The paper provides an international analysis of agricultural productivity. It relies on non-parametric methods to estimate a representation of technology. The analysis uses FAO annual data on agricultural inputs and outputs for 12 countries between 1960 and 1994. Productivity indexes are estimated using non-parametric methods. They show the evolution of agricultural productivity both over time and across countries. The empirical results illustrate the usefulness of the methodology as well as the limitations of current data.
Much interest has focused on the international analysis of agricultural productivity. Differences in agro-climatic conditions, human capital and infrastructure appear to contribute to a spread in agricultural productivity across countries. Following an early cross-country analysis by Bhattaccharjee (1995), a significant impetus in understanding the factors influencing international agricultural productivity was provided by Hayami (1969, 1970) and Hayami and Ruttan (1970), with updating by Nguyen (1979), Kawagoe and Hayami (1983), and Kawagoe, Hayami and Ruttan (1985). They stressed the influence of education and human capital on productivity growth. By conducting a separate analysis for developed countries and developing countries, Hayami and Ruttan (1970) implicitly assumed that technology may change with the level of development. This raises the question of how agricultural productivity varies across countries. In an attempt to explain cross-country differences in agricultural productivity, Evenson and Kislev (1975) emphasized the role of research, while Antle (1983) focused on the influence of infrastructure. These studies used econometric analyses relying on a Cobb-Douglas production function. They all faced both measurement problems (e.g. some variables are not consistently measured across countries) and multicollinearity problems that prevented a precise estimate of some parameters. In a recent survey of this literature, Mundlak (forthcoming) noted some significant differences in the empirical findings across studies. He concludes that previous research "provide clear evidence for the lack of robustness of the empirical results". One possible explanation may be the restrictive nature of the Cobb-Douglas production function. It is now well known that the Cobb-Douglas form is not "flexible" in the sense that it restricts a priori the Allen elasticities of substitution among inputs to be equal to one (Arrow et al., 1961). This suggests using more flexible functional forms. Another possibility may be that the production function varies across countries in ways that are more complex than acknowledged in the literature (Mundlak and Hellinghausen, 1982). Finally, it may be that data quality and measurement problems have adverse effects on the reliability of the empirical results.
In an attempt to develop more flexible analyses of production issues, two approaches have been proposed in the literature: the flexible parametric approach, and the non-parametric approach. The flexible parametric approach relies on a flexible parametric specification of the production function, cost function or profit function (Forsund, Lovell and Schmidt, 1980; Bauer, 1990). A parametric form is said to be flexible if it does not impose a priori restrictions on the Allen elasticities of substitution among inputs. For example, the translog specification proposed by Christensen and Jorgenson (1970) is "flexible" and has been commonly used in econometric analyses of production issues over the last two decades. The flexible parametric approach provides a consistent framework for investigating econometrically technology, production behaviour, and productivity growth. However, it still requires imposing parametric restrictions on the technology underlying production behaviour (Bauer, 1990). Alternatively, the non-parametric approach has been developed following the work of Afriat (1972), Hanoch and Rothschild (1972), Diewert and Parkan (1983), and Varian (1984). It has the advantage of imposing no a priori restriction on the underlying technology (Seiford and Thrall, 1990; Fare et al., 1985).
This research develops a non-parametric approach to production analysis to investigate international agricultural productivity. The analysis develops productivity indexes based on Shephard's (1970) input distance function and non-parametric representation of technology. The approach is applied to FAO annual data of inputs and outputs for twelve countries for the period 1960-1994. This illustrates the usefulness of the non-parametric approach in analysing and understanding international agricultural productivity. It also illustrates the complexity of measurement issues in productivity analysis, and the importance of data quality.
The paper is organized as follows. Section 2.2 reviews some basic concepts of production theory, relying on Shephard's distance function. It presents technical efficiency indexes that can be used to characterize the rate of technical change and productivity growth. In a first step toward the non-parametric estimation of these indexes, section 2.3 summarizes some results obtained by Afriat (1972) and Varian (1984) on non-parametric production analysis. Given a finite number of observations on production data, a key result is the derivation of representations of the underlying technology. Section 2.4 illustrates the usefulness of non-parametric methods in an application to international agricultural productivity analysis. The analysis relies on time-series cross-section data and generates non-parametric estimates of indexes providing information on the importance and nature of technical change and productivity growth in international agriculture over the last few decades.
This section briefly presents and reviews some index numbers that are useful in productivity analysis. It also sets up the notation for the rest of the paper. Consider a competitive firm choosing (x, y) where:
is a (n x 1) vector of inputs and is a (m x 1) vector of outputs. Using the netput notation (where outputs are positive and inputs are negative), the corresponding netput vector is (-x, y). Focus is on a general multi-output multi-input joint technology represented by the feasible set , where (-x, y) F. We assume throughout the paper that the production set F is non-empty, closed, convex, and negative monotonic.^{1}
Given some input-output vector (x, y), the technology F can be characterized by Shephard's input distance function:
(2.1)
Assuming that the maximization problem in 2.1 has a solution, the distance function involves a proportional rescaling of inputs x toward the frontier technology. Shephard (1970) has shown how the distance function D(x, y, F) characterizes the production technology. D(x, y, F) is linearly homogeneous in x, non-decreasing in x, and non-increasing in y. It has the following interpretations. First, (-x, y) F if and only if (Shephard, p. 67). Second, D = 1 implies that the inputs-outputs vector (x, y) is on the production frontier associated with the technology F, while D >1 (< 1) implies that the inputs-outputs (x, y) are below (above) the production frontier (Shephard, p. 67). As such, D(x, y, F) = 1 can be used as a representation of the multi-input, multi-output production frontier associated with technology F. And as long as D is finite, the outputs-inputs vector (x/D, y) is on the production frontier of technology F, where (x/D) is the (n´1) vector of input quantities x "radially rescaled" toward the production frontier.
Following the work of Debreu (1951), Farrell (1957), Farrell and Fieldhouse (1962) and Fare et al. (1985), technical efficiency can be defined as the minimal proportion by which a vector of inputs x can be rescaled while still producing outputs y. It follows that the distance function D in 2.1 can be used to define the following Farrell technical efficiency index
T(x, y, F) = 1/D(x, y, F). (2.2)
The technical efficiency index T satisfies (-x, y) F if and only if . Also, T(x, y, F) is homogeneous of degree (-1) in x, non-increasing in x, and non-decreasing in y. Moreover, T = 1 implies that the output-input vector (x, y) is on the production frontier of technology F, while T < 1 (> 1) means that the outputs-inputs are below (above) the production frontier. Then (x * T) is the (n x 1) vector of inputs quantities x that are radially rescaled toward the production frontier. Finding T = 1 identifies a situation of technical efficiency, where inputs-outputs (x, y) are on the production frontier. Alternatively, finding T < 1 identifies a situation of technical inefficiency where inputs-outputs (x, y) are below the production frontier. The technical inefficiency is motivated by the fact that, under technology F, the outputs y could have been produced by using less inputs x. In this context, (1 - T) can be interpreted as measuring the proportional reduction in all inputs x that could have been attained producing outputs y in a technically efficient way (given a radial reduction in inputs from x toward the production frontier). Alternatively stated, letting r be the (n x 1) vector of input prices for x, (1 - T) measures the proportional reduction in input cost r'x that could have been obtained by becoming technically efficient in the production of outputs y. Finally, finding T > 1 identifies a situation of "super technical efficiency" where inputs-outputs (x, y) are above the production frontier under technology T. Since technical feasibility is equivalent to , such a case is possible only under technological change. For example, this would correspond to a situation where inputs-outputs (x, y) are chosen under a technology F' that is better than technology F in the sense that (-x, y) F' F.
The input distance function D in 2.1 or the technical efficiency index T in 2.2 can be used to define productivity indexes. For that purpose, consider two situations: situation (a) corresponding to inputs-outputs (x^{a}, y^{a}) under technology F^{a}, and situation (b) corresponding to inputs-outputs (x^{b}, y^{b}) under technology F^{b}. Then, technological change between (a) and (b) can be measured using the following two productivity indexes
P(a, b) = D(x^{a}, y^{a}, F^{a})/D(x^{b}, y^{b}, F^{a}) = T(x^{b}, y^{b}, F^{a})/T(x^{a}, y^{a}, F^{a}) (2.3a)
and
P'(a, b) = D(x^{b}, y^{b}, F^{b})/D(x^{b}, y^{b}, F^{a}) = T(x^{b}, y^{b}, F^{a})/T(x^{b}, y^{b}, F^{b}). (2.3b)
Both productivity indexes 2.3a and 2.3b are ratios of distance functions D, or equivalently of technical efficiency indexes T. The index P(a, b) in 2.3a is the Malmquist productivity index proposed by Caves et al. (1982): it is the ratio of two distance functions evaluated under the same technology F ^{a}, but at different points (x^{a}, y^{a}) and (x^{b}, y^{b}). Alternatively, the index P'(a, b) in 2.3b is the ratio of two distance functions evaluated at the same point (x^{b}, y^{b}), but under the different technologies F^{a} and F^{b}. The index P' measures productivity in the sense that P' > 1 for all (x^{b}, y^{b}) implies that F ^{a} F ^{b}, meaning that technology F ^{b} is better than technology F ^{a} in the sense that the feasible set has expanded between situation (a) and situation (b). When (a) and (b) denote different time periods, this means that technological progress has taken place between time (a) and time (b). Then, (PI' - 1) can be interpreted as a measure of the rate of technological change. It reflects the proportional reduction in all inputs x (or in input cost r'x) that could have been attained producing outputs y using technology F ^{b} instead of F ^{a} (given a radial reduction in inputs from x toward the shifting production frontier).
In general, the two productivity indexes P in 2.3a and P'in 2.3b differ from each other. However, a special case of interest has been considered by Caves et al. (1982). It involves situations where the observed inputs-outputs (x^{a}, y^{a}) and (x^{b}, y^{b}) are always on the production frontier. In such cases,
D(x^{a}, y^{a}, F^{a}) = 1 = T(x^{a}, y^{a}, F ^{a}) and D(x^{b}, y^{b}, F ^{b}) = 1 = T(x^{b}, y^{b}, F ^{b}).
It follows that
P = P' = 1/D(x^{b}, y^{b}, F ^{a}) = T(x^{b}, y^{b}, F ^{a}). (3c)
This means that the two productivity indexes P and P' become identical, and productivity can be measured by the technical efficiency index T evaluated at (x^{b}, y^{b}) under technology F ^{a}. Finding P = P' = 1 would mean that switching between technology F ^{a} and F ^{b} generates no technical efficiency gain, and no productivity growth. Alternatively, finding P = P' > 1 for all (x, y) would imply that technology F ^{b} is better than technology F ^{a} in the sense that F ^{b} F ^{a}. In the case where P > 1, (P - 1) is a measure of productivity growth between (a) and (b). It gives the proportional increase in inputs x^{b} that would be required to produce outputs y^{b} using technology F ^{a} instead of F ^{b} (given a radial rescaling of inputs x^{b} toward the shifting frontier isoquant). Alternatively, letting r be a (n x 1) vector of input prices for x^{b}, (P - 1) measures the proportional increase in cost r' x^{b} that would be needed to produce y^{b} using technology F ^{a} instead of F ^{b}. And in the case where P < 1, (1 - P) is a measure of productivity growth between (b) and (a). It is the proportional decrease in inputs x^{b} (or in input cost r'x^{b}) that could be obtained in a technically efficient production of outputs y^{b} using technology F ^{a} instead of F ^{b} (given a radial rescaling of inputs x>^{b} toward the shifting frontier isoquant).
Note that alternative indexes have been discussed in the literature. For example, indexes similar to 2.1, 2.2 and 2.3 can be obtained by rescaling outputs instead of inputs (Shephard, 1970; Fare et al., 1985; Caves et al., 1982). Also, indexes that rescale both inputs and outputs have been proposed (Fare et al., 1985). In general, input-based indexes can differ from output-based indexes (Caves et al., 1982; Fare et al., 1985). The difference relates to the nature of returns to scale (Caves et al., 1982). So far, we have considered a general variable-return-to-scale (VRTS) technology F. This suggests that we also consider the constant-return-to-scale (CRTS) technology
(2.4)
The technology F _{c} generated by F in 2.4 is the smallest CRTS technology that contains F. It satisfies F F_{c}. The CRTS technology F_{c} has an interesting property. As shown by Caves et al. (1982) and Fare et al. (1985), under a CRTS technology, input-based and output-based technical efficiency indexes as well as productivity indexes are identical. In this case, the technical efficiency index T(x, y, F_{c}) can be interpreted as the largest scalar satisfying (-x, * y) F_{c}. Then, if T(x, y, F_{c}) > 1, (T - 1) measures the proportional increase in outputs y that could have been obtained using inputs x under technology F_{c}. And if T(x, y, F_{c}) < 1, (1 - T) is the proportional decrease in outputs y that would be generated by using inputs x under technology F_{c}.
In the context of (3c) where P = P' = T(x^{b}, y^{b}, F_{c}^{a}), finding P = P' > 1 under CRTS means that (P - 1) gives the proportional decrease in outputs y^{b} that would be required using inputs x^{b} under technology F ^{a} instead of F ^{b} (given a radial rescaling of outputs y^{b} toward the shifting production frontier). Alternatively, letting p be a (m x 1) vector of output prices for y, (P - 1) measures the proportional reduction in revenue p'y^{b} that would be generated by using inputs x^{b} under technology F ^{a} instead of F ^{b}. And finding P = P' < 1 under CRTS implies that (1 - P) is the proportional increase in outputs y^{b} (or in revenue p'y^{b}) that can be obtained by using inputs x^{b} under technology F ^{a} instead of F ^{b}. This is consistent with the intuitive interpretation of productivity growth, where (1 - P) (or (P - 1) if P > 1) is the proportional output change that cannot be explained by changes in inputs x and is thus attributed to technological change.
The interpretation of the production indexes T in 2.2 or P in 2.3 typically depends on the situation being considered. Such interpretations are discussed next.
Most industries exhibit technical progress over time. As a result, time series production data from any industry typically reflect some shift in technology across observations. As noted in equation 2.3b, the indexes T, P and P' become identical under the assumption that (x, y) are technically efficient. Then, T = P becomes the Malmquist productivity index discussed by Caves et al. (1982, p. 1407). Indeed, assuming that (x, y) is always on the production frontier,^{ }(1 - T) is a measure of the rate of technical change for inputs-outputs (x, y) compared to the reference technology F. In this context, if F represents some new technology, T < 1 means that the firm's old technology F' is not as good as the new technology F. Then, (1 - T) measures the proportional reduction in inputs or cost (or, under CRTS, the proportional increase in outputs or revenue) associated with changing technology from F' to F. Caves et al. (1982) investigated the relationship between the productivity index T in equation 2.3 and the Christensen-Jorgenson (CJ) productivity index commonly estimated in the literature (Ball, 1985). Under optimizing behaviour, constant return to scale, and a translog cost specification, Caves et al. (1982, p. 1408) showed that the CJ index can be written as [T(x^{b}, y^{b}, F_{c}^{a})/T(x^{a}, y^{a}, F_{c}^{b})]^{1/2}. They also investigated the impact of the scale elasticity (reflecting departures from CRTS) on productivity measurement.
Using cross section data, production analysis typically focuses on a set of economic units (firms or regions) at a given time within an industry. Assuming that the best available technology is available to all units within the industry, the concept of technical efficiency relates to the question of whether a firm uses the best available technology in its production process. This is precisely what is measured by the index T in 2.2, with F representing the best available technology. In this context, given an observed input-output vector (x, y), the Farrell index T(x, y, F) in 2.2 provides a simple measure of technical efficiency, with T £ 1. T = 1 implies that the firm is technically efficient and produces on the production frontier associated with technology F. Alternatively, T < 1 implies that the firm is not technically efficient as its inputs-outputs are below the production frontier
It has just been argued that the indexes T and P can be interpreted as measuring productivity change from cross section data, as well as technical change in time series data. But is it possible to isolate technical efficiency effects from productivity effects? Without a priori information on the sources of inefficiency or technical change, the general answer to this question is no. Indeed, knowing that a firm exhibits an index T < 1 does not distinguish between two alternative interpretations. On the one hand, the firm may be a late adopter of new technology F and, by still using an old technology, it appears below the production frontier of F. On the other hand, it is possible that the firm adopted the latest technology but is technically inefficient in its use. Thus, it is in general difficult to distinguish between the slow adoption of new technology and technical inefficiencies unrelated to technical progress. These difficulties have led many researchers to circumvent the problem by ignoring production inefficiency issues in productivity analysis (e.g. Binswanger, 1974; Ball, 1985), as well as ignoring technical change issues in efficiency analysis (e.g. Chavas and Aliber, 1993).
Yet, it seems reasonable to expect that, in most industries, a slow adoption of new technology can coexist with unrelated production inefficiencies. A way of dealing with this issue is to use panel data, which can allow for a simultaneous investigation of production efficiency and technical change. The cross-section part of the data can then be used to estimate time-specific production frontiers. And the time series part of the data can be used to evaluate the rate of productivity growth as the frontier technology shifts over time. Thus, the cross-section information can generate production efficiency indexes, while the time series information can yield indexes of technical change. Fare et al. (1994) give an example of this approach, which distinguishes empirically between productivity growth and efficiency change. However, this approach requires two conditions to be satisfied. First, at each time period, every unit in the cross-section sample must face a comparable production technology. Second, at each time period, there must be enough cross-section units observed on the production frontier (in order to provide a reliable empirical estimate of the production technology). Whether these conditions are satisfied or not would need to be assessed in the evaluation of the approach.
Given a set of production data, how can the indexes T or P just discussed be empirically estimated? This can be done using either parametric methods or non-parametric methods. This paper focuses on the use of non-parametric methods in production analysis. This section briefly reviews some key results on non-parametric production methods obtained by Afriat (1972), Hanoch and Rothschild (1972), Diewert and Parkan (1983), Varian (1984) and Banker and Maindiratta (1988).
Again, consider a competitive firm choosing the input-output vector (x, y) 0 under technology F. Assume that it behaves in a way consistent with the profit maximization hypothesis. Let p = (p_{1}, ..., p_{m})' > 0 denote the (m x 1) vector of output prices, and r = (r_{1}, ..., r_{n})' > 0 be the (n x 1) vector of input prices. Then, the firm production decisions are made as follows:
max_{y,x} {p' y - r' x: (-x, y) F }, (2.5)
The solution to 2.5 gives the profit maximizing output supplies and input demand correspondences denoted by y^{*}(p, r) and x^{*}(p, r).
Consider that the firm is observed making production decisions times. Let S be the set of these observations:
S = { 1, 2, ..., }. The t-th observation on production decisions is denoted by (y_{t}, x_{t}), with corresponding prices (p_{t}, r_{t}), t S. Economic rationality for production decisions is defined in terms of profit maximizing behaviour as stated in equation 2.5. It can be said that a production set F rationalizes the data {(y_{t}, x_{t}; p_{t}, x_{t}): t S} if y_{t} y^{*}(p_{t}, r_{t}) and x_{t} Î x^{*}(p_{t}, r_{t}), t S. A key linkage between observable behaviour and production theory is given next.
Proposition 1: (Afriat, 1972; Varian, 1984)
The following conditions are equivalent:
p_{t}' y_{t} - r_{t}' x_{t} p_{t}' y_{s} - r_{t}' x_{s}, t S, s S. (2.6)
Given 2.6, there exists a family of convex, negative monotonic production sets F that rationalizes the data in T according to 2.5, and satisfies F ^{i} F F ^{o}, where:
(2.7a)
and
(2.8)
Equation 2.6 states that the t-th profit (p_{t}' y_{t} - r_{t}' x_{t} ) is at least as large as the profit that could have been obtained using any other observed production decision (p_{t}' y_{s} - r_{t}' x_{s}), s S. It gives necessary and sufficient conditions for the data {(y_{t}, x_{t}; p_{t}, r_{t}): t S } to be consistent with profit maximization 2.5. This is useful as a means of testing the relevance of production theory in particular situations. Perhaps more importantly, proposition 1 provides a basis for recovering some representations of the underlying production technology. More specifically, it identifies a whole family of production sets that are consistent with the data and the profit maximization hypothesis. This family is bounded by F ^{i} in 2.7a and F ^{o} in 2.8. Proposition 1 states that F ^{i} in 2.7a gives the inner bound while F ^{o} in 2.8 is the outer bound representation of the underlying technology (Afriat, 1972; Varian, 1984). These representations are of considerable interest since they are empirically tractable and provide all the information necessary to conduct production analysis. The inner-bound representation F ^{i} in 2.7a has been commonly used in applied efficiency analysis, where it has been called data envelopment analysis or DEA. Note that it requires only data on input-output quantities. In contrast, the outer-bound representation F ^{o} in 2.8 requires data on both prices and quantities for inputs-outputs.
The Afriat-Varian non-parametric results reported in proposition 1 assume that all data points in S are consistent with the profit maximization hypothesis. However, this assumption is not always empirically satisfied. Thus, there is a need to extend the Afriat-Varian non-parametric analysis to allow for situations where profit maximizing behaviour does not hold for all observations in S. Such an extension was proposed by Banker and Maindiratta (1988). In the situation where equation 2.6 is not satisfied for all s, t S, Banker and Maindiratta proposed a method relying on the subset of data points that are consistent with profit maximization. This subset is given by:
(2.9)
Clearly, the criterion function in 2.9 always satisfies 0 for all t S. And = 0 only if there does not exist any data point s S such that p_{t}' y_{t} - r_{t}' x_{t} < p_{t}' y_{s} - r_{t}' x_{s} , i.e. such that equation 2.6 is violated. As a result, any observation in E S is necessarily consistent with profit maximization with respect to all data points in S. For this reason, Banker and Maindiratta call E the efficient subset of S. Banker and Maindiratta obtained the following results.
Proposition 2: (Banker and Maindiratta, 1988)
Assuming that E is non-empty, the following conditions are equivalent:
p_{t}' y_{t} - r_{t}' x_{t} p_{t}' y_{s} - r_{t}' x_{s}, t E, s S. (2.10)
Given 2.10, there exists a family of convex, negative monotonic production sets F that rationalizes the data in E according to 2.5, and satisfies (-x _{t} , y _{t} ) F for all t S, F ^{i} F F_{E}^{o}, where F ^{i} is given in 2.7a and
F_{E}^{o} = { (-x, y): p_{t}' y - r_{t}' x p_{t}' y_{t} - r_{t}' x_{t}, t E; x 0; y 0 }. 2.11
Note that proposition 2 reduces to proposition 1 when E = S. However, it allows for inconsistencies between the data in S and profit maximization whenever E is a proper subset of S. Note that such inconsistencies may arise because of technological progress across observations. In this case, proposition 2 would be particularly relevant in productivity analysis. Since the observations in E are consistent with profit maximization, their efficiency cannot be refuted by the data. In contrast, the observations that are in S but not in E are inconsistent with profit maximization. Moreover, F ^{i} in 2.7a and F_{E}^{o} in 2.11 can be used as inner bounds and outer bounds representations of the underlying technology F. In turn, such representations can be used to evaluate production efficiency and technical change. Note that the inner-bound representation F ^{i} is the same data envelopment analysis (DEA) representation 2.7a found in proposition 1. Again, it requires only data on input-output quantities. However, the outer-bound representation F_{E}^{o} in 2.11 differs from the one found in proposition 1, F ^{o} in 2.8. Again, evaluating F_{E}^{o} requires data on both prices and quantities for inputs-outputs.
This approach proposed by Banker and Maindiratta (1988) has one drawback. There are situations where the efficient subset E can be much smaller than the set S. This occurs when the number of observations in E (used to evaluate the outer-bound representation F_{E}^{o}) is significantly smaller than the number of observations in S (used to evaluate the inner-bound counterpart). This could be undesirable. For example, if E were to consist of only a few data points, the associated technology F_{E}^{o} would have few kinks, implying a relatively flat production frontier. Although not inconsistent with production theory, such a representation of the real world may be somewhat unrealistic. This has motivated Chavas and Cox (1997) to propose a modification of the Banker and Maindiratta approach that does not suffer from this drawback.
While the above analysis was presented under a general variable-return-to-scale (VRTS) technology F, it may be of interest also to consider the case of the constant-return-to scale (CRTS) technology F _{c} given in 2.4. Then, the results stated in propositions 1 and 2 can be appropriately modified. First, under CRTS, the inner-bound representation of technology given in 2.7a becomes
(2.7b)
which satisfies F ^{i} F_{c}^{i}. The set F_{c}^{i} in 2.7b is a convex, negative monotonic production set that rationalizes the data and satisfies (-x_{t} , y_{t}) F_{c}^{i} for all i S. It also satisfies the definition of CRTS: (- x, y) F_{c}^{i} for all > 0, indicating that a proportional rescaling of all inputs and outputs always remains feasible. The difference between 2.7a and 2.7b is slight: 2.7a restricts the weights to sum to one, while 2.7b does not (see Fare, Grosskopf and Lovell, 1985). Not requiring the weights to sum to one implies that the proportional rescaling of all inputs-outputs in 2.7a is unrestricted (with the proportionality factor being . This feasible rescaling of all inputs and outputs is precisely what characterizes a CRTS technology.
Second, the outer-bound representation given in 2.8 or 2.11 can also be appropriately modified under CRTS. It is well known that profit maximization and CRTS implies zero profit: p_{t}' y_{t} - r_{t}' x_{t} = 0. Then, following Afriat (1972), Varian (1984), and Banker and Maindiratta (1988), the outer-bound representations given in 2.8 and 2.11 would be obtained under CRTS, with the efficiency set E defined under the additional condition that p_{t}' y_{t} - r_{t}' x_{t} = 0. Alternatively, the analysis could be conducted under cost minimization and CRTS, as suggested by Varian. Again, the proposed modification discussed by Chavas and Cox (1997) could apply in this context as well.
The above discussion indicates practical ways of evaluating technology from production data, using non-parametric methods. Once the appropriate production technology is evaluated, it provides a basis for estimating the technical efficiency and productivity indexes T and P discussed in section 2.2. In fact, these two steps can be conveniently combined into a single step. Indeed, equations 2.7a, 2.7b, 2.8 or 2.11 are linear. Substituting these representations into 2.1, 2.2 or 2.3 generates simple linear programming problems (Fare, Grosskopf and Lovell, 1985). Since solving linear programming problems is not difficult, this means that the empirical implementation of the proposed methodology is fairly simple, as illustrated in the next section.
In this section, the usefulness of this approach applied to international agricultural productivity is illustrated. The analysis uses FAO annual data for inputs and outputs of 12 countries for the period 1960-1994. The 12 countries are: Brazil (BRA), Burkina Faso (BFA), China (CHN), France (FRA), India (IND), Madagascar (MDG), Mexico (MEX), Peru (PER), Poland (POL), Thailand (THA), Tunisia (TUN), and USA (USA). Outputs consist of two categories: crops and livestock. Inputs consist of four categories: fertilizer, labour, land and farm machinery. Fertilizer is measured by the total fertilizer weight. Labour is measured by rural population. Land is the area in cultivated land. And the number of tractors is used as a proxy variable for farm machinery. Note that these measured do not correct for possible effects of input quality changes on productivity.
Using FAO data, a non-parametric analysis of agricultural productivity both over time and across countries is presented. Since the FAO data only report quantity information, this investigation is focused on the inner-bound representation of technology given by 2.7a and 2.7b. This has two implications. First, it means that the analysis does not require any assumption about production behaviour. For example, it does not demand that inputs-outputs be chosen so as to maximize profit. This can be an advantage to the extent that production decisions may have more complex motivations than just profit maximization. For example, in an uncertain world, risk minimization may also be relevant. Second, by neglecting price information, little priori information is imposed on marginal rates of substitution. While diminishing marginal productivity is imposed (since the feasible sets F ^{i} or F_{c}^{i} are convex), the marginal physical product of any input in principle can vary between zero and infinite. Allowing for such a wide range of possibilities gives additional flexibility to the analysis. This enhanced flexibility should also reflect more accurately the information content of the data used in the analysis.
The empirical analysis proceeds in two steps. In a first step, a time series analysis of productivity for each country over the period 1960-1994 is conducted. This should provide useful information about agricultural technological change in each country over the last few decades. Note that this approach estimates a separate production frontier for each country. As a result, it does not require that the production technology is similar across country. On the one hand, this can be a significant advantage to the extent that agro-climatic conditions as well as physical and human capital can vary significantly across countries. While such factors can in principle be incorporated in the analysis, measuring them accurately is typically difficult. On the other hand, by associating each country with a different technology, this approach does not yield information on cross-country differences in technology and productivity.
In a second step, a cross-section analysis of productivity across countries is presented. To allow for the possibility that technology may change over time, the cross-section analysis is conducted for selected periods. This should provide useful information on cross-country productivity at a given time period as well as its evolution over time. However, as discussed in section 2.2, this requires two conditions to be met:
Starting start with the first step: a time series analysis is conducted for each country. Let (x_{t} , y_{t}) denote the t-th observation on inputs-outputs, t S, where S is the set of time series data for a given country. Using the VRTS representation of the technology F ^{i} given in 2.7a, how can the technical efficiency index T(x, y, F ^{i}) in 2.2 be evaluated? This can be obtained as follows
T(x, y, F ^{i}) = 1/D(x, y, F ^{i})
= Min_{} { : (- * x, y) F ^{i} }
= Min_{, } { : y_{t} y; x_{t} * x; = 1; 0, t S }, (2.12a)
for some x 0, y 0. In general, T(x, y, F ^{i}) satisfies 0 < T(x_{t}, y_{t}, F ^{i}) 1 for all t S. Expression 2.12a is a standard linear programming problem. As discussed in sections 2.2 and 2.3, it provides a convenient basis to evaluate productivity.
Alternatively, using the CRTS representation of the technology F_{c}^{i} given in 2.7b, how can the technical efficiency index T(x, y, F_{c}^{i}) in 2.2 be evaluated? This can be obtained as follows
T(x, y, F_{c}^{i}) = 1/D(x, y, F_{c}^{i})
= Min_{} { : (- * x, y) F_{c}^{i} }
= Min_{ ,} { : y_{t} y; x_{t} * x; 0, t S }, (2.12b)
for some x 0, y 0. Since F ^{i} F_{c}^{i}, it is clear that T(x, y, F_{c}^{i}) T(x, y, F ^{i}). In general, T(x, y, F_{c}^{i}) satisfies
0 < T(x_{t} , y_{t}, F ^{i}) 1 for all t S. Again, expression 2.12b is a standard linear programming problem that provides a convenient basis to evaluate productivity.
Equation 2.12b was used to estimate the technical efficiency index T(x_{t} , y_{t} , F_{c}^{i}) for each of the 12 countries, using time series data from t equals 1960 to 1994. In this context, for each country, F_{c}^{i} is a representation of the best CRTS technology that was available between 1960 and 1994. As discussed in section 2.2, these indexes provide a basis to evaluate productivity growth in each country. The results are presented in Table 2.1.
Table 2.1 shows that the technical efficiency indexes T vary from a low of 0.690 for Burkina Faso in 1977 to its upper-bound of one. In general, the T index tends to decrease during the 1960s, reach a minimum some time in the early or late 1970s, and rise to its upper-bound of one in the early 1990s. The decrease in technical efficiency found in most countries during the late 1960s appears somewhat puzzling. This could possibly be due to environmental degradation. But this is not consistent with the results obtained for Madagascar. While Madagascar has suffered significant soil erosion, deforestation and ecological deterioration over the last few decades, it exhibits one of the least amounts of technical inefficiency. Indeed, its lowest T index is 0.915 in 1979, which follows closely India (T equals 0.935 in 1979) and Thailand (T is 0.919 in 1979 and 1986). Thus, the decrease in productivity in the late 1960s seems hard to explain.
TABLE 2.1
Technical Efficiency Indexes T: Time Series Analysis Conducted for Each Country
year |
BRA |
BFA |
CHN |
FRA |
IND |
MDG |
MEX |
PER |
POL |
THA |
TUN |
USA |
1961 |
1.000 |
0.963 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1962 |
1.000 |
0.927 |
0.969 |
1.000 |
0.988 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
0.965 |
0.956 |
1963 |
0.951 |
0.914 |
1.000 |
0.949 |
0.988 |
0.985 |
0.962 |
0.966 |
0.972 |
1.000 |
1.000 |
0.941 |
1964 |
1.000 |
1.000 |
1.000 |
0.911 |
0.986 |
0.995 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
0.969 |
1965 |
1.000 |
1.000 |
1.000 |
0.959 |
0.965 |
0.922 |
1.000 |
0.950 |
0.988 |
1.000 |
1.000 |
0.900 |
1966 |
1.000 |
1.000 |
1.000 |
0.916 |
0.963 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
0.865 |
1967 |
0.954 |
0.979 |
1.000 |
0.908 |
0.961 |
1.000 |
0.984 |
0.990 |
1.000 |
0.958 |
1.000 |
0.858 |
1968 |
0.989 |
1.000 |
1.000 |
0.904 |
0.994 |
1.000 |
0.987 |
0.919 |
1.000 |
0.981 |
0.892 |
0.855 |
1969 |
1.000 |
0.999 |
0.953 |
0.870 |
0.993 |
0.991 |
0.910 |
0.950 |
0.990 |
1.000 |
0.828 |
0.842 |
1970 |
0.991 |
0.985 |
1.000 |
0.852 |
1.000 |
0.991 |
0.965 |
1.000 |
0.962 |
1.000 |
0.785 |
0.853 |
1971 |
0.975 |
1.000 |
1.000 |
0.860 |
0.986 |
0.948 |
0.990 |
1.000 |
0.932 |
0.949 |
0.935 |
0.869 |
1972 |
0.982 |
0.879 |
0.918 |
0.862 |
0.929 |
0.980 |
0.972 |
0.876 |
0.972 |
0.920 |
0.844 |
0.864 |
1973 |
0.965 |
0.706 |
0.968 |
0.911 |
0.976 |
0.972 |
0.988 |
0.898 |
1.000 |
1.000 |
0.924 |
0.815 |
1974 |
0.978 |
0.740 |
1.000 |
0.953 |
0.946 |
1.000 |
0.985 |
0.889 |
1.000 |
0.971 |
0.886 |
0.848 |
1975 |
0.890 |
0.801 |
0.965 |
0.953 |
0.983 |
1.000 |
0.930 |
0.899 |
0.987 |
1.000 |
0.964 |
0.833 |
1976 |
0.905 |
0.696 |
0.977 |
0.943 |
0.992 |
0.968 |
0.940 |
0.858 |
1.000 |
1.000 |
0.974 |
0.878 |
1977 |
0.900 |
0.690 |
0.872 |
0.919 |
1.000 |
0.940 |
1.000 |
0.843 |
0.970 |
0.940 |
1.000 |
0.868 |
1978 |
0.798 |
0.750 |
0.896 |
0.929 |
1.000 |
0.989 |
1.000 |
0.806 |
1.000 |
1.000 |
0.909 |
0.847 |
1979 |
0.803 |
0.797 |
0.840 |
0.970 |
0.935 |
0.906 |
0.947 |
0.830 |
1.000 |
0.919 |
0.763 |
0.875 |
1980 |
0.848 |
0.751 |
0.766 |
0.989 |
0.947 |
0.915 |
1.000 |
0.780 |
0.972 |
1.000 |
0.853 |
0.850 |
1981 |
0.872 |
0.759 |
0.814 |
1.000 |
0.986 |
0.944 |
0.971 |
0.854 |
0.897 |
0.990 |
0.868 |
0.899 |
1982 |
0.860 |
0.776 |
0.865 |
1.000 |
0.979 |
0.914 |
0.887 |
0.924 |
0.869 |
0.947 |
0.841 |
0.915 |
1983 |
0.919 |
0.827 |
0.872 |
0.994 |
1.000 |
0.919 |
0.907 |
0.916 |
0.893 |
0.984 |
0.901 |
0.876 |
1984 |
0.877 |
0.832 |
0.903 |
1.000 |
1.000 |
0.963 |
0.935 |
0.963 |
0.944 |
0.987 |
0.871 |
0.870 |
1985 |
0.979 |
0.945 |
0.964 |
0.981 |
1.000 |
0.940 |
0.994 |
0.983 |
0.944 |
1.000 |
0.978 |
0.900 |
1986 |
0.883 |
1.000 |
0.972 |
0.982 |
1.000 |
0.955 |
0.915 |
0.886 |
1.000 |
0.919 |
0.982 |
0.934 |
1987 |
0.967 |
0.955 |
0.876 |
0.992 |
1.000 |
0.968 |
0.937 |
0.937 |
0.961 |
0.921 |
1.000 |
0.922 |
1988 |
0.967 |
0.993 |
0.824 |
0.983 |
1.000 |
0.964 |
0.907 |
1.000 |
0.975 |
0.987 |
0.945 |
0.943 |
1989 |
1.000 |
0.948 |
0.831 |
0.963 |
1.000 |
1.000 |
0.896 |
1.000 |
0.989 |
1.000 |
0.966 |
0.917 |
1990 |
0.981 |
0.962 |
0.884 |
0.972 |
1.000 |
0.990 |
0.984 |
0.942 |
1.000 |
0.985 |
1.000 |
0.929 |
1991 |
1.000 |
1.000 |
0.880 |
1.000 |
0.988 |
0.996 |
0.963 |
1.000 |
1.000 |
1.000 |
1.000 |
0.941 |
1992 |
1.000 |
1.000 |
0.907 |
1.000 |
1.000 |
1.000 |
0.931 |
1.000 |
0.937 |
1.000 |
0.975 |
0.975 |
1993 |
1.000 |
1.000 |
1.000 |
0.997 |
1.000 |
1.000 |
0.977 |
1.000 |
1.000 |
0.974 |
1.000 |
0.957 |
1994 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
0.997 |
1.000 |
1.000 |
0.807 |
1.000 |
1.000 |
1.000 |
Also surprisingly, for each country, the technical efficiency index T in Table 2.1 starts with a value of one some time in the early 1960s and ends up with a value of one in the early 1990s. This suggests that, in general, the technology of the early 1990s is similar to the one in the early 1960s. In other words, the results in Table 2.1 suggests that, overall, there was little technological progress in agriculture over the last few decades. This seems counterintuitive: a priori, one would expect to find some significant technical progress over the last few decades (e.g. due to the green revolution). It is known that parts of the yield increases during the green revolution were associated with a significant rise in fertilizer and pesticides use. Caution should be exercised not to attribute such effects to technological change: they may simply correspond to a move along a given production frontier as more inputs are being used. However, significant and steady genetic progress has taken place in agriculture, for both crops and livestock. It is expected that at least part of this genetic progress would contribute directly to agricultural productivity growth in most countries. The results in Table 2.1 appear to be at odds with this conjecture. They are also at odds with previous estimates of productivity growth found in the literature. For example, there is strong evidence that agricultural productivity growth in the United States has been steady and large over the last few decades (Capalbo and Antle, 1988; Ball, 1985; Jorgenson and Gollop, 1992; Chavas and Cox, 1997; Chavas et al., 1997; Ball et al., 1997). Similar evidence has also accumulated in agriculture around the world (Evenson and Kislev, 1975; Pardey et al., 1991; Rosegrant and Evenson, 1992; Craig et al., 1997). Such discrepancies with the results reported in Table 2.1 appear puzzling. This issue is further discussed below.
Now, consider the second step: a cross-section analysis conducted for all 12 countries during specific periods. Changing the notation, let (x_{t} , y_{t} ) denote the t-th observation on inputs-outputs for the 12 countries during a given period, t S, where S is the corresponding data set. Then, the technical efficiency index T(x _{t}, y _{t}, F ^{i}) is given by equation 2.12a above under the variable-return-to-scale (VRTS) technology F ^{I}, and the technical efficiency index T(x, y, F_{c}^{i}) is given by equation 2.12b under constant-return-to-scale (CRTS) technology. Since F ^{i} F_{c}^{i}, it is clear that T(x, y, F_{c}^{i}) T(x, y, F ^{i}). Thus, in general, 0 < T(x _{t}, y _{t}, F_{c}^{i}) T(x _{t}, y _{t}, F ^{i}) 1 for all t S. Again, expression 2.12a or 2.12b is a standard linear programming problem that provides a convenient basis to evaluate productivity.
Equation 2.12b was used to estimate the technical efficiency index T(x _{t}, y _{t}, F_{c}^{i}) for all 12 countries for selected periods covering three consecutive years. The choice of three consecutive years was made on the grounds that technology might change, but slowly, over time. Thus the analysis is conducted separately for 11 different periods: 1962-64, 1965-67, ..., up to 1992-94. In this context, F_{c}^{i} is a representation of the best technology available across all 12 countries during each period. As such, it can be interpreted as characterizing the "international agricultural production function" during that period. As discussed in section 2.2, the estimated technical efficiency indexes can provide a basis to evaluate cross-country productivity and its evolution over time. The results are presented in Table 2.2 for the 11 periods, starting in 1962-64, and ending with 1992-94.
Table 2.2 shows that technical efficiency index T varies from a low of 0.626 for India in 1972 to its upper-bound of one. In general, the T index for India tends to be lower than any other state. Except for 1962, it is always less than one, and it stays in the range 0.6 to 0.7 for most of the 1970s and 1980s. No other country exhibits such a pattern. This seems difficult to explain. Table 2.2 shows that most countries are often found to be on the "international production frontier". For example, China and France exhibit a technical efficiency index equal to one in 26 of the 33 cases evaluated, or 78 percent of the time. For all countries except India and Tunisia, the technical efficiency index is equal to one at least 18 times out of 33, i.e. at least 54 percent of the time. Thus, there appears to be a sufficient number of observations located on the international production frontier to obtain a meaningful representation of the international production technology. The part that appears surprising in Table 2.2 is the fact that, except for India, the technical efficiency indexes are often close to one. As discussed in section 2.2, this suggests that agricultural technology is fairly uniform both across countries and over time. This is rather surprising since this analysis did not take into consideration cross-country variations in agro-climatic conditions, in infrastructure and in human capital. At a minimum, one would expect to find that agricultural productivity depends significantly on soil quality and climate. Thus, except for India, the lack of strong evidence of productivity differences across countries appears puzzling.
TABLE 2.2
Technical Efficiency Indexes T: Cross-Section Analysis Conducted
for Selected Three-Year Periods
1962 |
1.000 |
0.949 |
1.000 |
1.000 |
1.000 |
1.000 |
0.919 |
1.000 |
1.000 |
1.000 |
0.810 |
0.990 |
1963 |
0.975 |
0.957 |
1.000 |
1.000 |
0.961 |
0.990 |
0.823 |
0.943 |
1.000 |
1.000 |
1.000 |
1.000 |
1964 |
1.000 |
1.000 |
1.000 |
1.000 |
0.926 |
1.000 |
0.867 |
0.988 |
1.000 |
1.000 |
0.933 |
1.000 |
1965 |
1.000 |
1.000 |
1.000 |
1.000 |
0.957 |
0.973 |
0.834 |
1.000 |
1.000 |
1.000 |
0.930 |
1.000 |
1966 |
1.000 |
1.000 |
1.000 |
0.986 |
0.844 |
1.000 |
0.864 |
1.000 |
1.000 |
1.000 |
0.673 |
1.000 |
1967 |
0.966 |
1.000 |
1.000 |
1.000 |
0.721 |
1.000 |
0.863 |
1.000 |
1.000 |
1.000 |
0.732 |
1.000 |
1968 |
1.000 |
1.000 |
1.000 |
1.000 |
0.684 |
1.000 |
0.894 |
1.000 |
1.000 |
0.984 |
0.779 |
0.999 |
1969 |
1.000 |
1.000 |
0.980 |
0.990 |
0.681 |
0.996 |
0.904 |
1.000 |
1.000 |
1.000 |
0.553 |
1.000 |
1970 |
0.993 |
1.000 |
1.000 |
1.000 |
0.723 |
1.000 |
0.891 |
1.000 |
0.993 |
1.000 |
0.758 |
1.000 |
1971 |
1.000 |
1.000 |
1.000 |
1.000 |
0.681 |
0.985 |
0.932 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1972 |
1.000 |
0.966 |
0.986 |
1.000 |
0.626 |
1.000 |
1.000 |
0.969 |
1.000 |
0.932 |
0.903 |
1.000 |
1973 |
1.000 |
0.895 |
1.000 |
1.000 |
0.671 |
1.000 |
1.000 |
0.969 |
1.000 |
1.000 |
0.978 |
1.000 |
1974 |
1.000 |
0.955 |
1.000 |
1.000 |
0.720 |
1.000 |
1.000 |
1.000 |
1.000 |
0.974 |
0.893 |
0.952 |
1975 |
1.000 |
1.000 |
1.000 |
1.000 |
0.688 |
1.000 |
1.000 |
1.000 |
0.978 |
0.999 |
1.000 |
1.000 |
1976 |
1.000 |
0.935 |
1.000 |
1.000 |
0.700 |
0.981 |
1.000 |
1.000 |
1.000 |
1.000 |
0.892 |
1.000 |
1977 |
1.000 |
0.952 |
1.000 |
1.000 |
0.694 |
0.907 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1978 |
0.976 |
1.000 |
1.000 |
0.985 |
0.684 |
1.000 |
1.000 |
0.971 |
1.000 |
1.000 |
0.893 |
0.992 |
1979 |
0.991 |
1.000 |
1.000 |
1.000 |
0.650 |
1.000 |
0.978 |
1.000 |
0.982 |
0.928 |
0.888 |
1.000 |
1980 |
0.978 |
0.989 |
0.931 |
0.993 |
0.645 |
1.000 |
1.000 |
0.899 |
1.000 |
1.000 |
1.000 |
0.980 |
1981 |
1.000 |
1.000 |
0.949 |
1.000 |
0.680 |
1.000 |
1.000 |
0.974 |
0.895 |
1.000 |
0.792 |
1.000 |
1982 |
1.000 |
1.000 |
1.000 |
1.000 |
0.657 |
0.997 |
1.000 |
1.000 |
0.848 |
0.971 |
0.688 |
1.000 |
1983 |
1.000 |
1.000 |
0.942 |
1.000 |
0.657 |
1.000 |
1.000 |
1.000 |
0.806 |
1.000 |
0.947 |
0.984 |
1984 |
0.956 |
0.910 |
1.000 |
1.000 |
0.653 |
1.000 |
0.987 |
0.985 |
0.838 |
0.987 |
0.770 |
0.979 |
1985 |
1.000 |
1.000 |
1.000 |
0.992 |
0.658 |
1.000 |
1.000 |
1.000 |
0.827 |
1.000 |
0.906 |
1.000 |
1986 |
0.908 |
1.000 |
1.000 |
1.000 |
0.710 |
1.000 |
1.000 |
1.000 |
0.858 |
1.000 |
0.831 |
1.000 |
1987 |
0.999 |
0.924 |
1.000 |
1.000 |
0.693 |
1.000 |
1.000 |
0.939 |
0.895 |
0.957 |
0.972 |
0.992 |
1988 |
1.000 |
0.921 |
1.000 |
1.000 |
0.680 |
1.000 |
1.000 |
1.000 |
0.840 |
1.000 |
0.722 |
1.000 |
1989 |
1.000 |
1.000 |
0.972 |
0.996 |
0.761 |
1.000 |
0.923 |
1.000 |
0.839 |
1.000 |
0.619 |
0.980 |
1990 |
0.987 |
0.969 |
1.000 |
1.000 |
0.728 |
1.000 |
1.000 |
0.967 |
1.000 |
0.872 |
0.875 |
1.000 |
1991 |
1.000 |
0.954 |
1.000 |
1.000 |
0.735 |
1.000 |
1.000 |
1.000 |
1.000 |
0.946 |
1.000 |
1.000 |
1992 |
1.000 |
1.000 |
0.906 |
1.000 |
0.798 |
1.000 |
0.927 |
1.000 |
1.000 |
0.990 |
0.904 |
0.965 |
1993 |
1.000 |
0.978 |
1.000 |
0.996 |
0.801 |
1.000 |
0.974 |
1.000 |
1.000 |
0.879 |
1.000 |
0.957 |
1994 |
1.000 |
0.914 |
1.000 |
1.000 |
0.786 |
0.981 |
1.000 |
1.000 |
0.854 |
0.877 |
0.822 |
1.000 |
#1(/33) 22 18 26 1 24 18 23 20 20 7 22
The above results indicate that the agricultural sector may have been subject to little technical change and productivity growth both over time and across countries. This would suggest that most output changes (i.e. crop and livestock output) are due to changes in inputs (i.e. fertilizer, farm equipment, land and labour). As noted above, this surprising finding is at odds with the empirical literature on agricultural productivity. Is it possible to explain this discrepancy? Possible explanations can be linked to the methodology used as well as the data.
The results presented in Tables 2.1 and 2.2 are obtained from equation 2.12b, which relies on a CRTS representation of the technology F_{c}^{i}. What if equation 2.12a was used instead, relying on a VRTS representation of the technology? It has been seen above that T(x, y, F_{c}^{i}) T(x, y, F ^{i}) . Thus, the technical efficiency indexes T(x, y, F_{c}^{i}) reported in Tables 2.1 and 2.2 are in fact a lower bound to the ones one would obtain under VRTS. Given that the indexes T have an upper bound of one, using 2.12a under VRTS would generate higher technical efficiency index than the ones reported in Tables 2.1 and 2.2, and thus less evidence of productivity growth. It follows that, under VRTS, the evidence in favour of technical change and productivity growth would become even weaker!
Could it be that the results reflect a misspecification of the production technology? For example, this analysis does not control for agro-climatic conditions, for infrastructure, and for human capital. Incorporating such variables in the analysis would influence the results. This would amount to increasing the number of inputs in the production technology. How would that affect the analysis and findings? Increasing the number of inputs in equation 2.12a or 2.12b amounts to increasing the number of constraints without changing the number of variables optimized. Since 2.12a or 2.12b are minimization problem, this would imply in general an increase in the value of the optimized objective function. In other words, this would tend to increase the value of the technical efficiency index T toward one. Thus, introducing additional inputs in the analysis would further weaken the evidence in favour of technical change and productivity growth. Intuitively, these additional inputs would give new ways of explaining output changes, thus providing less evidence of technical change. This suggests that model misspecification is not a good candidate for explaining some of the surprising results.
Two possible explanations remain:
First consider the implications of using the inner bound representation F ^{i} of the technology. In principle, this specification is flexible in the sense that it does not impose restrictions on the possibility of substitution among inputs. It is also flexible in the sense that it imposes the basic concept of diminishing marginal productivity without requiring a parametric specification. Finally, as stated in propositions 1 and 2, it is consistent with the sample data. As such, it provides a simple representation of technology that is close to the data and the theory, with a minimum of ad hoc assumptions. Also, as illustrated by Fare et al. (1994), it can in principle generate evidence of significant productivity growth. All these characteristics appear quite positive, thus suggesting the general usefulness of the approach. If so, why is there not more evidence of agricultural technical change? As argued by Chavas and Cox (1997), it may be that the inner-bound and outer-bound representations discussed in propositions 1 and 2 do not provide tight estimates of the underlying technology. Then, there would be a fairly wide range of technologies that are consistent with the data and production theory. In this context, empirical searches (e.g. through the parametric testing of alternative functional forms) for a "true technology" may be futile. The non-parametric bounds identified in propositions 1 and 2 can help better assess the range of identification (or underidentification) of the underlying technology, and better evaluate the strength of the information that a particular data set can yield. While both inner-bound and outer-bound representations of technology are consistent with the data, it may be that the outer-bound representation is more realistic than the inner bound. The reason is that the inner-bound measure given in 2.7a allows the marginal physical product of inputs to vary between zero and infinity. While this is theoretically possible, it seems rather unlikely that real-world observations would cover such a wide range. In contrast, the outer-bound measures given in 2.8 or 2.11 constrain the marginal physical product to equal price ratios. This follows from the fact that, under profit maximization, the profit hyperplane must be tangent to the production frontier. As long as observed prices do not take extreme values, it would exclude the possibility of uncovering either "very small" or "very large" marginal physical products. In some sense, these exclusions may appear realistic. Thus, the use of price information in the outer-bound representations may in fact help obtain a more reasonable estimate of the production technology. This suggests a need to complement the cross-country productivity analysis presented here with an analysis based on outer-bound representations of the technology. This would require obtaining comparable price information on inputs and outputs both over time and across space. This seems to be a good topic for further research.
Second, consider the issue of data quality. As in any analysis, good data are required to obtain good results. In the context of this paper, this relates to measuring both input-output quantities as well as quality. As much as possible, input-output quantities should be obtained using superlative quantity indexes, i.e. quantity indexes that are exact indexes associated with a flexible function form. An example of a superlative index is the Theil-Tornqvist index commonly used in productivity analysis (Ball, 1985; Jorgenson and Gollop, 1992). This is a superlative index since it is an exact index associated with a translog production function. Note that, in this sample, fertilizer is measured at the total quantity of fertilizer used. This is not a superlative quantity index. This suggests that there are avenues to improve the measurement of quantity information used in this report. Also, whenever possible, quality changes should be taken into consideration. This may be relevant for most netputs. For example, given the spatial heterogeneity of land quality, it would be useful to correct land input for quality changes. This may be particularly relevant if the quantity changes involve marginal, less productive land. A similar argument would apply to fertilizer, labour or machinery. Thus, there are also avenues for improving netput measurement through quality adjustments. Implementing such changes across countries may be a significant challenge. However, it would help generate better data and thus improve the reliability of the empirical analysis. This is particularly crucial to the extent that quality concerns about current data may undermine the reliability of our results.
This paper proposes a non-parametric method to investigate the time-series and cross-section evolution of agricultural productivity. It relies on estimating technical efficiency and productivity indexes based on a non-parametric representation of the technology. The method is flexible in that it imposes little a priori restrictions on technology beyond diminishing marginal productivity. It is also easy to implement empirically: the technical efficiency and productivity indexes can be obtained as solutions of simple linear programming problems. This is illustrated in an empirical application to FAO data for 12 countries over the period 1960-1994. The analysis focuses on agricultural technology characterized by two outputs (crops and livestock) and four inputs (land, labour, machinery and fertilizer). It uses the inner-bound representation of the underlying technology in the estimation of technical efficiency indexes. The non-parametric approach discussed in this paper can help economists better assess the nature of the underlying technology, allowing a greater awareness of the strengths as well as limitations of the data, and a better evaluation of their informational content.
The empirical results indicate only weak evidence of agricultural technical change and productivity growth both over time and across countries. This would suggest that most of changes in agricultural outputs can be explained by corresponding changes in inputs. For example, crop output can rise due to an increase in fertilizer use, without requiring a shift in the production technology. But this is a rather surprising finding. Indeed, there is much evidence of strong productivity growth in agriculture over the last few decades. This reflects in part the large genetic progress that has taken place for both crops and livestock. Also previous literature has found much empirical evidence supporting agricultural technological progress. Such discrepancies with the empirical results are puzzling.
In an attempt to explain such differences, two elements are apparent. The first is the need to complement the analysis with an outer-bound representation of technology. Such a representation would require obtaining comparable price information on inputs-outputs both over time and across countries. Making use of price information would restrict a priori the possible range for the marginal physical products of inputs. As argued by Chavas and Cox (1997), such restrictions may give a better representation of technology and help uncover stronger evidence of technical change and productivity growth. Second, the findings may reflect data problems. Poor measurements of input-output quantities seem to contribute to the discrepancies between these results and previous literature on agricultural productivity analysis. If so, there is a need to refine and improve the measurement of inputs and outputs across countries. Addressing these issues presents significant challenges for future work.
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1 The set F is said to be negative monotonic if z F and z' z implies that z' F. That is basically a "free disposal" assumption.