In Chapter one we set in motion the purpose for this research and explain to the reader the essence of quantifying the amount the household is willing to pay for abating malaria both in the present and in the future. In this chapter we go a step further by reviewing literature in this area. This chapter is important because it provides the reader with a sort of ‘history’ into this area of research. It also gives the reader an opportunity to understand where our research stands vis-à-vis other researches in this area. Obtaining a value for the marginal effect of malaria on farmers’ technical efficiency is one of the ‘live wires’ on which precise estimates for our Willingness-To-Pay depend. We therefore want to start by reviewing literature in the area of efficiency measurement; afterwards, we will research into literature in the area of Willingness – To – Pay.

Before we go ahead we highlight the purpose of measuring technical efficiency to the reader.

Technical efficiency primarily enables one to understand the relationship between input used and the output (total harvested crop). It also enables us to measure the performance of individual farms in an industry as well as provide an index for the average performance of the overall industry. This then leads us to propose policy recommendations that could help shift the production frontier- the maximum attainable harvest from each input- of the farm closer to the industry frontier at the prevailing technology. As we progress in this research the reader will further appreciate this concept and the reason why it is one of the most talked about concepts in development/resource economics.

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At the moment, our aim is to examine some literature that relates to our area of research. We therefore start Section 2.1 by reviewing literature relating to the “poor but efficient” hypothesis of Schultz (1964). Section 2.2 reviews some agriculture-based literature on efficiency and health. In doing this we divide the study on inefficiency into two; the Frequentist (Section 2.2.1) and the Bayesian (Section 2.2.2) studies. Using another method of classification, we classify the study of efficiency into single output studies (Section 2.2.3) and multiple output studies (Section 2.2.4). This puts us in good standing to review literature on Willingness-To-Pay in Section 2.4.

## Productivity/Efficiency Studies in Agriculture

The “Poor but Efficient” Hypothesis

The huge volume of research on efficiency in agriculture draws motivation from Schultz (1964) book “Transforming Traditional Agriculture”. In the book he explains why rural farmers are efficient in the management and allocation of resources. He advances a hypothesis popularly called the “poor but efficient” hypothesis. Researchers try to verify this hypothesis quantitatively; in doing this, a lot of issues come to the fore, part of which is; the best way to measure productivity. Before the advent of the deterministic measure of productivity pioneered by Aigner and Chu (1968), and, Afriat (1972) researchers attempt to measure efficiency. Of great importance to us in this area are the works of Welsch (1965), Chennareddy (1967) and Lipton (1968) because they specifically test the validity of Schultz’s ‘poor but efficient’ hypothesis.

Chennareddy (1967) utilizes the linear regression analysis on a data of one hundred and four rice and tobacco farmers in South India using a Cobb-Douglas production function. His findings were in accord with Schultz hypothesis. He recommends that South Indian farmers should adopt modern technology and extension education in order to move to a higher frontier. Lipton (1986) [1] disagrees with this recommendation. He argues that if Schultz’s findings are correct then the rural farmers do not need any expert advice to improve their productivity in other words moving to a higher frontier should not be a problem for them. He further queries Schultz’s assertion believing that it only works under a neo-classical theory of perfect competition; he affirms that if Schultz uses linear programming to analyse his data his findings would show that the rural farmer is inefficient.

Welsch (1965) in his study on Abakaliki rice in Eastern Nigeria makes use of the linear regression to affirm that peasant farmers respond to economic inducement by allocating efficiently among several resources at their disposal. Hence, he supports Schultz’s hypothesis. One thing we want the reader to note in the above groups of literature is; the writers who concur with Schultz’s assertion use parametric techniques to arrive at their conclusion while Lipton (1968) employs a non-parametric linear programming technique that assumes at least one factor is not fully employed.

Just as the argument is about to cease, Sauer and Mendoza-Escalante (2007) involve themselves in it. Their work serves to reconcile these diametrically opposing schools of thought. It puts to use a parametric normalized generalized Leontief (GL) profit function technique to analyse joint production of Cassava flour and maize by small-scale farmers in Brazil. The small-scale farmers are allocatively efficient, they assert, but they show considerable inefficiency in the scale of operation. At this juncture, we remind the reader that our digression is intentional. Our aim is to show how Schultz’s assertion has brought an upsurge in the number of efficiency studies in agriculture with special focus on the developing economies of the world. We like to say that the work not only instigates research in development/resource economics but it also prompts research in anthropology and sociology (see Adams, 1986 and the review by Michelena, 1965 pp. 540-541).

Proper measure of productivity starts with Aigner and Chu (1968), Afriat (1972) and Richmond (1974) where they propose a deterministic method of frontier measurement. Though their studies are obsolete they however underscore the popularity of the Cobb-Douglas functional form in the early literature to show the relationship between input and output. Aigner, Lovell and Schmidt (1977), Meeusen and van den Broeck (1977), and, Battese and Corra (1977) [2] introduce the modern stochastic frontier analysis as we know it today simultaneously. Their model apart from incorporating the efficiency term into the deterministic model it also includes the effect of random shock, hence, the name ‘stochastic’. Lau and Yotopoulos (1971) also introduce a dual profit function model to measure efficiency but their method is not as popular in production analysis because it only yields efficiency measures for a group of farms while the frontier method gives efficiency values for individual farms in the industry (Førsund et al 1980).

The reader should note that the linear regressions of Chennareddy (1967) and Welsch (1965) give the shape of the technology of an average farm in the industry while the stochastic frontier model gives the shape of the technology of the most productive farm in the industry against which the efficiency of every other farm is measured (Coelli 1995). In other words, Chennareddy (1967) and Welsch (1965) use an average response model for their analysis.

The specification of a functional form and/or distributional assumption confers on a technique the nomenclature ‘parametric’ while the non-specification of a functional/distributional form confers on a technique the non-parametric nomenclature. The non-parametric nomenclature means, in the words of Koop (2003), you are “letting the data speak”. This he says is very difficult to achieve as even in the non-parametric system, just like in the parametric, one need to impose certain structure on a particular problem in order to achieve ones objectives.

The use of the Data Envelopment Analysis (DEA) (another technique is the Free Disposal Hull, FDH) overshadows every other technique in the non-parametric class. Charnes, Cooper and Rhodes (1978) introduce this technique and gave it the name as we know it today. The data envelopment analysis technique uses the linear programming method to generate a piece-wise envelop over the data points. The technique is widely used in technical efficiency studies but it has the shortcoming of not incorporating randomness in measuring efficiency. Also, the envelop curve is not everywhere differentiable. Our focus in this research is the parametric technique.

The parametric technique has progressed so much in the literature that there are now two different econometric schools of thought for estimating efficiency. The first school of thought are the Frequentists who dominate this field since its inception and the second school of thought are the Bayesians into which our research belongs.

The Frequentist Studies

The first set of Frequentist study is deterministic in nature and use the technological structure of the mathematical programming approach (see Aigner and Chu, 1968; Timmer, 1971; and, Førsund and Hjalmarsson, 1979 for exposition on mathematical/goal programming). Richmond (1974) introduces the Modified Ordinary Least Square (MOLS) approach to analyse the efficiency of Norwegian manufacturing industries specifying a Cobb-Douglas production function. Richmond (1974) is a modification of the Corrected Ordinary Least Squares (COLS) approach. Winsten (1957) introduces this model by assuming a distribution (such as half normal or exponential) for the disturbance term. The Corrected Ordinary Least Square technique involves a two step process. The first step involves the use of the Ordinary Least Squares to obtain consistent and unbias estimates of the marginal effect parameters; on the contrary, the intercept parameters are consistent but bias. The second step involves the shifting of the intercept upwards so the frontier envelops the data from above.

Greene (1980) takes Richmond (1974) work a step further as he assumes a gamma distribution for the random error term using the maximum likelihood approach. He uses the data from Nerlove (1963) which is a sample of one hundred and fifty five firms producing electricity in the United States in 1955. Apart from replicating the results of Aigner and Chu (1963), Greene (1980) tries to explain the statistical relevance of his model. The reader should note that Greene (1980)’s model is deterministic.

One of the early applications of the deterministic frontier were Shapiro and Müller (1977), Shapiro (1983), Belbase and Grabowski (1985). Shapiro and Müller (1977) attempt to estimate the technical efficiency of forty farms in Geita district of Tanzania. They follow Timmer (1971) method of analysing technical efficiency by applying the linear programming to a Cobb – Douglas production frontier. Their result which is similar to that of Chennareddy (1967) shows that the traditional farmer can improve his technical efficiency by adopting modern farming practices through easy access to information. This, they say, will be at the expense of non-economic costs like the farmer being branded “unsociable” by his community. Shapiro (1983) working in the same district as Shapiro and Müller (1977) tries to confirm the ‘poor but efficiency’ hypothesis but discovers the hypothesis may not be applicable in terms of peasant agriculture in Tanzania because their output could still be increased if all farmers had the same efficiency as the most efficient farmer in the sample. These assertions echo the conclusion of Lipton (1968). He uses the same model and method of analyses as Shapiro and Müller (1977).

Belbase and Grabowski (1985) introduce a technique that is different from the other two stated above. They apply the Corrected Ordinary Least Square (COLS) approach of Winsten (1957) on cross-sectional sample of farms in Nuwakot district of Nepal. They record an average technical efficiency value of 80% for joint production of rice, maize, millet and wheat. The average technical efficiency value for individual frontier calculation for rice and maize is given as 84% and 67% in that order. They find correlation between technical efficiency and other variables which are nutritional level, income and education. Technical efficiency is however not correlated with farming experience.

Some studies investigate the impact of certain agricultural policies on productivity. A priori one expects these policies to actually increase productivity but this is not always the case. One of such study; Taylor, Drummond and Gomes (1986) use a deterministic production function and discover the World Bank sponsored credit programme – PRODEMATA – did not impact positively on the technical efficiency of farmers in Minas Gerais, Brazil. Their result shows that there is no difference between the technical efficiency of farmers who participate in the programme and those that did not participate. This paper is one of the few that compare both the results of the Corrected Ordinary Least Square and the maximum likelihood approaches. Unexpectedly, the participant farmers in the PRODEMATA programmes have slightly lesser allocative efficiency than non-participant farmers. The researchers also favour Schultz’s hypothesis.

We want the reader to note that the deterministic frontier is still popular in the literature for example, Alvarez and Arias (2004) use Lau and Yotopoulos (1971) dual profit function model to measure the effect of technical efficiency on farm size using data from one hundred and ninety-six dairy farms in Northern Spain. They introduce technical efficiency as a parameter to estimate in a simple production function. They observe a positive relationship between technical efficiency and farm size after they control for output prices, input prices and quasi-fixed inputs. Also Amara et al (1999) use the deterministic frontier to discover the relationship between technical efficiency and the adoption of conservation technologies by potato farmers in Quebec. They found that farming experience and the adoption of conservation technologies have positive influence on technical efficiency.

Croppenstedt and Demeke (1997) use a fixed-random coefficients regression to analyse data for small-scale farmers growing cereal in Ethiopia. They observe that land size is a major constraint to crop production and large farms are relatively less productive than small farms other things being equal. They note that most of the farms are inefficient. They also observe inefficiency in the use of inputs especially labour and fertiliser. Share cropping is positively correlated to technical efficiency.

Karagiannis et al (2002) propose an alternative for separating technical change form time varying technical inefficiency. Their proposition uses the general formulation index to model technical change (Karagiannis et al 2002 cites Baltagi and Griffin 1988). They also model technical change as quadratic function of time. Their proposition does not assume any distributional assumption for the one sided stochastic error term. They then apply their proposition to the United Kingdom dairy sector from 1982 to 1992 using a translog production frontier. They obtain a mean technical efficiency value of about seventy-eight per cent for the dairy industry with this period.

One major disadvantage of the deterministic frontier model is that it over-values our inefficiency estimates. For example, Taylor and Shonkwiler (1986) discover the deterministic frontier gives over seventy per cent inefficiency while the stochastic frontier gives twenty per cent value for inefficiency.

At present, a lot of papers utilize the stochastic frontier model in their analysis. Coelli et al (2003) makes use of the stochastic frontier to calculate the total factor productivity for a panel data of crop agriculture in Bangladesh. The data contains thirty-one observations collected between 1960/61 and 1991/92 from 16 regions and the result reveals technical change is convex in nature with increase starting about the time of the introduction of the green revolution varieties in the 1970s. Technical efficiency reduces at an annual rate of 0.47 per cent during the period they investigate. This has an effect on the total factor productivity which declines at the rate of 0.23 per cent per annum with the rate of reduction increasing in later years. This, they say, raises questions of food security and increase in agricultural productivity in Bangladesh. They point out the non-use of price data in their analysis which makes their work different from other authors (Coelli et al; 2003 cites Pray and Ahmed, 1991, and, Dey and Evenson, 1991). Wadud and White (2000) compare the stochastic frontier approach with the data envelopment analysis and discover both methods indicate efficiency is significantly affected by irrigation and environmental degradation.

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There are a few papers that attempt to analyse technical, allocative and economic efficiencies at once in a single research. Bravo-Ureta and Pinheiro (1997) carry out a frontier analysis using the self-dual Cobb-Douglas production function to analyse farm data from Dominican Republic. They justify the use of the Cobb – Douglas production function because the method they adopt requires both the use of the production and cost frontiers. Their research is important because they use the maximum likelihood technique to emphasize the essence of not only estimating the technical efficiency but also, the allocative and economic efficiency. Another paper that follows in this light is that of Bravo-Ureta (1994) who attempts to measure the technical, allocative and economic efficiencies of cotton and cassava farmers in eastern Paraguay. He estimates economic efficiency for cotton and cassava farmers to be around forty per cent and fifty-two per cent respectively.

There could be spatial differences in the technical efficiencies of different farms based on ecological differences, farm size and interactions between these two variables. Tadesse and Krishnamoorthy (1997) set out to investigate this in their research on paddy rice farmers in the state of Tamil Nadu, India. They remark that the farmers still have opportunity of increasing their efficiency by seventeen per cent. They observe significant variation in the variation of mean technical efficiency in the four zones that make up Tamil Nadu. They use a two stage approach where the first task is to obtain farm-specific technical efficiency and then use a Tobit model to compare the differences in the technical efficiencies of each region and zone. Wang and Schmidt (2002) note a bias in the results obtained by this process and they went ahead to use the Markov chain Monte Carlo technique to prove that there is serious bias at every stage of the procedure.

Chen et al (2009) also examine the technical efficiency of farms in four regions of China. The four regions are North, North-East, East and South-West. They observe that different inputs need to be put to efficient use in the different regions. For example, inefficient use of industrial input is the main problem in the East while in the North it is capital. They assert that farms in the North and North-East are relatively more efficient than farms in the East and South-West. They recommend a change in the land tenure system to eliminate land fragmentation in China.

Other researchers have used the stochastic production frontiers to investigate the impact of government programmes on farmers’ efficiency. For example, Seyoum et al (1998) use the Battese and Coelli (1995) stochastic production function to compare between farmers that participate in Sasakawa-Global 2000 project and those who do not in Ethiopia. They collect twenty samples from two different districts (Keresa and Kombolcha) of eastern Ethiopia and show the difference in the levels of production in these two districts by use of a dummy for one district. The data is panel in nature which justifies their use of the Battese and Coelli (1995) model. Battese and Coelli (1995) [3] is a panel data extension of the Kumbhakar et al (1991) research work. Seyoum et al (1998) recommend that policy makers should expand the Sasakawa-Global 2000 project as farmers who participated have better output, productivity and efficiency than farmers that did not.

Still on the impact of government programmes on efficiency, Abdulai and Huffman (2000) look at the impact of the Structural Adjustment Programme on the efficiency of rice farmers in Northern Ghana using a stochastic profit function. Their results show rice producers in the region are highly responsive to market prices for rice and inputs. They support the introduction of the structural adjustment programme because it makes the farmers more market oriented. Also, Ajibefun and Abdulkadri (1999) find the Cobb-Douglas production function as being adequate to represent the efficiency of Nigeria’s National Directorate of Employment Farmers Scheme. They reject the half-normal distribution assumption for the inefficiency term. Ajibefun (2002) simulates the impact of policy variables on the technical efficiency of small-scale farmers in Nigeria. He discovers that increase in education level and the farming experience would significantly improve the small-scale farmers’ technical efficiency. Amaza and Olayemi (2002) investigate the technical efficiency of food crop farmers in Gombe State, Nigeria and arrive at similar mean technical efficiency as Ajibefun and Abdulkadri (1999). However, the difference between the minimum and maximum technical efficiency score for Amaza and Olayemi (2002) is seventy-six per cent while for Ajibefun and Abdulkadri (1999) is about sixty-six per cent.

Jara-Rojas et al (2012) look at the impact of the adoption of soil and water conservation practices on productivity and they discover a positive relation between soil and water conservation and technical efficiency. They discover that an enhancement of the technical efficiency also improves the net returns on investment.

The use of the stochastic frontier model to estimate the effect of health on farmers’ efficiency is also very important in the literature. Croppenstedt and Müller (2000) take up this challenge when they research into the role of the Ethiopian farmers’ health and nutritional status on their productivity and efficiency. They find that distance to the source of water as well as nutrition and morbidity affect agricultural productivity. Surprisingly, elasticities of labour productivity regarding their nutritional status are strong. They further affirm that this strong correlation continues with technology estimates and wage equations. However, they record considerable loss in production due to technical inefficiency even after accounting for health and nutrition of workers.

Ajani and Ugwu (2008) look at the impact of adverse health on the productivity of farmers living in the Kainji basin of North-Central Nigeria. Their study shows the health variable as being positive, large and statistically significant. They therefore conclude that health capital is an essential input in agriculture.

A paper that successfully combined the non-parametric technique of data envelopment analysis and an econometric model is Audibert et al (2003). They use a combination of the data envelopment analysis and the Tobit model to infer on the social and health determinants of the efficiency of cotton farmers in Northern Côte d’Ivoire. They use the high density of the malaria parasite in the blood of an individual as a proxy for the health of the household. They use a two step process; firstly, they use the data envelopment analysis to arrive at relative technical efficiency values and then they regress this efficiency scores against factors they think will affect efficiency. The ‘high density of malaria parasite in the blood’ variable enters the model at the second stage. Their results show that malaria greatly reduces farmers’ technical efficiency. They further conclude that it is intensity of infection by the disease that is important rather than its presence. Our research collects data on the prevalence of the disease in an area rather than just hospital reported cases; this we believe will give further credence to our results.

Ajani and Ashagidigbi (2008) use numbers of days of incapacitation as a proxy for malaria incidence in Oyo State, Nigeria. Surprisingly, they ran a normal linear regression to investigate the effects of malaria on agricultural productivity. Their analysis shows that age and days of incapacitation are insignificant statistically. Olarinde et al (2008) explore the factors that affect bee keepers’ technical efficiency in Oyo state, Nigeria. They observe that the bee keepers are efficient by about eighty-five per cent there is still room for to increase their efficiency by fifteen per cent. They point out that some of the farmers do not take bee-keeping as their main occupation. This, they say, is a major determinant of efficiency. Marital status is also another variable that affects technical efficiency, they note. They observe that a farmer who is single is likely to be more efficient than a married farmer.

Mochebelele and Winter-Nelson (2000) examine the effect of migratory labour (to mine fields in South Africa) on farm technical efficiency. They try to establish if migrant labour actually complement farm production or not. They establish that households with migrant farmers have higher production and are more efficient than households without migrant farmers.

In the use of the panel data for efficiency estimation, some researchers try to see if differences exist in efficiency values between the fixed effect model and the stochastic frontiers. Ahmad and Bravo-Ureta (1996) use panel data of ninety-six Vermont dairy farms between the periods 1971 to 1984. They carry out statistical tests to investigate the better model between the fixed effect model and the stochastic frontier model. The fixed effect model gave better results than the stochastic frontier model. Hence, they conclude that the fixed effect model needs to be considered in panel data analysis.

Reinhard et al (1999) estimate the technical and environmental efficiency of a panel of dairy farms. They assume the production of two outputs – dairy and excessive use of Nitrogen. They analyse their efficiencies separately. Their objective involves investigating whether farmers can both be technically and environmentally efficient. They also examine the compatibility of these two types of efficiencies. They obtain a mean output-technical efficiency of 0.894 while the input-oriented environment efficiency is 0.441. They remark that intensive dairy farming is both technically and environmentally more efficient than extensive dairy farming.

Reinhard et al (2000) examine comprehensive environmental efficiency in Dutch dairy farms. This paper is a continuation of Reinhard et al (1999) paper. In this paper, apart from surplus Nitrogen which they use in their earlier work, they also investigate excess use of phosphate and total energy use of these farms. They compare efficiency scores in the stochastic frontier analysis with the data envelopment analysis. The mean technical efficiency values for the two methods of analysis are different. The stochastic frontier has an output technical efficiency value of eighty-nine per cent while the data envelopment analysis has an efficiency value of seven-eight per cent. There is significant difference between their environmental efficiencies also. The stochastic frontier analysis records a value of eighty per cent while the data envelopment analysis records a value of fifty-two per cent. It is evident from the result of the two efficiencies that the stochastic frontier method over-values efficiency scores.

Before we close this section we refer the reader to a work by Strauss (1986). The work is important because it attempts to investigate the effect of nutrition on farm labour productivity in Sierra Leone. He uses an average response model to capture this effect. He estimates a Cobb-Douglas production function which accounts for simultaneity in input and calorie choice. His exercise shows calorie intake has significant impact on labour productivity. He, however, places a caveat on this result because individual-level nutrient and anthropometric data are not included in the analysis. His result supports the nutrition – productivity hypothesis to a great extent.

In the last few pages we attempt to explain to the reader the preponderance of the Frequentist method of analysing the stochastic frontier especially in agriculture. We emphasize the diverse uses of the parametric method of efficiency measurement in agriculture. We believe that other literature in agriculture will fall into one of the categories we peruse above. Next, we take a look at the Bayesian econometrist view. The reader should note how few the literature is compared to the Frequentist method. Also, for a thorough perusal of the literature from the Frequentist perspective we refer the reader to Bravo-Ureta et al (2007) Delete.

The Bayesian Studies

The works of van den Broeck, Koop, Osiewalski and Steel (1994); Koop, Osielwalski and Steel (1994, 1997); Koop, Steel and Osielwalski (1992), and, Fernández, Osiewalski and Steel (1997) herald the Bayesian technique for estimating the compose-error model.

van den Broeck, Koop, Osiewalski and Steel (1994) is a primer for estimating a Bayesian cross-sectional composed-error data. They resolve the problem of choosing the best functional form experienced in classical econometrics by mixing over a number of distributions. They use the Bayesian model averaging to average over the results of the Jondrow et al. (1982) and Greene (1990). In other words van den Broeck, Koop, Osiewalski and Steel (1994) solve the problem of choosing the better distribution between the two. They also carry out predictive inference on their results using the Monte Carlo technique of importance sampling.

In continuation of van den Broeck, Koop, Osiewalski and Steel (1994) work; Koop, Osielwalski and Steel (1994) show how to use the Gibbs sampling Monte Carlo method to arrive at estimates for the stochastic cost frontier model. They fit an asymptotically ideal price aggregator, non-constant returns to scale composed error cost frontier. They use Barnett, Geweke, and Wolfe (1991) method for generating the asymptotically ideal price aggregator (Koop, Osielwalski and Steel, 1994 cite Barnett, Geweke, and Wolfe 1999). They caution that care should be taken in the choice of functional form for frontier analysis. We believe the use of the Bayesian model Averaging technique should circumvent this problem. Also, they discover that imposing regularity condition on the price aggregator is found to reduce the spread of the “Müntz- Szatz” expansion.

Koop, Steel and Osielwalski (1995) essentially show how to draw the different parameters in the composed-error model using the Gibbs sampler. They provide an algorithm to draw the different parameters of choice in the composed-error model. They show the ease with which this can be done using the Gibbs sampler. They also note the use of 0.875 as an informative prior for the inefficiency value. van den Broeck, Koop, Osielwalski and Steel (1994) propose this value.

Fernández, Osiewalski and Steel (1997) introduce the Bayesian method for estimating panel data using a class of non- or partly-informative prior. They assert that using this type of priors for a cross-sectional data will make its posterior inference unreliable and inaccurate. This is because the total number of parameters in the entire model is larger than the sample size. They circumvent this problem in the panel data where the researcher can impose a structure on the inefficiency terms. Koop, Osielwalski and Steel (1997) take Fernández et al (1997)’