Research methodology

4.3.1 Theoretical framework

The chapter focuses mainly on measuring economic performance of diversified farms. The scope of measures of economic performance in this study includes economies of scale, output complementarity, and elasticity of substitution between inputs in light of increasing cost stress in farm production in Vietnam. Bravo-Ureta et al. (2007) define technical efficiency as a measure of management ability at a given level of technology and a source of productivity growth including technical change. They show that the enhancement of the decision-making process can derive the gain in technical efficiency, which is influenced by social and economic conditions such as education and farm characteristics. In contrast, technical change involves investments in research and technology.

In order to answer the research questions of this chapter, a multiple output and input production technology is required. In the study conducted by Paul and Nehring (2005), the authors used both input and output distance functions to evaluate the economic performance of US farms. Both output and input distance functions are capable of

dealing with multi-output technologies. The output-based measure estimates technical efficiency in terms of proportional expansion of outputs, given input being held constant. Whereas, the input-based measure estimates technical efficiency in terms of the proportional savings in inputs, given outputs being held constant. These measures can be interpreted in terms of output enhancement and cost savings, respectively. In this chapter, an input-oriented stochastic distance function is analyzed, instead of an output oriented distance function, for two major reasons: (i) the input distance function is unambiguously interpreted in terms of cost saving - one of key interests in this chapter in light of the rising costs in agricultural production due to the high inflation of the past decade in Vietnam (World Bank, 2011); (ii) the input-based measure allows this study to answer the research question about how family labour behaves in response to increasing cost stress.

This study applies the input distance function developed by Paul and Nehring (2005) to measure the economic performance of diversified farms. Because there is no access to cost data due to the unpriced nature of many inputs in this study, which is unable to calculate economies of scope relative to a cost function. Thus, output complementarity is, instead, calculated by an input distance function. This problem of unpriced inputs may explain why studies on diversification economies in developing countries use the approach of input distance function, instead of cost function (Coelli and Fleming 2004 for Papua New Guinea; Rahman 2009 and 2010 for Bangladesh). At the same time, the choice of a stochastic distance function approach can allow the separation of the random noise from technical inefficiency effects that is ignored in the data envelopment analysis by Dao and Lewis (2013).29 _{Several studies used the parameters of the estimated input}

distance function to estimate scale economies, technical efficiency and elasticity of substitution in diversified farms (Grosskopf et. al. 1995; Rahman 2010).

29_{Dao and Lewis (2013) also estimate the technical efficiency in rice-based crop diversification farms in} Vietnam. However, they only apply non-parametric regression as a data envelopment analysis. Kumbhakar and Lovell (2003) show that non-parametric methods cannot provide the determinants of technical inefficiency in stochastic production function.

Figure 4.1 Input distance function, over-utilization of X1, reproduced from

Grosskopf et al. (1995, p. 280)

In the study of stochastic frontier analysis, Kumbhakar and Lovell (2003) introduce the overview of the input distance function, which was firstly introduced by Shephard (1970).30 _{This function describes how much an input vector may be proportionally}

contracted with the output vector that is held fixed. This chapter uses the theoretical framework introduced by Kumbhakar and Lovell (2003), and Paul and Nehring (2005, p. 529). The input distance function D(x,y) is formally defined as:

𝐷(𝑥, 𝑦) = 𝑚𝑎𝑥 {𝜆; 𝜆 > 0,𝑥

𝜆∈ 𝐿(𝑦)} (4.1)

𝐿(𝑦) = {𝑥 ∈ 𝑅₊𝑁_{: 𝑥}, x can produce y (4.2)}

where x is a scalar, L(y) is the set of input requirement x, which is used to produce the output vector y. Figure 5.1 below describes the input distance function. As can be seen in Figure 5.1, the input vector x is feasible for output y, but y can be produced with the radically contracted input vector (x/λ*_{), and so D(x, y) =λ}*_{>1. D(x, y)=1 if and only if} the input bundle is an element of the isoquant of L(y). In addition, D(x, y) is non- decreasing, positively linear homogenous and concave in x, and increasing in y. Paul and Nehring (2005) show that the input distance function can provide the measure of technical efficiency because it allows for deviation (distance) from the frontier. Finally, there is a dual relationship between input distance function and cost function, which allow us to relate the derivatives of the input distance function to the cost function (Färe and Primon 1995).

30_{See further details of properties of input distance function in Kumbhakar and Lovell (2003).}

X2

X1

x/λ* x

According to Lovell et al. (1994), the imposition of a function form for D(x, y) cannot be directly estimated due to the unobserved value of the distance function. Lovell et al. (1994) thus suggest a way of solving this problem by exploiting the property of linear homogeneity of the input distance function as follows:

𝐷(𝜌𝑥, 𝑦) = 𝜌𝐷(𝑥, 𝑦), 𝜌 > 0 (4.3)

Assuming that x is a vector of dimension K and ρ=1/x1, where x1 denotes the (arbitrary chosen) first element of the input vector x, the Equation (4.3) is transformed in logarithmic form as:

𝑙𝑛𝐷_𝑖(𝑥, 𝑦) = 𝑙𝑛𝑥₁+ 𝑙𝑛𝐷(𝑥/𝑥₁ , 𝑦) (4.4)

Lovell et al. (1994) also show that the logarithm of the distance function in the Equation (4.4) measures the deviation of an observation (x, y) from the deterministic border of the input requirement set L(y), which is consist of two components according to the stochastic frontier literature. The first one describes random shocks and measurement errors. The second one corresponds to technical inefficiencies that are assumed to be stochastic and a non-negative random variable u. Conceptually the presence of inefficiencies can be evaluated by the distribution of management skills across the population of farm households using the same technology. These assumptions, thus, can be mathematically expressed as follows:

𝑙𝑛𝐷(𝑥, 𝑦) = 𝑢 − 𝑣 (4.5)

Substituting Equation (4.5) into Equation (4.4) gives:

−𝑙𝑛𝑥1 = 𝑙𝑛𝐷(𝑥/𝑥1, 𝑦) − 𝑢 + 𝑣 (4.6)

4.3.2 Functional form

To empirically estimate the distance function, a functional form must be specified. I select the translog functional form used by previous studies (Lovell et al. 1994; Grosskopf et al. 1995; Coelli et al. 1998; Paul et al. 2000; Irz and Thirtle 2004; Paul and Nehring 2005; Rasmussen 2010; Rahman 2010). The translog is a flexible function and it has some advantages in that it allows the elasticity of scale to change for various farm sizes. In addition, a flexible technology also allows for substitution effects in the function, which supports the answer to the research questions related to substitutability between inputs (Paul et al. 2000).

The translog input distance function with M outputs, N inputs of the farm household i is given by: 𝑙𝑛𝐷_𝑖 = 𝛽₀+ ∑ 𝛽_𝑛𝑙𝑛𝑥_𝑛 𝑁 𝑛=1 +1 2∑ ∑ 𝛽𝑛𝑘𝑙𝑛𝑥𝑛 𝑁 𝑘=1 𝑁 𝑛=1 𝑙𝑛𝑥_𝑘+ ∑ 𝛼_𝑚𝑙𝑛𝑦_𝑚 𝑀 𝑚=1 +1 2∑ ∑ 𝛼𝑚𝑙𝑙𝑛𝑦𝑚 𝑀 𝑙=1 𝑀 𝑚=1 𝑙𝑛𝑦_𝑙+ ∑ ∑ 𝛾_𝑚𝑛𝑙𝑛𝑦_𝑚 𝑁 𝑛=1 𝑀 𝑚=1 𝑙𝑛𝑥_𝑛 (4.7)

where Di measures the distance from (x,y) to the production function and denotes the unobservable value of the distance function. As the input distance function is linear homogenous in inputs, the parameters in Equation (4.7) must satisfy the following regulatory restrictions: ∑ 𝛽_𝑛 𝑛 = 1, ∑ 𝛽_𝑛𝑘 𝑘 = 0, ∑ 𝛾_𝑚𝑛 = 0 (𝑚 = 1, … , 𝑀) 𝑛 𝛽𝑛𝑘 = 𝛽𝑘𝑛 (𝑁, 𝐾 = 1, … , 𝑁); 𝛼𝑚𝑙 = 𝛼𝑙𝑚 (𝑚, 𝑙 = 1, … , 𝑀)

This chapter uses the approach of Lovell et al. (1994) and Coelli and Perelman (1999) in imposing these restrictions required for the homogeneity of degree of one in inputs

(∑7 𝛽_𝑛 = 1)

𝑛=1 by normalizing the function by one of the input, similar to Equations (4.3) and (4.4). As a result, Equation (4.7) is expressed as follows:

𝑙𝑛(𝐷_𝑖/𝑥_1𝑖) = 𝛽₀+ ∑ 𝛽_𝑛𝑙𝑛𝑥_𝑛 7 𝑛=2 +1 2∑ ∑ 𝛽𝑛𝑘𝑙𝑛𝑥𝑛 7 𝑘=2 7 𝑛=2 𝑙𝑛𝑥_𝑘+ ∑ 𝛼_𝑚𝑙𝑛𝑦_𝑚 4 𝑚=1 +1 2∑ ∑ 𝛼𝑚𝑙𝑙𝑛𝑦𝑚 4 𝑙=1 4 𝑚=1 𝑙𝑛𝑦_𝑙+ ∑ ∑ 𝛾_𝑚𝑛𝑙𝑛𝑦_𝑚 7 𝑛=2 4 𝑚=1 𝑙𝑛𝑥_𝑛= lnD(𝑥∗_{, 𝑦)} (4.8)

where 𝑥_𝑛𝑖∗ = 𝑥_𝑛𝑖/𝑥_1𝑖(∀𝑛, 𝑖), only n-1 inputs are not used for normalization. Substituting lnD with u and adding error term v to account for random noise, we end up with an estimating Equation (4.9). Substituting Equations (4.8) into (4.6), we have:

−𝑙𝑛𝑥1 = 𝑙𝑛𝐷(𝑥𝑛𝑖∗ , 𝑦) − 𝑢 + 𝑣 (4.9)

Paul and Nehring (2005) find that coefficient estimates from Equation (4.9) have opposite signs from those for a standard production or input requirement function. The authors introduce a method by reversing the signs of the equation in order to interpret

the measures from Equation (4.9) more similarly to those from the more familiar functions in the literature review:31

𝑙𝑛𝑥₁ = −𝑙𝑛𝐷(𝑥_𝑛𝑖∗ , 𝑦) − 𝑢 + 𝑣 (4.10)

Equation (4.10) is expressed as a stochastic distance function, which includes two error terms representing deviations from the frontier and random error. On the basis of a parameterisation of the distance function and distributional assumptions of error terms, Equation (4.10) can be estimated by the maximum likelihood methods, which have been extensively used in the stochastic frontier literature.32