A Net Profit Approach to Productivity Measurement, with an Application to Italy by Carlo Milana Istituto di Studi e Ana

1. A Net Profit Approach to Productivity Measurement, with an Application to Italy byCarlo MilanaIstituto di Studi e Analisi Economica, Rome, Italy • This presentation has been prepared for the OECD Workshop on productivity measurement, 16-18 October, 2006, Bern, Switzerland.

2. Contents I. Measurement problems with non-invariant index numbers II. Empirical evidence in Italy III. Finding a better approach with the normalized profit function IV. An application to Italy V. Conclusion

3. C(w,y) Invariant index number (with respect to y) Invariant index number (with respect to y) y C(w1,y1) C(w0,y1) C(w1,y1) C(w0,y1) C(w1,y0) C(w0,y0) C(w1,y0) C(w0,y0) B D C(w1,y0) C(w1,y1) Inverse of MFP Unit cost of productionwith constant returns to scale C(w,y) = c(w) · y Average cost ___ c(w1) c(w0) = = E A C(w0,y1) C(w0,y0) G F y y1 y0 Output

4. C(w,y) y Non-invariant index number (with respect toy) Effects on unit cost from diseconomies of scale Unit costs of production withnon-constant returns to scale C(w,y) = c(w) · g(y) Average cost C(w1,y0) C(w0,y0) C(w1,y1) C(w0,y1) > C(w1,y1) C(w1,y0) C(w0,y1) C(w0,y0) Inverse of MFP y y1 y0 Output

5. Törnqvist index number “Superlative” index numbersThe Translog-Törnqvist casewithC(w,y) = c(w) · g(y) Diewert (1976)has shown that if the cost function has a Translog functional form, y affects only the first-order terms in w, then Caves, Christensen, and Diewert (1982) have shown that In the case of homothetic separability in y, this price index is a pure price component of cost changes because, under the hypotheses made, the non-invariance elements of the Laspeyres- and Paasche-type economic indexes are completely offset in the geometric average procedure.

6. C(w,y) y C(w1,y0) C(w0,y0) C(w1,y1) C(w0,y1) > C(w1,y1) C(w1,y0) C(w0,y1) C(w0,y0) Inverse of MFP Unit cost of production withnon-constant returns to scale C(w,y) Average cost Non-invariant index number Diseconomies of scale y1 y y0 Output

7. Törnqvist index number “Superlative” index numbersThe Translog-Törnqvist casewith the general case ofC(w,y) Diewert (1976)has shown that if the cost function has a Translog functional form, y affects only the first-order terms in w, then Caves, Christensen, and Diewert (1982)have shown that Moreover, if the Translog cost function has also the second-order terms in w affected by y, then (see, Milana, 2005):

8. c(w1)/c(w0) The ratio falls into the interval between Paasche and Laspeyres index numbers. The ideal Fisher is just one of the points belonging to this interval. The “true” index may be equal to Homothetic case In the homothetic case we always have Paasche Ideal Fisher Laspeyres

9. General non-homothetic case • In the non-homothetic case economic index numbers are non-invariant • (this is because it is not possible to disentangle univocally the mutual effects • of variables) • If we deflate a nominal value by means of a non-invariant price index number • the resulting implicit quantity index is not in general homogeneous of degree 1 • (if, for example, the elementary quantities double, in general the quantity index • does not double). • This undesirable behaviour is related to an anomalous position of the “true” • index number with respect to the Laspeyres and Paasche index numbers.

10. In the nonhomothetic case, wemighthave the following reverse position • • • Laspeyres Ideal Fisher Paasche General non-homothetic case 1 0 C ( w , y ) Tr " True" Laspeyres type 0 0 C ( w , y ) Tr

11. General non-homothetic case Since a geometric average of two non-invariant economic index numbers is generally non-invariant with respect to reference variables, the “superlative” index numbers are also non-invariant in the non-homothetic case. While the price economic index number is linearly homogeneous by construction, in general the corresponding quantity index number fails to satisfy the linear homogeneity requirements in the non-homothetic case. (see, for example, Samuelson and Swamy, 1974, Diewert, 1983, p. 179). Samuelson and Swamy (1974, p. 576) observed that, in the general non-homothetic case, the corresponding quantity index obtained implicitly by deflating the nominal cost by means of the economic price index fails to satisfy the requirements of the linear homogeneity test. Samuelson and Swamy (1974, p. 570) noted: “[t]he invariance of the price index is seen to imply and to be implied by the invariance of the quantity index from its reference price base”.

12. Empirical evidence (I) Table 1. Alternative Measures of TFP Changes Based on Different Cost Functions (in percentage) All industries in the Italian economy Strong nonhomothetic changes

13. Empirical evidence (I) Table 1. (Continued) Alternative Measures of TFP Changes Based on Different Cost Functions (in percentage) All industries in the Italian economy

14. The Net Profit Approach (I) The basic idea is to find an unrestricted function where there are no reference variables. We build on the seminal research of Diewert and Morrison (1986) and Kohli (1990), who used the restricted revenue function to measure the terms-of-trade component of welfare change. We base our developments on the theory of profit functions. (See Lawrence J. Lau, “Profit Functions of Technologies with Multiple Inputs and Outputs”, Review of Economics and Statistics, August 1972, Vol. 54, no. 3, pp. 281-289.)

15. The Net Profit Approach (I) The function should exhibit some desirable properties, such asdifferentiability, homogeneity, and separability with respect to other variables. A possible candidate is thenet profit function Πt(p,w)which can be considered as a transformation function in the space of output and input prices for a given profit value. It is dual to thetransformation function Tt[y,(-x)]defined in the space of output and input quantities.

16. The Net Profit Approach (II) Let’s start with the simplest model of one output (y) and one input (x) of a price-taking firm producing under constant returns to scale and facing the output price (p) and the input price (w) in perfectly competitive markets. Productivity (TFP) is defined as

17. The Net Profit Approach (III) The aim is to provide a measure of the relative rate of technical change (productivity net of scale effect). Under constant returns to scale (no scale effect) and perfect competition, in a one-output, one-input model of production, the relative rate ofproductivity or technical change (TFPG0)between t=0 and t=1, as seen from the perspective of situation t=0,is Similarly, we could define the relative rate of change in TFP with respect to the comparison situation t=1.

18. The Net Profit Approach (IV) where k is the degree of RS. With CRS: k=1 In general, However, k is generally unknwon with w* = w / k if w*rather than w is observed. Normalized net profit function

19. The Net Profit Approach (V) Normalized net profit change Laspeyres-type relative pricechange component

20. The Net Profit Approach (VI) and Normalized net profit change If and then Relative price change component

21. Some empirical results (I) Cost-based measure of TFPG and its components Profit-based measure of TFPG and its components

22. Empirical resultsMain conclusions (I) Italy has had some special reasons to be concerned about productivity of the economy. The high public debt and the unresolved north-south regional divide require sustained growth in production. Many factors seem to constrain economic activities, including highly regulated markets and protective institutional setting in favour of incombents. While empirical studies have concluded that the US, for example, appear to have constant or slightly decreasing returns to scale thank to a relatively free capacity adjustment to the new opportunities of growth, decreasing returns to scale may be more dominant in Italy .

23. Empirical resultsMain conclusions (II) • One important element of productivity growth in Italy is technical change (TFP net of the scale economies or diseconomies). • Scale diseconomiesseem to affect the internal structure of production. • Non-homotheticity appear to prevail over the whole period 1970- 2003, except three years. • Non-constant returns to scale are not neutral, thus bringing about rather strong asymmetric changes in the composition of production and in the use of factor inputs.

24. Empirical resultsMain conclusions(III) Towards “TFP growth accounting” • The negative trend in productivity noted recently in this country, almost disappear with the proposed measure. • Future steps in our nonparametric productivity measurement will be towards the completion of “TFP growth accounting” by correcting our proposed measure for other main components, as for example, market power, cyclical behaviour, externalities, adjustments,technicaland organizational inefficiency. • Volunteers joining the company are welcome! • Critical comments are invited. • Carlo Milana • ISAE, Rome, Italy