Transcript ESTIMATING RANGES FOR PROFIT EFFICIENCY WHEN …

HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
SHADOW PRICE APPROACH TO
PRODUCTIVITY MEASUREMENT:
A Modified Malmquist Index
7/7/2015
Timo Kuosmanen
Thierry Post
Helsinki School of Economics
and Business Administration
FINLAND
E-mail: [email protected]
Erasmus University Rotterdam,
Faculty of Economics
THE NETHERLANDS
E-mail: [email protected]
HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Productivity change
= Output change / Input change
1
0
y /y
 1 0
x /x
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HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Productivity can improve through
- Technological development
- Improved operational efficiency
- Utilization of economies of scale and of
specialization
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HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Paasche output index
1 1
py
 1 0
py
Paasche, H. (1922): Über die Preisentwicklung der letzte Jahre nach den Hamburger
Börssennotierungen, Jahrbücher für Nationalökonimie und Statistik 23, 168-178
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HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Laspayers output index
0
1
p y
 0 0
p y
Laspayers, E. (1871): Die Berechnung einer mittleren Waarenpreisteigerung, Jahrbücher
für Nationalökonomie und Statistik 16, 296-314
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HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Fisher ideal output index
1/ 2
p y p y 
  0 0 1 0 
p y p y 
0
1
1 1
Fisher, I. (1922): The Making of the Index Numbers, Boston, Houghton Mifflin
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HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Törnqvist index
y
 

j 1  y
m
1
j
0
j




rj
Törnqvist, L. (1936): The Bank of Finland’s Consumption Price Index,
Bank of Finland Monthly Bulletin 10, 1-8
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HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Fisher ideal TFP index
p y
 0 0
p y


 w0 x1
 0 0
w x
0
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1
1/ 2
py 

1 0 
py 
1 1
1/ 2
wx 

1 0 
wx 
1 1
 p j yt
   j t
j 0 t 0  w x
1
1



t 1
2
HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Törnqvist TFP index
rj
y 





y
j 1 


ci
1
s
 xi 
 0 

i 1  xi 
m
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1
j
0
j
HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Why Fisher index?
-Intuitive: Accounts for the “value” of inputs and outputs
-Diewert (1992: JPA) proved that the Fisher ideal index
passes over 20 ‘axiomatic’ tests - more than any other
candidate considered
-A problem: Perfect price information required. E.g.
i) depreciating durable capital inputs,
ii) new inputs/outputs introduced in the target period,
iii) pricing non-market inputs/outputs.
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HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Malmquist index
1/ 2
 D (y , x ) D (y , x ) 
  0 0 0 1 0 0 
 D (y , x ) D (y , x ) 
0
1
1
1
1

j
1
1
  D ( y , x )
t
t
1

t 1/ 2
j 0 t 0
Malmquist. S. (1953): Index Numbers and Indifference Surfaces, Trabajos de Estatistica 4, 209-242
Caves, D.W., L.R. Christensen, and W.E. Diewert (1982): The Economic Theory of Index Numbers and the
Measurement of Input, Output and Productivity, Econometrica 50, 1393-1414
Färe, R., S. Grosskopf, M. Norris, and Z. Zhang (1994): Productivity Growth, Technical Progress, and
Efficiency Change in Industrialized Countries, Americal Economic Review 84(1), 66-83
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HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
The relationship of the Fisher and
the Malmquist indexes:
Diewert, W.E. (1992): Fisher Ideal Output, Input and Productivity
Indexes Revisited, Journal of Productivity Analysis 3(3), 211-248
The distance function D has a ‘flexible’ function form
=>
the Fisher and the Malmquist indexes are equivalent
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HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
The relationship of the Fisher and
the Malmquist indexes:
Färe, R., and S. Grosskopf (1992): Malmquist Productivity Indexes
and Fisher Ideas Indexes, The Economic Journal 102, 158-160
1) Production technology: Free disposability, Convexity, Constant Returns
to Scale
2) Allocative efficiency in terms of profit maximization (allows for the
Farrell type of technical inefficiency)
=>
the Fisher and the Malmquist indexes equivalent
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HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
The relationship of the Fisher and
the Malmquist indexes:
Balk, B. (1993): Malmquist Productivity Indexes and Fisher Ideas
Indexes: Comment, The Economic Journal 103, 680-682
-Does not suffice that (x0,y0) is allocatively efficient w.r.t. the prices and
technology of the base period, and (x1,y1) is efficient w.r.t. the prices and
technology of the target period.
-We need that (x0,y0) is allocatively efficient w.r.t. the prices and
technology of the target period and (x1,y1) efficient w.r.t. the prices and
technology of the base period!!?!!!
-Although the Fisher and the Malmquist indexes coincide only by accident,
they are reasonable first-order approximations of each other.
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HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Our research problem:
- Can we compute the exact value of the Fisher ideal index
without the price data?
- What is the minimal set of assumptions needed?
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HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
The output distance function:
D ( y, x) : Inf  ( y /  , x)  cmc (T )
t
t
Define the distance function instead of T w.r.t. cmc(T), where
cmc(T) = the smallest monotone convex cone that contains T.
Thus, the distance function has an equivalent dual formulation
 y y'
t
D ( y, x)  Sup 
 1 ( y' , x' )  T 
s m x x'
(  , ) 

t
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HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Output distance function:
y2
T0
A
Y0
O
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y1
HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Example:
y
cmc(T)
T
x
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HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Allocative efficiency:
Define the allocative efficiency in terms of the return-to-thedollar (the profit margin).
The shadow price cone:


y
y'
V t ( y, x) : (  ,  )  s m
 Dt ( y, x);
 1 ( y' , x' )  T t 
x
x'


Definition: Production vector ( y, x) is allocatively efficient
t
t
t
t
t
t




p
,
w
p
,
w

V
( y, x) .
T
with respect to technology
and prices
iff
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HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Intermediate result:
THEOREM 1 (‘The 1st Equivalence Theorem’):
The following conditions are equivalent:
1) Production vectors ( y , x ) and ( y , x ) are allocatively efficient
with respect to both prices  p , w  , p , w  and technologies
T ,T respectively.
2) The Malmquist index and the Fisher ideal index are equivalent.
0
0
1
0
0
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1
0
1
1
1
HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
The modified Malmquist index:
-We propose to correct for the Balk’s approximation errors by
ignoring the ‘irrelevant’ shadow prices. Assume for a moment
that unique shadow prices exist:
1
1

M (y , x )   D (y , x )
0,1
0,1
j 0 t 0
where
j
t
  yt
D (y , x ) 
sup

t
j
j
j
(  , )V ( y , x )   x
j
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t
t
t

t 1
2
,

 , t , j  0,1

HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Example 1:
y2
A
Y1
T0
T1
D
B
C
E
F
Y
O
0
y1
Figure 1: The conventional Malmquist index compares Y0 to the points F and E on
the technology frontiers, and Y1 to C and A. By contrast, the modified Malmquist
index compares Y0 to the points F and D on the iso-revenue surfaces, and similarly
Y1 to B and A.
7/7/2015
HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
A generalization:
-We propose an interval ‘estimator’:
 LF ( y 0,1 , x 0,1 ), U F ( y 0,1 , x 0,1 ) 
1/ 2
 D (y , x ) D (y , x ) 
UF (y , x )   0 0 0 1 0 0 
 D (y , x ) E (y , x ) 
1
0,1
1
1
0
1
1
0,1
1/ 2
 D (y , x ) E (y , x ) 
LF ( y 0,1 , x 0,1 )   0 0 0
1
0
0 
D
(
y
,
x
)
D
(
y
,
x
)

 y
E t ( y, x) 
inft
, t  0,1


.
(  , )V ( y , x )   x 
1
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1
1
t
1
1
.
.
HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Example 2:
y2
Y1
A
T1
T
0
C
B
D
Y0
O
y1
Figure 1: The case of non-unique shadow prices. The lower bound is obtained by
comparing Y0 to the point D and Y1 to A associated with the ‘least favorable’ shadow
prices. The uppe r bound compares Y0 to C and Y1 to B in light of the ‘most
7/7/2015favorable’ shadow prices.
HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
A test application with real data:
-Aggregate production data of 14 OECD countries for years
1970, 1975, 1980, 1985, 1990, 1994
-Variables:
Output: GDP (Mill. US$, 1990 prices).
Inputs: I) No. of employees.
II) The value of the capital stock (Mill. US$, 1990
prices).
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HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
GDP
Mean
AUS
BEL
CAN
DEN
FIN
FRA
GBR
ITA
JPN
NLD
NOR
SWE
USA
WGR
Entire
Sample
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St. Dev.
Growth
*
Labor
Mean
St. Dev. Growth
228109 62561
132311 26986
365282 95909
63803 13032
56515 13601
775771 173145
702660 128081
744208 171158
1719163 587729
189767 40869
57215 16347
112546 19381
4694372 1140329
932372 204628
0,16
0,12
0,16
0,11
0,13
0,13
0,10
0,13
0,21
0,13
0,18
0,09
0,13
0,12
5740
3022
9842
2093
1845
18360
22670
15125
45628
4302
1681
3833
90105
23802
769578 1231081
0,14
17718 23861
*
796
0,07
74
0,00
1909
0,11
177
0,04
157
0,00
1129
0,03
702 - 0,01
761
0,02
7247
0,09
265
0,03
218
0,07
261
0,01
15083
0,09
1467
0,03
0,06
Capital
Mean
St. Dev.
Growth
*
902801 275414
493663 134296
1623141 541238
369467 73453
336694 100584
2406982 620520
2390752 537426
2867872 785554
4864027 2635125
696085 148919
284115 94309
509968 121769
17235742 4382503
3596828 932064
0,19
0,17
0,20
0,12
0,18
0,16
0,13
0,16
0,42
0,13
0,21
0,14
0,15
0,16
2755581 4465362
0,18
HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
The empirical production frontier:
We fitted the Cobb-Douglas production function by the
Corrected Ordinary Least Squares (COLS).
2
R
Coefficients:
Constant
Labor
Capital
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1970
0,806
1975
0,817
1980
0,820
1985
0,822
1990
0,827
1994
0,862
12.354
0.739
9.368
0.682
7.831
0.648
5.904
0.585
9.609
0.671
13.087
0.721
(0.0017)
(0.0016)
(0.0018)
(0.0016)
(0.0017)
(0.0013)
0.261
0.318
0.352
0.415
0.329
0.279
(0.0016)
(0.0018)
(0.0019)
(0.0018)
(0.0018)
(0.0014)
HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
FIN'94
FIN'90
DEN'94
DEN'90
NOR'90
NOR'94
CAN'94
BEL'94
CAN'90
WGR'94
AUS'94 AUS'90
ITA'90
WGR'90
BEL'90
USA'94
NLD'90
GBR'94
NLD'94
USA'90
GBR'90
JPN'94
FRA'94
JPN'90
FRA'90
Capital
ITA'94
1990 frontier
1994 frontier
1990 data
1994 data
Labor
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SWE'94
SWE'90
HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
AUS
BEL
CAN
DEN
FIN
FRA
GBR
ITA
JPN
NLD
NOR
SWE
USA
WGR
7/7/2015
Geometric averages:
Standard Modified
1,0819
1,0431
1,1234
1,0657
1,0409
1,0143
1,0698
1,0456
1,1225
1,0687
1,1100
1,0655
1,1089
1,0618
1,1131
1,0655
1,1370
1,0336
1,1033
1,0672
1,0995
1,0476
1,0668
1,0327
1,0406
1,0230
1,0966
1,0525
Difference:
Minimum
0,0123
0,0333
0,0162
0,0074
0,0484
0,0262
0,0323
0,0380
0,0365
0,0017
0,0064
0,0304
0,0048
0,0173
Mean
0,0356
0,0557
0,0282
0,0254
0,0612
0,0411
0,0481
0,0467
0,0900
0,0334
0,0428
0,0419
0,0187
0,0413
Maximum
0,0545
0,0721
0,0428
0,0414
0,0908
0,0559
0,0733
0,0652
0,1443
0,0595
0,0579
0,0732
0,0343
0,0629
HELSINGIN KAUPPAKORKEAKOULU
HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION
Table 4: The annual gross ‘shadow wages’ per employee (in Thousands U.S. dollars
at 1990 prices and PPPs), assuming 10% depreciation rate for the Capital
1970
1975
1980
1985
1990
1994
AUS
31.6
28.5
28.2
23.7
35.5
48.1
BEL
29.6
26.9
27.8
25.2
39.7
58.3
CAN
37.5
30.0
27.7
23.6
36.3
51.3
DNK
40.3
35.0
31.9
24.8
38.5
53.6
FIN
33.5
30.4
30.1
26.0
43.1
71.3
FRA
26.0
23.5
22.9
19.7
30.5
42.3
GBR
21.0
18.4
17.6
16.0
25.1
37.1
ITA
37.9
32.8
31.0
27.9
45.2
65.8
JPN
12.2
14.8
17.0
15.6
27.9
40.0
NLD
33.8
30.7
28.7
25.0
36.4
49.2
NOR
32.3
29.3
28.5
25.1
41.1
53.2
SWE
28.5
23.8
22.4
18.8
29.9
47.2
USA
45.7
38.9
33.8
27.4
40.6
55.3
WGR
29.6
27.5
26.1
23.1
34.9
48.8
6/1990: 1 US$ = 3.6 FIM
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