Assessing production efficiency by farm size in Rwanda: a zero-inefficiency stochastic frontier approach

This study investigates the relationship between farm size and technical efficiency for maize 7 production in Rwanda. Since levels of technical efficiency tend to vary considerably across farms 8 in sub-Saharan Africa, with a mixture of both inefficient and fully efficient farms, the use of the 9 conventional stochastic frontier method is not appropriate. In this paper, we apply a zero- 10 inefficiency stochastic frontier method that manages both efficiency and inefficiency in the 11 studied sample. The average technical efficiency of maize farms for the full sample is estimated 12 at 0.64, demonstrating that maize output can be improved by approximately 36% without 13 increasing the proportion of farm inputs used. Regarding the relationship between farm size and 14 technical efficiency, the study results show a positive relationship between farm size and 15 technical efficiency for maize production in Rwanda. Thus, the enforcement of land reforms such 16 as land consolidation and enhanced aggregate productivity growth are needed. The results also 17 indicate that education, cooperative membership, extension services, access to credit, off-farm 18 income, land tenure, and livestock ownership have significant and positive effects on technical 19 efficiency. 20


Introduction
Increased agricultural productivity has been identified as a major solution that can lead to evidence from the literature indicates the presence of both inefficiency and full efficiency among  This study uses the ZISF model to investigate the relationship between farm size and TE 5 for maize production in Rwanda. In Rwanda's context, only three studies have attempted to 6 investigate the relationship between farm size and productivity (Ali and Deininger 2015; Ansoms  The remainder of this paper is structured as follows. Section 2 presents the background on 12 land problems in Rwanda. Section 3 presents the methods used with a detailed discussion of the 13 conventional SF and ZISF models. Section 4 provides a description of the data and summary 14 statistics. Section 5 presents the empirical results and a discussion. The final section concludes 15 with a summary of the study's findings and policy implications.  18 Land remains a valuable asset that determines to an exceptional level the social status and the  4 Land fragmentation and land scarcity are becoming significant threats to the improvement of 5 agricultural production and food security in Rwanda (Ntihinyurwa et al. 2019). In particular, 6 Rwanda is the most densely populated country in Africa, and a large number of the population 7 (about 83%) live in rural areas (Pritchard, 2013). 1 Rwanda's population pressure has also resulted 8 in smaller plots and fragmented landholdings (Bizoza 2014). Place (2009) argues that land 9 fragmentation constitutes a significant obstacle to agricultural development because it hinders 10 agricultural mechanization tools such as tractors and harvesters. Land fragmentation can also 11 discourage the adoption of irrigation technologies and other long-term investments on land that 12 are only profitable on a larger scale (Tran and Vu 2019). Given that plot size has been 13 diminishing over the years, land distribution in Rwanda is also highly unequal with a large 14 portion of land being owned by a minority of wealthier elite including politicians, business 15 people, and civil servants from urban areas (Musahara and Huggins 2005; Pritchard 2013). 16 Inequality in land distribution is a major policy issue for the government of Rwanda, as it may 17 influence land-related conflicts and poverty (Musahara 2006). 18 Bizoza (2014) also noted that population pressure had induced shifts in land use, resulting 19 in the cultivation of fragile marginal land on steep slopes and deforestation. In general, the 20 Ministry of Agriculture and Animal Resources (MINAGRI) acknowledges that considerable land degradation due to soil erosion has significantly reduced agricultural production (Minagri 2018   Moreover, the high population growth exerts severe pressure on arable land, which is a constraint

Methodology
According to Farrell (1957) classical proposition, TE is one of two components of total economic 1 performance and allocative efficiency (Coelli et al. 1998 in agricultural production studies, as the SF model can address statistical noise (outliers).  Consequently, we adopt the SF model in the present study. 13 14 15 The general form of the conventional SF model is as follows:

Stochastic production frontier model
where represents the output of farm , and (•) is the production function (e.g., Cobb-Douglas 17 or translog). denotes a vector of inputs, and is a vector of unknown parameters to be 18 estimated. Note that composite error term is made up of two components: 2 The TE of a farm reflects its ability to achieve the maximum output possible from a given set of inputs.
where is a random error term representing statistical noise due to unobserved factors beyond 1 the producer's control (e.g., weather fluctuation) and measurement errors. As noted by Coelli et inefficiency error term , which is assumed to follow a positive truncated normal distribution 5 (i.e., ≥ 0) with a mean of and variance of 2 , i.e., ~ + (0, 2 ). 6 The specification of technical inefficiency ( ) can then be written as: where denotes a set of farm-and household-specific covariates, and is a vector of parameters 8 to be estimated.
where represents the output of farm , is a vector of inputs, is a vector of unknown 13 parameters to be estimated, is the probability of a farm being fully efficient, and (1 − ) is the 14 probability of a farm being inefficient. The composed error term in the ZISF model is given by The density function of the convoluted error term of the ZISF model is defined as: where and are the normal probability density and normal cumulative distribution functions, 18 respectively, 2 = 2 + 2 , and = ⁄ .
For the estimation of the inefficiency function in the ZISF model, we adopt the approach 1 developed by Jondrow et al. (1982), which postulates that the conditional density function of 2 inefficiency given is zero with probability and truncated normal + ( * , * 2 ) with 3 probability 1 − . This function is expressed as: where * = − 2 2 ⁄ and * 2 = 2 2 2 ⁄ . From the specification in Equation (6), the 5 conditional mean estimator for inefficiency in the ZISF model is given by: Here, the measurement procedure entails the replacement of unknown parameters with their 7 maximum likelihood (ML) estimates, and error term should be replaced by its residuals ̂. In 8 addition, inefficiency in the ZISF model can be estimated by constructing the posterior estimates 9 of inefficiency, which are expressed as: where ̌ denotes the posterior estimate of the probability of full efficiency, which is written as: These posterior estimates of inefficiency are influenced by farm and household characteristics. 12 To test for zero inefficiency, we use the pseudolikelihood ratio (PLR) test. The PLR test to ensure flexibility in the estimated parameters and consistency with production economic 13 theory, we adopt the Cobb-Douglas functional form, which is specified as follows: where is the production output of maize, is the ℎ input of the ℎ farmer, and and Information on the socioeconomic characteristics of the households and on institutional and farm-20 specific characteristics was also collected. In general, agricultural production in Rwanda is not capital intensive because most of the 4 farmers do not use agricultural machinery (e.g., tractors) in their farming activities. Indeed, the 5 variable inputs included in our analytical model are land, labor, fertilizer, and seeds (see Table 1). 6 The land input is measured as the total farm size in hectares (ha) Table 2 show considerable differences between the three farm size 10 categories. Indeed, the average yield is highest (2285 kg/ha) for large-scale farms while small-11 scale farms have the lowest average yield (1766 kg/ha). In terms of production inputs used per 12 hectare, the average amount of fertilizers used on large-scale farms is higher than that used on 13 medium-and small-scale farms by approximately 9.3% and 22.5%, respectively. Similarly, on 14 average, large-scale farms use more maize seeds per hectare than medium-and small-scale farms. 15 Regarding the average use of human labor, large-scale farms use labor more intensively than 16 medium-and small-scale farms. 17 In our sample, the socioeconomic characteristics of households vary significantly across 18 farm size categories. The summary statistics reported in Table 2 indicate that educational 19 attainment is generally low (i.e., below six years of primary education). By comparison, large-  Table 2). Moreover, membership in agricultural cooperatives is higher among large-scale farmers 1 (62%) than for medium-(53%) and small-scale farmers (36%). In addition to the above variables, 2 other variables such as livestock ownership, land tenure, slope, and off-farm income exhibit clear

Results and Discussion
3 Table 3 presents parameter estimates of the production frontier and inefficiency effect functions 4 of the conventional SF and ZISF models. Table 3 also reports the results of the model 5 diagnostics. The PLR test result is statistically significant at the 1% level, implying the presence 6 of both inefficient and fully efficient farms in our sample. Additionally, the probability of being 7 fully efficient is 12.1% and statistically significant at the 1% level. Results of maximum 8 likelihood estimates of the coefficients for the stochastic production function indicate that the 9 parameters for all input variables are statistically significant, and the signs of all input 10 coefficients are positive, as expected, for both the conventional SF and ZISF models (see Table   11 3).

Partial production elasticities and returns to scale 2
In our empirical analysis, we estimate partial production elasticities to assess the responsiveness 3 of maize yields to varying levels of each of the classical inputs, ceteris paribus. The estimated 4 coefficients of the Cobb-Douglas production function are directly read as partial production 5 elasticities for the inputs (Coelli et al. 1998). The results listed in Table 3 indicate that maize 6 yields are more responsive to fertilizer inputs than to other production inputs (i.e., seeds, land,

Determinants of technical efficiency
22 Table 3 provides the coefficient estimates of factors affecting technical inefficiency. We analyzed

Technical efficiency estimates 1
The summary statistics of TE estimates for the full sample and for the three farm size groups are 2 reported in Table 4. The average TE score for all households in the sample is estimated at 0.64.

3
As we have estimated the output-oriented TE, the mean TE score of 0.64 implies that the maize sample of maize producers in Malawi.

10
Results for the relationship between farm size and TE are also presented in Table 4. The  In this study, we assessed the relationship between farm size and technical efficiency for the 4 maize production sector in Rwanda. In addition, we analyzed potential determinants of technical 5 efficiency for maize production systems in Rwanda. We employed the ZISF model for analytical 6 purposes and a cross-sectional dataset consisting of a sample of 351 household farmers operating 7 in the Eastern Province of Rwanda. The ZISF model is more advantageous than the conventional 8 SF model due to its ability to account for both inefficiency and full efficiency in a sample. 9 Our empirical results reveal that all key production inputs, i.e., fertilizer, labor, seeds, and 10 land, are statistically significant and have a positive effect on maize output. In particular, we 11 found that maize output is more responsive to fertilizer input than to other production inputs (i.e., 12 seeds, land, and labor). In terms of determinants of technical inefficiency, we found that  Data used for this study will be made available from the corresponding author up on request.

22
This study is a part of the Doctoral thesis by the first author. We thank the government of the