Study area
The study was conducted in the four main agroecological zones of Ghana, namely Northern Savannah, Transitional, Forest and Coastal Savannah zones. The Northern Savannah zone is located along the northeastern corridor of the Northern Region with a total land area of about 125,430 km^{2}. The tropical continental climate and Guinea Savannah vegetation type are seen in this area. The Transitional zone, which is located around the middle portion of the BrongAhafo Region and the northern part of Ashanti Region, covers a total land area of about 2300 km^{2}. The climate of the place is the wet semiequatorial type, while the vegetation is the Savannah woodland and a forest belt. The Forest zone, covering an area of about 135,670 km^{2}, is floristically divided into rain forest and semideciduous forest. The climate of the place is the semiequatorial type, while the vegetation is semideciduous forest zone with clay, sand and gravel deposits. The Coastal Savannah occupies about 20,000 km^{2} and comprises the HoKeta Plains, the Accra Plains and a narrow strip tapering from Winneba to Cape Coast.
Data collection
Farmlevel primary data on maize production for the 2014 cropping season were collected from 576 maize farmers using structured questionnaire. The study used [4]’s sample size determination formula in the determination of the appropriate sample size. That is
$$n = \frac{{t^{2} (p)\,(q)}}{{d^{2} }}$$
(1)
where n = sample size, t = value for selected alpha level of 0.025 in each tail = 1.96, p = proportion of population engaged in maize production activities, q = proportion of population who do not engage in maize production activities, and d = acceptable margin of error for proportion being estimated = 0.05.
According to the Ghana Living Standard Survey Report of the Fifth Round (GLSS 5), 41.5% of households who harvested staple and/or cash crops in the last twelve months before September 2008 were maize farmers [9]. Assuming 95% confidence level and 5% margin of error, the sample size was calculated as follows:
$$n = \frac{{1.96^{2} \times 0.415 \times 0.585}}{{0.05^{2} }} = 373$$
These procedures result in the minimum returned sample size. If a researcher has a captive audience, this sample size may be attained easily. However, since many educational and social research studies often use data collection methods such as surveys and other voluntary participation methods, the response rates are typically well below 100%. Salkind [26] recommended oversampling by 40–60% to account for low response rate and uncooperative subjects. The sample size was therefore increased by 54.5% to correct all probable anomalies that might occur, increasing the sample size to 576 maize farmers.
Multistage sampling technique was employed in the study. Two districts/municipalities were purposively selected in the first stage from each agroecological zone based on total maize production by Ghana’s districts/municipalities [24]. The second stage consisted of random sampling of nine (9) villages from each of the sampled districts. Finally, the third stage comprised a random sample of eight (8) maize farmers from a list of maize farmers in each of the villages with the help of agricultural extension agents. The data collected consisted of detailed information on the socioeconomic characteristics of the farmers, their inputs, outputs as well as prices of inputs and outputs.
Data analysis
Descriptive statistics were used to summarize the socioeconomic characteristics as well as quantities of inputs and outputs of the respondents. Also, the stochastic frontier production function was employed to analyse the determinants of maize output. Aigner et al. [1, 17] independently proposed the stochastic frontier production function. According to them, the stochastic frontier production function is defined by;
$$y_{i} = f(x_{i} ;\beta ) + e_{i} \quad {\text{where}}\;\, i = 1,2, \ldots ,N$$
(2)
$$e_{i} = v_{i}  u_{i}$$
(3)
where \(y_{i}\) represents the level of output of the ith maize farmer; \(f(x_{i} ;\beta )\) is an appropriate production function of vector, \(x_{i}\) of inputs for the ith maize farmer and a vector, \(\beta\) of parameters to be estimated. \(e_{i}\) is an error term which comprises two components, \(v_{i}\) and \(u_{i}\). \(v_{i}\) is a random error with zero mean, \(N(0; \sigma^{2} v)\), and is specifically associated with random factors like measurement errors in production as well as weather factors that the maize farmer cannot control and it is assumed to be symmetric and independently distributed as \(N(0; \sigma^{2} v)\), random variables and is independent of \(u_{i}\). Conversely, \(u_{i}\) is a nonnegative truncated half normal, \(N(0; \sigma^{2} v)\), random variable and is linked to farmspecific characteristics, which leads to the \(i{\text{th}}\) maize farm not achieving maximum production efficiency. \(u_{i}\) is therefore linked to the technical inefficiency of the maize farm and ranges from zero to one. However, \(u_{i}\) may have other distributions like exponential and gamma. N is the number of maize farmers that took part in the crosssectional survey. Technical efficiency of a maize farmer is the ratio of observed output to the frontier output, given the quantity of resources employed by the farmer. Technical inefficiency therefore refers to the margin with which the level of output for the farmer falls below the frontier output.
$${\text{Technical}}\,{\text{efficiency}} = {\text{TE}}_{i} = \frac{{y_{i} }}{{y_{i}^{*} }}$$
(4)
where \(y_{i}^{*} = f(x_{i} ;\beta )\), highest predicted value for the ith farm
$${\text{TE}}_{i} = {\text{Exp}}\,(  u_{i} )$$
(5)
$${\text{Technical}}\,{\text{inefficiency}} = 1  {\text{TE}}_{i}$$
(6)
The stochastic frontier production function can be estimated by the maximum likelihood estimation (MLE) technique. The technique makes use of the specific distribution of the disturbance term and is more efficient than corrected ordinary least squares [11]. Diagnostically, the generalized likelihood ratio test was used to determine whether the Cobb–Douglas or translog functional form fits the data collected from the maize farmers in this study better. The test allows evaluation of a restricted model with respect to an adopted model. The statistic associated with this test is defined as:
$$\lambda =  2\left[ {\ln \frac{{L\left( {H_{0} } \right)}}{{L\left( {H_{1} } \right)}}} \right] =  2\left[ {\ln L\left( {H_{0} } \right)  \ln L\left( {H_{1} } \right)} \right]$$
(7)
where \(L(H_{0} )\) and \(L(H_{1} )\) are the loglikelihood values of the adopted and the restricted models, respectively. The test statistic \(\lambda\) has approximately a Chisquare distribution with a number of degrees of freedom equal to the number of parameters (restrictions), assumed to be zero in the null hypothesis. When \(\lambda\) is lower than the corresponding critical value (for a given significance level), the null hypothesis cannot be rejected. The main hypothesis tested here is to find out whether the Cobb–Douglas functional form is an adequate representation of the maize production data collected, given the specification of the translog functional form. The test results showed that the translog functional form was more appropriate. Therefore, the translog functional form was adopted in this study. Theoretically, the stochastic frontier translog production function is specified as:
$$\ln y_{i} = \beta_{0} + \mathop \sum \limits_{k = 1}^{m} \beta_{k} \ln x_{ki} + \frac{1}{2}\mathop \sum \limits_{k = 1}^{m} \mathop \sum \limits_{j = 1}^{m} \beta_{kj} \ln x_{ki} \ln x_{ji} + v_{i}  u_{i}$$
(8)
where ln is natural logarithm, \(y_{i}\) is total output, \(x_{i}\) is vector of inputs, and \(ij\) are positive integers \((i \ne j)\). \(\beta\) is a vector of parameters to be estimated, and \(v_{i } \,{\text{and}}\, u_{i}\) have been defined above. The inefficiency model is also specified as:
$$u_{i} = \delta_{0} + \mathop \sum \limits_{m = 1}^{N} \delta_{m} z_{i}$$
(9)
where \(z_{i}\) is a vector of farmer characteristics and \(\delta\) is a vector of parameters to be estimated. STATA provides a joint estimation of the parameters in the stochastic frontier production function and those of variables in the inefficiency model as well as variance parameters. Empirically, the following stochastic frontier translog production function was estimated.
$$\begin{aligned} & \ln {\text{OUTPUT}}_{i} = \beta_{0} + \beta_{1} \ln {\text{SED}}_{i} + \beta_{2} \ln {\text{FET}}_{i} + \beta_{3} \ln {\text{PET}} + \beta_{4} \ln {\text{MAN}}_{i} + \beta_{5} \ln {\text{LAD}}_{i} \\ & \quad + \beta_{6} \ln {\text{LAB}}_{i} + \beta_{7} \ln {\text{HEB}}_{i} + \beta_{8} \ln {\text{CAP}} + \beta_{9} \ln ({\text{SED}})_{i}^{2} + \beta_{10} \ln ({\text{FET}})_{i}^{2} + \beta_{11} \ln ({\text{PET}})_{i}^{2} \\ & \quad + \beta_{12} \ln ({\text{MAN}})_{i}^{2} + \beta_{13} \ln ({\text{LAD}})_{i}^{2} + \beta_{14} \ln ({\text{LAB}})_{i}^{2} + \beta_{15} \ln ({\text{HEB}})_{i}^{2} + \beta_{16} \ln ({\text{CAP}})_{i}^{2} \\ & \quad + \beta_{17} (\ln {\text{SED}} \times \ln {\text{FET}})_{i} + \beta_{18} (\ln {\text{SED}} \times \ln {\text{PET}})_{i} + \beta_{19} (\ln {\text{SED}} \times \ln {\text{MAN}})_{i} \\ & \quad + \beta_{20} (\ln {\text{SED}} \times \ln {\text{LAD}})_{i} + \beta_{21} (\ln {\text{SED}} \times \ln {\text{LAB}})_{i} + \beta_{22} (\ln {\text{SED}} \times \ln {\text{HEB}})_{i} \\ & \quad + \beta_{23} (\ln {\text{SED}} \times \ln {\text{CAP}})_{i} + \beta_{24} (\ln {\text{FET}} \times \ln {\text{PET}})_{i} + \beta_{25} (\ln {\text{FET}} \times \ln {\text{MAN}})_{i} \\ & \quad + \beta_{26} (\ln {\text{FET}} \times \ln {\text{LAD}})_{i} + \beta_{27} (\ln {\text{FET}} \times \ln {\text{LAB}})_{i} + \beta_{28} (\ln {\text{FET}} \times \ln {\text{HEB}})_{i} \\ & \quad + \beta_{29} (\ln {\text{FET}} \times \ln {\text{CAP}})_{i} + \beta_{30} (\ln {\text{PET}} \times \ln {\text{MAN}})_{i} + \beta_{31} (\ln {\text{PET}} \times \ln {\text{LAD}})_{i} \\ & \quad + \beta_{32} (\ln {\text{PET}} \times \ln {\text{LAB}})_{i} + \beta_{33} (\ln {\text{PET}} \times \ln {\text{HEB}})_{i} + \beta_{34} (\ln {\text{PET}} \times \ln {\text{CAP}})_{i} \\ & \quad + \beta_{35} (\ln {\text{MAN}} \times \ln {\text{LAD}})_{i} + \beta_{36} (\ln {\text{MAN}} \times \ln {\text{LAB}})_{i} + \beta_{37} (\ln {\text{MAN}} \times \ln {\text{HEB}})_{i} \\ & \quad + \beta_{38} (\ln {\text{MAN}} \times \ln {\text{CAP}})_{i} + \beta_{39} (\ln {\text{LAD}} \times \ln {\text{LAB}})_{i} + \beta_{40} (\ln {\text{LAB}} \times \ln {\text{HEB}})_{i} \\ & \quad + \beta_{41} (\ln {\text{LAD}} \times \ln {\text{CAP}})_{i} + \beta_{42} (\ln {\text{LAB}} \times \ln {\text{HEB}})_{i} + \beta_{43} (\ln {\text{LAB}} \times \ln {\text{CAP}})_{i} \\ & \quad + \beta_{44} (\ln {\text{HEB}} \times \ln {\text{CAP}})_{i} + v_{i}  u_{i} \\ \end{aligned}$$
(10)
where OUTPUT is output of maize, measured in kilogramme per hectare (kg/ha), and it is the dependent variable; SED is quantity of seed used, measured in kilogramme per hectare (kg/ha); LANDSZ is area of land cultivated with maize, measured in hectares; LAB is quantity of labour employed in maize production, measured in mandays; CAP is capital used in maize farm, measured as depreciated charges on farm tools and implements in Gh¢; FET is quantity of fertilizer used in maize production, measured in kilogrammes per hectare (kg/ha); MAN is quantity of manure used in maize production, measured in kilogrammes per hectare (kg/ha); PET is quantity of pesticides used in maize production, measured in litres per hectare (L/ha); and HEB is quantity of herbicides used in maize production, measured in litres per hectare (L/ha).
According to [14, 30], for maize farmers to be efficient in their use of production resources, their resources must be used in such a way that their marginal value product (MVP) is equal to their marginal factor cost (MFC) under perfect competition. Therefore, the resource use efficiency parameter was calculated using the ratio of MVP of inputs to the MFC. According to [7, 10], the efficiency of resource use is given as:
$$r = \frac{\text{MVP}}{\text{MFC}}$$
(11)
where r = efficiency coefficient, MVP = marginal value product, and MFC = marginal factor cost of inputs.
$${\text{MFC}} = P_{x}$$
(12)
where \(P_{{x_{i} }}\) = unit price of input, say \(x\)
$${\text{MVP}}_{x} = {\text{MPP}}_{x} \cdot P_{y}$$
(13)
where \(y\) = mean value of output, \(x\) = mean value of input employed in the production of a product, \({\text{MPP}}_{x}\) = marginal physical product of input \(x\), and P
_{
y
} = unit price of maize output.
If \(\beta_{x}\) = output elasticity of input \(x\).
From the translog production function (Eq. 6),
$$\beta_{x} = \frac{\partial \ln Y}{\partial \ln X} = \frac{\partial Y}{\partial x} \cdot \frac{x}{Y}$$
$${\text{MPP}}_{x} = \frac{\partial Y}{\partial x} = \beta_{x} \frac{Y}{x}$$
(14)
\({\text{MPP}}_{x}\) = marginal physical product of input \(X\).
Therefore
$${\text{MVP}} = \frac{\partial Y}{\partial X} \cdot P_{y} = \beta_{x} \frac{Y}{X} \cdot P_{y}$$
(15)
Marginal value product (MVP) of a particular input is therefore calculated by the product of output elasticity of that input, the ratio of mean output to mean input values and the unit output price. On the other hand, marginal factor cost (MFC) of an input was obtained from the data collected on the unit price of that input. To decide whether or not an input was used efficiently, the following convention was followed in this study. If

\(r = \varvec{ }1\), it implies the input was used efficiently.

\(r > 1\), it implies the input was underutilized and therefore both output and profit would be increased if more of that input is employed.

\(r < 1\), it implies the input is overutilized and therefore both output and profit would be maximized if less of that input is employed [22].
Returns to scale were calculated by the sum of the output elasticities of the various inputs.
$${\text{Returns}}\,{\text{to}}\,{\text{scale}} = \mathop \sum \limits_{i} \frac{\partial \ln Y}{{\partial \ln X_{i} }} = \mathop \sum \limits_{i} \beta_{i}$$
(16)
where \(Y\) is output, \(X_{i}\) are inputs and \(\beta_{i}\) are output elasticities.