Memoire Online - The effect of land fragmentation on the productivity and technical efficiency of smallholder maize farms in Southern Rwanda

There are two approaches used to estimate technical efficiency: the one-step approach and the two-step approach. The two-step procedure using the stochastic frontier production function generally involves first estimating the production frontier then predicting the technical efficiency of each firm. In the second step, the predicted technical efficiency variable is regressed against a set of variables that are hypothesized to influence the firm's efficiency (Kalirajan, 1981).

However, the two-stage procedure lacks consistency in assumptions about the distribution of the inefficiencies. In step one, it is assumed that inefficiencies are independently and identically distributed in order to estimate their values. In step two, estimated inefficiencies are assumed to be a function of a number of firm-specific factors, violating this assumption (Coelli, 1996). To overcome this inconsistency, Kumbhakar et al. (1991) suggest estimating all the parameters in one step. In the one-step procedure, the inefficiency effects are defined as a function of the farm-specific factors and incorporated directly into the maximum likelihood (ML) estimate. This study used the single-step procedure.

In this study, a farm specific stochastic production frontier involving outputs and inputs was defined as follows:

Where is the maximum possible stochastic potential output from the i^th farm; is a vector of m inputs and are statistical random errors assumed to be distributed as . The production realized on the i^th farm can be modeled as follows:

Where is defined as a measure of observed TE of the i^th farm assuming that = 0. When takes the value zero, the i^th farm is technically efficient and realizes its maximum possible potential output. Thus TE can be defined as a ratio between the firm's realized output and the firm's stochastic/potential output as shown in equation 3:

Substituting equation (1) into equation (2) and taking logs on both sides gives:

Where denotes the production of the i^th farm (i = 1, 2,..., n); is a (1 x k) vector of functions of input quantities used by the i^thfarm; â is a (k x 1) vector of unknown parameters to be estimated; are random errors assumed to be independently and identically distributed with and they are independent of the . The is a one-sided error term representing the technical inefficiency (TIE) of farm i.

Subtracting from both sides of equation (4), the production of the i^thfarm can be estimated as:

Where is the natural logarithm of the predicted output of the i^th farm, is the natural logarithm of the i^th input is a set of parameters and is the measure of observed technical efficiency of the i^th farm.

Where is the natural logarithm of the output of the technically efficient farm, is the natural logarithm of the i^th input and is a set of parameters.

Then, from equations (5) and (6), computation of technical efficiency (TE) is given in equation 7:

Arguments in equation 7 are defined in equations 5 and 6. From equation 7, it follows that and when, then = 1 and production is said to be technically efficient.

The distribution of could be half normal with zero mean, truncated normal (at mean, ì), or based on conditional expectation of the exponential (). There are no a priori reasons for choosing a specific distributional form of because each has advantages and disadvantages (Kebede, 2001). The half normal and exponential distributions have a mode of zero, implying that most firms being analyzed are efficient. The truncated normal allows for a wide range of distributional shapes, including non-zero modes, but is computationally more complex (Coelli, 1996).

This study used the technical inefficiency model proposed by Battese and Coelli (1995), and defined the technical inefficiency effects as follows:

Where is a (1 x m) vector of explanatory variables associated with the technical inefficiency effects; ä is an (m x 1) vector of unknown parameters to be estimated; and are unobservable random variables. The parameters indicate the impacts of variables in z on technical inefficiency. The frontier model may include intercept parameters in both the frontier and the model for the inefficiency effects, provided the inefficiency effects are stochastic and not merely a deterministic function of relevant explanatory variables (Battese and Coelli, 1995).

Where is the variance of output conditioned on inputs. This says that the production uncertainty comes from two sources: pure random factors and technical inefficiency. Hence if , the proportion of uncertainty coming from technical inefficiency, is equal to zero, then it actually means there is no technical inefficiency. This can be used to test whether technical inefficiency is present in the firm. Further, the null hypothesis that the impact of the variables included in the inefficiency effects model in equation (8) on the TIE effects is zero is expressed by H0: ä ? = 0 , where ä ? denotes the vector, ä , with the constant term, , omitted (Battese and Broca, 1997).

3.2.1 Model specification

According to Battese and Coelli (1995), the functional form of the stochastic production frontier needs to be specified. In practice, both the Translog and the Cobb-Douglas forms are usually adopted. The Translog form is more flexible in permitting substitution effects among inputs, and is claimed to be a relatively dependable approximation to reality while the Cobb-Douglas form is simple and commonly used.

Kopp and Smith (1980) argue that functional specification has a discernible, though rather small, impact on estimated efficiency. Taylor et al. (1986) also argue that as long as interest rests on efficiency measurement and not on the analysis of the general structure of the production technology, the Cobb-Douglas production function provides an adequate representation of the production technology. Therefore, following Nguyen et al. (1996), a stochastic Cobb-Douglas production function was estimated because of its simplicity. We define the empirical form of the stochastic production function in equation 10:

The variables included in the stochastic production model and their expected signs are summarized in table 3.1.

Variable	Definition	Measurement unit	Effect
Land area planted with maize in season A (September 2008-February 2009)	maizearea	Hectares	+
Household size in season A (September 2008-February 2009)	hhsize	Number of persons in the household	+
Quantity of maize seed* used for maize production in season A(September 2008-February 2009)	seed	Kgs	+
Maize output in season A(September 2008-February 2009)	maizeout	Kgs	Dependent

*Maize seed includes both the improved and local varieties. However, Rwandan farmers generally use the improved variety.

Household size was used as a proxy for labor because larger households are always likely to have many people to participate in agriculture. During the survey, it was found out that hired labor is not so much used. Therefore observations with hired labor as outliers were excluded from the sample. All inputs in the Cobb-Douglas production function are expected to have a positive impact on maize output since an increase in each (or all of) the inputs can lead to increased output.

Where by is an error term which can be assumed to be distributed as truncated normal, half normal or exponential distribution. Note that instead of using indices (such as the Simpson index), single dimension indicators (number of plots per household, average plot size and average distance walked to reach a plot) were used to measure land fragmentation. This allowed for obtaining the explicit effect of each single dimension indicator on productivity and technical efficiency. The variables included in the technical inefficiency model and their expected signs are summarized in table 3.2.

Variable	Label	Measurement unit	Expected sign
Age of household head	Age	Years	+/-
Education level of household head	education	Years spent in school	+/-
Dependency ratio	dependratio	Dependency ratio	+
Number of plots per household	noplots	Number of plots owned by household	+/-
Plot size	plotsize	Hectare	+/-
Average distance from plots to homestead	avplotdist	Kilometers	+
Number of extension visits received by household in season A	Extension	Number of visits	-
Distance to the nearest market center	Distmkt	Kilometers	+
Dummy for land title	dummytitle	D=1 for have title, 0 otherwise	-
Dummy for agro-climatic zone	agroclimate	1 for Bwanamukali, 0 for Mayaga	-
Sex of the household head	Sex	1=Male, 0=female	+/-
TIE	Technical inefficiency		Dependent

Most studies have associated farmers' age and farmers' education with technical efficiency. Farmers' age and education are reported by many studies as having a positive effect on technical efficiency (Amos, 2007; Ahmad et al., 2002; Kibaara, 2005). Age may have a positive effect on technical efficiency if due to experience; older farmers tend to adopt better farming methods than young farmers. A higher level of education can lead to a better assessment of the importance and complexities of production decisions, resulting in better farm management. Educated farmers learn faster and utilize well extension information (Basnayake and Gunaratne, 2002).

In other studies the effect of age and education is ambiguous (Shuhao, 2005). Dependency ratio is reported to have significant negative effects on technical efficiency (Bagamba, 2007) while the farmers' gender (sex) can have ambiguous effects on technical efficiency (Tchale and Sauer, 2007).

Although studies by Amos (2007), Raghbendra, Nagarajan and Prasanna (2005), and Barnes (2008) found the relationship between land holding size and efficiency to be positive, a clear-cut conclusion on the influence of this variable on efficiency has not been reached as discussed in Kalaitzadonakes et al (1992) work. On the other hand, effect of the number of plots on efficiency has been hypothesized to be either negative (Raghbendra et al, 2005) or positive (Marara and Takeuchi, 2003) or ambiguous (Shuhao, 2005). It was hypothesized that the effect of number of plots on efficiency was ambiguous.

Distance from plots to residence is expected to negatively affect efficiency (Byiringiro and Reardon, 1996) Extension visits are expected to increase efficiency and distance to the nearest market is expected to reduce efficiency (Bagamba, 2007). Land ownership rights (possession of land titles) has been assumed to encourage soil conservation investments and therefore expected to increase productivity and efficiency (Musahara, 2006). This study expected agro-climate to have a negative effect on inefficiency since Bwanamukali is more fertile and receives more rainfall than Mayaga.

Age had been expected to have a quadratic effect on technical efficiency. However, results showed that age and its square had the same sign and were both not significant (once the square of age was included in the model) but age was significant (once the square of age was excluded from the model). Thus, the square term of age was excluded from the model.

The analysis of productivity for several crops can be made by regressing marginal value products against farm-specific and household specific characteristics. For a single crop, marginal physical products can be used (Byiringiro and Reardon, 1996). This study used marginal physical products as the dependent variable and all dimensions of land fragmentation as the independent variables. To analyze the productivity of smallholder maize farms, the following double-log regression model was specified:

Where is the natural log of the marginal product of land under maize, is the natural log of the size of the farm owned by a household, is the natural log of the total number of plots owned by the household, is the natural log of the average distance between households residences to plots and is the error term that is assumed to be independently and identically distributed with zero mean and constant variance. The interaction term was included to show what happens when a household has many/few farms that may be distant/close to each other.

The effect of land fragmentation on the productivity and technical efficiency of smallholder maize farms in Southern Rwanda

3.2 Theoretical considerations

3.2.1 Model specification