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...

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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 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]

Transcript of 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...

Page 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 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.

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]

Page 2: 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.

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

Productivity change= Output change / Input change

01

01

/

/

xx

yy

Page 3: 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.

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

Productivity can improve through- Technological development

- Improved operational efficiency

- Utilization of economies of scale and of specialization

Page 4: 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.

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

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

Page 5: 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.

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

Laspayers output index

00

10

yp

yp

Laspayers, E. (1871): Die Berechnung einer mittleren Waarenpreisteigerung, Jahrbücherfür Nationalökonomie und Statistik 16, 296-314

Page 6: 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.

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

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

Page 7: 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.

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

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

Page 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 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.

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

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

Page 9: 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.

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

Törnqvist TFP index

s

i

c

i

i

m

j

r

j

j

i

j

xx

y

y

10

1

10

1

Page 10: 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.

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

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.

Page 11: 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.

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

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

Page 12: 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.

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

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

Page 13: 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.

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

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:

Page 14: 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.

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:

Page 15: 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.

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

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?

Page 16: 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.

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

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'

'),(

),(

Page 17: 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.

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

Output distance function:

A

y2

y1

Y 0

T0

O

Page 18: 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.

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

Example:

x

y

T

cmc(T)

Page 19: 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.

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

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 .

Page 20: 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.

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

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.

Page 21: 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.

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

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

Page 22: 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.

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

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

Page 23: 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.

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

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

Page 24: 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.

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

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

Page 25: 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.

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

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).

Page 26: 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.

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

GDP Labor CapitalMean St. Dev. Growth* Mean St. Dev. Growth* Mean St. Dev. Growth*

AUS 228109 62561 0,16 5740 796 0,07 902801 275414 0,19BEL 132311 26986 0,12 3022 74 0,00 493663 134296 0,17CAN 365282 95909 0,16 9842 1909 0,11 1623141 541238 0,20DEN 63803 13032 0,11 2093 177 0,04 369467 73453 0,12FIN 56515 13601 0,13 1845 157 0,00 336694 100584 0,18FRA 775771 173145 0,13 18360 1129 0,03 2406982 620520 0,16GBR 702660 128081 0,10 22670 702 - 0,01 2390752 537426 0,13ITA 744208 171158 0,13 15125 761 0,02 2867872 785554 0,16JPN 1719163 587729 0,21 45628 7247 0,09 4864027 2635125 0,42NLD 189767 40869 0,13 4302 265 0,03 696085 148919 0,13NOR 57215 16347 0,18 1681 218 0,07 284115 94309 0,21SWE 112546 19381 0,09 3833 261 0,01 509968 121769 0,14USA 4694372 1140329 0,13 90105 15083 0,09 17235742 4382503 0,15WGR 932372 204628 0,12 23802 1467 0,03 3596828 932064 0,16EntireSample 769578 1231081 0,14 17718 23861 0,06 2755581 4465362 0,18

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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

The empirical production frontier:

We fitted the Cobb-Douglas production function by the Corrected Ordinary Least Squares (COLS).

1970 1975 1980 1985 1990 1994R2 0,806 0,817 0,820 0,822 0,827 0,862

Coefficients:Constant 12.354 9.368 7.831 5.904 9.609 13.087Labor 0.739

(0.0017)0.682

(0.0016)0.648

(0.0018)0.585

(0.0016)0.671

(0.0017)0.721

(0.0013)Capital 0.261

(0.0016)0.318

(0.0018)0.352

(0.0019)0.415

(0.0018)0.329

(0.0018)0.279

(0.0014)

Page 28: 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.

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

BEL'90NLD'90

USA'90

ITA'90AUS'90

WGR'90

FRA'90 J PN'90

SWE'90CAN'90

GBR'90

NOR'90

DEN'90FIN'90

AUS'94WGR'94BEL'94

NLD'94

FRA'94

GBR'94J PN'94

USA'94

ITA'94

CAN'94

NOR'94SWE'94

DEN'94

FIN'94

Labor

Ca

pita

l

1990 frontier

1994 frontier

1990 data

1994 data

Page 29: 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.

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

Geometric averages: Difference:Standard Modified Minimum Mean Maximum

AUS 1,0819 1,0431 0,0123 0,0356 0,0545BEL 1,1234 1,0657 0,0333 0,0557 0,0721CAN 1,0409 1,0143 0,0162 0,0282 0,0428DEN 1,0698 1,0456 0,0074 0,0254 0,0414FIN 1,1225 1,0687 0,0484 0,0612 0,0908FRA 1,1100 1,0655 0,0262 0,0411 0,0559GBR 1,1089 1,0618 0,0323 0,0481 0,0733ITA 1,1131 1,0655 0,0380 0,0467 0,0652JPN 1,1370 1,0336 0,0365 0,0900 0,1443NLD 1,1033 1,0672 0,0017 0,0334 0,0595NOR 1,0995 1,0476 0,0064 0,0428 0,0579SWE 1,0668 1,0327 0,0304 0,0419 0,0732USA 1,0406 1,0230 0,0048 0,0187 0,0343WGR 1,0966 1,0525 0,0173 0,0413 0,0629

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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

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

1970 1975 1980 1985 1990 1994AUS 31.6 28.5 28.2 23.7 35.5 48.1BEL 29.6 26.9 27.8 25.2 39.7 58.3CAN 37.5 30.0 27.7 23.6 36.3 51.3DNK 40.3 35.0 31.9 24.8 38.5 53.6FIN 33.5 30.4 30.1 26.0 43.1 71.3FRA 26.0 23.5 22.9 19.7 30.5 42.3GBR 21.0 18.4 17.6 16.0 25.1 37.1ITA 37.9 32.8 31.0 27.9 45.2 65.8JPN 12.2 14.8 17.0 15.6 27.9 40.0NLD 33.8 30.7 28.7 25.0 36.4 49.2NOR 32.3 29.3 28.5 25.1 41.1 53.2SWE 28.5 23.8 22.4 18.8 29.9 47.2USA 45.7 38.9 33.8 27.4 40.6 55.3WGR 29.6 27.5 26.1 23.1 34.9 48.8

6/1990: 1 US$ = 3.6 FIM