# Duality of technology and economic behavior in a data development analysis/assurance region context

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In 1957, Farrell decomposed overall efficiency of a production method into technical and price efficiencies, and provided the principles of measuring them. The principle of measuring technical efficiency is to continually choose the best base set of production methods to find the largest possible relative efficiency rating for a reference production method, i.e., a production method whose efficiency is being measured. For a data set of a few dozens of production methods, this easily requires thousands of arithmatic operations for a computer. A straightforward computation method was not available until 1978, when Charnes, Cooper and Rhodes developed sound mathematical foundations for measuring efficiency of decision making units (DMU), which thay called Data Envelopment Analysis (DEA). By using the definition of efficiency in natural science, they were able to define the efficiency measurement from a benefit-cost formulation as a linear program (LP). The name Data Envelopment Analysis suggests that in measuring efficiency, an efficient frontier is being constructed that envelops the data. The use of the term decision making units suggests that DEA applies to measuring efficiency of not-for-profit organizations in addition to for-profit firms. Since then it has attracted a wide range of contributions. As of June 1987, Seiford had catalogued 167 DEA contributions in a 1978-1986 bibliography. To date, DEA contributions have been predominantly in measuring technical efficiency. In 1985, Thompson et al. developed a concept which they called assurance region (AR) to apply in restricting the virtual multipliers, because most of them have no relation whatsoever to reasonable prices. This AR was applied in a post-DEA analysis to find the best site in Texas for a Super Conducting Super Colider. Since then, Thompson, Thrall and other collaborators have continued to refine the concept and applied it in other applications in the pursuit of working toward meaningful measures of overall economic efficiency. Here the virtual multipliers are opportunity cost-type weights for the inputs and outputs. These virtual multipliers are determined from value-free data of input and output combinations, as observed. Comparable measures made by economists are predominantly done using a parametric approach, in which an a priori functional form is assumed across all data points in arriving at an estimate of the efficient frontier. This dissertation's overall goal is to work soundly toward establishing linkage between a value base in a post-DEA analysis and a value-free base in a DEA analysis. This is accomplished by generalizing Shephard's duality theorem to profit and transformation functions. Then the formal definitions of cost (price), revenue, profit and allocative efficiencies, as reported by Lovell and Schmidt are extracted to give formal definitions of these efficiencies in a non-parametric post-DEA (DEA/AR) measurement setting.