Stochastic frontier model

The assumption of a production of a stochastic nature means that the level of production of a unit of production is limited superiorly by a stochastic frontier, which can be modeled as in Equation 1:

Where the error term is composed of two parts; a random perturbation v , symmetric that is assumed to be identically and independently distributed with mean 0, and u is a non-negative error term, which is distributed independently of v, following a one-tailed distribution²². The random component represents events that are not controllable by the CPU (climatic, social, economic, and political phenomena), while u collects the distance of each company to its stochastic frontier, representing a measure of technical inefficiency²³. Therefore, the Stochastic Production Frontier (SPF) is described by Equation 2:

For SPFs, the technical efficiency index for enterprise i can be calculated with Equation 3:

The SPF is first proposed in the 1970s of the last century^24,25 where they considered²⁴ the case in which u is semi-normally distributed, that is, u-N0,σu and v normally distributed. The implications at the conceptual level of PF being stochastic are very important for the interpretation of inefficiency. As Schmidt²⁴ says, “the farmer whose harvest is devastated by drought or a storm is unfortunate with our measure, but inefficient with the usual measure”. An important limitation of the first estimates of SPF is that only the average efficiency of the sample was calculated, and it was not possible to obtain a measure of the efficiency of each company. Later developments²⁶ managed to find a measure of individual efficiency using the conditional distribution of u in ε. The technical efficiency index for each firm i is:

The most commonly used measure of TE is the ratio of observed production and the corresponding stochastic production frontier, as in Equation 5:

This measure of technical efficiency takes a value between zero and one. It measures the output of the i-th CPU relative to the output that a fully efficient CPU could produce using the same input vector. The first step in calculating the TE is to estimate the parameters of the stochastic production frontier model:

Estimation of parameters

Because model 9.2 includes random terms; the symmetric error (v _i ) and a non-negative random variable (u _i ), the selected estimation method includes assumptions about both terms. Each v _i is distributed independently of each u _i and both are uncorrelated with the explanatory variables. Additionally, the noise component v _i is assumed to have properties identical to those of the classical linear regression model. The inefficiency component has similar properties except that it has a non-zero mean (u_i ≥0), so Ordinary Least Squares cannot be used. One solution is to make some distribution assumptions regarding the two error terms and estimate the model using the maximum likelihood (ML) method.

Half-normal model

ML estimators were obtained²⁴ under the following assumptions: vi=iidN(0,σv2) and ui=iidN+(0,σu2). This indicates that the v _i are normal random variables distributed independently and identically with means and variances zero and the u_i are semi-normal random variables distributed independently and identically with scale parameter. That is, the probability density function (pdf) of each u_i is a truncated version of a normal random variable that has zero mean and variance σu2.

The log-likelihood function was parameterized²⁴ for this half-normal model in terms of σ 2 = σ v 2 + σ u 2 and 𝜆 2 = 𝜎 𝑢 2 / 𝜎 𝑣 2 ≥0. If 𝜆=0, there are no technical inefficiency effects and all deviations from the frontier are due to noise. Using this parameterization, the maximum likelihood function is represented in Equation 6:

Where, y is an output vector; εi≡vi-ui≡Inqi-xi´β is the compound error term; and Φx is a cumulative distribution function (cfd) of the standard normal random variable evaluated at x.

The empirical analysis is based on the estimation of a Cobb-Douglas production function in which both output and inputs are expressed in logarithmic form (Equation 7), so that the estimated coefficients are interpreted as elasticities²⁷.

In this model, the dependent variable (Yi) is the value of cattle production of the CPUs. The explanatory variables are;

ARECAT is the area for cattle, in hectares owned by the BPU.

LA is the cost of the labor used in production.

ASS is the value of assets; value of machinery, equipment, and production facilities used in the cattle activity.

HEA is the expenditures in health; veterinary supplies and services.

FEED is the cost of feeding; cost of meadow maintenance and supplementary feeding.

Model of individual efficiencies

The estimated model of individual efficiencies (Equation 7) considers the measures of inefficiency estimated in the first stage as a dependent variable. Explanatory variables are a set of variables that hypothetically affect the performance of the CPU⁶. The literature reports as the most common explanatory variables the age of the head of the CPU, they level of schooling, experience in the activity under study, characteristics of the CPU, administration, and environmental factors, among the most cited^28-31. The multiple regression model was that described in Equation 8:

Where: Age is the age of the head of the CPU; Schoo is the level of schooling (in years) of the head of the CPU; Exper are the years of experience in the cattle activity; Admon is a dummy variable that takes the value of zero if the CPU does not have an administration system and one if they have an administration system; TA is technical assistance, zero if they did not receive technical assistance and one if they received the service. The variables of the stochastic frontier model and of the individual inefficiency model are showed in Table 2.

Results of the econometric model

The results of the stochastic frontier model, using the full sample, are shown in Table 3. The variables had the expected sign, according to economic theory. The positive sign means that increasing the use of the production factor increases production, while the magnitude of the coefficient accounts for the relative importance of each independent variable in explaining the dependent variable.

The variables LA, ASS, HEA, and FEED are significant at 5 %. Area for cattle (ARECAT) was also found significant³⁰ when studying factors influencing technical efficiency in southeastern Kenya in 2013; a 10 % increase in area for cattle resulted in a 29 % increase in cattle production. The LA variable was found to be significant by several authors^31,35,36. In a study in Botswana³⁶ conducted with four strata of producers, they found that increasing the amount of labor by 10 % increases producers’ profits by 15 % and 18 %, respectively. The ASS variable has not been identified as significant in the studies reviewed. In the present study, ASS has a positive effect on the production of cattle PUs, as expected by economic theory²⁰. The variables HEA and FEED were also reported as significant^14,30,31.

Regarding the fit of the model (7), the estimated stochastic production frontier showed a normal distribution of residuals (Shapiro-Wilks test), no serial correlation of errors (Durbin-Watson), no heteroscedasticity of variance and no autocorrelation or multicollinearity problems. In the values obtained from the general fitted model (Table 3), it was determined that cattle production presents increasing returns to scale (the sum of the coefficients is greater than the unit). To confirm this result, the test was performed for returns to scale, where a value of P= 0.03 < 0.05 was obtained, this causes the existence of constant returns to scale to be rejected⁶.

Regarding the inefficiencies of model 8, it was observed that the variance parameters of the maximum likelihood (ML) function are estimated from the total variance model defined as: σs2=σv2+σu2 and the estimated value in the model for the total variance ( 𝜎 𝑠 2 ) was 0.105. While the lambda value (%) resulted in 1.370, which shows that the variance of the efficiencies is greater than the variance of the random perturbations at 88 % ( 𝜆 2 −1) and the gamma value obtained from the relationship between the variances 𝛾= 𝜎 𝑢 2 / 𝜎 𝑠 2 states that 73.9 % of the total variance is explained by the variance of the inefficiencies.

The results of the stochastic frontier model for each stratum of cattle producers are shown in Table 4. Similar to the general model, the models for each estimated stratum showed normal distribution of residuals, no serial correlation of errors, no heteroscedasticity of variance, and no autocorrelation. The variables ARECAT, LA, ASS, HEA, and FEED are significant at 5 % in strata two and three.

In stratum 1, only HEA and FEED were significant. One possible explanation is that small producers have lower quality pastures, without agronomic management, use family labor, little specialized, and the value of their assets is very low, reflecting low-technified CPUs. The feed variable is the one that has the greatest weight in explaining the production of the CPUs for the three strata. The value of assets has twice as much relative weight in strata two and three than in strata one, which means that these CPUs not only have greater investment in assets, but that it is modern and generates greater productivity. The models for strata 2 and 3 show increasing returns to scale, but not the model of stratum 1 which has decreasing returns to scale. In this regard³⁷, in a study in the United States of America, it was found that as the size of the CPU increases, TE increases, which showed evidence of economies of scale. A possible explanation for the result of stratum 1 is that small producers have a low level of capitalization, low-skilled labor, and since they have little pasture area, they make intensive use, overexploiting the resource^38,39.

Frequency distribution of technical efficiency(TE) by UPG stratum

The TE range for cattle producers was between 0.50 and 0.95. Of the total of the 185 CPUs, 29 % have values between 0.50 and 0.70, 63 % between 0.71 and 0.90, and only 8 % TE values greater than 0.90. Table 5 shows that stratum 3 presents most of the values of 0.91 or more. In this regard⁴⁰, it was found that the CPUs with the largest number of animal units and the largest area for cattle presented the highest values of technical efficiency.

Results of individual inefficiencies

Table 6 shows the results of the individual inefficiencies model according to Equation (8). The significant variables, at different levels of significance, and with a negative coefficient, were Schoo, Exper, Admon and TA. The negative sign of the coefficients indicates an inverse relationship between the value of the explanatory variable and the value of the inefficiency. In this regard, previous studies^28,30,36 have reported results that support the results of this study. It was found that more years of schooling reduces inefficiency in values very similar to those reported in this research. Similarly, in the case of the Admon(^6,14,41) variable, they found an inverse relationship between having an administration system and inefficiency. For TA^28,30,41, they reported that receiving this service contributes to reducing the inefficiency of the CPUs. In the present study, Age is not significant, a result supported by what was found in the literature³⁰.

The above results suggest that reducing inefficiency should be addressed by providing public technical assistance services, an activity that, in Mexico, has been at very low levels since the nineties. In this regard, in a study on the use of livestock innovations in Sinaloa⁷, it was reported that only 3 % of the PUs receive technical assistance services, and of these, the CPUs represent only 19.3 %. Training in the management of the CPU, including administrative services, should also be a central aspect, in addition to the technological issues of cattle farming.

Results of the technical inefficiency model by CPU strata

Table 7 reports the results of the model of technical inefficiency by strata of CPUs. For stratum 1, Age and Exper are significant, but not Schoo, Admon and TA. The producers of this stratum have low schooling, 6 years on average, have experience, and most do not have administration systems and do not receive any type of technical assistance services. For stratum 2, Schoo, Exper and TA are significant. It was observed that the years of schooling increase significantly for the producers of this stratum. Finally, for stratum 3, four variables are significant. It should be noted that the values of the coefficients are in the range of 0.13 to 0.28, which shows an important effect of these variables to reduce inefficiency. Therefore, improving administration systems and the quality of technical assistance are aspects that can lead these CPUs to be highly efficient^14,30,41.