H E L S I N G I N K A U P P A K O R K E A K O U L U H E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N 06/07/22 SHADOW PRICE APPROACH TO PRODUCTIVITY MEASUREMENT: A Modified Malmquist Index Timo Kuosmanen Helsinki School of Economics and Business Administration FINLAND E-mail: [email protected]Thierry Post Erasmus University Rotterdam, Faculty of Economics THE NETHERLANDS E-mail: [email protected]
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H E L S I N G I N K A U P P A K O R K E A K O U L U H E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N.
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H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
04/10/23
SHADOW PRICE APPROACH TO PRODUCTIVITY MEASUREMENT:
A Modified Malmquist Index
Timo KuosmanenHelsinki School of Economics and Business AdministrationFINLANDE-mail: [email protected]
Thierry PostErasmus University Rotterdam, Faculty of Economics THE NETHERLANDSE-mail: [email protected]
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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Productivity change= Output change / Input change
01
01
/
/
xx
yy
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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Productivity can improve through- Technological development
- Improved operational efficiency
- Utilization of economies of scale and of specialization
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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Paasche output index
01
11
yp
yp
Paasche, H. (1922): Über die Preisentwicklung der letzte Jahre nach den HamburgerBörssennotierungen, Jahrbücher für Nationalökonimie und Statistik 23, 168-178
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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Laspayers output index
00
10
yp
yp
Laspayers, E. (1871): Die Berechnung einer mittleren Waarenpreisteigerung, Jahrbücherfür Nationalökonomie und Statistik 16, 296-314
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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Fisher ideal output index
2/1
01
11
00
10
yp
yp
yp
yp
Fisher, I. (1922): The Making of the Index Numbers, Boston, Houghton Mifflin
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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Törnqvist index
m
j
r
j
j
j
y
y
10
1
Törnqvist, L. (1936): The Bank of Finland’s Consumption Price Index,Bank of Finland Monthly Bulletin 10, 1-8
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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Fisher ideal TFP index
1
0
21
1
02/1
01
11
00
10
2/1
01
11
00
10
j
t
ttj
tj
xw
yp
xwxw
xwxw
ypyp
ypyp
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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Törnqvist TFP index
s
i
c
i
i
m
j
r
j
j
i
j
xx
y
y
10
1
10
1
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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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.
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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Malmquist index
1
0
1
0
2/1
2/1
001
111
000
110
),(
),(
),(
),(
),(
j t
tttj xyD
xyD
xyD
xyD
xyD
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 theMeasurement of Input, Output and Productivity, Econometrica 50, 1393-1414
Färe, R., S. Grosskopf, M. Norris, and Z. Zhang (1994): Productivity Growth, Technical Progress, andEfficiency Change in Industrialized Countries, Americal Economic Review 84(1), 66-83
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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The relationship of the Fisher andthe 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
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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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
The relationship of the Fisher andthe Malmquist indexes:
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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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.
The relationship of the Fisher andthe Malmquist indexes:
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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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?
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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The output distance function:
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
)(),/(:),( tt TcmcxyInfxyD
tt Txyx
y
x
ySupxyD
ms
)','(1'
'),(
),(
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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Output distance function:
A
y2
y1
Y 0
T0
O
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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Example:
x
y
T
cmc(T)
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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Allocative efficiency: Define the allocative efficiency in terms of the return-to-the-dollar (the profit
margin).
The shadow price cone:
ttmst Txyx
yxyD
x
yxyV )','( 1
'
';),(),(:),(
Definition: Production vector ),(xy is allocatively efficient
with respect to technology tT and prices ttwp, iff ttwp, ),(xyVt .
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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Intermediate result:THEOREM 1 (‘The 1st Equivalence Theorem’):The following conditions are equivalent:1) Production vectors ),( 00xy and ),( 11xy are allocatively efficientwith respect to both prices 00,wp , 11,wp and technologies 0T,1T respectively.2) The Malmquist index and the Fisher ideal index are equivalent.
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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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:
0 ,1 0 ,1( , )M y x 1 1 1
2
0 0
( , )tj t t
j t
D y x
,
w h e r e
( , ) ( , )
( , ) s u p , , 0 , 1j j j
tj t t
tV y x
yD y x t j
x
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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Example 1:
Figure 1: The conventional Malmquist index compares Y0 to the points F and E onthe technology frontiers, and Y1 to C and A. By contrast, the modified Malmquistindex compares Y0 to the points F and D on the iso-revenue surfaces, and similarlyY1 to B and A.
A
y2
y1
Y 1
Y 0
T0
O
B
T1
C
F
E
D
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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A generalization:
-We propose an interval ‘estimator’:
1 / 21 1 1 0 1 10 , 1 0 , 1
0 0 0 1 0 0
( , ) ( , )( , )
( , ) ( , )F
D y x D y xU y x
D y x E y x
.
1 / 21 1 1 1 10 , 1 0 , 1
0 0 0 1 0 0
( , ) ( , )( , )
( , ) ( , )
t
F
D y x E y xL y x
D y x D y x
.
( , ) ( , )( , ) i n f , 0 , 1
t
t
V y x
yE y x t
x
.
0,1 0,1 0,1 0,1( , ), ( , )F FL y x U y x
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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Example 2:
Figure 1: The case of non-unique shadow prices. The lower bound is obtained bycomparing Y0 to the point D and Y1 to A associated with the ‘least favorable’ shadowprices. The upper bound compares Y0 to C and Y1 to B in light of the ‘mostfavorable’ shadow prices.
A
y2
y1
Y 1
Y 0
T0
O
B
T1
D
C
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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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).
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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GDP Labor CapitalMean St. Dev. Growth* Mean St. Dev. Growth* Mean St. Dev. Growth*
H E L S I N G I N K A U P P A K O R K E A K O U L UH E L S I N K I S C H O O L O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N
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Table 4: The annual gross ‘shadow wages’ per employee (in Thousands U.S. dollarsat 1990 prices and PPPs), assuming 10% depreciation rate for the Capital