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Weekly UpdatesMeasuring quality attributes of raw materials and intermediate and final products is a common problem in many sectors of the economy. The importance of quality for agricultural production and commodity markets has long been documented, (1) but its treatment in an empirically tractable way has received little formal attention in the production literature. (2) To model and estimate quality with production data is challenging. Quality is multidimensional, and to make meaningful comparisons or policy recommendations, one often needs an aggregate measure of quality. In addition, given that different quality attributes often refer to a single production process, econometric estimation may be difficult due to endogeneity problems. (3)
In this study, we propose a linear programming methodology to measure the characteristics and composition of agricultural products and provide an application using data on grapes for wine production. We develop an aggregate measure of quality using a primal representation of a multioutput technology based on the directional distance function (Chambers, Chung, and Fare 1996). This measure can be used to compare firms' output taking into account a set of quality attributes. It is an improvement with respect to the industry standard practice of using a single attribute, for instance the sugar content in wine production.
This methodology allows one to evaluate also how quality attributes interact with quantity, making the quality indicators useful for regulation. In the United States, quality regulation is based on minimum standards enforced by Marketing Orders (see, e.g., Bockstael 1984; Nguyen and Vo 1985; Jesse 1987; Chambers and Weiss 1992). In contrast, the European Union typically controls quality through the imposition of a ceiling on the yields per unit of land. (4) Advocates of this regulation assert that by reducing production per acre, it is possible to increase quality. Arnaud, Giraud-Heraud, and Mathurin (1999) found that quantity restrictions can be welfare-enhancing if there is a trade-off between quality and quantity. If true, an output ceiling would benefit consumers and producers alike and should not be prosecuted by antitrust authorities (Canali and Boccaletti 1998). But although this trade-off may be theoretically plausible, it has received very little empirical attention. In this study, we investigate whether and to what extent aggregate quality and quantity are substitutes in wine grapes.
Our work is related in spirit to the hedonic pricing literature, which looks at how to adjust prices for increases in the quality of goods such as computers and cars (Triplett 1990). However, given that most hedonic studies use market prices to identify consumers' preferences, to the best of our knowledge, no hedonic study has estimated the production technology. Fixler and Zieschang (1992) provide one of the first attempts to incorporate quality attributes in a model of producer behavior. In their model, process and quality changes are outcomes of technologyptimization problem. The authors show how a market-determined price-characteristics locus can be used to adjust the Tornquist output- and input-oriented multifactor/multiple output productivity indexes of Caves, Christensen, and Diewert (CCD) (1982) for changes in input, output, and process characteristics. Using radial distance functions, they propose a quality-adjusted index as the product of a quality index and a CCD-type Tornqvist productivity index.
Fare, Grosskopf, and Roos (1995) study a panel of Swedish pharmacies and use the attributes together with ratios of distance functions to measure the service quality of each pharmacy by decomposing the Malmquist productivity change index into three components: quality change, technical change, and efficiency change. Jaenicke and Lengnick (1999) merge the soil science literature on soil-quality indexes with the literature on efficiency and total factor productivity indexes and develop a soil-quality index using radial distance functions.
A related approach uses the directional distance function. Chambers, Chung, and Fare (1996) introduce this generalization of the radial distance function to production economics, extending and adapting the translation functions of Kolm (1976) and Blackorby and Donaldson (1980), and Luenberger's (1992, 1994) benefit function. Directional distance functions are a more general way to represent technology and compare and measure input, output, and productivity aggregates (Chambers 2002). In addition, the directional distance function allows the comparison of different firms along a preassigned direction vector representing a numeraire or reference bundle.
In this study, we use the directional distance function to incorporate quality attributes into the technology and depart from the models reviewed above in the construction of an indicator expressed in difference form instead of an index in ratio form. Using data for Chardonnay and Merlot grapes, we propose two alternative measures of aggregate quality, based on different direction vectors or numeraires. Although the two indicators provide similar results, they are quite different when related to quantity. We find evidence of a trade-off between quantity and aggregate quality for both of the investigated grapes.
Notation and Model Specification
Following Chambers (1998, 2002), we construct an output aggregator expressed in difference rather than in ratio forms. This difference stems from the translation property of the directional distance function, which makes the Luenberger indicator translation invariant in outputs, in contrast to the property of homogeneity of degree zero in outputs of the Malmquist index coming from the linear homogeneity of the output distance function a la Shephard (1970).
An indicator based on directional distance functions allows comparison of firms based on the distance from the frontier along a preassigned and common (to all firms) direction, which can be chosen to reflect the preferences and needs of the buyer or downstream firm with respect to the quality attributes. Moreover, it may be the case that to be valuable to a downstream firm, the composition of the raw material has to be close to an "ideal" bundle of attributes preferred by the buyer. In other words, in some instances, the composition has to be well balanced, and some of the attributes have to be within a certain range. The choice of the direction allows one to take these aspects into account and evaluate the quality attributes produced by a pool of suppliers according to buyers' needs.
Let x [member of] [R.sup.N.sub.+] be a vector of inputs, y [member of] [R.sub.+], the output quantity, and s [member of] [R.sup.M.sub.+], a vector of quality attributes. We treat attributes as outputs and can think of the vector (y, s) as the output vector. The technology can be defined in terms of a set T [subset] [R.sup.N.sub.+] x [R.sub.+] x [R.sup.M.sub.+]:
T = {(x [memer of] [R.sup.N.sub.+], y [member of] [R.sub.+] s [member of] [R.sup.M.sub.+]: x can produce (y, s)}.
The technology consists of all output and attributes that are feasible for some input vector.
T satisfies the following properties (modified from Chambers 2002):
T.1: T is closed;
T.2: inputs are freely disposable: for x' [greater than or equal to] x, (x, y, s) [member of] T [??] (x', y, s) [member pof] T;
T.3: outputs (5) are freely disposable: for y' [less than or equal to] y and s' [less than or equal to] s, (x,y, s) [member of] T [??] (x, y', s') [member of] T; and
T.4: doing nothing is feasible: ([0.sup.n], 0, [0.sup.m]) [member of] T.
Following Chambers, Chung, and Fare (1996) and Chambers (2002), we define the directional technology distance function as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The reference vector ([g.sub.x], [g.sub.y], [g.sub.s]) establishes the direction over which the distance function is determined. The value [[??].sub.T] (x, y, s; [g.sub.x], [g.sub.y], [g.sub.s]) represents the maximal translation of the input and output vector in the direction of ([g.sub.x], [g.sub.y], [g.sub.s]) that keeps the translated input and output vector inside T. (6) It measures how many units of ([g.sub.x], [g.sub.y], [g.sub.s]), the numeraire or reference commodity, one firm would need to improve to move from an interior point to the frontier.
Chambers, Chung, and Fare (1996) derive the properties of this distance function and show that all known (radial) distance and directional distance functions can be depicted as special cases of the directional technology distance function. As a special instance, we propose the directional quality distance function
(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The value [[??].sub.Q](x, y, s; [0.sup.N], 0, [g.sub.s]) represents the maximal translation of the attributes vector in the direction of ([g.sub.s]) that keeps the translated output vector inside T. It thus represents how many units of the numeraire can be obtained by moving from an interior point to the boundary of T.
As a matter of comparison, it is useful to relate the directional quality distance function with the Shephard (1970) (radial) quality distance function
(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
which represents the minimum (technically, the infimum) that the quality bundle can be expanded and still be feasible. The Shephard distance function is related to the directional quality distance function when [g.sub.s] = s (i.e., when the direction is given by the firms' choices of quality attributes) by the following: (7)
(3) [[[??].sub.Q](x, y, s; [0.sup.N], 0, s) = 1/[D.sub.Q](x, y, s) - 1.
The Luenberger Quality Indicator
A quality indicator provides a summary measure of quality attributes. Comparing the input/output/attributes of a firm 1 to a reference firm 0, we can define the 1-technology Luenberger quality indicator as
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