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 2/5/2014 SHADOW PRICE APPROACH TO PRODUCTIVITY MEASUREMENT: A Modified Malmquist Index Timo Kuosmanen Helsinki School of Economics and Business Administration FINLAND E-mail: kuosmane@hkkk.fi Thierry Post Erasmus University Rotterdam, Faculty of Economics THE NETHERLANDS E-mail: gtpost@few.eur.nl

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 2/5/2014 Productivity change = Output change / Input change

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 2/5/2014 Productivity can improve through - Technological development - Improved operational efficiency - Utilization of economies of scale and of specialization

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 2/5/2014 Paasche output index

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 2/5/2014 Laspayers output index

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 2/5/2014 Fisher ideal output index

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 2/5/2014 Törnqvist index

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 2/5/2014 Fisher ideal TFP index

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 2/5/2014 Törnqvist TFP index

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 2/5/2014 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.

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 2/5/2014 Malmquist index

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 2/5/2014 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

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 2/5/2014 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 and the Malmquist indexes:

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 2/5/2014 Balk, B. (1993): Malmquist Productivity Indexes and Fisher Ideas Indexes: Comment, The Economic Journal 103, 680-682 -Does not suffice that (x 0,y 0 ) is allocatively efficient w.r.t. the prices and technology of the base period, and (x 1,y 1 ) is efficient w.r.t. the prices and technology of the target period. -We need that (x 0,y 0 ) is allocatively efficient w.r.t. the prices and technology of the target period and (x 1,y 1 ) 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 and the Malmquist indexes:

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 2/5/2014 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?

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 2/5/2014 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

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 2/5/2014 Output distance function:

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 2/5/2014 Example: x y T cmc(T)

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 2/5/2014 Allocative efficiency: Define the allocative efficiency in terms of the return-to-the- dollar (the profit margin). The shadow price cone:

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 2/5/2014 Intermediate result:

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 2/5/2014 The modified Malmquist index: -We propose to correct for the Balks approximation errors by ignoring the irrelevant shadow prices. Assume for a moment that unique shadow prices exist:

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 2/5/2014 Example 1:

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 2/5/2014 A generalization: -We propose an interval estimator:

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 2/5/2014 Example 2:

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 2/5/2014 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).

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 2/5/2014

27. 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 2/5/2014 The empirical production frontier: We fitted the Cobb-Douglas production function by the Corrected Ordinary Least Squares (COLS).

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 2/5/2014

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 2/5/2014

30. 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 2/5/2014 6/1990: 1 US$ = 3.6 FIM