Methodology and model specification

A multi-input multi-output yak production function is developed in the livestock husbandry sector in order to measure the production performance of yak production and to examine the impact of renting-in grassland on yak production and technical inefficiency. The livestock grazing system on the Qinghai-Tibetan Plateau is extensive, suggesting that we should rely on an approach which does not require behavioral assumptions. We adopt the stochastic distance

function approach instead of a deterministic approach in order to simultaneously accommodate random noise and systematic differences in technical efficiency.

2.3.1 Conceptual framework

The output distance function introduced by Shephard (1970) treats the inputs as given and looks at the potential proportional expansion of outputs, as long as the outputs are technologically feasible. Denoting a vector of inputs by 𝑥 = (𝑥₁, ⋯ , 𝑥_𝐾) ∈ ℜ^𝐾+ and a vector of outputs by 𝑦 = (𝑦₁, ⋯ , 𝑦_𝑀) ∈ ℜ^𝑀+, a feasible multi-input multi-output production technology can be defined using the output possibility set P(x), which can be produced using the input vector x: 𝑃(𝑥) = {𝑦: 𝑥 can produce 𝑦}. This is assumed to satisfy the set of axioms depicted in Färe and Primont (1996). The output distance function is defined as: 𝐷_𝑂(𝑥, 𝑦) = min {𝜇: 𝑦/𝜇 ∈ 𝑃(𝑥)}.

Since an output distance function Do (x,y) is defined in terms of the output set 𝑃(𝑥), satisfying certain properties, the output distance function is required to satisfy analogous conditions. As noted by Lovell et al. (1994), 𝐷_𝑂(𝑥, 𝑦) is non-decreasing, positively linearly homogeneous, convex in y, and decreasing in x. It should be clear from the definition and figures that 𝑃(𝑥) = {𝑦: 𝑥 can produce 𝑦} and that on the iso-quant, 𝑃(𝑥) = {𝑦: 𝐷_𝑂(𝑥, 𝑦) = 1} . If 𝐷_𝑂(𝑥, 𝑦) < 1, then (x, y) belongs to the production set 𝑃(𝑥), 𝐷_𝑂(𝑥, 𝑦) < 1 if y is located on the outer boundary of the output possibility set (Kumbhakar and Lovell, 2000).

In order to estimate the distance function in a parametric setting, a translog functional form is assumed. According to Coelli and Perelman (2000), the translog output distance function for the case of k inputs and m outputs is specified as:

𝑙𝑛𝐷_𝑜𝑖(𝑥, 𝑦) = 𝛼₀+ ∑ 𝛼_𝑚𝑙𝑛𝑦_𝑚𝑖+

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where i denotes the i_th household in the sample and T is the sample size. The restrictions required for linear homogeneity in outputs are:

∑^𝑀_𝑚=1𝛼_𝑚 = 1, 𝑚 = 1,2, ⋯ , 𝑀, ∑^𝑀_𝑚=1𝛼_𝑚𝑛 = 0, 𝑚, 𝑛 = 1,2, ⋯ , 𝑀, ∑^𝑀_𝑚=1𝛿_𝑘𝑚 = 0, 𝑘 = 1,2, ⋯ , 𝐾 and those required for symmetry are:

𝛼_𝑚𝑛 = 𝛼_𝑚𝑛, 𝑚, 𝑛 = 1,2, ⋯ , 𝑀, 𝛽_𝑘𝑙 = 𝛽_𝑙𝑘, 𝑘, 𝑙 = 1,2, ⋯ , 𝐾.

According to Lovell et al. (1994), the homogeneity implies that 𝐷_𝑂(𝑥, ϑ𝑦) = 𝜗𝐷_𝑂(𝑥, 𝑦), for any 𝜗 > 0. Hence, if we arbitrarily choose one of the outputs as the dominated output 𝑦_𝐷, and set 𝜗 = 1/𝑦_𝐷, we obtain 𝐷_𝑂(𝑥, 𝑦/𝑦_𝐷) = 𝐷_𝑂(𝑥, 𝑦)/𝑦_𝐷 . For the translog form, this provides: should be assumed to be exogenous. Similar arguments are confirmed in page 95 of the book Stochastic Frontier Analysis (Kumbhakar and Lovell 2000).

According to Aigner, Lovell and Schmidt (1977), the stochastic frontier model is obtained by adding a term 𝑣_𝑖 to capture noise. Thus the stochastic output distance function is:

−ln(𝑦_𝐷𝑖) = 𝑇𝐿(𝑥_𝑖, 𝑦_𝑚𝑖⁄𝑦_𝐷𝑖, 𝛼, 𝛽, 𝛿) + 𝑣_𝑖 − 𝑢_𝑖 (2-4)

As usual, the 𝑣_𝑖 term is assumed to be a two-sided random disturbance and is distributed as i.i.d. 𝑁(0, 𝜎_𝑣²). 𝑢_𝑖 is a random negative term derived from an independent distribution 𝑁(𝜇_𝑖, 𝜎_𝑢²), truncated above zero of the normal distribution with mean 𝜇_𝑖 and variance 𝜎_𝑢² (Battese and Coelli, 1988; 1995; 1996; Coelli, 1995; Coelli and Battese, 1996). The mean 𝜇_𝑖 is defined as:

𝜇_𝑖 = 𝑍_𝑖 ∗ 𝜏 (2-5)

where 𝑍_𝑖 is a vector of explanatory variables associated with the technical inefficiency effects which could include socioeconomic and farm management characteristics. 𝜏 is a vector of unknown parameter to be estimated. MLE could be used to estimate the parameters of the stochastic output distance function given appropriate distributional assumptions for 𝑣_𝑖 and 𝑢_𝑖 (Aigner, Lovell and Schmidt, 1977).

The production frontier is specified as follows:

y_𝑖 = 𝑓(𝑥_𝑖, 𝛽). exp (𝑣_𝑖 − 𝑢_𝑖) (2-6)

where, for all households indexed with a subscript i, the measure of technical efficiency of the i_th farm denoted by TE_i is defined as the ratio of the observed output to the corresponding potential output, written as:

𝑇𝐸_𝑖 =𝑓(𝑋_𝑖, 𝛽). exp (𝑣_𝑖− 𝑢_𝑖)

𝑓(𝑋_𝑖, 𝛽). exp (𝑣_𝑖) = exp(−𝑢_𝑖) = 𝐷_𝑜𝑖(𝑥, 𝑦) (2-7)

The predicted value of the output distance 𝐷_𝑜𝑖(𝑥, 𝑦) is not directly observable because 𝑢_𝑖 only appears as part of the composed error term 𝜀_𝑖 = 𝑣_𝑖 + 𝑢_𝑖. It may be obtained using the conditional expectation

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𝐷_𝑜𝑖 = 𝐸(exp(−𝑢_𝑖) |𝜀_𝑖) =1 −Φ(𝜎_𝐴− 𝛾𝜀_𝑖⁄ )𝜎_𝐴

1 −Φ(𝛾𝜀_𝑖⁄ )𝜎_𝐴 exp(𝛾𝜀_𝑖 + 𝜎_𝐴²⁄ ) 2 (2-8)

where 𝜎_𝐴 = √𝛾(1 − 𝛾)𝜎², 𝜎² = 𝜎_𝑣²+ 𝜎_𝑢², 𝛾 = 𝜎_𝑢²/𝜎², and Φ(∙) represents the distribution function of a standard normal random variable. Once the parameters of model (2-1) are estimated, it is both interesting and easy to calculate the meaningful elasticity.

2.3.2 Empirical specification

In agricultural economics literature, output is frequently treated as a stochastic variable because of weather conditions, diseases, and other exogenous random forces. We assume that the decision variables are fixed in the short term and that the production level follows common and reasonable assumptions when estimating production relationships in agriculture.

We therefore build the production frontier and auxiliary technical inefficiency model with a one-step approach.

y_𝑖 = 𝑓(𝑥_𝑖, 𝛽). exp (𝑣_𝑖 − 𝑢_𝑖) (2-9)

where yi denotes the vector of outputs. The first output describes the output of yak production, denoted by the amount of yak meat produced in the year. The second output denotes the revenue of the other outputs, including the revenue of Tibetan sheep, milk, yak hide, Tibetan sheep wool, and so on. T describes the sample size, which is equal to 197 households in this study.

𝑥_𝑖 is a vector of inputs of grassland area, labor, household capital, and initial yak.

𝛽 are technological parameters to be estimated for 𝑥_𝑖.

𝑣_𝑖 is a random error term, independently and identically distributed as 𝑁(0, 𝜎_𝑣²). It is intended to capture events beyond the control of the herdsman.

𝑢_𝑖 is a non-negative random error term, independently and identically distributed as 𝑁(𝜇, 𝜎_𝑢²), truncated above zero and intended to capture technical inefficiency in production. This is measured as the ratio of observed outputs to maximum feasible output.

According to the conceptual framework described above, the translog functional form for the parametric distance function of yak production for the two outputs and four inputs is written as follows:

−ln(𝑦_1𝑖) = 𝛼₀+ 𝛼₁ln(𝑦^2𝑖⁄ ) +𝑦_1𝑖 1

2𝛼₁₁ln(𝑦^2𝑖⁄ ) ln(𝑦𝑦_1𝑖 ^2𝑖⁄ ) + ∑ 𝛽𝑦_{1𝑖 𝑘}𝑙𝑛𝑥_𝑘𝑖

4

𝑘=1

+1

2∑ ∑ 𝛽_𝑘𝑙

4

𝑙=1

𝑙𝑛𝑥_𝑘𝑖𝑙𝑛𝑥_𝑙𝑖+ ∑ 𝛿_𝑘1𝑙𝑛𝑥_𝑘𝑖ln (𝑦^2𝑖⁄ )𝑦_1𝑖

4

𝑘=1 4

𝑘=1

+ 𝑣_𝑖 − 𝑢_𝑖

(2-10)

the empirical technical inefficiency model, as described in equation (2-7), is written in equation (2-11),

𝜇_𝑖 = 𝑧_𝑖 ∗ 𝜏 = 𝜏₀+ ∑ 𝜏_ℎ𝑧_ℎ𝑖

9

ℎ=1

(2-11)

where 𝑧_𝑖 is a vector of explanatory variables associated with the technical inefficiency effects, including household size, variables relating to livestock intensity and variables relating to grassland property rights of renting-in grassland. These variables are described in more detail in the following section. We used the maximum likelihood estimation method to estimate the

“one-step” model, which specifies both the stochastic frontier and technical inefficiency model.