Cobb-Douglas, Translog Stochastic Production Function and Data Envelopment Analysis in Total Factor Productivity in Brazilian Agribusiness

The objective of this paper is to apply a Cobb-Douglas, Translog Stochastic Production Function and Data Envelopment Analysis in order to estimate inefficiencies over time as well as respective TFP (Total Factor Productivity) sources for main Brazilian grain crops namely, rice, beans, maize, soybeans and wheat throughout the most recent data available comprising the period 2001-2006. The results indicate that, although positive changes exist in TFP for the sample analyzed, a decline in the use of technology has been evidenced for all grain crops in which it is observed a historical downfall in the use of inputs in Brazilian agriculture. KEYwoRDS: Total Factor Productivity, Stochastic Frontier, Data Envelopment Analysis.


INTRODUCTION
"Not all producers are technically efficient".As opposed to conventional microeconomic theory, such statement implies that not all producers are able to utilize the minimum quantity of required inputs in order to produce the desired quantity of output given the available technology.Similarly, not all producers are able to minimize necessary costs for the intended production of outputs.
From a theoretical point of view, producers do not always optimize their production functions.The production frontier characterizes the minimum number of necessary combinations of inputs for the production of diverse products, or the maximum output with various input combinations and a given technology.Producers operating above the production frontier are considered technically efficient, while those who operate under the production frontier are denoted technically inefficient.The Stochastic Frontier Analysis -SFA is an analytical approach that utilizes econometric (parametric) techniques whose models of production recognize technical inefficiency and the fact that random shocks beyond producers' control may affect the product.Differently from non-parametric approaches that assume deterministic frontiers, SFA allows for deviations from the frontier, whose error can be decomposed for adequate distinction between technical efficiency and random shocks (e.g.labor or capital performance variations).

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By the application of non-parametric methods as Data Envelopment Analysis -DEA, the Malmquist index is calculated by distance functions obtained by mathematical programming and allows for the absence of price information, utilizing physical quantities of multiple inputs and products instead.The main two components of the underlying index are technical change (innovation) and efficiency change ("catching up" effect towards the frontier).
The objective of this paper is to apply the Stochastic Frontier Analysis technique in order to estimate increase or decrease in inefficiencies through time, as well as the linear programming method Data Envelopment Analysis, namely the Malmquist index, for the analysis of change in TFP (Total Factor Productivity) in main Brazilian grain crops -rice, beans, maize, soybeans and wheat -throughout the 2001-2006 period.
Among observed results, even though there have been positive changes in main Brazilian grain crops, there have been a decline in the component referring to technological innovations for all Brazilian grain crops analyzed between the 2005/2006 period in which it is observed a general downfall in input usage in agriculture.

Stochastic Frontier Analysis -SFA
In the presence of inefficiencies, the Stochastic Frontier Analysis -SFA emerges as a theoretical and practical framework, whose objective is to contribute for the definition and estimation of production frontiers.SFA has been developed from remote influences but the literature that directly influenced the development of SFA has been the theoretical framework about production efficiency beginning in the decade of 1950 by Koopmans (1951), Debreu (1951) and Shephard (1953).Farrell (1957) was the first to measure production efficiency empirically.The use of linear programming by Farrell influenced research by Boles (1966), Bressler (1966), Seitz (1966) and Sitorus (1966) and eventually the development of Data Envelopment Analysis (DEA) by Charnes, Coopers and Rhodes (1978).The influence from Farrell is also definite for the works by Aigner and Chu (1968), Seitz (1971), Timmer (1971), Afriat (1972) and Richmond (1974) -direct collaborators for the SFA development.
This parametric method of stochastic frontier considers production frontier as a random shock.Differently from a non-parametric method such as DEA that assumes a deterministic frontier, the stochastic frontier allows for deviations from the frontier to represent both inefficiency and an inevitable statistic noise which intends to be a closer approach to reality given that observations normally involve a random walk.
The models of stochastic production frontier address technical efficiency and recognize the fact that random shocks beyond the control of producers may affect the production output.Therefore, in these models, the impact of random shocks (as labor or capital performance) on the product can be separated from the impact of technical efficiency variation.These models were simultaneously introduced by Aigner, Lovell and Schmidt (1977) and Meeusen and van den Broeck (1977).
This paper has followed the dominant functional specification in literature based on the works by Bettese and Coelli (1992Coelli ( , 1995)), in which it is formalized technical inefficiency in the production function of stochastic frontier for panel data.Thus, consider the following production function of a given state i: or sectors and years where it y is the vector representing produced quantities by the unit of production in period ; is the vector of inputs used by the unit of production in period ; is the vector of coefficients to be used that define the production technology.The OLS (Ordinary Least Squares) method provides a simple test for the identification of the presence of technical inefficiency in data.If , then .Thus, the error term is symmetric and data do not evidence technical inefficiency.However, if , then the distribution of is negatively symmetric and evidences of technical inefficiency in data exist.Thus, the term quantifies technical inefficiency or the distance in relation to the efficiency frontier.The most efficient estimate presents value 0 for .This suggests that the presence of technical inefficiency can be tested directly by the residuals of oLS.

Consider technical inefficiency as time-variant
when is positive, the value inside the brackets of the exponential term will become non-negative and ( will not be smaller than unity.This is the case in which .In other words, technical inefficiency will have decreasing effects through time (positive effect in technical efficiency over time).In case is negative, inefficiency will be increasing through time (also defined as persistent inefficieRiverancy).

Data Envelopment Analysis -DEA
A productively efficient firm is the one not able to increase its production unless some of its inputs are also increased.By the Malmquist index, such firm achieves an efficiency score of 1.Similarly, a productively inefficient firm obtains an efficiency punctuation smaller than 1.
Introduced by Caves et al. (1982) in its empirical usage, the Malmquist index do not require costs or income, being capable of measuring increase in TFP in a scenario of multiple products.For the Malmquist index definition, we assume that for each period of time, production technology models the transformation of inputs into products .
(3) (7) An equivalent form to express this index: (8) In the study concerning industrialized countries, Färe et al. (1994) observe that this decomposition allows for a measure in which one distinction exists between technical efficiency components ("catchingup") and technology change (innovation), given that previous works did not distinguish between these two components.
The ratios inside the brackets measure changes in technology dislocations to input levels and , respectively.Thus, changes in technology is measured as the geometric mean of these two components.The terms out of the brackets measure technical efficiency relative to and , capturing changes in efficiency over time, that is, whether production becomes closer (catching up effect) or more distant from the frontier.observe that if and (i.e., no input and product change between periods), the productivity index do not signalize any change: . Improvements in productivity result in Malmquist indexes greater than unity.Similarly, performance deterioration over time is associated with a Malmquist index smaller than unity.Besides, improvements in any of the Malmquist index components are also associated with values greater than unity of these components; and deterioration is associated with values smaller than unity.   .Given that it is located under , this firm is not efficient and its productive inefficiency is measured by the ratio .Similarly, the same firm in t+1, denoted by is efficient in relation to the frontier and its inefficiency measure is given by .
Given that this index captures the productivity dynamics by the incorporation of data from two different adjacent periods, reflects change in relative efficiency, while reflects changes in technology between periods and .As for the index and its components, values smaller than 1 indicate a decline in productivity (regression), while values greater than 1 indicate growth (progress).For the firm in the example, both components exceed 1.In terms of technical efficiency, the firm moved to a point closer to the contemporary relevant frontier, indicating that the production for this firm is converging to the frontier.In terms of technological change, the frontier, measured at levels of inputs and , moved between periods and ( (ALAM, 2001).

METhODOlOgy
From LSPA (Systematic Survey of Agricultural Production) of January 2007 released by the Brazilian Institute of Geography and Statistics (IBGE), data has been gathered for the main Brazilian grain crops -rice, beans, maize and wheat.Thus, obtained productions have been analyzed for each culture (outputs), as well as harvested area in acres for each crop (inputs) annually.
Additionally, from PAM (Municipal Agriculture Production) and the statistics available at the Ministry of Agriculture, data has been gathered on total quantity (in ton.) produced, harvested area (acres), agricultural credit (in Brazilian Real) and agricultural limestone (in ton.) for the period 2001 to 2006 for the 26 States of the Federation and the Federal District, allowing for the creation of regional dummies for the comparative analysis of total factor productivity.
Initially, data has been analyzed based on the stochastic frontier theory in order to verify gains or losses in efficiencies over time, expressed by the component and the estimated parameters of variables that explain technical inefficiency.Considering technical inefficiency as varying through time, if is positive, the value inside the brackets of the exponential term will become nonnegative and will not be greater than 1.This is the case in which .In other words, technical inefficiency will be decreasing over time (positive effect of technical efficiency through time).If is negative, inefficiency will be increasing.In case is null, it is observed technical inefficiency that does not vary over time (also referred as persistent inefficiency).The decomposition of Malmquist index assists in the determination of efficiency or inefficiency sources in a firm.
indicates technical progress.means the firm is approximating towards optimal scale in .

Stochastic Frontier Analysis for Brazilian Agriculture: Cobb-Douglas Production Function
The results related to the estimation of the stochastic frontier analysis according to a Cobb-Douglas production function is presented in Table 1.In the case of a Cobb-Douglas model, the significant variables were harvested area, agriculture credit and limestone -all assuming expected signs.The LR (Likelihood Ratio) statistic, which is a chi-square ( distribution under the null hypothesis that there has not been effects of technical efficiency, presents significant value to the 1% level, indicating effects of technical inefficiency in the model.
The greatest elasticity observed is that of harvested area.This indicates the intense relationship that exists between production and harvested area, independently of the utilization of other factors that, ceteris paribus, would contribute for productivity.The credit variable reveals the second major elasticity, confirming the importance of agriculture credit to cover costs and particularly, to execute investments which responds for the greatest share of the data analyzed.As expected, assuming positive and inferior elasticity in relation to the other relevant factors, limestone contributes for productivity by correcting sole acidity, which assumes a maximizing role for the potential of productivity already established by the other factors.

The estimate of parameter
, which measures the variability of the two sources of error (white noise disturbance and unilateral error), reached the level of 0.9469.This result means that about 95% of total variance of composed error of the production function is explained by the variance of the technical inefficiency term.This represents the importance of incorporating technical inefficiency in production function.
The term relative to technical inefficiency assumes a temporal pattern of behavior represented by the sign.In case this term is positive (negative), technical inefficiency will be decreasing (increasing) in time.If it assumes a null value, it is considered that technical inefficiency does not vary in time -also called persistent inefficiency.In the analysis, the term assumes negative value, which indicates that technical inefficiency in Brazilian agriculture, though not predominant, is increasing in time from 2000 to 2005.
Thus, the punctual reduction of inefficiency, which includes the concession of costs credit for income maintenance against fluctuations in prices and exchange rates, as well as investment credit for capital acquisition such as tractors and harvesters will certainly avoid the persistence of increasing inefficiency path in Brazilian agricultural productivity.

Stochastic Frontier Analysis for Brazilian Agriculture: Translog Production Function
Assuming a logarithmic transcendental (translog) technology, the parameters estimates of the production frontier and the technical inefficiency component are presented in Table 2.The statistically significant parameters at the level of 5% are essentially related to harvested area and agricultural credit, as well as the measures of regional technical inefficiency expressed by dummy variables.The LR (Likelihood Ratio) statistic presents significant value at 1% level, indicating effects of technical inefficiency in the model.
Analyzing the sample on the basis of stochastic frontier theory for the verification of gains or losses of efficiencies through time, it is observed that the component assumes negative sign and is significant at 5% level.Thus, technical inefficiency is increasing over time for the analyzed sample.It is important to emphasize that is unique for the analyzed sample.Thus, this component does not reveal productivity specificities for each state.
The coefficient for the mean of the error component relative to inefficiency, , is not statistically significant, indicating that the semi-normal distribution is more appropriate in relation to the normal truncated distribution ( ).
The positive sign of parameter indicates that the occurrence of technical progress.The indicator of technical inefficiency, , presents approximated value of 0,90.This result indicates that 90% of total composed error variance of the production function is explained by the variance of the technical inefficiency term.This reveals the importance to incorporate technical inefficiency in the production function.
In relation to the dummy variables parameters, they are all statistically significant to the 5% level.By having the Northeast region as reference for presenting a larger number of observations, it is verified that all the other regions are technically less efficient in relation to the reference region.Thus, by classifying according to the degree of increasing inefficiency, North region is followed by Southeast region, South and Center-west, respectively.

The coefficients
and indicate that the neutral part of technical progress has a positive effect over production.The signs of the coefficients , , and indicate, respectively, that the non neutral part of technical progress moves inversely with area, credit, defensives and accordingly with limestone.However, these parameters are not significant at the 5% level.That is, technical progress tends to diminish the usage of harvested area, agricultural credit, defensives and, on the other hand, is associated with the increase of limestone utilization.
In Table 3 are presented the statistical tests applied in order to verify the consistency of specific hypothesis related to the production function frontier adopted in the empirical model.The first null hypothesis relates to the adequacy test of Cobb-Douglas model relative to the less restrictive functional form expressed by the translog.Thus, it is tested the hypothesis that all the second order coefficients and the cross products are equal to zero.The value of the log likelihood ratio, 40.71, is greater than the critical value of the statistic with 5% significance level to the right.Duffy and Papageorgiou (2000) reject the Cobb-Douglas specification utilizing a data panel for 82 countries in a period of 28 years.Additionally, by examining the impact of production technology over technical efficiency, Kneller and Stevens (2003) reject the specification of the aggregate production function over the efficiency measures.Thus, translog production function constitutes a more flexible firm and is an approximation for any production frontier.The result of this test is presented on Table 2 rejects the specification in the form of a Cobb-Douglas function in favor of the translog specified model.
The second analysis refers to the joint significance tests of the parameters of the variables that explain technical inefficiency.The result rejects the hypothesis that the parameters are simultaneously equal to zero.
The last test examines the stability of the production frontier in relation to the time variable, that constitutes the presence or not of technology progress in the analyzed period.Thus, the result of the test accepts the null hypothesis that there have not been any of the known forms for the sample and the analyzed period.
According to the data of the analyzed period, it is observed that an amelioration of aggregate productivity exists over time.In a decreasing order, the Brazilian regions that represent greater relative degree of efficiency were the Northeast, North, Southeast, South and Center-west regions.This result points to the new Brazilian agriculture frontiers where the production of grain crops advances rapidly, followed by livestock activity.
Additionally, the most significant inputs that have contributed to Brazilian agriculture productivity were land factor, as well as agriculture credit.on the other hand, the inputs related to agricultural defensives and limestone were not significant to explain Brazilian agriculture productivity throughout the analyzed period.
According to the Economic Bulletin by IPEA (Brazilian Institute of Applied Economic Research), consid-ering the agriculture years 2000/2001 to 2004/2005, the sector has increased its debt in R$ 41,8 billion solely due to investment credit, constituting half of the total agriculture credit.The investment credit differs for not having annual cycle as credit destined for covering costs destined to cover normal expenses of production cycles.It is cumulative and exerts significant importance in the analysis of behavior in the agriculture sector.
For illustration purposes, considering only the agriculture years 2003/2004 and 2004/2005, the sector has contracted additional debt of R$20,9 billion, only on investment rubric -almost the same value of credit for costs which was, on average, R$ 22 billion (B.CONJ.IPEA, 2005).Thus, agriculture credit inserted in the model is an adequate and relevant proxy for the representation of machinery in the contribution of productivity in Brazilian agriculture.

Index
Utilizing data from LSPA/IBGE for the main grain crops in Brazilian agriculture -rice, beans, maize, soybeans and wheat -the following results have been obtained according to Table 5.Source: Elaborated by the authors.
In the period from 2001 to 2006, PTF in main Brazilian grain crops increased 24.6% according to calculation of Malmquist indexes.The component of this growth was the technical change index, which increased 27%.On the other hand, the component referring to efficiency change declined 1.9% during the period.Thus, it is considered that the effect of technology innovation during the period in study has been more expressive than effect in efficiency change for the analyzed crops.
Among analyzed crops, beans were the culture that underwent the greatest increase in TFP (+0.30%) during the observed period.In addition, it 5% is analyzed that the principal component of such TFP increase was technical change (+27), since growth in efficiency responded for only 2.7% of TFP elevation.Decomposing the EFCH index, it is verified that the indexes of pure efficiency change and scale change have responded for 50% of the underlying index increase, given the 1.3% growth for both.
Maize culture was the crop that obtained the second largest rise in TFP during the analyzed period.It is observed that its increase of 30% in the Malmquist index is predominantly due to technical change, which incurred an increase of 27%, similarly to global mean of data in study.However, the component of technical efficiency did not suffer a regress, but a 2.4% increase.Among its subcomponents, the elevation of the EFCH index occurred mainly due to change in pure efficiency, which progressed 2%, while the change in scale component suffered a 0.4% increase.
It is also observed that the soybeans culture suffered the third major increase in TFP (+26,2%), in which index of technical change was predominant over change in efficiency.However, in contrast to the soybeans culture, there has been a regress in change in efficiency (-0.6%).It is verified that such decline in this component occurs due to the regression of change in scale, which presented a 3% decline, and not because of alterations in the component refer-ring to scale change, which obtained a 2.4% growth during the study period.
Additionally, the wheat crop obtained the 4th major elevation in TFP (+21.8)throughout the observed period, in which effects occurred exclusively by effects in technical change, in which technology innovation is implicit, which in turn obtained a 27% progress, corresponding to the mean of crops in study.However, there has been efficiency change for this culture throughout the analyzed period (-4.1%).Among its components, the regression in this item occurred due to changes in scale, which suffered a decline of 4.2% during the years from 2001 to 2006.In different circumstances, an amelioration or stability in this index would have been verified, since the component referring to change in pure efficiency for the wheat crop obtained a 0.01% progress.
It is observed that, among main Brazilian grain crops, the rice culture suffered the smallest growth in TFP between 2001 and 2006 (+15.2%).Similarly to other crops, a progress in technical change (+27%) is observed.However, the regress in the index related with change in efficiency was the most expressive between the analyzed grains (-9.3%), being the unique culture to suffer a decrease in pure efficiency (-7%) and change in scale (2.5%).
Thus, all main Brazilian grain crops incurred in progress by the index referring to technical change.
In other words, it is observed the dislocation of the technology frontier, once detected that, on average, the product of a crop at is greater than the potential maximum product that could have been obtained at in relation of production factors of .
The component that negatively influenced in the Malmquist index performance verified in the cultures of rice, soybeans and wheat relates to change in efficiency.Given that this component of "change in efficiency", calculated in relation to CRS technology, can be decomposed in changes in "pure efficiency" and "changes in efficiency", it is observed that all crops experience regress either in "pure efficiency" only -change in efficiency in relation to the VRE frontier, as is the case of soybeans -or solely the regress in "scale change" -ratio between change in efficiency and change in pure efficiency, representing alterations in deviations between technologies CRS and VRE -as occurred with the wheat culture.However, exception is verified for rice culture in which both types of regress related to changes in efficiency occurred.
From Tables 6, 7 and 8, we verify the annual TFP evolution and its principal components -efficiency change (approximation to the frontier) and technical efficiency (innovation) -for each crop, analyzing changes particularly among the last studied periods (2005/2006).From the presented tables, it is observed that only the culture of beans increased in its TFP from 2005 to 2006.We analyze that efficiency changes surpassed the negative performance of the component related do innovation.on the other hand, maize culture was the sole crop besides beans that obtained an increase in the component related to change in efficiency.However, this progress did not compensate the decline suffered by the technical change index.
Thus, taking into account the hypothesis that firms mark-ups are positively related to productivity, the crops of rice, wheat, soy and beans incurred in the largest declines in mark-up in 2006 -mainly due to technological issues but also significantly affected by efficiency.For the last two periods, the largest and only mark-up increase is observed for beans.
Following Sumanth (1985), since the productivity has grown at a larger velocity (in this case, due to efficiency change) than other crops, mark-up increase for beans has been favored due to lower costs that are not entirely transmitted to consumers.
A general decline in technological change certainly affected mark-ups negatively for all grain crops in Brazilian agriculture.Therefore, the manner in which farmers could maintain and increase their mark-ups has been either by exports, since higher mark-ups can be charged due to the presence of trade costs, or increase in efficiency that affects mark-ups to all markets.Assuming that commodity producers are essentially price takers, total factor productivity is the ultimate form of cost decrease and opportunity for greater mark-ups both domestically and abroad.However, in 2006 in particular, technology has not been able to function as a tool for cost decrease and mark-up increase in the agriculture of grain crops in Brazil.
Common evidence to all analyzed cultures is the decline in technology component in 2006 as consequence of agriculture crisis and its indebtedness which affected mainly grain crops, thus interfering in the acquisition of inputs such as machinery and fertilizers that would represent technology innovation captured by this component in Malmquist index for increase in productivity.
According to IPEA, the increase in indebtedness occurred because of two conditions satisfied: excessively optimistic expectations in relation to the future and a generous supply of credit given the underlying business risk.The optimistic expectations were on the basis of increase in commodities prices that coincided with exchange rate devaluation, seen as a permanent phenomenon with the end of anchor currency in 1999 and Chinese economic growth.
on the other hand, the expansion of agricultural frontier especially in the Center-west for the grain crops was covered by credit from private and public banking institutions, as well as by product supplier firms.The crop expansion was associated with a high indebtedness of producers in the short and long term, inducing the financial system to restrict industry's access to new borrowings, interfering in maintenance of the current level of activities and, therefore, reflecting on acquisition of new technologies for productivity increase (B.CONJ. IPEA., 2005, 2006, 2006a, 2007)., 2005, 2006, 2006a, 2007).

CONClUSION
In this paper, the technique of Stochastic Frontier Analysis has been applied for the estimation of increase or decrease in inefficiencies through time, as well as the linear programming method Data Envelopment Analysis and Malmquist index for the analysis of TFP sources for the Brazilian crops of beans, maize, soybeans, wheat and rice -considered the main grain crops in Brazil -throughout the period that comprehends the years from 2001 to 2006, the most recent data available.
According to the Cobb Douglas model, we verify that the greatest elasticity observed is that of harvested area, followed by credit variable, confirming the importance of agriculture credit to cover costs and, particularly, to execute investments which responds for the greatest share of the data analyzed.As expected, assuming positive and inferior elasticity in relation to the other relevant factors, limestone contributes for productivity by correcting sole acidity, which assumes a maximizing role for the potential of productivity already established by the other factors.
In the stochastic frontier analysis for the Brazilian agriculture, assuming a Translog technology, it is observed no increase in aggregate productivity throughout the analyzed period.In a decreasing order, the Brazilian regions presenting the highest relative degree of efficiency were the Northeast, North, Southeast and Center-west.This results points to the new Brazilian agricultural frontiers where grain crop production advances rapidly, followed by livestock activity.
Additionally, the most significant inputs that contributed for Brazilian agriculture productivity were land factor, as well as the agriculture credit -the latter being an adequate and relevant proxy for representation of machinery in the contribution for Brazilian agriculture productivity.on the other hand, the inputs related to agricultural defensives and limestone were not significant to explain Brazilian agriculture productivity throughout the observed period.
on the other hand, the Malmquist indexes revealed clarifying results for independent crop analysis.In relation to the means throughout the study period, it is observed that major TFP changes occurred, in a decreasing order, for the cultures of beans, maize, soybeans, wheat and rice.Although the mean variations have indicated positive TFP changes, when analyzing changes between the years of 2005 and 2006, it is verified a decline in the component representing technology innovations for all major Brazilian grain crops, jointly with the loss of productive efficiency for all cultures, excepting beans and maize.
Thus, taking into account the hypothesis that firms mark-ups are positively related to productivity, the crops of rice, wheat, soy and beans incurred in the largest declines in mark-up in 2006 -mainly due to technological issues but also significantly affected by efficiency.However, only the beans crop assumed positive variation in its TFP, since it was the only culture among principal Brazilian grain crops in which efficiency gain surpassed the negative effect of technology use.The generalized decline in the technology component can be explained by the indebtedness crisis in agriculture that affected particularly grain crops in 2005/2006, generating a downfall in the employment of agriculture inputs and interfering negatively in the maintenance of current level of agriculture activities in Brazil and, especially, as observed for the grain cultures analyzed.

Volume 2 •
Number 2 • July -December 2009 20 distinct error components.The first term refers to the random part of error, with normal distribution, independent and identically distributed (iid), truncated in zero and variance [ v ~iid N(0, )].The second term concerns the part relating to technical inefficiency, constituting a deviation in relation to the production frontier (which can be inferred by negative sign and by restriction ).They are nonnegative random variables with normal truncated distribution, that is, non-null mean [ u ~N + ( m, 2 u s )].

Färe
distance function in relation to two different periods measure the maximum proportional change required in production to turn feasible in relation to technology in period .Caves, Christensen e Diewert (CCD) (1982) define Malmquist productivity as (5) In this formulation, technology at time is the reference technology.Alternatively, Färe et al. (1994) define a Malmquist index based on period as (6) In order to avoid an arbitrary benchmark, Färe et al. (1994) specify the Malmquist index for changes in productivity as the geometric mean of both CCD type Malmquist indexes:

Finally,
Färe et al. (1994)  highlight that, while the product of the components of efficiency change and technical change must, by definition, equal the Malmquist index, these components may be moving in opposite directions.For instance, a Malmquist index greater than unity, say 1.25 (which signalizes productivity gain), could have a component of efficiency change smaller than 1 (e.g.0.5) and a change in technology component greater than unity (e.g., 2,5).Alternatively,Alam (2001) expresses the Malmquist index in terms of distances throughout the y-axis, based on Figure1:

Figure 1 :
Figure 1: Malmquist Index In relation to Data Envelopment Analysis and the Malmquist index, Färe et al. (1994) discuss the usage of the VRS approach in the Malmquist index calculation.By calculating "change in efficiency" in relation to the VRS frontier, it is obtained the denominated "change in efficiency" and measured changes in production scale by the ratio between "change in efficiency" and "change in pure efficiency".Thus, the component change in efficiency (or technical efficiency) calculated in relation to technology with CRS can be decomposed in a component of change in pure efficiency (PEC, calculated in relation to the technology with VRS) and, in a component of change of scale efficiency (SEC), which represents changes in deviations between the CRS and VRS technologies.Thus, where which can be re-written in the following form: results obtained from gathered PAM and Ministry of Agriculture data for the proposed stochastic frontier model.

Table 1 -Cobb-Douglas Production Function
Source: Research results obtained from gathered PAM and Ministry of Agriculture data for the proposed stochastic frontier model.Note: *** significant to the 1% level; ** significant to the 5% level; * significant to the 10% level.