Environmental production technology
The standard producer theory's starting point is to define the technical link between inputs and outputs using production technology. Dairy farms produce both desirable (such as milk and meat) and undesirable byproducts (GHG emissions). Conventional technology can manage the desired outputs. Undesired outputs, on the other hand, require special attention in efficiency analysis. The set of environmental production technologies \(\left( \Psi \right)\) is therefore defined as follows:
$${\Psi } = \left\{ {\left( {x,y, b} \right){:}x \;{\text{can}}\;{\text{produce}}\;(\left. {y, b)} \right\}} \right.$$
(1)
where x, y, and b are the vectors of input, desirable output, and undesirable output, respectively. In the context of environmental production technology (Ψ)., it is crucial to mosdel the relationship between desirable and undesirable outputs. The environmental production technology set Ψ is assumed to satisfy three axioms

(a)
Null jointness, i.e. if \(((y,b) \in \Psi \;{\text{and}}\;b = 0\;{\text{then}}\;y = 0)\)

(b)
The set \(\Psi\) is a closed set and nonempty.

(c)
If \(\left( {\left( {y,b} \right) \in \Psi {\text{ and }}y^{\prime} \le {\text{y then }}\left( {y^{\prime},b} \right) \in \Psi } \right)\), that is, the technology set \(\Psi\) satisfies the free disposability of all inputs and outputs.
For details of other properties of the technology set, see Färe et al. (1985) and Chambers et al. (1996). An input or output possibility set can be used to illustrate environmental production technology (Färe et al. 2008). The input set (L) is then defined as follows:
$$L\left( y \right) = \left\{ {x:\left( {x,y,b} \right)\left. { \in \Psi } \right\}} \right.$$
(2)
Following Farrell (1957), the technology boundaries (input isoquants) of the technology set \({\Psi }\left( {x,y,b} \right)\) can be defined in terms of a radical as follows:
$$\partial L\left( y \right) = \left\{ {x:x \in L\left( {y,b} \right), \theta x \notin L\left( {y,b} \right), \left. {\forall \theta , 0 < \theta < 1} \right\}} \right.$$
(3)
Decisionmaking unit (DMU) or farms are efficient if they are within the boundaries of the input requirement set, that is, they are input efficient if \(x \in \partial L\left( y \right)\). On the other hand, DMUs are input inefficient if \(x \notin \partial L\left( y \right).\) Inputinefficient farms use more inputs to produce the same output compared to other inputefficient DMU. This is the case if the inefficient farms experience a slack in inputs.^{Footnote 1} In the directional distance function (Chambers et al. 1996), Eq. (3) can be represented as follows:
$$D_{I} \left( {y_{t} ,x_{t} ,b_{t} ,k_{t} ; t,\omega } \right) = \max \left\{ {\left( {\lambda :{\raise0.7ex\hbox{${x_{t} }$} \!\mathord{\left/ {\vphantom {{x_{t} } \lambda }}\right.\kern0pt} \!\lower0.7ex\hbox{$\lambda $}}} \right) \in \left. {L\left( {y_{t} ,x_{t} ,b_{t} ,k_{t} , t,\omega } \right)} \right\}} \right.$$
(4)
where \(y_{t}\) denotes the outputlevel targeted by a farmer, given, a vector of undesirable output \(b_{t}\), and a vector of initial capital stocks \(k_{t}\), and a vector of feasible variable inputs \(x_{t}\).
The feasible input set is represented by \(L\left( {y_{t} ,x_{t} ,b_{t} ,k_{t} , t,\omega } \right)\). \(\lambda\) is a scalar (\(\lambda \ge 1\)) assessing possible input reductions, with a minimum value of 1 corresponding to completely efficient production units. \(\omega\) indicates unobserved heterogeneity like farmeffects.
Equation (4) must meet certain regularity requirements, such as being nondecreasing in inputs, linearly homogenous, and decreasing in outputs. Following Lovell et al. (1994) normalizing all inputs by one of the inputs is a straightforward technique for applying the homogeneity constraint.
$${\raise0.7ex\hbox{${D_{I} \left( {y_{t} ,x_{t} ,b_{t} ,k_{t} ;t,\omega } \right)}$} \!\mathord{\left/ {\vphantom {{D_{I} \left( {y_{t} ,x_{t} ,b_{t} ,k_{t} ;t,\omega } \right)} {x_{1} }}}\right.\kern0pt} \!\lower0.7ex\hbox{${x_{1} }$}} = L\left( {y_{t} ,\mathop x\limits_{t} ,b_{t} ,\dot{K}_{t} ; t,\omega } \right)$$
(5)
where \(\mathop x\limits_{t}\) is a vector of input ratios with \(\mathop x\limits_{kt} = \frac{{x_{kt} }}{{X_{1} }}, \quad \forall k = 2, \ldots ,K;\) and \(\dot{K}_{t} = \frac{{K_{t} }}{{x_{1} }}\).
Equation (5) can be written in logarithm and a translog functional form as in Coelli et al. (2005) as
$$\ln D_{I} \left( {y_{t} ,x_{t} ,b_{t} ,k_{t} ;t,\omega } \right)  \ln x_{1} = {\text{TL}}\left( {\ln y_{t} ,\ln \mathop x\limits_{t} ,\ln b_{t} ,\ln \dot{K}_{t} , t,\omega } \right)$$
(6)
Rearrange Eq. (6) and add the random error term \((v_{it} )\) to make the distance function stochastic.
$$ \ln x_{1} = {\text{TL}}\left( {\ln y_{t} ,\ln \mathop x\limits_{t} ,\ln b_{t} ,\ln \dot{K}_{t} , t,\omega } \right) + v_{it}  \ln D_{I} \left( {y_{t} ,x_{t} ,b_{t} ,k_{t} ;t,\omega } \right)$$
(7)
where \(v_{it}\) is the noise and \(\ln D_{I} () = u_{it} \ge 0\) \(D_{I}\) measures the efficiency measure that is conditional on undesirable outputs which represent the relative excess in input factors due to ecoefficiency.
GTFP measurement and decomposition approaches
Previous research on efficiency has employed various methods for calculating and decomposing TFP (for details see O’Donnell 2010; Kumbhakar et al. 2015; Alem 2018). The Divisia index is commonly used as an easy way to track TFP growth (Kumbhakar et al. 2015). A recent measure of TFP change seeks to decompose TFP change into different sources. TFP was decomposed by Kumbhakar (1996) and Kumbhakar and Lovell (2000) into technical change (TC), scale change (SC), efficiency change (EC), and pricing change components. Several publications, for example, Brümmer et al. (2002), Karagiannis et al. (2004); and Kumbhakar and LozanoVivas (2005), decompose the TFP change into four major components.
TFP change (\(T\dot{F}P\)) denotes the difference between the rate of change of an output index (\(\dot{y}\)) and the rate of change of the index of an input \((\dot{x})\) (see Karagiannis et al. 2004). We follow the Divisia index for the productivity change decomposed into TEC, TC, SC, and Allocative efficiency change (AEC) components. In this study, y is the net dairy output that is the difference between desirable output (Y) minus undesirable outputs (b) then estimate the Green TFP change \(\left( {GT\dot{F}P} \right)\).
$$GT\dot{F}P = \dot{\user2{y}}  \dot{\user2{x}} \equiv \dot{\user2{y}}  \mathop \sum \limits_{j} S_{j} \dot{x}_{j}$$
(8)
where \(S_{j}\) is captures the expenditure share of input \(X_{j} \left( {S_{j} = w_{j} x_{j} /C} \right).\) \(C\) denotes the total cost (\(C = \mathop \sum \limits_{j} w_{j} x_{j}\)); and \(w_{ }\) is the vector of input price \(x_{j}\) (\(w_{ }\) = \(w_{1} \ldots \ldots w_{j} )\). As shown by Kumbhakar et al. (2014), by differentiating (8) totally, we get
$$GT\dot{F}P = {\text{TC}}  \frac{\partial u}{{\partial t}} + \mathop \sum \limits_{j} \left\{ {\frac{{f_{j} x_{j} }}{f}  S_{j} } \right\}\dot{x}_{j}$$
(9)
$$GT\dot{F}P = {\text{TC}} + \left( {{\text{RTS}}  1} \right)\mathop \sum \limits_{j} \lambda_{j} \dot{x}_{j} + \frac{\partial u}{{\partial t}} + \mathop \sum \limits_{j} \left\{ {\lambda_{j}  S_{j} } \right\}\dot{x}_{j}$$
(10)
where a dot above a variable denotes the rate of change for that particular variable. RTS = \(\mathop \sum \limits_{j} \frac{\partial \ln y}{{\partial \ln x_{j} }} = \mathop \sum \limits_{k = 1}^{4} \beta_{k}\) and \(\lambda_{j}\) is the elasticity of production for each input, i.e. \(\lambda_{j} = \frac{{\varepsilon_{j} }}{{{\text{RTS}}}}\), where \(\varepsilon_{j}\) are input elasticities defined at the input distance function \({\text{TL}}\left( {\ln y_{t} ,\ln \mathop x\limits_{t} ,\ln b_{t} ,\ln \dot{K}_{t} , t,\omega } \right)\).
Green total factor productivity change is the sum of technical change (TC), efficiency change (EC), scale change (SC), and allocative efficiency change \(\left( {{\text{AEC}}} \right)\), i.e. \(GT\dot{F}P\) = \({ }{\varvec{TC}} + {\mathbf{SC}} + {\mathbf{EC}} + {\text{AEC}}\). The GTFP change connected to the technology are \({\varvec{TC}} + {\varvec{SC}} + \user2{EC },\) which is the focus of this study.
The first source of the change in GTFP could be technical change (TC), which indicates that there is a change in the frontier. It is proof that best practices have improved because of the use of new technology. The improvement in the firm's capacity to utilise existing technology is the second factor contributing to the change in GTFP owing to efficiency change (EC). EC exhibits a move towards the frontier because of improved farm management, such as reduced resource wastage. With an intensive agricultural extension, inefficient farmers, lately adopting the currently available technology are improving efficiency (Alem 2018). The third component of GTFP is caused by a SC, which indicates movement approaching the frontier. SC illustrates how the company has evolved towards an operational size that is technologically feasible (Kumbhakar et al. 2015). The departure of input prices from the value of their marginal products in the allocation of inputs is captured by the AEC component of the GTFP change. Due to the lack of data on input prices at the farm level, AEC was not estimated for this study.
GHG emissions estimate
In the current study, the Intergovernmental Panel on Climate Change (IPCC 2006) methodology's Tier 2 approach is used, which incorporates countryspecific forecasts from the Norwegian Environment Agency (NIR 2020). The basic equation to calculate the emission factor for enteric fermentation is provided in IPCC 2006 as follows.
$${\text{The}}\;{\text{CH}}4\;{\text{emissions}}\;{\text{factors}}\;{\text{for}}\;{\text{dairy}}\;({\text{EF}}_{{\text{D}}} ) = \left( {\frac{{{\text{GE}}_{{\text{D}}} *Y_{{\text{m}}} *365 \;{\text{days/year}}}}{{55.65{\text{/Kg}}\;{\text{CH}}4}}} \right)$$
(11)
$${\text{GE}}_{{\text{D}}} = 137.900 + \left( {{\text{Milk}}\;305 \times 0.0250} \right)\;({\text{Concentrate}}\;{\text{proportion}} \times 0.281)$$
$$Y_{{\text{m}}} = 7.380{}({\text{Milk}}\;3050*0.00003)\;({\text{Concentrate}}\;{\text{proportion}}*0.0176)$$
where

\({\text{GE}}_{{\text{D}}}\) = gross energy intake for dairy farms, MJ/day

Y_{m} = methane conversion rate, %

The factor 55.65 (MJ/kg CH4) is the energy content of methane

Milk305 = Lactation output of energyadjusted milk at 305 days

Concentrate proportion is the percentage of concentrates in the diet as a whole calculated using net energy. Equation (11) takes an annual emission factor into account (365 days).
Methane yearly emissions resulting from manure management of dairy cattle (CH4 emissions Dairy FARM) in each farm were derived by multiplying the farmspecific emission factor \(({\text{EF}}_{{\text{D}}} )\) with the number of raised dairy cattle (\(N_{{\text{D}}}\)). Furthermore, countryspecific emissions factors for nondairy cattle (CH4 emissions, not dairy FARM) are derived by multiplying the number of nondairy cattle (\(N_{{{\text{notD}}}}\)) by the CH4 emissions from nondairy farms \(({\text{EF}}_{{\text{notD }}} )\), i.e.
$${\text{CH}}4\;{\text{emissions}}\;{\text{Dairy}}\;{\text{FARM}}\;{\text{Kg/year}} = {\text{EF}}_{{\text{D}}} *N_{{\text{D}}}$$
(12)
$${\text{CH4}}\;{\text{emissions}},\;{\text{not}}\;{\text{Dairy}}\;{\text{FARM}}\;{\text{Kg/year}} = {\text{EF}}_{{{\text{notD}}}} *N_{{{\text{notD}}}} .$$
(13)