Assessing the Technical Efficiency and Resilience of Rwandan Smallholder Farmers Participating in Small-Scale Irrigation Schemes

Abstract

In a number of developing countries, low productivity and technical inefficiency, with climate change looming in the background, remain a severe challenge for the agricultural sector, especially smallholder farmers. To enhance smallholder farmers’ livelihoods in terms of agricultural productivity while mitigating the adverse effects of climate change, improving technical efficiency in a sustainable manner is a promising option. One possible alternative is the use of solar-powered small-scale irrigation systems in areas vulnerable to climate change to ensure sufficient access to water. This study uses stochastic frontier analysis to analyze technical efficiency and its determinants among smallholder farmers who benefit from a solar-powered small-scale irrigation scheme in Gitaraga, Bugesera District, Rwanda. Similar smallholders from a neighboring village, who were not participating in the irrigation scheme, represent the control group. The results suggest that inputs such as land, water, and labor are positively correlated to agricultural productivity. Farmers participating in the irrigation scheme are 31.2 percentage points more technically efficient compared to non-participants, despite similar climatic conditions. Thus, relaxing the water constraint on arable land will increase agricultural productivity. Explanatory inefficiency determinants include years of farming experience and market access. Subsequently, policy makers should continue to support programs that improve smallholder access to sustainable irrigation schemes, other infrastructure, extension services, and upstream value chains, as well as markets.

1. Introduction

The agricultural sector in sub-Saharan Africa is crucial in achieving food and nutritional security, poverty alleviation, and the sustainable development goals (SDGs) [1,2,3]. However, the agricultural sectors face severe challenges stemming from external shocks such as climate change, pandemics like COVID-19, and international conflicts, e.g., the Ukraine–Russia war [4]. Muneza [5] argues that climate change will result in elevated average annual temperatures, prolonged dry seasons, and irregular rainfall in Rwanda. Ngango and Hong [6] associate unpredictable rainfall and more frequent droughts with low farm productivity in Eastern Africa. Low agricultural productivity is further exacerbated by a lack of adoption of improved inputs such as fertilizers, seeds, pesticides, irrigation schemes, credit, and extension services as well as other technologies such as renewable energy [1,2,3,6]. In Rwanda, 70% of the working-age population works in the agricultural sector, which is characterized by low productivity, leading to a high incidence of poverty and food insecurity [7]. The headcount ratio reveals that 57% of Rwanda’s population lives below the International Poverty Line of USD 1.90 in purchasing power parity per day, which is considerably higher than the sub-Saharan Africa average of 44% [8]. The Global Hunger Index (GHI) score for Rwanda in 2021 was 26 out of 50, ranking 98th out of 116 countries, which indicates a substantial incidence of hunger. Malnutrition, expressed by the percentage of children under the age of five who suffer from stunting in Rwanda, lies at 38%, above the world average of 25% and the sub-Saharan African average of 34% [8].

Studies reason that agricultural sector productivity reflects technical efficiency [4,9,10]. Technical efficiency can be understood as the ratio of the observed output to the maximum achievable output obtainable from the given inputs, or the ratio of the observed input to the minimum achievable input required to produce the output. Mulinga [11], Maniriho and Bizoza [12], and Ntabakirabose [13] argue that low technical efficiency in Rwanda is due to underutilization of high-quality inputs and technologies. Thus, enhancing technical efficiency in smallholder farming in a sustainable way is paramount, as it reduces wastage of productive resources and thus brings smallholders closer to their food production possibility frontiers [9]. This will improve smallholder livelihoods while reducing their vulnerability to external and internal shocks.

Unsurprisingly, the use of irrigation technologies is widely recognized as an effective measure in improving agricultural productivity among smallholder farmers in sub-Saharan Africa because it builds resilience to the issue of erratic rainfall and drought. Furthermore, the use of irrigation ensures all-year-round crop production in the tropics and sub-tropics irrespective of season. However, investments in irrigation technologies, especially small-scale solar off-grid systems, suffer from high up-front costs and limited load usage [14]. It is also necessary to explore the use of renewable energy as a source of power for irrigation, and move away from conventional fossil fuel, in an attempt to mitigate climate change. This could be in the form of agrophotovoltaics with solar installations. Ngango and Hong [6] assessed the impact of small-scale irrigation technologies on farm productivity and observed higher yields for small-scale irrigators relative to non-irrigating farmers in Rwanda. Despite the positive results, a mere 18,000 ha out of 589,000 ha of land is enrolled in irrigation schemes, prompting the Rwandan government to subsidize small-scale irrigation technologies (SSITs) since 2014 [6]. This creates the need for more studies highlighting the potential of smallholder irrigation technologies, especially given its potential to sustainably improve smallholders’ technical efficiency and livelihoods.

This study aims to contribute to the literature on smallholder agricultural productivity in sub-Saharan Africa by analyzing the impact of small-scale irrigation on the technical efficiency of smallholder farmers. Furthermore, its insights are useful in understanding the deployment of renewable energy such as agrophotovoltaics in powering irrigation systems, given access to sustainable water resources (i.e., not depleting groundwater resources).

Methodologically, the analysis is based on a stochastic frontier approach (SFA) and a unique dataset, relating to 50 smallholder farmers participating in the solar-powered small-scale irrigation project in Gitaraga (irrigators) and 30 non-participating farmers (non-irrigators) from the nearby village of Gatwe in Rwanda. The survey took place between September and November 2021. The solar-powered Rwanda Small-Scale Irrigation Project is implemented by TU eMpower Africa e.V. “see: https://tu-empower-africa.org/ (accessed on 24 November 2024)”, a non-profit organization associated with the Technical University of Munich (TUM), Germany. Through the installation of a solar-powered irrigation system, the project aims to mitigate the adverse impacts of low and erratic precipitation, thus improving technical efficiency, which will increase agricultural production and resilience and ultimately reduce food and nutrition insecurity. The results suggest that the average technical efficiency of participating and non-participating smallholder farmers in the solar-powered small-scale irrigation schemes was 74.5% and 43.3%, respectively. This implies that the former group is more technically efficient than the latter.

The paper is structured as follows. Section 2 presents the materials and methods used for the analysis. Section 3 presents the results of the analysis. Section 4 provides an insightful discussion of the results before the paper is concluded in Section 5.

2. Materials and Methods

2.1. Study Area

Primary data were collected in the Eastern Province of Rwanda. Rwanda is Africa’s second-most densely populated country, with 538 inhabitants per km² and a population of 13.3 million with an annual growth rate of 2.31% in 2021 [7]. Approximately 20% of the population lives in the capital of Kigali and other cities, while 80% reside in rural areas. Approximately 89% of rural households are employed in the agricultural sector [15]. Rwanda records average temperatures between 16 °C and 24 °C and annual rainfall is between 2000 mm in the northwest and 400 mm in the east. Rwanda has two rainy seasons from February to May and mid-September to December and two dry seasons from June to mid-September and January to February. The soils are typically fragile and acidic, which limits the choice of crops for cultivation.

The Eastern Province is located in the country’s low-altitude zone, at approximately 1400 m. Within the Eastern Province, the survey sites (Gitaraga and Gatwe) are situated in the Bugesera District, which extends south of Kigali and is characterized by extensive swamps and marshlands (see Figure 1). With a semi-arid climate and average daily temperatures between 25 °C and 30 °C, as well as low average annual precipitation of 400 mm to 800 mm, the region is regarded as the hottest and driest in the country. Therefore, it is prone to droughts, with increasing water shortages due to unpredictable rainfall [16].

2.2. Rwanda Small-Scale Irrigation Project

TU eMpower Africa e.V. implemented the Rwanda Small-Scale Irrigation Project in the village of Gitaraga, Bugesera District from October to December 2021. This project installed an 11.7 kW photovoltaic system, powering two pumps to draw water from a nearby swamp into water storage consisting of a pond and a water tank on top of the hill for irrigating the local farmland with no adverse environmental effects on groundwater levels. Figure 2 and Figure 3 show the photovoltaic system. Due to the hill’s slope of about 10%, a gravity irrigation system is in place, with pipes distributing the water from the pond and/or tank to a central point in the fields.

2.3. Sampling Technique and Sampling Size

Primary and secondary data were collected through survey work as well as desk research, respectively. Between September and November 2021, treatment and control groups were surveyed to generate primary quantitative data in Bugesera District of the Eastern Province, Rwanda. The treatment group comprised 50 small-scale farming irrigators, who represented 100% of the Rwanda Small-Scale Irrigation Project in Gitaraga. All of them were also members of the Cooperative Ubumwe Iterambere Abahinzi Mwogo (CUBIAM). A total of 2 observations had to be dropped from Gitaraga due to incomplete or missing information.

The control group, i.e., farmers from Gatwe, who did not participate in the solar-powered irrigation scheme showed similarities with the treatment group in terms of location, distance to the marshland, and socio-demographics. These similarities make the two groups comparable, which was considered crucial to minimize the risk of bias in the observation [18]. While the treatment group represented 100% of the farmers receiving access to the newly installed solar-powered irrigation system, the sample of the control group was derived through a non-probability sampling technique (purposive sampling) based on expert opinion [19,20]. Mosley [18] argues that such techniques may result in sample bias, and include motivational problems, leading to ambiguous or wrong answers. To mitigate this risk, Mosley [18] suggests working with a skilled and sympathetic enumerator. Thus, a college-educated, experienced local enumerator was contracted for this study. For non-probability sampling techniques, there are no defined rules concerning sample sizes [20]. Secondary data were retrieved from peer-reviewed academic articles for data complementarity.

2.4. Stochastic Frontier Analysis Model

This study uses a stochastic frontier analysis (SFA) approach (see Coelli et al. [21]) to estimate technical efficiency levels among smallholder farmers participating in a small-scale irrigation scheme and a control group of non-participating farmers. According to Kumbhakar, Wang, and Horncastle ([22], p. 30), a production function can be defined as a “mathematical representation of the technology that transforms inputs into output”. SFA is a parametric stochastic approach and differentiates between the impact on agricultural output by random shocks and inefficiencies in production. This differentiation makes the SFA a common tool for estimating technical efficiency in production economics compared to other methods such as Data Envelopment Analysis (DEA), which regards all deviation from the production frontier as inefficiency. The production function for a single output can be written as:

$$y=f\left(x_1, x_2,\dots \dots ,x_i\right)\equiv f\left(\mathrm{x}\right)\mathrm{with} i=1, \dots , m$$

(1)

where the function $f(.)$ postulates the technology for the input–output relationship and is characterized as finite, real-, and single-valued. The variable $x_i$ represents the vector of unknown input parameters ($i=1, \dots , m$). The formal representation of the stochastic frontier production function is written as:

$$\ln y_j=x_i\mathsf{\beta }+\mathsf{\epsilon }$$

(2)

where $y_j$ is the output of the j-th observation (i.e., decision-making unit, in this case a smallholder farmer), $x_i$ is the vector of the applied inputs, and β is a vector of unknown input parameters. A frontier production function is estimated for all decision-making units (j $=1, \dots , N$) in the regression analysis. A disturbance term, $\epsilon$, equal to the deviation from the production possibility frontier, is determined for each observation $j$. The disturbance term $\epsilon$ is divided into the two error components, $v_j$, which captures the random shocks, and $u_j$, which reflects the technical inefficiency term [21]. The error term is written as:

$${\epsilon }_j=v_j-u_j$$

(3)

Equations (2) and (3) result in:

$$\ln y_j=x_i\mathsf{\beta }+v_j-u_j$$

(4)

The random error component $v_j$ can take on both positive and negative values, which is why the output can fluctuate above and below the frontier. These random fluctuations include weather effects. However, the inefficiency error component $u_j$ can only take values greater than or equal to zero. Figure 4 illustrates the scenarios for $v_j$ and $u_j$, for which the functional form first has to be selected. There are various functional or algebraic forms which depict relationships between economic variables, such as the Cobb–Douglas or the translog forms. The latter is a second-order flexible form, meaning it has sufficient parameters to provide a second-order differential approximation to any function at a single point. According to Coelli et al. [21], the model with the higher log-likelihood value should be used, which can be retrieved by conducting a likelihood-ratio test.

In the example from Figure 4, a simple input–output Cobb–Douglas production function is depicted and written as:

$$\ln y_j={\beta }_0+{\beta }_1\mathrm{l}\mathrm{n}{x}_i+v_j-u_j$$

(5)

or

$$y_j=\exp \left({\beta }_0+{\beta }_1\mathrm{l}\mathrm{n}{x}_i+v_j-u_j\right)$$

(6)

or

$$y_j=\exp \left({\beta }_0+{\beta }_1\mathrm{l}\mathrm{n}{x}_i)·\exp \left(v_j\right) ·\mathrm{e}\mathrm{x}\mathrm{p}(-u_j\right)$$

(7)

The frontier in Figure 2 assumes diminishing returns to scale, and two representative decision-making units A (B) are considered to generate output $y_{iA}$ ($y_{iB}$) with input $x_{iA}$x ($x_{iA}$), respectively. The input is displayed on the x-axis and the output on the y-axis. Without any inefficiency effects, $u_A=0$ and $u_B=0$, the frontier output would be as follows:

$$y_{A|B}^{\ast }=\exp \left({\beta }_0+{\beta }_1\mathrm{l}\mathrm{n}{x}_{iA|B}+v_{A|B}+u_{A|B}\right)$$

(8)

In this scenario, the frontier output for A would be above the deterministic production function (point 1 in Figure 2), and the frontier output for B would be below it (point 3 in Figure 2). Points 2 and 4 in Figure 2 include inefficiencies in the calculation for A and B, respectively, and, thus, can be expressed by the following equation:

$$y_{A|B}^{\ast }=\exp \left({\beta }_0+{\beta }_1\mathrm{l}\mathrm{n}{x}_{iA|B}+v_{A|B}-u_{A|B}\right)$$

(9)

The observed output can only be above the frontier if the random error component $v_j$ is positive and larger than the inefficiency error component $u_j$. At point 2, the actual output of A lies below the deterministic production function, since the sum of the random error $v_A$ and the inefficiency error component $u_A$ is negative ($v_A-u_A<0$).

In output-oriented terms, technical efficiency can be defined as the ratio of the observed output $\widehat{y_j}$ of the $j$-th observation to the maximum possible stochastic frontier output, and written as:

$${TE}_j={\displaystyle \frac{\widehat{y_j}}{\exp \left({\beta }_0+{\beta }_1\mathrm{l}\mathrm{n}{x}_i\right)\ast \exp \left(v_j\right)}}$$

(10)

or

$${TE}_j={\displaystyle \frac{\exp \left({\beta }_0+{\beta }_1\mathrm{l}\mathrm{n}{x}_i)\ast \exp \left(v_j\right) \ast \mathrm{e}\mathrm{x}\mathrm{p}(-u_j\right)}{\exp \left({\beta }_0+{\beta }_1\mathrm{l}\mathrm{n}{x}_i\right)\ast \exp \left(v_j\right)}}$$

(11)

or

$${TE}_j=\mathrm{e}\mathrm{x}\mathrm{p}(-u_j)$$

(12)

The technical efficiency of the decision-making units (i.e., the smallholder farmers) can only take values between one and zero, with the value of one being considered fully technically efficient. Since the calculation of technical efficiency is based on the stochastic frontier output and not on the deterministic frontier output, technical efficiency cannot adopt values greater than one even if the observed output is above the deterministic frontier. Before estimating technical efficiency, which is one of the objectives of this study, the unknown parameters β of the stochastic production function (Equation (2)) must be determined. To estimate the combined error term, an a priori assumption is made about the distribution of the two error components, $u_j$ and $v_j$. For the inefficiency component u, relatively simple distributions are frequently used, such as the half-normal or the truncated normal distribution. For the error component, v, assumptions are that it is typically distributed with zero mean and the variance ${\sigma }_v^2$. Beyond that, it is assumed that $u_j$ and $v_j$ are independent of each other and identically distributed across all decision-making units, i.e., smallholder farmers [21].

2.5. Inefficiency Effects Model for Cross-Sectional Data

For efficiency analyses, it may not only be of interest to assess the pure levels of efficiency or inefficiency but also to understand the influencing factors. Reifschneider and Stevenson [23] wrote one of the early studies analyzing the relationship between the inefficiency degree and the potential factors of inefficiency, i.e., inefficiency explanatory variables. A SFA estimates these explanatory variables in two steps. In the first step, the production frontier and the technical efficiency are estimated. Then, in the second step, the influence of exogenous variables on the inefficiency is examined. However, this two-step procedure has been recognized as biased due to contradictory assumptions of identically distributed inefficiency effects in the frontier [22]. More recently, a one-step estimation method has become common, which simultaneously estimates the parameters of the stochastic frontier (β-parameters) and the parameters of the inefficiency model (δ-parameters). The δ-parameters are used to show the influence of the explanatory variables (z-variables) on the inefficiency. Positive values of the δ-parameters indicate that the associated explanatory variable negatively influences technical efficiency. There are different techniques to estimate the parameters, such as the corrected ordinary least squares (COLS) model or the maximum likelihood estimation (MLE). MLE is often the preferred option [21], which is why this technique is used in this analysis. The inefficiency effects model makes it possible to examine the influence of exogenous variables on inefficiency for cross-sectional data. Thus, based on the cross-sectional data nature of this study, the reduced form regression is written as:

$$y_j=\mathrm{e}\mathrm{x}\mathrm{p}({\beta }_1\mathrm{l}\mathrm{n}{x}_i+v_j-u_j)$$

(13)

The parameters are similar to those in Equation (4). $v$ represents the random shock and is assumed to be normally distributed across decision-making units $j$ with zero mean and the variance ${\sigma }_v^{2}N(0,{\sigma }_v^2)$. $v$ is independent of $u.$ $u$ represents the non-negative random technical inefficiency and can be assumed to be half normally distributed with the ${mean}_j=z_{kj}\delta$, (with $k=1,\dots ,o$) and the variance ${\sigma }^{2}N({mean}_j,{\sigma }^2)$. $z_{kj}$ represents a vector of inefficiency explanatory variables of each decision-making unit, whose impact on technical inefficiency is assessed. δ represents a vector of unknown coefficients of $z_k$. ${\sigma }^2$ is written as:

$${\sigma }^2={\sigma }_u^2+{\sigma }_v^2$$

(14)

while

$$\gamma ={\displaystyle \frac{{\sigma }_u^2}{{\sigma }_u^2+{\sigma }_v^2}}$$

(15)

γ indicates what share of the overall variance can be ascribed to the inefficiency component $u_j$. The model assumes a functional relationship between technical inefficiency $u_j$, the inefficiency explanatory variables $z_k$, and its vector of coefficients δ, which can be written as:

$$u_j=z_{kj}\delta +w$$

(16)

where $w$ represents the latent random variables. Equation (12), the technical efficiency (TE) for the $j$-th observation (i.e., DMU, in this case a smallholder farmer) is defined by the following equation and written as:

$${TE}_j=\exp \left(-u_j\right)=\mathrm{e}\mathrm{x}\mathrm{p}(-z_{kj}\delta -w)$$

(17)

2.6. Statistical Description

The SFA is based on eight inefficiency explanatory variables ($z_k$), four input variables ($x_i$), and one output variable ($y_j$); see Figure 5. The input variables, the explanatory variables, and the output variables are presented in

Table 1. Description of input variables ($x_i$).

		Gitaraga			Gatwe
	Mean	Min	Max	Mean	Min	Max
Land (m2)	1862	100	13,000	2640	100	12,500
Labor (RWF)	54,596	1	291,601	6614	1	80,001
Water (RWF)	125,246	1	528,001	54,641	1	422,401
Seeds (RWF)	3672	1	27,001	4829	1	24,001

Notes: Local currency and used monetary unit Rwandan francs (RWF), 1 EUR = 1160.78 RWF [20].

,

Table 2. Description of inefficiency explanatory variables ($z_k$).

Gitaraga [1] (n = 48)
	Mean	sd	Min.	Max.
Gender (1 = male, 0 = female)	0.58	0.49	0	1
Market (1 = sales, 0 = own consumption)	0.68	0.46	0	1
Age (in years)	41.00	12.00	22	79
Experience (in years)	20.00	16.00	1	67
Family size (number of members)	5.00	3.00	1	16
Education (in years)	5.00	3.00	0	12
Extension (days of training)	3.00	13.00	0	78
Gatwe [0] (n = 30)
	Mean	sd	Min.	Max.
Gender (1 = male, 0 = female)	0.53	0.50	0	1
Market (1 = sales, 0 = own consumption)	0.73	0.44	0	1
Age (in years)	42.00	13.00	23	84
Experience (in years)	15.00	15.00	1	60
Family Size (number of members)	5.00	2.00	2	14
Education (in years)	5.00	3.00	0	6
Extension (days of training)	0.96	2.00	0	10

Notes: Local currency and used monetary unit Rwandan francs (RWF), 1 EUR = 1160.78 RWF [20]; two observations from Gitaraga (n = 50) were dropped due to incomplete or missing information.

, and

Table 3. Description of output variables.

Monetized Yield (in RWF)	Gitaraga (n = 48)			Gatwe (n = 30)
	Mean	Min.	Max.	Mean	Min.	Max.
Agricultural production	341,365	7000	1,964,000	119,943	1800	550,000
Crop production	271,159	7000	1,710,000	111,443	1800	540,000
Livestock	70,206	0	507,000	85,000	0	100,000

Notes: Local currency and used monetary unit Rwandan francs (RWF), 1 EUR = 1160.78 RWF [20]. Two observations from Gitaraga (n = 50) were dropped due to incomplete or missing information.

, respectively.

The SFA consists of four input variables ($x_i$), eight inefficiency explanatory variables ($z_k$), and one output variable ($y_j$) (see Figure 5). The inputs and inefficiency explanatory variables are described in detail in subsequent sections. The variables were selected based on the challenges of the Rwandan agricultural sector and follow other efficiency analyses identified in sub-Saharan Africa [24,25,26].

2.7. Input Variables ($x_i$)

Table 1 presents details of the inputs and respective production factor variables. Land refers to both irrigated and non-irrigated arable land measured in square meters (m²). On average, the farm size in Gitaraga was 1862 m² (0.2 ha), 29% smaller than farms in Gatwe, where the average size was 2640 m² (0.3 ha). Most farmers in this sample can be termed smallholders as they cultivated less than 2 hectares (ha). Farmers hire daily labor to support them in seeding and harvesting. The average annual expenses for hired laborers were RWF 6614 (EUR 5.70) in Gatwe, and eight times higher in Gitaraga, with an average of RWF 54,596 (EUR 47). The water variable denotes the annual expenditure that 56% of farmers in Gitaraga and 37% of farmers in Gatwe pay for operating a water pump fueled by diesel. The water pumps are used to distribute water to the fields through pipes. Typically, these water pumps are used for a period of 24 weeks per year during the dry seasons. In Gitaraga, three (6%) of the 48 treatment farmers in the final sample owned water pumps, which they also rented out to their peers. In Gatwe, none of the 30 farmers owned such a water pump, but eleven farmers (37%) regularly rented water pumps from service providers in nearby towns. The farmers bear the rent and the costs for the petroleum to power the pumps. The price of petroleum in Gitaraga and Gatwe was RWF 1175 (EUR 1.01) and RWF 1200 (EUR 1.03) per liter, respectively. In Gitaraga, the average yearly expenses for irrigation were RWF 125,246 (EUR 107) compared to RWF 54,641 (EUR 47) in Gatwe. About 77% and 17% of farmers in Gitaraga and Gatwe, respectively, produced their own seeds. Thus, the annual average seed costs in Gitaraga were lower (RWF 3672; EUR 3.16) compared to Gatwe (RWF 4829; EUR 4.16).

2.8. Inefficiency Explanatory Variables ($z_k$)

Table 2 describes the explanatory variables for production inefficiency. Location is a dummy variable, with one for the treatment group in Gitaraga and zero for the control group in Gatwe (see Figure 5). A total of 48 (62%) of the farmers were from Gitaraga and 30 (39%) from Gatwe. Men account for 58% and 53% of surveyed farmers in Gitaraga and Gatwe, respectively. In Rwanda, 72% of the farm households are male-headed [27]. Market access was included as a dummy variable, with one signifying that farmers had access to a spot market or retailers and zero the opposite. Thus, zero implies that the farmers adhere to subsistence agriculture. A higher number of farmers in Gitaraga practiced subsistence agriculture compared to those in Gatwe. The mean age of respondents from both villages was similar, at 42 years. In Rwanda, 77% of smallholders are aged 35 to 54 years, making it the predominant age group for farmers [28]. Nevertheless, Rwanda’s population is young, with an average of 20 years, while only 5% are older than 60 years [29]. Farming experience, measured in years, served as a proxy for knowledge, with respondents in Gitaraga having approximately five years more experience than those in Gatwe. On average, farming households in the sample had more than five members, which is close to the average in the east with 5.5 members. Education refers to the number of years spent in a formal schooling system. The sampled farmers had, on average, five years of schooling. Extension service was measured as the number of days farmers had received some form of training, e.g., in seed production, agricultural techniques, and postharvest handling/management. Farmers in Gitaraga had had three days of training, compared to those in Gatwe with just one day.

Farmers participating in the irrigation scheme in Gitaraga, on average, cultivated two crops, while the non-participants in Gatwe were more specialized, cultivating an average of one crop. Farmers in Gitaraga and Gatwe were observed to cultivate crops such as beans, eggplants, maize, soybeans, cabbage, rice, beetroot, amaranth, taro, tomatoes, bell peppers, chilies, sorghum, groundnuts, sweet potatoes, cassava, peas, and bananas.

2.9. Output Variable ($y$)

A single output variable, namely agricultural production, was estimated as the total of the gross income in RWF from all crop and livestock production for each smallholder farmer. Kumbhakar et al. ([22], p. 61) and Coelli et al. ([21], p. 156) endorse this option of combining several outputs into one variable if they can be reasonably grouped. The aggregated output variable can simply be written as:

$$y_{j total }=y_{j crop production }+y_{j livestock production }$$

(18)

where $y_{j crop production}$ and $y_{j livestock production}$ represent the products of crop and livestock production in quantities multiplied by their respective market price for each smallholder farmer. Table 3 indicates that crop production had a higher revenue stream compared to livestock production in Gitaraga and Gatwe. The annual gross income from agricultural output of the treatment group from Gitaraga was almost three times (RWF 341,365; EUR 294) that of the control group from Gatwe (RWF 119,943; EUR 103).

3. Results

3.1. Stochastic Frontier Estimation

The stochastic frontier parameter estimation determines the factors influencing the output, i.e., the effect of the input variables on agricultural production, with all parameters being logarithmized.

Table 4. Stochastic frontier model of agricultural production based on the input variables ($x_i$) using translog.

Variable	Coefficient	Standard Error
Land (m2)	0.264 ***	0.10
Water (RWF)	0.061 ***	0.02
Labor (RWF)	0.055 **	0.03
Seed (RWF)	0.025	0.03
Constant	9.280 ***	0.70
N	78

Notes: *** p < 0.01, ** p < 0.05; two observations from Gitaraga were dropped due to incomplete or missing information.

provides an overview of the translog model. We note that all results should be interpreted as correlatives since the study design does not allow for establishing causality.

The results in Table 4 indicate that inputs used in the production function are inelastic, implying that a 1 percent increase in all inputs will lead to a less than 1 percent increase in agricultural output. The positive output elasticity for land is the highest and implies that the larger the farm size, the higher the output. More precisely, an increase in land area by one square meter will raise agricultural production by approximately 0.26 percent, ceteris paribus. Similarly, inputs such as labor, water, and seed positively correlate to output, with seed being the only statistically insignificant variable.

3.2. Heteroscedasticity

The regression model follows the homoscedasticity assumption that the variance of the composed error term $\epsilon$ is estimated to be ${\sigma }^2$. If this rule is violated, the composed error term does not have a constant variance, and the model becomes heteroscedastic, leading to invalid results of significance. The Breusch–Pagan test is used in this study for variables in the stochastic frontier model of agricultural production. The test reveals a p-value of 0.47, which is greater than 0.05. Thus, the null hypothesis of a constant variance cannot be rejected. This implies the presence of homoscedasticity, as observed in Figure 6.

3.3. Multicollinearity

Multicollinearity is a situation in which two or more independent variables are highly correlated. Multicollinearity can be quantified for each independent variable using the Variance Inflation Factor (VIF). A VIF above ten indicates multicollinearity between the parameters [30]. The mean VIF estimated for the present data was 1.96, far below the critical threshold of ten. For instance, the VIF for age and experience were 4.39 and 4.61, respectively, which are also below the critical threshold of ten. Therefore, both variables were eligible to be included in the model.

3.4. Technical Efficiency

This study found that 88% of farmers in Gitaraga (treatment location) and 30% of farmers in Gatwe (control location) were operating at a technical efficiency of above 80%. The combined technical efficiency of farmers in both groups was 62.5%. The mean technical efficiency of smallholders participating in the small-scale irrigation scheme in Gitaraga was 74.5%, which is 31.2 percentage points higher than that of the control group in Gatwe, with 43.3% technical efficiency. This result suggests that farmers in the treatment group who allocate their resources efficiently can produce 25.5% more output with the same inputs. The results of the technical efficiency analysis are presented in

Table 5. Technical efficiency of farmers from the Gitaraga, Gatwe, and combined groups.

	Combined		Gitaraga		Gatwe
TE Range (%)	Frequency		Frequency		Frequency
	Absolute	Percent	Absolute	Percent	Absolute	Percent
0	5	6.41	-	-	5	16.67
20	9	11.54	2	4.17	7	23.33
40	8	10.26	3	6.25	5	16.67
60	5	6.41	1	2.08	4	13.33
>80	51	65.48	42	87.50	9	30.00
Total	78	100.00	48	100.00	30	100.00
Mean TE	62.5%		74.5%		43.3%
Std. Dev.	26.8%		15.2%		30.2%
Min. TE	0.0		0.2		0.0
Max. TE	0.8		0.8		0.8

Note: TE = technical efficiency; two observations from Gitaraga were dropped due to incomplete or missing information.

.

3.5. Determinants of Technical Inefficiency

The explanatory variables in the technical inefficiency model were regressed against the dependent variable, namely the technical inefficiency term, $u_j$, to analyze the factors influencing technical efficiency, and the results are presented in

Table 6. Determinants ($z_k$) of technical inefficiency in agricultural production among farmers from Gitarage and Gatwe in Rwanda.

Variable	Coefficient	Standard Error
Location (1 = Gitaraga, 0 = Gatwe)	−1.40 ***	0.47
Age (in years)	0.11 ***	0.04
Experience (in years)	−0.06 **	0.03
Gender (1 = male, 0 = female)	−1.30 **	0.55
Family Size (members)	−0.16 ***	0.10
Education (in years)	0.23 **	0.10
Market (1 = sales, 0 = own consumption)	−1.21 **	0.47
Extension (days of training)	0.02	0.02
Constant	9.28 ***	0.7
Sigma $\left(\sigma u\right)$	−4.36	2.79
Sigma $\left(\sigma v\right)$	−0.29 *	0.16
N	78

Note: See Table 1 for a description of the variables. Two observations from Gitaraga were dropped due to incomplete or missing infromation *** p < 0.01, ** p < 0.05, * p < 0.1.

. There is an inverse relationship between the explanatory variables and technical efficiency: positive values of the variables’ coefficients signify a negative effect on technical efficiency.

The results indicate that farmers with access to the irrigation scheme in Gitaraga (treatment location) exhibit higher levels of technical efficiency than those in Gatwe (control location). Older farmers are less efficient, which may be a result of reduced physical productivity and/or a general aversion vis-à-vis innovation adoption. However, experience in certain farming practices can reduce technical inefficiency. The gender variable indicates that women were less technically efficient compared to men. A possible reason could be that women, relative to men, have minimal access to the extension training services necessary for improved agricultural production. Family size can contribute to technical efficiency, as those with more family members are more technically efficient, while education does not influence technical efficiency. However, as the smallholder farmers had only had 5 years of formal school education on average, this level of schooling may have been insufficient to impact their technical efficiency. Market access variable, indicating spot market participation or sales to retailers, are all positively correlated with technical efficiency. Thus, farmers who are integrated in the market are more technically efficient than subsistence farmers. Revenues from market integration could improve their willingness and capability to adopt innovations.

4. Discussion

A stochastic frontier model was used to estimate the technical efficiency of farmers from Gitaraga, who participate in a solar-powered small-scale irrigation, and farmers from Gatwe, who constitute the control group. On average, small-scale irrigating farmers using solar energy to power irrigation in Gitaraga were more technically efficient than farmers in Gatwe. The mean technical efficiency for Gitaraga was 74.5% compared to 43.3% for Gatwe. The combined mean technical efficiency for both farmer groups was 62.5%, leaving substantial room for bringing smallholder farmers closer to their production frontier.

In this study, several input parameters affect agricultural output. The size of the farm in terms of arable land has a positive relationship to output. This result is in line with findings by Akamin et al. [31], who analyze the technical efficiency of vegetable farmers in Cameroon. Ngango and Hong [32] reported a positive correlation between farm size and output for maize farmers in Rwanda. While these studies found a positive association between increasing farm size and agricultural output, Ngango and Hong [32] propose land consolidation to increase production based on economies of scale (also see [33,34] for examples from other regions in sub-Saharan Africa). However, Ingabire [35] advocates for improving technology adoption in Rwanda to increase productivity rather than solely expanding arable land size, especially as Rwanda is already one of the most densely populated countries in Africa.

This study accounts for hired labor costs, which is a more accurate measurement of farm labor [21], and found that output was positively affected by increasing hired labor. Theriault and Serra [36] confirm these findings for cotton and maize producers in Mali, Burkina Faso, and Benin. Ngango and Kim [37] also observed a positive relationship between hired labor and output for small-scale coffee farmers in the Northern Province of Rwanda.

The results of this study suggest that higher expenditures for irrigation (as a proxy for more irrigation) are associated with higher output. Kelemework [38] found a positive relationship between water and output too, measured in the number of times farmers irrigate their farmland, in the context of production in Ethiopia. Similarly, Hengsdijk et al. [39] reported that small-scale irrigators had higher yields relative to non-irrigating farmers. This may explain why the government of Rwanda started to promote small-scale irrigation schemes over the last decade within its agricultural policy initiatives [15]. Yet, most often, irrigation is driven by pumps fueled by diesel. The use of agrophotovoltaics as an energy source for irrigation is still rather novel, combining the advantage of reducing the vulnerability due to too little rainfall and mitigating climate change. Yet it entails the risk that farmers may use agrophotovoltaic energy, once established, to exploit surface and/or groundwater resources beyond sustainable levels. Thus, farmers need to be made aware of this risk, which may undermine their long-term sustainable farming goals.

Seeds expenditure was positively correlated with agricultural production but was not statistically significant. Nevertheless, seed expenditure implies access to improved seeds. Rao et al. [40] found positive correlations between seed expenditure and agricultural output. Ola and Menapace [41] argue that limited access to quality inputs such as seeds is the most important constraint on farmers entering high-value markets. Cioffo et al. [42] argue that agronomic and environmental risks are imposed by hybrid seeds and that Rwandan farmers find it impossible to replicate seeds, which undermines agricultural circular economy. Thus, the availability of seeds at affordable prices and local multiplication of improved seeds should be a priority. In Rwanda, farmers are supported with improved seeds through the Crop Intensification Program, which introduced hybrid seeds in 2007 [27], but farmers in the study area, at the time of data collection, were not served by the program.

We turn our attention to the inefficiency explanatory variables. The location of farmers was found to significantly affect technological efficiency. Farmers in Gitaraga were technically more efficient than the control group in Gatwe. This is likely due to the former participating in the small-scale irrigation technology scheme. This statement is supported by Ngango and Hong [25], who reported that irrigation positively affected the land productivity of smallholder farmers in Rwanda.

The results suggest that the age of farmers is negatively correlated to technical efficiency. This result is in line with the results of Oyetunde-Usman and Olagunju [43] and Anang et al. [44] for Nigeria and Ghana, respectively. A reason for this could be that older farmers prefer to stick to well-known traditional but inefficient methods of production. However, this result contradicts the findings of Mulinga et al. [45], who observed that increasing age reduces technical inefficiency among farmers in Rwanda.

This study found that experience reduces technical inefficiency among farmers in Rwanda. Similarly, Paul [46] observed that technical efficiency increases with more years in potato farming in Rwanda.

Gender influences productivity in the sense that women were less technically efficient than men. This is in line with the results of Ngango and Hong [25] and Anang et al. [44] for women in mixed farming systems in Rwanda and Ghana, respectively.

The analysis suggests that family size influences technical efficiency. Larger families are more technically efficient than families with fewer heads. This is similar to the results of Paul [46] in Rwanda. Paul [46] argues that the higher demand for food to feed the family motivates larger families to strive for higher output. Similarly, Oyetunde-Usman and Olagunju [43] found that larger households in Nigeria were more technically efficient. Obviously, the composition of the family in terms of active persons will make a difference. If the share of dependent household members (those below 15 years of age and those above 65) is large, it may reduce technical efficiency.

This study found that the higher the level of education (measured in years of schooling, the average being 5 years), the more technically inefficient a farmer becomes. This mirrors results by Anang et al. [44] and Paul [46] who report a negative correlation between education and technical inefficiency among farmers in Ghana and Rwanda. While Paul [46] explains this finding by pointing to the general low level of education among Rwandan farmers, Anang et al. [44] show that educated farmers in Ghana devote less time and effort to farming due to high opportunity cost, which could lower their efficiency. It should be noted that while education might increase farmers’ ability to process information, the technical skills acquired in agricultural activities require hands-on training different from that received through formal schooling [47].

The markets considered in the study area are spot markets and access to retailers. Farmers with access to markets had significantly higher technical efficiency compared to farmers without. Market access is considered a key element for minimizing production losses and generating steady incomes [48]. Furthermore, access to both input and output markets may facilitate higher productivity [33]. Efficiency gains may also originate through regular knowledge exchange between producers and value chain actors. Rao et al. [40] argue that farmers selling to (modern) value chains have access to crucial information. In this context, it is interesting to mention that a contract farming agreement between farmers of the cooperative CUBIAM, to which the sampled farmers in Gitaraga belong, and an Indo-French company was sealed in 2021. Under this agreement, the farmers are expected to cultivate hybrid chilies on parts of their farms to supply the export market [49]. Given the price and market assurances, production diversification to chilies is likely to have promoted efficiency gains [40,49]. Ton et al. [48] argue that contract farming can improve smallholder farmer income in the range of 23–54%. Thus, contract farming may enhance the Rwandan Government’s plans to transform agriculture from subsistence farming to a market-oriented food sector. Contract farming helps to achieve this transformation, while on a micro-level also contributing to the overall goal of the solar irrigation project, which is livelihood improvement for the 50 farmers and their families in Gitaraga.

5. Conclusions

Small-scale and sustainable irrigation systems improve efficiency and, ultimately, agricultural productivity among smallholders vulnerable to climate change in Rwanda. To minimize the contribution of smallholder agriculture to climate change, it is important to ensure that the source of energy to power small-scale irrigation systems is based on renewables such as solar and wind power (where applicable). At the same time, these renewable energy sources ought not to be misused to deploy surface or groundwater resources for irrigation as sometimes observed.

This study found that smallholder farmers in Gitaragwa participating in the solar-powered small-scale irrigation scheme were on average more technically efficient (TE = 74.5%) than non-irrigating farmers in Gatwe (TE = 43.3%). Farming experience was beneficial for reaching higher technical efficiency levels. This finding is important when promoting the adoption of solar-powered irrigation systems. For instance, these farmers could be pioneer (lead) farmers and train other peers on the use of irrigation and agrophotovoltaics. Furthermore, policymakers should support programs that improve irrigation practices and governance of irrigation infrastructure. This could involve acting as guarantor in credit applications for the purchase of agricultural innovations and agrophotovoltaics. Another way of increasing the adoption of solar-powered small-scale irrigation is through an effective and knowledge-based public extension service. Thus, policymakers need to strengthen the knowledge base of public extension service staff on practical agriculture innovations, technologies for smallholders, and the governance of water resources for sustainable agricultural practices.

Furthermore, this study found that access to markets had a positive effect on technical efficiency. To this end, policymakers should improve access to upstream markets by creating an enabling environment, e.g., tax breaks for agribusiness that lowers the entry barrier and thus widens market participation. In all of these areas, the needs of women farmers should be addressed, as they are crucial for food production but are usually less technically efficient than male smallholders.

This study comes with several limitations. First, the sample size is relatively small. This is mostly due to the small population of farmers participating in the solar-powered irrigation scheme serving as the treatment locale. Therefore, the small sample size restricts the statistical power of our model. While we took precautions to ensure robust results, we acknowledge the limitation posed by the small sample size. Second, we also acknowledge the lack of causal inference in our study. This is partly due to the data limitation as well as the study objectives. We aimed to provide correlative results that will provide useful insights for future research examining factors that drive technical efficiency in the context of renewable energy-powered irrigation schemes. This limitation also ensured that the SFA model used in the study was overly simplistic since it did not account for several socioeconomic and environmental variables that could address potential endogeneity issues.