Edward C. Jaenicke is an assistant professor in the Department of Agricultural .... Larson and Pierce; Smith, Halvorson, and Papendick; Granatstein and ...
A Soil-Quality Index and its Relationship to Efficiency and Productivity Growth Measures: Two Decompositions Edward C. Jaenicke and Laura L. Lengnick Abstract: This paper reconciles two notions of soil-quality indexes with the economic concepts of technical efficiency and productivity growth. An example uses data from the U.S. Department of Agriculture's experimental fields in Maryland and data envelopment analysis techniques to estimate a soil-quality index consistent with the notion of technical efficiency. Common regression techniques shed additional light on the role of individual soil-quality properties in a very restricted linear approximation of the estimated soilquality index. Key words: data envelopment analysis, efficiency, productivity, soil-quality index
Revised December 1998 Submitted to the American Journal of Agricultural Economics
Edward C. Jaenicke is an assistant professor in the Department of Agricultural Economics and Rural Sociology at the University of Tennessee. Laura L. Lengnick is an agronomist with the Agricultural Research Service, U.S. Department of Agriculture. The authors wish to thank Bob Chambers and Rolf Färe for discussions about the article’s central topic, and two anonymous reviewers for their suggestions.
A Soil-Quality Index and its Relationship to Efficiency and Productivity Growth Measures: Two Decompositions
Edward C. Jaenicke and Laura L. Lengnick
This paper reconciles two notions of soil-quality indexes with the economic concepts of technical efficiency and productivity growth. An example uses data from the U.S. Department of Agriculture's experimental fields in Maryland and data envelopment analysis techniques to estimate a soil-quality index consistent with the notion of technical efficiency. Common regression techniques shed additional light on the role of individual soil-quality properties in a very restricted linear approximation of the estimated soilquality index.
Key words: data envelopment analysis, efficiency, productivity, soil-quality index
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A Soil-Quality Index and its Relationship to Efficiency and Productivity Growth Measures: Two Decompositions Soil quality has historically been linked to soil productivity, the ability to promote plant or crop growth. More recently, the definition has expanded to include the soil's other functions, namely, promoting health and buffering environmental impacts (National Research Council; Warkentin). Doran and Parkin suggest soil quality plays a role in six broad functions: food and fiber production, erosivity, groundwater quality, surface water quality, air quality, and food quality. Under an expanded definition, soil quality is the ability to produce safe and nutritious crops in a sustained manner over the long term, and to enhance human and animal health, without impairing the natural resource base or harming the environment (Parr et al.). No matter its precise definition, soil quality is determined by complex interactions of physical (e.g., texture, water-holding capacity, soil depth); chemical (e.g., acidity, cation-exchange capacity); and biological (e.g., organic matter content, micro-organism activity, earthworm population) properties. Despite the abundance of component research on these properties' relations to farming practices and crop yields, no generally accepted criteria exist for aggregating these soilquality properties into a single index. A soil-quality index is desirable because individual soil properties, in isolation, may not be sufficient to quantify changing soil conditions. Soil organic matter, for example, is one of the single best indicators of soil quality (Granatstein and Bezdicek; Rasmussen and Collins; Romig et al.). However, significant biological, chemical, and physical differences can exist between two soils with the same organic matter. Granatstein and Bezdicek cite studies from Australia, Canada, and Ohio that show
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organic carbon levels are not necessarily correlated with the amount of water stable aggregates, which is another indicator of soil quality. This type of research points out that a soil-quality index must acknowledge that individual indicators can move independently and may be complements or substitutes with each other. A soil-quality index can be an important tool in economic analysis of agricultural systems and in agro-environmental policy design. Parr et al. suggest a number of specific uses for a soil-quality index, including: assessing the impact of management practices on soil degradation and soil conservation; providing a basis for land capability classification and for eligibility in USDA's Conservation Reserve Program and cost-sharing programs; aiding in setting land prices, loan values, and tax assessments; and providing information on simulating and predicting environmental change. Ultimately, a soil-quality index would help estimate the monetary returns from “investing” in soil quality. A soil-quality index would also help separate the productivity effects of technological advances from changes in soil quality. Walker and Young (1986a and 1986b) suggest that advances in agricultural technology have been land complementary (that is, technology interacts positively with topsoil depth). The ability to construct a broad-based soil-quality index can stimulate re-assessment of this issue. One purpose of this article is to merge the soil science literature on soil-quality indexes with the literature on efficiency and total-factor productivity indexes. The soil science literature often focuses on which soil properties may be important index components, but says little about how to determine index weights on individual components. Recent economic research has shown how economic indexes can be decomposed into various components (see for example Färe, Grosskopf, and Lovell; and
3
Färe, Grosskopf, and Roos). Our main goal, therefore, is to employ these decomposition techniques to isolate a theoretically preferred soil-quality index. A secondary goal is to approximate the relative importance of individual soil properties in the overall index. Before describing two decompositions and presenting an empirical example, we first summarize the current research on soil-quality indexes. Two Categories of Current Research on Soil-quality Indexes Several research teams have proposed various properties as candidates for a soil-quality index (see for example Kennedy and Papendick; Doran and Parkin; Karlen and Stott; Larson and Pierce; Smith, Halvorson, and Papendick; Granatstein and Bezdicek; Arshad and Coen). This body of research presents models of soil-quality measurement that generally fit into one of two categories: static models and comparative-static models. Static models explain soil quality at one point in time. For instance, Doran and Parkin define a soil-quality index (S) as a function of Q soil-quality attributes: (1)
S = f(s1, s2, ..., sQ),
where s1 through sQ represent individual attributes of soil quality. Similarly, Karlen and Stott suggest a model where soil quality with regard to soil erosion by water is the weighted score of four critical soil functions: accommodating water entry, transporting and absorbing water, resisting degradation, and supporting plant growth. These static models examine soil quality in relation to some ideal soil, but do not address how soil quality may change over time. Comparative-static models examine the change in soil quality. Larson and Pierce (p.37) state that "it is well known that soils vary in quality and that soil quality changes in
4
response to use and management." Larson and Pierce define the comparative change in soil quality between time 0 and time t as (s Q ,t − s Q ,0 ) (s1,t − s1, 0 ) ,L, s1,0 s Q,0 dS = f . dt dt
(2)
Here the key diagnostic is the temporal change in S, an aggregator function of soil properties. Both the static and comparative-static approaches share a common problem: They implicitly or explicitly require individual soil properties to be weighted in some general function. Yet these weights, if given, are generally not based on statistical tests that might show the relative contribution each property makes to the soil's main functions, broadly defined. Karlen and Stott acknowledge that the problem may be even more complicated when they suggest that these weights might differ for various soil functions. This problem highlights the fact that the soil-quality aggregator function is not based on theory, either agronomic or economic. Soil Quality's Role in Static Efficiency Measurement Technical efficiency reflects a comparison between observed production and bestpractice (or "frontier") production. If this comparison is examined at a single point in time, then it reflects only static efficiency. For example, this comparison may be defined as the ratio of observed output to maximum potential output from a given set of inputs, at a given time period. In its most general form, this notion of efficiency accounts for all producible outputs and all types of inputs. For example, off-farm chemical runoff may be included in the set of farm-produced outputs and soil quality may be included in the set
5
of inputs. Omitting an important input or output may dramatically affect this comparison.1 The questions of exactly which outputs and inputs should be included, and how they should be included, have essential implications on the soil-quality models discussed above. Potentially important components in a model of agricultural production include: management inputs, weather conditions, soil-quality properties, crop outputs, and farmproduced environmental outputs such as sequestered carbon or chemical runoff. A production model should be able to account for these factors. For computational ease, describe these components of crop production as follows: Let y ∈ ℜ+M be an Mdimensional vector of annually produced crop outputs such as corn or wheat; let x ∈ ℜ+N be an N-dimensional vector of crop inputs, some under management control such as fertilizer and labor, and some not under management such as weather; let z ∈ ℜ+U be a U-dimensional vector of farm-produced environmental outputs, some good such as sequestered carbon, and some bad such as chemical leachate; and let s ∈ ℜ+Q be a Qdimensional vector of soil-quality properties such as topsoil depth, acidity, bulk density, nutrient levels, or total organic carbon. At a given time, the production system can be characterized by an output correspondence (x, s) → Y(x, s) ⊆ ℜ(M+U) , where Y(x, s) = {(y, z): (x, s) can produce (y, z) }. The producible-output set Y(x, s) is a representation of the production technology and contains information about how the various inputs interact to produce the two types of output. It is the set consisting of all combinations of crop outputs and farm-produced environmental outputs that can be produced by a given bundle of crop inputs and initial
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soil properties.2 The output set Y(x, s) is a static representation of the technology for two reasons: First, it is static because it characterizes production for a given time period; and second, production in the given period is not linked to any other time period. The static nature of production explicitly suggests that soil quality is exogenously predetermined. If it is to be useful in modeling crop production, the output set must satisfy certain axiomatic properties. Färe and Chambers each present a number of these axioms for a general output set. These axioms address the impossibility of producing something from nothing, the possibility of choosing not to produce, the ability to produce convex combinations of inputs and outputs, and the extent to which a producer can "dispose" of unwanted outputs or use more than sufficient inputs. Many of the axioms discussed by Färe or Chambers require little or no modifications to be applicable to the set Y(x, s). However, the axioms concerning output and input disposability should be given special consideration because of the special nature of Y(x, s), which may include undesirable outputs in the vector z (or undesirable inputs in the vector s). Free output disposability allows a producer to throw away or reduce outputs without cost. It may be impossible, however, for producers to reduce environmental pollution without incurring some cost. Weak output disposability says that a producer can reduce farm-produced environmental outputs, but only by simultaneously (and proportionally) reducing other crop outputs.3 Figure 1 shows two possible producible-output sets for the case where a single crop output (y) and a single environmental output (z) are produced. Weak disposability of outputs, reflected by the output set in Figure 1 labeled 0DEC, is useful in the case where z is an undesirable output (e.g., phosphorus runoff). If the output bundle at D is producible with the input bundle (x, s), weak disposability guarantees that all points
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along OD are also producible. On the other hand, free output disposability, reflected by the output set labeled 0ABC, is useful where z is desirable (e.g., sequestered carbon). If the output bundle at B is producible with (x, s), free disposability guarantees that all points along AB are also producible. The output set Y(x, s), along with the appropriate set of axioms, can be used to construct single-valued functions that also characterize the production technology. Shephard's “output distance function” represents the largest proportional increase (formally described as a radial expansion) of all crop and environmental outputs still producible with a given bundle of crop and soil-quality inputs.4 The output distance function is a generalization of the standard production function and happens to be a natural measure of technical efficiency. In fact the reciprocal of the output distance function is sometimes called the Farrell output efficiency measure. The concept of the output distance function in two dimensions is illustrated in Figure 1. At point F, the largest radial expansion that is on or within 0ABC is given by the distance 0B/0F. The value of Shephard's output distance function is the reciprocal of this distance (i.e., 0F/0B), which is less than one. The value of the output distance function is less than or equal to one for all points that are elements of the output set Y(x, s). Note that if the output set 0DEC is used, the value of the output distance function at point F would be 0F/0E. The output distance function treats inputs as fixed and finds the largest radial expansion of all outputs such that the expanded outputs are still producible with a given input bundle. In other words, it measures the relative distance between a particular
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production bundle and the output set frontier. The producer has implicit control over all crop and environmental outputs. Define the radial output distance function as: D(y, z; x, s) = inf {θ > 0: (y/θ, z/θ) ∈ Y(x, s) },
(3)
where the inf operator finds the set’s infimum, the greatest lower bound. Färe lists distance function properties that follow upon the axioms of output sets like Y(x, s). For example, D(y, z; x, s) is nondecreasing in outputs y and z, nonincreasing in inputs x and s, and linearly homogeneous in y and z. The value of D(y, z; x, s), which is between zero and one for observed (feasible) production units, is a natural measure of technical efficiency. As the value of the distance function increases, efficiency improves. Decomposition of efficiency and soil quality If (3) is multiplicatively separable in soil quality and other outputs and inputs, then D(y, z; x, s) can be decomposed into two components (see Färe, Grosskopf, and Tyteca; or Färe, Grosskopf, and Roos): D(y, z; x, s) = S(s) · D(y, z; x) .
(4)
The function S(s) is an aggregator function reflecting overall soil quality, and D(y, z; x) is an output distance function with soil quality omitted. In other words, S(s) is a natural soil-quality index. Like D(y, z; x, s), the S(s) function is mathematically nonincreasing in s. Soilquality indexes of the type proposed by Doran and Parkin are typically constructed to be nondecreasing in beneficial soil attributes. If this property continues to be desirable in a soil-quality index, we can examine the reciprocal of S(s). While S(s) and its reciprocal could be considered the endpoint of the decomposition, they can also be regressed on individual soil properties to give a very restricted estimate of the individual contributions
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to the index. For example, if 1/S(s) can be approximated by a linear weighting of the individual contributions, one could estimate the linear function 1/S(s) ≅ β0 + β1s1 + β2s2 + ... + βQsQ + ε ,
(5)
where ε might be assumed to be a normally distributed error term. In this case, the β's represent the weights on individual soil-quality components. Note that (5) corresponds to a special (linear) case of (1). The decomposition shows that the static notion of soil quality given by (1) is a component of a broadly defined efficiency measure. Moreover, the weights on individual soil attributes in (5) can be approximated by ordinary statistical methods. Computation methods In this section, we present one method for estimating both types of distance functions in (4), thereby allowing us to calculate 1/S(s).5 The method is a nonparametric data envelopment analysis (DEA) approach where observed data are used to construct an approximation of the output set (Färe, Grosskopf, and Lovell). We assume that there are K plot-level observations on outputs and inputs. For each observation k' = 1, …, K, the first distance function in (4) is estimated using the following optimization program:6
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D * (y k ' , z k ' , x k ' , s k ' ) = [max θ > 0 : θ ,λ
K
(i) θy km' ≤ ∑ λk y km , m = 1,..., M k =1 K
(ii) θz ku ' ≤ ∑ λk z ku ,
(6)
u = 1,..., U
k =1
K
∑λ x
(iii)
k
k =1 K
∑λ s
(iv)
k k q
k =1
K
∑λ
( v)
k n
k
≤ x kn ' ,
n = 1,..., N
≤ s qk ' , q = 1,..., Q
≤ 1, λk ≥ 0 ] −1
k =1
In (6), λ1 through λK are intensity variables that allow the optimization program to choose convex combinations of the data. The first four constraints in (6) impose free disposability on outputs and inputs. Together these constraints require the DEA approximation to be the free disposal, convex hull of the data. If necessary, these constraints can be modified to reflect weak disposability following methods in Färe, Grosskopf, and Lovell. For example, if z includes undesirable outputs, weak disposability can be imposed by changing the second constraint to K
(6ii´)
θz ku ' = ∑ λk z ku , k =1
for pertinent elements of z. If s includes bad soil-quality inputs such as the degree of soil compaction or acidity, weak disposability can be imposed by changing the fourth constraint to K
(6iv´)
∑λ s k =1
k k q
= δ s qk ' , δ ∈ (0,1],
for pertinent elements of s. The fifth and last constraint in (6) allows the approximated technology to satisfy non-increasing returns to scale. Also called the subhomogeneous
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case, non-increasing returns to scale allows all radial contractions, but not radial expansions, to belong to the approximated technology. Rather than the cases of constant returns or variable returns, it is presented here to be consistent with the empirical example that follows. In that example, elements of the X vector are constant across all K observations so the third constraint in (6) trivially reduces to the non-increasing returns constraint. The second distance function in (4), D(y, z; x), is estimated using a similar program that omits soil quality. Once these two distance functions are estimated, for each observation the soil-quality component 1/S(s) can be calculated according to (4). The resulting soil-quality index can then be used to identify individual component weights in a very restricted approximation of the index according to (5). Soil Quality's Role in Productivity Growth Measurement
When a single output is produced by a single input, productivity is defined simply as the ratio of the output quantity to the input quantity. When the output grows faster over time than the input, this ratio increases and productivity grows.7 This simple definition, however, gets more complicated for cases, like the one considered here, where there is more than one input or output. Total-factor productivity measures require some means of aggregating all inputs and outputs. The distance function provides a natural means for this aggregation and can serve as the basis for productivity measurement in a multioutput, multi-input setting. Caves, Christensen, and Diewert show that the ratio of distance functions from two time periods can serve as a productivity index, which they call an output-oriented Malmquist index. They also show that the Malmquist index can be related to the Törnqvist index, an exact and superlative index (Diewert). Färe et al.
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(1994) develop a Malmquist-type index that can be represented as the geometric mean of two distance function ratios, where both the technology and the inputs and outputs are time-dated. Their Malmquist-type index, applied to the agricultural production technology described above, is given by: 1/2
(7)
D t ( y t +1 , z t +1 ; x t +1 , s t +1 ) D t +1 ( y t +1 , z t +1 ; x t +1 , s t +1 ) M( t , t + 1) = . t t +1 D ( y t , z t ; x t , s t ) D ( y t , z t ; x t , s t )
Productivity increases from period t to t+1 if (7) is greater than one and decreases if it is less than one. Two of the distance functions that compose (7) are "mixed-period" distance functions measuring the distance between a production bundle observed in one time period and the set frontier from a different time period. These mixed-period distance functions help isolate the impact of time's passage on the production technology.8 Decomposition of productivity and soil quality
Färe, Grosskopf, and Roos present a method that is useful, under certain circumstances, in decomposing the Malmquist index in (7) to reveal a soil-quality component. If the distance functions in (7) are multiplicatively separable in soil-quality attributes and other inputs and outputs, then again we can write the distance function as (4). Under these circumstances, the Malmquist-type index in (7) can be rewritten as: (8i)
t t +1 t +1 t M(t, t+1) = S (s t +1 ) D (ty t +1 , z t +1 ; x t +1 ) ⋅ S (s t +1 ) D t +1( y t +1 , z t +1 ; x t +1 ) t t +1 S (s t ) D ( y t , z t ; x t ) S (s t ) D ( y t , z t ; x t )
1/ 2
1/2
,
(8ii)
t +1 t +1 t t = S t( s t +1 ) S t +1( s t +1 ) D ( yt t +1 , z t +1 ; x t +1 ) D t(+1y t +1 , z t +1 ; x t +1 ) D (y t , zt ; x t ) S ( s t ) S ( s t ) D (y t , zt ; x t )
(8iii)
= ∆S(st, st+1) · M(t, t+1) ,
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1/ 2
,
The first term in (8ii), labeled in (8iii) as ∆S(st, st+1), is the comparative-static soil-quality index. The index M(t, t+1) is a modified Malmquist productivity index constructed without the soil-quality component. The soil-quality index ∆S(st, st+1) can be calculated for each observation if both M(t, t+1) and M(t, t+1) are known. As before, ∆S(st, st+1) can be used as the dependent variable in a regression estimating a very restrictive form of ∆S(st,st+1). For example, if ∆S(st, st+1) is approximated by linearly weighting the proportional changes in soil-quality inputs, one can estimate, (9)
∆S(st, st+1) ≅ β 0 + β 1
(s1,t − s1,t +1 ) s1,t
+ β2
( s 2 , t − s 2 , t +1 ) s2 ,t
+ L βQ
(s Q ,t − s Q ,t +1 ) sQ ,t
+ε,
where again ε is assumed to be a normally distributed error term. Note the similarities between (9) and (2), which is Larson and Pierce's definition of comparative soil-quality change. This last decomposition shows that the comparative-static measure of soil-quality change is a component of a broadly defined productivity-growth measure. The added benefit of this decomposition is that the weights on individual soil-quality attributes can be approximated statistically. Calculation of ∆S(st, st+1) is again possible using DEA techniques. Given an appropriate data set, the Malmquist index M(t,t+1) and the modified index M(t, t+1) can be calculated using the same DEA techniques described above. An Empirical Example of a Static Efficiency Decomposition
This section applies the efficiency decomposition to data from the Farming Systems Project at the U.S. Department of Agriculture/Agricultural Research Service's Beltsville Agricultural Research Center (BARC) in Beltsville, MD, which has extensively sampled
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soils and yields associated with three years of no-till corn production on a 40-acre field. By combining several data sets collected at different spatial resolutions, 1994 crop-yield data and data collected on a number of soil biological, physical and chemical indicators of soil quality (Table 1) were examined for their use in a decomposition analysis. Although no definitive soil-quality index has been accepted by soil scientists, the soils data set reported in Table 1 represents a potential subset of the "minimum data set" of soil-quality indicators as proposed by several soil-quality researchers (Doran et al.). In preparation for conducting the decomposition analysis, soil variables were analyzed for correlation. The full data set was revised to include only variables representing chemical, physical and biological indicators that were not highly correlated with other soil variables. This reduced data set included: as a proxy biological indicator the carbonnitrogen ratio (C/N); as physical indicators the bulk density and water-holding capacity; and as chemical indicators the available phosphorus (P) and potassium (K). The reduced data set also included two additional soil chemical measures that are not commonly suggested as part of a soil-quality index: acidity and available magnesium (Mg). These seven variables comprise s, the vector of soil-quality inputs. The greatest degree of correlation among the elements of s is the interaction between K and available P, which has a correlation coefficient of 0.43. No-till corn yield is the sole element of the output vector y. A description of this variable and the seven soil-quality variables in s is presented in Table 1 (in bold). Agronomic theory suggests that two elements of s, acidity and density, adversely affect crop production and, therefore, are modeled by (6ii´) in the DEA framework to reflect weak disposability. Management inputs and weather are constant in the input vector x;
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that is, land area, labor, applied fertilizer, rainfall, and growing-degree days are identical across all sampled plots (call the constant input vector xc). Because these variables are K
identical across all observations, the third constraint in (6) trivially reduces to ∑ λk ≤ 1 , a k =1
result that dictates the imposition of non-increasing returns to scale (i.e., the fifth constraint). Because no information is available on environmental outputs, the calculated soil-quality index will reflect only the soil’s crop-production function and not its environmental functions. Put simply, there are no elements in the vector z. The reduced data set can be used to estimate the DEA-based distance functions in (4), and ultimately to approximate the weights in a static soil-quality index specified by (5). Unfortunately, the BARC data will not allow estimation of the distance functions in (8) because soil and yield sampling were not spatially consistent over the three-year period. Therefore, an empirical application of the comparative-static soil-quality index using the BARC data is not feasible. Table 2 summarizes the 271 individual calculations of D(y; xc, s), D(y; xc), S(s), and 1/S(s). On average, technical efficiency is understated by approximately 2.7% when measured by D(y; xc) instead of D(y; xc, s) (0.94631 versus 0.97344 from Table 1 using the geometric averages). Thus, accounting for soil quality helps explain, on average, about half of the inefficiency found in individual plots. Figure 2 shows how 1/S(s) is distributed over the range zero to one. Figure 3 presents contour maps comparing actual corn yields and the calculated soil-quality index relative to the position of the field. The vertical and horizontal borders on the two panels represent the "northings" and "eastings" of the field’s position
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expressed in meters. The top panel describes the field's crop yields. Field positions above the maximum contour line represent corn yields above 10,700 kg/ha. Yield contour lines decrease in 200 kg/ha increments. The bottom panel shows the field's soilquality index contours. Here, field positions in the extreme northeast corner have a soilquality index of 1.0. Soil-quality contours decrease in 0.025 increments. Because crop yields and soil-quality index numbers are highly correlated (i.e., they have a correlation coefficient of 0.68), the two maps in Figure 3 are related.9 For example, the portion of the field with the lowest soil-quality scores corresponds to the portion with the lowest crop yields. However, the maps also show that there is generally less variation in the soil-quality index map than the corn-yield map and that most of the variation occurs on low-yielding, poor quality soil. For this example, therefore, the soil-quality contours may prove more useful than the yield contours in targeting the poorest soil. Table 3 presents the results of approximating the soil-quality index by a linear weighting of individual soil attributes by an OLS regression.10 The coefficients for each of the regressors -- acidity, available phosphorus (P), the carbon/nitrogen ratio (C/N), density, potassium (K), magnesium (Mg), and water-holding capacity -- are statistically significant at the 95 percent confidence level. On a unitless basis, the relative importance of the soil attributes in the soil-quality index is given by β q (
σ$ q ) , where βq is the σ$ S
estimated coefficient, σ$ q is the standard deviation of the attribute and σ$ S is the standard deviation of the dependent variable. Table 3's results suggest that water-holding capacity plays the strongest role in the index and should be given a linear weight of more than 0.50, which is nearly twice as large (in absolute value) as the next largest weight. In
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relative importance, water-holding capacity is followed by acidity, magnesium, available phosphorus, the carbon/nitrogen ratio, and potassium. With the exception of magnesium and acidity, all of these variables can be found on proposed lists of minimum data sets for soil quality (Arshad and Coen; Doran and Parkin; Granatstein and Bezdicek; Kennedy and Papendick; Larson and Pierce). Acidity is closely related to pH, which is on many proposed lists. However, the influence of magnesium in determining soil quality was unexpected, and may be related to the exceptional growing conditions in the 1994 season. In an environment characterized by optimal timing and amounts of rainfall as well as fertile soils high in plant-available nitrogen and potassium, magnesium may have become a limiting factor to corn grain production. Of course, the results presented here are sitespecific to BARC's field and practices. Concluding Comments
We believe this article presents a timely solution to the ongoing research problem of constructing a soil-quality index. In doing so, it reconciles two notions of soil-quality indexes with the economic concepts of technical efficiency and productivity growth. The article demonstrates that when efficiency- and productivity-growth measures include information on soil quality (and farm-produced environmental outputs), two types of soilquality indexes can be decomposed in a manner consistent with economic theory. An example uses data from experimental fields at USDA/ARS's Beltsville Agricultural Research Center in a DEA framework to estimate a static soil-quality index that is consistent with the notion of technical efficiency. To shed additional light on the role of individual soil-quality properties, common regression techniques find that, in descending order, water-holding capacity, acidity, magnesium, available phosphorus, the
18
carbon/nitrogen ratio, and potassium should be given the greatest weights in a very restricted linear estimate of the soil-quality index. The efficiency and productivity-growth decompositions presented in this article roughly correspond to static and comparative-static soil-quality models found in current soil science and agronomy literature. The static model implicitly assumes that management decisions do not measurably affect soil-quality attributes over a short time period. The comparative-static model not only assumes that changes over time in soil attributes can be measured, but also that these changes can form the basis for measures of sustainable management (Larson and Pierce). However, the comparative-static model is silent on how these changes occur and is, therefore, not truly dynamic. This last statement highlights this article's most important caveat. A truly dynamic model might map how soil-quality attributes transit from one time period to another by treating both initial soil attributes as inputs to production and the terminal soil attribute levels as outputs from production. Furthermore, a truly dynamic model might address how soil quality moves relative to a long-run equilibrium as management practices change.11 To move towards this type of dynamic model, one might turn to two sources: dynamic DEA models such as Färe and Grosskopf's general model or Chambers and Lichtenberg's specific model of sustainable agricultural production, which treats soil quality as both an input to and an output from production, and adjustment cost models, which treat capital as a quasi-fixed factor that is only partially responsive to management decisions. The adjustment cost model, which has been applied to aggregate agricultural production by Vasavada and Chambers, and by Luh and Stefanou, roughly resembles McConnell's soil conservation model when soil quality is treated as a capital good. With
19
the proper data set covering soil data matched with production prices and quantities, the adjustment-cost model might be used to isolate the soil-quality component much like Luh and Stefanou isolate technical change, scale, and disequilibrium components in aggregate agricultural productivity growth. The decompositions developed in this article may facilitate the inclusion of soil quality (defined to reflect multiple soil functions) in the adjustment cost model.
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Arshad, M.A., and G.M. Coen. "Characterization of Soil Quality: Physical and Chemical Criteria." Am. J. of Alternative Agriculture 7 (Nos. 1 and 2, 1992):25-31. Benjamin, D. "Can Unobserved Land Quality Explain the Inverse Productivity Relationship?" J. Develop. Econ. 46 (February 1995):51-84. Bhalla, S.S. "Does Land Quality Matter?" J. Develop. Econ. 29 (July 1988):45-62. Bhalla, S.S. and P. Roy. "Mis-Specification in Farm Productivity Analysis: The Role of Land Quality." Oxford Econ. Pap. 40 (March 1988):55-73. Caves, D.W., L.R. Christensen, and W.E. Diewert. "The Economic Theory of Index Numbers and the Measurement of Input, Output, and Productivity." Econometrica 50 (November 1982):1393-1414. Chambers, R.G. Applied Production Analysis: A Dual Approach. New York: Cambridge University Press, 1988. Chambers, R.G., Y. Chung, and R. Färe. "Benefit and Distance Functions." J. Econ. Theory 70 (August 1996):407-419. Chambers, R.G., and E. Lichtenberg. “Economics of Sustainable Farming in the MidAtlantic.” Final Report to the USDA/EPA ACE Program. Department of Agricultural and Resource Economics, University of Maryland, December 1995. Chavas, J.-P. and T.L. Cox. "On Nonparametric Supply Response Analyses." Am. J. of Agr. Econ. 77 (February 1995):80-92. Diewert, W.E. "Exact and Superlative Index Numbers." J. Econometrics 4 (May 1976):115-145. Doran, J.W., and T.B. Parkin. "Defining and Assessing Soil Quality." In Defining Soil Quality for a Sustainable Environment: SSSA Special Publication Number 35, J.W. Doran, D.C. Coleman, D.F. Bezdicek, and B.A. Stewart, eds. Madison, WI: Soil Science Society of America, 1994. Doran, J.W., M. Sarrantonio, and M.A. Liebig. "Soil Health and Sustainability." In Advances in Agronomy, D.L. Sparks, ed. San Diego, CA: Academic Press Inc., 1996. Färe, R. Fundamentals of Production Theory. New York: Springer-Verlag, 1988.
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Färe, R., and S. Grosskopf. Intertemporal Production Frontiers: With Dynamic DEA. Boston: Kluwer Academic Publishers, 1996. Färe, R., S. Grosskopf, B. Lindgren, and P. Roos. "Productivity Developments in Swedish Hospitals: A Malmquist Index Approach." In Data Envelopment Analysis: Theory, Methodology, and Application, A. Charnes, W.W. Cooper, A.Y. Lewin, and L.M. Seiford, eds. Boston, MA: Kluwer Academic Publishers, 1994. Färe, R., S. Grosskopf, and C.A.K. Lovell. Production Frontiers. New York: Cambridge Univ. Press, 1994. Färe, R., S. Grosskopf, C.A.K. Lovell., and S. Yaisawarng. "Derivation of Shadow Prices for Undesirable Outputs: A Distance Function Approach." Rev. Econ. Statist. 75 (1993):374-380. Färe, R., S. Grosskopf, and D. Tyteca. "An Activity Analysis Model of the Environmental Performance of Firms -- Application to Fossil-Fuel-Fired Electric Utilities." Ecological Econ. 18 (August 1996):161-175. Färe, R., S. Grosskopf, and P. Roos. "Productivity and Quality Changes in Swedish Pharmacies." Int'l. J. Production Econ., forthcoming. Granatstein, D., and D.F. Bezdicek. "The Need for a Soil Quality Index: Local and Regional Perspectives." Am. J. of Alternative Agriculture 7 (Nos. 1 and 2, 1992):12-16. Greene, W.H. Econometric Analysis, Second Edition. New York: Macmillan Publishing, 1993. Karlen, D.L., and D.E. Stott. "A Framework for Evaluating Physical and Chemical Indicators of Soil Quality." In Defining Soil Quality for a Sustainable Environment: SSSA Special Publication Number 35, J.W. Doran, D.C. Coleman, D.F. Bezdicek, and B.A. Stewart, eds. Madison, WI: Soil Science Society of America, 1994. Kennedy, A.C., and R.I. Papendick. "Microbial Characteristics of Soil Quality." J. Soil and Water Conserv. 50 (May-June 1995):243-248. Larson, W.E., and F.J. Pierce. "The Dynamics of Soil Quality as a Measure of Sustainable Management." In Defining Soil Quality for a Sustainable Environment: SSSA Special Publication Number 35, J.W. Doran, D.C. Coleman, D.F. Bezdicek, and B.A. Stewart, eds. Madison, WI: Soil Science Society of America, 1994.
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Luh, Y-H., and S.E. Stefanou. "Productivity Growth in U.S. Agriculture Under Dynamic Adjustment." Am. J. of Agr. Econ. 73 (November 1991):1116-1125. McConnell, K.E. "An Economic Model of Soil Conservation." Am. J. of Agr. Econ. 65 (February 1983):83-89. National Research Council. Soil and Water Quality: An Agenda for Agriculture. Washington, D.C.: National Academy Press, 1993. Parr, J.F, R.I. Papendick, S.B. Hornick, and R.E. Meyer. "Soil Quality: Attributes and Relationship to Alternative and Sustainable Agriculture." Am. J. of Alternative Agriculture 7 (Nos. 1 and 2, 1992): 5-11. Rasmussen, P.E., and H.P. Collins. "Long-term Impacts of Tillage, Fertilizer, and Crop Residue on Soil Organic Matter in Temperate Semiarid Regions." Advances in Agronomy 45 (1991):93-134 Romig, D.E., M.J. Garlynd, R.F. Harris, and K. McSweeney. "How Farmers Assess Soil Health and Quality." J. Soil and Water Conserv. 50 (May-June 1995):229-236. Sampath, R.K. "Farm Size and Land Use Intensity in Indian Agriculture." Oxford Econ. Pap. 44 (July 1992):494-501. Shephard, R.W. Theory of Cost and Production Functions. Princeton, NJ: Princeton Univ. Press, 1970. Smith, J.L., J.J. Halvorson, and R.I. Papendick. "Multiple Variable Indicator Kriging: A Procedure for Integrating Soil Quality Indicators." In Defining Soil Quality for a Sustainable Environment: SSSA Special Publication Number 35, J.W. Doran, D.C. Coleman, D.F. Bezdicek, and B.A. Stewart, eds. Madison, WI: Soil Science Society of America, 1994. Vasavada, U. and R.G. Chambers. "Investment in U.S. Agriculture." Am. J. of Agr. Econ. 68 (November 1986):123-137. Warkentin, B.P. "The Changing Concept of Soil Quality." J. Soil and Water Conserv. 50 (May-June 1995): 226-28. Walker, D.J., and D.L. Young. "Assessing Soil Erosion Productivity Damage." In Soil Conservation: Assessing the National Resources Inventory, Volume 2. Washington, DC: National Research Council, National Academic Press, 1986a. Walker, D.J., and D.L. Young. "The Effect of Technical Progress on Erosion Damage and Economic Incentives for Soil Conservation." Land Econ. 62 (Feb. 1986b):83-93.
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Figure 1: An Output Set and Distance Function for a Single Crop Output and Environmental Output
24
Figure 2: Histogram of 1/S(s)
25
Figure 3: Corn Yield and Soil-Quality Index Contour Maps for the USDA/ARS Field Crop yield contours
Soil quality contours
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Table 1: Summary of Soil and Crop Variables from BARC, 1994.
Study Variables* (Bold = those used in calucations)
Units
Data Type†
Sample Statistics Mean Range# Std. Dev.
Soil Variables Biological -C -N - C/N
g/100 g soil g/100 g soil -
point point point
1.45 2.76 0.018 0.097 0.014 0.001 15.2 19.4 2.9
Physical - Density - WHC - Ap Depth
g soil/cc soil g water/cc soil cm
map unit map unit point
1.33 0.32 24
0.015 0.003 0.015 0.002 21.5 3.6
Chemical - pH -P - Acidity -K - Mg - CEC
kg/ha meq/100 g meq/100 g meq/100 g meq/100 g
point point point point point point
6.8 189 0.63 0.25 1.32 7.0
1.6 683 2.7 0.72 1.9 7.0
kg/ha
point est.
10211 2240
Crop Variable - Grain Yield
0.31 70 1.0 0.006 0.017 0.065 408
*
Variable definitions: Ap depth - depth of plowed surface layer of soil; C - total organic soil carbon; N - total soil nitrogen; C/N - ratio of C to N; Density - mass of soil per unit volume; WHC - total soil water storage capacity; pH - concentration of hydrogen ions in the soil solution; Acidity - concentration of exchangeable hydrogen ions; P, K, Mg - plant available soil phosphorus, potassium and magnesium, respectively; CEC - soil cation exchange capacity; Grain Yield - corn grain yield at 15.5 % moisture. †
Data for this project were pooled from three different data sets: point - analysis of soil samples taken on regular 25 m square grid across the study site; map unit - from soil measurements taken on representative samples within each map unit at the study site; point est. - geostatistical estimates of grain yield at each soil sample grid point based on grain yield in 25 m by 10 m plots taken on a regular 25 m square grid across the study site. There were 271 data points used in the analysis.
#
Range = maximum value - minimum value.
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Table 2: Summary statistics for a static efficiency decomposition
No. obs. arith. avg. geo. avg. std. dev. min mode max
D(y; xc, s)
D(y; xc)
S(s)
1/ S(s)
271 0.97387 0.97344 0.02863 0.87088 1.00000 1.00000
271 0.94709 0.94631 0.03782 0.79225 0.98739 1.00000
271 1.02915 1.02867 0.03219 1.00000 1.04568 1.26223
271 0.97257 0.97213 0.02864 0.79225 0.95279 1.00000
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Table 3: Approximation of 1/S(s) by OLS regression
Soil Attribute (sq)
βq
Acidity Avail. P C/N Density K Mg Wat.Hold.Capac. (Constant)
-0.008133 0.000053 0.001282 0.051958 0.034479 0.023045 0.420518 0.703450
Adjusted R2
0.502046
βq (
σˆ q σˆ S
)
-0.277140 0.129509 0.129314 0.096004 0.127902 0.234360 0.512643
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t-stat -5.923 2.642 2.923 1.936 2.449 5.102 10.991 16.211
ENDNOTES 1
For example, Benjamin, Sampath, Bhalla, and Bhalla and Roy show the adverse effects
of omitting land quality in estimating the observed inverse relationship between farm size and productivity in a developing country setting. 2
The production technology could be represented equivalently by an input-requirement
set, which is the set of all input combinations able to produce a given output bundle, or a feasible-production set, which is the set of all feasible input-output combinations. For a single output technology, the frontier of the feasible production set would look similar to the yield response functions described in Walker and Young (1986a and 1986b). 3
The axiom of weak disposability of outputs has allowed researchers to estimate the
shadow value of environmentally undesirable outputs. For example, Färe et al. (1993) recovered the shadow price of undesirable outputs in the pulp and paper industry after imposing weak disposability on undesirable outputs. 4
Non-radial distance functions, such as the “directional distance function” of Chambers,
Chung, and Färe, may characterize production equally as well, especially when outputs are weakly rather than freely disposable. 5
One could also estimate the distance functions econometrically.
6
This approximation corresponds to Chavas and Cox's inner approximation.
7
Here it is assumed that output and input quality remain constant over time.
8
Färe et al. (1994) also show that the right-hand side of (7) can be decomposed into two
separate terms, an efficiency-change term and a technical-change term. 9
The present application nearly resembles a special case where the soil-quality index is
equivalent to a simple measure of relative yields. From (4), S(s) equals relative yield if 30
(i) D(y, z; x, s) equals one for all observations and (ii) D(y, z; x) equals the ratio of observed yield to maximum yield. Condition (i) holds if soil quality explains away all technical inefficiency so that D(y, z; x, s) always equals one. Condition (ii) holds if z is unobservable and x is constant for all observations. For the present observation, note that (ii) holds but (i) does not. 10
The soil-quality index 1/S(s) is bounded above by 1.0 and below by 0.0. Because only
one observation reaches the upper bound and none reach the lower, however, an OLS regression yields virtually identical coefficients to a double-censored MLE regression. The index could also be treated as a proportion, and transformed and estimated according to Greene's suggested method (p.653). The magnitude and interpretation of the estimated coefficients in this regression will change; however, the relative magnitudes and levels of significance are comparable to the OLS and MLE results. 11
An anonymous reviewer points out that certain cropping practices (such as growing a
green manure, which involves incorporating a legume cover crop directly into the soil) have both short-run and long-run impacts on soil quality.
31
