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Efficiency of the Indian leather firms: some results obtained using the two conventional methods

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Abstract

Indian leather industry has massive potential for generating employment and achieving high export-oriented growth. However, its economic performance has not been assessed much till date. The present paper attempts to fill in this gap by examining technical efficiency (TE) of individual leather producing firms for some years since the mid-1980’s. Analyzing the industry’s firm-level data through the two conventional tools, viz., data envelopment analysis and stochastic frontier analysis, the paper observes a significant positive association between a firm’s size and its TE, but no such clear relation between a firm’s age and TE. It also finds significant variation in TE across firms in different groups of states as well as under different organizational structures and observes some technological heterogeneity across states. Although, non-availability of panel data does not allow one to assess effects of economic reforms on the performance of the Indian leather firms, the average firm-level TE, however, seems to be on an increasing path, except for downswing in the immediate post-reform years.

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Notes

  1. For instance, export of leather and leather manufacturers (including leather footwear, leather travel goods and leather garments) went up from US$59 million in 1960–1961 to US$493 million in 1980–1981 and thereafter to US$1449 million 1990–1991 and further to US$2323 million in 2004–2005 (Government of India 2004–2005).

  2. Banerjee and Nihila (1999) and Mohapatra and Srivastava (2002) discuss in detail the various stages of the organizational structure of the industry like the stage of collection of raw materials, the stage of marketing etc.

  3. For instance, India’s share in the world bovine animal population was the highest (about 19%) in 2005, while the shares of Brazil, China and USA in this year were respectively 13%, 9% and 6%.

  4. In addition, a number of leather development programs have been initiated in the recent past, e.g., a UNDP assisted National Leather Development Program (NLDP). The Phase I of this program was carried out during 1992–1998 for upgrading the training systems for the design and manufacture of footwear, garments and leather goods while its second phase—called the Small Industries Development and Employment Program (SIDE-NLDP) was conducted during 1998–2002, the objective being alleviation of poverty and building up of linkages between the organized and unorganized sectors. To complement these programs another scheme titled Indian Leather Development Program (ILDP) started operation in 1992 to bridge critical gaps in infrastructure for integrated development of this industry, to undertake investment/trade development activities and build up an information base for leather industry. Productivity improvement programs have also been launched for upgrading the manufacturing processes of footwear in the organized sector. A scheme for tannery modernization was launched under ILDP in 2000 to provide the financial help to the Indian tanneries so as to enable these units to adopt cleaner process technologies which would abate/reduce pollution.

  5. There is a separate efficiency concept, viz., allocative or price efficiency, which refers to a firm’s ability to combine inputs and outputs in optimal proportions, given their respective prices and production technology. See Coelli et al. (1998, pp. 134–140), Lovell (1993, pp. 40) and Kumbhakar and Lovell (2000) for detailed discussion. We shall, however, consider only technical efficiency in the present study.

  6. It may be noted that the distribution of the estimated inefficiency variable, the \( \hat{u}_{i} \)—estimated on the basis of the observed sample—may not look like the one assumed for the population. This is demonstrated in a recent study by Wang and Schmidt (2009). In view of this observation, we have derived the theoretical density function of the \( \hat{u}_{i} \) for the distribution we have assumed here and have shown this distribution also diagrammatically in Appendix 3.

  7. Using Jondrow et al. (1982) technique, firm specific estimates of technical efficiency are given by \( E[\exp ( - u_{i} )\left| {\varepsilon_{i} } \right.] = \left\{ {{{\left[ {1 - \Upphi \left( {\sigma_{*} - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {1 - \Upphi \left( {\sigma_{*} - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)} \right]} {\left[ {1 - \Upphi \left( { - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {1 - \Upphi \left( { - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)} \right]}}} \vphantom{\frac{1}{2}}\right\}\,\exp \left( { - \mu_{i}^{*} + \left( {{{\sigma_{*}^{2} } \mathord{\left/ {\vphantom {{\sigma_{*}^{2} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)} \right) \), where \( \varepsilon_{i} \equiv v_{i} - u_{i} ,\mu_{i}^{*} \equiv {{\left( {\mu_{i} \sigma_{v}^{2} - \varepsilon_{i} \sigma_{u}^{2} } \right)} \mathord{\left/ {\vphantom {{\left( {\mu_{i} \sigma_{v}^{2} - \varepsilon_{i} \sigma_{u}^{2} } \right)} {\left( {\sigma_{u}^{2} + \sigma_{v}^{2} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\sigma_{u}^{2} + \sigma_{v}^{2} } \right)}},\sigma_{*}^{2} \equiv {{\sigma_{u}^{2} \sigma_{v}^{2} } \mathord{\left/ {\vphantom {{\sigma_{u}^{2} \sigma_{v}^{2} } {\left( {\sigma_{u}^{2} + \sigma_{v}^{2} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\sigma_{u}^{2} + \sigma_{v}^{2} } \right)}}\,{\text{and}}\,\Upphi ( \cdot ) \) is the standard normal distribution function (see Kumbhakar and Lovell 2000, p. 86).

  8. This variable is also used as an input in the frontier production function. This is in line with the practice followed by others (see Battese and Broca 1997; Bhandari and Maiti 2007; Huang and Liu 1994; Lundvall and Battese 2000; Wang and Schmidt 2002).

  9. Note that the problem for firm k is to find that set of weights λ ik ’s (i = 1,…, N) which makes the convex combination of the observed outputs of the different firms (\( \sum\nolimits_{i} {Y_{i} \lambda_{ik} } \)) as large as possible relative to its own observed output (Y k ) (constraint (i) and the objective function), subject to the requirement that the same convex combination of the input vectors of these firms does not exceed its own actual use of inputs, X k (constraint (ii)). Obviously, its own (observed) input bundle and output [i.e., φ = 1 and \( \{ \lambda_{kk} = 1\,{\text{and}}\,\lambda_{ik} = 0\quad {\text{for}}\,{\text{all}}\,i \ne k\} \)] provides one feasible solution to this problem. Hence, the optimal value, φ M k , cannot be less than unity. In case of multi-product firms, Y i and Y k in the first inequality will each be a vector (of appropriate dimension), and not a scalar as is being assumed here. The meaning of the term ‘proportional increase in output’ (vector) will be quite clear then.

  10. While the objective function remains unchanged, the set of constraints will now read as follows: (i) \( \sum\nolimits_{i \in g} {Y_{i} \,\lambda_{ik} \ge \varphi Y_{k} } \), (ii) \( \sum\nolimits_{i \in g} {X_{i} \,\lambda_{ik} \le X_{k} } \), (iii) \( \sum\nolimits_{i \in g} {\lambda_{ik} = 1} \), and (iv) \( \lambda_{ik} \ge 0\quad {\text{for}}\,{\text{all}}\,i \in g \).

  11. Suppose [\( \tilde{\varphi },\{ (\tilde{\lambda }_{ik} ),i \in g\} ]\} \) is a feasible solution to (P o k ). Then a feasible solution to (P M k ) is given by \( [\tilde{\varphi },\{ (\lambda_{ik} ),k = 1,\ldots,N,\quad {\text{where}}\,\lambda_{ik} = \tilde{\lambda }_{ik} ,\;{\text{for}}\,i \in g\,{\text{and}},\lambda_{ik} = 0\;{\text{for}}\,{\text{all}}\,i \notin g\} ] \). Hence, the optimal value of (P o k ), viz., φ o k , can be no greater than that of the firm’s meta-frontier problem, (P M k ), viz., φ M k .

  12. Battese et al (2004) calls it technology gap ratio. Note that such a ratio may also be defined separately for each firm k as the ratio of TE M k to TE O k .

  13. As shown in Battese and Rao (2002) and Battese et al. (2004), a similar exercise can also be done in the context of SFA to find TCR’s of firms. We have not done that exercise in the present study.

  14. The definitions of the various concepts like ex-factory value, fixed asset, manday etc are as used by the CSO. It would have been very useful if we had the panel data over a number of years. However, the lack of sufficient information did not allow one to construct a panel data set from this source.

  15. To describe our procedure for obtaining the two state dummy variables, we started by fitting a frontier function with ten dummy variables to differentiate firms located in ten major leather producing states. However, given such a large number, some state dummies came up with non-significant coefficients. We, therefore, sought to reduce the number of state dummies by merging (firms from a number of) states/groups of states into a fewer number of groups, following the criterion that the dummy coefficients for the states/groups of states merged had the same sign as well as values close to each other at the preceding exercise (i.e., the exercise before merging). We went on reducing successively the number of groups of states in this way. At each stage we also examined whether such a process of merging had any significant effect on the regression results, by conducting likelihood ratio tests for the null hypotheses that the dummy coefficients (at the preceding stage) for the states/groups of states merged were equal. If such hypotheses were not rejected, we proceeded on our task of merging so as to get a smaller number of groups. Continuing this exercise we finally arrived at three groups of firms—those located in West Bengal/Tamil Nadu, those located in the four northern states and those located in the remaining states of India.

  16. The estimated values of generalized LR statistic under H 0 are given in the second row of the table shown at the top of this page (the corresponding critical value at 1% level is 13.28).

  17. The estimated values of generalized LR statistic under H 0 are given in the fourth row of the table referred in footnote 16 (the corresponding critical value at 1% level is 6.64).

  18. We can also test joint nullness of these two sets of parameters, i.e., H 0: β3 = β13 = β23 = β33 = δ02 = 0 which, as may be inferred from the results of the earlier two tests, fails to get rejected. The estimated values of generalized LR statistic under H 0 are given in the last row of the table referred in footnote 16 (the corresponding critical value at 1% level is 15.09).

  19. As shown in Battese and Broca (1997), given the type of distribution assumed here for the u i , the elasticity of TE i with respect to an explanatory variable, say X i , is given by \( \left\{ {{\frac{1}{{\sigma_{u} }}}\left[ {{\frac{{\varphi \left( {{{\mu_{i} } \mathord{\left/ {\vphantom {{\mu_{i} } {\sigma_{u} }}} \right. \kern-\nulldelimiterspace} {\sigma_{u} }} - \sigma_{u} } \right)}}{{\Upphi \left( {{{\mu_{i} } \mathord{\left/ {\vphantom {{\mu_{i} } {\sigma_{u} }}} \right. \kern-\nulldelimiterspace} {\sigma_{u} }} - \sigma_{u} } \right)}}} - {\frac{{\varphi \left( {{{\mu_{i} } \mathord{\left/ {\vphantom {{\mu_{i} } {\sigma_{u} }}} \right. \kern-\nulldelimiterspace} {\sigma_{u} }}} \right)}}{{\Upphi \left( {{{\mu_{i} } \mathord{\left/ {\vphantom {{\mu_{i} } {\sigma {}_{u}}}} \right. \kern-\nulldelimiterspace} {\sigma {}_{u}}}} \right)}}}} \right] - 1} \right\}\left( {{\frac{{\partial \mu_{i} }}{{\partial \ln \,X_{i} }}}} \right) \), where X i is either I i (size) or Age i , φ(·) is the standard normal density function and ∂μ i /∂ln X i is to be computed from the estimated relation.

  20. Theoretical expression of the elasticity of TE with respect to any inefficiency-explaining variable is shown in footnote 19. Observe that it varies from one firm to another. Let the estimated values of μ i and σ u be denoted by \( \hat{\mu }_{i} \) and \( \hat{\sigma }_{u} \), respectively and let ∂μ i /∂ln X i be computed from the estimated relation (2) where X i is the inefficiency-explaining variable. The only problem is to compute φ(·)/Φ(·) for each i. This can, however, be easily solved by writing a simple command in any modern statistical/mathematical package and we have done it using MATLAB package. For example, the corresponding MATLAB command is “normpdf(*)/normcdf(*)” where (*) may be either \( \left( {\left( {{{\hat{\mu }_{i} } \mathord{\left/ {\vphantom {{\hat{\mu }_{i} } {\hat{\sigma }_{u} }}} \right. \kern-\nulldelimiterspace} {\hat{\sigma }_{u} }}} \right) - \hat{\sigma }_{u} } \right) \) or \( \left( {{{\hat{\mu }_{i} } \mathord{\left/ {\vphantom {{\hat{\mu }_{i} } {\hat{\sigma }_{u} }}} \right. \kern-\nulldelimiterspace} {\hat{\sigma }_{u} }}} \right) \). To obtain this expression for N number of observations in a single step the corresponding MATLAB command to be used is “normpdf(*). /normcdf(*)” where (*) must be an (N × 1) vector which needs to be defined clearly in the program.

  21. Since we are unable to construct firm-level panel data from the data set we have at our disposal and since benchmark frontier (i.e., the one relative to which the individual firms are evaluated) itself changes from one period to another in view of separate cross-sectional analysis for each individual year, our findings of increasing average TE over time may appear to be a tentative one. However, to address this issue in a more satisfactory way with the given data, we have sought to construct a representative panel comprising average values of the variables for 20 separate groups of firms. Carrying out this analysis, we get a panel of 100 (time series) observations on 20 (hypothetical) average firms and we find the estimated TE of each such group to rise consistently over the period from 1984–1985 through 2002–2003, thereby corroborating our observation given above. A detailed description of this exercise is given in Appendix 1.

  22. This was done by deflating their figures by suitable price indices. In particular, output figures have been deflated by the wholesale price index of manufactured products and the figures of intermediate inputs, by the (simple geometric) mean of the wholesale price index of non-food primary articles and that of the group of fuel, power, light and lubricants.

References

  • Aigner DJ, Lovell CAK, Schmidt P (1977) Formulation and estimation of stochastic frontier production function models. J Econ 6(1):21–37

    Google Scholar 

  • Banerjee N, Nihila M (1999) Business organization in the leather industry of Calcutta and Chennai. In: Bagchi AK (ed) Economy and organization: Indian institutions under the neoliberal regime. Sage Publications, New Delhi

    Google Scholar 

  • Banker RD (1984) Estimating the most productive scale size using data envelopment analysis. Eur J Oper Res 17(1):35–44

    Article  Google Scholar 

  • Banker RD, Charnes A, Cooper WW (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manage Sci 30(9):1078–1092

    Article  Google Scholar 

  • Battese GE, Broca SS (1997) Functional forms of stochastic frontier production functions and models for technical inefficiency effects: a comparative study for wheat farmers in Pakistan. J Prod Anal 8(4):395–414

    Article  Google Scholar 

  • Battese GE, Coelli TJ (1988) Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data. J Econ 38(3):387–399

    Google Scholar 

  • Battese GE, Coelli TJ (1993) A stochastic frontier production function incorporating a model for technical inefficiency effects. Econometrics and applied statistics working papers series, 69, University of New England

  • Battese GE, Rao DSP (2002) Technology gap, efficiency and a stochastic meta-frontier function. Int J Bus Econ 1(2):87–93

    Google Scholar 

  • Battese GE, Rao DSP, O’Donnell CJ (2004) A meta-frontier production function for estimation of technical efficiencies and technology gaps for firms operating under different technologies. J Prod Anal 21(1):91–103

    Article  Google Scholar 

  • Bhandari AK, Maiti P (2007) Efficiency of Indian manufacturing firms: textile industry as a case study. Int J Bus Econ 6(1):71–88

    Google Scholar 

  • Bhandari AK, Ray SC (forthcoming) Technical efficiency in the Indian textiles industry: a nonparametric analysis of firm-level data. Bulletin of Economic Research

  • Bhavani TA (1991) Technical efficiency in Indian modern small scale sector: an application of frontier production function. Indian Econ Rev 26(2):149–166

    Google Scholar 

  • Charnes A, Cooper WW, Rhodes E (1978) Measuring the efficiency of decision making units. Eur J Oper Res 2(6):429–444

    Article  Google Scholar 

  • Coelli TJ, Rao DSP, Battese GE (1998) An introduction to efficiency and productivity analysis. Kluwer Academic Publishers, Boston

    Book  Google Scholar 

  • Goldar B (1985) Unit size and economic efficiency in small scale washing soap industry in India. Artha Vijnana 27(1):21–40

    Google Scholar 

  • Goldar B, Renganathan VS, Banga R (2004) Ownership and efficiency in engineering firms: 1990–91 to 1999–2000. Econ Polit Wkly 39(5):441–447

    Google Scholar 

  • Government of India (2004–05) Economic survey, Ministry of Finance

  • Huang CJ, Liu JT (1994) Estimation of a non-neutral stochastic frontier production function. J Prod Anal 5(2):171–180

    Article  Google Scholar 

  • Jondrow J, Lovell CAK, Materov IS, Schmidt P (1982) On the estimation of technical inefficiency in the stochastic frontier production function model. J Econ 19(2/3):233–238

    Google Scholar 

  • Kodde DA, Palm FC (1986) Wald criteria for jointly testing equality and inequality restrictions. Econometrica 54(5):1243–1248

    Article  Google Scholar 

  • Kumbhakar SC, Lovell CAK (2000) Stochastic frontier analysis. Cambridge University Press, Cambridge

    Google Scholar 

  • Lall SV, Rodrigo GC (2001) Perspectives on the sources of heterogeneity in Indian industry. World Dev 29(12):2127–2143

    Article  Google Scholar 

  • Little IMD, Mazumdar D, Page JM Jr (1987) Small manufacturing enterprises: a comparative analysis of India and other economies. Oxford University Press, Oxford

    Google Scholar 

  • Lovell CAK (1993) Production frontiers and productive efficiency. In: Fried HO, Lovell CAK, Schmidt SS (eds) The measurement of productive efficiency: techniques and applications. Oxford University Press, New York

    Google Scholar 

  • Lundvall K, Battese GE (2000) Firm size, age and efficiency: evidence from kenyan manufacturing firms. J Dev Stud 36(3):146–163

    Article  Google Scholar 

  • Marshall A (1920) Principles of economics, 8th edn. MacMillan, London

    Google Scholar 

  • Meeusen W, van den Broeck J (1977) Efficiency estimation from Cobb-Douglas production function with composed error. Int Econ Rev 18(2):435–444

    Article  Google Scholar 

  • Mohapatra KM, Srivastava K (2002) Leather goods industry in Kanpur: casteism, religion and business interlinkings. Econ Polit Wkly 37(39):4029–4035

    Google Scholar 

  • Mukherjee K, Ray SC (2004) Technical efficiency and its dynamics in Indian manufacturing: an inter-state analysis. Department of Economics working paper series, 18, University of Connecticut

  • Neogi C, Ghosh B (1994) Intertemporal efficiency variations in Indian manufacturing industries. J Prod Anal 5(3):301–324

    Article  Google Scholar 

  • Nikaido Y (2004) Technical efficiency of small-scale industry: application of stochastic production frontier model. Econ Polit Wkly 39(6):592–597

    Google Scholar 

  • Page JM Jr (1984) Firm size and technical efficiency: application of production frontiers to Indian survey data. J Dev Econ 16(1/2):129–152

    Article  Google Scholar 

  • Penrose ET (1959) The theory of the growth of the firm. Basil Blackwell, Oxford

    Google Scholar 

  • Ramaswamy VK (1994) Technical efficiency in modern small-scale firms in Indian industry: applications of stochastic production frontier. J Quant Econ 10(2):309–324

    Google Scholar 

  • Stinchcombe AL (1965) Social structure and organizations. In: March JG (ed) Handbook of organizations. Rand McNally, Chicago

    Google Scholar 

  • Wang H-J, Schmidt P (2002) One-step and two-step estimation of the effects of exogenous variables on technical efficiency levels. J Prod Anal 18(2):129–144

    Article  Google Scholar 

  • Wang WS, Schmidt P (2009) On the distribution of estimated technical efficiency in stochastic frontier models. J Econ 148(1):36–45

    Google Scholar 

Download references

Acknowledgments

We would like to thank the two anonymous referees and an Associate Editor of the Journal for their detailed comments and suggestions which help us greatly in revising and improving the paper. The usual disclaimer applies.

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Authors and Affiliations

  1. Centre for Development Studies, Prasanth Nagar Road, Ulloor, Trivandrum, Kerala, 695011, India

    Anup Kumar Bhandari

  2. ICRA Ltd., FMC Fortuna, A-10 & A-11, 3rd Floor, 234/3A, A. J. C. Bose Road, Kolkata, 700020, India

    Anup Kumar Bhandari

  3. Economic Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata, 700108, India

    Anup Kumar Bhandari & Pradip Maiti

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  1. Anup Kumar Bhandari
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  2. Pradip Maiti
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Correspondence to Pradip Maiti.

Appendices

Appendix 1

The Appendix 1 reports some additional analyses to corroborate our findings on the inter-temporal behavior of firm-level TE’s. We have noted that TE measures for the sample years given in the last row of Table 3 point to an increasing time-trend in the average firm-level TE. However, one might argue that these findings are not of much significance, since the benchmark frontier (i.e., the frontier relative to which firm-level TE’s are measured) itself changes from one period to another so that these TE measures are not strictly comparable over time. Obviously, there would have been no problem of interpretation, if we had firm-level panel data, i.e., time series data on outputs and inputs of a fixed set of firms. But such a panel cannot be constructed from the CSO data set, as firms cannot be identified therein. However, one could construct a panel of firms in an indirect way. That is what we shall try to do here and carry out our analysis on this constructed panel. We describe our approach below.

Let us consider the data for a particular year. We first arrange the different firms in ascending order of their sizes (size being measured by the amount of intermediate inputs used by a firm) and then club these firms into twenty fractile groups, viz., the group containing the smallest 5% firms, that containing the next 5% firms and so on up to the final group containing the largest 5% firms. For each fractile group we first compute the average value of each of output and various inputs across all firms in the group and then take these average values to correspond to a hypothetical firm, to be called the average firm. Thus we have data on twenty (average) firms for a given year. We carry out this exercise for each of the sample years 1984–1985, 1989–1990, 1994–1995, 1999–2000 and 2002–2003. (We ignore 2 years, viz. 1985–1986 and 1990–1991. The reason is simply to have (more or less) equal time interval between the sample years which is a precondition for the construction of any panel data set). Thus we have a panel of 100 observations on 20 (hypothetical) average firms. Since ASI data give values of output and inputs at current prices, we had to convert these to values at constant prices before classifying firms in the various fractile groups.Footnote 22 We then estimate a time-varying TE model (taking the regression equation to be the same as the one reported in Table 3, except for the dummy variables) with this panel data and obtain estimate of individual TE for each of the twenty firms for each year. The estimated TE of each average firm is seen to have risen over time. In fact, the mean TE of the panel of firms is found to have gone up from 49.32 per cent in 1984–1985 to 71.55 per cent in 2002–2003. The year-wise estimates are given in Tables 10 and 11.

Table 10 Individual TE score (in %) of each of 20 groups of firms over years
Full size table
Table 11 Estimated translog stochastic frontier production function with hypothetical (average) panel data
Full size table

The above clearly points to an increasing trend of TE over time. Thus our observation in the text that the overall performance of the Indian leather manufacturing firms has improved over time is seen to be rather robust.

Appendix 2

See Fig. 1a, b.

Fig. 1
figure 1figure 1

a Scatter diagrams of two series of firm TE’s computed from two alternative stochastic frontiers—one with FA as one explanatory variable (vertical axis) and the other excluding it (horizontal axis). b Histograms sowing proportions of firms (vertical axis) in different class intervals of technical efficiency scores (horizontal axis)

Full size image

Appendix 3

Here we show some further results in the light of Wang and Schmidt (2009). In our paper we assume that u i  ∼ idN +(μ i  = z i δ, σ 2 u ) and v i  ∼ iidN(0, σ 2 v ). Now writing ɛ i  ≡ v i  − u i we get the joint density function of (u i , ɛ i ) is

$$ \left\{ {{\frac{1}{{\sigma_{u} \sqrt {2\Uppi } }}} \times {\frac{1}{{\left[ {1 - \Upphi \left( { - {\frac{{z_{i} \delta }}{{\sigma_{u} }}}} \right)} \right]}}} \times \exp \left[ { - \frac{1}{2}\left( {{\frac{{u_{i} - z_{i} \delta }}{{\sigma_{u} }}}} \right)^{2} } \right]} \right\} \times \left\{ {{\frac{1}{{\sigma_{v} \sqrt {2\Uppi } }}}\exp \left[ { - \frac{1}{2}\left( {{\frac{{\varepsilon_{i} + u_{i} - 0}}{{\sigma_{v} }}}} \right)^{2} } \right]} \right\} $$

Therefore, marginal density of ε is given by

$$ \begin{aligned} f_{\varepsilon } (\varepsilon_{i} ) = & {\frac{1}{{2\Uppi \sigma_{u} \sigma_{v} }}} \times {\frac{1}{{\left[ {1 - \Upphi \left( { - {\frac{{z_{i} \delta }}{{\sigma_{u} }}}} \right)} \right]}}} \times \int\limits_{0}^{\infty } {\exp \left[ { - \frac{1}{2}\left( {{\frac{{u_{i} - z_{i} \delta }}{{\sigma_{u} }}}} \right)^{2} - \frac{1}{2}\left( {{\frac{{\varepsilon_{i} + u_{i} }}{{\sigma_{v} }}}} \right)^{2} } \right]} \,du_{i} \\ & = {\frac{{\exp \left[ { - \frac{1}{2}\left( {{\frac{{z_{i} \delta + \varepsilon_{i} }}{{\sqrt {\sigma_{u}^{2} + \sigma_{v}^{2} } }}}} \right)^{2} } \right]}}{{\sqrt {2\Uppi \left( {\sigma_{u}^{2} + \sigma_{v}^{2} } \right)} }}} \times {\frac{{\left[ {1 - \Upphi \left( { - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)} \right]}}{{\left[ {1 - \Upphi \left( { - {\frac{{z_{i} \delta }}{{\sigma_{u} }}}} \right)} \right]}}} \\ \end{aligned} $$

and

$$ f_{u} (u_{i} \left| {\varepsilon_{i} } \right.) = {\frac{{\exp \left[ { - \frac{1}{2}\left( {{\frac{{u_{i} - \mu_{i}^{*} }}{{\sigma_{*} }}}} \right)^{2} } \right]}}{{\sigma_{*} \sqrt {2\Uppi } \left[ {1 - \Upphi \left( { - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)} \right]}}},\quad u_{i} \ge 0 $$

Hence,

$$ \begin{aligned} \hat{u}_{i} = & E(u_{i} \left| {\varepsilon_{i} } \right.) = \int\limits_{0}^{\infty } {u_{i} \,f_{u} (u_{i} \left| {\varepsilon_{i} } \right.)} {\text{d}}u_{i} = \mu_{i}^{*} + \sigma_{*} {\frac{{\varphi \left( { - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)}}{{\left[ {1 - \Upphi \left( { - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)} \right]}}} \\ & = h(\varepsilon_{i} )({\text{say}}) \Rightarrow \varepsilon_{i} = h^{ - 1} (\hat{u}_{i} ) = g(\hat{u}_{i} )({\text{say}}) \\ \end{aligned} $$

where, as defined in the text,

$$ \mu_{i}^{*} = {\frac{{z_{i} \delta \sigma_{v}^{2} - \varepsilon_{i} \sigma_{u}^{2} }}{{\sigma_{u}^{2} + \sigma_{v}^{2} }}}\,{\text{and}}\,\sigma_{*}^{2} = {\frac{{\sigma_{u}^{2} \sigma_{v}^{2} }}{{\sigma_{u}^{2} + \sigma_{v}^{2} }}}. $$

Now,

$$ \hat{u}_{i} = h(\varepsilon_{i} ) = \mu_{i}^{*} + \sigma_{*} {\frac{{\varphi \left( { - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)}}{{\left[ {1 - \Upphi \left( { - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)} \right]}}} = \mu_{i}^{*} + \sigma_{*} \lambda \left( { - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)\quad {\text{where}}\,\lambda (s) = {\frac{\varphi (s)}{{\left[ {1 - \Upphi (s)} \right]}}}. $$

So,

$$ {\frac{{\partial \hat{u}_{i} }}{{\partial \varepsilon_{i} }}} = - {\frac{{\sigma_{u}^{2} }}{{\sigma_{u}^{2} + \sigma_{v}^{2} }}} + \sigma_{*} \times \lambda^{\prime}\left( { - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right) \times {\frac{\partial }{{\partial \varepsilon_{i} }}}\left( { - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right) = \gamma \left[ {\lambda^{\prime}\left( { - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right) - 1} \right] $$

where, as in the text,

$$ \gamma = {\frac{{\sigma_{u}^{2} }}{{\sigma_{u}^{2} + \sigma_{v}^{2} }}}\,{\text{and}}\,\lambda^{\prime}(s) = - s\lambda (s) + \lambda^{2} (s). $$

Therefore,

$$ \begin{aligned} f_{{\hat{u}}} (\hat{u}_{i} ) = f_{\varepsilon } (g(\hat{u}_{i} )) \times \left| {g^{\prime}(\hat{u}_{i} )} \right| = & {\frac{{\exp \left[ { - \frac{1}{2}\left( {{\frac{{z_{i} \delta + \varepsilon_{i} }}{{\sqrt {\sigma_{u}^{2} + \sigma_{v}^{2} } }}}} \right)^{2} } \right]}}{{\sqrt {2\Uppi \left( {\sigma_{u}^{2} + \sigma_{v}^{2} } \right)} }}} \times {\frac{{\left[ {1 - \Upphi \left( { - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)} \right]}}{{\left[ {1 - \Upphi \left( { - {\frac{{z_{i} \delta }}{{\sigma_{u} }}}} \right)} \right]}}} \times \left| {{\frac{1}{{\gamma \left[ {\lambda^{\prime}\left( { - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right) - 1} \right]}}}} \right| \\ & = {\frac{{\exp \left[ { - \frac{1}{2}\left( {{\frac{{z_{i} \delta + \varepsilon_{i} }}{{\sqrt {\sigma_{u}^{2} + \sigma_{v}^{2} } }}}} \right)^{2} } \right]}}{{\sqrt {2\Uppi \left( {\sigma_{u}^{2} + \sigma_{v}^{2} } \right)} }}} \times {\frac{{\Upphi \left( {{\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)}}{{\Upphi \left( {{\frac{{z_{i} \delta }}{{\sigma_{u} }}}} \right)}}} \times \left| {{\frac{1}{{\gamma \left[ {\lambda^{\prime}\left( { - {\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right) - 1} \right]}}}} \right| \\ & = {\frac{{\exp \left[ { - \frac{1}{2}\left( {{\frac{{z_{i} \delta + \varepsilon_{i} }}{{\sqrt {\sigma_{u}^{2} + \sigma_{v}^{2} } }}}} \right)^{2} } \right]}}{{\sqrt {2\Uppi \left( {\sigma_{u}^{2} + \sigma_{v}^{2} } \right)} }}} \times {\frac{{\Upphi \left( {{\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)}}{{\Upphi \left( {{\frac{{z_{i} \delta }}{{\sigma_{u} }}}} \right)}}} \times \left| {{\frac{1}{{\gamma \left[ {\left( {{\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)\left\{ {{{\varphi \left( {{\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)} \mathord{\left/ {\vphantom {{\varphi \left( {{\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)} {\Upphi \left( {{\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)}}} \right. \kern-\nulldelimiterspace} {\Upphi \left( {{\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)}}} \right\} + \left\{ {{{\varphi \left( {{\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)} \mathord{\left/ {\vphantom {{\varphi \left( {{\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)} {\Upphi \left( {{\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)}}} \right. \kern-\nulldelimiterspace} {\Upphi \left( {{\frac{{\mu_{i}^{*} }}{{\sigma_{*} }}}} \right)}}} \right\}^{2} - 1} \right]}}}} \right| \\ \end{aligned} $$

since φ(·) is symmetric and \( \left\{ {1 - \Upphi \left( {{{ - \mu_{i}^{*} } \mathord{\left/ {\vphantom {{ - \mu_{i}^{*} } {\sigma_{*} }}} \right. \kern-\nulldelimiterspace} {\sigma_{*} }}} \right)} \right\} = \Upphi \left( {{{\mu_{i}^{*} } \mathord{\left/ {\vphantom {{\mu_{i}^{*} } {\sigma_{*} }}} \right. \kern-\nulldelimiterspace} {\sigma_{*} }}} \right) \). We have also plotted this density of \( \hat{u} \) against \( \hat{u} \) for each of the sample years we have considered (See Fig. 2).

Fig. 2
figure 2

Scatter diagrams of probability density of \( \hat{u} \) (vertical axis) against values of \( \hat{u} \) (horizontal axis) for different sample years

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It may be noted that for any given year Fig. 1b gives histogram of estimated values of TE while Fig. 2 shows density of firms against estimated values of u. Thus they are two alternative frequency distributions of firms; on the horizontal axis the latter measures the value of \( \hat{u} \) and the former measures the value of \( TE( \cong \exp ( - \hat{u})) \). Therefore, they will look different and the skewness of the two distributions will be exactly the opposite.

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Bhandari, A.K., Maiti, P. Efficiency of the Indian leather firms: some results obtained using the two conventional methods. J Prod Anal 37, 73–93 (2012). https://doi.org/10.1007/s11123-011-0219-1

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