2.1 Harrod–Domar (HD) model and financing gap measurement
As posited by Easterly (1999), and recently applied by Tang et al. (2018), Bermejo Carbonell and Werner (2018), van der Merwe and Dodd (2019), Harrod–Domar growth model has been employed in international financing institutions (IFIs). Chenery and Strout (1966) gave the definitive statement of the Financing Gap model in their Two-Gap model that aid will “fill the temporary gap between investment ability and saving ability”. The usual ICOR formulation determines investment requirements for a given growth target.
Easterly (1999) noted that the model has two important features viz. (A) investment requirements to achieve a given growth rate are proportional to the growth rate by a constant known as the Incremental Capital Output Ratio (ICOR) and (B) aid requirements are given by the “Financing Gap” between the investment requirements and the financing available from the sum of private financing and domestic saving. Moreover, he referred to this model as “Financing Gap Model” for short, because, according to him, its most important use is to determine financing shortfalls. He further noted that (A) and (B) imply the following testable assumptions: (1) aid will go into investment one for one, and (2) there will be a fixed linear relationship between growth and investment in the short run. The constant of proportionality is one over the ICOR.
The shortcomings of the Harrod–Domar approach are well noted in the study of Hussain (2000). These, he stated, centre on two closely related problems. The first is the inaccuracy of estimating the resource gap to achieve a target rate of growth and the second is the failure of the basic Harrod–Domar relationship to predict growth rates. With regard to the former, he noted that if the economy is working below capacity, which is typical in most developing countries such as Nigeria, the true value of the ICOR cannot be computed with any degree of precision, and definitely not with the precision suggested by the equations. Also, he noted that the Harrod–Domar approach assumes that all additional growth in income is attributed to the increments of capital. The approach overstates the productivity of capital and understates the ICOR based on the fact that other factors contribute to growth.
However, Geda et al. (2009) observed that there are a number of considerations that still make the Harrod–Domar (HD) framework attractive for policy, which includes: (1) it deals with short-run planning problems, while most growth models that have theoretical appeal and some degree of sophistication deal with long-run growth. They noted that this distinction is very important in application because it is about an economy reaching its equilibrium or steady state over a certain period of time, or to be specific, zero per capita growth or GDP growing at the rate of population growth. (2) The lack of alternative models that can fit the needs of policymakers and practitioners like development banks, especially in dealing with short to medium-term financing needs. (3) The HD approach provides a useful benchmark—a first-order approximation to the complicated task of estimating financing needs for development. It allows a check on consistency across the macroeconomic balances as well as sectoral investment programmes. They finally concluded that HD may continue to be relevant when time and resources are limited.
In analysing the empirical validity of HD in the African context, Easterly (1999) found no empirical basis to support the 44 predictions of the HD in over 138 countries for the 1950–1992 period. In the same vein, Bermejo Carbonell and Werner (2018) also found that the Spanish EU and euro entry have had no positive effect on growth. The findings call for a fundamental rethinking of methodology in economics. However, Geda et al. (2009) were unable to replicate Easterly’s findings. Setting aside issues of model specification and others, they attempted to re-examine these relationships for a sample of 12 African countries and their results actually suggested a strong support for HD predictions with the exception of two countries. They found significant relationships between growth and investment for the 10 countries when a constant is added in the OLS regression. They noted that this is because the HD model assumes no constant term in the relationship between growth and investment (proportionality) and that once they imposed a zero constant on the regressions, it turned out that all countries exhibit a strong and positive short-term relationship between investment and growth.
They also found the relationship between aid and investment to be positive, and in most cases, significant. Although they agreed with the argument that HD ignores diminishing returns to aid, they however stated that the existence of diminishing returns implies that the straightforward HD projections will underestimate the actual resource requirements. In summary, Geda et al. (2009) stated that the African Development Bank (AfDB), as well as other institutions, continue to use various methodologies to estimate resource requirements for developing countries. They noted that any of these methodologies has its own limitations in relation to empirical application to country-specific and context-specific circumstances. However, they affirmed that estimates generated from simple models like the HD turn out to be very consistent with estimates generated by more sophisticated methodologies.
2.2 Conceptual and analytical frameworks
For this study, a stochastic frontier analysis (SFA) framework was used to assess the technical efficiency of rice production in the study area. The basic stochastic frontier production function of rice production can be expressed as;
$$Y_{i} = f(X_{i} ;\beta )\exp (v_{i} - u_{i} ).$$
(1)
where \(Y_{i}\) denotes the quantity of rice produced by \(i{\text{th}}\) farm \((i = 1,2, \ldots N),\)\(X_{i}\) is a vector of production inputs of the \(i{\text{th}}\) farm, and \(\beta\) is a \((kx1)\) vector of unknown parameters to be estimated. \(v_{i}\) is a stochastic noise distributed symmetrically with mean zero and unknown variance \(N\left( {0,\left. {\sigma_{\text{V}}^{2} } \right)} \right.\) (Aigner et al. 1977). \(u_{i}\) are systematic and non-negative random variables which are responsible for farmers technical inefficiency in production and are obtained by truncation (at zero) of normal distribution with mean \(z_{i} \partial\), and variance \(\sigma^{2} .\)\(z_{i}\) is a vector of covariates explaining technical inefficiency associated with farm production and, δ is a vector of unknown parameters (Battese and Coelli 1995).
In line with the frontier production function as specified in Eq. (1), the study define technical efficiency of the \(i{\text{th}}\) rice farm as the ratio of the observed rice mean output, given the values of production inputs (\(X_{i}\)) and its assumed technical inefficiency effects (\(u_{i}\)), to corresponding potential output if there was non-existence of technical inefficiency \((u_{i} = 0)\) in rice production. The technical efficiency of a \(i{\text{th}}\) farm can, therefore, be expressed as:
$${\text{TE}}_{i} = \frac{{f(Y_{i} /u_{1} ,X_{i} )}}{{f(Y_{i} /u = 0,X_{i} )}} = \exp ( - v_{i} ),$$
(2)
where \({\text{TE}}_{i}\) indicates technical efficiency score which is constrained within the interval (0, 1). The value of 1 indicates a fully technically efficient farm and the value of 0 implies a fully technically inefficient farm. Following the single stage approach proposed by Caudill and Ford (1993), the study parameterized the variance of the pre-truncated of the inefficiency error term \(u_{i}\). This is to explore how socioeconomic and policy variables influence rice farmers’ performance (Kumbahkar and Lovell 2000). The inefficiency effect (\(u_{i}\)) can be specified as:
$$u_{i} = z_{i} \delta + \theta_{i} ,$$
(3)
where \(z_{i}\) is (mx1) vector exogenous variables explaining rice farmers’ technical inefficiency, such as age, farming experience, off-farm income, household size, membership in farmers’ association), \(\delta\) is (1xm) vector of parameters to be estimated, and \(\theta_{i}\) is an error term of the inefficiency effect.
The Cobb–Douglas production function model used to represent the production of rice is specified as
$$\ln Q_{i} = \ln \beta_{0} + \sum\limits_{j = 1}^{5} {\beta_{i} \ln Z_{i} } + \left( {v_{i} - u_{i} } \right),$$
(4)
where \(Q_{i}\) represents value of rice output, \(Z_{i}\) represents the conventional inputs usually used in rice production namely, quantity of labour used, farm size, insecticides, herbicides and quantity of seeds planted.
For this study, four main hypotheses were tested, viz; (i). There is no inefficiency effect in rice production; (ii) the coefficients of the square values and the interaction terms in translog have zero values; (iii) exogenous factors are not responsible for the inefficiency term (\(u_{i}\)), and (iv) there is no heteroscedasticity in both the stochastic (\(v_{i}\)) and inefficiency error terms (\(u_{i}\)). The results of the four hypotheses were tested using the generalized likelihood-ratio test statistic specified as:
$$LR\left( \varOmega \right) = - 2\left[ {\left\{ {\ln L\left( {H_{0} } \right)\left. {} \right\}} \right. - \left\{ {\ln L\left( {H_{1} } \right)\left. {} \right\}} \right.} \right],$$
(5)
where \(L(H_{0} )\) and \(L(H_{1} )\) represent the likelihood functions under null and alternative hypotheses, respectively. Following Coelli (1995), if the null hypothesis is rejected, then the test statistic (λ) has a Chi-square distribution of the degree of freedom defined as the difference between the estimated parameters under (H1) and (H0). However, if the null hypothesis is accepted, then the asymptotic distribution involves a mixed Chi-square distribution. The results of the four null hypotheses tested are presented in Table 2.
2.3 Harold–Domar growth model
According to Geda et al. (2009) regarding the continuous relevance and usefulness of the HD model in estimating financing gap, this study employed an adapted form of the HD model to estimate the financing (credit) gap of smallholder rice farmers in Southwestern Nigeria. However, in order to place all the producers on a desirable efficiency level (growth rate) and cater for the issue of efficient use of investment, the growth rate in the HD model is substituted with the production frontier. Thus, this study is based on the assumption that: credit amounts required by rice farmers to produce at the frontier level are directly proportional to the production frontier by a constant known as the Incremental Capital Output Ratio (ICOR). In the same vein, it is assumed that credit (finance) requirements of the farmers are given by the “Financing Gap” between the credit amount required to produce at the frontier level and the finance available to them at present.
$$Y^{*} = \frac{1}{c}\varPhi ,$$
(6)
where \(Y^{*}\) = production frontier (technical efficiency), \(\frac{1}{c}\) is the reciprocal of the incremental capital output ratio (ICOR) given as \(c = \frac{\varPsi }{\varphi }\), where \(\varPsi\) is the annual investment in rice production and \(\varphi\) represents annual increase in output of rice produced \(\varPhi\) = amount required to produce at the frontier level. The ICOR is hypothesized to be a measure of the inefficiency with which credit is used. The adapted H–D model is thus hinged on the condition that the credit is used for the purpose of rice production. As posited by Bifarin et al. (2010), if production credit is invested on the farm, it is however, expected to lead to higher levels of output, but in case the credit is not accessed on time, it may, more often than not, lead to misapplication of funds. Hence, the expected impact of such funds will not be felt on the farm. If, however, the credit is invested in consumption purpose as peculiar to smallholder farmers, credit will likely not lead to an improvement in the efficiency level.
2.4 Study area and source of data
The study was carried out in the southwestern part of Nigeria consisting of the Lagos, Ogun, Oyo, Osun, Ondo and Ekiti States, collectively known as the South-West geographical zone of Nigeria. The area lies between the longitude 2° 311 and 6° 001E and the latitude 6° 211 and 8° 371N, with a total land area of about 77,818 km2. It is bounded in the east by the Edo and Delta States, in the north by Kwara and Kogi States, in the west by the Republic of Benin and in the south by the Gulf of Guinea. The climate of South-West Nigeria is tropical in nature and characterized by wet and dry seasons. The mean temperature ranges between 21 and 34 °C, while the annual rainfall ranges between 150 and 3000 mm. The wet season is associated with the southwestern monsoon wind from the Atlantic Ocean, while the dry season is associated with the northeastern trade wind from the Sahara Desert. The vegetation in South-West Nigeria is made up of fresh water swamp and mangrove forest at the belt, the low land in forest stretching inland to the Ogun and part of the Ondo states, with the secondary forest stretching towards the northern boundary by the derived and southern Guinea savannas (Agboola 1979).
A multistage sampling technique was used to select the respondents for the study from June to July, 2017. The first stage involved a typical case-purposive selection of three states, Ekiti, Ondo and Osun States located in the same agro-ecological area as shown in Fig 1. In the second stage, four local government areas (LGAs) were then selected from each state, based on the predominance of smallholder rice farmers in these areas, using a typical case-purposive sampling. In the third stage, five villages were randomly selected from each of the four LGAs. Following Tesfahunegn et al. (2016), at 95% confidence level and 5% margin of error, the sample size for the study was determined using the sample determination formula as described by Cochran (1977), allowing for six smallholder rice farmers to be selected from each of the 5 villages earlier selected to give 360 respondents interviewed for the study. Data were collected by means of a pre-tested, well-structured questionnaire by trained and experienced enumerators who have good knowledge of the farming systems and speak the local language in collaboration with the Agricultural Development Programme (ADP) agents in each state. Information sought were on respondents’ socio-economic characteristics, inputs and output in rice production and as well as the costs of and returns on rice production.