Assessing Technical Efficiency of Quebec Dairy Farms

Assessing Technical Efficiency of Québec Dairy Farms Msafiri D. Mbaga1, Robert Romain2, Bruno Larue3 and Luc Lebel4 1

Centre for Research in the Economics of Agri-Food (CREA), Université Laval, Québec, Québec (Currently, Research Economist with Agriculture and Agri-Food Canada, Ottawa, Resarch and Analysis Directorate). 2 Corresponding author, CREA and Department of Agricultural Economics and Consumer Sciences, Université Laval, Québec, Québec. 3 CREA and Department of Agricultural Economics and Consumer Sciences, Université Laval, Québec, Québec. 4 CREA and Department of Wood Science and Forestry, Université Laval, Québec, Québec. Received October 2001, accepted November 2002 The purpose of this paper is twofold. Our first objective is to measure the level of technical efficiency of Québec dairy farms. Our second objective is to gauge the robustness of our results with respect to the selection of a functional form and of a distribution for the inefficiency index. We estimate efficiency frontiers for Cobb-Douglas (C-D), translogarithmic (TL) and generalized Leontief (GL) production functions with half-normal, truncated normal and exponential distributions. Our results, based on likelihood dominance criterion (LDC) indicate that the GL production technology dominates the other two functional forms, and this ranking is robust to changes in the distribution of the inefficiency index. Efficiency scores and ranks are highly correlated for all the functional forms and distributions. The differences in the mean levels of efficiency are statistically significant across functional forms and distributions, although the magnitude of the difference is minuscule. The very high mean level of efficiency and the low standard deviation confirms that Québec dairy farms are very homogenous in terms of getting the most from their inputs. This is not surprising, given that the sector has been very stable policywise and that it has been difficult for dairy farmers to expand. To augment the comparisons, results obtained from data envelopment analysis (DEA), are added to the analysis. In this case, the correlation coefficients between DEA and parametric specifications are found to be very low. L’objectif de cet article est double. Le premier est de mesurer le niveau d’efficacité technique des fermes laitières au Québec. Le second est d’évaluer si nos résultats sont robustes quant aux choix de la forme fonctionnelle et de la distribution de l’indice d’inefficacité technique. Nous estimons une frontière d’efficacité technique en utilisant des fonctions de production de formes Cobb-Douglas (C-D), translogarithmique (TL) et Léontief généralisée (GL) en utilisant, pour l’indice d’inefficience, les distributions semi-normale, normale tronquée et exponentielle. Selon le critère de sélection LDC (Likelihood Dominance Criterion), nos résultats indiquent que la technologie de production de type GL domine les deux autres et ce, pour les trois distributions de l’indice d’inefficacité. Les indices d’efficacité ainsi que leur rang sont fortement corrélés pour toutes les formes fonctionnelles et les distributions analysées. Les différences entre les niveaux moyens d’efficacité technique sont significatives quoique l’ampleur des différences soit très faible. Un niveau moyen élevé d’efficacité et un faible écarttype confirment que les entreprises laitières du Québec sont très homogènes en ce qui a trait à l’utilisation maximale de leurs intrants. Ce résultat n’est pas surprenant puisque la politique laitière a été très stable depuis une trentaine d’années et la croissance de la taille des entreprises a été difficile. Afin d’enrichir les comparaisons des différentes mesures, nous avons également estimé les indices d’efficacité avec la méthode DEA (Data Envelopment Analysis). Les résultats montrent que les indices d’efficacité obtenus avec cette méthode sont très peu corrélés avec ceux des méthodes paramétriques. Canadian Journal of Agricultural Economics 51 (2003) 121–137

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INTRODUCTION The dairy sector is the third largest farm sector in Canada, accounting for about $3.8 billion in 1998, which is approximately 14% of total farm cash receipt (Canadian Dairy Commission 2000). Because of continuous structural adjustment, dairy farms have decreased in number and increased in size. In 2000, there were 20,600 dairy farms in Canada, a number that represents an annual decline of roughly 6% over the previous five-year period. Farm size varies considerably by province, but overall size has been growing (Romain 2001). The average herd size in 1999 of 57.1 cows per farm, while small in comparison with what is observed in the United States, is nevertheless much larger than the 38.2 cows per farm observed in 1989 (GREPA 2000). The province of Québec is the most important milk-producing province in Canada, with over 46% of total manufacturing milk, and it has experienced somewhat smaller structural adjustments than other provinces. Over the same five-year period, the number of producers decreased by 4.1% annually to 10,300 farms in 1998, and the average annual growth rate in farm size was 4.5%. The average size of dairy farms in Québec (46.5 cows in 1998 according to GREPA 2000) is even lower than the Canadian average. Supply management has made it difficult for farm operators to increase herd sizes. Our hypothesis is that the homogeneity of farm size and the stability of the policy environment have brought about a high degree of homogeneity in technical efficiency.1 During the past 10 years, there have been challenges under the North American Free Trade Agreement and the World Trade Organization against elements of Canada’s supply management system. These challenges were launched by Canada’s main trade partner, the United States. At the eve of a new round of multilateral negotiations, there is increased pressure to modify the Canadian supply management system in which the dairy sector has been operating since the mid-1960s so issues related to productivity and/or efficiency are of interest to farm managers and policymakers. The overall objective of this study is to measure and assess the robustness of technical efficiency for the milk production sector of the province of Québec, both in relative and absolute terms across the industry based on a 1996 cross-section of 1,143 dairy farms. Our results show that the small Québec farms exhibit similar levels of technical efficiency. Consequently, the structural change that would follow a hypothetical phasing out of Canada’s supply management policy would not be driven by technical efficiency disparity across farms, but more likely by variables like the farm operator’s age and capacity to borrow. This conjecture is based on the belief that that farm size would have to increase to levels closer to what is observed in the United States if supply management were to be phased out. Doubling or tripling the size of one’s farm would be very costly and dairy producers who are close to retirement age and/or with limited borrowing capacity may not want to embark on such ambitious expansion projects. Measuring efficiency is not straightforward, and research efforts have been devoted to the improvement and refinement of the techniques used in such pursuits over the past several years.2 Still, practitioners confront several issues, like choosing between the parametric and nonparametric approaches (i.e., stochastic production frontier (SPF) versus data envelopment analysis (DEA)). If the former is chosen, the researcher must then choose a functional form as well as a distribution for the inefficiency index. Both approaches have distinct advantages and limitations. In this paper, we first address the issue of choosing a dominant functional form from three competing popular alternatives. Second, the robustness of the efficiency scores to the

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choice of a distribution for the inefficiency index is assessed. Our results show that the generalized Leontief production technology emerges as the dominant functional form, and this ranking is robust to changes in the distribution of the inefficiency index. The efficiency scores are very highly correlated for all three functional forms and three distributions of the inefficiency scores that are analyzed. The differences in the mean levels of efficiency are statistically significant across functional forms/distributions, but the magnitude of the differences is very small. To enrich the comparisons, results obtained from DEA, which circumvent the need for distributional assumptions, are added to the analysis. In this case, the correlation coefficients between the efficiency scores from DEA and any parametric specification are very low. For our data set, we can conclude that the choices of a functional form and a distribution are almost inconsequential relative to the choice between the parametric and nonparametric approaches. The rest of the paper is organized as follows. The next section summarizes methodological issues associated with the parametric and nonparametric approaches. This is followed by a brief presentation of the models investigated in this study. The fourth section presents the empirical results while the last section provides a summary of the analysis as well as some managerial and policy implications. ISSUES IN EFFICIENCY MEASUREMENT TECHNIQUES The SPF approach requires the specification of a production technology by selecting from a small pool of functional forms. In many studies, the choice of the functional form appears to be arbitrary, but Cobb-Douglas or a flexible form, like translogarithmic, is generally adopted. Several flexible functional forms have been proposed (Griffin et al 1987), but the translogarithmic is by far the most popular alternative even though there seems to be a consensus that the dominance of one functional form over others is dataset specific (Kumbhakar and Lovell 2000). Another requirement of the SPF approach is the choice of a distribution for the inefficiency scores. Again, it seems that most practitionners do not invest much time and effort in choosing a particular distributional form. Finally, the SPF approach is suited only for single-output technologies. A multi-output case can only be studied if the various outputs can be aggregated into a single aggregate output (Coelli et al 1998). Though the nonparametric DEA approach is not without flaws, it does not suffer from the above shortcomings (Kalaitzandonakes et al 1992). In addition, Gong and Sickles (1992) find that DEA can outperform parametric frontier functional forms when the selected functional form is significantly different from the actual Data Generating Process (DGP) and when inefficiency is heavily correlated with the regressors. Recent advances relying on bootstrapping make it possible to perform limited statistical tests on DEA results. Unfortunately, the bootstrap is too restrictive and hard to implement. Second, the deterministic nature of DEA makes it also very sensitive to extreme values, which are crucial in the determination of the frontier. Third, the results are greatly influenced by the arbitrary choice of explanatory variables. Tauer and Hanchar’s (1995) Monte Carlo simulations suggest that measures of efficiency are more sensitive to changes in the number of products and inputs than to changes in the number of observations. Finally, Coelli et al (1998) show that many firms could spuriously appear on the DEA frontier when the sample is small and there are many inputs. DEA and SPF also share limitations. Specifically, the technical efficiency/efficiency score of an individual firm is defined as the ratio of the observed output to the corresponding

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frontier output, conditional on the levels of inputs used by that firm. Therefore, technical efficiency is a relative concept, not an absolute one, even when applied at the firm level. Furthermore, mean efficiency scores from two samples only reflect the dispersion of the firms’ level of efficiency within each sample. As a result, a comparison of two mean scores conveys no information about the efficiency of the firms in one sample relative to the firms in the other sample. ALTERNATIVE APPROACHES TO TECHNICAL EFFICIENCY MODELING Stochastic Frontier Models The seminal contributions of Aigner, Lovell and Schmidt (1977) and of Meeusen and van den Broeck (1977) have inspired many studies on efficiency over the last two decades. The general stochastic frontier production function is specified as: Yi = Xi b + Vi – Ui,

i = 1,…, N

(1)

where: Yi = the output of the ith farm Xi = a (1 × K) vector of inputs for the ith farm β = a (K × 1) vector of unknown parameters Vi = iid~N(0, σV2 ) Ui’s = independently distributed, nonnegative, unobservable random variables associated with technical inefficiency in production such that, for a given technology and input levels, the observed output falls short of its potential.3 Specific functional forms and inefficiency distributions must be chosen to estimate the model defined by Eq. 1. The most common functional forms encountered in the literature are the parsimonious Cobb-Douglas function and the translogarithmic (TL) form, which offers the advantage of being a second-order Taylor series expansions to an arbitrary technology. The Cobb-Douglas (C-D) functional form has been frequently used because of its simplicity. This simplicity comes at the cost of imposing strong restrictions on substitution possibilities. Second-order Taylor series expansions (or flexible forms) impose fewer restrictions on the technology, but their estimations are susceptible to multicollinearity, and sometime, to low degrees of freedom problems (Berndt 1991). Like the TL, the generalized Leontief (GL) function is also flexible. It is commonly used in the estimation of cost functions and input demands, but it is not as popular in the estimation of efficiency frontiers. The TL and GL forms along with the C-D, are estimated and compared in this study. The TL and GL stochastic frontier production functions are respectively defined by Eqs. 2 and 34: n

ln Yi = β 0 +

∑

n

β j ln X ji +

j =1

∑ ∑ β jk ln X ji ln Xki + Vi − Ui

∑

j =1

(2)

j ≤ k k =1

n

Yit = β 0 +

n

n

β j X ji +

n

∑ ∑ β jk j ≤ k k =1

where Yi and Xi = the output and inputs for the ith firm.

X ji Xki + Vi − Ui

(3)

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The C-D technology is nested in the TL technology (i.e., when all bjk = 0). The Vi’s are assumed to be independent and identically distributed normal random variables with mean zero and variance σ2V. With respect to the distribution of the inefficiency scores, Ui, three distributions are compared: the half-normal, truncated-normal and exponential distributions. The choice of variables was inspired from previous dairy efficiency studies (e.g., Tauer and Belbase 1987). The dependent variable is total milk output per cow, adjusted for butterfat content, expressed in hectolitres, as in Romain and Lambert (1995). Five independent variables have been retained: concentrate, forage, labor, capital and genetic endowment. The quantity of feed concentrate is measured in kilograms consumed per cow while the quantity of forage is expressed in kilograms consumed per cow. Labor is expressed in total units of full-time dairy farm worker in a year normalized on a per-cow basis, and capital is defined by the total value of assets in dollars per cow. Finally, the average weight of dairy cows is considered to be a reliable proxy for the genetic quality of the herd.5 Data Envelopment Analysis Data envelopment analysis (DEA) is the commonly used nonparametric mathematical programming approach to frontier estimation. It posits that the efficiency of a decision-making unit (DMU) is measured relative to the efficiency of all the other DMUs subject to the restriction that all DMUs are on or below the frontier.6 DEA was first developed under the assumption of constant return to scale (CRS). In 1984, Banker, Charnes and Cooper (1984) extended the CRS model to allow for variable returns to scale. The optimization problem is stated as follows: MinΨℜ Ψ such that –yi + Yℜ ≥ 0 Ψxi – Xℜ ≥ 0 N1′ℜ = 1 ℜ≥0

(4)

where: N1′ℜ = 1 = the convexity constraint Ψ = a scalar containing the efficiency score for the ith firm or DMU ℜ = a N × 1 column vector of variable weights xi, yi = input and output vectors for the ith firm. X and Y represent the matrices of input and output levels for all firms in the sample while N1 is a N × 1 vector of ones. The variable returns to scale specification is the most commonly used specification in empirical applications.7 The value of Ψ must satisfy the restriction 0 ≤ Ψ ≤ 1, with a value of 1 indicating a point on the frontier and hence a technically efficient firm. The DEA optimization problem Eq. 4 is solved N times to obtain a value of Ψ for each firm. EMPIRICAL RESULTS Data Sources and Descriptive Statistics The data used in this study are compiled from the Agritel database. This database is collected by the Federation of Management Clubs in the province of Québec and contains detailed

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Table 1. Sample summary statistics of Québec dairy farms Variablea

N

Mean

Std. dev.

Min

Max

Nonmaize region: Number of cows Milk outputa (hL/cow) Concentrate (kg/cow) Foragea (kg/cow) Labora (person/cow) Assets (capital) (person/cow) Weight (kg/cow)

821 821 821 821 821 821 821

46.5 78.6 2656 7738 0.0302 5264 596

20.4 11.0 705 150630 0.0075 1544 31

14 39.3 488 1014 0.014 1626 475

205 113.2 6537 121 0.071 10874 725

Maize region: Number of cows Milk outputa (hL/cow) Concentrate (kg/cow) Foragea (kg/cow) Labora (person/cow) Assets (capital) ($/cow) Weight (kg/cow)

322 322 322 322 322 322 322

48.2 80.2 2690 7451 0.0287 5339 596

17.2 10.1 598 1482 0.0063 1428 28

22 43.6 236 3457 0.016 2494 487

172 114.6 4330 14122 0.050 12647 674

aIndicates that the difference between the means of the two regions is statistically significant at the 5% level. Source: Agritel – Data base.

socio-economic information on about 2,200 farms. However, the sample used in this study is restricted to a 1996 cross-section of 1,143 farms specialized in dairy production (i.e., farms that generated at least 80% of their 1996 revenue from dairy activities). Moreover, the sample is divided into two subgroups because climatic and soil conditions are heterogeneous in the province. The justification for this segregation is that the two groups face different constraints likely to affect efficiency measures. The first subgroup includes farms located in areas not suitable for maize production while the second subgroup is made up of the remaining farms. Hereafter, these subsamples are referred to as the nonmaize and maize regions. Table 1 presents descriptive statistics for both regions. The results of the Student t-tests on the difference of means between the two regions support the hypothesis that the two subsamples are significantly different in many ways even though they are similar in terms of number of cows. On the production side, average yield per cow, adjusted for butterfat content, is slightly higher in the maize region than in the nonmaize region. Conversely, the quantity of forage fed to cows is higher in the nonmaize region and so is labor utilization. Finally, utilization of concentrate and assets per cow, as well as average weight per cow are not statistically different between regions. Choosing Between Alternative Parametric Models Because the three functional forms investigated in this study are not all nested, the likelihood dominance criterion (LDC) proposed by Pollak and Wales (1991) is used to identify a dominant functional form.8 The main advantage of LDC is that it does not require the estimation of a (arbitrary) composite hypothesis. As such, it is very easy to implement. Moreover,

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Table 2:.Log likelihood values for alternative functional forms and distributions Half normal

Truncated normal

Exponential normal

Nb parameters

Nonmaize regiona: C-D 770.96 (– 2803.49) 770.39 (–2804.06) 775.44 (– 2799.01) TL 800.32 (–2774.13) 800.84 (–2773.61) 804.24 (– 2770.21) GL – 2712.42 –2712.54 –2710.09

6 21 21

Maize regionb: C-D TL GL

6 21 21

aNumbers bNumbers

362.59 (–1046.60) 362.42 (–1046.77) 362.56 (1046.63) 372.67 (–1036.52) 372.00 (–1037.19) 372.24 (1036.95) –1033.30 –1033.34 –1033.15

in parentheses are log likelihood adjusted by the Jacobian term – Σ ln Output = 3574.45. in parentheses are log likelihood values adjusted by the Jacobian term – Σ ln Output =

1409.19.

accepting or rejecting both competing hypotheses is not nearly as frequent with the LDC as it is with conventional nonnested testing procedures. When the number of parameters in both hypotheses differs, the LDC has a small range where it is indecisive. When the number of parameters is the same in both competing models, the model with the higher log likelihood value is selected as dominant, which is implied by the statistically sound dominance ordering (Pollack and Wales 1991). The LDC is consistent with the classical statistical approach to hypothesis testing, which is not the case with neither of the popular Akaike’s and Schwarz’s information criteria. Table 2 presents the log likelihood values for the three functional forms.9 The numbers in parentheses are the log likelihood values adjusted by the appropriate Jacobian term in order to adjust for the different measures of the dependant variable between models.10 Table 3 reports the ranking results for the alternative functional forms and efficiency distributions. When the TL and C-D are compared, the results are different according to the region. The TL is dominant in the nonmaize region and this result holds for all three distributions of the inefficiency scores. The C-D dominates the TL in the maize region for two distributions while the test is inconclusive when the exponential normal distribution is used. However, when the GL functional form is compared to the TL (nonmaize) or the C-D (maize), the GL emerges as the dominant functional form for all three distributions. Furthermore, variations in the log likelihood values for alternative distributions of the inefficiency scores are trivial when a particular functional form is considered. Therefore, at least with this dataset, the choice of the distribution does not affect the selection of the functional form. The previous two results lead naturally to two questions. First, does the choice of the functional form and of the associated distribution of the efficiency scores significantly affect the ranking of efficiency scores? Second, are the levels of efficiency scores significantly affected? Table 4 presents the correlation coefficients of the efficiency scores as well as the rank correlation coefficients for all nine specifications in the nonmaize region.11 The correlation coefficients are generally higher for ranks than for efficiency levels, but both are nevertheless

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Table 3. Ranking of the alternative model specificationsa Region Nonmaize Maize

Functional Half Truncated Exponential formsb normal (A) normal (A) normal (A) TL vs. C-D GL vs. TL TL vs. C-D GL vs. C-D

29.36 –61.71 9.65 –13.3

30.44 –61.07 9.58 –13.43

28.8 –60.12 10.11* –13.48

(B)

(E)

Decision

9.91

11.23

9.91

11.23

TL GL C-D GL

aA

= LTL – LC-D or LTL – LGL; B = [ C(N2 + 1) – C(N1 + 1)]/2; E = [C(N2 – N1+ 1) – C(1)]/2. critical chi-square values (C’s) are evaluated at the 5% significance level. *Indecisive. bThe

Table 4. Correlation and rank correlation coefficients of efficiency scores (nonmaize region) C-D– HN Correlation: C-D–HN C-D–TRN C-D–EXPN TL–HN TL–TRN TL–EXPN GL–HN GL–TRN GL–EXPN DEA Mean Scores Rank correlation: C-D–HN C-D–TRN C-D–EXPN TL–HN TL–TRN TL–EXPN GL–HN GL–TRN GL–EXPN DEA

C-D– C-D– TRN EXPN

TL– HN

TL– TRN

TL– EXPN

GL– HN

GL– TRN

GL– EXPN

1

0.972 1

0.974 0.998 1

0.959 0.928 0.928 1

0.920 0.952 0.952 0.956 1

0.935 0.955 0.954 0.973 0.996 1

0.955 0.941 0.940 0.981 0.961 0.968 1

0.955 0.940 0.9397 0.979 0.959 0.966 0.999 1

0.925 0.953 0.954 0.949 0.988 0.988 0.969 0.968 1

1

0.999 1

1 0.999 1

0.962 0.961 0.962 1

0.962 0.961 0.962 0.999 1

0.964 0.962 0.964 0.999 0.998 1

0.950 0.948 0.950 0.967 0.970 0.966 1

0.948 0.947 0.948 0.965 0.968 0.964 1 1

0.951 0.949 0.952 0.970 0.971 0.970 0.999 0.999 1

DEA

0.401 0.317 0.303 0.476 0.351 0.379 0.357 0.353 0.308 1 0.91050.9495 0.9395 0.9136 0.9599 0.9407 0.9463 0.9473 0.9591 0.9500

0.532 0.550 0.526 0.590 0.583 0.583 0.458 0.454 0.464 1

HN = Half normal distribution; TRN = Truncated normal; and EXPN = Exponential normal.

very high. The lowest correlation coefficient for levels is 0.920 for the C-D (half normal)–TL (truncated normal) combination, while for ranks the lowest correlation coefficient is 0.947 for the C-D (truncated normal)–GL (truncated normal) pair. The highest correlation coefficient

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Table 5. Summary statistics for the efficiency scores from the GL function for alternative distributions of the inefficiency scores and for DEA Nonmaize region Half normal Meana Std. dev. Minimum Maximum

0.9447 0.0259 0.7894 0.9891

(%) < 0.85 0.85 0.85 to 0.90 5.72 0.90 to 0.95 41.29 > 0.95 52.13

Truncated Exponential normal normal 0.9473 0.0240 0.8013 0.9891 (%) 0.61 4.63 38.73 56.03

0.9591 0.0254 0.6974 0.9914 (%) 0.97 1.95 19.00 78.08

Maize region DEA

Half normal

0.9215 0.0466 0.786 1

0.9481 0.0254 0.8374 0.9866

(%) 6.33 28.01 38.37 27.28

(%) 0.62 4.97 40.06 54.35

Truncated Exponential normal normal 0.9488 0.0248 0.8416 0.9866 (%) 0.62 4.66 39.44 55.28

0.9659 0.0201 0.8455 0.9902 (%) 0.31 1.24 11.49 86.96

DEA 0.95 0.041 0.805 1 (%) 0.62 11.49 36.96 50.93

aFor both regions, when the efficiency scores are compared with matched paired parametric and nonparametric tests (t-test and Wilcoxon signed rank test), the mean efficiency scores from the exponential distribution are higher than those of the other two distributions, while those of the truncated normal are higher than those of the half normal.

for efficiency levels is 0.999 for the GL (truncated normal)–GL (half normal) combination while several pairs of efficiency levels have a rank correlation coefficient of 0.999. The range of our correlation coefficients is tighter than the one found in Kumbhakar and Lovell (2000). They calculated rank correlation coefficients (using Greene’s 1990 results) ranging from 0.7467 (exponential and gamma) to 0.9803 (half normal and truncated normal). This evidence of strong correlation suggests that the choices of a functional form and of a distribution may not be critical, at least in our technical efficiency investigation.12 However, Ritter and Simar (1997) also found that the choice of the distribution is largely immaterial and argued for the use of a relatively simple distribution. In contrast, Giannakas et al (2000) estimated low and even negative rank correlation coefficients across alternative functional forms from a sample of Greek olive farms. The mean efficiency scores and the other statistics, reported in Table 4, confirm our hypothesis about the homogeneity of Québec dairy farms in terms of technical efficiency. A more complete analysis is presented in Table 5 for the dominant functional form. We conducted parametric and nonparametric matched pair tests (t-test and Wilcoxon signed rank test) at the 1% level between the means of the efficiency scores. For both regions, all distributions yield consistent results. Hjalmarsson et al (1996) reported similar results. The exponential distribution yields the largest mean values while the half normal yields the lowest. Hence, the choice of the distribution affects the level of the efficiency scores, but the magnitude of the difference is very small in this study. Table 5 also reports the percentage of farms that are classified in different ranges of technical efficiency. The exponential distribution leads to a higher number of farms with efficiency scores higher than 0.95 than do the other two distributions (i.e., 78% and 87% for the

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nonmaize and maize regions respectively, compared with 52–56% for the other two distributions). This result is surprising considering that the mean efficiency scores are not much different among distributions. This suggests that the technical efficiency level of a particular farm could vary widely across competing distributions. For example, in the nonmaize region, the least efficient farm was ranked the same by all three distributions, but its level of efficiency varied from 0.6974 (exponential normal) to 0.8013 (truncated normal). Parametric Versus Nonparametric Comparisons The DEA model is used with the same dataset to estimate the efficiency scores of Québec dairy farmers.13 As seen in Table 4, all correlation coefficients as well as rank correlation coefficients between the DEA scores and those of the parametric models are low. The correlation coefficients are all under 0.5, while the rank correlation coefficients are all under 0.6. This implies that the two approaches cannot be used interchangeably. The summary statistics for the efficiency scores derived from the DEA model are reported in Table 5. Average efficiency scores are 0.9215 for the nonmaize region and 0.95 for the maize region, with standard deviations of 0.0466 and 0.041, respectively. These mean efficiency scores are within the range reported by Cloutier and Rowley (1993) in their DEA study of Québec dairy farms. For the maize region, the DEA average score is close to those generated by the GL function, but it is somewhat lower for the nonmaize region. An important difference between the DEA and the GL results relates to the distribution of the scores, especially in the nonmaize region. With the DEA model, about 66% of the farms are classified in the 90–100% efficiency range, while more than 93% of the farms fall in this bracket with the GL function, irrespective of the choice of distribution. The difference in the results is not so drastic in the maize region. Still, it must be emphasized that, considering the low correlation coefficients, the subclasses of efficiency would include different farms. Other authors have also identified discrepancies in the results emanating from similarly specified DEA and stochastic frontier models. Hjalmarsson et al (1996) report negative correlation coefficients between efficiency scores obtained from inter temporal DEA and SPF models. Singh et al (2000) also report contradictory results between the two methodologies and they hypothesized that the choice of a restrictive functional form (C-D) may be responsible for the differences in their results. Our results indicate that the two methodologies generate different results even when flexible functional forms are used.14 Therefore, Singh et al’s warning to practitioners “to think carefully about the likely impact of methodological choice upon their results and to use more than one method if they suspect that it may have some influence” (p. 25) remains pertinent. Elasticity Results The output elasticities have important managerial and policy implications. These elasticities are reported in Table 6 for the dominant functional form and three distributions of the inefficiency scores. Table 6a reports the elasticities evaluated on the production frontier, while Tables 6b reports them at the predicted values of output; that is, assuming no change in the average efficiency level.15 Because the average level of efficiency is high in this study, both sets of elasticities (based on frontier and predicted outputs) are very similar. Therefore, predicted output and frontier output do not differ much in magnitude. Another general result is that the average

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Table 6a. Output elasticities (frontier) from the generalized Leontief functional form Elasticity with respect to: Concentrate A

B

Forage C

Exponential distribution: Maize 0.153 0.232 0.201 Nonmaize 0.162 0.168 0.171 Half normal distribution: Maize 0.133 0.268 0.200 Nonmaize 0.141 0.174 0.163 Truncated normal distribution: Maize 0.131 0.271 0.201 Nonmaize 0.144 0.174 0.165

A

B

Labor C

A

B

Assets (capital) C

A

B

C

–0.020 0.089 0.046 –0.043 0.190 0.100 0.269 0.094 0.180 –0.067 0.119 0.034 –0.028 0.109 0.058 0.005 0.322 0.189 –0.027 0.097 0.045 –0.053 0.201 0.098 0.237 0.103 0.173 –0.063 0.115 0.033 –0.024 0.104 0.056 0.001 0.316 0.184 –0.037 0.106 0.045 –0.053 0.203 0.098 0.236 0.104 0.173 –0.061 0.114 0.033 –0.023 0.102 0.055 0.002 0.316 0.185

A = Mean elasticity for the high 10% input use. B = Mean elasticity for the low 10% input use. C = Mean elasticity for the entire sample. Table 6b. Output elasticities (predicted) from the generalized Leontief functional form Elasticity with respect to: Concentrate A

B

Forage C

Exponential distribution: Maize 0.158 0.240 0.208 Nonmaize 0.168 0.176 0.179 Half normal distribution: Maize 0.140 0.286 0.211 Nonmaize 0.148 0.185 0.173 Truncated normal distribution: Maize 0.137 0.289 0.212 Nonmaize 0.151 0.185 0.174

A

B

Labor C

A

B

Assets (capital) C

A

B

C

–0.021 0.093 0.048 –0.045 0.197 0.104 0.279 0.097 0.186 –0.070 0.124 0.035 –0.029 0.114 0.061 0.005 0.338 0.197 –0.028 0.103 0.048 –0.056 0.213 0.104 0.249 0.109 0.182 –0.066 0.122 0.035 –0.025 0.110 0.060 0.001 0.338 0.195 –0.039 0.112 0.048 –0.056 0.214 0.104 0.247 0.110 0.182 –0.065 0.120 0.035 –0.024 0.102 0.058 0.002 0.337 0.195

A = Mean elasticity for the high 10% input use. B = Mean elasticity for the low 10% input use. C = Mean elasticity for the entire sample.

elasticity coefficients are not sensitive to the choice of the distribution of inefficiency scores. Similar findings are reported by Kim and Schmidt (2000), who compared two distributions (half normal and exponential) using three different datasets. Their estimated parameters (for a Cobb-Douglas) were almost identical for both distributions. This accumulation of empirical evidence suggests that the choice of the distribution may not have an important impact on the elasticity estimates. For comparison purposes, Table 7 presents the average frontier out-

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Table 7. Output elasticities (frontier) for the translogarithmic and Cobb-Douglas functional forms Elasticity with respect to: Concentrate EXPN HN TL: Maize Nonmaize C-D: Maize Nonmaize

Forage

TRN EXPN HN

Labor TRN EXPN HN

Assets (capital) TRN EXPN HN

TRN

0.206 0.210 0.215 0.049 0.050 0.052 0.107 0.103 0.104 0.173 0.172 0.171 0.179 0.175 0.175 0.032 0.042 0.032 0.059 0.058 0.057 0.190 0.188 0.188 0.217 0.219 0.217 0.036 0.034 0.033 0.105 0.101 0.099 0.170 0.169 0.168 0.172 0.170 0.171 0.052 0.054 0.063 0.054 0.055 0.056 0.196 0.193 0.180

HN = Half normal distribution; TRN = Truncated normal; and EXPN = Exponential normal.

put elasticities for the TL and the C-D functional forms. Surprisingly, the elasticity coefficients do not vary across functional forms, in contrast with the comparisons found in Shumway and Lim (1993) and Giannakas et al (2000). Table 6a and 6b also report the average output elasticity coefficients for the low and the high input users (10%). For concentrate, forage and labor, the elasticities decrease with input use in both regions, which is an anticipated result under the profit maximization assumption. However, in the maize region, the elasticity with respect to capital increases with input use. This could be explained by an underutilization of capital in the maize region. Alternatively, this result could reflect significant differences in technology. Because the unit used to measure the capital input (dollars) does not distinguish between technologies/types of capital equipment, the higher elasticities observed at higher levels of input may simply reflect the choice of a more expensive underutilized technology/type of capital.16 The high value of production quota licenses under the supply management system undoubtedly contributes to the capacity utilization problem.17 Finally, for forage and labor in both regions, high input users have much lower elasticity coefficients than low input users, and these coefficients are close to zero and even negative in the case of forage. This phenomenon reflects a management practice used by several producers that let the cows eat as much as they want. As a result, too much forage is being used and allocative efficiency is likely to be low. With respect to labor, high input use could be attributed to several factors. For example, in spite of a small number of cows on the farm, it is relatively common for both husband and wife to work full time on the farm. In these cases, the opportunity cost (and the productivity) of labor is relatively low. The financial constraint stemming from the high quota prices under the supply management policy makes it difficult to increase the size of the herd and this might explain the persistence of low labor productivity on Québec dairy farms. Limited off-farm employment opportunities also tend to depress the marginal product of labor on the farm. SUMMARY AND IMPLICATIONS This paper measures technical efficiency of two groups of dairy farms in the province of Québec, Canada. The actual production process is unknown and three commonly used func-

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tional forms, as well as three alternative potential distributions for the inefficiency scores, are analyzed and compared using the stochastic frontier analysis framework. For comparison purposes and completeness of the study, the data envelopment analysis is also used as an alternative methodology to estimate production efficiency. The likelihood dominance criterion (LDC) selects the generalized Leontief (GL) functional form over the Cobb-Douglas and the translogarithmic (TL) functions. The differences between the likelihood values of the GL frontiers estimated under alternative distributions for the inefficiency scores are insignificant. To assess the impact of the selection of a functional form and of a distribution for the inefficiency scores on the derived efficiency measures, the correlation and rank correlation coefficients are calculated for pairs involving the nine models. The correlation coefficients are high, but the rank correlation coefficients are even higher in most cases. More specifically, all rank correlation coefficients exceed 0.947, while 0.92 constitutes a minimum for the correlation coefficients. It can be concluded that for this dataset, neither the choice of a functional form nor the choice of a distribution for the inefficiency scores is critical in the production of a ranking of the level of efficiency of individual dairy farms. The average efficiency scores calculated from all models are high and are characterized by relatively little dispersion. This is what was expected given the stability of Canada’s dairy policy and the policy’s incidence on farm size. In the maize region for example, all reported scores exceed 91%. For the dominant functional form, the average scores vary between 0.9447 and 0.9659 depending upon the region and the distribution of the inefficiency scores. Parametric and nonparametric tests performed on the differences in these scores are significant at the 1% level, but the magnitudes of the differences are small. Therefore, with this dataset, the choice of the distribution of the inefficiency scores is not critical in the estimation of both the rank and the level of technical efficiency. However, further analysis across distributions of the efficiency scores shows that, even if the differences in the means are small across alternative distributions, the level of technical efficiency of a given farm may vary significantly from one distribution to another. Comparisons between the rank, the level and the distributions of the inefficiency scores obtained from parametric estimation and those obtained from DEA demonstrate significant discrepancies between the methodologies. This result has been reported in other studies, but it is certainly a cause for concern as researchers are left wondering which set of results, if any, is reliable. The average output elasticities from all nine models are compared and the results show that neither the functional form nor the distribution of the inefficiency scores significantly impact on the magnitudes of the elasticities. The main results pertaining to the dominant functional form are: • elasticities are not sensitive to the distribution of the error term • they are relatively low (i.e., 0.2 or less for all inputs in both regions). Further analysis of the elasticity coefficients for the high and the low input users shows that for most inputs, in both maize and nonmaize regions, the elasticity coefficients for the high input users are relatively low, and are even negative for labor and forage. Although these factors are offered in quasi-fixed supply and as such have low opportunity costs, the negative elasticities nevertheless point out that there are significant allocative inefficiencies on these farms. A final result worth of mention is that, in the maize region, there seems to be an under-

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utilization of capital. This result could be attributed to a problem with the measurement of this input (dollars), which does not distinguish between alternative technologies/types of capital. Alternatively, the capacity utilization problem could be driven by inhibiting effect of high quota prices on output expansion. . Three points need to be emphasized concerning the high levels of technical efficiency obtained for Québec dairy farms. Firstly, increased technical efficiency could not increase farm output by much. Québec dairy farmers have managed small farms for quite some time and have become quite proficient at it. Secondly, the reported high levels of efficiency do not imply that output per cow would remain at the same level if a major change in the policy environment were to take place; Knutson et al (1997) show that the optimal level of output per cow should be lower under a quota system. Since output per cow and herd size are significantly higher in the U.S., a first extension to this research would be to determine whether U.S. and Québec farms of the same size are equally close to the frontier. Finally, the high levels of efficiency obtained in this study lead us to conjecture that the structural change that would likely follow a hypothetical phasing out of Canada’s supply management policy would not be driven by technical efficiency disparity across farms, but more likely by variables like the farm operator’s age and capacity to borrow. A second extension to this research would be to identify the determinants of efficiency to ascertain whether the profile of an efficient operator is aligned with the profile of operators capable and willing to operate under a different policy environment. NOTES 1Intuitively,

homogeneity in size facilitates extension activities and the learning transmitted from successful producers to less efficient ones. 2Measurement of efficiency has a long and rich history, dating back to Koopmans (1951), Debreu (1951) and Farrell (1957). More detailed discussions of the evolution of modern measurement of efficiency techniques are provided by Färe, Grosskopf and Lovell (1985, 1994), Battese (1992), Lovell (1993), Greene (1999) and more recently by Kumbhakar and Lovell (2000). 3If Y and X were defined as the logarithms of output and the inputs respectively, then Eq. 1 would be i i a Cobb-Douglas production function. 4Some authors refer to Eq. 3 as the square root function (see Griffin et al 1987) or the generalized linear function (Diewert 1973). We retain the name used by Fuss et al (1978) and Anderson et al (1996). 5The quality of this proxy could be questioned on the ground that the weights of dairy cows differ significantly across breeds. However, the Holstein breed is by far the most popular one in Québec. This is reflected in our sample,which contains only Holstein herds. Dairy science literature (e.g., Schmidt, Van Vleck and Hutjens 1988, 63–77) indicates that weight is correlated with the capacity to eat,which is in turn directly related to a cow’s ability to produce milk. 6Details on DEA can be found in Seiford and Thrall (1990), Lovell (1993), Ali and Seiford (1993), Lovell (1994), Charnes et al (1995) and Seiford (1996). 7A detailed discussion of the DEA variants is found in Coelli (1996). 8The LDC can be summarized as follows. The two competing hypotheses are H1 : C-D is dominant and H2 : TL is dominant. We define N1, N2 as the number of parameters in the C-D and TL functions respectively and C(.) as the critical value of the chi-square distribution with (.) degrees of freedom. Ranking functional forms based with the LDC is done as follows: • H1 is preferred to H2 if LTL – LC-D < {C(N2 + 1) – C(N1 + 1)}/2. • The test is inconclusive if: {C(N2 – N1+ 1) – C(1)}/2 > LTL – LC-D > {C(N2 + 1) – C(N1 + 1)}/2. • H2 is preferred to H1 if: LTL – LC-D > {C(N2 + 1) – C(N1 + 1)}/ 2. When N1 = N2, the model with the larger log likelihood value is preferred. 9The maximum likelihood estimation of all equations was performed with LIMDEP 7.0.

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10The

dependent variables in the C-D and TL functions are in logs while this is not the case for the GL. To compare and hence rank the three functional forms, the log likelihood function values have to be adjusted by the appropriate Jacobian transformation term. The necessary adjustment to the C-D and TL log likelihood values is to subtract Σ ln Y from their value (Davidson and MacKinnon 1993, 491). 11The results for the maize region are similar to the ones reported and are omitted because of space consideration. They can be obtained from the authors upon request. 12Interestingly, a conclusion to the effect that the levels of the efficiency scores differed but that the rankings were similar was reached by Bravo-Ureta and Rieger (1990) when comparing alternative methods of estimating production frontiers. 13The results were generated by the computer program DEAP Version 2.1 developed by Tim Coelli (1996) from the Centre for Efficiency and Productivity Analysis at the University of New England, Australia. DEAP is available at: www.une.edu.au/febl/EconStud/emet/cepa.htm. 14As pointed out by a Journal reviewer, further research is needed to identify conditions under which DEA and stochastic frontier models provide divergent results. It seems most important to determine whether one approach dominates the other or under what conditions one approach dominates the other (i.e., data generating process characteristics, sample sizes, variance ratios …). 15Frontier output elasticities are always larger than predicted output elasticities. 16Though a common practice, measuring capital in dollars is not ideal because it may bias efficiency scores. Unfortunately, it was not possible to construct an alternative measure for sensitivity purposes. 17The necessary investment in quota for a cow that yields 8,000 kg of milk per year is about $20,000. To put this price in perspective, a “good” cow generates a gross revenue of (80 hL per year * $53/hL) $4,240.

ACKNOWLEDGMENT The authors appreciatively acknowledge the financial support of FCAR.

REFERENCES Aigner, D. J., C. A. K. Lovell and P. Schmidt. 1977. Formulation and estimation of stochastic frontier production function models. Journal of Econometrics 6 (July): 21–37. Ali, A .I. and L. M. Seiford. 1993. The mathematical programming approach to efficiency analysis. In The Measurement of Productive Efficiency: Techniques and Applications, edited by H. O. Fried, C. A. K. Lovell and S. S. Schmidt, pp. 120–59. New York: Oxford University Press. Anderson, D. P., T. Chaisantikulawat, A. Tan Khee Guan, M. Kebbeh, Lin, N. and C. R. Shumway. 1996. Choice of functional form for agricultural production analysis. Review of Agricultural Economics 18: 223–31. Banker, R. D., A. Charnes and W. W. Cooper. 1984. Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science 30: 1078–92. Battese, G. E. 1992. Frontier production functions and technical efficiency: A survey of empirical applications in agricultural economics. Agricultural Economics 7: 185–208. Bendt, E. R. 1991. The Practice of Econometrics: Classic and Contemporary. Boston, MA: AddisonWesley. Bravo-Ureta, B. E. and L. Rieger. 1990. Alternative production frontier methodologies and dairy farm efficiency. Journal of Agricultural Economics 41: 215–26. Brummer, B. 2001. Estimating confidence intervals for technical efficiency: The case of private farms in Slovenia. European Review of Agricultural Economics 28: 205–306. Canadian Dairy Commission. 2000. Available at: www.cdc.ca. November 2000. Charnes, A., W. W. Cooper, A. Y. Lewin and L. M. Seiford. 1995. Data Envelopment Analysis: Theory, Methodology and Applications, Boston, MA: Kluwer Academic Publishers. Cloutier, L. M. and R. Rowley. 1993. Relative technical efficiency: Data envelopment analysis and Quebec’s dairy farms. Canadian Journal of Agricultural Economics 41: 169–76.

136

CANADIAN JOURNAL OF AGRICULTURAL ECONOMICS

Coelli, T. J. 1996. A Guide to DEAP Version 2.1: A Data Envelopment Analysis (Computer) Program. CEPA Working Paper 96/08. Armidale, NSW, Australia: University of New England, Department of Econometrics. Coelli, T., D. S. Prasada Rao and G. E. Battese. 1998. An Introduction to Efficiency and Productivity Analysis. Boston, MA: Kluwer Academic Publishers. Davidson, R. and J. G. MacKinnon. 1993. Estimation and Inference in Econometrics. New York, NY: Oxford University Press. Debreu, G. 1951. The coefficient of resource utilization. Econometrica 19: 273–92. Diewert, W. E. 1973. Functional forms for profit and transformation functions. Journal of Economic Theory 6: 284–316. Färe, R., S. Grosskopf and C. A. K Lovell. 1985. The Measurement of Efficiency of Production. Boston, MA: Kluwer Academic Publishers. Färe, R., S. Grosskopf and C. A. K Lovell. 1994. Production Frontiers. Cambridge, UK: Cambridge University Press. Farrell, M. J. 1957. The measurement of productive efficiency. Journal of the Royal Statistical Society, Series A 120 (3): 253–90. Fuss, M., McFadden and Y. Mundlak. 1978. A survey of functional forms in the economic analysis of production. In Production Economics: A Dual Approach to Theory and Applications, edited by M. Fuss and D. McFadden. Amsterdam, The Netherlands: North-Holland Publishing Co. Giannakas, K., K. C. Tran and V. Tzouvelekas. 2000. On the choice of functional from in stochastic frontier modeling. Working Paper. Saskatoon, SK: University of Saskatchewan, Department of Economics. Gong, B. H. and R. C. Sickles. 1992. Finite sample evidence on the performance of stochastic frontiers and data envelopment analysis using panel data. Journal of Econometrics 51: 259–84. Greene, W. H. 1990. A gamma-distributed stochastic frontier model. Journal of Econometrics 46: 141–64. Greene, W. H. 1999. Frontier production functions. In Handbook of Applied Econometrics, Volume II: Microeconomics, edited by M. H. Pesaran and P. Schmidt. Oxford, UK: Blackwell Publishers. GREPA. 2000. Les faits saillants laitiers québécois, 1999. Québec, QC: Université Laval, Département d’économie agro-alimentaire et ses sciences de la consommation. Available at: http://alpha.eru.ulaval.ca/grepa/ Griffin, R. C., J. M. Montgomery and M. E. Rister. 1987. Selection functional form in production function analysis. Western Journal of Agricultural Economics 12 (2): 216–27. Hjalmarsson, L., S. C. Kumbhakar, and A. Heshmati. 1996. DEA, DFA and SFA: A comparison. Journal of Productivity Analysis 7 (July): 303–27. Kalaitzandonakes, N. G., Shunxiang Wu, and Jian-chun Ma. 1992. The relationship between technical efficiency and firm size revisited. Canadian Journal Agricultural Economics 40: 427–42. Kim, Y., and P. Schmidt. 2000. A review and empirical comparison of bayesian and classical approaches to inference on efficiency levels in stochastic frontier models with panel data. Journal of Productivity Analysis 14 (September): 91–111. Knutson, R. D., R. Romain, D. P. Anderson and J. W. Richardson. 1997. Farm-level consequences of Canadian and U.S. dairy policies. American Journal of Agricultural Economics 79: 1563–72. Koopmans, T. C. 1951. An analysis of production as an efficient combination of activities. In Activity Analysis of Production and Allocation, edited by T. C. Koopmans. Cowles Commission for Research in Economics, Monograph13. New York, NY: Wiley. Kumbhakar, S. C. and C. A. K Lovell. 2000. Stochastic Frontier Analysis: Cambridge, UK: Cambridge University Press. Lovell, C. A. K. 1993. Production frontiers and productive efficiency. In The Measurement of Productive Efficiency: Techniques and Applications, edited by H. O. Fried, C. A. K. Lovell and S. S. Schmidt, pp. 3–67. New York, NY: Oxford University Press.

ASSESSING TECHNICAL EFFICIENCY OF QUÉBEC DAIRY FARMS

137

Meeusen, W. and J. van den Broeck. 1977. Efficiency estimation from Cobb-Douglas production functions with composed error. International Economic Review 18: 435–44. Pollak, R. A. and T. J. Wales. 1991. The likelihood dominance criterion: A new approach to model selection. Journal of Econometrics 47: 227–42. Ritter, C. and L. Simar. 1997. Pitfalls of normal-gamma stochastic frontier models. Journal of Productivity Analysis 8 (May): 167–82. Romain, R. and R. Lambert. 1995. Efficacité technique et coûts de production dans les secteurs laitiers du Québec et de l’Ontario. Canadian Journal of Agricultural Economics 43: 37–55. Romain, R. 2001. The future of the dairy industry in Canada. In Advances in Dairy Technology, Striving for Sustainability, vol. 13, pp. 337–53. Edmonton, AB: University of Alberta, Department of Agricultural, Food and Nutritional Science. Seiford, L. M., and R. M. Thrall. 1990. Recent developments in DEA: The mathematical approach to frontier analysis. Journal of Econometrics 46: 7–38. Seiford, L. M. 1996. Data envelopment analysis: The evolution of the state of the art (1978–1995). Journal of Productivity Analysis 7: 99–138. Shumway, R. C. and H. Lim. 1993. Functional form and U. S. agricultural production elasticities. Journal of Agricultural and Resource Economics 18 (2): 266–76. Singh, Satbir, Tim Coelli and Euan Fleming. 2000. Performance of dairy plants in the cooperative and private sectors in India. CEPA Working Paper 2/2000. Armidale, NSW, Australia: University of New England, Department of Econometrics. Tauer, L. W. and K. P. Belbase. 1987. Technical efficiency of New York dairy farms. Northeastern Journal of Agricultural and Resource Economics 16: 10–16. Tauer, L. W. and J. J. Hanchar. 1995. Nonparametric technical efficiency with K firms, N inputs, and M outputs: A simulation. Agricultural and Resource Economic Review 24:185–89.