AN INPUT DISTANCE FUNCTION APPROACH TO THE MEASUREMENT OF TECHNICAL AND ALLOCATIVE EFFICIENCY: WITH APPLICATION TO INDIAN DAIRY PROCESSING PLANTS

Transcription

1 AN INPUT DISTANCE FUNCTION APPROACH TO THE MEASUREMENT OF TECHNICAL AND ALLOCATIVE EFFICIENCY: WITH APPLICATION TO INDIAN DAIRY PROCESSING PLANTS by Tim Coelli School of Economics, Universy of Queensland Satbir Singh Department of Agriculture, Beribagh, Sahranpur, U.P., India and Euan Fleming School of Economics, Universy of New England. Draft: 0//003 Abstract In this paper we describe how one can measure technical and allocative efficiency relative to a stochastic input distance function. This method avoids many of the problems that can afflict the cost and/or production frontier approaches, such as nonoptimising behaviour, limed price variation, regressor endogeney, and the single output restriction. The method is illustrated using survey data on private and cooperative Indian dairy processing plants. Our empirical results indicate that the private firms are not more cost-efficient than the cooperative firms, and also show that the introduction of reforms to encourage the entrance of new private sector firms did not have the expected posive effect upon cost efficiency in this industry. JEL: C6, C, D0 Key words: distance function, technical efficiency, allocative efficiency, Indian dairy processing, cooperatives

2 . Introduction The measurement of the cost efficiency of a firm, and s decomposion into technical and allocative components, has been the subject of a number of papers since the seminal work of Farrell (957). Farrell suggested that the degree to which a firm was operating above minimum cost could be divided into one part resulting from the use of input quanties in the wrong proportions, given the prevailing prices (allocative inefficiency), and another part resulting from operation below the production frontier (technical inefficiency). Farrell showed how these various efficiency components could be measured, given that one had an estimate of the production technology. The calculation of these efficiency components is a fairly straightforward exercise. However, the estimation of the production technology has proven to be more challenging. Many possible estimation methods have been proposed over the years. The two most popular methods have been the stochastic frontier analysis (SFA) method proposed by Aigner, Lovell and Schmidt (977) and Meeusen and van den Broeck (977), and the data envelopment analysis (DEA) method, developed by Boles (966), Afriat (97) and Charnes, Cooper and Rhodes (978). SFA involves the use of econometric methods, while DEA is a linear programming technique. DEA has the advantage that one does not need to choose a specific functional form or distributional form for the error terms, while SFA has the advantage that attempts to account for the effects of data noise and allows the conduct of standard statistical tests. In this paper we utilise the SFA method because we believe that data noise is an important issue, especially in cases when allocative efficiency is of interest. This is because DEA forms a frontier by building a piece-wise linear surface over the top of the sample data. If there is data noise associated wh a few frontier points, this will not only cause the frontier to shift out and overstate the mean level of technical efficiency, will also distort the shape of the estimated frontier, leading to allocatively efficient points being shifted some distance from their correct posions. To be precise, Farrell (957) used the terms price efficiency and economic efficiency instead of allocative efficiency and cost efficiency, respectively. However, in this paper we use the terminology that is in common use today. See Coelli, Rao and Battese (998) for a discussion of the various frontier estimation methods and their relative mers.

3 For further discussion of the relative mers of DEA and SFA methods, see Coelli, Rao and Battese (998). The papers by Aigner, Lovell and Schmidt (977) and Meeusen and van den Broeck (977) showed how one could estimate a stochastic production frontier and measure (mean) technical efficiency. The issue of allocative and cost efficiency measurement was later addressed by Schmidt and Lovell (979), who described how one could estimate a Cobb-Douglas stochastic cost frontier and then use dualy to derive the implic production frontier. Wh these two frontiers, one could then measure cost efficiency and technical efficiency, and calculate allocative efficiency residually. 3 In some cases, the direct estimation of a cost frontier may not be practical or appropriate. It will not be practical when input prices do not differ among firms. It will not be appropriate when there is systematic deviation from cost-minimising behaviour in an industry, for example when polical, union or regulatory factors cause shadow prices to deviate from market prices in a systematic way. In this suation, the dualy between the cost and production functions break down, and the resulting bias in the cost frontier estimates will make the cost efficiency calculation and decomposion biased as well. One solution to this problem is to obtain a direct estimate of the primal production technology, and then derive the implic cost frontier. 4 For example, Bravo-Ureta and Rieger (99) estimated a Cobb-Douglas stochastic production frontier, and then derived the implic cost frontier. However, one particular inconsistency in the Bravo- Ureta and Rieger (99) approach is that a production function is estimated when one is clearly assuming that the input quanties are decision variables. This leaves the approach open to the cricism that simultaneous equations bias may afflict the production function estimates, 5 and may explain why has not been widely adopted. 6 3 The cost frontier methods of Schmidt and Lovell (979) were subsequently extended to the more flexible translog functional form by various authors, such as Greene (980) and Schmidt (984). These new methods avoided the restrictions inherent in the Cobb-Douglas functional form, but at the cost of introducing considerable complexy to the modelling exercise. For a thorough review of this lerature, see Kumbhakar and Lovell (000, chapter 4). 4 Another possible solution is to estimate some form of shadow cost function, where systematic deviations from allocative efficiency are explicly modelled. This can be a complex exercise, which often requires that a number of simplifying assumptions must be made in order to produce a tractable model for estimation. For examples of this approach see Atkinson and Cornwell (994a,b) and Balk (998). 5 See Marschak and Andrews (944) for an explanation of this problem.

4 3 In this paper, we propose a new approach involving the estimation of an input distance function that avoids all of the above problems. It does not require price information that varies across firms, is robust to systematic deviations from costminimising behaviour, and does not suffer from simultaneous equations bias when firms are cost minimisers or shadow cost minimisers. 7 The input distance function can also easily accommodate multiple outputs and hence has an addional advantage relative to the production function, which is restricted to a single output variable. The remainder of this paper is organised into three sections. In section, the analytical framework is developed. In section 3 we provide an application of these methods to survey data on Indian dairy processing plants. The final section provides a summary and some conclusions.. Methodology The production technology of a dairy plant may be described using input sets, L(y), representing the set of all (K ) input vectors, x, which can be used to produce the (M ) output vector, y. That is, L(y) = {x: x can produce y}. () The production technology is assumed to satisfy the usual axioms, such as convexy and strong disposabily. 8 The input distance function, d(x,y), is then defined on this input set as: d(x,y) = sup{ρ: (x/ρ) L(y)}, () where ρ is a posive scalar. The distance function, d(x,y), is non-decreasing, posively linearly homogeneous and concave in x, and non-increasing and quasiconcave in y. The value of the distance will be equal to one or greater than one if the input vector, x, is an element of the feasible input set, L(y), ie. d(x,y) if x L(y). 6 Another drawback of the Bravo-Ureta and Rieger (99) approach is that involves the use of the Cobb-Douglas functional form, which is a restrictive functional form. That is, imposes unary elasticies of substution and constant production elasticies across all firms. In the empirical exercise in this paper we find that the more flexible translog functional form is not a statistically significant improvement over the Cobb-Douglas functional form. However, this is unlikely to be the case in all data sets. 7 See Coelli (000) for the proof of this result. 8 See Färe and Primont (995) for further details.

5 4 The distance function has a value of uny if x is located on the inner boundary of the input set. Figure provides an illustration of an input distance function, where two inputs, x and x, are used to produce output, y. The isoquant, SS /, is the inner boundary of the input set, reflecting the minimum input combinations that may be used to produce a given output vector. In this case, the value of the distance function for a firm producing output, y, using the input vector defined by point A, is equal to the ratio, OA/OB. Figure : The input distance function and the input set x S A B L(y) S / 0 x The log form of a Cobb-Douglas input distance function, for the case of one output, K inputs, N firms and T time periods, is specified as: 9 lnd = αln y K + δ + β ln x j j= j, i=...n, and t=...t, (3) where y is the output quanty, x j is the j-th input quanty, ln represents a natural logarhm, and α, δ and β j are unknown parameters to be estimated. 9 The application in this paper uses panel data on Indian dairy processing plants, which involves information on one output and four inputs. Hence, we define a model for panel data wh one output and K inputs. The extension to the case of M outputs and/or cross-sectional data is straightforward.

6 5 Imposing the restriction for homogeney of degree + in inputs upon this function, K j β j =, we obtain the estimating equation: ln x K = αln y + δ + K β j j ln(x where ε j x K = v ) + ε u., (4) Observe that in the above equation we have defined ln d = ε = v -u to indicate that the distance term may be interpreted as a tradional SFA disturbance term. That is, the distances in a distance function (which are the radial distances between the data points and the frontier) could be due to eher noise (v ) or technical inefficiency (u ). This is the standard SFA error structure. To allow us to test for the significance of efficiency change over time, we utilise the time-varying technical efficiency stochastic frontier model, defined by Battese and Coelli (99). That is, we assume that the v are i.i.d. N(0,σ v ) and u =u i [exp(η(t-t))], where u i is i.i.d. N(µ,σ u ). Given these distributional assumptions, the values for unknown parameters can be obtained by the method of maximum likelihood. 0 The input-orientated technical efficiency (TE) scores are then predicted using the condional expectation predictor: TE = E[(exp(-u ) ε )], (5) proposed by Battese and Coelli (99). Once the parameters of the Cobb-Douglas input distance function have been estimated, we can derive the corresponding parameters of the dual cost function. The Cobb-Douglas cost function is defined as: K ln c = b + b ln p + a ln y, (6) 0 j= j j 0 Alternative models of time-varying technical efficiency have been proposed by Kumbhakar (990), Cornwell, Schmidt and Sickles (990) and Lee and Schmidt (993). These models allow a more flexible pattern of technical efficiency change relative to the single-parameter structure used in this study. However, given the short time frame in this study (five years), we took the decision that the extra complexy of these latter models was not warranted in this instance.

7 6 where c is the cost of production, p j is the j-th input price, and b 0, b j and a are unknown parameters. Using the first order condions for cost minimisation, p K p j β jx K = x j x K =, j=,,...,k-, (7) β x K j we can show (see proof in Appendix ) that the parameters of the cost and input distance functions are related as follows: b j = β j, j=,,,k. a = α, and K j= ( β ) b 0 = δ β j ln j. Once we have estimated the parameters of the input distance function, we can predict the technical efficiency scores using equation (5). We can then predict the technically efficient input quanties as: T j xˆ = x TÊ, j=,,,k. (8) j The cost-efficient input quanties are predicted by making use of Shephard s Lemma, which states that they will equal the first partial derivatives of the cost function: c = ĉ bˆ p, j=,,,k, (9) C xˆ j = p j j j where ĉ is the cost prediction obtained by substuting the estimated parameters into (the exponent of) equation (6). Thus, for a given level of output, the minimum cost of production is xˆ C. p, while the observed cost of operation of the firm is x. p. These two cost measures are then used to calculate the cost efficiency (CE) scores for the i-th firm in the t-th year: 3 In the case when there are M outputs in the model, the output coefficients are a m =-α m, m=,...,m. The dot in this expression denotes vector multiplication. 3 This cost efficiency score could also be obtained using predictions from the cost function in equation (6). However, the advantage of our approach is that the vectors of technically efficient and allocatively efficient input quanties are also calculated.

8 7 C xˆ.p C Ê =. (0) x.p Then, following Farrell (957), allocative efficiency (AE) can be calculated residually as: A Ê = CÊ / TÊ. () Each of these three efficiency measures take a value between zero and one, wh a value of one indicating full efficiency. The above method is illustrated in Figure, for the case of a two-input technology. As in Figure, the line SS is the isoquant corresponding to a particular output level, y. 4 In Figure, the input vectors x, T xˆ and C xˆ are represented by the points A, B and C, respectively. Point A is the observed input vector, point B is the technically efficient input vector, and point C is the cost-efficient input vector, which is the point at which the isocost line is at a tangent to the isoquant. An isocost line, which has a slope reflecting the relative input prices, is drawn through each of these three data points. These three isocost lines are labelled x, xˆ. p and xˆ. p, respectively.. p T C Figure : The decomposion of cost efficiency x S A B L(y) x. p 0 C S / xˆ xˆ T. p C. p x 4 Note that in this study we estimate a stochastic input distance function. Hence the posion of this isoquant may vary from firm to firm (even when the output level is the same) as a consequence of the random error term, v, which will be added to the intercept parameter, δ.

9 8 3. An Application to Indian Dairy Processing Plants Prior to 99, dairy processing in India involved a mixture of private and cooperative plants. However, the industry was dominated by plants that were run as cooperatives. This was a result of a government policy that prevented private operators from setting up in areas where cooperative plants were established. However, in 99 the government introduced new regulations that made easier for private companies to establish processing plants. This was done to stimulate improved efficiency in the dairy processing industry. The change in regulations resulted in the establishment of over 00 new private plants in the 99/99 period. 5 The purpose of this empirical analysis is to assess the impact of the new government policy on this industry. The data in this study are taken from two states, Punjab and Haryana, in the northern region of India. The northern region is the largest milk-producing region in India. A questionnaire was used to collect annual data on 3 dairy plants, comprising 3 cooperative plants and 0 private plants, between 99/93 and 996/97. The resulting panel data set involves a total of 00 observations. 6 One output variable and four input variables are used in estimating the input distance function. The output variable is a Fisher multilateral index of the outputs produced by these firms. The main outputs are ghee (a solid material used instead of cooking oil), fluid milk and milk powder. These products contribute 38%, 3% and 8% to total revenue. The remaining % of revenue is derived from flavoured milk, butter, milk cake, sweets, paneer (unprocessed cheese), baby food and various other minor products. All plants produce two or more products. We observe that all plants produce ghee, while 7 and 9 plants produce milk powder and liquid milk, respectively. 7 5 See Singh (000) for further discussion of this reform process. 6 The sampling method used was to approach all plants in these two regions for data. Thus, was a census. All cooperative plants provided the requested data. The privately owned plants were not keen to cooperate, however sufficient information was derived from the annual reports of most of these companies. We could not get access to the annual reports of a few private companies, but we do not expect this non-response to introduce any systematic bias in our sample. However, if there is any bias, is likely to be an upward bias in the mean efficiency of the private firms (based on an expectation that those firms that did not supply copies of their annual reports are likely to be the more inefficient firms). If bias of this type does exist, would not cause us to question the main conclusion derived from our empirical results, that the private firms are not more efficient than the cooperative firms. 7 The decision to include only one aggregate output variable in our model was primarily due to the fact that we wished to conserve degrees of freedom. It was also influenced, to some extent, by the observations of Klein (953, p7) who noted that the Cobb-Douglas function imposes a functional

10 9 The four input variables are raw materials, labour, capal and other inputs. The raw materials input involves raw milk and, in some cases, a small amount of milk powder or butter. Labour expendure is taken directly from the firms accounts. Capal expendure was measured using the reported costs of depreciation, repairs and maintenance, and interest. The other inputs expendure variable includes all other costs of dairy production inputs (eg. administration, fuel, power and insurance). Price information was used to derive implic output quanties and implic input quanties, by dividing revenues and costs by appropriate price indices. These price indices were obtained from the Directorate of Economics and Statistics, Government of India, New Delhi. These price indices are discussed in detail in Appendix. Summary statistics for the sample input and output quanties are given in Table. From this table we note that the private plants are, on average, over twice the size of the cooperative plants. This is evident from the fact that the means of output and inputs (wh the exception of labour) are more than twice as large in the private plants. Table : Sample means* Plants Raw materials Labour Capal Other inputs Output Cooperative Private Total * All values are in millions of 996/97 Rupees. The maximum likelihood estimates of the parameters of the Cobb-Douglas input distance function are reported in Table. One issue of some concern to us in this paper is the degree to which the use of the Cobb-Douglas functional form may be form that is convex to the origin in the output dimensions, which is hence not consistent wh prof maximisation. This could be a problem in some cases. However, is not a problem when there is only one output variable (as in our study). Furthermore, even if there is more than one output variable, one could argue that the Cobb-Douglas provides a first-order approximation to the slope of the production possibily surface at the mean of the data. Hence, if one is focussing on cost minimisation issues, and not prof maximisation, the model should still provide a reasonable approximation to the underlying technology, especially in small samples where degrees of freedom are limed. 8 See Singh (000) for further detail on the data used in this application.

11 0 imposing unwarranted restrictions upon the production technology. To address this question we have also estimated a translog input distance function, 9 and used a likelihood ratio test to test the significance of the ten extra parameters which differentiate from the Cobb-Douglas functional form. The log-likelihood function (LLF) value for the estimated translog model is 93.98, and the likelihood ratio (LR) test statistic, to test the null hypothesis of the Cobb-Douglas versus the alternative hypothesis of the translog, is ( )=5.74. This value is less than the 5% Chi-square crical value (for ten degrees of freedom) of 8.3. Hence, we do not reject the null hypothesis and conclude that the extra complexy of the translog is not warranted in this instance. The estimates of the coefficients of the inputs in Table are significant wh expected signs. The coefficient of output is less than one in absolute value, indicating increasing returns to scale in this industry. 0 The estimated coefficient of the raw material input is the largest, at According to our survey data, raw material contributes 8 per cent of the total cost of operation and is the major expendure component. Many past studies (eg. Singh and Kalra, 980; Arora and Bhogal, 996) make similar observations. The estimated coefficients of labour and capal are 0.3 and 0.045, respectively. The coefficient of the other inputs variable is calculated via the homogeney restriction, and is found to be equal to The estimate of the variance ratio parameter, γ, is and is significant at the % level. This suggests that the technical inefficiency effects are significant in the stochastic frontier model. Therefore, the ordinary least squares estimation of this function (ie. a model assuming no technical inefficiency) would not be an adequate representation of the data. This observation is confirmed by conducting a likelihood ratio test, to test the null hypothesis of the OLS model against the alternative hypothesis of the frontier model reported in Table. The LLF for the OLS model is 64.0, providing a LR test statistic of The corresponding 5% mixed-chi-square crical value (taken from Table in Kodde and Palm, 986) is equal to for three degrees of freedom. Thus, the null hypothesis (the OLS model) is rejected in 9 See Coelli and Perelman (000) for a description of the translog input distance function. 0 Note that the standard returns to scale elasticy, which is regularly reported in production function studies, is equal to the inverse of the negative of this value. That is, /0.9=.098.

12 favour of the alternative hypothesis corresponding to the stochastic frontier model. This suggests that there is significant technical inefficiency in this industry. Table : Maximum-likelihood estimates Variable Parameter Coefficient Standard error Constant α 9.749*** 0.6 Raw Material β 0.677*** 0.03 Labour β 0.3*** 0.04 Capal β ** 0.0 Other Inputs # β *** 0.03 Output δ -0.9*** 0.0 Variance parameters: s u v σ = σ + σ 0.034* 0.0 γ = σ u σ 0.797*** 0.40 s µ η -0.05** 0.04 LLF 86. * Asterisks indicate significance levels: *** % level, ** 5% level and * 0% level. # The estimate of β 4 was obtained via the homogeney restriction. Its standard error was approximated using a second-order Taylor series expansion. The trend parameter for the inefficiency effects, η, is negative and significant at the 5% level. This indicates that there has been a significant decrease in technical efficiency over this five-year period, contrary to expectations. It was expected that the reform process would have led to improvements in technical efficiency, not the converse. Note that our model assumes that there has been no technical change over this five-year period. Based on our knowledge of the technologies used in the plants in the sample, this is a reasonable

13 The average technical, allocative and cost efficiency scores are reported in Table 3. These results suggest that private plants are not as cost-efficient as their cooperative counterparts. This difference is mostly due to extra allocative inefficiency in the private plants. To see if these differences are statistically significant, we tested the null hypothesis that the mean efficiency of private plants was equal to the mean efficiency of cooperative plants, versus the alternative hypothesis that they differed. The test statistic used was the t-ratio for the case of two large independent samples (see Kvanli, Guynes and Pavur, 996, p37). The calculated test statistics were 0.46,.04 and.37, for TE, AE and CE, respectively. All of these values are less than the 5% crical value of the Standard Normal distribution of.96. Hence, we conclude that the mean efficiencies of private and cooperative plants are not significantly different. This result is contrary to what the government officials may have expected, but consistent wh what has been reported in some recent empirical comparisons of private and cooperative organisations in other industries. For example, see the analysis of the Costa Rican coffee-processing sector by Mosheim (998). Table 3: Mean efficiencies* Dairy Plants Technical efficiency Allocative efficiency Cost efficiency Cooperatives (0.083) Private (0.083) Total (0.083) (0.07) (0.056) (0.07) (0.084) 0.76 (0.05) (0.08) * Sample standard deviations are presented in parentheses under each sample mean. The mean allocative efficiencies reported in Table 3 indicate that some inputs are being used in incorrect proportions. To investigate which inputs are being over- or under-used, we can calculate the ratio of the technically efficient input quanty over assumption. Furthermore, we conducted some preliminary analysis in which a time trend was included in the model. The resulting coefficient was negative and insignificantly different from zero at the 5% level, supporting our expectation that technical change was not present in this industry over this time period.

14 3 the cost-efficient quanty (for each input and for each observation). The means of these ratios are given in Table 4, where we observe that plants have tended to overutilise raw materials and capal inputs, and under-utilise labour and other inputs. In particular, we note that the private firms are under-utilising labour to a greater extent than the public firms. This could perhaps be a consequence of greater use of imported labour saving methods and technologies in private plants. These methods may be cost-efficient in developed countries, where wages are high, but this may not translate to low-wage countries such as India. Table 4: Input usage ratios* Dairy plants Raw material Labour Capal Other inputs Cooperatives Private Total * Note that a value greater than one indicates overuse. Also, note that these are weighted averages, where the weights are the input quanties. The government expected that the introduction of new private firms would encourage competion and hence improve efficiency levels over time. The changes in the efficiencies of cooperative and private plants over the five-year study period are presented in Table 5. Average cost efficiencies decline by approximately 5% over this period. This decline was primarily due to a decline in technical efficiency. This decline is contrary to the effect that the government would have hoped to achieve from liberalization. When we first saw this decline in technical efficiency, we suspected that may be a consequence of capacy utilisation problems, brought about by the sudden One should provide a word of warning about the use of the term allocative efficiency (AE) in this study. We have measured allocative efficiency among all factors of production. That is, we have not distinguished between capal and non-capal inputs. Thus, some of the measured AE could actually reflect errors in forecasting the future relative prices of capal and non-capal inputs, made when capal investment decisions were taken at some time in the past. Furthermore, some of the measured AE could reflect input ratios that may appear inefficient in the short run but which are actually efficient in the long run, given particular predictions of future levels of prices and production. Thus, one could argue that the AE measures reported in this study are likely to be a lower bound estimate of the abily of the plant managers to select an optimal input mix.

15 4 introduction of a number of new plants into the industry. However, we then recalled that milk plants in India only collect approximately 0% of the fresh milk produced in India. The remaining 90% is utilised in the informal sector, where individual farmers or small operations process a small amount of milk for local consumption. Thus, is unlikely that capacy utilisation is contributing to the observed decline in technical efficiency. Table 5: Mean efficiencies over time Cooperatives Private Year Technical efficiency Allocative efficiency Cost efficiency Technical efficiency Allocative efficiency Cost efficiency 99/ / / / / Concluding comments The analysis of cost efficiency in the Indian dairy industry, reported in this study, was designed to shed light on the effects of the reform process. Two questions were of particular interest. First, has the performance of the cooperative plants improved since market liberalization was introduced? Second, are the private plants more efficient than the cooperative plants? The answer to both questions, based on the empirical evidence in this paper, is no. This suggests the reforms have not achieved the desired results. The main contribution of this paper is methodological. We have proposed a new input distance function approach to the calculation and decomposion of cost efficiency. This new approach avoids many of the problems inherent in the existing production frontier and cost frontier approaches. Namely, does not require price information that varies across firms, is robust to systematic deviations from cost-

16 5 minimising behaviour, and does not suffer from simultaneous equations bias when firms are cost minimisers or shadow cost minimisers. Furthermore, the input distance function approach can also accommodate multiple outputs, as opposed to the production frontier approach, which is limed to a single output. However, having listed a number of the advantages of this new method, we should also remind the reader of s main shortcomings, and outline some possible areas of future work. The main concern we have wh the method outlined in this paper is the use of the Cobb-Douglas functional form. This is a restrictive functional form, which assumes unary elasticies of substution and fixed production elasticies. We found that these restrictions were appropriate for the sample data used in the paper, but is unlikely to be the case for all data sets. Hence, one might find necessary to use a more flexible functional form, such as the translog, in some cases. The main down-side of the use of the translog (and other flexible functional forms, such as the quadratic or generalized Leontief) is that they are not self-dual, and hence is not possible to derive the implic cost function as done in this paper wh the Cobb-Douglas functional form. Hence, one would not be able to identify the costminimizing input quanties using simple calculations, as is done in this paper. In fact, to calculate the cost-minimizing input quanties relative to a translog production technology, one is required to solve a non-linear optimization problem for every data point in the sample. This is a messy and time-consuming exercise. In addion, one can also run into problems when the estimated translog function does not satisfy the standard regulary properties (convexy and monotonicy) at all sample data points. This is not an unusual occurrence. One is unable to solve the cost-minimising problem for these data points, and must hence find a way to reestimate the production technology wh these restrictions imposed whout sacrificing the flexibily of the translog form. This is not a simple exercise. Some years ago, we estimated a translog stochastic production frontier using these dairy plant data, and found that convexy and monotonicy condions were violated at some data points. This led to the development of a Bayesian method to estimate the model wh the required regulary condions imposed. This method is described by Cuesta, et al (00), for the case of the translog production frontier. We intend to extend this approach to the case of the input distance function in future work.

17 6 References Afriat, S.N. (97), Efficiency Estimation of Production Functions, International Economic Review, 3, Aigner, D.J., C.A.K. Lovell and P. Schmidt (977), Formulation and Estimation of Stochastic Frontier Production Function Models, Journal of Econometrics, 6, -37. Arora, V.P.S. and T.S. Bhogal (996), Integrated Milk Cooperatives in North-West Uttar Pradesh: Organisation, Functioning and Performance, Indian Journal of Agricultural Economics, 5, Atkinson, S.E. and C. Cornwell (994a), Parametric Estimation of Technical and Allocative Efficiency wh Panel Data, International Economic Review, 35, Atkinson, S.E. and C. Cornwell (994b), Estimation of Output and Input Technical Efficiency using a Flexible Functional Form and Panel Data, International Economic Review, 35, Balk, B. (998), Industrial Price, Quanty, and Productivy Indices: The Micro- Economic Theory and an Application, Kluwer Academic Publishers, Boston. Battese G.E. and T.J. Coelli (99), Frontier Production Functions, Technical Efficiency and Panel Data: Wh Application to Paddy Farmers in India, The Journal of Productivy Analysis, 3, Boles, J.N. (966), Efficiency Squared - Efficient Computation of Efficiency Indexes, Proceedings of the 39th Annual Meeting of the Western Farm Economics Association, pp Bravo-Ureta, B.E. and L. Rieger (99), Dairy Farm Efficiency Measurement using Stochastic Frontiers and Neoclassical Dualy, American Journal of Agricultural Economics, 73, Charnes, A., W.W. Cooper and E. Rhodes (978), Measuring the Efficiency of Decision Making Uns, European Journal of Operational Research,,

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20 9 Appendix : Dualy derivation In this appendix we derive the relationship between the parameters of a Cobb-Douglas input distance function and the implied dual cost function. The proof is provided for the case of one output and two inputs, but can be easily extended to the M-output/Kinput case. Consider the Cobb-Douglas input distance function in log form ln d = α ln y + δ + β, (A) ln x + β ln x where homogeney implies β + β =, and all notation is as defined in the text. If we set the distance to one we obtain the equation of the production surface α ln y + δ + β ln x + β ln x = 0. (A) Now, making ln x the subject of this equation, we can obtain the partial derivative ln x ln x β = β, (A3) or equivalently x x x x β = β. (A4) Using the first order condions for cost minimisation we obtain x x β = β x x p = p, (A5) or equivalently p = x β p x β 0. (A6) Wh this equation and the cost identy p x + px = c, (A7) we have two equations and two unknowns. Next we solve these two equations for x and x and then substute these expressions into equation (A) to derive the cost function. Rearranging equation (A6) we obtain

21 0 x p c x = +. (A8) p p Substuting equation (A8) into equation (A6) and exploing the homogeney condion (β + β = ) we obtain, after some algebra x cβ =. (A9) p Substuting equation (A9) into equation (A8) we obtain x cβ =. (A0) p Now, substuting equations (A9) and (A0) into equation (A), and again using the homogeney condion, we obtain the cost function ln c = α ln y δ ( β. (A) lnβ + β lnβ ) + β ln p + β ln p Q.E.D. The proof for the M-output/K-input case follows in the same way. The main differences are that we utilise K- first order condions in equation (A5) and end up solving K equations in K unknowns in equations (A6) and (A7), which is best done using matrix algebra.

22 Appendix : Price Indices In this appendix we present details on the price indices used in the empirical analysis in this paper. These price indices, wh two exceptions, were obtained from the Directorate of Economics and Statistics, Government of India, New Delhi. The first exception is the deflator used for other costs, which is a general index of wholesale prices in Indian manufacturing, obtained from the Economic Advisor, Ministry of Industry, New Delhi. The other exception is the wage index, which was obtained from the Labour Bureau, Simla, India. The wage indices for 995 and 996 were not available at the time the analysis was conducted. Hence, the CPI was used to approximate the wages indices in these years. The price indices reported here, wh the exception of the wage price indices, are price indices for India as a whole. Thus is assumed that prices do not vary significantly across regions, nor across firms whin regions. This is believed to be a reasonable assumption for these firms. However, should be kept in mind that if any firms do face differing prices (perhaps due to geographical disadvantage) the resulting efficiency scores will be biased. Table A: Price indices Item Milk Ghee Butter Whole milk powder Skim milk powder Baby food Malted food Dairy products Machinery Other costs Wages - Haryana Wages - Punjab