8011(10214)

1ˆiy

2ˆiy

3ˆiy

1iy

2iy

3iy

Actual and optimal unused assets