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A Dynamic Network Production Model for Bangladeshi Banks Seyd Akther 1, Hirofumi Fukuyama 1* and...
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Transcript of A Dynamic Network Production Model for Bangladeshi Banks Seyd Akther 1, Hirofumi Fukuyama 1* and...
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
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