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Essays in production theory : efficiency measurement and comparative statics Mendoza, Maria Nimfa F. 1989

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ESSAYS IN PRODUCTION THEORY: EFFICIENCY MEASUREMENT AND COMPARATIVE STATICS By Maria Nimfa F. Mendoza B. Sc. (Applied Mathematics) University of the Philippines M . Sc. (Statistics) University of the Philippines A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES ECONOMICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1989 © Maria Nimfa F. Mendoza, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date iSOe^-jU^ 3 V , / < ? ?9 DE-6 (2/88) Abstract Nonparametric linear programming tests for consistency with the hypotheses of technical effi-ciency and allocative efficiency for the general case of multiple output-multiple input technolo-gies are developed in Part I. The tests are formulated relative to three kinds of technologies — convex, constant returns to scale and quasiconcave technologies. Violation indices as summary indicators of the distance of an inefficient observation from an efficient allocation are proposed. The consistent development of the violation indices across the technical efficiency and allocative efficiency tests allows us to obtain comparative measures of the degrees of technical inefficiency and pure allocative inefficiency. Constrained optimization tests applicable to cases where the producer is restricted to optimizing with respect to a subset of goods are also proposed. The latter tests yield the revealed preference-type inequalities commonly used as tests for consis-tency of observed data with profit maximizing or cost minimizing behavior as limiting cases. Computer programs for implementing the different tests and sample results are listed in the appendix. In part II, an empirical comparison of nonparametric and parametric measures of technical progress for constant returns to scale technologies is performed using the Canadian input-output data for the period 1961-1980. The original data base was aggregated into four sectors and ten goods and the comparison was done for each sector. If we assume optimizing behavior on the part of the producers, we can reinterpret the violation indices yielded by the efficiency tests in part I as indicators of the shift in the production frontier. More precisely, the violation indices can be considered nonparametric chained indices of technical progress. The parametric measures of technical progress were obtained through econometric profit function estimation using the generalized McFadden flexible functional form with a quadratic spline model for technical progress proposed by Diewert and Wales (1989). Under the assumption of constant ii returns, the index of technical change is defined in terms of the unit scale profit function which gives the per unit return to the normalizing good. The empirical results show that the parametric estimates of technical change display a much smoother behavior which can be attributed to the incorporation of stochastic disturbance terms in the estimation procedure and, more interestingly, track the long term trend in the nonparametric estimates. Part III builds on the theory of minimum wages in international trade and is a theoreti-cal essay in the tradition of analyzing the effects of factor market imperfections on resource allocation. The comparative static responses of the endogenous variables — output levels, employment levels of fixed-price factors with elastic supply and flexible prices of domestic re-sources — to marginal changes in the economy's exogenous variables — output prices, fixed factor prices and endowments of flexibly-priced domestic resources -— are examined. The effect of a change in a fixed factor price on other flexible factor prices can be decomposed Slutsky-like into substitution and scale effects. A symmetry condition between fixed factor prices and flexible factor prices is obtained which clarifies the concepts of "substitutability" and "com-plementarity" between these two kinds of factors. As an illustration, the model is applied to the case of a devaluation in a two-sector small open economy with rigid wages and capital as specific factors. The empirical implementation of the general model for the Canadian economy is left to more able econometricians but a starting point can be the sectoral analysis performed in Part II. iii Table of Contents Abstract ii List of Tables ix List of Figures xi Acknowledgement xiv Overview 1 I Nonparametric Measures of Technical and Allocative Efficiency 9 1 Introduction 10 2 Three Alternative Nonparametric Specifications of Technology 13 2.1 Description of technologies 13 2.2 Construction of approximating convex sets 18 3 The Measurement of Technical Inefficiency for a Convex Technology 24 3.1 Definition of technical efficiency 24 3.2 Test for technical efficiency (test 1) 28 3.3 Some notes on the technical efficiency test 32 3.3.1 Specifying the efficiency direction vector 7 32 3.3.2 Complications due to the free disposability assumption 33 3.3.3 An area of further research 35 3.4 Dual interpretation 36 iv 3.5 Special case: assuming concavity of an individual technology function 39 4 The Measurement of Technical Inefficiency for a Convex Conical Technology 43 4.1 Definition of and test for technical efficiency (test 2) 43 4.2 Dual interpretation 47 4.3 Special case: assuming linear homogeneity of an individual technology function . 50 5 The Measurement of Technical Inefficiency for a Quasiconcave Technology 52 5.1 Definition of and test for technical efficiency (test 3) 52 5.2 Dual interpretation 60 6 The Nonparametric Measurement of Allocative Inefficiency 62 7 The Measurement of Allocative Inefficiency for a Convex Technology Assum-ing Partial Profit Maximization 67 7.1 Allocative efficiency test (test 4) . . . 67 7.2 Some results 70 7.2.1 Technical efficiency proposition 70 7.2.2 Relative efficiency proposition 71 7.2.3 Nonnegativity of ef 71 7.2.4 Comparability of ef and 6* 71 7.2.5 Special cases: K = 0,1,N-1,N 72 7.2.6 Restricting u* to be nonnegative 74 8 A LeChatelier Principle for Measures of Allocative Inefficiency 76 9 The Measurement of Allocative Inefficiency for a Convex Conical Technology Assuming Partial Profit Maximization 78 9.1 Allocative efficiency test (test 5) 78 9.2 Some results 80 v 10 The Measurement of Allocative Inefficiency for a Quasiconcave Technology Assuming Partial Profit Maximization 83 10.1 Allocative efficiency test (test 6) 83 10.2 Some results 85 11 The Measurement of Allocative Inefficiency Assuming Complete Profit Maximization 87 11.1 The convex technology case (test 7) 87 11.2 The convex conical technology case (test 8) 90 11.3 The quasiconcave technology case (test 9) 95 12 The Efficiency Tests with the No Technological Regress Assumption 98 13 Conclusion 100 II Nonparametric and Parametric Measures of Technical Progress: A n Appli-cation to the Canadian Input-Output Data 104 14 Introduction 105 15 A Nonparametric Approach to Measuring Technical Progress 110 16 A Parametric Approach to Measuring Technical Progress 126 17 A Comparison of Nonparametric and Parametric Estimates of Technical Progress 132 III Comparative Statics for a Small Open Economy in the Presence of Factor Price Rigidity 142 18 The Model 143 vi 18.1 Introduction 143 18.2 Model formulation, a constrained GNP problem 144 18.3 First-order conditions 147 18.4 Fundamental matrix equations 148 19 Comparative Static Results 152 20 A n Illustration: Devaluation with Rigid Wages in a Specific-Factors Model 161 21 Towards Econometric Implementation 168 21.1 Policy implications 168 21.2 The shadow price of a fix-price factor 170 21.3 Econometric estimation 174 Bibliography 178 Appendices 183 A Incorporating the No Technological Regress Assumption with the Efficiency Tests 183 B The Canadian Input-Output Data, 1961-1980 186 B. l The data base 186 B.2 Data manipulations at the sectoral level 188 B.2.1 Output and intermediate input prices and quantities 188 B.2.2 Missing observations for 1980 intermediate inputs 189 B.2.3 Imports and inventories 189 B.2.4 Tax adjustment of output and input prices 192 B.2.5 Capital rental prices , . . 196 B.2.6 List of retained variables 208 vii B.3 Aggregation 209 B. 4 The constructed data sets . . ' 220 C Computer Programs for Implementing the Nonparametric Efficiency Tests 233 C. l Introduction to the computer programs 233 C.2 Listings of computer programs 235 C.2.1 Subroutines TESTQ, TESTPQC and TESTPQ . 235 C.2.2 Sample main calling programs 256 C.3 Sample results using sector I (resources) data 265 C.3.1 Technical efficiency test results 266 C.3.2 Allocative efficiency test results (assuming partial profit maximization) . 276 C.3.3 Allocative efficiency test results (assuming complete profit maximization) 290 D Results of the Sectoral Profit Function Estimation 304 viii List of Tables 4.1 Two input-constant output example for test 2 45 15.2 Test 8 violation indices for the four sectors I l l 15.3 Nonparametric indices of technical progress A' for the four sectors 114 15.4 Test 8' violation indices for the four sectors 118 15.5 Divisia productivity change indices TJJ for the four sectors 120 15.6 Fisher productivity change indices rF for the four sectors 121 16.7 Model specification, R2 and log likelihood values for the four sectors 128 16.8 Parametric indices of technical progress A' for the four sectors 129 19.9 Factor intensity configurations and resulting effects on the flexible factor prices 156 B.10 Data items for each of the 37 sectors 187 B . l l Inconsistencies in the value of intermediate output and the sum of values of the good used as intermediate inputs, 1980 190 B.12 Imputed values for missing observations for 1980 intermediate inputs 191 B.13 Average proportion of market price received by the producers in sectors with the most output price distortions 194 B.14 Sectoral 20-year average internal rates of return 203 B.15 A comparison of calculated internal rates of return and the McLeod, Young and Weir's 10 industrial bond yield average 206 B.16 List of aggregated sectors 219 B.17 List of goods 220 B.18 The data for sector I, the resources sector 221 ix B.19 The data for sector II, the manufacturing sector (export market-oriented) . . . 223 B.20 The data for sector III, the manufacturing sector (domestic market-oriented) . 225 B.21 The data for sector IV, the services sector 227 B.22 Capital data series based on an exogenous bond rate, sector I 229 B.23 Capital data series based on an exogenous bond rate, sector II 230 B.24 Capital data series based on an exogenous bond rate, sector III 231 B. 25 Capital data series based on an exogenous bond rate, sector IV 232 C. 26 Summary of the different efficiency tests, informational requirements and options 236 D. 27 Values of the exogenous parameters of the estimated models 306 D.28 Parameter estimates, standard errors and t-values, sector I 315 D.29 Parameter estimates, standard errors and t-values, sector II 317 D.30 Parameter estimates, standard errors and t-values, sector III 319 D.31 Parameter estimates, standard errors and t-values, sector IV 321 D.32 Own price elasticities and 1971 cross-price elasticities, sector I 323 D.33 Own price elasticities and 1971 cross-price elasticities, sector II 324 D.34 Own price elasticities and 1971 cross-price elasticities, sector III 325 D.35 Own price elasticities and 1971 cross-price elasticities, sector IV 326 D.36 Five-year interval cross-price elasticities, sector I 327 D.37 Five-year interval cross-price elasticities, sector II 329 D.38 Five-year interval cross-price elasticities, sector III 331 D.39 Five-year interval cross-price elasticities, sector TV 333 x List of Figures 2.1 Illustration of the three classes of technologies in the single output-single input case 17 2.2 Construction of free disposal convex hulls under the assumptions of a convex technology and a constant returns to scale technology 20 2.3 Construction of approximating level sets for a quasiconcave technology . . . . 22 3.4 Convex hull of observations from a single output-single input technology, an example 25 4.5 The conic convex hull for a two input-const ant output example 46 5.6 Technical efficiency for a quasiconcave technology function, single output-single input case 57 6.7 Allocative efficiency for a two-good convex technology 63 7.8 Production possibilities set T with u* > 0 75 15.9 Test 8 violation indices £(, sector I 112 15.10 Test 8 violation indices , sector II 112 15.11 Test 8 violation indices £jT, sector III 113 15.12 Test 8 violation indices , sector IV 113 15.13 Nonparametric indices A f , sector I 115 15.14 Nonparametric indices A' , sector II 115 15.15 Nonparametric indices A' , sector III 116 15.16 Nonparametric indices A' , sector TV 116 15.17 Comparative behavior of A' , T1D and rjr, sector I . . 122 xi 15.18 Comparative behavior of A' , T1D and TF, sector II 123 15.19 Comparative behavior of A' , r|j and rF, sector III 124 15.20 Comparative behavior of A' , r|j and TF, sector IV 125 16.21 Parametric indices A1, sector I 128 16.22 Parametric indices A' , sector II 130 16.23 Parametric indices A f , sector III 130 16.24 Parametric indices A' , sector IV 131 17.25 Comparative behavior of A' and A f , sector I 133 17.26 Comparative behavior of A* and A*, sector II 133 17.27 Comparative behavior of A' and A f , sector III 134 17.28 Comparative behavior of A' and A' , sector IV . 134 17.29 Residual plot for equation 1, sector I 135 17.30 Residual plot for equation 2, sector I 136 17.31 Residual plot for equation 3, sector I 136 17.32 Residual plot for equation 4, sector I 137 17.33 Residual plot for equation 5, sector I 137 17.34 Residual plot for equation 6, sector I 138 17.35 Residual plot for equation 7, sector I 138 17.36 Residual plot for equation 8, sector I 139 17.37 Residual plot for equation 9, sector I 139 17.38 Divisia indices of output and input growth in the resources sector (I) 141 B.39 The behavior of the weighted average of the internal rates of return and the McLeod, Young and Weir's 10 industrial bond yield average 207 B.40 Output prices for the resources sectors (I) 211 B.41 Output prices for the export market-oriented manufacturing sectors (II) . , .212 xii B.42 Output prices for the domestic market-oriented manufacturing sectors (III), subgroup 1 213 B.43 Output prices for the domestic market-oriented manufacturing sectors (III), subgroup 2 214 B.44 Output prices for the services sectors (IV), transportation subgroup 215 B.45 Output prices for the services sectors (IV), communications and utilities sub-group 216 B.46 Output prices for the services sectors (IV), construction, trade, f.i.r.e., and commercial services subgroup 217 xiii Acknowledgement I thank my major adviser, Erwin Diewert, for his intellectual and personal generosity. Without his support, I would not have made it through my very rough sailing of graduate school. I am fortunate to have such a brilliant adviser and working with him has been one of my life's greatest pleasures. I thank James Roumasset who has been a personal mentor and who made it possible for me to study at UBC. I am grateful to Terry Wales for the computer programs for the nonlinear optimization routines; Ken White for econometric help and, moreover, for pushing me though at times I resented it; and Ken Hendricks for reading the draft of my thesis and raising good questions in my departmental thesis defence. The meticulous review of my work by my external examiner, Ulrich Kohli of the University of Geneva, and his insightful comments and suggestions are deeply appreciated. Sincere thanks go to Alan and Esther Chung for providing office space, computer help, enlightening discussions and dinner invitations. I thank my friends who have not given up on me, including those who constantly berated me to finish; and my family, especially my mother and sister Divina, for their emotional and financial support. This dissertation is dedicated to J.P. Mendoza; I am proud to be his daughter. xiv Overview This dissertation is a collection of three essays in production theory. The major focus of this research is the development of measurement methods in production analysis rather than the evaluation of a specific economic policy issue. The first essay (part I) proposes nonparametric linear programming tests for consistency of observed data with the hypothesis of productive efficiency at the firm level. The empirical performance of a nonparametric measure of technical progress vis-a-vis a parametric measure of technical progress is examined in the second essay (part II) in an application to the Canadian input-output data. An efficiency test developed in part I is used to obtain the nonparametric measures of technical progress and in addition, econometric profit function estimation assuming an explicit functional form is carried out to yield the parametric measures of technical progress for four aggregated sectors covering the Canadian economy. Whereas the first two essays analyze the economic behavior of price-taking producers facing technological constraints, the third essay (part III) takes a more general view of the economy by linking the different sectors through a trade-theoretic approach to produc-tion theory. In this framework, the assumptions of price-taking behavior and technological constraints are still maintained for individual producers and the constraints on factor endow-ments for the domestic economy are explicitly taken into account. Specifically, part III looks at the effects of factor price rigidities on resource allocation. The assumption of productive efficiency or optimizing behavior plays a central role in eco-nomics. For example, in duality theory, Hotelling's lemma and Shephard's lemma giving the derivative properties of profit functions and cost functions, respectively, derive from the as-sumption of optimizing behavior of producers. Given the technological constraints determining the feasible output-input combinations and the market constraints as reflected in the prices of 1 Overview 2 goods, the producer is assumed to maximize some objective function. The production plan cho-sen by the producer is then technically efficient; that is, the plan is some point at the boundary of the firm's production possibilities set. Loosely speaking, technical efficiency implies that the firm is producing maximal output given input levels and minimizing input use given output levels. Moreover, if at the prevailing market prices, the producer is maximizing the value of the objective function describing its behavioral goal (profit maximization, cost minimization or some variant of restricted profit maximization), then the production plan is ahocatively efficient. Since technical inefficiency entails some waste of resources and, consequently, an economic cost, then an ahocatively efficient production plan must necessarily be technically efficient in some sense. Given the observed production plans and prices faced by firms, it would be of interest to test whether these data are consistent with the hypothesis of optimizing behavior. The efficiency tests proposed in part I can be used to empirically test whether the regularity conditions often imposed in production theory hold. This study also defines violation indices as indicators of the degree of departure of an observed production plan from the maintained efficiency hypotheses. Since the art of economic theorizing requires some level of abstraction to get meaningful results, the pattern of violation indices may be explained by some unaccounted factors like technical change, differences in firm sizes and regulation costs and hence, may suggest to the researcher depending on prior beliefs and subjective interpretation modifications regarding model and variable specification. In this function, the efficiency tests can then serve as a tool for exploratory data analysis. In their straightforward use in efficiency studies of a cross-section of firms in an industry, the violation indices measure the economic loss due to inefficiency and as such can be of practical importance. Since the proposed efficiency tests also recover efficient allocations relative to a production plan rated inefficient, the efficiency tests can also be useful for prescriptive purposes in efficiency studies. • The major contributions of part I to the field of nonparametric efficiency measurement are: (1) the generalization of technical efficiency and allocative efficiency tests to handle multiple output-multiple input technologies; (2) the definition and consistent development across the Overview 3 efficiency tests of violation indices or efficiency loss measures which are invariant to units of scale in the measurement of goods; and (3) the formulation of constrained optimization or partial profit maximization tests applicable to cases where the producer is restricted to optimizing with respect to a subset of goods. There are three general sets of tests proposed: technical efficiency tests using quantity data only, constrained optimization or partial profit maximization tests using quantity data and a subset of price information, and unconstrained optimization or complete profit maximization tests using both quantity and price data. Each of the above sets of tests is developed relative to three kinds of technologies — convex, constant returns to scale and quasiconcave technologies. A basic problem in empirical production analysis is that the underlying technology is gen-erally unknown. In the nonparametric approach taken in this study, a deterministic production frontier is assumed. Based on observed quantity data, the approximation technique uses the mathematical tools of convex analysis. The regularity conditions imposed on the unknown technologies facilitate the description of the technologies using closed, convex sets exhibiting free disposal. The true production possibilities sets in the convex and convex cone case, and the true upper level sets in the quasiconcave case are approximated by the convex hulls of ob-served data. Thus, our approach is essentially an elaboration of the original pioneering works of Farrell (1957) and Farrell and Fieldhouse (1962). In the context of multiple output technologies and due to the complications brought by the assumption of free disposability which plays an important role in the theory of efficiency measurement, it is necessary to give a precise definition of technical efficiency and introduce the use of an efficiency direction vector specifying the goods with respect to which we would like to measure the inefficiency. The definition of technical efficiency used in this study is conditional on the efficiency director vector chosen. The violation index for a given technical efficiency test gives the equiproportionate increase in outputs and decrease in inputs indexed in the efficiency direction vector needed for an observation to satisfy the technical efficiency hypothesis. Input based and output based measures can be obtained by specifying the efficiency direction vector Overview 4 r appropriately. The violation indices for the technical efficiency tests are equiproportionate loss measures; the required adjustment indicated by the violation index may move an inefficient production plan in the interior of the production possibilities set to a free disposal region at the boundary of the production possibilities set. For the allocative efficiency tests, a profit function or a restricted profit function is posited to describe the economic objective of the firm. The formulation of the constrained optimization tests encompass a wide range of behavioral descriptions among which are full profit maxi-mization, revenue maximization, cost minimization and variable profit maximization. A set of reference goods is specified to obtain the violation index which measures the value of goods in this set lost due to allocative inefficiency. With a consistent specification of the efficiency direction vector in the technical efficiency test, the violation index in the allocative efficiency test can be decomposed into components due to technical inefficiency and due to pure allocative inefficiency. A LeChatelier principle proposition for measures of allocative inefficiency is ob-tained which says that given a fixed reference set of goods, the violation index cannot decrease as the number of goods (or prices) with respect to which the producer can optimize increases. This proposition is consistent with the notion that the more goods a producer can freely vary, the higher profits can be. In the limiting cases of full profit maximization, the constrained optimization tests involving the solution of linear programming problems yield violation indices which can alternatively be calculated using revealed preference-type inequalities. Appendix A lists the modified efficiency tests incorporating the no technological regress assumption. Computer programs for implementing the different efficiency tests and sample results using the Canadian input-output data (described in appendix B) are given in appendix C. The interpretation of the empirical results in appendix C is meant to illustrate the concepts discussed in part I and hence, the violation indices are interpreted as measures of inefficiencies. This interpretation may be considered more applicable to cross-section data on firms.rather than the kind of time series data used. Like those of statistical tests, the results of the efficiency tests can be used for several purposes depending on the economic problem under study, and Overview 5 their interpretation can be subject to some element of arbitrariness depending on extraneous information possessed by the researcher. It must be noted that the efficiency tests are developed under joint assumptions on the technology and the optimizing behavior of the producers. If we assume optimizing behavior on the part of the producers, the violation indices obtained from the proposed efficiency tests in part I can be reinterpreted as indicators of the shift in the production frontier. In cross section data on firms' production plans, the violation indices can be used as multifactor productivity measures with smaller violation indices indicating better productivity performance. With time series data, the violation indices as shown in part II can be interpreted as chained indices of technical progress or productivity growth. Since productivity improvements mean greater effectiveness in utilizing resources to produce output goods, the efficiency gain can have an impact on economic agents, both producers or owners of factors of production and consumers, in terms of higher real incomes and standards of living. Interest in productivity measurement therefore has practical significance. In part II, an empirical comparison of nonparametric and parametric measures of technical progress for constant returns to scale technologies is performed. The Canadian input-output data for the period 1961-1980 are used. The data are aggregated into four sectors: resources, export market-oriented manufacturing, domestic market-oriented manufacturing, and services sectors. Each sector is assumed to have a single output technology with output from the other sectors entering the production process as intermediate inputs. A ten-good model is used for the resources sector and a nine-good model for each of the three other sectors. The data is described in appendix B. Measures of technical progress are obtained for each sector. With the assumption of constant returns, the index of technical change is defined in terms of the unit scale profit function which gives the per unit return to the scaling or normalizing good. The nonparametric measures of technical change are based on the violation indices calculated using the unconstrained optimization test for a convex conical technology developed in part I. For each sector, the pattern of shift in the production frontier indicated by the nonparametric index of technical progress is consistent with that yielded by the Divisia and Fisher productivity Overview 6 change indices. The parametric measures of technical change are obtained through economet-ric profit function estimation using the generalized McFadden flexible functional form with a quadratic spline model for technical progress proposed by Diewert and Wales (1989b). The sectoral profit function estimation procedure and results, together with some interpretation of the price elasticities of output supply and input demand, are discussed in Appendix D. The empirical results show that the parametric estimates of technical change display a much smoother behavior which can be attributed to the incorporation of stochastic disturbance terms in the estimation procedure. However, the parametric estimates of technical change do track the long term trend in the nonparametric estimates. Both nonparametric and parametric methods use the same information: price and quantity data. The nonparametric method has the advantage of not imposing any functional form in characterizing the production technology and is computationally much simpler. The nonparametric estimates in this study, obtained without introducing stochastic disturbance terms, do not possess statistical properties needed for hypothesis testing in the context of statistical inference. In part III, we move to a more general view of the production sector of the domestic economy by linking the different sectors and explicitly modeling the constraints on factor endowments in a small open economy model. The constrained GNP maximization approach of Neary (1985) and the fundamental matrix equation of production theory as developed by Diewert and Wood-land (1977) and Diewert (1982) are adapted to analyze the effects of factor price rigidities on resource allocation. We investigate in what sense the Tobin-Houthakker conjecture in consumer theory that says a reduction in the ration of one good will increase the consumption of unra-tioned substitutes and decrease the demand for unrationed complements hold in production. In a trade-theoretic approach to production theory, the analogous question is how do rigid factor prices affect other flexible factor prices. The comparative static responses of the endogenous variables — output levels, employment levels of fixed-price factors with elastic supply and flex-ible prices of domestic resources — to marginal changes in the economy's exogeneous variables — output prices, fixed factor prices and endowments of flexibly-priced domestic resources — Overview 7 are examined. The effect of a change in a fixed factor price on flexible factor prices can be decomposed into a pure substitution term and a scale effect term. Hence, factor intensities which determine the scale effect term, in addition to substitution possibilities among factors, play a role in determining this response of the flexible factor prices. However a similar Tobin-Houthakker conjecture occurs when we look at the symmetry condition between fixed factor prices and flexible factor prices and their respective employment levels and endowments. In this interpretation, the essense of "substitutability" and "complementarity" differs from that as reflected in the price elasticities of output supply and input demand obtained in the parametric sectoral analysis in part II. As an illustration, the model is used to analyze the effects of a devaluation in a two-sector small open economy with rigid wages and capital goods as specific factors. This exercise highlights the mechanism by which currency devaluation works as an employment stimulation policy tool. In the presence of wage rigidity, the devaluation leads to a more than proportional increase in returns to the sector-specific capital that makes possible the output and national income effects. The effectiveness of the devaluation as an employment policy tool depends on the wage elasticity of labor demand in the economy which is a function of both substitution terms and factor intensities at the sectoral level. To conclude this overview, we note some major assumptions and limitations of the ap-proaches taken in this research. The competitive market framework have been utilized in all three essays. The prices we use are assumed to be outside the control of the individual pro-ducers. In the empirical comparison of nonparametric and parametric measures of technical progress in part II and the trade-theoretic model used to derive the comparative static results in part III, the sectoral technologies are assumed to exhibit constant returns to scale. These simplifying assumptions enable us to ignore the complications due to the possibility of mar-ket power of firms and the necessity of modeling market structure, entry and exit of firms and demand conditions arising from the consumption side of the economy. Also, the models used are static, and dynamic considerations such as adjustment costs, price expectations and Overview 8 uncertainty, etc. are not taken into account. The thrust of this research is efficiency which is concerned with the allocation of resources among alternative uses. Economic policies and undertakings must ultimately address the satisfaction of human needs and wants and this can involve equity and distributional concerns. The efficiency measurement methods can be useful, however, in evaluating possible efficiency-equity tradeoffs. I. Part I Nonparametric Measures of Technical and Allocative Efficiency Chapter 1 Introduction Nonparametric linear programming tests for consistency with the hypothesis of productive efficiency at the firm level for the general case of multiple output-multiple input technologies are developed in this study. Given the technology, prices and a behavioral description of the firm as embodied in its economic objective function, productive efficiency1 at the firm level is defined as utilizing the optimal combination of outputs and inputs given the constraints faced by the firm. We consider three general technologies — convex, constant returns to scale and quasiconcave technologies. Productive efficiency subsumes technical efficiency which we define as producing at some point at the boundary of the production possibilies set for the convex and convex conical technologies and of the upper level set for a quasiconcave technology. Violation indices as summary indicators of the distance of an inefficient observation from an efficient allocation are proposed. Alternatively, the violation indices can be interpreted as measures of economic loss due to inefficiency. As is the case with most nonparametric programming approaches to efficiency analysis, productive efficiency is measured in the following tests relative to the best-practice technology. Hence, in addition to the violation indices, we can recover efficient allocations corresponding to an inefficient one. This feature of the tests can be useful for prescriptive purposes. This study draws on the earlier works in nonparametric production analysis in economics and data envelopment analysis (DEA) in operations research. The pioneering works in non-parametric production analysis were done by Farrell (1957) and Farrell and Fieldhouse (1962) and latter important contributions were made by Afriat (1972), Hanoch and Rothschild (1972), 1 P r o d u c t i v e efficiency is a l t e r n a t i v e l y referred to as economic efficiency or a l l o c a t i v e efficiency. 10 Chapter 1. Introduction 11 Diewert and Parkan (1983), Varian (1984a) and Fare, Grosskopf and Lovell (1985).2 An intro-ductory survey to data envelopment analysis can be found in Charnes and Cooper (1985) and Charnes, Cooper, Golony and Seiford (1985). The major contributions of this study to the field of nonparametric efficiency measurement are the generalization of technical efficiency and allocative efficiency tests to handle the multiple output-multiple input technologies; the definition and consistent development, across technical efficiency and allocative efficiency tests, of violation indices as measures of inefficiency; and the formulation of constrained optimization tests applicable to cases where the producer is restricted to optimizing with respect to a subset of goods. The symmetric treatment of output goods and input goods facilitates the handling of multiple output-multiple input technologies. The resulting violation indices, being equiproportionate loss measures, are invariant to the units of measurement of goods. As such, the violation indices are in the spirit of Debreu's (1951) concept of coefficient of resource utilization whose inverse measures the economic loss due to inefficient utilization of resources. The consistent development of the violation indices allows us to obtain comparative measures of the degrees of technical inefficiency and pure allocative inefficiency. The constrained optimization tests, using price and quantity data, turn out to be the more general formulations for testing for overall productive efficiency; they yield the revealed preference-type inequalities commonly used as tests for consistency of observed data with profit maximizing or cost minimizing behavior as limiting cases. Empirically, the efficiency tests developed can be used for several purposes depending on the economic problem at hand, and the prior beliefs of and subjective interpretation of the results by the researcher. Since the tests are developed under joint assumptions on the technology and the optimizing behavior of the producer, the tests can be used in efficiency studies of firms (as is the focus in DEA) or in testing for the regularity conditions of production theory (as is 2 A closely related approach i s the e s t i m a t i o n of f r o n t i e r p r o d u c t i o n f u n c t i o n s a n d f r o n t i e r cost functions. A survey of this approach can be fou n d i n Forsund, L o v e l l a n d S c h m i d t (1980) a n d an example of e s t i m a t i n g a fr o n t i e r cost f u n c t i o n is given by K o p p an d D i e w e r t (1982). T h e p r o g r a m m i n g approach involves an o p t i m i z a t i o n per observation while the regression approach uses a single o p t i m i z a t i o n over a l l observations. T h e s t a t i s t i c a l approach has the advantage of being capable of m o d e l l i n g stochastic disturbances outside the c o n t r o l of the firm. We do not pursue the s t o c h a s t i c approach i n t h i s thesis. Chapter 1. Introduction 12 the focus in nonparametric production analysis). In the latter, the tests can be used as a data exploratory tool prior to parametric estimation, and hence serve as a complementary tool to statistical or econometric analysis. It is possible that the pattern of the calculated violation indices can be explained by some "left-out" variable or factor not taken into consideration. Examples are technical progress in time series data, firm sizes in cross-section data or simply an unaccounted factor like cost incurred to satisfy environmental regulations. In chapter 2, we describe the three general technologies relative to which the various effi-ciency tests are developed; we also lay out how convex sets are nonparametrically constructed from observed data to approximate the unknown technologies. In the next three chapters (3, 4 and 5), technical efficiency is given a precise definition and the technical efficiency hypothesis, test and violation index for each of the three technologies are described. Chapter 6 introduces the concept of allocative efficiency and illustrates the decomposition of the violation index into its technical inefficiency and pure allocative inefficiency components. Chapters 7 to 10 de-scribe the allocative efficiency tests assuming constrained optimizing behavior, that is, when the producer can optimize only with respect to a subset of goods. Chapter 11 shows that for unconstrained optimization or complete profit maximization, the linear programming tests reduce to a comparison of inqualities. Chapter 12 illustrates how the previous tests can be modifed to account for technical progress. We conclude in chapter 13 with a summary and a mention of some limitations of the study. Chapter 2 Three Alternative Nonparametric Specifications of Technology 2.1 Description of technologies In this study, measures of firm productive inefficiencies are developed relative to the technology of the firm under consideration. We posit three kinds of general technologies: convex, con-stant returns to scale and quasiconcave technologies. We consider a technology with N goods described by its production possibilities set, say T, containing all the feasible z G RN vectors attainable by the given technology. We treat output goods and input goods symmetrically; outputs axe measured positively and inputs are measured negatively. With respect to inputs and outputs, we require the following (innocuous) assumptions for feasibility and boundedness: 1. z = (zi, Z2,.. •, zjv) G T and z,- > 0 implies there exists at least one j such that Zj < 0, that is, the net production of a positive amount of an output always requires the net use of another good as an input, and * 2. outputs are finite for all finite inputs. The following regularity conditions are imposed on the different technologies. • Conditions I (for a convex technology): T is a subset of RN (the iV-dimensional Euclidean space) and is (i) closed; (ii) exhibits free disposal: z G T, z' < z =>• z' G T; and (iii) convex. 13 Chapter 2. Three Alternative Nonparametric Specifications of Technology 14 • Conditions II (for a convex constant returns to scale technology): T is a subset of RN and is (i) closed, (ii) exhibits free disposal, (iii) convex, and is (iv) a cone: z £ T, X > 0 => Xz € T. We generally describe the technology by means of a production possibilities set. In some instances, it is helpful to describe the technology in terms of a production function. We first single out a particular good, say good n. We then describe the technology by a function / " such that *n = f"(zi, Z2, . . . , Z„_l, Z n + i , . . . , ZN) = f"(z") where z" is the z vector with the nth component dropped and /n(z") denotes the maximum amount of net output of good n that can be produced given zn. The interpretation of / " depends on whether zn is an output or input: 1. If good n is an output, then / " is a production function; that is, /"(z") gives the maximal amount of output n that the technology can produce given amounts z,- of other outputs to produce and amounts — Zj of inputs available where i denotes an output subscript and j denotes an input subscript.1 2. If good n is an input, then / " is a factor requirements function; that is, fn(zn) gives the negative of the minimum amount of input n required to produce outputs z,- given that other inputs — Zj are available. The regularity conditions for a quasiconcave technology will be defined over the individual good technology functions / " , n = 1,2,..., TV. l I f the vector z" is not consistent w i t h any net o u t p u t for good n, define fn(zn) = —oo. Chapter 2. Three Alternative Nonparametric Specifications of Technology 15 • Conditions III (for a quasiconcave technology): The function fn is a real-valued function of TY — 1 variables defined over a nonempty, closed, convex subset of RN~* and is (i) continuous from above, (ii) nonincreasing, and (iii) quasiconcave. In conditions III, part (i) means that the upper level sets L(zn) = {zn : f"(zn) > zn} are closed; part (ii) means that higher production of other outputs and/or decrease in availability of other inputs cannot increase the output of good n if zn > 0, or decrease the input requirements for good n\izn < 0; and part (iii) implies that the upper level set L(zn) is convex. Note that the given regularity conditions describe the technologies by means of convex sets that are closed and exhibit free disposal. We exploit these convexity properties of the technologies through the use of the mathematical tools of convex analysis to derive various efficiency tests. Since efficient points must he on the boundary of the relevant convex set, the continuity property or equivalently the closure property of the convex sets ensure the recovered efficient points do belong to the pertinent technology set. The free disposal property offers a theoretical justification for the technical feasibility of inefficient observations. Among the three general technologies, the quasiconcave technology is least restrictive; in-dividual technology functions / n can have nonconvexities exhibiting flats or increasing returns to scale. A convex technology must have diminishing or constant returns to scale; individual technology functions fn corresponding to a convex technology must be concave. A constant returns to scale technology is the most restrictive; individual technology functions /" must be positively linearly homogeneous and concave. Also, a convex constant returns to scale technol-ogy implies convexity of its production possibilities set, and the latter implies quasiconcavity of the individual technology functions /", ~n = 1,2,.. .,N. The differences among the three general technologies are illustrated in figure 2.1 for a single output (z2)-single input (zi) case where z-i = f2(z\) and f2 is a production function (describing the technology with respect to Chapter 2. Three Alternative Nonparametric Specifications of Technology 16 output z2), z2 > 0, z\ < 0. The production possibilities set T, a subset of RN, is convex if and only if z' £ T, z" £ T and 0 < A < 1 implies Az' + (1 — X)z" £ T. If, in addition, T is closed under nonnegative scalar multiplication, that is, z £ T, A > 0 implies Xz £ T, then T is a convex cone and exhibits constant returns to scale. The individual technology function / " is quasiconcave if and only if its upper level sets L(zn) = {zn : f"(z") > zn} are convex. Corresponding regularity conditions on the individual technology functions /" , n = 1,2,.. ,,N can be derived when the production possibilities set is either a convex set or a convex cone. Thus the following conditions on / " are equivalent to our old conditions I and II on the production possibilities set T. • Conditions I' (for a convex technology): The function / " is a real-valued function of N — 1 variables which is defined over a nonempty, closed convex subset of . K ^ - 1 and is (i) continuous, (ii) nonincreasing, and (iii) concave. • Conditions II' (for a convex constant returns to scale technology): The function / " is a real-valued function of N — 1 variables which is defined over a nonempty, closed convex subset of RN~l (which is a cone) and is (i) continuous, (ii) nonincreasing, (iii) concave, and (iv) positively linearly homogeneous. Concavity of fn is a more restrictive condition than quasiconcavity. Concavity rules out interior flats and increasing returns in the technology function. The function / " defined over a convex subset of RN_1, say C, is concave if and only if znl G C, zn2 £ C, 0 < A < 1 implies fn(Xznl + (1 - X)zn2) > Xfn(znl) + (1 - X)fn(zn2). Figure 2.1: Illustration of the three classes of technologies in the single output-single input case: (a) quasiconcave technology, (b) convex technology, and (c) convex cone technology Chapter 2. Three Alternative Nonparametric Specifications of Technology 18 Equivalently, the function / " is concave if and only if its hypograph Hn defined by Hn = {z: ze RN, fn(zn) > zn} is convex. If, in addition, / " is positively linearly homogeneous, that is, for every zn 6 C, A > 0, we have f"(\zn) = Xf"(Xz"), then the hypograph of / " is a convex cone. We have the following proposition relating conditions I' and II" to conditions J and II, respectively. • Proposition: The production possibilities set T satisfies conditions 1(11) if and only if for all goods n, n = 1,2,..., N, the individual technology functions / " satisfy conditions I' (II"). Moreover, the hypographs H" of /" , n = 1,2,..., N are identical and are given by T, that is, Hn = T for all n, n = 1,2,..., N. The regularity conditions on / " given by conditions I' imply that the hypograph H" of the function / " is closed, exhibits free disposal and is convex. It is trivial to show that if z 6 T, then z € Hn; and if z € Hn, then z 6 T. Since this must hold for all goods n, then it follows that the hypographs Hn, n = 1,2,...,N are all identical to the production possibilities set T described by conditions I. The first part of the proposition naturally follows. The equivalence of conditions II and II' for a constant returns to scale technology can be analogously shown. In our subsequent discussion, we shall work mainly maintaining the assumptions of condi-tions I and II where the concept of a production possibilities set is used to describe the convex and constant returns to scale technologies; and with conditions III where the concept of a technology function is used to describe a quasiconcave technology. The preceding discussion will be helpful in showing that the asymmetric choice of a good in the multiple output case is equivalent to focusing on the particular production function that corresponds to that good. 2.2 Construction of approximating convex sets The true technology is generally unknown and the information at hand is data on the amounts of the outputs produced and the inputs used in the production process. Suppose we have J observations; we denote by zl, z2,..., zJ the observed iV-dimensional quantity vectors. The Chapter 2. Three Alternative Nonparametric Specifications of Technology 19 J observations may pertain to a cross-section of production units whose underlying technology may be assumed relatively homogeneous or to a particular production unit at various time periods. A particular observation denoted by the vector z3 = (z\, z32,..., z3N) has components z3n > 0 if good n is a net output, z3n < 0 if good n is a net input, and z3n = 0 if good n is neither produced nor used by the production unit j. Given the quantity data {z3 : j = 1,2,..., J} we approximate the relevant convex sets by constructing the free disposal convex hull of the observed data in a manner satisfying the given regularity conditions on the unknown technologies. For a convex technology satisfying conditions J , the production possibilities set T is approximated by the set T\ defined by f1(z\z2,...,zJ) = {z: £ / = i A ' V >z, £ / = i V = 1 , Xj > 0, j = 1,2,..., J) . (2.1) For a constant returns to scale technology, we construct the free disposal conic convex hull of the observed data to approximate the unknown production possibilities set T satisfying conditions II. In this case, the approximating convex set T-j will be f2(z\z2,...,zJ) = {z: £ / = 1 A > V > z, Xj > 0, j = 1,2,...,J}. (2.2) Since a convex cone is closed under addition and nonnegative scalar multiplication, the re-striction 2~2j=i ^J = 1 is dropped in (2.2). Figure 2.2 illustrates the construction of the N-dimensional free disposal convex hulls for the two-good case (N = 2) under the assumptions of conditions I and conditions II. Let the first good be an output and the second be an input. For a quasiconcave technology, it is necessary to single out one good to play an asymmetric role. The N—1 dimensional convex set approximation to the upper level set corresponding to a particular level of good n, say z„, is constructed in the following manner. Define the index set I(zn) = {j: zl > zn, j = 1,2,...,J} (2.3) containing the observation indices whose values of good n is at least as large as zn. If I(zn) is not empty, then the free disposal convex set approximation to L(zn) = {zn : fn(zn) > zn] is Figure 2.2: Construction of free disposal convex hulls under the assumptions of a convex tech-nology and a constant returns to scale technology: (a) observed production plans, (b) free disposal convex hull, and (c) free disposal conic convex hull Chapter 2. Three Alternative Nonparametric Specifications of Technology 21 given by L(zn) defined as follows: L(zn; z\ z2,..., zJ) = {zn : £j€7r. Xzn> > zn, Zjeip X3 = 1, A' > 0 j ' £ I(zn)} (2.4) where znj is the vector z3 without its nth component, and zn is a z vector without its nth component; that is, z" is an N — 1 dimensional vector. If the index set given in (2.3) has only one element, say j*, then the corresponding convex set L(zn) defined in (2.4) reduces to L(zn) = {zn : znj* > zn} which is an N — 1 dimensional orthant with the origin translated to z"3*; the set L(zn) is closed, convex and exhibits free disposal. If the index set I(zn) in (2.3) is empty, then the set L(zn) is empty; trivially, the empty set is closed and convex. Figure 2.3 illustrates the construction of the sets L(zn) for a three good (N = 3) technology. Let the first good be an output (z\ > 0) and the second and third goods be inputs (z2 < 0, z$ < 0). We construct upper level sets for the output good whose levels are denoted beside the observed points. For output level z\ = 9, the constructed level set L(9) is the set of all input combinations (z2,z3) northeast of the only observed production plan with output z\ = 9. By definition, the approximating level set for output level Z\ = 8 is also X(9) though no production plan is observed to produce an output level z\ = 8. Since the quasiconcave technology function is assumed to be nonincreasing, then any input combination which can yield an output level of z\ =9 can also produce a lower output level of Z\ = 8. In this case, the production function f1 would display a flat region. The constructed level set corresponding to an output level of z\ = 5 is denoted by L(5). Note that the level sets corresponding to output levels z\ > 5 are contained in X(5), that is, L(z\) C L(5) for Z\ > 5. The sets f^z1, z2,..., zJ) and f2(zl,z2,...,zJ) defined in (2.1) and (2.2), and L(zn; z1, z2, ..., zJ) defined by (2.3) and (2.4) are, by construction, closed, convex and exhibit free dis-posal. Hence, these constructed sets also satisfy the regularity conditions we imposed on the unknown technologies. The sets have the additional properties of being polyhedral since they are generated by a finite number of points and of being the smallest convex sets containing the Chapter 2. Three Alternative Nonparametric Specifications of Technology 22 11(9) = 1(8) £ " I I I I ' I I I I 96 I I I I I I I I I L(5)-L(4) *4 I ' H I " I I I I I I I I I I L(2) = i(l) M I I I I I I M I I I I I I I Figure 2.3: Construction of approximating level sets for a quasiconcave technology Chapter 2. Three Alternative Nonparametric Specifications of Technology 23 observed data and satisfying their respective regularity conditions. Therefore, the boundaries of the constructed production possiblities set in the convex case, and of the upper level set in the quasiconcave case are piecewise linear. This implies that our approximation assumes a constant marginal rate of substitution or transformation of goods between observed (data) points on the boundary of the sets. The degree of substitution or transformation between goods may then be underestimated; a greater number of data points or observations on the frontier will yield a closer approximation to the actual degree of substitution. Since inefficiencies will be measured in terms of the distance of an observed point relative to a point on the boundary of the set, our methodology will err more on the side of underestimating the true value of inefficiency and towards acceptance of the efficiency hypothesis. Chapter 3 The Measurement of Technical Inefficiency for a Convex Technology 3.1 Definition of technical efficiency The concept of technical efficiency is based on maximizing the production of output goods given a vector of input goods or minimizing the levels of input use given target output levels. For a convex technology, the technically efficient allocations must lie on the boundary of its production possibilities set. The free disposal assumption lends a complication on how an observed allocation z-7 lying on the boundary of the constructed convex hull T\ will be deemed technically efficient. This problem is addressed in the following discussion. Consider figure 3.4 showing the constructed convex hull of some observed points generated by a single output (z2)-single input (z\) technology. An interior point, like c, has an open neighborhood around it contained in T\ and there are an infinite number of directions and paths to adjust the allocation c to the boundary of the constructed production possibilities set T\. Clearly, the point c is technically inefficient since more output z2 could have been produced given the amount of input — z 1 ? or a lower level of input (—z\) could have been used to produce the given level of output Z2. There are numerous ways of measuring the degree of inefficiency of the allocation c. What is desired is an indicator of the relative distance of an inefficient allocation to a point on the boundary of the production possibilities set. For the point c, we can measure the inefficiency by the relative amount of input wasted and hence, we can compare c with the point c1. The ratio cc 1/c/ gives the proportional decrease in input use needed for the efficient production of the given level of output eg. Alternatively, we can compare c with the point c2 which gives the maximal amount of output that could be produced given the input level og. The corresponding inefficiency measure is c2c/c2g which gives 24 Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 25 Figure 3.4: Convex hull of observations from a single output-single input technology, an example Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 26 the proportional increase in output needed for the efficient use of the given amount of input og. It is also possible to move the point c to the boundary through a simultaneous adjustment in output and input levels; the point c can be compared with c3. In this case, an aggregate measure of the degree of adjustment in both output and input needed to make the allocation c technically efficient is obtained as follows. We ask what proportional reduction in input utilization along with a proportional increase of the same magnitude in output is required to move the point c onto the efficient frontier. In order to obtain this measure of the inefficiency of observation c graphically, let hg equal the distance og along the input axis. Now draw a straight line joining h to c and extend this line until it hits the frontier at c3. The inefficiency measure in this case is ig/hg = kf/fo. Note that this composite inefficiency measure is smaller than our earlier input based measure of inefficiency c 1c/c/ = c1c/hg ( since ig is less than c1c ), and it is also smaller than our earlier output based measure of inefficiency c2c/cg = c2c/fo (since kf is less than c2c). However, all three measures of inefficiency have the very important property of being invariant to scale changes in the units of measurement. In the one output-one input case, the above three measures are the only ones we consider. Unfortunately, the three alternative measures can yield conflicting results. Consider the ob-served points a and b which lie on the free disposal sections of the boundary of the constructed production possibilities set. The production plan a uses minimal input for its given level of output but is output-inefficient; more output could have been produced using the given input level. If we set the direction for efficiency adjustment along the input axis, the inefficiency measure would be zero. If we set the adjustment direction in terms of the output good, we obtain a positive inefficiency loss measure a1a/al > 0 . There is no allocation strictly northeast of the point a, and so, if we measure inefficiency by specifying simultaneous positive adjust-ments in the input-output levels we obtain zero inefficiency. The production plan b which is output-efficient but input-inefficient can be analogously analyzed. The foregoing discussion suggests that technical efficiency can only be defined conditional Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 27 on the direction of efficiency adjustment desired. Let E be the subset of goods with respect to which we want to measure inefficiency. Define an iV-dimensional vector 7 such that its nonzero entries correspond to the goods in E. Corresponding to each good rc, n = 1,2,..., 7Y, we set 7 „ equal to 0, —1 or +1 in the following manner: if good n is an output, choose 7 „ = 0 or +1; if good n is an input, choose 7 „ = 0 or —1; if good n is sometimes a net output and sometimes a net input, choose 7 n = 0. Hereafter, we call the vector 7 as an efficiency direction vector. Following are some examples which illustrate how this vector may be specified. • case (i): to measure inefficiency by the amount of output n lost due to inefficiency or which could have been produced if the firm were efficient: set jn = 1, ~/k = 0 for k ^ n. • case (ii): to measure inefficiency by the amount of input n wasted due to inefficiency or which could have been saved if the firm were efficient: set 7 „ = — 1,7^ = 0 for k ^ n. • case (iii): to measure inefficiency with respect to all outputs: set 7 „ = 1 if good n is an output, and 7 n = 0 otherwise. This case may be applicable when we want to test whether a firm is producing maximal output, or when we test later for overall productive efficiency using price information as well and the firm's behavior can be characterized as revenue- maximizing. • case (iv): to measure inefficiency with respect to all inputs: set 7 „ = —1 if good n is an input, and 7 „ = 0 otherwise. This case may be applicable when we want to test whether a firm, given its output levels, is using minimal inputs or when the firm's behavior can be characterized as cost-minimizing. The waste measure obtained in this case corresponds to Debreu's (1951) concept of coefficient of resource utilization. • case (v): to measure inefficiency with respect to variable goods; firms may have some fixed output levels or fixed factors: set 7 „ = 1 if good n is a variable output, 7 „ = — 1 if good n is a variable input, and 7 „ = 0 if good n is a fixed output or input. Let us now formally define technical efficiency for a production plan z. Suppose the produc-tion plan 2 is generated by a convex technology described by a production possibilities set T Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 28 satisfying conditions J . We define the production plan z to be weakly technically efficient for the technology set T if and only if z is a boundary point of T, that is, z belongs to the frontier of T and is not an interior point. We also require a definition of technical efficiency with respect to the goods in the set E; we term this technical ^-efficiency. The production plan z is technically ^-efficient for T if and only if 1. z £ T, and 2. there is no z' € T such that z' = (z'1,zf2,..., z'N) > z = (zi, z2,..., z/v) and z'n > zn for all n £ E. Hence, by definition, the production plan z is technically J5-efficient (or technically efficient conditional on the efficiency direction vector 7 ) if and only if there exists no alternative pro-duction plan z € T that is a strict Pareto improvement1 to z with respect to the goods in set E. Note that if z is technically E-efficient for T, then z cannot be an interior point of T, and hence, the production plan'2 is, by necessity, weakly technically efficient. 3.2 Test for technical efficiency (test 1) An objective of this study is to develop empirical measures of inefficiency. An indicator of the distance of an observed production plan relative to an efficient allocation on the boundary of the unknown production possibilities set is desired. Given the quantity data {z3 : j — 1,2,..., J) we construct the convex hull of the data points to approximate the true technology set under the assumption that they were generated by an underlying technology satisfying conditions I. If an observation z3 is technically ^-efficient, then by our definition, no strictly Pareto improving adjustments in the goods contained in the set E is feasible. If we let 6j denote the violation index or loss measure for observation j, then we would like 6j to equal zero in this case. l T h e str o n g P a r e t o c r i t e r i o n is used for t e c h n i c a l efficiency for a convex technology because the boundary of the c o n s t r u c t e d p r o d u c t i o n p o s s i b i l i t i e s set has free d i s p o s a l regions; s t r i c t m o n o t o n i c i t y does not h o l d g l o b a l l y at i t s boundary. Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 29 A technical efficiency test must be devised such that an observation z*, given the data {z-7 : j = 1,2,..., J} and an efficiency direction vector 7 , is uniquely determined to be either efficient or inefficient, but not both. We lay out our technical efficiency hypothesis for a convex technology and the proposed test for consistency of the data {z-7 : j = 1,2,..., J) with the hypothesis. In addition, a measure of inefficiency for observation i , denoted by 6*, is proposed. As a violation index, 6* is zero if observation i is consistent with the hypothesis, and positive, otherwise. As a measure of the degree of deviation of observation i from the maintained hypothesis, 6* gives the magnitude of the proportional adjustment in the goods in the set E needed to make the observation i efficient; the necessary adjustment to the vector z' is given by the 6*1 z', where 7 is the chosen efficiency direction vector diagonalized into an NxN matrix with the elements of 7 running down the main diagonal and having zeroes elsewhere. We now formally state our first efficiency hypothesis as follows. • Technical Efficiency Hypothesis I (for a convex technology): The data {z-7 : j = 1,2,..., J} are generated by an underlying technology satisfying conditions I and are technically in-efficient. In order to determine whether a given set of data {z-7 : j = 1,2,..., J) is consistent with the technical efficiency hypothesis 7, it turns out to be necessary to choose the efficiency direction vector 7 so that the following condition is satisfied: 7 nz£ > 0 f o r n e £,.7 = 1,2,...,J. (3.1) This condition implies that the choice of goods n belonging to the set E is restricted to goods consistently either a net output or a net input across all observations. If the data {z-7 : j = 1,2, ...,J} and the efficiency direction vector 7 satisfy condition (3.1), then the technical efficiency hypothesis I can be tested by solving J linear programming problems contained in Test 1 below. • Test 1 Chapter 3, The Measurement of Technical Inefficiency for a Convex Technology 30 1. For each observation i, i = 1,2, . . . , J , solve the following linear programming sub-problem i: where 7 is the efficiency direction vector 7 diagonalized into a matrix. The informa-tional requirements to solve the linear programming problem (3.2) are the quantity data {z3 : j = 1,2,..., J] and the efficiency direction vector 7 . 2. Consider the following consistency condition: Suppose 6* is the optimized objective function for the ith. subproblem (3.2), i = 1 , . . . , J . If condition (3.1) holds and 6* — 0 for all i, i = 1,..., J , then the data {z3 : j = 1,2,..., J} are consistent with technical efficiency hypothesis I for a convex technology. If the condition (3.1) holds and 6* > 0 for some i , then observation z' violates technical efficiency hypothesis I and the data {z3 : j = 1,2,..., J) are not consistent with this hypothesis. Caution is warranted in interpreting a violation of the hypothesis. Note that the linear programming test is a joint test for the regularity conditions on the unknown technology and technical efficiency. There are two major areas of possible use of the efficiency tests: efficiency studies of production units and testing for regularity conditions on production functions.2 Re-searchers in the first area can interpret a violation as being due to technical inefficiency, i.e., a failure to produce maximal output and/or to use minimal inputs. On the other hand, re-searchers in the second area often assume that a convex technology exists, and so a violation may be interpreted as a failure of curvature conditions on the technology. In the formulation of the linear programming problem (3.2), the search for a point on the boundary of T\ is restricted to Pareto improvements on z* through the terms 7„zJ, > 0, n = 1,2,..., iV and S{ > 0. The set of feasible Pareto improving allocations, including those in the weak and strong Pareto sense, is restricted to points z* such that z* is on the boundary 2Allocative efficiency tests, later discussed, are more pertinent to testing for regularity conditions on profit and cost functions which assume optimizing behavior on the part of the producers. maXff.>o1Ai>o,...,A-'>o{*«- : E/=i X 3 z i > z' + 6 n z ' , E . :/=i ^ = 1} = v (3.2) Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 31of Ti and z* = z«' + fjzi > z{ (3.3) for some r > Ojv and f is the r vector diagonalized into a matrix. For each (z*; z', 7 ) , we define a distance indicator function as 6i(z*; z\ 7 ) = min„{r„ : n e ^ z ' z z ^ f7z'', r > OAT} > 0. (3.4) There remains the problem of choosing a particular z* among the set of feasible (strong and weak) Pareto improving production plans with respect to the goods in E. For example, in figure 3.4 we can see that if 7 has no zero components, there are an infinite number of allocations on the boundary of Ti , including c1 and c2, that are northeast of c. With our definition of technical ^-efficiency, we would like to be positive, or equivalently, rn > 0 for all n € E, if ever there exists a feasible production plan z* which is a strict Pareto improvement with respect to the goods in E. A search over all feasible z* can be done such that we obtain 6* defined by 6* = max.z*{6i(z*; z', 7 ) : z* is on the boundary of T\ and z* = zl + rjz' for some r > OAT). (3.5) Hence, 6* > 0 if the production plan z', conditional on 7 , is technically inefficient, and 6* = 0 otherwise. Since z* belongs to T\, we can express z* as z* = Fj/=i f ° r some A 1 > 0,..., XJ > 0 such that yj/=i AJ = 1, and from (3.3) and (3.4), we have z* > z* + S^z*. Therefore, the feasibility region in terms of A = (A1, A 2 , . . . , XJ)T can be expressed as {A : £ / = 1 \iz> > zi + 6az\ £ / = 1 \J = 1, A > 0j}. (3.6) To ensure z* is on the boundary of T\, the linear programming formulation (3.2) jointly de-termines z* and 6*, or equivalently, A* and 6*. If z* — ^2j=i Xi*zi is not on the boundary of Ti , then 6* is not maximal since z* would be in the interior of T\ and there would always exist for any 7 a strictly Pareto-improving allocation. Hence, at the optimal solution to the subproblem (3.2), the allocation z* = Sj=i Xi*zi has to be at the boundary of T\. Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 32 To recapitulate, the optimized value 6* serves both as an indicator of the consistency of a particular observation with the technical efficiency hypothesis and of the distance of the observed z' from the boundary of its hypothesized production possibilities set given an efficiency direction. As a violation index, the magnitude of 6* is a measure of the degree of departure of z' from the maintained hypothesis. If 6* is positive, the violation can be due either to inappropriate assumptions on the underlying technology or to pure inefficiency in the sense of failure to produce maximal outputs and utilize minimal inputs. In production efficiency studies where more weight is placed on the second factor, 6* can be interpreted as a loss measure giving the equiproportionate amount of a specified bundle of goods wasted due to technical inefficiency. Since 6* measures proportional changes, it is invariant to scale in the measurement of goods. 3.3 Some notes on the technical efficiency test 3.3.1 Specifying the efficiency direction vector 7 Two extreme examples of specifying the efficiency direction vector 7 for the iV-good model are discussed. First, let the vector 7 have zero components except for a good n. Then the efficiency test reduces to the question: Under the regularity conditions for a convex technology, is zn maxi-mal for zn1 If good n is an output, technical inefficiency requires that output zn be the maximal amount of good n that the firm can produce given it must produce at least as much other out-puts and has available inputs at levels (z\, z2,..., 2„_i, 2 „ + i , . . . , ZN). If good n is an input, then technical ^-efficiency requires the — zn to be the minimal amount of input n required to produce at output levels and other input availability indicated by (21, z2,..., zn-\,zn+\,..., ZN). In the one output (y)-one input (x) case, let (21,22) = (y, —x). If 7 = (1,0)T, then technical efficiency requires that output level y be maximal for x; if more output can be produced given x, then (zi,z2) = (y,—x) is technically jB-inefficient (or technically inefficient conditional on 7 ) . If 7 = (0, — 1)T, then technical efficiency requires x to be the minimal (22 be maximal) amount of input required to produce y; if y could have been produced with a lesser amount of input, then Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 33 (^ 1)^ 2) = (y, — x) is technically ^-inefficient. At the other extreme, we can let 7 have no zero components. In this case, the linear programming subproblem (3.2) reduces to testing whether z* is on the boundary of T\. If 6* > 0, then z* is in the interior of Ti . Otherwise, if 6* = 0, then the production plan z' is weakly technically efficient. Whether an observation i is deemed technically efficient or not generally depends on the efficiency direction vector 7 . If 2' is an interior point of Ti , then z' will always be judged to be technically inefficient independently of 7 . If 2* is on the boundary of T\ and does not lie on any free disposal region of the boundary, then z' will always be technically efficient independently of 7 . If z' is on the boundary of Ti but lies on a free disposal region at the boundary, then detecting a violation at observation i will be sensitive to the specification of the efficiency direction vector 7 . The magnitude of 6* will generally differ with 7 since the latter determines the efficient point on the boundary of T\ with which z' is compared. The ranking of observations i, i = 1,2,..., J by 6* will generally vary from one 7 specification to another. 3.3.2 Complications due to the free disposability assumption Solution of the linear programming problem (3.2) yields two possibly different technically i n -efficient production plans relative to z*; let us denote by 2* and 2** the allocations 2* = 2' + Sfjz* and 2** = J2j=i X**zi. If observation i is technically inefficient (or technically effi-cient given the corresponding efficiency direction vector 7 ) , then 8* = 0 and 2* = z' + S*jz' = 2'. With the inequality constraint in the linear programming problem (3.2), the vector 2** = J2j=i ^*zi may be different from 2* = 2'. Since at the optimal solution, at least one of the constraints corresponding to the goods n € E must be binding (otherwise, <5* is not maximal), then 2** = 2* = z\ for at least a good n £ E. Hence, 2** does not offer z' a strictly Pareto improving adjustment in the direction of 7 and the minimum proportional adjustment needed for 2' to attain 2** is still given by 6* = 0. The divergence between 2 " and 2* is a sufficient indicator that 2* = 2' is in the free disposal region of some good n not necessarily in E. It is Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 34 only sufficient because the solution of the linear programming problem (3.2) does not guarantee that z** = E/=i X3*Z3 be on the relative interior of and away from any free disposal region at the boundary of T\, and z** may equal z*. The free disposability occurs in the good n where J2j=1 AJ*z£ > z'n. The production plan z**, in itself, is technically inefficient. Suppose observation i is technically ^ -inefficient, that is, 6* > 0. If we look at the subspace of Tj generated by conditioning on the values of z'n, n g E, then z' will be an interior point in this subspace.3 Hence, there will always exist an equiproportionate adjustment vector, given by *^ I z ' n I where 6* > 0 and n G E, that will bring the observed point z' to the boundary of this subspace. The point on the boundary of this subspace attained by the equiproportionate adjustment in the goods n G E is z* = z' + 6*1 z'. The vector z* is technically inefficient and the relative distance of z' is indicated by 6*. The procedure of obtaining 6* by maximizing the minimum of ratios r„ = (z* — z'n)/ \ z'n |, n G E yields 6* = rn for all n G E and the technically efficient point generated is z*. The optimized value 6* can then be interpreted as an equiproportionate waste measure; it gives the equiproportionate increase in outputs and decrease in inputs belonging to E which could have been achieved if the firm were technically inefficient. Again, the point z** = ^/=i X3*z3 may differ from z* = z* + 6*-yz' if z* is in a free disposal region at the boundary of T\ for some good n not necessarily in E. The production plan z** yields a strict Pareto improvement on z' for the goods in E and is itself technically in-efficient. In contrast to z*, the allocation z** can be attained from z' by adjustment in the goods n G E that is not necessarily equiproportionate. We can recover the adjustment ratios r** = (z** — zln)/ | z*n |, n = 1,2,..., N. The lower bound for the adjustment ratios of the goods in E is given by 6*; hence, 6* gives the minimum proportional increase in outputs and decrease in inputs belonging to E required for z* to be technically inefficient. If there is variation in r**, the ratios can be indicative of sources and varying degrees of inefficiencies in the output-input mix in the production plan z*. It must be cautioned though that the test procedure we have 3 If the set E contains all ./V goods, then z' lying on the free disposal region with respect to all JV dimensions of Ti is equivalent to z' being an interior point of 7\. Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 35 does not guarantee that 2**, as well as 2*, not be in any of the free disposal regions of the boundary of T\, and 2** = 2* is possible. 3.3.3 An area of further research An avenue of further research is suggested by our analysis. The definition of technical efficiency can be weakened such that a production plan z' is defined to be technically efficient conditional on the efficiency direction vector 7 if and only if there exist no production plan z' € T that is a weak Pareto improvement on z' with respect to the goods in E.4 The linear programming problem (3.2) has to be reformulated such that the point on the boundary 2** = yj/=i A J*z J does not lie on any free disposal region with respect to the goods in E. Thus, if 2' lies on a free disposal region at the boundary of T\ with respect to at least one good in E, then z* given 7 will be deemed technically inefficient. If all N goods are in E, such a reformulation of the test will pick any allocation in a free disposal region, whether in the interior of Ti or at the boundary, to be technically inefficient. The suggested modifications may require redefinition of the violation index 6* and the reformulation of the linear programming problem (3.2) may have to explicitly incorporate some form of maximizing the slack variables. If the search for a boundary point can ensure that with respect to the goods in E it does not lie on a free disposal region, then the divergence between 2** = X)/=i Xi*zi and the observed z' may be useful in identifying and measuring varying degrees of input and output inefficiencies. In the field of operations research, a related approach termed data envelopment analysis or DEA (see, for example, Charnes and Cooper (1985) and Charnes, Cooper, Golany and Seiford (1985)) attempts to address the problem of correcting for free disposability at the boundary by the maximization of slack variables. Though the economic theory underlying their approach, for example in describing the technology they are analyzing, does not seem to be as rigorous as that found in the nonparametric analysis literature in economics, some insights may be gained from this body of work. 4 We may still want to retain the concept of conditionality on the vector 7 to handle fixed goods. Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 36 3.4 Dual interpretation The dual to the linear programming problem (3.2) is minq>oN^i{-ziTq + Ui : ziTjq > 1, -z&q + pi >0,j = 1,2,...,J}. (3.7) If we interpret the dual variables q > ON as a vector of nonnegative prices, then zlTq is the profit obtained for the production plan z* at prices q. Suppose q* > ON and P* solve problem (3.7). By the duality theorem in linear programming 8* — —z*Tq* + p.*. If observation i satisfies our technical efficiency hypothesis, then 8* = —z,Tq* + u* = 0. Together with the constraints in problem (3.7), this implies ^ = z«V>^'V,;' = i , 2 , . . . , J (3.8) and ziTjq* > 1. (3.9) Hence, the observed quantity vector z' solves the profit maximization problem n(q*) defined as ir(q*) = maxz{q*Tz : z E Ti} = p* (3.10) where p* (unrestricted in sign) is the optimal profit at at prices q* given the technological constraints. This condition is equivalent to z' being a boundary point of T\. The price normal-ization (3.9) and assumption (3.1) imply that there exists at least one n € E such that q* > 0, that is, at least one good n in E is not at free disposal. The above analysis derives from the supporting hyperplane theorem for convex sets. A dual definition of technical efficiency for a convex technology is obtained. • Definition. Given a convex technology set T satifying conditions J , a production plan z g T is technically efficient relative to an efficiency direction vector 7 or technically ^-efficient where E is the set of coordinate axis indices corresponding to the nonzero components of 7 , if and only if, there exists a semipositive price vector q* > 0jv such that q* > 0 for at least One good n belonging to set E , and at prices §*, the production plan z is a profit maximal choice. Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 37 The optimal price vector q* > ON can be related to Koopmans' efficiency prices. If 7 has no zero component then a production plan z is technically efficient if and only if z is at the boundary of the production possiblities set. As Koopmans (1951, p.462) has noted, "the price concept established does not in any way presuppose the existence of a market or of exchanges of commodities between different owners". The technical efficiency prices q* may differ greatly from actual market prices of the goods. Suppose a violation of the technical efficiency hypothesis is detected at observation i, that is, 8* = —z,Tq* + p* > 0. Then at the optimal solution q* > ON, p.* unrestricted in sign, we have Pi > ziTq*; (3.11) Mi > ziTQ*, J = 1,2,. . . ,J; and (3.12) ziTiq* = 1. (3.13) Equation (3.13) follows from the complementary slackness condition in linear programming; since 8* > 0, then the constraint ztTjq > 1 must be binding at q*. Since one of the J inequalities in (3.12) must hold with strict equality, inequalities (3.11) and (3.12) imply p*{ = maxj{ziTq* : j = 1,2,..., J} > ziTq*. (3.14) Then, the violation index 8* can be expressed as 8\ = (max^'V : j = 1,2,..., 7} - ziTq*)/ziTjq* = max, j ( Z J ~T^9* : j = 1,2,..., j | (3.15) > 0. Hence, the violation index 8* in its dual interpretation measures the distance of the observation i from some profit maximal choice, say zi*, j* € {1,2,..., J} in terms of the shortfall from the optimal profit p* with the goods in E as the reference goods. The profit maximal choice zJ* is a function of prices q* and p* (dual to z* and 8*) jointly determined by the linear programming problem (3.7). Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 38 Also, by the complementary slackness condition in linear programming, it follows that \i*(-ziTq* + u*) = 0ior j = 1,2,...,J. (3.16) If u* > zjTq* for an observation j at the optimal solution for subproblem i, then the corre-sponding dual variable must be zero, that is, AJ* = 0. In this case, the particular observation j carries zero weight in the optimal convex combination X)/=i X3*z3 which is a point on the boundary of Ti relative to which the current observation zl is being compared. By the same reasoning, if 8* > 0 and X3* > 0 for an observation j at the optimal solution for subproblem i, then zjTq* = u* and observation j is relatively efficient to z' and, if assumption (3.1) holds, is itself technically ^ -efficient. In the DEA literature, the observations j with corresponding X3*s positive are termed "evaluators" of observation i. It is possible to obtain 8* = 0 and XJ* > 0 for some j ^ i and z3 ^ z'; in this instance, observation i is likely to be in a free disposal region at the boundary of the production possibilities set and *^ = 0 for at least a good n. Nevertheless, the observations z' and z3 s with positive X3*s must he on the same supporting hyperplane with efficiency prices given by q*, or equivalently, in DEA terminology, along the same facet of the convex hull. By looking at the dual formulation, we can also infer why the efficiency test yields measures only of relative efficiency. At the optimal solution of the dual formulation (3.7), at least one of the constraints —zjTq* -f u* > 0, j = 1,2,..., J must be binding for the objective function to be at a minimum. If the constraint is binding for some observation fc, then u* = zkTq* and there exist prices q* > ON with q* > 0 for at least one good n belonging to E such that zk is a profit maximal production plan. By the dual definition of technical efficiency, observation k is technically E-efficient. With the assumption 7„z£ > 0 for all n € E, then the optimized value 8£ will be zero when the subproblem (3.2) is solved for observation fc.5 The result of the test at observation i establishes the possible existence of a price vector q* which can support zk as *Note that 61 = —zkTq* + nl is homogeneous of degree one in q* and n'k. Hence, if at observation « the optimal price vector is q* > ON and zkTyq* > 1, then with the assumption that 7 n z£ > 0 for all n € E, a price vector ql > ON (and fil), which is a scalar multiple of q* (and /i*), exist such that zkTyql > 1 and remains zero. Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 39 an efficient allocation. Hence, given the data set {zJ : j — 1,2,..., J}, at least one observation will be rated technically efficient, that is, 6* = 0 for at least one j, j = 1,2,..., J. Intuitively, the above result follows from the fact that the convex hull of the observed data points is being used to approximate the production possibilites set. As such, inferences about efficiency are relative to the "best-practice" technology. The primal formulation (3.2) of the efficiency test utilizes the concept of an "internal repre-sentation" of a convex set (see, for example, Rockafeller (1970)). Using Caratheodory's theorem, the points of a convex set can be expressed as a convex combination of a subset of points in the set. On the other hand, the dual formulation (3.7) uses the concept of an "external representa-tion" of a convex set as the intersection of some collection of half-spaces. For the (polyhedral) convex hull of a set of points, only a finite number of points and a finite number of half-spaces is required. Whereas the primal formulation works on quantity space (through A), the dual formulation works on the price space. If the linear programming subproblems are solved for the whole data set {zl : i = 1,2,..., J) , then the primal formulation implicitly constructs an "inner approximation" or "inner bound" (Bazaraa and Shetty (1979, p.60), Diewert and Parkan (1983, p.139), Varian (1984b, p.60)) to the true convex production possibilities set. In the dual formulation, the observations which pass the test and their corresponding hyperplanes trace an "outer approximation" or "outer bound" to the unknown convex production possibil-ities set. The "inner approximation" and "outer approximation" generally differ but coincide at the observed points lying on the boundary of either constructed sets. 3.5 Special case: assuming concavity of an individual technology function In some instances, the focus of interest is the technology with respect to a particular good. As mentioned earlier in the description of various technologies, the technology function / " for a singled-out good n can be interpreted either as'a production function if good n is an output, or as a factor requirement function if good n is an input. Assuming fn is continuous, nonincreasing and concave, then technical efficiency requires zn to be the maximal amount of net output of Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 40 good n which can be produced given z", that is z„ = / " (z n ) . If good n is an input, then technical efficiency requires — zn to be the minimal amount of input n required to produce outputs given availability of other inputs at levels indicated by zn = (z\, z2,..., z n _ i , z n + i , . . . , ZN). The technical efficiency hypothesis for a concave technology function / " and the correspond-ing efficiency test follow. • Technical Efficiency Hypothesis I' ( for a concave technology function /"): The data {z J : j = 1,2,..., J} are generated by an underlying technology satisfying conditions I and are technically ^-efficient where E = {n} contains only the nth coordinate axis index. The first part of the hypothesis implies that the technology function / " satisfies conditions / '; the second part implies z3n = / n ( z n j ) , j = 1,2,..., J . Note that the hypothesis assumes that the underlying production possibilites set is convex. To test the technical efficiency hypothesis / ' , perform test 1 using the linear programming formulation (3.2) with the efficiency direction vector specification: jn = 1 or — 1, accordingly, if good n is an output or input; 7^ = 0 for k ^ n. If 8* = 0 for i = 1,2,..., J , then the data are consistent with the technical efficiency hypothesis / ' . As illustrated in figure 3.4, detecting a violation at a particular observation can be sensitive to the choice of the good n. If E = {1}, then the production plan a is technically efficient and b technically inefficient. If E = {2), then we obtain opposite results. In some sense, both production plans are inefficient because more output could have been produced at a and less inputs could have been used at b. Both production plans a and b are at the boundary of the constructed production possibilites set but in the free disposal regions. Therefore, to test for simultaneous maximality of outputs and minimality of inputs, test 1 has to be performed for n = 1,2,.. .,N with E = {n} at a time. Observations which pass the N test then satisfy the DEA definition of efficiency (Charnes and Cooper, 1985, p.72): "100 % efficiency is attained for any DMU (decision-making unit) only when (a) none of its outputs can be increased without either Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 41 (i) increasing one or more of its inputs or (ii) decreasing some of its outputs. (b) none of its inputs can be decreased without either (i) decreasing some of its outputs or (ii) increasing some of its other inputs." Next it is shown that the formulation of test 1 is a generalized version of the Diewert-Parkan (1983, p.141) efficiency test for a concave technology function / " in a multiple output-multiple input context. The Diewert-Parkan test where a good n is asymmetrically chosen and using our notation is : (i) For i — 1,2,..., J , solve the linear programming subproblem i: maxAi> 0,„.,^>o{E/=i *j4 • E/=i Xzni > E/=i * = ^} = /"(«•')• (3-17) (ii) If z'n = f"(z') for i = 1,2,..., J , then the data {z1 : j = 1,2,..., J) are consistent with the technical efficiency hypothesis I' for some / " satisfying conditions If z'n < f"(z') for some i, then the data is not consistent with the technical efficiency hypothesis I' for any / " satisfying conditions (iii) The violation index can be defined as A' = (/"(z') — z'n)/ | z'n | > 0 which gives the proportional increase in z'n needed to put z' on the efficient frontier. Note that the Diewert-Parkan formulation (3.17) implicitly assumes that the underlying produc-tion possibilities set is convex. The linear programming subproblem (3.2) for the case E = {n} can be rewritten as maxff|.>oiAi>o,...^ >o{*.- = E/=i > E/=i »4 > A + ^7n4, E / = 1 A > = 1} = ^ . (3.18) At the optimal solution for (3.18), the constraint E/=i ^J*zt% ^ zn + ^ilnZxn has to be binding; otherwise, a higher value for 6* can still be obtained. In the case of a single good being Chapter 3. The Measurement of Technical Inefficiency for a Convex Technology 42 asymmetrically chosen, let us redefine <5; as 6, = U = X ; " (3.19) and rewrite (3.18) as { y^J \j — z* T f — - : E/=i > E/=i A J ; = 1 • (3- 2 °) lnzn ) Since d>i is a monotonic function of J2j=i ^zii f ° r a hxed zj,, problem (3.18) reduces to maxA 1>O i... i A,> 0{£/=i Xj4 : E;=i > *"*', E;=i ^ j = 1} (3-21) which is the Diewert-Parkan linear programming subproblem (3.17). From definition (3.19), it is easy to see that at the optimal solution for the problems (3.17) and (3.18), our consistency conditions (6* = 0 implies efficiency, 6* > 0 implies inefficiency) are equivalent to parts (ii) and (iii) of the Diewert-Parkan test with A' = 6*. If the data {z3' : j = 1,2,..., J) pass the technical efficiency test for a concave technology function /" , then a nonparametric (piecewise linear) concave function /"*. satisfying condi-tions I' and defined by f *(z") = maxAl>0 i... )AJ>0{E/=i A>'*£ : E/=i > A E/=i = 1} (3-22) can be recovered. The function fn* defined by (3.22) gives the boundary of the convex produc-tion possibilities set T\ along the nth dimension. Chapter 4 The Measurement of Technical Inefficiency for a Convex Conical Technology 4.1 Definition of and test for technical efficiency (test 2) A comparison of the regularity conditions I and II on the production possibilities sets shows that a constant returns to scale (CRS) technology is a special case of a convex technology. As such, the definition of technical efficiency for a convex technology must necessarily hold in the CRS case. A production plan z is technically ^-efficient if and only if there exists no alternative plan z' € T, where T is the production possibilities set, that is a strict Pareto improvement to z with respect to the goods in set E. Again, it is necessary to choose the goods in set E such that they all have nonzero quantity values, or equivalently, to specify the efficiency direction vector such that 7„4 > 0 for n£ E,j = 1,2,..., J. (4.1) If the data {z3 : j = 1,2,..., J} and the efficiency direction vector 7 satisfy condition (4.1), then the following technical efficiency hypothesis for a convex constant returns to scale technology can be tested using test 2 given below. • Technical Efficiency Hypothesis II (for a convex conical technology): The data {z3 : j = 1,2,..., J) are generated by an underlying technology satisfying conditions II and are technically i n e f f i c i e n t . To test the above hypothesis, given the data {z3' : j = 1,2,..., J} and the efficiency direction vector 7 corresponding to set E, test 2 which involves solving J linear programming subproblems is performed. • Test 2 43 Chapter 4. The Measurement of Technical Inefficiency for a Convex Conical Technology 44 1. For each observation i, i = 1,2, . . . , J , solve the following linear programming sub-problem i: max5i.>0iAl>0 xj>0{8i : E;=i W > **' + Silt* 4} = (4-2) where 7 is the efficiency direction vector 7 diagonalized into a matrix. 2. Consider the following consistency condition: If condition (4.1) holds and 6* = 0 for all i, i = 1,..., J, then the data {z3 : j = 1,2,..., J} are consistent with the technical efficiency hypothesis II for a convex constant returns to scale technology. If condition (4.1) holds and <5* > 0 for some i, then observation z' violates technical efficiency hypothesis II and the data {z3 : j = 1,2,..., J} are not consistent with this hypothesis. Test 2 is more restrictive than test 1; the regularity conditions II require the production possibilities set to be a cone in addition to being convex. Geometrically, the technically efficient points for a CRS technology must lie along rays at the boundary of the production possibilities set and furthermore, these rays must pass through the origin. Given the same 7 , an observation may pass the convexity test but fail the CRS test. An observation which passes test 2 must necessarily pass test 1. The constructed production possibilities set for a convex technology Ti is contained in the set T2 constructed under the assumption of a convex CRS technology. Since Ti C T2, as can be seen in figure 2.2, then it is possible that points on the boundary of Ti he in the interior of T-j. Hence, the violation index 6* for observation i obtained with test 2 cannot be less than the value of 6* obtained with test 1, given the same efficiency direction vector 7 . We can expect at least as many observations violating the technical efficiency hypothesis of test 2 as against that of test 1. The foregoing discussion suggests an indirect way to test for returns to scale, at least, the possible existence of decreasing returns as opposed to constant returns. Given a 7 configuration, tests 1 and 2 for a convex technology and a CRS technology, respectively, can be performed. If the violation indices 6*' significantly increase going from test 1 to test 2, then the data can be interpreted as being more consistent with the convex technology assumption. How large an Chapter 4. The Measurement of Technical Inefficiency for a Convex Conical Technology 45 observation 8* when 8* when 6* when no. 7 = (- l ,0 ,0) r 7 = ( - l , - l , 0 ) T 7 = ( - l , - l , l ) T (1) (2) (3) (4) (5) 1 (-1,-4,1) 0.000000 0.000000 0.000000 2 (-2,-2,1) 0.000000 0.000000 0.000000 3 (-4,-1,1) 0.000000 0.000000 0.000000 4 (-5,-1,1) 0.200000 0.000000 0.000000 5 (-3,-2,1) 0.333333 0.142857 0.076923 6 (-2,-3,1) 0.250000 0.142857 0.076923 7 (-9,-2,1) 0.777778 0.500000 0.333333 Table 4.1: Two input-constant output example for test 2 increase in 6* is indicative of a departure from the CRS assumption is left, at this stage, to the subjective judgment of the researcher. In cases where the number of goods N > 3, it is possible that at the optimal solution for subproblem (4.2), the points E/=i X3*Z3 and (z' + 8*-yz') are different. This result can be similarly interpreted in terms of the complications brought about by the free disposability assumption as in the convex case. In this instance, the point (z' + 8*jz*) lies on a free disposal region at the boundary of the conical convex hull of the observed data {z3 : j = 1,2,...,/}. To illustrate this possibility, we perform test 2 for a two-input (21,22), constant output (23) example taken from Chang and Guh (1988). The data and results of test 2 for different 7 configurations are listed in table 4.1. Figure 4.5 illustrates the data; in three-dimensional space, the boundary of the conical convex hull of the observed data would consist of the points along the rays drawn from the origin and passing through the points at the boundary of the level set for z3 = 1. From column 5 of table 4.1, the results show that the observations which are at the boundary of the conic convex hull of the observed data are 1, 2, 3 and 4. The same observations are at the boundary of the level set corresponding to 23 = 1 as verified by the results in column 4 and shown in figure 4.5. However, the test at 7 = (-1,-1,0) r indicates that observation 4 is at best weakly efficient; this is shown by the linear programming results for this particular observation. At 7 = (-1, - l , 0 ) r and i = 4, the optimal solution of subproblem (4.2) yields = 0.000000 Chapter 4. The Measurement of Technical Inefficiency for a Convex Conical Technology 46 Figure 4.5: The conic convex hull for a two input-constant output example Chapter 4. The Measurement of Technical Inefficiency for a Convex Conical Technology 47 and A3* > 0 with A-*'* = 0 for j ^ 3; hence, z* = z' + Sf-yz' = z4 but z** = J^j=i ^'z3 - z3. Note that observation 3, not 4, enters the optimal basis for observation 4, as indicated by a positive A3*; the difference between z* = zx + S^jz' and z** = J2j=i X3*z3 indicates that free disposability occurs with respect to good 1. If test 2 is performed with 7 = (—1,0,0)r, then we ask the question whether, assuming a CRS technology, input 1 is minimal given output level is at least at Z3 and — z2 of input 2 is available. The earlier results on the weak efficiency of observation 4 are supported by the results of test 2 with 7 = ( — 1,0,0)T. If observation 4 is an interior point of the conic convex hull of the observed data, detecting a violation at observation 4 will be invariant to the specification of the efficiency direction vector 7 . 4.2 Dual interpretation The dual formulation to the linear programming subproblem (4.2) is given by m i n g > 0 „ { - z * T g : ziTjq > 1, -z3'Tq > 0, j = 1,2,..., J} (4.3) and consists of a price normalization constraint and J homogeneous linear inequalities. Hence, the "outer approximation" to the production possibilities set is also a polyhedral convex cone. The dual variables in (4.3) can be similarly interpreted as in the convex case given by (3.7). Under the regularity conditions for a constant returns to scale technology, maximal profit u* is restricted to zero. A dual definition of technical efficiency for a CRS technology is obtained. • Definition. Given a convex conical technology set T satifying conditions II, a production plan z G T is technically efficient relative to an efficiency direction vector 7 or technically Inefficient where E is the set of coordinate axis indices corresponding to the nonzero components of 7 , if and only if, there exists a semipositive price vector q* > 0;v such that q* > 0. for at least one good n belonging to set E , and at prices q*, the production plan z yields zero profit. Observations which pass the CRS efficiency test can be interpreted as operating at a tech-nically optimal scale in the sense that they have exploited possible increasing returns and Chapter 4. The Measurement of Technical Inefficiency for a Convex Conical Technology 48 decreasing returns in the technology. Even if the underlying technology is everywhere convex, we can interpret the observations passing test 2 as consistent with scale efficiency in the con-text of "most productive scale size" of Banker (1984); that is, observations that fail test 2, and hence are "scale-inefficient", can move to an efficient point by either a more than propor-tionate increase in output when inputs are increased, or a more than proportionate decrease in inputs when outputs are decreased. Alternatively, scale efficiency can be viewed as consis-tent with the zero-profit long-run competitive equilibrium and hence, is socially desirable (Fare, Grosskopf and Lovell, 1985) if some price mechanism can be implemented such that the optimal "efficiency" prices can prevail. If an observation i fails test 2, that is, 6* = —z,Tq* > 0, then the production plan z' yields a loss at prices q*. The following relationships hold as well: 0 > ziTq*; (4.4) 0 > ziTq*, j = 1,2,..., J; and (4.5) ziTjq* = 1. (4.6) Since at least one of the J inequalities in (4.5) must hold with strict equality, equations (4.4) and (4.6) imply 0 = maxj{ziTq* : j = 1,2,..., J} > ziTq* = -6?. (4.7) Then the violation index 6* can be expressed as 0 — ziTa* Sf = .* q > 0. (4.8) In its dual interpretation, the violation index 6* measures the distance of the observed z' from the zero-profit allocation at prices q* in terms of the loss incurred at z' with the goods in E as reference goods. The relationship of the value of the current production plan z,Tq*, the value of the basket of reference goods zxTxjq*, and the violation index 6* is more apparent in the following rearrangement of equation (4.8): ziTq* + 6?(ziTjq*) = 0. (4.9) Chapter 4. The Measurement of Technical Inefficiency for a Convex Conical Technology 49 If 7 is specified such that 6* is an output based measure, then z*T,yq* — 1 is the implicit revenue obtained with the production plan z' if prices were q*. Equation (4.8) in this case would reduce to "cost" «: = T T - ^ ; - 1 > 0 (4.10) revenue where the cost and revenue concepts are only implicit in the sense that q* are efficiency prices and not necessarily prevailing market prices. Equation (4.10) says 8* is the excess of cost over revenue measured as a proportion of revenue. Alternatively, from (4.9), we can say that 8* gives the equiproportionate increase in outputs needed to attain zero profit. For an input based 8*, we can interpret z'T/yq* = 1 as the cost incurred with production plan z' if prices were q*. In this case, equation (4.8) reduces to "cost"-"revenue" „ * = > 0 <4-n) which gives the excess of cost over revenue as a proportion of cost. Equivalently, an input based 6* gives the equiproportionate reduction in inputs needed to attain zero profit. The question arises whether the equiproportionate adjustment 6*jz* is technically feasible. The answer is yes; the adjustment moves the observation z* to a boundary point which necessar-ily belongs to the approximating convex cone. As discussed earlier, the obtained allocation (zl + 8*xfzt) may he on a free disposal region at the boundary of the constructed production possibilities set. At the optimal solution of subproblem i and if assumption (4.1) holds, the evaluators of observation i, which themselves are technically ^-efficient, can be identified by positive A3*s. When an output variable is zero at z', it is possible to obtain optimal \3*s to be all zero for observation i implying the shutdown point is the relatively efficient allocation given the efficiency direction vector 7 . Since the free disposal conic convex hull of the observed data is being used to approximate the unknown technology, then given the data set {z3 : j = 1,2,.. .,</}, at least one observation j will be rated technically ^-efficient with 8* = 0. Chapter 4. The Measurement of Technical Inefficiency for a Convex Conical Technology 50 4.3 Special case: assuming linear homogeneity of an individual technology func-tion As in the convex technology case, corresponding conditions on the individual technology func-tions /", n = 1,2,..., N when the production possibilities set is described by a convex cone satisfying conditions II can be obtained. The regularity conditions imposed on /" are given by conditions 77'; that is, the individual technology function /" is continuous, nonincreasing, con-cave and positively linearly homogeneous. Since the convex conical technology is a special case of a convex technology, the definition and test for technical /^ -efficiency are analogous. Tech-nical efficiency for a linearly homogeneous technology function /" requires zn be the maximal net output given the subvector of other goods z". Formally, the technical efficiency hypothesis and test for a linearly homogeneous function /" are as follow. • Technical Efficiency Hypothesis //' ( for a linearly homogeneous concave technology func-tion /"): The data {z3 : j = 1,2,..., /} are generated by an underlying technology sat-isfying conditions // and are technically /^ -efficient where E = {n} contains only the nth coordinate axis index. The first part of the hypothesis implies that the technology function /" satisfies conditions //'; the second part implies z^ = fn(zn3), j = 1,2,...,/. To test the efficiency hypothesis //' for the data set {z3 : j = 1,2,...,/}, perform test 2 using the linear programming formulation (4.2) with the efficiency direction vector specification: 7„ = 1 or — 1, accordingly, if good n is an output or input; 7& = 0 for fc ^  n. If 6* = 0 for i = 1,2,...,/, then the data are consistent with the technical efficiency hypothesis //'. If 6* > 0 for some i, then a violation of the technical efficiency hypothesis //' occurs at observation i. If the data set {z3 : j = 1,2,..., /} passes the test, the recoverable nonparametric function /"* satisfying conditions //' can be defined by fn*(z") = maXAl>oi„.,A,>0{£/==i X'4 : £/=i Xz* > zn}. (4.12) This test forces the technically efficient observations with 6* — 0 not to be on the free disposal section of the boundary of the conic convex hull at least with respect to the nth good. Chapter 4. The Measurement of Technical Inefficiency for a Convex Conical Technology 51 If the above test is performed with respect to all goods, that is, E = {n}, n = 1,2,.. . , TV, then observations which pass all N tests are boundary points not lying on any free disposal region of the conic convex hull of the observed data and hence, are DEA-efficient. Chapter 5 The Measurement of Technical Inefficiency for a Quasiconcave Technology 5.1 Definition of and test for technical efficiency (test 3) The efficiency tests for a quasiconcave technology require choosing a particular good n and focusing on its individual technology as described by the function fn assumed to satisfy con-ditions III. The function / " is assumed to be continuous from above, nonincreasing and quasiconcave. Geometrically, the definition of technical efficiency is analogous to that in the convex case. Whereas in the convex case technical efficiency is defined relative to the boundary of the production possibilities set in iV-dimensional space, in the quasiconcave case technical efficiency is defined relative to the boundary of the upper level set in N — 1 dimensional space corresponding to a particular level of the singled-out good n. Hence, technical efficiency in the latter is conditional on a truncated efficiency direction vector, denoted by 7 " , which is the full vector 7 with the nth component deleted. Let us denote the corresponding set containing the indices of the coordinate axes of the goods with respect to which we want to measure inefficiency as En. A formal definition of technical efficiency for a quasiconcave technology follows. If the technology function is / " , define the upper level set L(z„) = {zn : / n(z") > zn}. A production plan z, which has the nth component equal to z„ and the remaining components written as the vector z n , is technically £n-emcient if and only if there exists no alternative vector z" belonging to the upper level set L(zn) which is a strict Pareto improvement on z" with respect to the directions in E". If the production plan z is not technically ^ "-efficient, then there exist z" 6 L(zn) such that zjt > zk for all fc e En and f(z") > z„. In the case of a concave technology function, the definition of technical efficiency requires 52 Chapter 5. The Measurement of Technical Inefficiency for a Quasiconcave Technology 53 z„ = /"(z n) and zn to be maximal for z"; passing the corresponding efficiency test does not preclude the possibility that z" is not ^"-maximal for z„. In figure 3.4, if we perform the convexity test (test 1) with the efficiency direction vector 7 = (0,1)T or equivalently with E = {2}, then the observed point b passes the test implying output z 2 is maximal given input — zi at point b. For technical efficiency relative to a quasiconcave technology, we require z" to be maximal for z n at least with respect to the goods n £ En. In the one output (z2)-one input (zi) case and focusing on the production function f2 where z 2 = / 2 (zi) , this definition requires the input level —z\ to be minimal for the output level z 2. Given the observed points b and d at the same output level in figure 3.4, clearly the point b does not use minimal input for the given output level and hence, is technically £2-inefficient where E2 — {1}. The definition of technical efficiency for a quasiconcave technology in this study, after taking into account the sign convention for inputs, is consistent with the usual cost minimization concepts and tests which often assume a quasiconcave technology. This study offers a generalized version of the efficiency tests to handle multiple outputs and multiple inputs. We now state our technical efficiency hypothesis and test for a quasiconcave technology function / " . • Technical Efficiency Hypothesis III (for a quasiconcave technology): The data {(z£, z"J) : j = 1,2,..., J} are generated by an underlying technology function / " satisfying condi-tions III and are technically JE7n-efficient. To test the above hypothesis, given the data {z3 : j — 1,2,..., J} and the truncated efficiency direction vector 7 " , test 3 below is performed. It involves solving J linear programming sub-problems where each problem corresponds to an observation. Since upper level sets are used to describe the quasiconcave technology, it is necessary to construct an index set containing the observations with values for z„ greater than or equal to its current value z'n. In constrast to tests 1 and 2 where the full range of observations 1,2,..., J is used in each linear programming subproblem, only a subset of observations is used in each subproblem in test 3. Again, we shall choose the goods k indexed in the set En and the efficiency direction coefficients 7fc so that the Chapter 5. The Measurement of Technical Inefficiency for a Quasiconcave Technology 54 following condition holds: 7k4 > 0 for k e En, j = 1,2,..., J. (5.1) The technical efficiency test for a quasiconcave technology follows. • Test 3 1. For each observation i, i = 1,2,..., J, (a) define an index set I" = {j: z3n>z'n,j = 1,2,...,J}, and (5.2) (b) solve the following linear programming subproblem i: m^,>o,Ai>o,i € I^6i : X 3 z n j > *"*' + tiT*™, Zjeip » = i} = sr (5.3) where 7n,zn',z"-7 are the original vectors with their nth row (corresponding to good n) deleted, and 7 " is the vector 7 " diagonalized into an (JV — l)x(N -1) matrix with zero off-diagonal elements. 2. Consider the following consistency condition: If condition (5.1) holds and 6* = 0 for all i, i = 1,..., J, then the data {z3 : j = 1,2,..., J} are consistent with the techni-cal efficiency hypothesis III for a quasiconcave technology. If condition (5.1) holds and 6* > 0 for some i, then observation zl violates the technical efficiency hypothe-sis III and the data {z3 : j — 1,2,..., J} are not consistent with this hypothesis. Next, we show that the consistency condition above necessarily follows from the technical efficiency hypothesis III. Suppose the technical efficiency hypothesis III holds. Then, for each zln, i = 1,2,..., J, there exists a closed convex upper level set L(zln) = {zn : fn(zn) > z'n} which we approximate by the convex hull L(z'n; z1, z2,..., zJ) as defined in (2.3) and (2.4). If j G 7", then z3n > zln and fn(zn3) > / n(z m) using (5.2) and the technical efficiency hypothe-sis 777. By the quasi con cavity of /", we obtain fn{zni) < /"(Eie/r. A**"') (5.4) Chapter 5. The Measurement of Technical Inefficiency for a Quasiconcave Technology 55 where A-7*, j* £ J" are optimal for subproblem (5.3). At the optimal solution to (5.3), we also have from the linear programming constraint zni + g ^ n z n i < £ \J*Z»3. (5.5) It follows from the nonincreasingness property of /" that /"(EjgJ* X*znj) < f"(zni + 6tfnzni). (5.6) Equations (5.4) and (5.6) yield /"(*"*') < /n(E;G/rA i**B J ') ^ /"(*"*' + « r 7 n O - (5-7) Suppose 6* > 0. Then, from the positivity of 7fczJ. for k £ the following inequalities hold: 4 < '(l + ^ 7 J b)4 forkeE", (5.8) which impUes by the nonincreasingness property of /" that fn(zni) > fn(zni + S^nzni). (5.9) There are two possible cases for equation (5.7): case (i): fn(zm) < fn(zm + 6*j"z"'). Clearly, equation (5.9) leads to a contradiction and hence, we have a violation of the technical efficiency hypothesis 17/ at observation i. case (ii): fn(zni) = fn(zni + 6fjnzni). By assumption (5.1), jnzni ^ 0>-i. Hence, if S* > 0 and case (ii) holds, then znt is not technically Inefficient. The point ( z m+£*7 "z m ) belongs to -L(zJ,; z1, z2,..., zJ) C L(zln) such that (5.8) holds; output(input) levels for goods indexed in E" can be increased(decreased) and still yield the same level of z'n. Hence, a violation of the technical efficiency hypothesis 777 occurs at observation i. Therefore, if technical efficiency hypothesis 777/ holds, then 6* = 0 for i = 1,2,..., J. Chapter 5. The Measurement of Technical Inefficiency for a Quasiconcave Technology 56 Suppose 6* — 0 for all observations i = 1,2,..., J. Then the data {z-7 : j = 1,2,...,/} can be rationalized by the function /n*(z") defined by /"*(z") = max,-{4 : i = 1,2,..., J and i such that £ , e / n \ j z n i > zn, Eje/r. V = 1, A J > 0, j <E / n ) (5.10) which satisfies condition III. Geometrically, if 6* = 0 then observation zx must lie on the boundary of the level set L(z*n) and there is no alternative z" € L(z'n) that is a strict Pareto improvement to znt with respect to the goods indexed in E". Hence, 6* = 0 implies that observation i satisfies the definition of technical ^"-efficiency for a quasiconcave technology function / " . The function /n*(z") defined in (5.10) is illustrated for the single output (z2)-single in-put (zi) case in figure 5.6. Suppose the true production function satisfying conditions III is given by f2 where z2 = / 2 ( z i ) ; the heavy dots denote the observed production plans. Based on these observed points, the production function that can be recovered as defined in (5.10) is given by f2*, a step function.1 Note that the function f2* is continuous from above, nonincreasing and quasiconcave. Not all the points on the graph of f2* are technically efficient; the definition of technical efficiency for a quasiconcave technology function reduces to —z\ being minimal for output z2. The point b is technically J?2-inefficient since L(z2) = L(z2) = {z\ : —oo < Z\ < c} and z\ = d is in the interior of this ray. Observations which pass test 3 though must necessar-ily lie on the boundary of the production possibilities set approximation to the quasiconcave technology. If it is desired to redefine technical efficiency such that "frontier points" on the graph of /"* are deemed technically efficient, then test 3 can be modified appropriately. This alternative definition, which Hanoch and Rothschild (1972) use, poses the question whether z n is maximal for z" that is, whether z n = /"(z") given the quantity subvector z" and / " is a quasiconcave 'The p r o d u c t i o n p o s s i b i l i t i e s set corresponding to this step f u n c t i o n is s i m i l a r to the p o l y h e d r a l p r o d u c t i o n set used by Deprins, S i m a r and T u l k e n s (1984) i n t h e i r t h i r d m ethod of measuring t e c h n i c a l efficiency based on the sole a s s u m p t i o n of i n p u t a n d o u t p u t d i s p o s a b i l i t y . T h i s method corresponds to our case of measuring t e c h n i c a l inefficiency assuming a quasiconcave technology. T h e y describe the p r o d u c t i o n p o s s i b i l i t i e s set as "also p o l y h e d r a l , b u t not necessarily a convex one (p.251)". Figure 5.6: Technical efficiency for a quasiconcave technology function, single output-single input case Chapter 5. The Measurement of Technical Inefficiency for a Quasiconcave Technology 58 technology function satisfying conditions III. Let us rephrase the technical efficiency hypothesis for a quasiconcave technology as follows. • Technical Efficiency Hypothesis 177' (for a quasiconcave technology): The data {(z£, znj) : j = 1,2,..., J) are generated by an underlying technology function /" satisfying condi-tions III and z3n = fn(zni) for all j = 1,2,..., J. The corresponding efficiency test is test 3 with the following modifications. 1. For each observation i, i = 1,2,..., J, (a) redefine the index set I" in (5.2) as I" = {j j = 1,2,..., J}, and (5.11) (b) solve the following linear programming subproblem i which is (5.3) now defined over an unrestricted variable: max*.-,;u>oj € : Ejg/j. XJZ"J * *ni + Silnzni, E ; e / r » = 1) = Sf. (5.12) 2. The consistency condition is changed to: If condition (5.1) holds and 8* < 0 for all i, i = 1,..., J, then the data {z3 : j = 1,2,..., J) are consistent with the technical efficiency hypothesis III' for a quasiconcave technology. If condition (5.1) holds and 8* > 0 for some i, then observation z' violates the technical efficiency hypothesis III' and the data {z3 :j = l,2,...,J} are not consistent with this hypothesis. The level set implicitly used in (5.12) will include subdata only on observations with values for z n strictly greater than the current value z*n. With this revised definition of technical efficiency, an observation z* passes the test if and only if the observed subvector z m is external to the level set constructed using the index set ip defined in (5.11); in this case, 8* < 0. If observation i fails the test, then 8* > 0 gives the equiproportionate increase in output goods and decrease in input good contained in En needed to bring z' to the efficient frontier. In figure 5.6, the point b passes the test since z\ is maximal for z\ = d; geometrically, the point d Chapter 5. The Measurement of Technical Inefficiency for a Quasiconcave Technology 59 is external to the ray {zi : — oo < Z\ < /}. The point g is technically 752-inemcient since z\ — h is in the interior of the ray {z\ : — oo < z\ < c}; the violation index will be 6* = ch/oh. Alternatively, the technical efficiency hypothesis III' or Hanoch-Rothschild hypothesis can be tested without the aforementioned modifications by performing the original test 3 repeatedly (JV—1 times). Single out a good fc, fc / n, at a time and specify the efficiency direction vector 7fc such that Ek = {n}, that is, fk has only one nonzero element which corresponds to good n. If the violation indices for all JV —1 tests at observation z, i = 1,2,..., J, are zero, then the data are consistent with the technical efficiency hypothesis III' for a quasiconcave technology. If the violation index is positive for some observation i in one of the JV —1 tests, then observation z1 violates the technical efficiency hypothesis III'. For the rest of this study, we revert to our original definition of technical .^-efficiency for a quasiconcave technology function / " . Whether an observation is technically efficient or not is not invariant to the choice of the good n to play an asymmetric role. The sensitivity of the results arises if the observed vector z' happens to lie on either vertical or horizontal flats at the boundary of the approximating production possibilities set. Note that under our regularity conditions / ' , II' and III, a quasiconcave function can have fiats in the relative interior of its graph, while a concave function can only have flats at the global maximum of its graph. In figure 5.6, if we focus on the factor requirement function f1 where zx = f1^), then point b will be technically /^-efficient (but J3 ^ inefficient) since z\ = I is at the boundary of the level set {z2 : 0 < zi < I}. Given the input level —z\, the output level z\ = I is maximal. In test 3, the magnitude of the violation index 6* will depend on the choice of the good n and the efficiency direction vector j". At the optimal solution to subproblem (5.3), there can also be a divergence between the points £ j e / n \i*zni and (z m + 6*7nzm). Such a divergence can be interpreted analogously as in the convex and convex cone cases as due to (z m + 6*jnz"') lying on the free disposal region at the boundary of the relevant convex hull. The underlying production set approximation for a quasiconcave technology, say T 3 , is smaller than those of the convex (T\) and convex cone (T2) Chapter 5. The Measurement of Technical Inefficiency for a Quasiconcave Technology 60 technologies. In particular, f3 C f i C TV Across the three technical efficiency tests and given identical efficiency direction vectors such that E" = E, the violation index 6* cannot decrease as we go from test 3 (quasiconcavity test) to test 1 (convexity test) to test 2 (CRS test). Hence, if the value of 8* significantly increases going from test 3 to test 1, for En = E, then the result can be indicative of possible increasing returns or fiats in the true underlying technology; in short, a departure from the convexity assumption. Since conditions III for a quasiconcave technology are quite weak, violations of test 3 can be indicative more of a failure to maximize production of outputs or minimize input use. Test 3 can be repeatedly performed with respect to each good n, n = 1,2,..., N. If an observation i passes all the N test, then observation i does not lie on any flat portion at the boundary of the production possibilities set approximation. In the empirical implementation of the test, the choice of the good n to play the asymmetric role can depend on the economic problem at hand. This arbitrariness holds true as well in the specification of the efficiency direction vector 7 or 7 " . The researcher may want to focus on the production of a particular output. The objective of the analysis may be on the efficiency of some input, for example, labor. In some cases, output based measures or input based measures are desired. Nevertheless, the multiple output-multiple input framework offered in this study is amenable to a wide range of applications. 5.2 Dual interpretation Another perspective on the technical efficiency concept and test for a quasiconcave technology function / " can be gleaned from the dual to the primal linear programming subproblem (5.3): min gn>o J V_ 1,w{-z"*T9n + : * - T 7 V > 1, -znjTqn + IH > 0, j € I?} (5.13) If we interpret the dual variables qn > Ojv-i as a vector of nonnegative prices, then — zn,Tqn is the net cost of producing zln at prices qn. Therefore, if there exist prices qn* such that g£ > 0 Chapter 5. The Measurement of Technical Inefficiency for a Quasiconcave Technology 61 for at least one k g E" and znx solves the following net cost minimization problem: C(<T,4) = min2n{-<rTz" : /"*(*») > z« } = -tf (5.14) where /"* is defined by (5.10),. then z' is technically i5n-efncient for some / " satisfying condi-tions III. The optimal net cost at prices qn* is given by —p*. A dual definition of technical efficiency for a quasiconcave technology function / " is obtained. • Definition. Given a quasiconcave technology function / " satisfying conditions III, a production plan z is technically efficient relative to an efficiency direction vector 7 " , or technically -Coefficient where E" is the set of coordinate axis indices corresponding to the nonzero components of 7 " , if and only if, there exists a semipositive price vector qn* > 0/v_i such that q£ > 0 for at least one good k belonging tothe set En, and at prices q"* and net output level zn, the subvector zn is a cost minimal choice. This definition uses the concept of Koopmans' efficiency prices and requires the possible exis-tence of such prices, not necessarily prevailing market prices, to support a production plan. Alternatively, as in the convex case, the violation index 6* can be expressed as With our sign convention for inputs and outputs, the violation index 6* measures the distance of the subvector zm from the cost minimizing allocation, say z"J*, in terms of the excess "cost" of <5* gives the number of units of the basket of goods in En, evaluated at the shadow prices q"*, lost due to inefficiency. (5.15) incurred at znt relative to that at znj* with the goods in En as reference goods. The magnitude Chapter 6 The Nonparametric Measurement of Allocative Inefficiency Allocative efficiency is sometimes referred to as economic efficiency. Given data on quantities and relevant prices, the behavioral goal of the firm as modeled by an objective function, and the firm's technological constraints, we would like to test whether the firm is using the optimal mix of outputs and/or inputs at the given prices. As will be shown, allocative efficiency subsumes technical efficiency. Two general cases are discussed: the first, when plant managers are not able to optimize with respect to some inputs or outputs; and the second, when complete and accurate price information is available and plant managers can freely vary all goods. The former can be interpreted as a case of constrained optimization; the latter, as unconstrained optimization. The former is also pertinent to short-run considerations when some factors are fixed; the latter can be associated with longer-run optimization when all inputs and outputs are variable. The concept of allocative efficiency is illustrated for a two-good (N = 2) convex technology in figure 6.7. The observed production plans are denoted by heavy dots and T\ is the convex hull of the observed points. Let us focus on the point a which is in the interior of T\ and hence, is technically inefficient independently of 7, the efficiency direction vector. Suppose an output based violation index is desired and we set 7 = (0,1)T or E = {2}. The proportional increase in output Zi needed to bring a to the point b at the boundary of Ti is given by 6* = ba/ae = gh/hf. The slashed isoprofit line through the points a and b have slope —ql/q^ where qf and q% solve the dual linear programming subproblem (3.7). Suppose producer a faces prices p = (pi,P2)T- Then, the profit-maximizing production plan, given prices p, is the point c where the isoprofit line P with slope —p\jp2 is tangent to T\. 62 Figure 6.7: Allocative efficiency for a two-good convex technology Chapter 6. The Nonparametric Measurement of Allocative Inefficiency 64 The point b is technically efficient but does not yield maximal profits at the given prices; the isoprofit line with slope —p\/p2 passing through 6 is lower than the line P. A gauge of the distance between the point a and the relatively (given the prices producer a faces) allocatively efficient production plan c is desired. Since the concept of allocative efficiency is price dependent, the violation index should also reflect the shortfall in profits at a relative to that in c. Another criterion we desire for the allocative efficiency violation index is consistency or comparability with the technical efficiency violation index 6* defined earlier by (3.2) in chapter 3. To satisfy this criterion, we use the same goods indexed in the set E as the reference goods. In figure 6.7, a measure of allocative inefficiency at a in terms of output loss is given by the vertical distance between the points a and d. The point d, though not technically feasible given Ti , lies on the same isoprofit line as c. The resulting proportional loss measure, say £*, would be da/ae. The following relationship holds between the two violation indices: The allocative efficiency violation index e* can be decomposed into its components in the following manner: with 6* due solely to technical inefficiency and (e* — S*) due solely to failure to respond effi-ciently to prices. The latter source of inefficiency we term as "pure" allbcative inefficiency. A proportional increase in output at a by S* brings us to the boundary of Ti and hence, the out-put loss due to pure allocative inefficiency given by (e* — S*)z2 can be considered hypothetical. Therefore, for prescriptive purposes, an efficiency test has to be devised that will also yield the allocatively efficient point c. In terms of the reference good z2, the violation index £* is also equal to kl/li (= da/ae). In this form, the index can be expressed as > 0. Chapter 6. The Nonparametric Measurement of Allocative Inefficiency 65 which has as numerator the distance between the two parallel isoprofit lines with slope —p\/p2 passing through c and a, and as denominator the distance between the two parallel isoprofit line through a and e. Hence, the violation index £* gives the proportion of the value of the reference goods, pTja = P2CL2, lost due to allocative inefficiency (both pure allocative inefficiency and technical inefficiency). At prices p = (p\,p2)T, actual profits at a must be increased by £*(p2«2) to attain allocative efficiency (or economic efficiency ), that is, pTc - pTa + el(pTja). Graphically, it is apparent that pTc — pTa kl ^ _ q*Tc — q*Ta _ gh a pT,ja li a q*T/ja hf where q*T = (q^q^) corresponds to the slope of the "efficiency price" line Q. In the next chapters, the more general formulations of the allocative efficiency tests in the context of partial profit maximization are developed. Suppose plant managers are not able to optimize with respect to some inputs or outputs, or we simply do not have accurate price data for some goods. The constrained optimization tests are applicable in the short run when the firm has some fixed factors, such as capital equipment and structures. The behavior of the firm can be described as maximizing a restricted or variable profit function. The following tests are also helpful in evaluating the economic performance of nonprofit entities, regulated firms and government enterprises. Market prices may not exist for their output (which can be undesirable like pollution) or output levels may be mandated by law; even if output prices exist, they may be set beyond market forces. However, they are most likely competitive in the inputs market. An appropriate description of the behavior of such producers is cost minimization. Alternatively, some researchers have defined efficiency in the context of producing maximal output (which can be multi-valued) given a vector of inputs; the corresponding behavioral description for allocative efficiency testing might be revenue maximization. As well, the constrained optimization tests that follow offer a general framework wherein the unconstrained, optimization or complete profit maximization tests can be derived as special cases. As in the analysis of technical efficiency, Chapter 6. The Nonparametric Measurement of Allocative Inefficiency 66 the allocative efficiency tests are developed relative to a postulated general technology assumed to satisfy the appropriate regularity conditions. Suppose the firm can optimize with respect to K goods where 1 < K < N. Let us denote this set of goods by S. Hence, the relevant prices for a particular firm, say firm i, are p'n > 0, n G S. We also specify a set E C S containing the desired reference goods with respect to which we want to measure the efficiency loss. We retain the assumption | z^ |> 0 for all n G E and j = 1,2,..., J. Let us denote by ef the violation index for the allocative efficiency hypothesis at observation i and given the prices of the goods in 5. Hence, if ef = 0, then observation i is allocatively efficient with respect to the goods in set 5; otherwise, ef > 0 implies observation i is allocatively inefficient with respect to the goods in S. The value of the reference goods is the price-weighted sum of outputs and inputs (indexed positively) in the set E. The violation index ef gives the proportion of this basket of goods or, more accurately, the value of this basket of goods given up due to economic inefficiency. In this sense, the violation index is also termed a loss measure. Chapter 7 The Measurement of Allocative Inefficiency for a Convex Technology Assuming Partial Profit Maximization 7.1 Allocative efficiency test (test 4) Suppose the true technology has a convex production possibilities set satisfying conditions I. Given quantity and price data, {z-7 : j = 1,2,..., J} where z-7 = (z\, z%,..., zJN) and {p^ : n € 5, j = 1,2,..., J}, we want to test whether the production plan z' is constrained alloca-tively efficient at prices p'n > 0, n € S. The efficiency test involves solving a linear programming problem given below for each observation i. • Test 4 1. For each observation i, i = 1,2,..., J , solve the following linear programming sub-problem i and define the violation index ef by maxAx>0 A^>0{En65P,n(E/=l *4) • £/=! ^ £ > 4, » 2 5, £j=l A-7 = 1} (7.1) = En€S^4 + ef 4 I- (7-2) Note that rewriting the optimized objective function in (7.1) as (7.2) defines the violation index for the z'th observation. 2. Consider the following consistency condition: If ef = 0 for all i, i = 1,2,..., J , then all J observations are allocatively efficient with respect to the goods in set S. If ef > 0, then observation i is not allocatively efficient with respect to the goods in set S. 67 Chapter 7. The Measurement of Allocative Inefficiency for a Convex Technology . . . 68 Under the assumption of conditions I for the technology and given quantity data, a pro-duction possibilities set approximation is given by f1(z\z\...,zJ) = {z: £ / = 1 A ' V >z,E/=i^' = l , A ' > 0 , j = l,2,...,J}. (7.3) For any nonnegative price vector p > ON and z £ Ti, there exist A-7 > 0, j = 1,..., J such that £ / = 1 AJ = 1 and PT(E/=i XJz3) * VTz. (7.4) The unconstrained profit maximization problem for producer i facing a complete set of prices p' > ON can be formulated as maxA1>0,...,A,>o{p,T(E/=i : E/=i *j = 1} (7-5) = piTz* >p,Tz3,j = 1,2 , . . . ,J where z* = Ej=i AJ'*zJ' is the unconstrained allocatively efficient point and A-7*, j = 1,2,..., J , solve the linear programming problem (7.5). If producer i cannot optimize with respect to good n, n g" 5, then we add the constraint J2j=i ^3zn ^ z\n n & S- This constraint implies that output must be at least as large as z'n if z'n > 0, or input must be at most equal to — z'n if z'n < 0. This is just the free disposal assumption applied to good n, n # S. As will be seen more clearly in the saddlevalue problem approach to this constrained optimization problem, the prices for good n, n £ S are irrelevant to the firm. Hence, the constrained optimization problem for a convex technology can be formulated as given in (7.1). A violation index ef > 0 can also be obtained. The optimal production plan at prices {pln : n £ 5} is zs* with components E/=i ^ *zn for n £ S and z'n for n £ S where X3*, j = 1,2,..., J solve (7.1). In contrast to the point (z* + 6*jzf), the vector zs* is not necessarily a Pareto improvement on z' except possibly for the goods not in 5. The optimal partial profit given S is J2nesPnzn* while partial profit at the current production plan is £nesPn z n- Their difference is given by efEnecPnl zxn |. Whereas Chapter 7. The Measurement of Allocative Inefficiency for a Convex Technology . 69 the adjustment Sf-yz' is always technically feasible, the adjustment ef \ z'n | for all n G E is not necessarily feasible. Note too that the problem formulation (7.1) uses firm-specific producer prices. In a cross-section of firms, actual prices faced by producers may vary from market prices due to, for example, transportation costs or tax distortions in the economy. In time series data, relative prices of goods may change. The Lagrangian function corresponding to the left-hand side problem in (7.1) is <KK ?,A0 = E»esPJ.(E/=i XJ4) + £„<zs<7n(£/=1 A>*j - 4) + p(l - £/=i V) (7.6) where A > Oj, q > ON-K and p is unrestricted. The Kuhn-Tucker necessary optimality condi-tions corresponding to (7.6) are: for; = l,2,...,j: AJ* > 0, (7.7) X*(T,nesPn4 + EngS9n4 ~ P*) = 0; for n g- S: and Ej=iXJ*4-4 > o, q'n > 0, (7.8) <£(£/=! A ' *4 -4) = 0; 1 - E i i AJ'* = 0 (7.9) where A* > 0j solves (7.1), and q* > 0N-K a n d P* unrestricted are Lagrange multipliers such that <f>(\, q", p*) < <f>(\*, q*, p*) < <£(A*, q, p) Chapter 7. The Measurement of Allocative Inefficiency for a Convex Technology . 70 for all A > 0j, q > 0/v-A' a n ( i n unrestricted. The vector q* can be interpreted as the shadow prices for the goods not in 5 and u* as the optimal shadow profit. At the optimal production plan zs* the value of the shortfall in output and excess supply of inputs not in 5 at shadow prices q* is zero. For convenience, let us term a production plan z that is allocatively effi-cient with respect to the goods in set 5 as allocatively 5-efficient. Hence, the plan zs* with components J2j=i AJ*z£ for n £ S and z'n for n ^ 5 is allocatively 5-efficient. 7.2 Some results 7.2.1 Technical efficiency proposition The allocatively 5-efficient production plan zs* is technically 5-efficient and does not lie on any free disposal region with respect to the goods in 5. This proposition implies that zs* is on the boundary of the production possibilities set; the second part follows from the positivity of prices for goods in 5. There exist prices q* > Ojv with q* = p'n > 0 for n G 5 and q* > 0 for n g" 5 such that zs* is a profit maximal choice. By the dual definition of technical efficiency for a convex technology, the vector zs* is technically 5-efficient. The proof of the proposition uses the saddlevalue problem approach discussed earlier. From the complementary slackness condition in (7.7) and summing across all j's, we obtain E„€sK(E/ = i A***) + E„^5^(E/=i A>4) - /i*E/=i A>' = o which implies £ „ € s r t , ( £ i = x A>*z£) + £ „ e s « = u* (7.10) using (7.9) and the complementary slackness condition in (7.8). It then follows that zs* is technically E-efficient for E = S and E — {n} for any n G 5. Therefore, we obtain the proposition above. Chapter 7. The Measurement of Allocative Inefficiency for a Convex Technology . 71 7.2.2 Relative efficiency proposition There exists at least one observation j (j £ {1,2,..., J} and j = i is possible) which is relatively efficient to observation i at given prices p'n, n £ 5 yielding the same shadow profit level p*; this observation has the corresponding A-7* positive, that is, AJ* > 0. From (7.9), at least one of the \i*s must be positive and from (7.10) and the complementary slackness condition in (7.7), the corresponding observation zi yields the same shadow profit level as zs* at prices p*n, n £ S and q*,nfS. This proposition implies that observations with positive A-7* lie on the same facet of the convex hull as zs*. The facet is identified by p* and the prices p'n, n £ S and q*, n g" S. In the DEA-terminology, these observations are evaluators of observation i. 7.2.3 Nonnegativity of ef The violation index for allocative 5-efficiency, ef, is nonnegative; that is, ef > 0. Note that A' = 1 and A-7 = 0 for j ^ i is feasible for problem (7.1). Hence, z' is feasible but not necessarily optimal for problem (7.1). It follows that E»€SJ>*n(E/=l »*4) > EneSPn4 (7.11) and £?ZneEPn\Zn l>0. By assumption, pn > 0 and | z'n \ > 0 for n £ E; thus, ef > 0. If ef = 0, then (7.11) must hold with equality and z' is allocatively 5-efficient. If ef > 0, then z' does not yield maximal partial profit with respect to the goods indexed in 5. 7.2.4 Comparability of ef and 8* If the technical and allocative efficiency tests for a convex technology (tests 1 and 4, respectively) are performed for observation i with identical specifications of the reference set E such that 75 C S, then the resulting violation indices are comparable and moreover, ef > 8*. Suppose Chapter 7. The Measurement of Allocative Inefficiency for a Convex Technology . . . 72 test 1 is performed; the resulting point (z' + S*jzl), with components (z'n + S*\z*n\) if n 6 E and z'n if n g" E, is feasible for the partial profit maximization problem (7.1). The optimal \3s for problem (3.2) obtained in test 1 are feasible but not necessarily optimal for problem (7.1) since ECS implies S C E.1 Hence, £ „ e s K ( £ / = i A ' ' * z £ ) > E„ esP*„(4 + %7«4) where XJ*, j = 1,2,..., J , solve (7.1). By definition from (7.1), the inequality can be rewritten as Since YlneEPnl zn I > 0, it follows that ef > St. (7.12) With a consistent specification of the efficiency direction vector 7 in test 1 and the set E in test 4 , the allocative efficiency violation index can be decomposed into its components: ef = S: + (ef-6t). (7.13) The term S* measures the waste due purely to technical inefficiency, or loosely defined, the failure to produce at some point on the boundary of the production possibilities set. The second term (ef — S") can be attributed to pure allocative inefficiency, or the failure to respond efficiently to the prevailing prices of the goods in S. The relative magnitude of the two terms may depend on the choice of goods in the set E. In empirical applications, the researcher must be guided by the economic problem under consider-ation. Extraneous information, other than price and quantity data, can be helpful in modeling the behavior of the firm. 7.2.5 S p e c i a l cases: K = 0,1,N-1,N (1) K — 0. In the absence of market prices for goods, as the case may be in controlled economies, the objective function in the linear programming subproblem (7.1) becomes vacuous. Hence, 'The sets 5 a n d E are the complements of S and E, respectively. Chapter 7. The Measurement of Allocative Inefficiency for a Convex Technology . . . 73 the allocative efficiency test is not applicable in this instance. The producer would not face a price-dependent objective function to guide its behavior. As Koopmans (1951) has noted, the notion of efficiency in the context of such an economy reduces to that of technical efficiency, that is, producing at the boundary of the production possibilities set. The choice of production plan must still be subject to technological feasibility considerations. The appropriate test for efficiency in the case K = 0 is the technical efficiency test which uses quantity data only. (2) K = 1. If there is only one good, say good n, with a relevant market price, then the allocative efficiency test 4 is equivalent to the technical efficiency test 1 with the reference set E = {n}. The revenue maximization or cost minimization problem, as the case may be depending on whether good n is an output or input, in (7.1) would entail merely the search for the maximal quantity of good n given the quantities of the other goods. Mathematically, the solution of the linear programming subproblem (7.1) is equivalent to the solution of the Diewert-Parkan technical efficiency formulation (3.17) which we have shown to test for the special case of technical /^ -efficiency where E = {n}. Both test 4 (with K = l) and test 1 with E = {n} will yield identical values for their violation indices, that is, 6* = ef. Intuitively, this result follows because the existence of a market price for a good ensures that with respect to this good, an optimizing producer will not be at a free disposal region of the production possibilities set. (3) K = TV —1. If there is only one good not in 5, then the partial profit maximization problem reduces to a net cost minimization problem if the excluded good is an output, or to a net revenue maximization problem if the excluded good is an input. The special case with K — JV—1 may be applicable to instances wherein the firm has actually several fixed goods which can be aggregated to form a Leontief or Hicksian composite good. For a convex technology, the appropriate test for consistency with the net cost minimization (or net revenue maximization) hypothesis is still test 4. The usual cost minimization tests found in the literature and which involve the simpler "revealed preference"-type inequalities (see, for example, Diewert and Parkan (1983) and Var-ian (1984a)) assume a quasiconcave technology. A similar test will be derived later for testing Chapter 7. The Measurement of Allocative Inefficiency for a Convex Technology . . . 74 for unconstrained optimization under the assumption of a quasiconcave technology. If the true technology is convex, then a test using these inequalities may underestimate the degree of in-efficiency. Intuitively, the production possibilities set approximation asssuming a quasiconcave technology is contained by that for a convex technology. Given the same efficiency direction, the distance of an observed point to the quasiconcave frontier cannot be larger than the distance to the boundary of the convex technology set. More rigorously, in the linear programming formulation for a convex technology given in (7.1), the constraint for a good n, n g- S is J2J=i ^zn ^ z%n where £ / = i AJ = 1 does not restrict the positive X3*s to observations where z^ > z'n. In contrast, for a quasiconcave technol-ogy, an upper level set for z'n is constructed and the corresponding constraint is J2jeir> X3 z3n > z'n where J^jeir AJ = 1. (4) K = TV. When K = TV, plant managers can freely vary all goods and complete and accurate price data are available. In this case, the producer has an unconstrained optimization problem. (Implicitly, the producer still remains bounded by technological feasibility consider-ations.) The test for full profit maximization, assuming a convex technology, is test 4 with S = {1,2,..., TV}. As will be shown in a later chapter, the test in this case reduces to a comparison of inequalities; the resulting test we number as test 7. Both tests, either solving the linear programming problem (7.1) or comparing inequalities, yield identical values for the violation index ef where S contains all the TV goods. 7.2.6 Restricting u* to be nonnegative As can be seen from the Lagrangian function (7.6), the shadow profit u* can be negative, zero or positive for a convex technology. Hence, negative profits or losses can be consistent with profit maximization for a convex technology. If the shadow profit p* is restricted to be nonnegative, then this is equivalent to having Y^j=i ^ < 1 or to assuming that Ojv 6 T. The production possibilities set approximation is given by T defined by f(z\ z\ ...,zJ) = {z: £/=1 \3z3 > z, £/=1 \i < 1, X3 > 0, j = 1,2,..., J}. (7.14) Chapter 7. The Measurement of Allocative Inefficiency for a Convex Technology . 75 Figure 7.8: Production possibilities set T with p.* > 0 As illustrated in figure 7.8 for a two-good technology, the technology described by T in (7.14) is assumed to be conical near the origin and display decreasing returns elsewhere. Fare, Grosskopf and Lovell (1985, p.181) describe T in (7.14) as the smallest closed star-like set containing the observed z3s; only radial contraction to the origin is allowed. The test for allocative 5-efflciency assuming a convex technology and restricting shadow profits to be nonnegative can be done in two ways. Perform test 4 with the linear programming problem (7.1) modified to either maxA 1> 0 A J > 0 {E„65PJ , (E /= i »4) • E/=i » 4 >z'n,n?S, £ / = 1 A> < 1} (7.15) or maxA 1>0 A,>o{E„654(E/=oA J4): E/=oAJ'4 > 4, « t S, ZJj=0» = 1} (7.16) where z° = 0;v. Problem (7.15) exphcitly includes the restriction E/=i A-7 < 1 while prob-lem (7.16) includes the origin in the convex hull. The violation index ef will have the same value using either formulation. Chapter 8 A LeChatelier Principle for Measures of Allocative Inefficiency As the number of goods K with respect to which the producer can optimize increases, the difference between the firm's optimized objective function value and its actual realized value cannot decrease. Given a fixed reference set E, the violation index ef defined in (7.1) cannot decrease as the number of goods in S increases. This result is consistent with the conventional wisdom that the unrestricted or long-run profit is always at least as large as the restricted or short-run profit. We prove the proposition by induction. Suppose we increase by one the number of elements in set S containing the goods with respect to which producer i can optimize; denote the new set by S1. Corresponding to the set S', the number of constraints in (7.1) is reduced by one. Let us rewrite problem (7.1) for these two cases: :/=i A j 4) : £ / = i X'zi >z'n,n? S, £ / = i \* = 1); (8.1) (8.2) maxA 1>0 AJ>0{E„es'P'n(E/=i A '4) : E;=i *4 > 4 , » £ S', E/=i » = 1}-It can be shown that 5 A' + 1(A") - E „ e s ' t f , 4 > 5 A ' ( A * ) - E n € s K 4 - (8.3) The proof is as follows: gK+1(\n - E„G5<K4 = E „ € s ' K ( E / = i A'**4) - E„ 6 5'PJ ,4 > E„ 6 S'PJ,(E/=1 AJ'*^) - E„€5'PJ,4 76 Chapter 8. A LeChatelier Principle for Measures of Allocative Inefficiency 77 since A-7* is feasible but not necessarily optimal for problem (8.2), = EnesPi,(E/=i »*4) + rir+i(£/=i v**ic+i) — Y,n£SPnzn - P'K+1 ZK+1 where p%x+\» Z'K+I a r e * n e P r i c e a n ( l quantity of the (A' + l)th good added to the set S to obtain S' = gK(n - E „ e s K 4 + Pk + i (E/=i -> PA'(A*) - £ „ e s K 4 since > 0 by assumption and (E/=i XJ*zJK+1 - zK+1) > 0 from (8.1) and n = (K +1)#S. Thus, we obtain inequality (8.3). If we divide both sides of (8.3) by the value of the refer-ence goods £ „ e £ P j j | z'n |, then we have ef > ef where ef and ef are the violation indices corresponding to sets S' and S, respectively. Chapter 9 The Measurement of Allocative Inefficiency for a Convex Conical Technology Assuming Partial Profit Maximization 9.1 Allocative efficiency test (test 5) Suppose the true technology has a production possibilities set that is a convex cone and satisfies conditions II. Let 1 < K < N — 1, that is, the set 5 is nonempty and contains at most N — 1 goods. The test for allocative S-efnciency of the production plans z' for prices p'n > 0, n € S and i = 1,2,..., J is given below. • Test 5 1. For each observation i, i = 1,2,..., J , solve the following linear programming sub-problem i and define the violation index ef by m a x A 1 > 0 ) . . . ) A J > 0 { £ „ e S K ( £ / = 1 A ^ ) : £ / = i A'z£ > z'n, n £ 5} (9.1) Note that rewriting the optimized objective function in (9.1) as (9.2) defines the violation index for the ith. observation. 2. Consider the following consistency condition: If ef = 0 for all i, i = 1,2,..., J , then all J observations are allocatively efficient with respect to the goods in set S. If ef > 0, then observation i is not allocatively efficient with respect to the goods in set S. 78 Chapter 9. The Measurement of Allocative Inefficiency for a Convex Conical Technology . . . 79 The production possibilities set approximation for a constant returns to scale technology satisfying conditions II is T2 defined as f2{z\z\...,zJ) = {z: £/ = 1A'V >z, A'>0,j = l,2,...,J}. (9.3) Since any z € T2 can be scaled up or down along a ray through the origin and still be in T2, output levels are unbounded in T2. In contrast, output levels are bounded from above in the set 7j defined by (7.3) for a convex technology satisfying the regularity conditions I. This difference can be seen in figure 2.2. With constant returns, if a producer can make positive profits, he can theoretically scale up the production plan and obtain infinite profits. On the other hand, profits are bounded from below since z = (bv is in its production possibilities set. Since we observe only finite quantity vectors, the meaningful profit-maximizing choice for a firm having a constant returns to scale technology is a zero profit allocation. However, if the firm has found such a zero profit production plan, the producer can scale his operations up or down by a nonnegative factor and be indifferent to the scale of operations. We therefore further assume that the firm arbitrarily fixes the level of at least one good to determine the scale of its operations. Hence, the restriction Ii < N - 1. The corresponding Lagrangian function is q) = £„€sK(£/=i Xizi) + £„^9n(£/ = 1 Xjzn - zn) (9.4) and the necessary Kuhn-Tucker optimality conditions are for j = l,2,...,j: A> > 0, (9.5) A''*(£„ esM + E « « s « ) = 0; and for n g S: Chapter 9. The Measurement of Allocative Inefficiency for a Convex Conical Technology. .80 > 0, (9.6) -zn) = 0. Again, the Lagrange multiplier q* > 0, n € 5 can be interpreted as the shadow price of good n. If good n is an input and is in excess supply, then q* = 0. If good n is an output more of which could have been produced, then q* = 0. If q* > 0, the converse holds. With a convex cone technology, the optimal shadow profit u* which appears in (7.7) is restricted to zero. Note that The solution to the linear programming problem (9.1) can be similarly interpreted as in the convex case. Where the convex cone results differ will be discussed next. There are three possible solution configurations for (9.1). They are: (i) A* > 0j and finite, that is, A* has a finite positive value for at least one j; (ii) A* = 0j; or (iii) A* is unbounded. If ef > 0, then the allocatively 5-efficient production plan at the given prices is zs* with components E/=i X3*z3n for n € 5 and zln for n g" 5. After correcting for free disposability in the goods not in 5, an allocation z** where z** = Ylj=i X3*z} can be obtained. As elaborated in the discussion of the technical efficiency tests, the linear programming formulation we have does not guaranteee detection of free disposability along all N dimensions. The first two cases yield a finite zs*. The first case results in nontrivial allocations for zs*and z**. The second case yields z** = O^ r or the shutdown point. The third case implies that at prices p'n, n € 5, the shadow profit is infinite. Having an input not in 5 ensures a finite solution. If all the goods not in 5 are output goods, it is possible, though it does not necessarily follow, that A* would be unbounded. 9.2 Some results The relative efficiency proposition for testing for allocative 5-efficiency assuming the technology is described by a convex cone satisfying conditions II is as follows. Assuming a finite optimal solution to the linear programming problem (9.1), either (i) there exist at least one observation j Chapter 9. The Measurement of Allocative Inefficiency for a Convex Conical Technology... 81 (j G {1,2,..., J} and j = i is possible) which is relatively efficient to observation i at given prices p'n > 0, n G S yielding a zero shadow profit;1 or (ii) the optimal choice for the producer, given the prices p'n > 0, n G 5, is the shutdown point. Case (i) obtains when A* is finite and nonzero, that is A* > fj/v. The proof uses the complementary slackness condition in (9.5); the geometric interpretation is analogous to that in the convex case. Case (ii) obtains when the optimal solution A* is the zero vector, that is, A* = 0j. Assuming a finite optimal solution to the linear programming solution (9.1), the technical efficiency and LeChatelier principle propositions and the results on the nonnegativity of ef and the comparability of ef and 6* hold as well in the constant returns to scale case as in the convex case. When a firm having a constant returns to scale technology can optimize with respect to all goods but one, that is, K = N — 1, we consider the producer as having an unconstrained allocative efficiency problem. As discussed earlier, at least one good is held fixed to determine the scale of operations. Test 5 can be performed to test for allocative efficiency. An alternative test (test 8), involving a comparison of inequalities, is given in a later chapter. At K = N — 1, both tests yield identical values for ef. Let (ef)convex and (ef)cRS be the violation indices obtained froms tests 4 and 5, respectively, for a given set S of size K where 1 < K < N — 1; prices p*n > 0, n G S; and a fixed reference set E. It can be shown that the violation index obtained under the convex conical technology assumption cannot be smaller than that obtained under the convex technology assumption, that is, (ef)cRS > (ef)convex- (9.7) The proof is simple. Suppose A-7, j = 1,2,..., J solve (7.1) and A-7*, j = 1,2,..., J solve (9.1). Then, the X3s are feasible but not necessarily optimal for problem (9.1). Hence, EnesPUEU XJ*4) > E„ €5PJ,(E/=i A ^ ) , (9-8) lThis observation has the corresponding AJ* positive (AJ* > 0 ) . Chapter 9. The Measurement of Allocative Inefficiency for a Convex Conical Technology ... 82 that is, the optimal partial profit assuming a constant returns technology cannot be smaller than the optimal partial profit assuming a convex technology. Since the right-hand side terms of (7.1) and (9.1) are identical except for the violation indices, the result follows. Intuitively, this result arises because the convex production possibilities set approximation T\ is contained in the convex conical production possibilities set approximation TV Chapter 10 The Measurement of Allocative Inefficiency for a Quasiconcave Technology Assuming Partial Profit Maximization 10.1 Allocative efficiency test (test 6) Suppose the true technology has nonconvexities in its production possibilities set. We assume that the individual technologies can be described by quasiconcave functions satisfying condi-tions III. Single out a good n with respect to whose technology we would like to test for allocative efficiency. Let the set 5" contain the K goods (where 1 < K < N — 1) with respect to which the producer can optimize. We assume Sn does not contain good n. Given prices p'm > 0, m € S" and quantity data {z3 : j = 1,2,..., J) , we would like to test whether the production plans z3, j = 1,2,..., J are allocatively S"-efficient, that is, the z3s are allocatively efficient with respect to the goods in the set 5". Specify a set E C 5" containing the desired reference goods. The efficiency test follows. • Test 6 1. For each observation i, i = 1,2,..., J , (a) define an index set I? = {j •• 4>4,j = 1,2,..., /}, and (10.1) (b) solve the following linear programming subproblem i and define the violation index ef by max A J > 0 ) i € / i r .{E m € S np| n (i: j e / r Xjz3m) : £ j € / r X3z3m >zm,m^Sn, £ i € j r V = l } (10-2) 83 Chapter 10. The Measurement of Allocative Inefficiency for a Quasiconcave Technology . . . 84 = Emes»pinzL + ef EmefiPml ZL I- (10.3) Note that rewriting the optimized objective function in (10.2) as (10.3) defines the violation index for the ith observation. 2. Consider the following consistency condition: If ef = 0 for all i, i = 1,2,..., J , then all J observations are allocatively efficient with respect to the goods in set S". If ef > 0, then observation i is not allocatively efficient with respect to the goods in set Sn. The regularity conditions III for a technology function / " imply the existence of closed convex upper level sets. The level set approximation for z'n is given by L(zn; z\z2,...,zJ) = {zn : £ i € / r X3z»3 > zn, Zjei? A' = 1, A' > 0, j € J ? ) } (10.4) where I" is defined in (10.1). For any nonnegative price vector p" > 0JV-I and zn £ L(zn), there exist X3 > 0, j £ I" such that X3 = 1 and PnT(T,jeir>X3zn3) > pnTzn. (10.5) The unconstrained optimization problem for producer i facing prices pn* > 0jv-i can be formu-lated as maxAJ>o,i€/in{p",T(Ei€/r XJZ"J) •• Eieif ^ = 1, j £ /?} (10.6) — pTliT gTl* > pniTzn3\ j £ If, where zn* = Ej-gTj* AJ'*^nj' and A*", j * £ J / 1 solve (10.6). If zn > 0, or good n is an output, then subproblem (10.6) can be interpreted as a net cost minimization problem, that is, finding zn £ L(zn) such that the net cost of producing z'n is minimal given prices pm. If z'n < 0, or good n is an input, then subproblem (10.6) can be seen as one of maximizing the net return to this input. With either interpretation, the objective function is a partial or restricted profit function. Chapter 10. The Measurement of Allocative Inefficiency for a Quasiconcave Technology . . . 85 If the firm is unable to optimize with respect to some goods, then additional constraints are introduced to subproblem (10.6). The objective function, which remains a partial profit function, is appropriately modifed as well. Formulation (10.2) then obtains for the constrained optimization problem. Since T\ in (7.3) and L in (10.4) have the same property of being closed convex sets, the results of test 6 can be interpreted analogously as those of test 4 for a convex technology. 10.2 Some results Keeping in mind that the analysis uses level sets in TV—1 dimensional space, we obtain technical efficiency, relative efficiency and LeChatelier principle propositions similar to those in test 4 for a convex technology. The results on the nonnegativity of ef and the comparability of ef and hold where the violation indices are obtained under the quasiconcavity assumption. An alternative test (test 9) for the limiting case K = N — 1 or unconstrained optimization for a quasiconcave technology is presented in the next chapter. If, for notational simplicity, we define Sn = S, then it follows that, for given prices p'm > 0, m G S and a fixed reference set E, (ef)cRS > (sf) convex ^ (£,' ^quasiconcave (10.7) where the ef are obtained from tests 5, 4 and 6, respectively. Suppose X3, j G I" are optimal for problem (10.2). Then, X3, j G I" and X3 = 0, j g" I? are feasible but not necessarily optimal for problem (7.1). The second inequality in (10.7) follows. The first inequality in (10.7) is given in (9.7). The values of the partial profit functions will be related in the following manner: where the X3*s and X3s are optimal for the constant returns to scale problem (9.1) and the convex problem (7.1), respectively. The allocative efficiency tests 4, 5 and 6, as well as tests 7, 8 and 9 to be presented in the next chapter, are developed under the joint hypotheses of the regularity conditions on the technology Chapter 10. The Measurement of Allocative Inefficiency for a Quasiconcave Technology . . . 86 and of efficiency. A positive violation index can be attributed to either of two sources. If the researcher gives lesser weight to inefficiency, then significant differences in the values of the violation indices in (10.7) can be interpreted in the context of the curvature conditions on the technology. Chapter 11 The Measurement of Allocative Inefficiency Assuming Complete Profit Maximization 11.1 The convex technology case (test 7) Suppose complete and accurate price data are available and firms can freely vary all goods. The following tests for unconstrained allocative efficiency are limiting cases of the correspond-ing tests for constrained optimization. The saddlevalue problem approach is used to derive the complete profit maximization tests; they can alternatively be shown using revealed preference type arguments (see Diewert and Parkan (1983, pp.150-152), Hanoch and Rothschild (1972), Varian (1984a, pp.57-64; 1984b)). The unconstrained optimization tests and their correspond-ing constrained optimization versions yield identical values for the violation indices for a fixed set of reference goods. The following tests are simpler to implement since they do not require solving linear programming problems and involve only a comparison of inequalities. The informational requirements are identical for the constrained and unconstrained opti-mization tests. Price and quantity data and the specification of a set E of reference goods are required. Using the notation in the technical efficiency tests, the set E corresponds to an JV-dimensional vector 7 with nonzero components corresponding to the goods in E. Let us denote by e* the violation index at observation i given the price vector p* = (p\,p*2, • • • ,PN)T ^ OAT. If e* = 0, then observation i is (unconstrained) allocatively efficient relative to the technology posited; e* > 0 implies otherwise. In this chapter, we give the formulas for the violation index e* which is ef when the size of the set S is at its limiting value. To test for allocative efficiency assuming the underlying technology satisfies conditions I, the violation index e* is defined in test 7 given below. 87 Chapter 11. The Measurement of Allocative Inefficiency Assuming Complete . . . 8 8 • Test 7. Suppose K = N, the number of elements in set S. Calculate the violation index e* defined as ^ E m a x ^ ^ . ^ : j = 1,2,...,J ^  . (11.1) The proof of the equivalence of test 7 and test 4 where K = N is given below. Suppose K — N. Then the linear programming subproblem i in (7.1) reduces to maxA1>0,...iV>0{p'T(E/=1 \izi) : £/=1 M = 1} (11.2) = P,T^' + efE„eBP ,„l4l where the set E is a subset of the TV goods chosen as the reference goods. The Kuhn-Tucker necessary optimality conditions are for j = 1,2,..., j: piTzj - p* < 0, \'m > 0, (11.3) and l-£j=iAJ'* = 0. (11.4) From (11.3) and (11.4), we obtain maxi{p'JV :j = l,2,...,J} = p*. (11.5) Equation (11.4) ensures equality for at least one j. Using the complementary slackness condition in (11.3) and summing across all j's, we have P iT(EJj=i^zJ) = PiTzi + £?Zn€EPn\4 I using (11.2) = p* using (11.3) = maxj{p*'IV : j = 1,2,..., J) using (11.5). (11.6) Chapter 11. The Measurement of Allocative Inefficiency Assuming Complete 89 An expression for ef, using (11.6), is then er = s _ maxjjp^z3 : j = 1,2,..., J} - piTzi fp ' T (2J-z i ) : j = 1,2,...,J). (11.7) If we denote by e* the value of ef when K — N, then (11.1) follows. Suppose j * solves (11.1). Then, at prices p', a profit maximal production plan is z3'. The vector z3* is not necessarily a Pareto improvement on z', that is, it does not follow that z3* > z'. By moving from z' to z3*, (full) profits can be increased by e*(p'T,yz'), which is the value lost due to inefficiency. The possibility of free disposability lends a complication in the analysis of technical efficiency and constrained optimization. If a full range of (nonzero) market prices exist and the firm can optimize with respect to all goods, then the allocatively efficient points will not lie on any free disposal section of the production possibilities set. Note the similarity between e* in (11.1) and 8* in (3.15). For a convex technology and given a fixed set E of reference goods, the following relationship among the different violation indices holds: <5* < ef < et (11.8) where E is contained in S. The first inequality is given in (7.12) and the second follows from the LeChatelier principle proposition. Inequality (11.8) implies that profits can be higher the greater the number of goods the producer can vary and the lesser the restrictions on the direction of adjustment. A decomposition of the violation index e* is possible: e? = + (ef-S;) + (e!-ef). (11.9) Equation (11.9) can be interpreted accordingly; the last term (e*-ef) measures the waste due to failure to optimize with respect to the goods not in S. Suppose E = S contains all the input goods. Then, the violation index 6* gives the pro-portional reduction in inputs needed to bring z' to a point on the boundary of the production possibilities set. The magnitude of ef gives the proportional reduction in cost needed for z* to Chapter 11. The Measurement of Allocative Inefficiency Assuming Complete 90 be consistent with cost minimizing behavior given input prices and current output production levels. Minimal cost can be obtained by being more technically efficient and by altering the input mix; this may entail increasing the use of some inputs and decreasing other inputs. Even if the firm is cost minimizing, its output mix may not yield maximal profits at prevailing prices. By being able to vary all goods, inputs and outputs, the possible increase in profits is equivalent to savings of e* of the initial cost incurred by the firm. The proposed tests can then be applicable to efficiency studies for an industry with regulated firms, and private firms subject to less regulations and whose output prices may be more reflective of market conditions. The behavior of regulated firms can often be described as cost minimizing while that of private firms as profit maximizing. Though the existence of public or regulated enterprises can be justified by other economic and social objectives, it is still enlightening to have a measure of productive efficiency losses in weighing the tradeoffs among possibly conflicting objectives of the firm. The analysis also indicates a reason why government-controlled or state-run firms or any firm, for that matter, insulated from some market price mechanism, may be at a disadvantage when forced to compete in a world market economy. If the ratio of profits to cost or revenue is small, as is the usual case in empirical applications, the magnitude of the violation index e* in (11.1) obtained with the set E containing all the input goods will not differ much from that obtained with the set E containing all the output goods. As profit approaches zero, the ratio of the input based e* to the output based e* approaches 1.0. Generally, the magnitude of the violation indices proposed in this study, as well as the ranking of observations by the violation index, will depend on the goods chosen to be in the reference set E. 11.2 The convex conical technology case (test 8) For the unconstrained optimization test assuming a constant returns to scale technology sat-isfying conditions II, a good n is chosen to determine the scale of operations of the firm. We pick the normalizing good n such that either zn > 0 for all j, j = 1,2,. . . , J, that is, good n Chapter 11. The Measurement of Allocative Inefficiency Assuming Complete 91 is consistently an output, or zn < 0 for all j, j = 1,2,..., J , that is, good n is consistently an input. The violation index £* is calculated as described in test 8. • Test 8 1. Choose a normalizing good, say good n. 2. Define the normalized truncated vectors znj, j = 1,2,..., J where z"3 - T ? ( u - 1 0 ) I zn I where | z3n | is the absolute value of the normalizing good at observation j, and the zero vector z"° = Ojv_a. (11.11) 3. Calculate the violation index £* as e;=maxAP™T{zn3-*"'} : j = 0,1,..., j) (11.12) I 2ZmeEPm\zm\ J if either (i) good n is an input, or (ii) good n is an output and max.j{pniTzni : j = 0,1,..., J} < 0. Otherwise, set e* to be unbounded. The proof of the equivalence of test 8 and test 5 when K = N — 1 is given below. Suppose K = N — 1. Then the linear programming subproblem (9.1) reduces to max A 1 > 0 ) . . . | V > 0 {p"« T (E/ = iA J 2"i) : £ / = i X3z3n > z'n} (11.13) = pniTZni + efEmeEPrn\zim\ where the set E is a subset of the N — 1 goods in S and good n is held fixed. The Kuhn-Tucker necessary optimality conditions are for j = 1,2,..., j: pniTznj + ^ < o, A> > 0, (11-14) \i*(p"iTznj + = 0; Chapter 11. The Measurement of Allocative Inefficiency Assuming Complete 92 and £ / = 1 A ^ - z « > 0, q* > 0, (11.15) £ ( £ / = ! * j * z n - z'n) = 0. From equation (11.14), we obtain 0 > m a x j { p n i T z n i + q n z n : j = 1,2,..., J}. (11.16) Let us assume a finite optimal solution to (11.13) exists. If a AJ* > 0, then there exists at least one j , j € {1,2,..., J}, such that equality holds. There is no £ j = 1 A-7 = 1 restriction so the optimal A* can be the zero vector. Hence, (11.16) can be rewritten as 0 = m a x J { p n i T z " i + q n z n : j = 0,1,2,..., J } . (11.17) where z n 0 = 0N-\. Using (11.14) and (11.15), we obtain the relation PniT(ZUXJ*zni) + « = 0. (11.18) By construction in (11.13), the above equation can be reexpressed as p n i T z m + e?ZmeEPm\zm I + q * z n = 0. (11.19) It is harmless to divide each term in (11.19) by \z'n \ > 0 and each of the elements inside the brackets in (11.17) by their respective | z n \ > 0. If good n is an input, then equating (11.17) and (11.19) after the normalization of variables yields = m a x j { p n i T z n i - q* : j = 0,1,..., J] (11.20) Chapter 11. The Measurement of Allocative Inefficiency Assuming Complete 93 where maxj{pn,T z"J : j = 0,1,..., J} > 0 since the zero vector is a feasible choice and where normalized values are denoted by zm = zm/zn. Hence, there will always exist a q* > 0 such that (11.20) holds. Hence, the violation index ef can be calculated as g f 5 m " ' { L U i 4 i ; j = M - - i ( 1 1 - 2 1 ) If good n is an output and a finite solution to (11.13) exists, then equating (11.17) and (11.19) yields 0 = Pn'TZni + efT.meEP\n\Z,m\ + Q'n = maxj{pniTzni + q* : j = 0,1,..., J} (11.22) where maxj{pniTznJ : j = 0,1,..., J} < 0. (11.23) If (11.23) does not hold, then infinite profits can be made by firm i by scaling up production. In this case, the optimal solution to (11.13) is unbounded and there exists no q* > 0 such that (11.22) holds. Hence, if (11.23) does not hold, set ef to be unbounded since the maximal profit is unbounded. Otherwise, the violation index ef, after algebraic manipulation of (11.22), is also given by (11.21). If we let = ef when K = N-l, then (11.12) obtains. The violation index e* above is similar in form to the violation index for the unconstrained optimization test in the convex case given by (11.1). In contrast, the calculation of the violation index e* in test 8 uses normalized variables, includes the zero vector, and can possibly have an unbounded solution. The results of test 8 can be similarly interpreted. Instead of full profit maximization, the analysis for the constant returns case is done in terms of normalized profit maximization. Assuming a finite solution to the corresponding linear programming problem, let j* solve (11.12). From equations (11.20) and (11.22), the following equations are obtained: q*zn* = maxj{pniTznJ : j = 0,1,..., J) = pniTznj* (11.24) Chapter 11. The Measurement of Allocative Inefficiency Assuming Complete 94 for good n an input, and - q*z3n* = maxj{pniTzni : j = 0 ,1 , . . . , J} = pniTznj* (11.25) for good n an output. Therefore, the normalized profit maximization problem is equivalent to finding a production plan such that, consistent with the zero profit condition, there exists a shadow price for good n such that good n receives the maximal return if it is an input, or the cost of producing good n is minimal if it is an output. Any scalar multiple of z3* belongs to the production possibilities set and yields the same zero shadow profit. Assuming z3* is not a zero vector, one such allocation is z3 = z3*(z'n/zn*) which is the efficient vector yielded by the linear program in test 5. Identical values for the violation indices are obtained from tests 5 and 8 , and j* enters the optimal basis with X3* > 0 in test 5. The partial profit values in test 5 differ from the normalized profit values in test 8 by a factor of | z'n |. With full price data on p = (pi,... ,PN)T >• 0jv, the usual profit maximization test for a constant returns to scale technology is a joint test of the profit maximization conditions for a convex technology and the zero profit condition (see Diewert and Parkan (1984, pp.151-152), Varian (1984a, p.586)). That is, z' is consistent with profit maximization if and only if piTzi > piTz3, j = 1,2,..., J; (11.26) and piTzi = 0. (11.27) Real world data usually fails the zero profit condition (11.27) unless the data were manipulated to impose it. Even if accounting methods ensure this condition, there is a problem of the meaningfulness of the constructed prices. Zero profit does not imply that the allocation is profit maximal at the given prices. Also, it is difficult to obtain a violation index that captures simultaneously departures from the profit maximization and zero profit conditions, as we have done in this study. However, the cost of constructing.our efficiency measure is that we ignore Chapter 11. The Measurement of Allocative Inefficiency Assuming Complete 95 price information on one good. This may be innocuous if we can accept the idea that in most production processes, there is at least one factor or good, which can be intangible, for which a market price does not exist. The returns to this good is imputed to the good chosen to determine the scale of operations, or what we term here as the normalizing good. Some theoretical models assume constant returns to scale. Empirical implementation of these models then requires that relations (11.26) and (11.27) hold, or are not significantly violated inorder to estimate a profit function using the N prices. If the violation indices obtained in test 8 are small, then at most two additional goods, one an input and the other an output can be introduced into the model. Let p = (p\,... , P A T ) T , Z = (z\,..., ZN)T and assume good N + l is an input or z/v+i < 0, and good N + 2 an output or ZN+2 > 0. Let the price and quantity of these goods at observation j be defined in the following manner: If piTzJ > 0, then let z3N+1 — -1 , = p,Tz3, and z3N+2 = 0-If p3Tz3 < 0, then let z3N+2 = 1, p/v + 1 = -p3Tz3, and z}N+1 = 0. UpiTz3' = 0, then let zJN+1 = z3N+2 = 0. Instead of the normalizing good in test 8 absorbing the losses or profits, we let good n enter the profit function estimation at its market price and let a hypothetical output good absorb the losses and a hypothetical input good absorb the profits. Alternatively, the profit function estimation can be done without introducing hypothetical goods by redefining the price of the omitted nth good (the normalizing good) such that the zero profit condition (11.27) holds. 11.3 The quasiconcave technology case (test 9) Assuming the technology function / " satisfies conditions III, the violation index £* measuring unconstrained allocative inefficiency is defined in test 9. • Test 9 Chapter 11. The Measurement of Allocative Inefficiency Assuming Complete . 96 1. Single out a good n with respect to whose technology we would like to perform the test. 2. Define the index set I? as I? = {j: 4 > zn,j = l,2,...,J}. (11.28) 3. Calculate the violation index e* as { nniT( z"J — 2N , N) 1 — : j € If* \ . (11.29) The proof of the equivalence of test 9 and test 6 when K = N — 1 is analogous to that of the corresponding tests under convexity (tests 7 and 4) and hence, is omitted. The detection of allocative inefficiency at a particular observation can depend on the good singled out. This sensitivity of the result arises because a quasiconcave function can have "flats". An example is the case z3n> zn, z"3 = znt for some j € J", z"' is cost minimal for z'n, and p' =p3. The linear programming problem (10.2) for unconstrained optimization with a quasiconcave technology reduces to a net cost minimization problem if good n is an output, and to a net revenue maximization problem if good n is an input. The magnitude of e* in (11.29) for a quasiconcave technology is not comparable with the £* in (11.1) for a convex technology. If we single out a good n and let 5" be the set of all goods except good n (implying K = N — l), then comparable indices and their ranking are given by the following inequality relation (assuming a fixed reference set E C 5): {e*)cRS > {ef)convex > (e*)quaaiconcave (11.30) where (S*)CRS, (£f)convex and {e*)qua8iconcave are obtained using tests 8, 4 and 9, respectively. Inequality (11.30) follows from (10.7). The unconstrained optimization test for a quasiconcave technology function /" can be per-formed with respect to each good n, n = 1,2,...,N. If an observation i passes all TY tests, then producer i with production plan z' is responding efficiently to the full set of prices Chapter 11. The Measurement of Allocative Inefficiency Assuming Complete 97 p' = Q » i , p 2 , • • • ,PN)T ^ Unlike in the convex case, recourse is made to singling out a good at a time and performing N separate tests because of possible nonconvexities in the pro-duction possibilities set. The general technology can exhibit increasing returns and hence, the usual tangency condition between the price hyperplane and the production possibilities set for profit maximization does not hold, as does in the convex case. In the allocative efficiency tests, both for constrained and unconstrained optimization, pro-ducers are assumed to be price takers with respect to the K goods belonging to the set S. A quasiconcave technology can have increasing returns which may be inconsistent with pure competition. If a firm exerts some degree of market power over prices of two or more of the TV goods, the appropriate allocative efficiency test is test 6 which involves solving linear pro-gramming problems. More stringent tests on whether the price and quantity data are consistent with some form of market structure other than pure competition can be devised. Varian (1984a) offers a test for consistency with the hypothesis of profit maximization under monopolistic be-havior. Since there is no general theory of imperfect competition, specific price setting rules have to be posited. Factors like strategic variables (price or quantity), behavior (cooperative or noncooperative), firm entry and exit conditions, demand conditions, etc. can be taken into account. A decomposition of e* in (11.29) into its components due to technical inefficiency (6*) and pure allocative inefficiency due to failure to optimize, given the relevant prices, with respect to goods in Sn (ef — 6*) and with respect to goods not in Sn (e* — ef) can be made. The comparability of e* in (11.29) as a measure of allocative efficiency with 6* in (5.3) for technical efficiency for a quasiconcave technology is apparent in the dual expression for 6* in (7.12). Chapter 12 The Efficiency Tests with the No Technological Regress Assumption With time series data such that the observation index j corresponds to time, violations of the preceding efficiency hypotheses can be due to technological progress. Over time, the produc-tion possibilities set of the firm can be growing due to increasing effectiveness in the use of production inputs as a result of research and development, innovation, etc. or due to increasing availability of resources such as capital, labor or raw materials. Hence, if an observation j is deemed relatively efficient to observation i, j > i, it is possible that the production vector z3 is not feasible at period i. It may then be desired to incorporate the no technological regress assumption with the efficiency tests. To deal with technical change, we follow the Diewert-Parkan (1983, pp.153-157) method of comparing the current observation only with the earlier observations. Suppose there is no technological regress, that is, T3 C T* if j < i. To illustrate how the efficiency tests can be modified to take into account the added assumption of no technological regress, we show the revised technical efficiency test for a convex technology. First, we formally state the technical efficiency and no technological regress hypothesis. • Technical Efficiency and No Technological Regress Hypothesis I (for a convex technol-ogy): The data {z3 : j = 1,2,...,/} are generated by an underlying technology satisfying conditions i" and are technically /^ -efficient. Furthermore, the production possibilities sets at periods i and j, j < i, are such that T3 C T 1 . Again, we choose the goods in E such that the efficiency direction vector 7 satisfies condi-tion (3.1). To test the above hypothesis, given quantity data and the efficiency direction vector, we perform the following test. 98 Chapter 12. The Efficiency Tests with the No Technological Regress Assumption 99 • Test 1' 1. For each observation i , i = 1,2,..., J, solve the following linear programming sub-problem i: max5,.>o,Ai>0>...,A.>o{<5. : £J=iA J V * + ^z\ Y.)=^3 = 1} = (12-1) where 7 is the efficiency direction vector 7 diagonalized into a matrix. 2. Consider the following consistency condition: Suppose 8* is the optimized objective function for the ith subproblem (12.1), i = 1,..., J. If condition (3.1) holds and 8* = 0 for all i , i = 1,..., J , then the data {zJ : j = 1,2,..., J} are consistent with the technical efficiency and no technological regress hypothesis I for a convex technology. If the condition (3.1) holds and 8* > 0 for some i, then observation z' violates the technical efficiency and no technological regress hypothesis I and the data {z3 : j = 1,2,..., J} are not consistent with this hypothesis. In contrast to test 1, the linear programming subproblem (12.1) constructs the convex hull of only the first i data points {z1, z2,..., z'} instead of the whole set of J observations. The modified test is then less restrictive. Since modifying the rest of the efficiency tests to account for technical change is a straight-forward exercise, the list of changes in the various efficiency tests are relegated to appendix A. The same trick can be adapted for cross-section data when it is desired to divide the set of firms into subgroups for purposes of comparison. Productivity may be a function of the size of the firm as measured by output or installed capital. Depending on the purpose of the study, the researcher may want to take into account, if present, the disparity in firm sizes. Chapter 13 Conclusion In summary, nonparametric linear programming tests for consistency with the hypotheses of technical efficiency and allocative efficiency are proposed. The tests are formulated relative to three kinds of general technologies — convex, constant returns to scale and quasiconcave technologies. For the technical efficiency tests (tests 1, 2 and 3), the informational requirements are quantity data on production plans and an efficiency direction vector which corresponds to the subset of goods with respect to which inefficiency is to be measured. Relative to the general technology posited, technical efficiency is given a precise definition wherein being a boundary point of the production possibilities set is a necessary but not sufficient condition. The efficiency tests yield violation indices 6*, invariant to scale in the measurement of goods, which give the equiproportionate adjustment in goods, as specified in the efficiency direction vector, needed for a particular observation to satisfy the hypotheses. Alternatively, the violation indices can be interpreted as measures of the loss or waste due to failure to maximize production of outputs or minimize the use of inputs. Dual interpretations of the technical efficiency definitions and tests are also presented to link the analysis to Koopmans' notion of efficiency prices and the concepts of profit maximization and cost minimization. With price and quantity data, tests for allocative efficiency are proposed. Given a behavioral description of the firm as embodied by an objective function and the technological constraints under which the firm is operating, allocative efficiency entails the use of the optimal mix of out-puts and inputs. For unconstrained optimization, the tests (7, 8 and 9) reduce to a comparison of inequalities. For a convex technology, a production plan z' is allocatively efficient if piTzi > p i T z J J = 1,2,. . . ,J; 100 Chapter 13. Conclusion 101 that is, if at prices p' > OAT, the production plan z* yields the largest profit when compared to z }, j = 1,2,..., J. For a constant returns technology, z' is allocatively efficient, given z'n, if there exists a shadow price q* > 0 for good n such that 0 = p niTz ni + q*zn > p" iTz"3 + q*nzin, j = 0,1, 2,..., J where z° is the shutdown point; that is, (z n',z'n) yields the maximal profit of zero at prices (p"'T,qn). For a quasiconcave technology, z' is allocatively efficient, given z'n, if pniTzni > pniTznj^ j <z Jp; that is, z"' yields maximal net revenue for z'n among the alternative allocations in the upper level set of z'n. In the unconstrained optimization tests, the objective functions of the linear programming problems take the form of an unrestricted profit function in the convex case, a normalized profit function (whose value is the shadow price of the normalizing good) in the constant returns case, and a net revenue (or negative net cost function) in the quasiconcave case. The unconstrained optimization tests are special cases of the more general linear program-ming tests for constrained optimization. If some goods are fixed or their competitive prices do not exist, then the constrained optimization tests are given by tests 4, 5 and 6. The objective functions of these linear programming problems are partial profit functions, restricted versions of the unconstrained forms. The allocatively efficient allocations are technically efficient in the sense that they he on the boundary of the relevant convex set, production possibilities set or upper level set as the case may be, and with respect to the goods with positive market prices, do not he on the free disposal region. The violation indices e* for unconstrained optimization and ef for constrained optimization give the proportion of the value of goods in the reference set lost due to allocative inefficiency. The waste is measured relative to the optimal value of the objective function. A number of future areas of research are indicated by the limitations of the foregoing analysis. The violation indices proposed in this study have been interpreted so far to be Chapter 13. Conclusion 102 measures of the degree of violation of the maintained efficiency hypotheses. A deterministic approach has been taken and the possibility of measurement errors ignored. The researcher may have prior reasons to suspect data errors and the efficiency test can be alternatively used to explore how "good" or "bad" the data is before parametric estimation of production, cost or profit functions. To address the problem of measurement error, Hanoch and Rothschild (1972) suggest some ad hoc ways to modify the tests and Varian (1985) offers a procedure in the context of statistical hypothesis testing. The assumption of nonincreasingness or free disposal in our regularity conditions for the various technologies rules out the possibility of congestion. Firms may produce multiple outputs, some of which like pollution may be undesirable. In the framework of this study, undesirable output goods, if measurable, can be treated as negative outputs and hence would enter the tests as inputs. If indeed congestion is present, our technical efficiency and constrained efficiency tests would tend to overestimate the true violation index since the constructed convex set would be larger than the true production possibilities set. A revision of the technical efficiency tests to satisfy the DEA definition of efficiency would also yield an "unbiased" violation index since the chosen efficient points would not be in any free disposal region of the boundary of the constructed convex set. If market prices do not exist for the "undesirable outputs", then the constrained optimization tests can be performed. If (positive) market prices do exist, then these prices ensure that the allocative efficiency tests would yield the optimal amounts of the undesirable outputs. For further studies on nonparametric efficiency analysis with congestion, the reader is referred to Fare, Grosskopf and Lovell (1985) and Fare, Grosskopf, Lovell and Pasurka (1989).1 In the real world too we may observe negative profits or firms producing at a loss. This can sometimes be explained by dynamic considerations such as expectations, strategic behavior and adjustment processes (such as entry and exit of firms). The basically static approach adopted in this study abstracts from these complications. In the tradition of nonparametric 'The above authors also use h y p e r b o l i c efficiency measures i n contrast to the e q u i p r o p o r t i o n a t e adjustment measures defined by the v i o l a t i o n indices i n t h i s study. Chapter 13. Conclusion 103 production analysis in economics, the framework in this study can be extended to testing for other regularity conditions on the technology, such as homotheticity and separability, and to forecasting behavior. These extensions may require solving quadratic or nonlinear programming problems instead of linear programming problems. Part II Nonparametric and Parametric Measures of Technical Progress: An Application to the Canadian Input-Output Data 104 Chapter 14 Introduction Nonparametric and parametric measures of technical progress are compared in this study. The nonparametric measures of technical progress are obtained by a reinterpretation of the viola-tion indices yielded by the efficiency tests of part I. In this context, we assume optimizing behavior on the part of the producers and the violation indices as indicators of the shift in the production frontier. The parametric measures of technical progress are obtained through econometric profit function estimation explicitly incorporating technical change variables. The time variable is used as a proxy for the technical change variables and is introduced additively through a quadratic spline subfunction developed by Diewert and Wales (1989b). The sym-metric generalized McFadden functional form, also introduced by Diewert and Wales (1987), is used in the profit function estimation. For convenience (though unnecessary), constant returns to scale is assumed. Annual Canadian input-output data for the period 1961-1980 are used. Measures of tech-nical progress are estimated for four aggregated sectors: I. resources sector; II. manufacturing sector, export market-oriented; III. manufacturing sector, domestic market-oriented; and the IV. services sector. We assume the sectors have single output technologies. The resources sector (I) has ten output and input goods; they are 105 Chapter 14. Introduction 106 (I! (2; (3; (4; (5; (e: (7; (s: (9; (10: resource goods (from sector I), manufactured goods (from sector II), manufactured goods (from sector III), service goods (from sector IV), imports, labor inventories, land. machinery and equipment structures, and For the resources sector (I), the output good is good 1 and the other nine goods are all input goods. The other three sectors (II, III and IV) each has nine output and input goods; since the quantity data on structures and land for these sectors are proportional in the original database, these two goods have been aggregated as a Leontief composite good which we have termed as "structures" (good 9) in the current data set. The capital rental prices for the capital inputs (goods 7-10) are calculated using internal rates of return. A detailed description of the construction of the data set for the empirical estimation performed in this study can be found in appendix B. To calculate the nonparametric measures of technical progress, the allocative efficiency test for a convex conical technology assuming complete profit maximization (test 8 in part I) was performed for each of the four sectors with structures as the normalizing good. For each sector, a unit scale profit function was also estimated with structures as the scale variable. The measure of technical progress is then defined as the marginal change in the return to this factor (structures) due to the passage of a time period divided by the value of some basket of reference goods. For a particular sector, let there be N + 1 goods (JV -f 1 = 10 for sector I, and JV + 1 = 9 for sectors II, III and IV). Let the (TV + l)th good be the normalizing good or the good chosen to be Chapter 14. Introduction 107 the scale variable. In the allocative efficiency test and the profit function estimation performed in this study, the price of this good is not used. However, we use quantity data on this good; denote the quantity of this good by z^+\. For the quantity data, outputs are indexed positively and inputs are indexed negatively. Denote the price vector of the first TY goods at period t as p* = (p\,P2, • • • ,PN)T ^ ®N and the quantity vector at period t as zl = (z[, z\,..., zN)T. Let n(pl, t) be the period t unit scale profit function. If we denote the period t unit scale production possibilities set containing the feasible z vectors when | z;v+i |= l 1 by Slr then the unit scale profit function ^(jf-^i) is defined as Under the assumption of constant returns to scale and finite production plans, the optimal profit defined over the whole set of TV +1 goods is zero. Hence, if an input is chosen as the scale variable, then the unit scale profit function defined by (14.1) gives the per unit return to the normalizing or scaling input. If a capital good which can be considered a fixed factor is chosen as the scale variable, then the unit scale profit function in (14.1) can be interpreted as a short-run or restricted or variable (unit scale) profit function defined over the first TV variable goods. Mathematically, we define an index of technical progress at period t, say A' , as where E is the set containing the indices of the chosen reference goods and m ^ TV + 1, and zlm = zlml | zlN+1 | where | zlN+1 | is the absolute value of the quantity of good TV + 1, the normalizing good, at period t. Note that the numerator in (14.2) can be positive, zero or negative while the denominator is always positive. Hence, A* > 0 implies technical progress at period t and A 4 < 0 implies technical regress at period t. 1 Since a convex cone is closed under a d d i t i o n a n d nonnegative scalar m u l t i p l i c a t i o n , then the p r o j e c t i o n of the p r o d u c t i o n p o s s i b i l i t i e s set i n (N + l ) - d i m e n s i o n a l space i n t o the u n i t scale p r o d u c t i o n p o s s i b i l i t i e s set i n A f-dimensional space is harmless. Tv(p\t) = maxz{pTz : z 6 S1}. (14.1) dn(p',t)  dt (14.2) Y,meEPm I Zm I Chapter 14. Introduction 108 The numerator in equation (14.2) is the marginal change in the per unit return to the normalizing (or scaling) good N + 1. If we multiply the numerator and denominator by | zN+l |, we obtain A1 = V m - (14-3) SEPm I *m I The numerator in (14.3) gives the marginal change in the return to the normalizing good N + 1 at period t due to the passage of a time period. Alternatively, the numerator in (14.3) can be interpreted as the change in the value of outputs minus the change in the value of inputs other than the normalizing good due to the passage of a time period. The denominator in (14.3) has unsealed quantity variables. For the empirical exercise, we focus on the private domestic production sector and the economy's output goods 1^ 1 are chosen as the reference goods. At the sectoral level, some of these goods enter as intermediate inputs. If A' > 0, then the index of technical progress gives the proportional increase in the sector's contribution to the economy's value of output goods available for final demand (including exports) due to technical change occuring in that sector. The index of technical change defined by equation (14.2) uses price information. If we define a multifactor productivity measure as the ratio of a quantity index of output goods to a quantity index of input goods, then to measure technical change it is imperative to net out relative price effects as reflected in the substitution of goods. Both the nonparametric approach using the allocative efficiency tests developed in part I and the parametric approach using either profit or cost function econometric estimation are able to isolate these relative price effects. In the efficiency tests, a deterministic approach is taken and the residual unexplained component of the multifactor productivity measure is all attributed to technical change. Hence, we can expect the estimates of the index of technical progress obtained using the nonparametric approach to be quite erratic or to display a greater degree of variability. On the other hand, the stochastic approach requires the specification of the distribution of the disturbance terms. In the profit function estimation performed in this study, the usual error structure — errors are distributed normally with zero mean and covariance structure £ — for Chapter 14. Introduction 109 simultaneous equations is assumed. The quadratic spline subfunction used in modeling technical change attempts to be less restrictive and have greater capability than the simple quadratic subfunction in capturing the ups and downs of the rates of technical change. However, compared to the nonparametric efficiency tests, we can expect the profit function estimates of the index of technical progress to display a smoother behavior. The empirical exercise that follows aims to investigate the comparative behavior of the estimates of the index of technical progress obtained under the nonparametric and parametric approaches. Chapter 15 A Nonparametric Approach to Measuring Technical Progress The dual representations of production technologies — either through profit or cost functions — assume optimizing behavior on the part of the producers. As was noted in part I, the efficiency tests developed are based on the joint hypotheses on the technology and the behavior of the producer. Hence, a positive violation index can be due to either of two sources. For the sectoral aggregated data we are using, the attribution of positive violation indices to failure of the producers to respond optimally may be suspect. We can interpret the positive violation indices as due to violations of the technology assumptions; in particular, that no technical change is occuring and the true underlying technology remains constant from time period to another. If indeed technological change is occuring, then the production possibilities set may be expanding or shrinking over time. Since we assume constant returns to scale, the allocative efficiency test assuming complete profit maximization for a convex cone technology (test 8 in part I) was performed for each of the four sectors.1 Neither the shutdown point nor an unbounded solution (since an input, structures, was used as the normalizing good) was obtained as the optimal allocation for any observation; hence, the allocative efficiency unconstrained optimization violation index £ j f (see equation (11.12), part I) can be expressed as £t = m a x > f ^ .gt | . i = 1,2,• • • , T (15.1) I. L,m£EPm \Zm\ ) where z3' = (z{, z3,,..., zN)T = z-'/ | z3N+1 |, and T = maximum time index. The violation indices using test 8 in part I for the four sectors are listed in table 15.2 and are plotted in 'Test 8 was also performed for the d a t a set using c a p i t a l r e n t a l prices based on an exogenous bond rate. T h e results even i n magnitudes of the v i o l a t i o n indices i n a l l four sectors do not differ s i g n i f i c a n t l y f r o m those reporte d here for the d a t a set using c a p i t a l r e n t a l prices based on i n t e r n a l rates of r e t u r n . 110 Chapter 15. A Nonparametric Approach to Measuring Technical Progress 111 violation index ej" using test 8 year ,2 sector I sector II sector III sector IV 1961 0.10542 0.15834 0.14341 0.13116 1962 0.05616 0.12018 0.10799 0.10106 1963 0.02377 0.10041 0.10086 0.08981 1964 0.02448 0.08306 0.08327 0.06938 1965 0.02328 0.07228 0.07604 0.05787 1966 0.00000 0.08498 0.07670 0.04788 1967 0.06135 0.08632 0.08497 0.04601 1968 0.03270 0.06943 0.07018 0.03466 1969 0.01889 0.04745 0.06054 0.03407 1970 0.03312 0.07279 0.06436 0.02983 1971 0.03515 0.06656 0.05363 0.02982 1972 0.03247 0.05035 0.03759 0.02076 1973 0.00000 0.03097 0.01830 0.01434 1974 0.04741 0.02919 0.02238 0.01648 1975 0.10403 0.04823 0.04690 0.01560 1976 0.10567 0.02656 0.02105 0.00425 1977 0.12384 0.00984 0.00738 0.01277 1978 0.15356 0.00000 0.00318 0.01393 1979 0.16163 0.01241 0.00000 0.00596 1980 0.18950 0.02804 0.00096 0.00000 Table 15.2: Test 8 violation indices £j for the four sectors figures 15.9-15.12. The violation indices £j" can be interpreted as chained indices of technical change. Let us denote the nonparametric estimate of the period t index of technical progress defined in equation (14.2) (or equation (14.3)) by A' . The nonparametric index of technical progress A* can then be defined as At = l i_±L _ i (15.2) f * — F * (! + £?)" The values of A* for the four sectors for the period 1962-1980 are listed in table 15.3 and are plotted in figures 15.13-15.16. Chapter 15. A Nonparametric Approach to Measuring Technical Progress 112 0.20 Figure 15.10: Test 8 violation indices ej, sector II Chapter 15. A Nonparametric Approach to Measuring Technical Progress 113 0.20 0 . 2 0 Figure 15.12: Test 8 violation indices ej, sector IV Chapter 15. A Nonparametric Approach to Measuring Technical Progress 114 nonparametric index A' year ,2 sector I sector II sector III sector IV 1962 0.04664 0.03406 0.03196 0.02733 1963 0.03164 0.01797 0.00648 0.01033 1964 -0.00069 0.01602 0.01624 0.01910 1965 0.00118 0.01006 0.00672 0.01088 1966 0.02328 -0.01171 -0.00061 0.00953 1967 -0.05780 -0.00123 -0.00762 0.00178 1968 0.02774 0.01579 0.01383 0.01097 1969 0.01355 0.02099 0.00909 0.00057 1970 -0.01377 -0.02362 -0.00359 0.00412 1971 -0.00196 0.00584 0.01019 0.00002 1972 0.00259 0.01543 0.01545 0.00888 1973 0.03247 0.01880 0.01894 0.00633 1974 -0.04526 0.00172 -0.00399 -0.00211 1975 -0.05128 -0.01817 -0.02341 0.00087 1976 -0.00149 0.02111 0.02532 0.01130 1977 -0.01617 0.01656 0.01357 -0.00841 1978 -0.02576 0.00984 0.00418 -0.00115 1979 -0.00695 -0.01225 0.00318 0.00793 1980 -0.02343 -0.01521 -0.00096 0.00596 Table 15.3: Nonparametric indices of technical progress A' for the four sectors Chapter 15. A Nonparametric Approach to Measuring Technical Progress 115 0.06 -0.06 1 1—i i i | i i 1 — i 1 i r — i r 1960 1965 1970 1975 1980 year Figure 15.13: Nonparametric indices A' , sector I 0 . 0 6 - 0 . 0 6 i — i i i — | — i — i — i — r — | — i — i — r — i — | — i i i — r 1960 1965 1970 1975 1980 year Figure 15.14: Nonparametric indices A' , sector II Chapter 15. A Nonparametric Approach to Measuring Technical Progress 116 0.06 -0.06 i i i i | i i i i | — i i 1960 1965 1970 year i i i i 1975 1980 Figure 15.15: Nonparametric indices A' , sector III 0.06 -0.06 1980 Figure 15.16: Nonparametric indices A' , sector IV Chapter 15. A Nonparametric Approach to Measuring Technical Progress 117 The results obtained using the nonparametric index A* indicate that technological regress occured during the following years in the given sectors: for sector I: 1964, 1967, 1970, 1971, 1974-1980; for sector II: 1966, 1967, 1970, 1975, 1979, 1980; for sector III: 1966, 1967, 1970, 1974, 1975, 1980; and for sector IV: 1974, 1977, 1978. These findings are given support by the corresponding efficiency test incorporating the no technological regress assumption, test 8' in appendix A, which was also performed for each of the four sectors. The results are given in table 15.4; the values of are given only for the years violating the efficiency and no technological regress assumption of test 8'. Almost the same set of years is discerned by the nonparametric index A* (obtained by unchaining the violation indices z*t from test 8) and the efficiency and no technological regress test 8' to be periods where technological regress occured. For most years where the results conflict either the nonparametric index A' is small or the violation index t\ for test 8' is small. Next we compare the nonparametric index of technical progress A f with total factor produc-tivity measures based on the index number approach. Divisia and Fisher indices of productivity change were calculated for each of the four sectors of the Canadian economy. In the economic approach to index numbers based on the assumption of optimizing behavior on the part of the producers, Caves, Christensen and Diewert (1982) and Diewert (1989b) have shown that the Divisia (or Tornqvist or translog) and the Fisher productivity change indices, being exact for some flexible functional form, are superlative. The Fisher ideal quantity index has the added property of being the only function that satisfies some twenty tests considered desirable in the axiomatic approach to index number theory (Diewert, 1989b). The econometric computer package SHAZAM by K.J. White (1987) was used to compute the productivity indices. For each sector, chained output and input Divisia and Fisher ideal quantity indices are obtained for each year t. The output quantity indices are defined over the four output goods (goods 1^ 1) with quantities of intermediate inputs indexed negatively. The input quantity Chapter 15. A Nonparametric Approach to Measuring Technical Progress observation relatively year,t violation index efficient to t sector I: 1964 0.001795 1963 1967 0.061346 1966 1968 0.032703 1966 1969 0.015616 1966 1970 0.031200 1966 1971 0.016859 1966 1972 0.009744 1966 1974 0.047410 1973 1975 0.104030 1966 1976 0.105673 1966 1977 0.123843 1966 1978 0.153557 1966 1979 0.161633 1966 1980 0.189503 1966 sector II: 1966 0.007065 1965 1967 0.003043 1965 1970 0.021252 1969 1971 0.009695 1969 1975 0.012834 1974 1979 0.012405 1978 1980 0.028041 1978 sector III: 1967 0.000555 1966 1974 0.003566 1973 1975 0.025523 1973 1980 0.000963 1979 sector IV: 1977 0.009799 1976 1978 0.011078 1976 1979 0.001898 1976 Table 15.4: Test 8' violation indices for the four sectors Chapter 15. A Nonparametric Approach to Measuring Technical Progress 119 indices (with input quantities now indexed positively) are defined over the primary inputs: goods 5-10 for the resources sector and goods 5-9 for the other sectors. Output growth and input growth indices for year t are calculated by dividing the current year fs quantity index by its lagged value. A productivity change index is then defined as the ratio of the output growth index to the input growth index minus one; that is, , output growth index . ' . T = • — ~ — ~ r^—; 1 (!5-3) input growth index where T1 is the productivity change index for year t. If r* > 0, then there occurs a productivity improvement at period t or an outward shift of the production frontier from year t — 1 to year t. If r* < 0, then no productivity improvement occurs at period t. Let us denote by T1D the Divisia productivity change index for year t and by TF the Fisher productivity change index for year t. The values of T1D and TF for the different sectors are listed in tables 15.5-15.6. As reflected too in figures 15.17-15.20 where their curves coincide, the Divisia and Fisher productivity indices are almost identical. The magnitude of the nonpara-metric index A* is not expected to be directly comparable to the magnitudes of the productivity indices and TF since the technical progress index A' (equation (14.2)) and productivity in-dex r* (equation (15.3)) are defined differently. The interpretation of A' , rlD and TF as measures of the shift in the production frontier (or its dual representations) rests on the assumption of optimizing behavior on the part of the producers. The consistency of the patterns of behavior of the three indices, as illustrated in figures 15.17-15.20, lends support to the reinterpretation of the violation indices e% in (15.1) as chained indices of technical progress in the context of the time series data used. Phapter 15. A Nonparametric Approach to Measuring Technical Progress 120 Divisia productivity change index rlD year,t sector I sector II sector III sector IV 1962 0.11498 0.04388 0.06869 0.03527 1963 0.06345 0.03174 0.01168 0.01737 1964 -0.00307 0.02517 0.03573 0.02765 1965 0.00475 0.02290 0.02055 0.01813 1966 0.04396 -0.01614 0.00786 0.02019 1967 -0.09872 0.00678 -0.00185 0.00997 1968 0.04993 0.03716 0.04377 0.02325 1969 0.02422 0.04411 0.02634 0.00492 1970 -0.02681 -0.04532 0.00216 0.00878 1971 0.00658 0.02272 0.03059 0.00635 1972 0.00992 0.02952 0.03502 0.01614 1973 0.06629 0.04119 0.04242 0.01274 1974 -0.07655 0.01089 -0.00846 -0.00055 1975 -0.06721 -0.02821 -0.04545 0.00428 1976 -0.00233 0.04215 0.06297 0.01766 1977 -0.01805 0.03516 0.03131 -0.01253 1978 -0.04178 0.01991 0.00926 -0.00095 1979 0.01316 -0.02534 0.00726 0.01171 1980 -0.01039 -0.03163 -0.00282 0.00875 Table 15.5: Divisia productivity change indices rj-, for the four sectors Chapter 15. A Nonparametric Approach to Measuring Technical Progress 121 Fisher productivity change index rF year,i sector I sector II sector III sector IV 1962 0.11499 0.04388 0.06871 0.03575 1963 0.06345 0.03173 0.01168 0.01737 1964 -0.00307 0.02517 0.03573 0.02765 1965 0.00475 0.02290 0.02055 0.01813 1966 0.04397 -0.01614 0.00787 0.02018 1967 -0.09868 0.00677 -0.00186 0.00996 1968 0.04994 0.03715 0.04377 0.02324 1969 0.02423 0.04411 0.02634 0.00492 1970 -0.02682 -0.04533 0.00214 0.00878 1971 0.00639 0.02268 0.03057 0.00569 1972 0.00991 0.02951 0.03503 0.01614 1973 0.06692 0.04119 0.04263 0.01274 1974 -0.07661 0.01095 . -0.00842 -0.00053 1975 -0.06723 -0.02828 -0.04553 0.00427 1976 -0.00232 0.04214 0.06297 0.01765 1977 -0.01805 0.03516 0.03130 -0.01252 1978 -0.04178 0.01991 0.00926 -0.00093 1979 0.01317 -0.02534 0.00727 0.01173 1980 -0.01035 -0.03164 -0.00283 0.00874 Table 15.6: Fisher productivity change indices rF for the four sectors Chapter 15. A Nonparametric Approach to Measuring Technical Progress 122 0 . 12 0 . 1 0 -0 . 0 8 -0 . 0 6 0 . 0 4 S 0 . 0 2 T J C >-> 0 . 0 0 o P - 0 . 0 2 H - 0 . 0 4 - 0 . 0 6 H - 0 . 0 8 - 0 . 1 0 H nonporomet r l c Divisia Fisher - 0 . 1 2 - I — i — i — i — i — r — i — i — i — i — i — r — i 1 — r i 1 1 r 1960 1965 1970 year 1975 1980 Figure 15.17: Comparative behavior of A*, T1D and TF, sector I Chapter 15. A Nonparametric Approach to Measuring Technical Progress 123 0 . 12 0 . 10 H 0 . 0 8 0 . 0 6 H 0 . 0 4 H 2 0 . 0 2 H C I" 0 . 0 0 u 3 P - 0 . 0 2 H - 0 . 0 4 -- 0 . 0 6 -- 0 . 0 8 -n o n p o r a m e t r l c D i v i s i a _ _ F i s h e r - 0 . 10H 0 . 12 H r — i 1 r 1960 t —i 1 1 1 — | 1 1 1 r — | 1 1 1 r | 1965 1970 1975 1980 year Figure 15.18: Comparative behavior of A ' , TLD and TF, sector II Chapter 15. A Nonparametric Approach to Measuring Technical Progress 124 0.12 0 . 10 A 0 . 0 8 H 0 . 0 6 H 0 . 0 4 H i> 0 . 0 2 1 •o c •I* 0 . 0 0 o 3 p - 0 . 0 2 - 0 . 0 4 - 0 . 0 6 -i - 0 . 0 8 - 0 . 1 0 - 0 . 1 2 n o n p o r o m e t r i c D i v l s l o F i s h e r " i 1 1 1 1 1 — r — l 1 — I 1 1 1 1 1 1 1 — i r 1960 1965 1970 year 1975 1980 Figure 15.19: Comparative behavior of A' , rjj and rF, sector III Chapter 15. A Nonparametric Approach to Measuring Technical Progress 125 0 . 12 0 . 1 0 -0 . 0 8 -0 . 0 6 -0 . 0 4 in S 0 . 0 2 •o c I" 0 . 0 0 u 3 p - 0 . 0 2 - 0 . 0 4 - 0 . 0 6 - 0 . 0 8 - 0 . 10 n o n p o r o m e t r i c D i v i s i a F i s h e r - 0 . 1 2 | i i — i — i — | — i — p 1960 1965 i | — i — i — i — i — | — i i > — p 1970 1975 1980 year Figure 15.20: Comparative behavior of A' , T1D and rF, sector IV Chapter 16 A Parametric Approach to Measuring Technical Progress In the parametric approach to estimating the index of technical progress defined by equa-tion (14.2), a functional form is posited for the unit scale profit function. The symmetric gener-alized McFadden flexible functional form with a quadratic spline model for technical progress, proposed by Diewert and Wales (1989b), is used. This model for the unit scale profit function has the property of being TP (technical progress) flexible. The unit scale profit function takes the following form: v ( p , t ) = h(p) + d ( p , t ) (16.1) where and h(p) = p T b 1 + 1 ( P T B P \ 2\a Tp ) d ( p , t ) = < p T b 2 t + l p T b 3 t 2 p T b 2 t + l p T b 3 t 2 i + p T b 3 ( t _ t i ) h + ip?V( t - i 1 ) 2 for t < t i ; for t\ < t < t2; (16.2) (16.3) p T p t + l p T p t l + p T p i t i - t ^ h + 1 2 P T b 4 ( t 2 - t 1 ) 2 + p T P ( t - t 2 ) t 1 + pTb\t-42)(t2 - h ) + \ p T b 5 ( t - t 2 f i o i t 2 < t < T. The exogenous parameters of the model are a = (ai, a 2 , . . . , CXN)T > ON in (16.2) and the break points in the quadratic spline model fx and t2 in (16.3). The endogenous parameters are the TV-dimensional vectors b1, b2, b3, b4 and b5, and the TVxTV matrix B. Convexity of the profit 126 Chapter 16. A Parametric Approach to Measuring Technical Progress 127 function in prices is imposed by restricting the B matrix in (16.2) to satisfy the restrictions: B = BT and B is positive semidefinite. For identification, a reference vector p* is chosen such that Bp* = 0N. The Diewert-Wales (1989b) method of specifying the a vector was followed. The price vec-tor for 1971 was arbitrarily chosen as p*. A system of N equations corresponding to output supply and input demand equations for each sector was estimated by maximum likelihood esti-mation using nonlinear optimization routines. To aid convergence, the semiflexible estimation technique (Diewert and Wales, 1988) wherein the rank of the B matrix is restricted was used. For each sector, several runs of the model using different break points t\ and t2 were performed to search for those which yield higher values of the likelihood function. Table 16.7 presents the R2 and log likelihood values, the number of parameters estimated, the rank of the B matrix and the values of t\ and t2 for the final models selected for the four sectors.1 Denote the parametric estimator of the index of technical progress (given in (14.2)) obtained using the functional form defined by equations (16.1)-(16.3) as A*. The numerator in this estimator is then drc(p,t) _ dd(p,t) dt dt (16.4) pTb2 + pTb3t for t < tj; pTb2 + pTbH1 + pTb\t -tx) for ti < t < t2; pTb2 + pTbHx + pTb4(t2 -t1) + pTb5(t-t2) for t2 < t < T. The economy's output goods 1-4 are used as the reference goods in the denominator of (14.2). The values of A f for the four sectors are listed in table 16.8 and are plotted in figures 16.21-16.24. : A more d e t a i l e d d e s c r i p t i o n of the e s t i m a t i o n procedure a n d li s t s of the parameter estimates and price el a s t i c i t i e s of o u t p u t s u p p l y and i n p u t demand, together w i t h some i n t e r p r e t a t i o n , can be fou n d i n appendix D. Chapter 16. A Parametric Approach to Measuring Technical Progress 128 R2 values* equation sector 1 2 3 4 1 resource goods (I) 0.956 0.910 0.984 0.977 2 manufactured goods (II) 0.966 0.979 0.959 0.968 3 manufactured goods (III) 0.990 0.993 0.980 0.987 4 service goods (IV) 0.995 0.985 0.993 0.998 5 imports 0.928 0.988 0.962 0.988 6 labor 0.888 0.911 0.889 0.993 7 inventories 0.957 0.970 0.982 0.875 8 machinery and equipment 0.997 0.999 0.998 1.000 9 land 0.999 log likelihood 512.133 381.420 357.068 277.785 number of parameters 66 65 65 65 rank of B 3 5 5 5 t\,t? 1967,1975 1965,1975 1967,1975 1970,1976 "The R2 values are calculated as one minus the residuals to the variance of the output supply ratio of the variance of the or input demand variable. Table 16.7: Model specification, R2 and log likelihood values for the four sectors c D SZ o "5 u c X o c 0.06 0.04 -£ 0 . 0 2 -0 .00 - 0 . 0 2 -- 0 . 0 4 O s e c t o r I - 0 . 0 6 I i — i i i | — i i i i — | — i i 1960 1965 1970 1975 1980 year Figure 16.21: Parametric indices A' , sector I Chapter 16. A Parametric Approach to Measuring Technical Progress 129 parametric index A* year ,2 sector I sector II sector III sector IV 1961 0.01782 0.02746 0.01516 0.01793 1962 0.01838 0.01842 0.01229 0..01631 1963 0.01576 0.01202 0.01102 0.01498 1964 0.01208 0.00656 0.00958 0.01333 1965 0.00919 0.00233 0.00896 0.01197 1966 0.00712 0.00370 0.00883 0.01086 1967 0.00159 0.00469 0.00926 0.01010 1968 0.00054 0.00539 0.00930 0.00923 1969 -0.00057 0.00576 0.00901 0.00830 1970 -0.00366 0.00705 0.00902 0.00733 1971 -0.00483 0.00853 0.00899 0.00633 1972 -0.00797 0.00890 0.00820 0.00537 1973 -0.00939 0.00928 0.00714 0.00442 1974 -0.01337 0.00982 0.00625 0.00319 1975 -0.01720 0.01074 0.00629 0.00201 1976 -0.01490 0.00677 0.00605 0.00074 1977 -0.01241 0.00294 0.00548 0.00181 1978 -0.00973 -0.00051 0.00516 0.00291 1979 -0.00671 -0.00427 0.00480 0.00418 1980 -0.00087 -0.00797 0.00367 0.00557 Table 16.8: Parametric indices of technical progress A* for the four sectors Chapter 16. A Parametric Approach to Measuring Technical Progress 130 0.06 0.04 -•o £ 0.02 « c o 0.00 o o i - 0 . 0 2 -« -0.04 --0.06 O s e c t o r II o o o o-i — i i i — i — r 1960 1965 - i — i — | — i — i — i — i — | i i — i — r 1970 1975 1980 year Figure 16.22: Parametric indices A' , sector II 0.06 0.04 -X V X) £ 0.02 • c o 0.00 o a J -0.02 o -0.04 O s e c t o r III o o o o o o o o o Q G H ) -0.06 - j — i — i — r — i — | -1960 1965 " i — | — i — r 1970 year 1975 1980 Figure 16.23: Parametric indices A' , sector III Chapter 16. A Parametric Approach to Measuring Technical Progress 131 c o o o 0.06 0.04 -O sector IV £ 0.02-0.00 -0.02 -0.04 -—0.06 | i i i i | i i i i 1960 1965 i i i r 1970 year 1975 1980 Figure 16.24: Parametric indices A ' , sector IV Chapter 17 A Comparison of Nonparametric and Parametric Estimates of Technical Progress The comparative behavior of the nonparametric measure of technical progress A* and the parametric measure of technical progress A* for the four aggregated private domestic sectors of the Canadian economy for the period 1961-1980 is illustrated in figures 17.25-17.28. Since the parametric approach incorporates a stochastic disturbance term in modeling the unit scale profit function, the estimates of the parametric index A' display a much smoother behavior. The interesting feature of figures 17.25-17.28 though is that the parametric index A' seems to track the long term trend in the nonparametric index A*. Both approaches then seem to isolate relative price effects similarly. While the nonparametric approach attributes the residual all to technical change, the parametric approach takes into account the possibility of random noise in the empirical relationship. To illustrate how the parametric approach smoothens the behavior of the index of technical progress, the residuals1 for the nine equations of the parametric model for sector I are plotted against the graphs of the technical progress indices A* and A1 in figures 17.29-17.37. We choose sector I, the resources sector, because it displays the most volatility in the nonparametric measure A1, and in the profit function estimation had the most computational difficulties.2 The magnitude of A f (or e*) may be due partly to allocative inefficiency; there has been a much rapid rise in prices in this sector relative to the rest of the economy and the violation indices £* may reflect problems in adjusting to price changes. The equations correspond to 'The residuals are the differences between actual and predicted values of output supply or input demand, not unit scale output supply or input demand. See Appendix D, equation ( D . l l ) for the specification of the stochastic disturbance term. 2Convergence of the nonlinear optimization routines was not obtained with models using one break point for the quadratic spline representation of technical progress; even with two break points, the sector I model required the most trials of different values for ti and <2 since false convergence was often encountered. 132 Chapter 17. A Comparison of Nonparametric and Parametric Estimates . 133 0.06 0 . 0 4 -I 0 .02 c 2 o .oo o c o « - 0 . 0 2 -- 0 . 0 4 -- 0 . 0 6 n o n p a r a m e t r i c p p r o r w t r t c I 1 I I 1960 1965 i i i 1970 year 1975 1980 Figure 17.25: Comparative behavior of A' and A' , sector I o> c o JC o "5 o « 0.06 0 .04 £ 0 .02 0 .00 _£ - 0 . 0 2 -- 0 . 0 4 -n o np o r a m • tj Ic poroirf trie 0.06 - j — i — i — i — i — | — i — i — i — i — | — r — i — i — i — | — i — \ — i — i — J 1960 1965 1970 1975 1980 year Figure 17.26: Comparative behavior of A ' and A' , sector II Chapter 17. A Comparison of Nonparametric and Parametric Estimates . 134 0.06 0.04 -£ 0.02-0.00 -0.02 --0.04 --0.06 no nporom ••flc porom«tfic T — I 1 1 | I 1960 1965 i I i i i i I i i i i 1970 1975 1980 year Figure 17.27: Comparative behavior of A' and A', sector III 0.06 0.04 -x £ 0.02 « o> £ I 0.00 o o i -0.02 o 0> -0.04 -poronrtrlc -0.06 -j—i—i i i—|—i—i—i i | i i—i—i—| i i i i 1960 1965 1970 1975 1980 year Figure 17.28: Comparative behavior of A' and A', sector IV Chapter 17. A Comparison of Nonparametric and Parametric Estimates 135 0 . 0 6 D "w Figure 17.29: Residual plot for equation 1, sector I output supply and input demand equations for the nine goods listed in table 16.7. Note that for each equation, the magnitude of the residuals from the parametric estimation is likely to be higher at time periods where the peaks and troughs of the nonparametric index A* occur or where the difference between the values of the indices A* and A* is higher. An advantage of the nonparametric approach is that no functional form is imposed on the data. Another is its low computational cost. The nonparametric index A' is based on the efficiency test violation indices £* which require merely a comparison of inequalities; even if linear programming problems have to be solved, these are computationally much easier to solve compared to the nonlinear optimization needed in profit or cost function estimation. A disadvantage of the nonparametric approach is its possible sensitivity to extreme observations and measurement error. Also, since no specification about stochastic disturbance terms is made, the nonparametric estimates have no statistical properties and hence cannot be subjected to the usual statistical hypothesis testing.3 *See Varian (1985) though on how a pseudo-statistical test can be performed to take into account measurement Chapter 17. A Comparison of Nonparametric and Parametric Estimates . 136 0 . 0 6 - 0 . 0 4 -- 0 . 0 6 n o n p o r o r n « t r l c p o > o m » t r t c 1960 1965 Figure 17.30: Residual plot for equation 2, sector I Figure 17.31: Residual plot for equation 3, sector I Chapter 17. A Comparison of Nonparametric and Parametric Estimates ... 137 Figure 17.32: Residual plot for equation 4, sector I X o C « O ) c o X o "5 o 'E x u 0 . 0 6 0 . 0 4 0 . 0 2 0 . 0 0 - 0 . 0 2 -- 0 . 0 4 -- 0 . 0 6 n o n p o r o m « 1 r l c x \ p o r a m * 1 r l c \ h X v X x \ :' — ' '. x; .' x ; • X i t r x 1960 1965 1970 year 1975 0 . 2 5 0 . 0 0 J - 0 . 2 5 1980 Figure 17.33: Residual plot for equation 5, sector I Chapter 17. A Comparison of Nonparametric and Parametric Estimates 138 0.06 TJ t> O) c o X u c 0.04-0.02 0.00 £ -0.02 --0.04 --0.06 nonporgmvtrfc \ parametric X \ (\ (^ X X X x x X \ \ V '; i X • i X \ ! X '. x; '; / X 1960 1965 1970 year 1975 o. 10 0.08 o TJ t> -0.08 -0 . 10 1980 Figure 17.34: Residual plot for equation 6, sector I a> T) _c o o> c o X u "5 o 0.06 0.04 -0.02 -0.00 -0.02 --0.04 -0.06 1960 nonp •> poro x oromvtric mvtrlc X x v : x x • ; '. .' X » t < i y x X y> \ ;x "\ / 0.04 o 0.00 -g "« h -0.01 -0.02 -0.03 -0.04 1965 1970 year 1975 1980 Figure 17.35: Residual plot for equation 7, sector I Chapter 17. A Comparison of Nonparametric and Parametric Estimates ... 139 0 . 0 6 V XI c 0 . 0 4 -0 . 0 2 c x 0 . 0 0 o "5 o J - 0 . 0 2 u - 0 . 0 4 -- 0 . 0 6 1960 Figure 17.36: Residual plot for equation 8, sector I Figure 17.37: Residual plot for equation 9, sector I Chapter 17. A Comparison of Nonparametric and Parametric Estimates 140 A two-stage procedure is suggested as a possible computational alternative to getting a parametric estimate of the index of technical progress A' . First, obtain the nonparametric indices A' and then fit a functional form such as (16.4) on the obtained A1 (after adjusting for the denominator in A'). The parametric estimation part would then involve a fewer number of parameters and hence would be less computationally burdensome. The foregoing discussion has focused on the techniques of measuring technical progress. As an aside, it may be worthwhile to comment on the technical regress findings for the resources sector. The Divisia indices of output growth and input growth for the resources sector are graphed in figure 17.38. To obtain these Divisia growth indices, a Divisia quantity index of the first four goods, the output goods of the production sector of the economy (with the quantities of intermediate inputs indexed negatively) and a Divisia quantity index of the primary inputs (goods 5-10) were first constructed per year. The Divisia growth index is then the ratio of the current year quantity index to the previous year quantity index. Note that, as seen in figure 17.38, the rate of input growth in the resources sector has been quite steady over the years while the output growth index displays much greater fluctuations. This is in marked contrast to the manufacturing sectors (II and III) and the services sector (IV) where the patterns of the Divisia indices of output and input growth follow each other quite closely. As mentioned earlier, the resources sector experienced a considerable output price increase in the 1970s. The low price elasticities for this sector, discussed in appendix D, may be indicative of a low degree of flexibility in this sector. However, this is at most a partial explanation since the divergence of the patterns of output and input growth is also observed in the 1960s. Other explanations have been offered in the literature. One is that to model producer behavior for the resources sector which is subject to output price volatility it may be necessary to incorporate intertemporal dynamics such as price expectations and uncertainty. Another is that the observed technological regress is due to resource depletion and an appropriate model for measuring multifactor productivity should correct for the quality of the resource stock (see errors. Chapter 17. A Comparison of Nonparametric and Parametric Estimates . 141 1 .2 1 . 1 5 -ou tpu t g r o w t h input g r o w t h b c > U 0.95 -1 .05 -1 . 1 -1 0.9 1960 1980 year Figure 17.38: Divisia indices of output and input growth in the resources sector (I) for example, Lasserre and Ouellette (1988)). Related to the resource depletion hypothesis, more of the other inputs may then be required to maintain production levels in this sector as the quality of the resource stock deteriorates. Also, exploration and development of new resources have to be undertaken which entails employment of nonproduction workers. There can also be a considerable lag between exploration and development, which may require large outlays of capital investment, and actual production. Because of the nature of the capital goods in this sector, it may be difficult to adjust factor inputs in the short run. Adjusting the capital data for capacity utilization may address this latter problem. Whether taking into account the phenomenon of resource depletion in measuring technical change in the resources sector requires shifting into a dynamic model of producer behavior and more than data adjustment necessitates further investigation. Part III Comparative Statics for a Small Open Economy in the Presence of Factor Price Rigidity r 142 Chapter 18 The Model 18.1 Introduction In the theory of consumer behavior under conditions of quantity constraints, it has been shown by Latham (1980) and Neary and Roberts (1980) that the Tobin-Houthakker conjecture that a reduction in the ration of one good will increase the consumption of unrationed substitutes and decrease the demand for unrationed complements may not hold. The derivatives of the rationed demand functions can be decomposed Slutsky-like into income and substitution effects. The income effect arises from the divergence between the individual's shadow or demand price for the rationed good (termed "virtual price" by Neary and Roberts) and its actual price. In the context of a trade-theoretic approach to production theory, an analogous question can be posed. In a small open economy, how do factor price rigidities affect resource allocation? In particular, how do changes in these fixed or rigid factor prices affect other flexible factor prices? The role of factor substitution in the response of flexible factor prices to exogenous changes in fixed factor prices has to be examined. It can be shown that since a fix-price factor acts as a negative output in a constrained gross national product (GNP) maximization problem, a Stolper-Samuelson type generalization can be obtained. If the domestic resources are used as inputs to production, as in the usual case treated in international trade theory, an increase in a fixed factor price leads to at least one flexible factor price decreasing. The response of flexible factor prices can be decomposed into a pure substitution term and a scale effect term. Furthermore, if we have local factor price equalization in the sense that factor prices are independent of changes in their endowments, then substitution possibilities among fix-price factors and flexible-price factors, even if they 143 Chapter 18. The Model 144 exist, will not affect the response of the flexible factor prices to changes in the fixed factor prices. In this case, there is solely a scale effect. In this study, the constrained GNP function is used to derive the comparative statics for the production sector of a small open economy with both fixed and flexible factor prices. An illustration of the model through a simple analysis of the effect of a devaluation with rigid wages in a two-sector specific-factors model is presented. We conclude with a discussion of how the model can be empirically estimated, the associated problems in its econometric implementation, and the possible policy implications that can be derived. 18.2 Model formulation, a constrained GNP problem In contrast to flex-price domestic resources with given endowments and inelastic supply, fix-price domestic resources will have a variable or elastic supply. In the standard full-employment general equilibrium models, factors with excess supply have a zero price. In the real world, these zero prices are rarely, if ever, observed. Hence, the framework to be used in the following model hopes to get around this theoretical deficiency and implicitly assumes that unemployment of domestic resources is due to factor price rigidities. Specifically, the fixed prices of these domestic resources are set above their market clearing levels.1 The constrained GNP maximization approach adopted here follows that of Neary (1985). In his comparative statics exercises, Neary compared a small open economy with some fix-price productive factors having perfectly elastic supply to a hypothetical unconstrained economy (with all factor prices flexible) with the endowments of factors set at their utilization levels in the constrained economy. Some important results are: the change in the employment levels 'For a domestic resource for which the economy has a given endowment, it is important to make this assump-tion so that its supply remains elastic. In contrast, imports used as intermediate inputs to production in a small open economy can be considered a fix-price factor but need not be constrained in this manner; its supply will always be elastic. The model to be used is in the genre of the "fixprice method" proposed by Hicks (1965). This formulation does not imply that "fixed prices" remain the same from period to period. In the sense that prices do not necessarily adjust in response to demand and supply disequilibrium in the short run, price determination becomes exogenous to the model. Signalling of excess demand or excess supply for price adjustment over the longer run is not precluded. As Hicks has pointed out, though the fixprice method tends to be macroeconomics-oriented, it is important to decipher the workings of individual markets. Chapter 18. The Model 145 of- the fix-price factors due to a marginal increase in their prices is the inverse of the change in those factor prices induced by a marginal change in their endowments in the corresponding flex-price economy; factor price rigidities lead to an increase in the economy's price-output responsiveness and a reduction in the responsiveness of the remaining flexible factor prices to changes in their endowments; and an equiproportionate decrease in the fixed factor prices will raise the value of gross national product (assuming factor immobility across countries). A more direct approach to the comparative statics exercises for this constrained GNP max-imization problem is the use of the fundamental matrix equation of production theory as devel-oped by Diewert and Woodland (1977) and Diewert (1982). This approach is more amenable to empirical implementation and does not require direct comparison between constrained and unconstrained economies. The model also allows joint production and intermediate inputs. The development of the model relates the behavior of individual firms or industries to the aggregate economy's maximization problem. Let there be M constant returns to scale sectors with industry scales z = (z\, z2,.. •, ZM)T > OM producing K net outputs of internationally traded goods y = (j/i, y2, • • •, VK)T w i th given world prices p = (p\,p2,.. -,PK)T "> ®K- There are Q + N domestic resources of which the first Q factors / = (Zi, l2,..., IQ)T have fixed prices r = (r 1 ; r 2 , . . . , TQ)T > OQ and the last N factors have initial endowments v = (vi, v2,..., VN)T with associated prices w = (w\,w2,..., wjsj)'r ~> ON- The exogenous variables are the world prices of internationally traded goods p, the fixed prices of the variable-supply factors r, and the endowments of the flex-price factors v. The endogenous variables are the industry scale variables z, the employment levels of the fix-price factors /, the prices of inelastically supplied factors w, and the net outputs of internationally traded goods y. Let the feasible set of net outputs for the mth sector be described by Tm = {zmCm : zm > 0} where Cm is a nonempty, closed, convex set of feasible net outputs when industry m operates at unit scale. At the industry level, the output and input prices are parametric. Let inputs be indexed with a negative sign and outputs with a positive sign. Thus, the unit scale profit Chapter 18. The Model 146 function for sector m can be defined as irm(p,r,w) = maXafi,c{p • c — r • b - w • a : (c, —b, —a) € Cm}, (18.1) where, for convenience, we have defined the vectors a, b and c to be nonnegative. The unit scale production possbilities set Cm can be recovered as Cm = {(c, -b, -a) : p-c-r-b-w-a < 7rm(p, r,u>) for every p > 0K, r > OQ, W > 0^ }. (18.2) If 7rm is differentiable, we obtain, by Hotellings's lemma, the optimal unit scale input-output coefficients: cm(p,r,w) = V p 7 r m (p,r, W ) , (18.3) -bm(p,r,w) = Vrirm(p,r,w), (18.4) -am(p,r,w) = VwTtm(p,r,w). (18.5) The industry profit function 7 r m is nondecreasing in p, nonincreasing in r and w, homogeneous of degre one, convex and continuous in (p, r, w) for p >^ 0A', r 0Q, W >> Ojv-Given world prices p, fixed factor prices r and resource endowments v, the economy's ag-gregate profit maximization problem is max(cm _bm,-arn)ec^,m=i,2,..,Mmax^>oM{2Zm=i(P- ~ r • bm)zm • £ 2 f = 1 a m * m < v} (18.6) = m a x( cm i_ km i_ am) e Cm i m = l i 2 , . . . , w [ m a x , { E m=i ( p -cm -r- bm)zm : » - E!=i«^m > ON, Z > 0M}] (18.7) = max (cm _ 6 m -a-)ec-,m=i,2,...,M[maxi>oAfminto>oJV{Em=i(P • cm - r • bm)zm + w(v-Z™=1a™zm)}} (18.8) = max2>oMminu;>oJV{Em=i7rm(P5r,w)zm + w • v} (18.9) = 7r(p,r,t>), (18.10) Chapter 18. The Model 147 where we have used the saddlepoint theorem for linear programs to transform the constrained maximization problem in (18.7) to the saddlepoint problem in (18.8), and definition (18.1) to go from (18.8) to (18.9). The pure dual form of the economy's profit maximization problem is 7r(p, r, v) = minw>oN{w • v : nm(p, r, v) < 0, m = 1,2, M). (18.11) The economy's profit maximization problem ir(p, r, v) can be considered the constrained (or restricted) GNP function. It yields the value of aggregate output minus the cost of fix-price factors.2 The value of w(p, r, v) would also be the sum of payments to flex-price domestic resources. In contrast, the usual unconstrained GNP function gives the value of output which equals the sum of all factor payments in the economy. The model, so far, has been discussed in the context of the Q fix-price factors as domestic resources. However, the elements of the variable factor supply vector I = (/i,/2,..., IQ)T can be imports used as inputs in the production of internationally traded goods. With the small country assumption, the prices of imports are exogenous. As well, price controls on nontraded output (produced goods) can be modeled by lumping these goods with internationally traded goods. Note that the small country assumption plays an important role in the dichotomy of goods and factors. For ease of interpretation in subsequent discussion, the vector / will be treated as fix-price domestic resources used as inputs in the production of internationally traded goods y. 18.3 First-order conditions Equation (18.9) is a concave programming problem. Suppose z* ]> 0M, W* > 0./v solve (18.9) when the exogenous variables are set at p*,r*,v*. The first-order conditions for an interior solution yield the zero-profit conditions and flex-price resource exhaustion conditions: xm(p*,r*,w*) == 0, m = 1 ,2 , . . . ,M; (18.12) Em=i^w^m(p*,r*,w*)zm + v* = 0N. (18.13) 2The constrained G N P function ir(p, r, v) corresponds to Neary's factor-price constrained revenue function where fix-price factor employment levels are interpreted as "negative outputs" sold at their fixed prices. Chapter 18. The Model 148 Define the matrices of optimal input-output coefficients using (18.3)-(18.5) as A = [au,a2*,...,aM*] = [ - V ^ p V , W% - V W T T 2 ( P * , r*, w"),..., - V„,7r M ( p * , r*, to*)]; (18.14) B EE [61*, 6 2 * , . . . , bM*] EE [ - V r 7 r V , r*, to'), - V r 7 r 2 ( p * , r% to*),..., - V r 7 r M ( p ' , r*, «;•)]; (18.15) C EE [ C 1 * , ^ , . . . , ^ ] — [ V p7r 1 ( p * , r * , « ; * ) , V p7r 2 ( p * , r * , 1 i ; * ) , . . . , V p7r M ( p * , r * , U ; * ) ] (18.16) where A, B and C are of dimensions NxM, QxM and KxM, respectively. The first-order conditions (18.12) and (18.13) can be rewritten in matrix form, with the use of equations (18.1) and (18.14)-(18.16), as p*TC-r*TB-w*TA = Ojf, (18.17) Az* = v*. (18.18) The economy's net output vector y and employment levels for fix-price factors / are given by y(p*,r*,v*) EE E ^ = A V p7r m ( p * , r * , « ; ( p * , r * , U * ) ) 2 m ( p * , r * , t ; * ) , (18.19) / ( p W ) EE - E ^ = 1 V r7r m ( p * , r * , u ; ( p * , r « , V * ) ) 2 m ( p * , r * , i ; * ) , (18.20) or in matrix notation, y* = Cz*, (18.21) /* = Bz*. (18.22) 18.4 Fundamental matrix equations If we assume that the unit profit functions wm are twice continuously differentiable, total differ-entiation of (18.12), (18.13) and (18.20) yields the following equations relating the endogenous Chapter 18. The Model 149 variables w, z and / to the exogenous variables v,p and r -SWW A ONXQ AT OMXM OMXQ -SRW B —IQ dw dz -dl IN S OMXN CT —BT OQxN -rp dv dp dr (18.23) (18.24) where are the aggregate substitution matrices. Since the industry unit scale profit functions are convex in (p, r, w), the symmetric matrices Spp, Srr and Sww are positive semidefinite. The linear homogeneity of nm implies the following: SppP* + Sprr* + Spww* = 0K; (18.25) Srpp* + Srrr* + Srww* = OQ; (18.26) SwpP* + Swrr* + Swww* = 0N. (18.27) To obtain the comparative static responses of w, z and / to changes in v, p and r, we apply the implicit function theorem. The inverse of the (TV + M + (J)-dimension square matrix in the left hand side of (18.23) is required. Define the submatrix G as G = (18.28) AT OMXM Assume that the NxM matrix A has rank M < N, that is, the number of flex-price resources N is at least as great as the number of industries M . 3 Furthermore, assume that the matrix SWw + AAT is positive definite.4 The positive semidefiniteness of Sww, the rank condition on A, and the positive definiteness of SWw + AAT are necessary and sufficient conditions for the existence of the inverse of G. Define this matrix to be ^ WW A -1 D E AT OMXM E T F (18.29) 3 Since the rank of a matrix is the number of linearly independent rows or columns in that matrix, the rank condition on A, rank(A) = Af, requires distinct industry or sectoral classification. 4 The positive definiteness of Sww + AAT implies that Sww + AAT has a positive determinant and hence is of rank N. Since AAT is of rank M, the flex-price factor substiution matrix Sww has rank at least equal to N — M. Chapter 18. The Model 150 The properties of G~x are developed in the appendix of the work by Diewert and Wood-land (1977). If we denote the Qx(N + M) matrix H as H = [SRW, —B], then the inverse of the left hand side matrix in (18.23) can be expressed as G 0(TV+M)IQ -H -IQ -1 t-1 o (N+M)xQ -HG-1 -IQ (18.30) Hence, we obtain the comparative static response matrix Vvw VpU; V r u) V „ z V p z V r z vv/ vp/ vrz G 1 °(7V+M)xQ -HG-1 -IQ D ET Stup S ujr IN ®MXN CT —BT OQXN SRP SR E F ONxQ QMxQ -(SRWD — BET) -(SRWE - BF) -IQ D DSWP + ECT ET Jwp ^wr (18.31) -(SRWD - BET) ETSWP + FCT -[{SRWD - BET)SWP +(SRWE - BF)CT +SRP] To interpret the above equations, it is helpful to list down the conditions satisfied by the matrices D, E and F that appear in (18.29). As given in the appendix of Diewert and Wood-land (1977), these are: IN OMXN CT —BT 0QXN $rp $rr DSWR - EBT ETSWR - FBT -{SRWD - BET)SWT +(SRWE - BF)BT -SWWD + AET = IN ATD = OMXN (18.32) (18.33) Chapter 18. The Model 151 ATE = IM (18.34) -SWWE + AF = 0NxM (18-35) D = DT (18.36) F = FT (18.37) F = ETSWWE (18.38) where D is a negative semidefinite matrix and F is a positive semidefinite matrix. Chapter 19 Comparative Static Results 1. The matrices V„tu = D and V„z = ET have the usual interpretation as given in Diewert and Woodland (1977) with N as the number of flex-price domestic resources. Local factor price equalization obtains when the number of industries is equal to the number of flex-price domestic resources, that is, M = N. In this case, the matrix A is square of dimension N and rank N, its inverse A - 1 exists and from (18.33), D becomes a zero matrix. This yields X?vw = [ ^ J = O^xN which means flexible factor prices are independent of small changes in endowments.1 Also, if M = N, the domestic resources exhaustion equation (18.18) can be rewritten as z* = A_1v*, meaning the industry scales are completely determined by the input coefficient matrix A and the initial endowment vector v*. In this case, the fix-price factors, having an elastic supply, do not seem to impose constraints on the scale of production. Hence, V„z = = A - 1 . 2. Flexible factor prices w are now homogeneous of degree one in (p, r). The homogeneity property of w(p, r, v) can be seen as follows: [Vpw]p* + [Vrw] r* = (DSwp + ECT)p* + {DSwr - EBT)r* from (18.31) = D{Swpp* + Swrr*) + E(CTp* - BTr*) = -DSwww* + EATw* using (18.27) and (18.17) = (—DSWW + EAT)w* = INw* using (18.32) = w*. 'Generally, even if M ^ N, it has been shown by Neary (1985) that the response of flexible factor prices to changes in endowments is less negative relative to the unconstrained case where all domestic resources have flexible prices. In this sense, the economy has a greater tendency towards "local" factor price equalization. 152 Chapter 19. Comparative Static Results 153 We then have [Vpw]p* + [Vrw]r* = w*. (19.1) For A > 0, w(Xp, Ar, v) = Xw(p,r,v). A proportional increase in internationally traded goods prices (such as through currency depreciation) does not necessarily lead to a proportional increase in flexible factor prices if there are fix-price factors. With respect to the endowment vector v, the flexible factor prices still retain the property of homogeneity of degree zero, that is, for A > 0, w(p, r,Xv) = w(p, r, v). 3. Industry scale functions z(p, r, v) are no longer homogeneous of degree zero in world prices p. We have [VPz]p* dp p* = - [V r z]r* (19.2) Equivalently, industry scales are homogeneous of degree zero in (p, r). For A > 0, z(Ap, Ar, v) = z(p, r, v). Intuitively, since a proportional change in prices p and r lead to a proportional change in w (as given in result 2), relative prices in the economy do not change. Hence, there is no real effect and industry scales remain unchanged. Industry scales remain homogeneous of degree one in v, that is, z(p, r, Xv) = \z(p, r, v) for A > 0. Equivalently, [V„z]u* = v* = z*. Since the fix-price factors are in elastic supply, the industry scales or production levels are determined or limited by the exhaustion of inelastically supplied (flex-price and fully employed) factors. 4. Rybczynski-type and Stolper-Samuelson type generalizations still obtain in the analysis of V„z and Vpw. 5. a) The response of flexible factor prices to marginal changes in the fixed factor prices is given by 'dw' 8T = DSwr - EB1 dw' \dZ] dv. BT using (18.31). (19.3) A Slutsky-type decomposition of the total effect can be made: the first term measures the pure substitution effect between fix-price factors and flex-price factors, and the second term Chapter 19. Comparative Static Results 154 measures the scale effect. The scale effect can be explained as follows: changing, say, a fixed factor price leads to a change in the employment level of that elastically supplied factor such that its value of marginal product equals its new fixed price; therefore, the vector of domestic resources available for productive activities changes. The first term in the right hand side of (19.3) will be zero if either SWR — ONXQ or D = ONXN-In the first case, there is no substitution between fix-price factors and flex-price factors or equivalently, there is no change induced in the input coefficient matrix A by a change in fixed factor prices. In the second case, the number of industries is equal to the number of flex-price factors (M — TV), the flexible factor prices w are completely determined by world output prices p and local factor price equalization holds; changes in the matrix A will have no effect on prices w. In either of these cases, the total effect will be due purely to changes in scale. b) Stolper-Samuelson type generalizations obtain with the fixed factor prices rq acting like a world output price pk and the fix-price factor behaving like a "negative" internationally traded good. If we premultiply equation (19.3) by AT, we obtain 'dw' = ATDSwr - ATEBT - -BT using (18.33) and (18.34) (19.4) Transposing, we have dw Or -B. (19.5) If the matrix A is nonnegative and B contains at least one positive element in the gth row, that is, all flex-price domestic resources are inputs or nonproduced and at least one industry uses T the gth fix-price factor as an input, then the qth row of dw dr dw\ dw2 dWN drq ' drq ' ' drq (19.6) has at least one negative element. If rq increases, at least one flexible factor price will decrease. A weakness of the fundamental matrix approach, probably due to its generality, is that it does not yield more specific results on which flex-price factor will gain or lose. Let us try the Chapter 19. Comparative Static Results 155 special case of K = M = 2 internationally traded goods (assuming a single output for each industry), N — 2 flex-price domestic resources and Q = 1 fix-price domestic resource. Since we have M = N, the matrix D is a zero matrix and the response of flexible factor prices to marginal changes in fixed factor prices would be solely a scale effect. Substitution possibilities among domestic resources will not affect w\ hence, we would suspect relative factor intensities to determine the price responses. For this special case, equation (19.3) reduces to Ir1 dtV2 = -EBT since D = 0 2 x 2 = —(A~1)TBT using (18.34) = -(BA~l)T where B = [b\,b\} and A = ai at at, A where A =|A|= a\a?, — a\a\. (19.7) a\b\ — a^] a\b\ - a\b\ Let us assume that the matrices A and B are positive, that is, all domestic resources are used as inputs. From the previous analysis, at least one flexible factor price will decrease. In (19.7), we can sign the following quantities: , U 2 A^O if a\>a\ a f < a 2 jb\>0 if a}2>b\ al<b\ (19.8) \bj>0 if b\ya\ 6 2 < a 2 ' The six possible factor intensity configurations and the resulting effects on the flexible factor prices are given in table 19.9. An examination of the table indicates that the factor that varies the most in its input coefficient ratio from that of the fix-price factor tends to gain with an increase in the fixed factor price. Vice-versa, the more similar the flex-price factor in the manner it is allocated to the different productive activities (as measured by relative factor intensities), the greater the tendency for this factor to decrease in price with an increase in the fixed factor price. Chapter 19. Comparative Static Results 156 factor intensities A E E | A | ^ (= - f t ) + + -nL > b± > s i «? b? «S + - -«! °? b? - - + a l b? a l - - -+ - + ^ ^ al - + -Table 19.9: Factor intensity configurations and resulting effects on the flexible factor prices c) From equations (19.3) and (18.31), we obtain the following symmetry condition: V r w = DSwr - EBT = [SrwD - BET)T where Srw = STr = - [V„Z] T . (19.9) This implies — — or the change in the nth flexible factor price due to a marginal increase in the qth. fixed factor price is equal in magnitude but opposite in direction to the change in the employment level of the gth fix-price factor due to a marginal increase in the endowment of the nth flex-price resource. Suppose we redefine factor complementarity and substitutability in terms of V„Z: the nth flex-price factor and qth fix-price factor are complements if J^2- > 0, that is, an increase in the endowment of the nth flex-price factor induces a higher employment level for the fix-price factor; otherwise, if ^ < 0, then they are substitutes. In this context, we can get a result that is analogous to the Tobin-Houthakker conjecture: an exogenous increase in the fixed price of a factor will reduce the price of its complements and increase the price of its substitutes.2 2This interpretation is merely another description of factor price behavior and begs the question of explaining Chapter 19. Comparative Static Results 157 Note that the symmetry condition in equation (19.9) holds in general whereas the cor-responding symmetry between flex-price factor and output variables as described in equa-tion (19.10) below holds only under more restrictive conditions. The possibility of intermediate inputs and joint production complicates the relationship. The presence of intermediate inputs is innocuous if the local factor price equalization condition holds, that is, if factor prices are independent of changes in endowments; this nullifies any effect of input substitution on factor prices. Specifically, we obtain the symmetry condition Vpw = [Vvz]T = [Vvy]T (19.10) only if C — IK and K = M (single output technologies)3 and either SWP = 0/VxA' (no interme-diate inputs) or M = N (local factor price equalization) hold. 6. The analysis of the matrix V„Z = —(SRWD — BET) yields a Rybczynski-type result: if A is nonnegative and B contains at least one positive element in the gth row, that is, all flex-price resources are inputs or nonproduced and at least one industry uses the gth fix-price factor, then the gth row of V„Z: r d l " 0 l " --- ^ 1 (19.11) .dvi' dv2' ' dvfj. has at least one positive element. In this case, there exists at least one flex-price factor whose increase in endowment leads to greater utilization of the variable supply factor q.4 This result is obtained by substituting (19.9) into (19.5) to yield dv A = B. (19.12) Again, the total effect can be decomposed Slutsky-like into two sources: a pure substitution effect and a scale effect. In the context of marginal changes in the endowment vector v, the condition SRW = 0QXJV implies that the B matrix does not change with respect to changes in the mechanism behind price changes. What is of interest are the underlying determinants of factor price behavior, as we have attempted to identify in isolating substitution and scale effects. 3With single output technologies, that is C = IK and K = M, we have the equality of the scale and output variables: zm = ym-4If each fix-price factor is used by at least one industry, then in the modified sense of substitution as defined in result 5c, each fix-price factor has at least one complement among the flex-price factors. Chapter 19. Comparative Static Results 158 flexible prices w; the scale effect is due to a change in the vector of domestic resources available for productive activities. For the special case discussed in result 5b where K = M = N = 2 and Q — 1, the signs of the elements of V„Z: v / = [—1 — \£h. 2h] = Lf^i ci9 " ~ [dv\ [dvi''dv2i I 9ri ' dr2 J can be obtained from the previous table by using the symmetry condition (19.9). The total effect is solely a scale effect. Note that will be negative (substitutes in the modified sense of result 5c) if their associated factor intensity ratios are not adjacent to each other (in the inequalities); otherwise, it is positive. 7. The response of industry scales z to changes in the fixed factor prices r is given by dz = E Swr — FB . (19.14) This equation can again be interpreted as a Slutsky-type decomposition. The first term accounts for the effect of substitution between fix-price factors and flex-price factors. The second term corresponds to the scale effect arising from a change in the utilization of the fix-price factors. If there is no substitution between fix-price factors and flex-price factors (SWR — ONXQ), as in the case when fix-price factors enter production in fixed coefficients, then where F is an MxM positive semi definite matrix. Premultiplying by B, we obtain 'dz B dr = -BFBT, a negative semidefinite matrix which implies .dZm EM Ln m=l°q EM in m = l 0 i drq  dz„ dr; < 0, q = 1,2,...,Q and = -FBT (19.15) (19.16) (19.17) 8. The change in the employment levels of fix-price factors due to a change in world prices p is given by VPZ di dp. -(SrwD - BET)Swp - (SrwE - BF)C2 III 11 ,W ^ ^ — - Srp -f-'rp \dl] \dz] dv. dr. CT using (18.31) (19.18) ] Chapter 19. Comparative Static Results 159 The first two terms on the right hand side measure the pure substitution effects: -Srp, the direct effect due to substitution between internationally traded goods and fix-price factors and [j^ j Swp, the indirect effect due to substitution between internationally traded goods and flex-price factors. The third term can be interpreted as a scale effect. There is no clear-cut result on the definiteness property of the matrix VPZ. Equation (19.18) can be rewritten as VPZ — - S r p — [Srw, ~B] D E Q uwp E T F CT (19,19) Let us consider the usual case of no joint production and no intermediate inputs: M = K, C - IK, S w p = OiVxA', Srp = OQXK- Equation (19.18) reduces to VPZ = -(SrwE-BF) = -[V rz]T using (19.14) — ~[^ry]T since y = z when M - K and C = IK- (19.20) drn (19.21) In this case, we obtain the symmetry condition dig _ dzk dpk drq If we further assume S r w = ®QxN as discussed in result 7, postmultiplication of (19.20) by BT yields dyk di .dp. BT = BFBT = - B dy dr (19.22) a positive semidefinite matrix. 9. The response of fix-price factor employment to changes in their price is given by dr VJ = dr = -(SrwD - BET)Swr + (SrwE - BF)BT - Sr = S r r + = —Srr ~\~ \dn .dv. C ^>wr + dz] dr. 'dw Swr + \d z' . dr dr_ Srw 'dw' . dr. id .d BT using (18.31) BT using (19.9) (19.23) Chapter 19. Comparative Static Results 160 The total effect can be decomposed into a direct substitution effect among fix-price factors, an indirect substitution effect between flex-price factor and fix-price factors and a scale effect. Equation (19.23) can be rewritten as V rZ = -[SrwD - BET, IQ] WW *~>WT (19.24) DSwr - EBT IQ where the symmetric matrix of aggregate substitution terms in the right hand side is positive semidefinite. Hence, the matrix V„/ is negative semidefinite. This implies that -gf- < 0; an increase in the fixed price of a factor reduces or at most does not change the employment level of that factor. If SWR — ONXQ as discussed in result 7, then equation (19.23) reduces to Vr7 = -SRR- BFB (19.25) where both matrices, SRR and BFBT, are positive semidefinite. It can also be shown that / is homogeneous of degree zero in (p, r), that is, /(Ap, Ar, v) = l(p,r,v) for A > 0. Hence, V p/p* + V r / r * = OQ. 10. The usual comparative static results apply to V pt/ and Vvy. In the response of output to changes in fixed factor prices, we get V r j / = dy dw dz dr (19.26) obtained by differentiating equation (18.19). Equation (19.26) can be rewritten as yry = Spr + Spw(DSwr - EBT) + C(ETSwr - FBT). (19.27) A definiteness property for V r y cannot be established. The terms on the right hand side of equation (19.27) can be interpreted accordingly. 11. The homogeneity properties of w,z,l and y and the definiteness properties of V p y and V r / can also be established from the definition and convexity properties of the constrained GNP function w(p, r, v) as given in equations (18.6) and (18.11). Chapter 20 An Illustration: Devaluation with Rigid Wages in a Specific-Factors Model In recent years, there has been a resurgence in interest on the type of labor unemployment generated by real wage rigidity rather than by Keynesian aggregate demand deficiency. Cur-rency devaluation is an alternative employment stimulation policy. The effects of a devaluation are examined in the context of a two-sector specific factors model.1 It has been argued by Neary (1978) that capital sectoral specificity need not necessarily have only a short run in-terpretation. In the medium run or long run, capital equipment is generally not physically transferred from one sector to another and capital stock adjustments take the form of varying investments relative to depreciation. Suppose we have two single-output sectors (M = K = 2) with output y\ and y2 and associated world prices p\ and p2. Each sector uses a sector-specific input, say capital; imports and labor as fix-price factors; and output from the other sector as an intermediate input. Let l\ and l2 be the total quantity of imports and labor, respectively, utilized by the economy; let r\ and r2 be their exogenous prices. The endowments of sector-specific capital are v\ for sector 1 and v2 for sector 2; their endogenous prices are w\ and w2, respectively. The configurations of the unit input-output coefficient matrices A, B and C and of the substitution matrix Sww are as follow: A = a\ 0 %' 1 -c\ ; B = ; c = . 0 a 2 b\ b2 - c 1 1 'Early analysis of the specific-factors model were done by Samuelson (1971), Jones (1971), Mayer (1974), Mussa (1974) and Neary (1978). Samuelson (1971) shows that free trade will tend to partial, though not complete, factor price equalization between countries. With factor specificity, anti-Rybczynski and anti-Stolper-Samuelson outcomes explain factor owners' responses to trade policies (such as workers supporting protection for capital-intensive industries and opposing liberal immigration policies) as short-run reactions. 161 Chapter 20. An Illustration: Devaluation with Rigid Wages in a Specific-Factors Model 162 c = J Wl Wl 0 5, (20.1) — = A > 0; — 0; dv-i = dv2 — 0. (20.2) where a] > 0, bj > 0, c3 > 0, = 1,2) are scalars. A devaluation can be modelled as a proportional increase in output prices (pi,P2) a n d import price (ri) such that dpi _ dp2 Pi P2 We assume that wages do not change and the capital stocks remain constant. The effects of a devaluation on capital returns, output, employment, imports and national income are discussed next. 1. Returns to sector specific capital. The changes in capital returns Wi and w2 due to the devaluation are given by d w i = (P^-P2c\-Tlb\\x d w 2 = (p2-Pic\-nbl\x = r2b\ -f- w\a\ r2b\ + w2a\ 'r2b\ + ti>i J A > Xwi > 0; (20.3) T-^- + w2) A > Xw2 > 0. \ 2 / \ • 2 / \ 2 / In full employment models, a devaluation has a neutral effect since domestic factor prices increase proportionally. In contrast, we now obtain a more than proportional increase in capital returns in both sectors since the specific factors absorb the gain that could have accrued to labor under full employment. Hence, the higher the value-added content in a sector, the greater the gains to capital owners in that sector arising from the devaluation. Export promotion by currency devaluation will not be as beneficial to capital owners in sectors with low value added, such as those in assembly of completely-knocked-down parts. 2. Output. The nonproportional increase in capital returns (w\ and w2) makes possible the output effects given by dyi 'SWlr7r2 + S, WlWl 'P\ ~ P2c\ - rxb\ - wia\ r2b\ Chapter 20. An Illustration: Devaluation with Rigid Wages in a Specific-Factors Model 163 (20.4) dyi r2 + S 102 «J2 Su>2 T2 r 2 "t" ' p2 ~P\c\- rxb\ - w2a\ W2W2 By the positive semidefiniteness of the substitution matrix Sww, its elements SWlWl and SW2W2 are nonnegative. The output of a particular sector will increase if labor and capital in that sector are substitutes; that is, Smr2 < 0 for sector 1 and SW2r2 < 0 for sector 2. Labor and capital are not necessarily substitutes in the presence of intermediate inputs and imports. However, it is still possible to obtain a positive output effect if the degree of complementarity between labor and capital is sufficiently small. Note that the output effects in equations (20.4) can be rewritten as d dyi = dy2 = b\ dwi \a\ d dwo b\ (Ar 2 ) ; (Ar 2 ) . (20.5) Hence, as long as the labor-capital ratio in a sector increases, the devaluation leads to an expansion of that sector. The reason why the change in output can be expressed as a function of wages r2 can be seen in the following discussion. 3. Employment. From the comparative statics exercise (result 9), we know that the employment levels of fix-price factors / are homogeneous of degree zero in the exogenous prices (p, r); that is, [Vp/]p* + [VrZ] r* — OQ. For our example here, we then have (20.6) Together with the negative semidefiniteness of the matrix V rZ, the homogeneity property can be used to derive the change in employment dl2: dpi ap2 Pl + dr\ 0 dh . 5pi 9p2 . .  P2 . OTl OT2 r2 0 dU dh, . dl2 , dl2, •5—dpi + -^—dp2 + -K—drx dpi dp2 dri Chapter 20. An Illustration: Devaluation with Rigid Wages in a Specific-Factors Model 164 = A dl2 dl2 , dpi dp2 dr2 (Ar 2) > 0. (20.7) Hence, labor employment increases or at the least does not change with a devaluation. The proportional change in employment depends on the wage elasticity of labor demand, as can be seen in the expression f - " (g£) <»•« The above derivation also shows that a devaluation of A > 0 is equivalent to a proportional decrease in the wage rate of A: dv2 dpi = dp2 = drx = 0; = — A; dv\ = dv2 = 0. T2 (20.9) This suggests how devaluation is a policy instrument for reducing real wages and increasing employment. Empirical estimation of the wage elasticity of labor demand will indicate whether exchange rate adjustments are an effective employment policy tool. 4. Imports. Similarly, the change in quantity of imports due to the devaluation is given by where dU (20.10) dh dr2 ai + $wi r i Swi wi i b\\ b\ l + I Sw2ri bl\ b\ JU>2W2 2 (20.11) A priori, we cannot sign the direction of change in imports dh- It will be opposite in sign to the wage elasticity of imports. Let us analyze the change in imports due to a wage rate change (f^ ) • The import response can be decomposed into three components of which the latter two can be attributed to changes in capital returns (K>I,U>2) induced by the wage rate change. Chapter 20. An Illustration: Devaluation with Rigid Wages in a Specific-Factors Model 165 The first component as measured by the term — Srir2 is the direct effect due to the substi-b1 b2 tution between labor and imports. The second component Sw,r2-\; + SW2r2-^ is the indirect effect due to substitution between labor and capital in the two sectors which leads to a change ^j-, . The third component is the scale effect due to a change in labor employment. This last component can be reexpressed as c _ c 1±\1± + ( Q _ c M l M •JwiTi Ju)iu)i 1 I 1 i \ Jw2ri Ju>2W2 2 I „2 \ " 2 / "2 )a\ ( b\\ bl + 'bt d\V2 [a2. bz2. (20.12) Let us assume that labor and imports are substitutes (5 r i r 2 < 0) and labor and capital are substitutes (SWir2 < 0 and SW2r2 < 0, which imply both sectors expand). Then the first component in equation (20.11) is positive while the other two components are negative. Hence the direct substitution effect has to predominate the combined indirect substitution and scale effects for imports to increase with an increase in wages (or decrease with a devaluation). The preceding discussion on the mechanism behind the import response to wage changes is an illustration of the LeChatelier principle (see, for example, Samuelson (I960)). The wage elasticity of imports in the case of rigid wages (unemployment) will be different from that in a flexible-wage (full employment) scenario. The change induced in capital returns in the rigid wage case underlies the difference. 5. National income. Ignoring international factor payments and abstracting from the absence of indirect taxes, we can express the gross national (or domestic) product GNP net of capital depreciation as the sum of domestic factor payments or national income, NI: NI = r2l2 + wivi + w2v2. (20.13) The change in national income due to the devaluation is given by d(NI) - r2dl2 + Vidwi + v2dw2 > 0 (20.14) which is positive. Hence, national income increases because employment increases or at the Chapter 20. An Illustration: Devaluation with Rigid Wages in a Specific-Factors Model 166 least does not fall (dl2 > 0, equation (20.7)) and capital returns increase (dw\ > 0, dw2 > 0, equation (20.3)). The devaluation resulting in an increase in national income is not surprising. Initially, we have a rigid wage above the market clearing level. The devaluation reduces the degree of distortion in the factor markets caused by wage rigidity. It is equivalent to a reduction in the real wage. Even if the wage elasticity of labor demand is low (and hence, the devaluation does not lead to a significant increase in employment), gains still accrue to capital owners, particularly in sectors with high value added. Empirical estimation will yield information on the magnitude of the income distribution repercussions of devaluation. We can exploit the symmetry condition between prices and quantities of fix-price and flex-price factors {js^ = equation (19.9)) to predict the employment effects of investment in our specific-factors model. We have shown that a devaluation is equivalent to a decrease in wages and that it leads to increases in returns to capital in both sectors. Hence, we have for any sector m, Q^JJL < 0; that is, a wage increase will decrease capital returns. By the symmetry condition, we have ^j 2 - > 0; that is, capital investment will increase labor employment. In terms of magnitudes, the greater the gains of a devaluation to capital owners (or equivalently, the greater the loss to them due to a wage increase and which is likely to occur in higher value-added sectors), the greater the employment effects of investment. The specific-factors model in this section can be generalized to any M = K > 2 sectors and the comparative statics results of a devaluation obtained will remain robust. A two-sector model was used to highlight sectoral impacts of a devaluation. Note that the substitutability between intermediate inputs (other than imports) have not been so far mentioned in the discussion of our specific-factors model. The model has been rigged, by having an equal number of sectors and flex-price factors (capital), to yield "local" factor price equalization; that is, competitive prices of domestic primary factors are independent of changes in their endowments. Technically, we would have M = N, the number of sectors equal to the number of flex-price factors. As noted in our comparative statics result 1, we have the matrix Vvw = D = ONXN- Hence, the Chapter 20. An Illustration: Devaluation with Rigid Wages in a Specific-Factors Model 167 substitutability between intermediate goods and flex-price factors would not affect the factor prices w. In equation (18.31), the first term in the expression.Vpw; = DSwp + ECT is a zero matrix. The mechanism described in this section can be considered the impact effect of a devaluation on resource allocation. There would be a general tendency towards these effects as long as there is a less than proportionate increase in wages. In reality, a devaluation can trigger domestic inflationary pressures and the final outcome with respect to employment of a devaluation will depend on the sensitivity and speed of real wage adjustments to exchange rate changes. Chapter 21 Towards Econometric Implementation 21.1 Policy implications Aside from making explicit the mechanism by which exogenous shocks, such as changes in output and import prices or increases in capital stock, generate general equilibrium effects on resource allocation, econometric implementation of the model yields empirical information useful to policy makers. With wage rigidity, one important estimate that can be derived is the wage elasticity of labor demand; if this turns out to be small, then wage flexibility policies may not be very effective in promoting employment. A substantial decrease in real wages would be needed to induce greater employment. In this case, devaluation as an employment-stimulating policy tool would not make much difference though it can still have income distribution repercussions. Also, a low elasticity of employment with respect to real wages is favorable to labor unions which can extract wage premiums at little cost in terms of employment. If unemployment is due to real wage rigidity, then labor employment will not be a function solely of wages. The relative importance of the determinants of employment such as wages, out-i put and import prices and capital stock changes can be evaluated. With the decline of the dollar in the United States from 1985 to 1987, there has been a revival of and increased employment in the manufacturing industries, particularly the export sectors. Some have conjectured that it was not as much the weak dollar but the capital stock readjustments (accelerated scrapping, modernization and retooling of the factories, etc.) that took place which contributed more to the competitive turnaround of the U.S. economy. Since imports are inputs to production as well, changes in their prices will affect the compet-itive returns to domestic primary factors. With the small open economy assumption, imports 168 Chapter 21. Towards Econometric Implementation 169 have exogenous prices; hence, the behavior of (endogenous) prices of domestic primary factors in response to changes in import prices is also described by our "revised" Tobin-Houthakker conjecture (the decomposition of the response to substitution and scale effects and the clarifi-cation of the "substitutability" and "complementarity" definitions). The relationship between imports and returns to durables or capital, and imports and labor employment can be examined. In the period 1961-1980, the relative share in value terms of imports in production increased while that of labor decreased in Canada. Information on the price elasticities of output can be useful in studying the effects of free trade on Canada. A problem is estimating the magnitude of the expected decrease in prices of traded goods arising from a tariff reduction; hence, empirical estimation of resultant output changes is not a straighforward exercise. Actual prices reflect not only explicit tariffs and do-mestic taxes but also the price effect of other market distortions, trade (quotas, standards and content requirements, etc.) as well as nontrade (marketing boards, interprovincial trade restric-tions, etc.). For example, Canadian food processors say that free trade will not significantly decrease the prices of eggs, poultry, milk and the like which are subject to marketing boards. Some of them argue that currency devaluation will be more beneficial to them than a reduction of tariffs, since under the latter, their competitivenesss vis-a-vis the U.S. market is hampered by the relatively higher prices of their input commodities or raw materials. Therefore, the price effect of a tariff reduction cannot be assessed independently of other nontariff distortions in the economy. Another complicating factor is the nonuniformity in application of tariffs. In Canada, tariff rates are governed by several multilateral trade agreements; hence, for the same class of goods different rates may apply depending on the country of origin of the imports.1 A weakness of the modeling approach taken in this study is the naive way of introducing factor price rigidity. Though there can be varying causes, models endogenizing price rigidity for a domestic resource is a possible area of future research. Also, social welfare considerations, other than factor income distribution, are ignored. Nevertheless, this study can generate partial 1 Rates may differ according to most-favored nation tariff, general preferential tariff, British preferential tariff, U.K. (United Kingdom) and Iceland tariff, etc. Chapter 21. Towards Econometric Implementation 170 information useful in assessing equity-efficiency tradeoffs of government policies impinging on the production sector of the economy. 21.2 The shadow price of a fix-price factor The theoretical framework discussed assumes that the fix-price production factors have elastic supplies. Generally, in disequilibrium, there can be excess demand for these factors. Labor shortages can occur. Quotas or foreign exchange controls can limit the inflow of imported raw materials. In this case, the production side of the economy is rationed and gives rise to the divergence of the shadow prices of these factors from their market prices (or fixed prices). The model can be reformulated to account for this possibility. Given an endowment I of the fix-price factors,2 the economy's aggregate profit maximization problem becomes max ( c m _bm _ am ) e Cm i m = 1 > 2 i . . . t M m a x 2 > o M{Em=i ( P -cm - r- bm)zm : L i L i ^ m < v, Y;%=ibmzm < 1} = max ( c m _bm,_a".)eC".,m=i,2,...,M[max2{^=1(p • cm - r • bm)zm : V ~ £ n f = l « m Z m > 0N, I- Em=l&m-*m > 0Q, Z > 0M}} = max( cm i_ 6m i_ am) e Cm i m = 1 ) 2 i... )M [ m a x 2>o M niin^>o^, < T>o Q{Z)m=i(P • c m - r • bm)zm + w-(v- ^ = 1 a m z m ) + a-(l- Y%=ibmzm)}} using the saddlepoint theorem for linear programs, = max 2>oMmin t u> 0 w , ( T>o Q{Em=i 7 r m(P 5 r. w) 2m + w • v + a • 1} using definition (18.1), = f(p,r,v,l) = min„,>oN,<T>oQ {w-v + a -l : irm(p,r + o,w) < 0, m = 1,2,..., M}. 2For labor, the assumption of fixed endowment is a simplification. This assumes that each worker offers a fixed number of hours of work. It may be argued that the economy is not endowed with a certain amount of labor but has an endogenous supply of this factor. That is, the workers have given time endowments and preferences defined over leisure and consumption goods. Hence, a more complete treatment of labor would simultaneously consider the production side and consumption side of the economy. Chapter 21. Towards Econometric Implementation 171 This maximization problem TT(p, r, u, /) explicitly incorporates the constraint on the fix-price factors. Corresponding to these constraints, a new vector of Lagrange multipliers o > OQ is introduced. These Lagrange multipliers cr can be thought of as the wedge between the shadow price or virtual price of the fix-price factors and their market (or fixed) prices. This inter-pretation becomes clearer upon examination of the first-order conditions and complementary slackness conditions. Suppose z* >• O M , W* 3> OAT, O* > OQ solve the above problem when the exogenous variables are set at p*, r*, v*, P. Then the following conditions hold: 7r m (p* ,r* + o*,w*) = 0, m = 1,2, . . . , M ; Em=i V r-7r m (p*, r* + a*, w*)z*m + P > OQ where r' = r + a; ^•EliV r-x r a(j>',^ + ^ »*)4 + n = o. Note that the nonnegativity of the Lagrange multipliers cr, the feasibility constraint and comple-mentary condition on the fix-price factors imply that for each of these factors q, q — 1,2,..., Q, M d 7 r " ( p * , 7 - * + < T * , 0 2^m=l £ r / Zm 'q 9 = 0. By Hotelling's lemma, a - d r , q is the optimal unit scale input requirement of fix-price factor q in sector m when faced with the price r* + o*. The above condition can be rewritten as < ft - £m=l&r*m) = 0, q = 1, 2, • • •, Q. For each fix-price factor, there are three possible cases. Case 1. <r* = 0 and I* — Em=i^g"*zm = 0. In this case, the price of factor q is set at its market-clearing level and r'* = r*. There is no divergence between its shadow price and its market price. The fix-price equilibrium is a Walrasian equilibrium. Walrasian demand and supply of this factor are equal. Chapter 21. Towards Econometric Implementation 172 Case 2. a* — 0 and lq — £ „ 1 = 1 i ™ * z J 1 > 0. In this case, there is excess supply of the fix-price factor and for the producer, the market price it faces is the fixed price r*. The resource constraint on this factor is ineffective and can be ignored as was done in the theoretical framework discussed in the text. This is the case for involuntary unemployment in the labor market. The producers or demanders of labor are not rationed but the consumers or suppliers of labor are compelled to consume more leisure than they desire. From the viewpoint of the whole economy, the divergence of the shadow price of labor from the prevailing wage arises from the consumption side of the economy. If the wage is rigid at too high a level, the quantity constraint on the amount of leisure consumed/labor supplied imposed on the suppliers of labor (given constant time endowments) comes from the production side of the economy. The comparative statics for the production side of an economy in this regime was carried out in the text. Case 3. a* > 0 and Fq — Em=i&™*-Jm = 0- I n this case, there is excess demand for the fix-price factor at the fixed price r* and its shadow price is above this fixed price by the amount a* > 0 implicitly defined by the equation _ M dnm(p*,r' + <r*,w*) _ r 2^m=l Qr, m V This expression for the virtual price r* = r* + a* is a standard result in the microeconomic literature on quantity rationing.3 The virtual price or shadow price is the price that would induce the unconstrained producers to employ of factor q. This virtual price is not necessarily what will prevail in a Walrasian equilibrium because the flexible factor prices w* can have different configurations in these two regimes. Empirical implementation of the comparative statics for a production economy in this regime is more involved because of the unobserved Lagrange multipliers a*. Whereas in case 2, the sec-toral profit functions depend only on observables z*, p*, r* and w*, in this case the shadow price 3See Latham (1980, pp.310-311), Neary and Roberts (1980, p.30), Deaton (1981, p.59), Lee and Pitt (1986, p.1238), and Lee (1986, p.301), among others. Chapter 21. Towards Econometric Implementation 173 wedge cr* is a latent variable.4 Virtual price functions and rationed demand functions corre-sponding to some functional forms have been analytically derived by Neary and Roberts (1980) for the linear expenditure system and by Deaton (1981) for an extended version of the linear expenditure system and a rationed almost ideal demand system. The empirical findings of the latter study indicate that treating housing as a ration explains much of the inhomogeneity observed with unrationed demand functions obtained in earlier studies. Theoretically, a comparison between the observed market price and the virtual price deter-mines the classification of sample observations into periods of excess supply or excess demand. This procedure is equivalent to determining whether a* is zero or positive. A theoretical work on the use of virtual prices in the specification of inverse demand and supply functions as an alternative to the conventional econometric disequilibirium models with supply and demand functions and min conditions can be found in Lee (1986). This new approach has the advan-tage of more computationally tractable likelihood functions without necessarily the multiple integrals of the usual multimarket disequilibrium models. However, other problems in likeli-hood maximization endemic to disequilibrium models remain. Among them are: coherency conditions for nonlinear systems, multiple local maxima, and unboundedness of the likelihood function when some variances of the disturbances approach zero.5 From the above discussion, it can be inferred that the comparative statics discussed in the text should be performed only for periods where there is excess supply of the fix-price factors. However, information from sectoral profit functions estimated from time series data are inputs to the comparative static response matrix. Hence, ideally, for the periods when there is excess demand for the fix-price factor, its shadow price should be used instead of the observed market price. Otherwise, an error in variables is made which may bias the parameter estimates for the sectoral profit functions. 4This shadow price wedge corresponds to the virtual tax in exchange economies considered by Cornielje and van der Laan (1986). sSurvey articles on econometric disequilibrium models can be found in Quandt (1982) and Maddala (1983, 1986). Chapter 21. Towards Econometric Implementation 174 21.3 Econometric estimation The major interest in this study is the development of a trade-theoretic approach in the analysis of resource allocation in the presence of unemployment due to real wage rigidity. Factor mar-ket imperfections, in general, have aroused interest among trade theorists. The study of real wage rigidity, in particular, has been couched in terms of the theory of minimum wage rates in international trade. In contrast to the large country 2x2 model used by Brecher (1974a, 1974b) in his pioneering works, this study uses the small country assumption; otherwise, the complications due to terms of trade effects have to.be addressed. Additionally, the restriction that the number of flexibly priced factors be at least as great as the number of industries is im-posed to avoid specialization in production. Later works in this area were done by Dixit (1978), Schweinberger (1978), Muellbauer and Winter (1980), Neary (1985) and Flug and Galor (1986). The essence of "minimum wages" in trade theory is quite different from that in labor economics; it has to do more with the degree of flexibility, especially downwards, of labor prices relative to other factor prices. Recent developments in fix-price models can help enrich the "theory of minimum wage rates" in trade literature. However, fix-price theory, as well as the theory of minimum wage rates, is subject to the major criticism of being unable to explain price rigidities and hence, its policy prescriptions are suspect. Without a theory of price determination (for the fix-price good), it has weak predictive power. The implicit contract theory and efficiency wage theory in labor economics are attempts to explain the dual phenomenon of rigid wages and unemployment. Appeal is sometimes made to the existence of socioeconomic and institutional factors, such as labor unions, minimum wages and unemployment insurance, to explain wage rigidity. It seems there can be a multitude of causes of inertia in wage adjustment, some of which are sector-specific. In his 1979 presidential address to the American Economic Association, Solow (1980) argues that the absence of an acceptable theory to explain the failure of the labor market to clear does Chapter 21. Towards Econometric Implementation 175 hot necessarily negate the premise that wages do not move flexibly to clear the market. Re-jecting studies that explain the then existing unemployment as voluntary leisure resulting from intertemporal substitution of future work with perceived higher discounted value of earnings, he counters: It is thus legitimate to wonder why the unemployed do not feel themselves to be engaged in voluntary intertemporal substitution, and why they queue up in such numbers when legitimate jobs of their usual kind are offered during a recession. (Solow, 1980, p.7) His hunch is the real wage elasticity of labor demand is low6 and with a low elasticity of labor demand, together with other institutional factors, sellers of labor services are likely to resist downward wage adjustments. Moreover, the labor market cannot be modeled just like any goods market, say the buying and selling of cloth. The labor market responds to both pecuniary and nonpecuniary elements. Unemployment and real wage rigidity is a contentious issue which this thesis does not aim to resolve. Empirical analysis is left to more able econometricians. At most, some suggestions and possible problems are outlined here. A review of published empirical works on testing for excess supply of labor using disequi-librium models and on estimation of natural rates of unemployment indicates that there could have been excess demand for labor in Canada and the United States during the years 1965-1969 and 1972-1974. In the years of excess demand for labor, the production sector is rationed and the producer shadow price of labor diverges from the market wage. Hence, in the sectoral profit estimation the shadow price of labor should be used instead of the market wage for these years. Furthermore, the comparative statics exercise outlined in this study should not be performed 6 A survey of empirical studies on aggregate employment-wage elasticity can be found in Hamermesh (1986). Estimates of the magnitude of the aggregate long-run constant-output labor demand elasticity for developed countries for the period 1950 onwards lie in the range 0.15-0.50. The estimates are based on the Hicksian formula for the own-wage elasticity of labor demand at constant output, r]u, which for the two-factor case is given by T / H = —(1 — s)a < 0 where s is the labor share in total revenue and a is the elasticity of substitution between capital and labor. Chapter 21. Towards Econometric Implementation 176 for these years. The theoretical framework discussed can be applied to annual input-output data. The first step is the estimation of sectoral unit scale profit functions Trm(p, r, w), m = 1,2,..., M as de-fined in equation (18.1). Flexible functional forms providing a second-order approximation with constant returns to scale and technical change variables can be used in specifying the sectoral profit functions ir™. The dimension of the estimation model will be limited by the number of observations at hand. Hence, sectoral aggregation may be necessary. With semifiexible estimation (Diewert and Wales, 1988), the number of goods in the model can be increased. The theoretical model has assumed constant returns to scale and competitive behavior. Im-posing constant returns when otherwise may bias the parameter estimation of sectoral profit functions. In the absence of a general theory of imperfect competition, one has to posit par-ticular market structures and pricing behaviors to handle increasing returns and imperfect competition (see, for example, Helpman and Krugman (1985)). It is recognized that scale economies are a potential source of gains from trade liberalizaiton. For Canada, cost of pro-tection estimates range from 0.5 to 2 percent of GNP under the competitive constant returns assumption; incorporating possible scale economies yields estimates of 8 to 12 percent (Harris, 1984). Price distortions are another source of economic waste. By using actual prices of inputs and outputs faced by the producer, the inefficiency due to non-Pareto optimality of prices can be captured. Let us redefine the price vector as h = (pi,P2,---,PK,r1,r2,...,rQ,w1,w2,...,wN)T and the quantity vector as * = (yi,y2,---,yK,-h,-h,---,-lQ,-vi,-v2,...,-vN) T. Let r m be a K + Q + N vector of price wedges faced by sector m. Then the actual price faced by sector m is hm = h + Tm. '" Chapter 21. Towards Econometric Implementation 177 For estimation, the sectoral profit function nm should take the more general definition wm(hm) = maxxm{hm • xm : xm = (ym, -lm,-vmf € Sm} where the elements of xm are indexed positively(negatively) for outputs(inputs) and Sm is the constant returns to scale production possibilities set of sector m. Other than labor, imports can be considered a fix-price factor. Our small country assump-tion implies that import prices are exogenous. Appelbaum and Kohli (1979) have tested the small open economy hypothesis with respect to Canada-United States trade. Their study indi-cates that Canada does not exert significant power in its imports market but does in its export markets. The results of the sectoral profit function estimation can then be used to calculate the comparative static response matrix as given in equation (18.31). 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Winter, 1980, Unemployment, employment and exports in British manufacturing, a non-clearing market approach, European Economic Review 13, 383-409. Bibliography 182 [60] Mussa, M., 1974, Tariffs and the distribution of income: the importance of factor speci-ficity, substitutability, and intensity in the short and long run, Journal of Political Econ-omy 82, 1191-1203. [61] Neary, J.P., 1978, Short-run capital specificity and the pure theory of international trade, Economic Journal 88, 488-510. [62] Neary, J.P., 1985, International factor mobility, minimum wage rates, and factor-price equalization: a synthesis, Quarterly Journal of Economics 100, 551-570. [63] Neary, J.P. and K.H.S. Roberts, 1980, The theory of household behavior under rationing, European Economic Review 13, 25-42. [64] Ostensoe, L.A., 1986, Production analysis for applied general equilibrium modelling, Hon-ors thesis (Department of Economics, University of British Columbia, Canada). 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[73] Statistics Canada, 1987, Fixed capital flows and stocks (historical) 1936-1983, Cata-logue 13-568 (Minister of Supply and Services, Ottawa). [74] Varian, Hal R., 1984a, The nonparametric approach to production analysis, Economet-rica 52, 579-597. [75] Varian, Hal R., 1984b, Microeconomic analysis (W.W. Norton, New York). [76] Varian, Hal R., 1985, Non-parametric analysis of optimizing behavior with measurement error, Journal of Econometrics 30, 445-458. 1 [77] White, K.J. , 1987, SHAZAM: A general computer program for econometric methods (version 5), American Statistician 41, 80. Appendix A Incorporating the No Technological Regress Assumption with the Efficiency Tests Listed below are the modifications in the various efficiency tests needed to incorporate the no technological regress assumption. The changes involve merely a change in the range of the observation indices used in constructing the relevant convex sets. • Test 1'. Technical Efficiency and No Technological Regress Test for a Convex Technology: Modify the linear programming subproblem (3.2) to maxs.>o,Ai>o,...1A.->o{fc = Ej=i*j*j > **' + <W, £J=1A> = 1} = Sf. • Test 2'. Technical Efficiency and No Technological Regress Test for a Convex Conical Technology: Modify the linear programming subproblem (4.2) to max5i>0iAi>o,...,A<>o{^' : £ j = i ^ V > zl + Sijz'} = 6*. • Test 3'. Technical Efficiency and No Technological Regress Test for a Quasiconcave Technology: Modify the index set defined by (5.2) to lT = {j: zn> zn,j = 1,2,...,i}. • Test 4'. Allocative Efficiency and No Technological Regress Test for a Convex Technology Assuming Partial Profit Maximization: 183 Appendix A. Incorporating the No Technological Regress Assumption 184 Modify the linear programming subproblem (7.1) to maxA 1>0 A .> 0 {E„ e sPi ,(Ei = iA J4) : E J - i ^ > 4 , » ? 5, EJ=i>J' = 1} = £n € SPn4 + ef J2neEPn\4 I-• Test 5'. Allocative Efficiency and No Technological Regress Test for a Convex Conical Technology Assuming Partial Profit Maximization: Modify the linear programming subproblem (9.1) to maxAi>0 i... i A.>0{E„e 5K(E;= 1A>zJ) : Y.%j=i^4 > 4> n £ S} = £n€SPJ,4 + ^ En€EPn\4 I-• Test 6'. Allocative Efficiency and No Technological Regress Test for a Quasiconcave Technology Assuming Partial Profit Maximization: Modify the index set defined by (10.1) to = *£> 4 , J' = 1 . 2 , - . . , « } • • Test 7'. Allocative Efficiency and No Technological Regress Test for a Convex Technology Assuming Complete Profit Maximization: Redefine the violation index defined in (11.1) to (piTfzj_zi\ ] e? = max,-| piT^zi : j = l , 2 , . . . , i j . • Test 8'. Allocative Efficiency and No Technological Regress Test for a Convex Conical Technology Assuming Complete Profit Maximization: Redefine the violation index defined in (11.12) to Appendix A. Incorporating the No Technological Regress Assumption 185 • Test 9'. Allocative Efficiency and No Technological Regress Test for a Quasiconcave Technology Assuming Complete Profit Maximization: Modify the index set defined by (11.28) to I? = {3-- 4><,j = l,2,...1i}. In each test, if the violation index (6*, ef or £*) is zero for all observations i, i = 1,2,..., J , then the data are consistent with the efficiency hypothesis being tested. Otherwise, if the violation index is positive for some observation i, then a violation of the efficiency hypothesis occurs at this observation. Appendix B The Canadian Input-Output Data, 1961-1980 B.l The data base The original data base used in this study is the 1961-1980 annual Canadian input-output accounts compiled by Cas (1984) for use in preliminary studies on multifactor productivity measures for the Input-Output Division of Statistics Canada. This data base draws from both published and unpublished data sources at Statistics Canada. It is generally comparable to published Canadian input-output tables at the "M" level of aggregation. Whereas the latter gives commodity by industry transaction accounts, the present data base has been modified into an industry by industry classification. It covers the domestic private industrial sectors for which the profit-maximizing model is applicable though some industries, for example, broadcasting and rail transportation, are heavily subsidized by the government. Dummy industries, used for routing goods whose precise commodity content are unknown, and which appear in published accounts are not included in the present data base. Due to problems obtaining data on its capital stock, the "post office" industry was also removed from this data set. Input-output data for 37 sectors of the Canadian economy are given. For each sector, there are 57 data items: 5 output variables, 37 intermediate input variables, 9 primary input variables and 6 financial variables. These data items are listed in table B.10. Each variable is given in values of current and constant (1961 or 1971) dollars from which implicit price deflators can be derived; constant dollar values can then be used as quantities. The output variables are classified according to end use. Intermediate inputs pertain only to purchases of current output of domestic industries by domestic industries. Primary inputs are termed "primary" in the sense that the origin of these goods are external in time or boundaries 186 Appendix B. The Canadian Input-Output Data, 1961-1980 outputs: 1. gross output 2. intermediate output 5. final demand 4. exports 5. re-exports intermediate inputs from sectors: 1. agriculture and fishing 2. forestry 3. mines, quarries and oil wells 4. food and beverages 5. tobacco products 6. rubber and plastic products 7. leather 8. textiles 9. knitting mills 10. clothing ' 11. woods 12. furniture and fixtures 13. paper and allied industry 14. printing, publishing and allied industries 15. primary metals 16. metal fabricating 17. machinery 18. transportation equipment 19. electrical products 20. nonmetallic mineral products 21. petroleum and coal products 22. chemical and chemical products 23. miscellaneous manufacturing 24. construction 25. air transporation, other utilities and transportation 26. railway transportation and telegraph 27. water transportation motor transportation urban and suburban transportation storage 31. broadcasting 32. telephone 33. electric power 34. gas distribution 35. trade 36. finance, insurance and real estate 37. commercial services 28 29 30 primary inputs: 1. imports, competitive 2. imports, noncompetitive 3. government goods 4. labor 5. raw inventories 6. finished inventories 7. machinery and equipment (M&E) 8. structures (S) 9. land financial variables: 1. commodity indirect taxes 2. subsidies 3. other indirect taxes 4. royalties 5. capital consumption allowance, M&E 6. capital consumption allowance, S Table B.10: Data items for each of the 37 sectors Appendix B. The Canadian Input-Output Data, 1961-1980 188 relative to current production of the 37 domestic business sectors; hence, the inclusion of imports and government goods in this group of inputs. In contrast to published Canadian input-output accounts where imports are included in the use and final demand matrices, the present data base has netted out this component of the intermediate input matrices. The land data series was constructed by Cas, Diewert and Ostensoe (1986). The "financial" variables are used to adjust prices of outputs and factors so as to reflect actual prices received or paid by the producers. B.2 Data manipulations at the sectoral level All prices and quantities obtained are normalized such that 1961 prices equal 1.0 and expendi-tures or nominal values (current dollar values in $1000 Canadian) are preserved. B.2.1 Output and intermediate input prices and quantities Each of the 37 sectors is assumed to produce a single output with market prices pi, i = 1,2,...,37. These prices pi are obtained as implicit price deflators calculated as the ratio of current dollar value to constant dollar value of gross output: current dollar value of gross output of sector z p- = 2 £ t = l 2 . . . 37. ' constant dollar value of gross output of sector i' >'•••> For price consistency, the prices of intermediate inputs are also set equal to their corresponding output prices. That is, if we let pij be the price of a unit of good i used in the production of good j, then p tj = pi, i,j = 1,2,..., 37. The corresponding quantities of intermediate inputs are adjusted accordingly. Let g,j be the quantity of good i used to produce good j; then current dollar value of input i used in sector j q i j = £ J-. Pi For the estimation of profit functions, the concept of net output is used. Hence, gross output quantities have to be adjusted by subtracting the quantity of intermediate inputs coming from the same sector. Denote the net output by sector i as g t ; then qi = (constant dollar value of gross output of sector i) — qn. Own input prices and quantities are set to zero: pn = qn = 0. Appendix B. The Canadian Input-Output Data, 1961-1980 189 B.2.2 Missing observations for 1980 intermediate inputs Prior to doing data manipulations on the data base, a check for missing observations was done. There are 23 missing observations for current dollar (as well as constant dollar) values for intermediate inputs in 1980. Due to differences in commodity by industry configurations and the netting out of imports in the use matrix for the present data base, cross-checking with published input-output accounts is precluded. Hence, an ad hoc procedure was used to fill in the missing observations. For five sectors (3, 15, 16, 22, 23) in 1980, there are significant differences between the value of the intermediate output variable and the sum of the values of the good used as intermediate inputs (as given by £ | I i Pijqij, i — 3,15,16,22,23). These differences are listed in table B . l l . Except for miscellaneous manufacturing, the figures indicate that a substantial proportion, 8% to 23%, of intermediate output may have been unallocated to various users. For miscellaneous manufacturing, the 1980 value of intermediate output was revised so that the ratio of inter-mediate output to gross output is identical to that of 1979. To impute values for the missing observations, the share of the using sector j of the value of intermediate output of sector i (pijqijlvalue of intermediate output of sector i) in 1979 is assumed to carry over to 1980. The 1979 values and the 1980 imputed values for the missing ovservations are listed in table B.12. Once the imputed current dollar values for the missing observations for the intermediate inputs are obtained, the relevant price and quantity variables, pij and can easily be calculated as outlined in the previous section. B.2.3 Imports and inventories The attribution of imports and inventories to different categories can be considered questionable. Whereas all sectors have positive competitive import values for all years 1961-1980, only eight sectors (4,6,8,20,22,23,35,37) have positive noncompeting import values for all years. Eleven other sectors (3, 7,9,10,13-16, 18,19,21) have positive noncompeting values for one to nineteen years. The other 18 sectors have zero noncompeting imports for all years. Of the total value Appendix B. The Canadian Input-Output Data, 1961-1980 190 sector i value of intermediate . output* (1) value of the good used as intermediate inputs* (2) difference (l)-(2) (3) percentage discrepancy (3)/(l) (4) 3 mines, etc. 15 primary metals 16 metal fabricating 22 chemical, etc. 23 misc. manufacturing 17554720 9798036 8150517 6995275 1066220 16121327 8392645 6282122 6149460 1074116 1433393 1405391 1868395 845815 -7896 8.17% 14.34% 22.92% 12.09% -0.74% * Figures are in current do lars. Table B . l l : Inconsistencies in the value of intermediate output and the sum of values of the good used as intermediate inputs, 1980 of imports, both competitive and noncompetitive used by the whole economy (the 37 sectors), noncompeting imports account only for 2.3% to 6.1%. Except for three sectors (2, 3, 37), all sectors have positive values for raw inventories input. The absence of raw inventories in forestry and in the mines, quarries and oil wells sectors is disturbing. The resource and manufacturing sectors (1-23) have positive finished inventories. The rest — construction, transportation, communication, utilities and services sectors (24-37) — have zero finished inventories. Of the group with both positive values for raw and finished inventories (sectors 1, 4-23), all have identical data series for the years 1961-1970; for agriculture and fishing the implicit price deflators for the two data items are the same for all years. For the whole economy, raw inventories as a proportion of the value of total inventories has decreased from 68% in 1961 to 55% in 1980; the share of finished inventories has increased from 32% in 1961 to 45% in 1980. In the light of the above data deficiencies, it has been decided to merge imports and inven-• tories into single categories. The price and quantity variables for imports, PM and qM, were calculated as follows: P M = NCM + NXM RCM + RXM ' qM = RCM + RXM Appendix B. The Canadian Input-Output Data, 1961-1980 191 1979 value for 1980 imputed value for originating intermediate input intermediate input sector i using sector j PijQij PijQij mines, 3 mines,etc. 880393 1008678 quarries, 18 transportation 29951 34315 equipment and oil wells 22 chemical,etc. 340748 390400 total 1251092 1433393 primary 3 mines,etc. 185221 157207 metals 18 transportation 1342057 1139073 equipment 22 chemical,etc. 128555 109111 total 1655833 1405391 metal 3 mines,etc. 119078 128917 fabricating 15 primary metals 176037 190582 16 metal fabricating 811038 878050 19 electrical products 160451 173708 22 chemical,etc. 125582 135958 23 misc. manufacturing 50303 54459 35 trade 158132 171198 37 commercial services 125179 135522 total 1725800 1868395 chemical and 15 primary metals 122519 133806 chemical 18 transportation 108047 118000 equipment products 19 electrical products 143078 156258 23 misc. manufacturing 112959 123365 35 trade 71816 78432 37 commercial services 216051 235954 total 774470 845815 miscellaneous 3 mines,etc. 20440 22102 manufacturing 18 transportation 44715 48351 equipment 22 chemical,etc. 28486 30803 total 93641 101256* * This figure has been revised as described in the text and hence differs from the entry in column 3 of table B . l l . Table B.12: Imputed values for missing observations for 1980 intermediate inputs Appendix B. The Canadian Input-Output Data, 1961-1980 192 where NCM = current dollar or nominal value of competitive imports, NXM = current dollar or nominal value of noncompetitive imports, RCM = constant dollar or real value of competitive imports, and RXM = constant dollar or real value of noncompetitive imports. The price and quantity variables for inventories, pi and g j , were calculated analogously: p i = NRI + NFi RRI + RFI ' qi = RRI + RFI where Nm = current dollar or nominal value of raw inventories, Npi = current dollar or nominal value of finished inventories, RRI = constant dollar or real value of raw inventories, and RFI = constant dollar or real value of finished inventories. B.2.4 Tax adjustment of output and input prices The relevant output and factor prices for profit function estimation are "the output prices that reflect the revenue actually received by the firm and the input prices that reflect the actual cost paid by the firm for the use of the inputs in the production process" (Diewert, 1980a, p.479). Output prices should reflect payments, after subsidies and royalties, to the producer, that is, Pi = Pi + si ~ rVi Appendix B. The Canadian Input-Output Data, 1961-1980 193 where p* = tax adjusted output price, Pi = market output price (initially obtained as implicit price deflator), 5,- = per unit subsidy receipts, ryi = per unit royalty payments, N{ = current dollar value of gross output, N* = current dollar value of gross output plus subsidy minus royalty, and JV* —r = proportion of the market price actually received by the producer. iv,1 All sectors received some form of subsidy for at least some years. The data reflect royalty payments from only five sectors: forestry; mines, quarries and oil wells; woods; electric power; and trade (the latter for only 1976-1980). Note that we might have expected an industry like petroleum and coal products to have positive royalty payments. The data on royalty payments can be considered questionable. The way the national accounts are constructed, royalties on natural resources are classified as an industry under the finance, insurance and real estate sector. Ideally, these royalty payments should be considered as taxes levied on the use of government-owned natural resources or as rental prices for a separate capital aggregate for land or a natural resource factor of production. Given the data, we list in table B.13 the sectors whose producer prices p* deviate, on the average, by more than 1% from their market price p,-. Hence, for broadcasting, an additional 48 cents was given by the government as subsidy for every dollar received in the marketplace. It must be noted though that the subsidy rate has been consistently decreasing over the years but still remained relatively high compared to other sectors. On the other hand, seven to eight cents of every dollar received by the producer in forestry and the mining sectors went back to the government as royalties. Prior to 1974, there was hardly any price distortion as measured by jy* jjr in the petroleum and coal products sector; thereafter, the subsidy ranged from a minimum of 5 cents for every dollar in 1978 to 26 cents for every dollar in 1975 and in 1980. A similar Appendix B. The Canadian Input-Output Data, 1961-1980 194 sectors average minimum (year) maximum (year) 1 agriculture and fishing 1.036 1.008 (1963) 1.064 (1975) 2 forestry 0.931 0.896 (1979) 0.957 (1976) 3 mines, etc. 0.925 0.831 (1978) 0.976 (1961) 21 petroleum, etc. 1.054 * * 26 rail transportation, etc. 1.082 1.035 (1971) 1.163 (1975). 27 water transportation 1.019 1.006 (1974) 1.039 (1977) 31 broadcasting 1.478 1.355 (1980) 1.564 (1961) 34 gas distribution 1.042 * * * See text for interpretation for the petroleum and coal products sector and the gas distribution sector. Table B.13: Average proportion of market price received by the producers in sectors with the most output price distortions increase in subsidy rates occured in the gas distribution sector. Pre-1974, the subsidies were negligible; the subsidy rate was 5 cents for every dollar in 1974 and increased to a maximum of 14 cents for every dollar thereafter. The increase in subsidy rates to the petroleum and coal products and gas distribution sectors can be interpreted as policy responses to the oil price shock that occured in 1974-1975. The tax-adjusted output prices p* we obtained are the relevant prices we seek for profit TV* function estimation. However, we must be cautious in interpreting the value of as an indicator of relative sectoral price distortions. Keeping in mind the data imperfections, we interpret these numbers as measuring the effect of explicit royalties and subsidies per se on actual producer revenue. There are other price and nonprice distortions or policy instruments which may have more significant effects on producer prices, that is, directly on our "unadjusted" output prices p,-. Margins arise between the value received by the producer to cover his cost of production and the value paid by the purchaser. At the producer side, we have to adjust for subsidies and royalties. Trade, transportation and delivery margins have already been recorded as output of the relevant sectors in the original data base; hence, the initial unadjusted output prices p,- are those levied at the final stage of production. As well, these trade, transportation and delivery Appendix B. The Canadian Input-Output Data, 1961-1980 195 margins appear separately as intermediate inputs in the accounts of the purchasing sector. Another margin that has to be accounted for is the value of commodity indirect taxes paid by the using sectors for their intermediate inputs.1 Since the data base gives only the total value of commodity indirect taxes paid by each sector, an ad hoc assumption has to be made on how to distribute them over the relevant inputs. We assume an equal commodity tax rate for all intermediate inputs and imports in a particular sector. Let tj denote the commodity tax rate in sector j. We define tj as the ratio of the value of commodity indirect taxes in sector j to the sum of the values of intermediate inputs from other sectors and the value of imports: _ value of commodity indirect taxes paid by sector j 3 Hi^jPijQij + PMQM Then, the tax-adjusted input prices p*-, j = 1,2,..., 37 and p*M are Pij = Pij(l + *j) = Pi(l + tj), PM = PM(1 + tj). With this tax imputation, the total cost to the purchasing sector, say sector j, for its intermedi-ate inputs from other domestic sectors and imports is equal to the value of the inputs assessed at the initial prices pij, PM plus the value of commodity indirect taxes paid by the purchasing sector: T,lLiP*jQij + PM QM = (YJiliPijQij + PMQM) + (value of commodity indirect taxes paid by sector j). The commodity tax rates tj were lowest in the manufacturing sectors (4-23, except 20) and electric power industry; it was usually less than 1% to about 1.5% in these sectors. The rates, ranging from 4% to 18%, were generally higher in the construction and air, motor and 1 Commodity indirect taxes include federal taxes (excise duties, excise taxes and federal sales taxes, oil export charge, petroleum levy, Canadian ownership charge, natural gas and gas liquids tax, and air transport tax), provincial taxes (amusement tax, fuel tax, profits of liquor commissions, liquor gallonage tax, Quebec Hydro levy, and sales tax on gas, electricity, telephone and telegraph), local amusement tax, other provincial and local sales taxes, and import duties. Appendix B. The Canadian Input-Output Data, 1961-1980 196 urban transportation sectors. The rest of the industries, usually resource and service sectors, have rates averaging 3% to 6%. In reality, commodity taxes are commodity-specific; hence, the sectoral differences in our calculated effective commodity tax rates tj can be explained by differences in sectoral input requirements. B.2.5 Capital rental prices Definition of capital service and capital rental price The capital input variables are taken to be the durable goods: inventories, machinery and equipment, structures and land (k = 1,2,3 and 4, respectively). The data base gives values of capital stock in current and constant dollars. For a particular capital good, say k, asset price p£ and quantity qff were obtained as For profit function estimation, the relevant price for a durable input is the capital rental price or user cost of a flow of capital services used in the production process. As noted by Jorgenson and Griliches (1967), though asset prices may be considered the discounted value of all future capital services across capital goods, they are not proportional to their service prices (what we terms here as rental price or user cost) because different capital goods can have different economic and physical depreciation rates and asset price appreciation rates (capital gains or losses). Denote the rental price for capital good k as pjf*. Following Diewert (1980a), we define the rental price p£ * before corporate income tax as current dollar value of capital stock k constant dollar value of capital stock k' qk = constant dollar value of capital stock k. _ rkPk + (1 + rk)rkp^ + 6kpk - (pk - pf) 1 + rjb where pk = rental price for capital good k, Appendix B. The Canadian Input-Output Data, 1961-1980 197 p£ = asset price (or purchase price) for capital good k, Pk = next period's expected asset price for capital good k, rk = rate of return for capital good k, Tk = property tax rate for capital good k, and 6k EE combined depreciation and obsolescence rate for capital good k. Reiterating Jorgenson and Griliches' point, we can see from the above formula that asset prices and rental prices are proportional to each other (and can be econometrically safe to be sub-stituted for each other) only under some restrictive conditions. One case is when there are static expectations (pk = Pk) a n ^ the discount, property tax and depreciation rates (rk,rk,6k) are identical across all capital goods and across all sectors. Ideally, the rental prices must be adjusted to reflect the actual cost to the producers after corporate income taxes. Due to problems in matching financial data on corporate income taxes collected on a company basis and input-output data collected on an establishment basis, this adjustment cannot be carried out in this study.2 This implies that the rental prices we obtain may not be capturing the distortionary effect of policies such as investment tax credits and accelerated depreciation tax rules. Previous attempts at incorporating nonstatic expectations of future asset prices by Osten-soe (1986) and Fortin (1988) yielded negative and erratic capital rental prices. Drawing from these studies but still acknowledging that the problem of modelling capital gains still remain, we henceforth assume static expectations for future asset prices: pk = p £ . We also assume that in a sector with more than one type of capital, the rate of return r& is equal across asset types. The reason for this assumption was expressed by Jorgenson, Gollop and Fraumeni (1987, p.145): It is with respect to each asset's nominal rate of return that economic agents choose the optimal mix of capital stocks, altering the composition of capital input until all 2 An example of the incorporation of corporate tax liabilities in the calculation of capital rental prices using United States data can be found in Jorgenson, Gollop and Fraumeni (1987, p.128). Appendix B. The Canadian Input-Output Data, 1961-1980 198 nominal rates of return are equalized across asset classes. Consequently, measured property compensation should be allocated among assets on the basis of equality of the nominal rate of return for all assets. With these two assumptions, the capital rental formula reduces to PP = r + (14- r)rfc + 6k 1 + r where r is the rate of return or discount rate identical across asset types in a particular sector. Implicitly, we assume that capital service flows are proportional to capital stocks, that is, the rate of capital utilization is constant over time. Hence, we take the variable qjf as the relevant quantity measure for our profit function estimation. The problem of estimating capital stocks and rental prices with consistent patterns of relative efficiency is discussed by Jorgenson, Gollop and Fraumeni (1987). Calculation of capital rental prices For the empirical implementation of the capital rental price formula, we further assume that inventories of raw materials and finished goods and machinery and equipment capital stocks are not levied property taxes and that inventories and land have zero depreciation rates. In short, we assume T\ = r 2 = 0 and Si = 64 = 0. A detailed description of the data on the capital series on machinery and equipment, and structures can be found in the Statistics Canada Catalogue 13-568: Fixed Capital Flows and Stocks (Historical), 1936-1983 compiled by the Construction Division. Our data on machinery and equipment include capital items charged to operating expenses; structures include both building and engineering construction. The current and constant dollar value of capital stocks are mid-year net stock values based on the difference between the cumulative value of past gross investment and cumulative depreciation. The depreciation or capital consumption allowance data are also mid-year values and are based on the straight-line depreciation rule. Since the quantity variable qjf has already netted out depreciation for the same year, we would like a quantity variable, say qjf*, gross of depreciation. We define the adjusted variable Appendix B. The Canadian Input-Output Data, 1961-1980 199 as qfc + (capital consumption allowance in constant dollars Qk = \ for capital good k), k = 2,3 q£\ k = i,4. This correction leads to lower depreciation and property tax rates especially for sectors having assets whose specified service lives are shorter. For example, machinery and equipment in forestry, construction, and air and motor transportation sectors have service lives of 10 years or less whereas in other sectors the service lives range from 15 to 35 years. Our data on other indirect taxes, which are not commodity-specific, are assumed to comprise mainly of property taxes levied on fixed capital: structures and land. The property tax rates are defined as the ratio of the value of other indirect taxes paid to the value of structures and land in a given sector: other indirect taxes in current dollar paid by sector £ ' — 3 4 E l „„K„K» ' — ' k=3Pk 0, k = 1,2 where pj^Qk* = current dollar value of structures and land in given sector, respectively. The twenty-year average of the property tax rates for most sectors ranges from 1% to 5%. It was slightly higher (7-10%) in knitting mills, clothing, furniture and fixtures, and miscellaneous manufacturing. It was less than 1% for the petroleum and coal products, electric power and transportation sectors except for motor transportation. The property tax rates rk obtained for the construction (24) and motor transportation (28) industries are substantially higher, averaging 25% and 34%, respectively. These values may indicate some data anomalies pertaining to these sectors. As noted by Fortin (1988), the taxes for these two industries may include payments for permits and licenses. If this is the case, then the asset prices of fixed capital in these sectors have to be adjusted to reflect these payments; in an efficient market we would expect these payments to be capitalized in the purchase price. The ratio of the capital consumption allowance to the value of capital stock yields the Appendix B. The Canadian Input-Output Data, 1961-1980 200 effective depreciation rates. We define the rates 6k as ^ k — 2} 3 The capital stock and consumption allowance data for machinery and equipment, and structures are based on the straight-line depreciation rule. We would then expect the depreciation rates Sk to be inversely related to the service life specified for that asset located in a particular sector. The estimates for the depreciation rates are quite stable over the years. The depreciation rates for machinery and equipment are lowest (about 4%) in the electric power and gas distribu-tion sectors where these assets, excluding capital items charged to operating expenses, have the longest specified service life of 35 years. Most sectors have depreciation rates from 5% to 10%. Some sectors (1, 2, 6, 7, 23-25, 28, 29, 37) where most machinery and equipment have service lives of 15 years or less have average depreciation rates between 12% and 16%. Having longer service lives of at least 20 years to at most 75 years, structures have lower depreciation rates. The sectoral averages for the depreciation rates of structures range from 2% to 7%, with most clustering in the 3-5% range. It is theoretically unclear which discount rate r is to be used. In practice, either an internal rate of return or an exogenous bond rate is used. Diewert (1980a) argues that the relevant interest rate should reflect the firm's actual borrowing and lending rates. The choice of an exogenous bond rate can be justified on the grounds that firms make their decisions based on ex ante rates of return and the exogenous bond rates would be more reflective of the opportunity cost of capital. In the same manner that equality of rates of return determine the optimal mix of capital goods in a particular sector, then the discount rate should also be identical across sectors if there is relatively smooth capital adjustment in the economy. Additionally, calculated internal rates of return may be negative and tend to be more erratic compared to an exogenous bond rate. The use of sector-specific internal rates of return is consistent with the assumption of con-stant returns to scale technologies; it enables us to impose the equality of value of output and Appendix B. The Canadian Input-Output Data, 1961-1980 201 factor payments. It seems, too, with static expectations, the sectoral reallocation of capital may respond to ex post user costs. As in the labor market where sectoral wage dispersion does not seem to be an unreasonable phenomenon, the rates of return for capital assets can vary across sectors. The choice of the relevant discount rate may be related to the problem that the industries operate in a world subject to risks and uncertainty and are in a constant flux of adjustment. To see how both rates perform empirically, we calculate sector-specific internal rates of return and compare them to an exogenous bond rate. First, for each sector i, i = 1,2,.. .,37, we calcualate the surplus value 5; which is the net value added by capital assets after variable costs: Si = surplus value for sector i = P*Qi ~ T,%iP*jiQji ~ PMQM - PGQG - PLQL where po = unit price of government goods used in sector i current dollar value of government goods used in sector z constant dollar value of government goods used in sector i' qc = quantity of government goods used in sector i = constant dollar value of government goods used in sector i, PL = unit price of labor input in sector i current dollar value of labor input in sector i constant dollar value of labor input in sector i qi = quantity of labor input in sector i -, and = constant dollar value of labor input in sector i. We then equate capital income to the surplus value: Si = ZUiPk'tf*-Given our rental price formula for pj£*, we can solve for the internal rate of return for sector i, Appendix B. The Canadian Input-Output Data,, 1961-1980 202 say r,-, as Si -Pi- Di Ai + Pi - Si where Pi property taxes on capital assets paid by sector i k=\TkPk Ik other indirect taxes paid by sector i, Di value of depreciation of capital assets in sector i Y - 4 c A ' A ' * 22k=ibkPk qk capital consumption allowances in sector i, and Ai value of capital assets in sector i 2^k=iPk Qk • The sectoral averages and standard deviations of internal rates of return over the period 1961-1980 are listed in table B.14. Beforehand, let us examine some inconsistencies in the calculated internal rates of return. The rail, water and urban transportation sectors (26, 27, 29) have either near zero or negative internal rates of return. In these sectors, the surplus value which we equate to capital income is too small and in the years with negative returns cannot cover property taxes and depreciation costs.3 For the urban transportation sector, the surplus was negative for 1972 to 1980. For selected years, negative internal rates of return were also obtained for forestry (1975, 1976), primary metals (1976) and petroleum and coal products (1969, 1976, 1978). In these years, there was a considerable drop in their surplus values. Since the negative internal rates of return were generally small in magnitude4 except for the urban transportation sector, for subsequent empirical estimation we set these rates to zero. The negative internal rates of return may, after all, reflect inefficiencies in these sectors or slow adjustment to external shocks. 3From our earlier discussion, the property tax rates and depreciation rates for these sectors seem reasonable. 4Magnitudes were usually less than 0.009. Appendix B. The Canadian Input-Output Data, 1961-1980 203 sector internal rates of return proportion of economy's capital stock, average* average* standard deviation 1 Agr k Fish 0.105517 0.032098 0.070233 2 Forest 0.052536 0.028257 0.005529 3 Mines,etc 0.141935 0.034892 0.062376 4 Food k Bev 0.192317 0.013683 0.022483 5 Tobacco 0.189888 0.045612 0.002066 6 Rub k Plas 0.184638 0.078319 0.003381 7 Leather 0.083946 0.039032 0.000903 8 Textile 0.087968 0.026585 0.006552 9 Knitting 0.145049 0.060775 0.001012 10 Clothing 0.308656 0.096094 0.001858 11 Woods 0.110226 0.080586 0.007315 12 Furn k Fix 0.239743 0.051399 0.001424 13 Paper,etc 0.084048 0.048859 0.023639 14 Printing 0.235046 0.050968 0.004458 15 Pri Metals 0.057638 0.028322 0.022070 16 Metal Fab 0.208812 0.045787 0.009212 17 Machinery 0.244392 0.081077 0.005495 18 Transport 0.163221 0.062689 0.013601 19 Elec Prod 0.208085 0.051838 0.007770 20 Non Metal 0.130039 0.033322 0.007547 21 Pet k Coal 0.023998 0.029733 0.012484 22 Chemical 0.095060 0.040507 0.017164 23 Misc Manuf 0.210496 0.053974 0.002618 24 Construct 1.658639 2.451802 0.009476 25 Air Tran 0.032037 0.007964 0.034234 26 Rail Tran -0.002551 0.006324 0.038919 27 Water Tran 0.004475 0.012749 0.010452 28 Motor Tran 1.348575 0.397836 0.002959 29 Urban Tran -0.057397 0.045573 0.004450 30 Storage 0.062577 0.027002 0.003574 31 Broadcast 0.174301 0.050592 0.001809 32 Telephone 0.069288 0.009257 0.036151 33 Elec power 0.026961 0.006564 0.099529 34 Gas distr 0.090609 0.017178 0.007848 35 Trade 0.180829 0.038916 0.057177 36 F.I.R.E. 0.051077 0.008432 0.358857 37 Comm serv 0.358631 0.081720 0.023375 * Averages are over 20 years. Table B.14: Sectoral 20-year average internal rates of return Appendix B. The Canadian Input-Output Data, 1961-1980 204 A more serious cause for concern are the high internal rates of return obtained for construc-tion (24) and motor transportation (28). They were generally greater than 0.50, and greater than 1.0 for 1973-1980 in the construction sector. For the motor transportation sector, they range from 0.74 to 2.01. These results may be indicative of data problems for these sectors; recall from our earlier discussion that the property tax rates for these sectors are suspiciously high, averaging 25% and 34%, respectively. It is possible that there is undervaluation of capital assets in these sectors. A possible source of error is in the accounting of leased or rented components of a sector's capital stock which are recorded as a primary input in the finance, insurance and real estate sector. This would not pose a problem in our gross output profit function estimation if these leased capital is properly recorded as intermediate inputs in the using sectors. Table B.14 also gives the 20-year average proportion of the economy's value of capital stock for the different sectors. The entries in the last column are the means of annual proportions of the total capital stock as given by A i / Y ^ L i Ai. The bulk of the economy's capital stock is in the finance, insurance and real estate sector which accounts for 33% to 40% of the total.5 The construction sector is relatively large compared to other domestic sectors; the value of its output is about 10% of the sum of the value of output of all sectors (P24?24/Z)?=iP*9«')-6 But, as seen in table B.14, it has only about 1% of the economy's total capital stock.7 A significant proportion of its capital input must be recorded as primary input in the finance, insurance and real estate sector. The same data error must be underlying the high internal rates of return in the motor transportation sector, though this is not as immediately clear. This sector is smaller with its output accounting for about 2% of the sum of the value of output of all sectors (P289W£3IiP*<Zi) 5It must also be noted that the finance, insurance and real estate sector includes "owner occupied dwellings" as a sub-industry. Since the purpose of this study is the modelling of production technologies of profit-maximizing business sectors of the economy, the inclusion of owner occupied dwellings which can be considered a consumer durable can be questionable. 6The denominator £;IiP<9' *s a 8 r o s s output concept and hence does not give the private gross domestic product which nets out intermediate inputs, imports and government goods. 7The output of the construction industry in the data base includes the output of construction labor in other industries; all inputs, except for the capital stock, have been properly adjusted. Hence, the capital stock of sectors engaged in own-account construction activities will tend to be overestimated. The extent to which this possible source of error contributes to the low capital stock data recorded in the construction industry is unknown. Appendix B. The Canadian Input-Output Data, 1961-1980 205 but its recorded total capital stock is only about 0.3% of the economy's total stock. Nevertheless, a closer examination of the capital stock data in the construction and motor transportation sectors is warranted. For subsequent empirical estimation, the internal rates of return for these two sectors are set at the economy-wide averages, as discussed below. Ignoring the aforementioned anomalies in the calculated internal rates of return, we note that within sectors the estimated rates are erratic in the sense that there generally are no consistent increasing or declining trends except for a few sectors. Internal rates of return were 10% or more in the 1960's, and less than 10% in the 1970's in the chemical and gas distribution sectors (22, 34). In the petroleum and coal products and commercial services sectors (21, 37), there is a declining trend in internal rates of return which is more marked in the latter sector. From table B.14, we can infer that the higher the mean sectoral internal rate of return, the variability of rates in that sector tends to be greater. The food and beverages sector (4) and finance, insurance and real estate sector (36), probably due to its huge capital stock, have quite stable internal rates of return. Probably due to government regulation, the internal rates of return has very low variability in the air transportation, telephone and electric power industries (25, 32, 33). To facilitate a more direct comparison with an external bond rate, the annual average and variance of sector-specific internal rates of return, weighted by the sector's proportion of the economy's total capital stock value, were calculated. We define the annual weighted average and variance of internal rates of return across sectors, say f,- and V(rt-), respectively, as Note that as discussed earlier, negative internal rates of return have been set to zero, and for the construction and motor transportation sectors, we have set r2\ = 7*28 = r,-. The results are given in table B.15. For an external bond rate, following Fortin (1988), we use the annual average of McLeod, Young and Weir's monthly 10 industrial bond yield average (also listed in Appendix B. The Canadian Input-Output Data, 1961-1980 206 internal rates of return weighted McLeod, Young and weighted standard Weir's 10 industrial year average deviation bond yield, average 1961 0.083871 0.069202 0.0550 1962 0.091580 0.071394 0.0544 1963 0.094619 0.071858 0.0537 1964 0.097758 0.077412 0.0549 1965 0.096656 0.079609 0.0563 1966 0.094136 0.074457 0.0643 1967 0.083479 0.071633 0.0703 1968 0.089054 0.074986 0.0787 1969 0.087531 0.075099 0.0866 1970 0.079076 0.065497 0.0922 1971 0.080637 0.073217 0.0840 1972 0.083002 0.079482 0.0831 1973 0.093489 0.090409 0.0842 1974 0.087360 0.084356 0.1001 1975 0.074704 0.071649 0.1073 1976 0.071822 0.068574 0.1058 1977 0.066134 0.057062 0.0972 1978 0.068757 0.061839 0.0996 1979 0.075095 0.072375 0.1074 1980 0.074180 0.068493 0.1311 Table B.15: A comparison of calculated internal rates of return and the McLeod, Young and Weir's 10 industrial bond yield average table B.15). The behavior of the two discount rates is illustrated in figure B.39. The two series exhibit different time trends; while internal rates of return show a general decline, the exogenous bond rate has a strong upward trend. The external bond rate, being a nominal interest rate, follows closely the behavior of inflation rates. Theoretically, anticipated inflation should be accounted for by the capital gains term in our general rental price formula. With our static expectations assumption, the rental prices we would obtain using the external bond rates will be biased upwards particularly in the 1970's decade of high inflation rates. This fact highlights the importance of modelling and incorporating asset price expectations in our capital rental price formula. On the other hand, the use of static expectations when capital markets are relatively efficient may be justifiable in calculating internal rates of return that Appendix B. The Canadian Input-Output Data, 1961-1980 207 Figure B.39: The behavior of the weighted average of the internal rates of return and the McLeod, Young and Weir's 10 industrial bond yield average Appendix B. The Canadian Input-Output Data, 1961-1980 208 yield reasonable indicators of returns to equities. Figure B.39 also shows that the volatility of internal rates of return, in the aggregate, is not unreasonable. The range of its mean is even narrower than the range of the external bond rate. As discussed above, the choice of an internal rate of return or an exogenous bond rate can have different implications on the rental price behavior. Profit function estimation, particualrly using full information maximum likelihood methods, can be sensitive to misspecification. For these reasons, two capital rental price series were constructed using the two discount rates. B.2.6 List of retained variables After doing the data manipulations on the original data base, we retain only a subset of variables prior to goods and sector aggregation. The list of retained price and quantity variables for each of the 37 sectors is as follows: For i = 1,2,..., 37: 1. net output: p*, 2. intermediate inputs: p*-, qji, j = 1,2,..., 37 3. primary inputs: i) imports: pM, qM ii) labor: p*L, qL iii) capital services: a) inventories: p^*, q^* b) machinery and equipment: p2*, q2* c) structures: p$*, qlf* d) land: pf*, q?* In subsequent econometric estimation of profit functions, intermediate inputs and primary inputs are treated symmetrically as goods having prices exogenous to the sector. In this manner, we do not impose separability restrictions between intermediate inputs and primary inputs j Appendix B. The Canadian Input-Output Data, 1961-1980 209 which are implicitly assumed with value-added production or profit functions. A graphical plot of prices of output, imports and labor for each sector shows that the price of labor services has increased the most over the twenty-year period. For each sector, we have two sets of capital rental price series using different discount rates: an internal rate of return and an exogenous bond rate. A discussion of some of the output variables in the original data base and government goods dropped in our condensed list is inorder. For each sector, the sum of intermediate output and final demand equals gross output. Intermediate output is purchased by domestic producers as input to production. Final demand is output that ultimately goes for consumption, investment, government purchases and exports. To obtain the gross private domestic product, we add the final demand outputs and subtract the imports of the 37 domestic business sectors. It is the sum of sectoral value added, in our case, returns to labor and capital sevices, that is termed "domestic product" in the usual national accounting framework. Government produced goods and services8 used as inputs in production comprise less than 1% of the value of primary inputs for the whole economy. As such, its omission can be deemed harmless. B .3 Aggregation Under the assumption of single-output industries, four sectors or output goods are specified. To aggregate the original 37 sectors into four general industries, a heuristic approach was taken. Theoretically, aggregation of goods is justified on the basis of either Hicksian separability where restrictions are on the domain of producer prices or functional separability where restrictions are imposed on the production technologies. Testing for functional separability is a separate study in itself and hence is precluded in our empirical analysis. Though the groupings of the 37 sectors were guided as well by traditional industry classifications, an examination of the plots of their output prices (see figures B.40-B.46) indicates that within group price variation 8These government goods include postal services and the like but exclude output of other crown corporations (e.g. Air Canada, Via Rail, CN Rail, CBC) which are recorded as output of the relevant business sectors. Appendix B. The Canadian Input-Output Data, 1961-1980 210 is less than that between groups. Hence, our sector aggregation can be justified by appealing to Hicks aggregation theorem that says goods whose prices move proportionally act as a composite good. The aggregate industries and their subsectors are: I. resources sector (1) agriculture and fishing (2) forestry (3) mines, quarries and oil wells (21) petroleum and coal products II. manufacturing, export-oriented sector9 (11) woods (13) paper and allied industry (15) primary metals (17) machinery (18) transportation equipment III. manufacturing, domestic market-oriented sector (4) food and beverages (5) tobacco products (6) rubber and plastic products (7) leather (8) textiles (9) knitting mills (10) clothing (12) furniture and fixtures (14) printing, publishing and allied industries (16) metal fabricating (19) electrical products (20) nonmetallic mineral products (22) chemical and chemical products (23) miscellaneous manufacturing IV. services sector (24) construction (25) air transportation, other utilities and transportation (26) railway transportation and telegraph information from the output variables classified according to end use in the original data set was used to select these subsectors. Appendix B. The Canadian Input-Output Data, 1961-1980 211 Figure B.40: Output prices for the resources sectors (I) Appendix B. The Canadian Input-Output Data, 1961-1980 p r t c e 212 5.5 -5 -4.5 4 -3.5-2.5 -1 .5 0.5-uoods(11) • p o p e r , e t c ( 1 3 ) • p_rc_metaLs_l_l 5_2 O mo.chi.nerg 117) A t r a n s p o r t ( 1 8 ) 1980 Figure B.41: Output prices for the export market-oriented manufacturing sectors (II) Appendix B. The Canadian Input-Output Data, 1961-1980 213 pruce 5.5^ 4.5 3.5 2.5H i . s H H O.S-\ I960 • food&bev(4) • tobacco(5) • rub&plqs (6J_ O Leather(7) A textLle(8) X k n l t t l n g ( 9 ) 1965 1970 year 1975 1980 Figure B.42: Output prices for the domestic market-oriented manufacturing sectors (III), sub-group 1 Appendix B. The Canadian Input-Output Data, 1961-1980 price 214 5.5-5-• f u r n m x ( 1 2 ) 4.5-• printing!14) _ • metal fob (_16 ) O e l e c prodt19) 2 ^_ A non metol(20) X chemical(22) O r 1960 1965 1970 y e a r 1975 1980 Figure B.43: Output prices for the domestic market-oriented manufacturing sectors (III), sub-group 2 Appendix B. The Canadian Input-Output Data., 1961-1980 215 6-1 pruce 5.5H 5 H 4.5H 3.5-1 3 H 2.5 2H 1 . 5 H 0.5-1 ]960 • olr(25) • roLl(26) • water(27) O motor(28) A urban(29) 1965 1970 year 1975 1980 Figure B.44: Output prices for the services sectors (rV), transportation subgroup Appendix B. The Canadian Input-Output Data, 1961-1980 216 p r t c e 5.5-4.5 3.5-2.5-2-1.5-1 -0.5 I sioroqe(30) • broodccist (31) • telephone (_32)_ O elec power(33) A qos distr(34.) 1960 1965 1970 year 1975 1980 Figure B.45: Output prices for the services sectors (rV), communications and utilities subgroup Appendix B. The Canadian Input-Output Data, 1961-1980 217 Figure B.46: Output prices for the services sectors (rV), construction, trade, f.i.r.e., and com-mercial services subgroup Appendix B. The Canadian Input-Output Data, 1961-1980 218 (27) water transportation (28) motor transportation (29) urban transportation (30) storage (31) broadcasting (32) telephones (33) electric power (34) gas distribution (35) trade (36) finance, insurance and real estate (37) commercial services Aggregation was carried out in two stages: first, at the sectoral level across goods (interme-diate inputs) and second, across sectors to yield our aggregate industries. Prices were obtained as chained Divisia (Tornqvist or translog) indices; quantities were obtained residually such that expenditures are preserved.10 The Tornqvist discrete approximation to the continuous Divisia index is superlative; it is exact for the flexible translog aggregator function. Being a superla-tive index, it has also been shown to possess approximate consistency-in-aggregation property (Diewert, 1980a). For each aggregated sector, we have two sets of capital data based on different discount rates. Rental prices based on internal rates of return are more erratic and generally lower than those based on the external bond rates. This difference is more marked for inventories and land which have either zero property tax or depreciation rates. There was considerable variation in internal rates of return among the original 37 sectors; on the other hand the same external bond rate was applied to all the sectors. Thus, it is not unexpected that rental prices obtained using the external bond rates have a smoother behavior over time and even for a particular capital good, the trends of rental prices among the different aggregated sectors are almost identical. Due to the manner by which prices and quantities were obtained in our aggregation pro-cedure, the quantities of capital input also differ according to the discount rate used. At the level of 37-sector disaggregation, the quantities are identical across the two capital data sets 1 0 The econometrics computer program SHAZAM by K.J . White (1987) was used to compute the Divisia indices. Appendix B. The Canadian Input-Output Data, 1961-1980 219 I. resources sector II. manufacturing sector, export market-oriented III. manufacturing sector, domestic market-oriented IV. services sector Table B.16: List of aggregated sectors which differ only in the rental prices. The implied capital service bill (rental price x quantity of capital input) is therefore conditional on the discount rate as well. Since in the aggregation procedure the capital service bill or expenditure is preserved, different rental prices would then yield different values for the quantity. The above discussion underscores the possibility that our profit function estimation can be sensitive to the specification of the capital variables. Due to the manner by which the land data was constructed by Cas, Diewert and Osten-soe (1986), the land quantity data except for the agriculture and fishing sector at the 37-sector level of disaggregation are in fixed proportions to the quantity of structures (approximately 0.22 to 0.26). Hence, after the sectoral aggregation to four general industries was performed, structures and land were aggregated into a Leontief composite good for the manufacturing and services sectors (II, III and IV) in the following manner: (new)g3*'* = q** + q? *, where the right hand side variables are as described in the list of retained variables at the sectoral level, and the new price variable is obtained such that the value of expenditure is preserved. Hereon, for convenience, we shall refer to the Leontief aggregate of structures and land in the aggregated sectors II, III and IV as simply structures. For the resources sector (I), we retain separate variables for structures and land. In summary, we have four aggregated industries (table B.16) and ten goods (table B.17). For each sector, the first four goods are either a net output or an intermediate input. The last Appendix B. The Canadian Input-Output Data, 1961-1980 220 (1) resource goods (from sector I) (2) manufactured goods (from sector II) (3) manufactured goods (from sector III) (4) service goods (from sector IV) (5) imports (6) labor • (7) inventories (8) machinery and equipment (9) structures (10) land Table B.17: List of goods six goods are primary inputs. The land price and quantity data are effectively set to zero for the manufacturing and services sectors (II, III and IV). For subsequent empirical analysis, quantities of net outputs are indexed positively and quantities of net inputs are indexed negatively. For scaling purposes, the quantities are further divided by a factor of 106; hence, the value of expenditure which can be obtained by multiplying the relevant price and quantity variables are in units of billion Canadian (current) dollars. Prices are normalized such that 1961 prices are equal to 1.0. B.4 The constructed data sets The constructed sectoral accounts with the capital rental prices based on internal rates of return are listed below. The constructed capital data series based on an exogenous bond rate are also listed separately for each of the four sectors. Appendix B. The Canadian Input-Output Data, 1961-1980 221 Table B.18: The data for sector I, the resources sector Sector I: Resources net output resources exp-manuf year quantity price quantity price quantity price 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 6.388490 7.052499 7.549368 7.826248 8.172144 8.801898 8.426695 8.951132 9.282653 9.401631 9.970642 10.329735 11.729394 11.345067 10.910657 11.313760 11.483928 11.481984 12.255849 12.640729 1.000000 1.037613 1.036749 1.038113 1.061550 1.114239 1.139288 1.148554 1.167532 1.206057 1.201196 1.287679 1.586959 2.163120 2.492335 2.580402 2.795015 3.121217 3.699945 4.411511 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.108087 0.111440 0.121395 0.137002 0.151658 0.174431 0.179564 0.190346 0.180709 0.196889 0.195240 0.189899 0.223233 0.235692 0.221426 0.233222 0.255290 0.252023 0.282798 0.271277 1.000000 1.014777 1.029922 1.045107 1.067568 1.094045 1.118140 1.146075 1.181067 1.225520 1.252829 1.288753 1.378919 1.608616 1.811454 1.937699 2.085647 2.273325 2.600041 2.923751 dom-manuf services imports year quantity price quantity price quantity price 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 0.629793 0.668620 0.695051 0.743054 0.777908 0.872301 0.898229 0.897794 0.886964 0.908480 0.912856 0.940359 1.165134 1.182529 1.162251 1.210588 1.221552 1.237485 1.305133 1.360189 1.000000 1.009155 1.026388 1.038356 1.050673 1.083464 1.099892 1.114265 1.153143 1.175219 1.208091 1.267690 1.411643 1.692004 1.901332 1.959881 2.066206 2.260185 2.562756 2.873773 1.025314 1.126051 1.172576 1.248063 1.335080 1.405787 1.443741 1.494201 1.525226 1.551686 1.603493 1.703694 2.012527 2.103291 2.122066 2.295185 2.432256 2.545984 2.784181 2.973667 1.000000 1.009202 1.022710 1.038945 1.062627 1.095994 1.140893 1.183047 1.236926 1.297293 1.350673 1.408225 1.492519 1.673627 1.872227 2.079581 2.244244 2.407110 2.551099 2.872205 0.651119 0.577858 0.589565 0.591657 0.604773 0.645203 0.641647 0.705593 0.771333 0.797622 1.133501 1.269822 1.438189 1.378212 1.372453 1.342869 1.257737 1.261508 1.299690 1.233598 1.000000 1.233829 1.268691 1.309422 1.320841 1.370586 1.370002 1.379412 1.356426 1.385346 1.144341 1.189303 1.357069 2.893564 3.429959 3.556358 3.964575 4.336803 5.532236 8.194973 Appendix B. The Canadian Input-Output Data, 1961-1980 ooo Table B.18 (continued) Sector I: Resources labor inventories cap-M&E year quantity price quantity price quantity price 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1.526986 1.537920 1.488821 1.478606 1.504062 1.474743 1.464122 1.400979 1.349746 1.371695 1.353306 1.293386 1.360092 1.414700 1.382598 1.384431 1.399334 1.408160 1.472186 1.513216 000000 155035 216110 204332 295214 493066 496646 630398 808434 906155 095066 .299937 907178 291578 631992 020559 408647 879079 181565 845244 0.257658 0.236531 0.250657 0.270415 0.259018 0.249352 0.256565 0.252045 0.260671 0.294792 0.301009 0.276501 0.196503 0.169916 0.150509 0.153134 0.166939, 0.145943 0.120251 0.105367 1.000000 1.673956 1.857383 1.611608 1.690091 2.217558 1.430552 1.541310 1.573871 1.308617 1.253800 1.483027 3.510525 5.003183 5.085024 4.018019 3.427975 4.297119 5.529375 6.126964 0.974477 0.967728 0.979927 1.008983 1.045854 1.096940 1.166594 1.229138 1.271160 1.295795 1.320508 1.371592 1.457358 1.563316 1.690316 1.841283 2.004779 2.126939 2.238255 2.303775 cap-S land year quantity price quantity price 1.051897 1.130091 1.200532 1.274241 1.356734 1.451109 1.553798 1.651811 1.747637 1.847144 1.954259 2.049686 2.131857 2.221274 2.319290 2.427455 2.532432 2.664501 2.812549 3.075338 000000 062735 120759 127154 116906 175998 080166 117851 125036 116097 079182 207907 843159 269770 309336 314483 485781 597678 439241 859592 0.157218 0.169682 0.182722 0.198536 0.217224 0.238384 0.261559 0.281879 0.296819 0.306257 0.325562 0.353363 0.371761 0.398255 0.446428 0.497528 0.545635 0.615000 0.705863 0.818887 1.000000 1.055788 1.123692 1.180153 1.180537 1.249582 1.125995 1.200644 1.146463 1.098177 0.948264 1.127055 2.022788 2.624213 2.804449 2.728358 2.763854 2.875689 3.891534 •4.578141 1.000000 1.158633 1.231795 1.230761 1.252861 1.344356 1.154664 1.184329 1.195877 1.164084 1.155752 1.224451 1.620626 1.942379 2.068841 1.998528 2.001755 2.279105 2.777311 3.146931 Appendix B. The Canadian Input-Output Data, 1961-1980 223 Table B.19: The data for sector II, the manufacturing sector (export market-oriented) Sector II: Manufacturing, Export Market-Oriented net output resources exp-manuf year quantity price quantity price quantity price 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 7.089804 7.756834 8.462628 9.485412 10.567053 11.235968 11.725869 12.831917 13.987605 13.567631 14.234698 15.544878 17.314678 18.041398 16.492716 17.809943 18.486470 19.701988 20.284628 19.557889 1.000000 1.015587 1.027039 1.041505 1.062243 1.089246 1.112759 1.141234 1.176512 1.214419 1.236027 1.284886 1.399653 1.632355 1.835973 1.956790 2.128178 2.346463 2.700821 3.002165 1.341709 1.383495 1.428701 1.553258 1.648344 1.665766 1.743833 1.838889 1.950108 2.099769 1.979726 2.112818 2.337245 2.185011 1.692250 1.850811 1.765179 1.779699 1.934461 2.001963 1.000000 1.022700 1.038229 1.056743 1.086871 1.128543 1.171423 1.211578 1.249376 1.271914 1.291096 1.353306 1.586489 2.031106 2.414300 2.710048 3.038581 3.392082 3.966979 4.513225 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 dom-manuf imports year quantity price quantity price quantity price 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 0.568828 0.671022 0.770343 0.882642 0.999005 1.085627 1.115533 1.207253 1.335115 1.275334 1.388669 1.427852 1.559194 1.590350 1.498918 1.579308 1.592931 1.704987 1.712680 1.725187 1.000000 1.000327 1.005779 1.017522 1.035669 1.055387 1.075190 1.083365 1.110239 1.142737 1.168522 1.199486 1.276929 1.533877 1.735936 1.837759 1.945536 2.099040 2.384405 2.714167 0.906523 0.982276 1.068108 1.197216 1.345723 1.483687 1.577666 1.683154 1.798719 1.799884 1.840433 1.991768 2.147521 2.340701 2.259968 2.389705 2.477600 2.688620 2.940197 3.015241 1.000000 0.997330 0.997897 1.007568 1.023561 1.046020 1.084100 1.123447 1.173175 1.231258 1.276767 1.328342 1.402412 1.566645 1.755429 1.942656 2.095455 2.256407 2.413208 2.700301 0.977387 1.081217 1.228443 1.468789 1.708937 1.919632 2.007965 2.459309 2.691873 2.595263 3.044751 3.485332 3.963157 4.313688 4.081083 4.369601 4.568010 4.882679 5.078234 4.482947 1.000000 1.081267 1.094631 1.113372 1.125834 1.156208 1.200146 1.219124 1.254088 1.294618 1.240952 1.269917 1.341706 1.558066 1.865408 1.947160 2.194062 2.507265 2.807190 3.141109 Appendix B. The Canadian Input-Output Data, 1961-1980 224 Table B.19 (continued) Sector II: Manufacturing, Export Market-Oriented labor inventories cap-M&E year quantity price quantity price quantity price 1961 2.119785 1.000000 0.139830 1.000000 0.617652 1.000000 1962 2.241733 1.020438 0.140142 1.285324 0.635518 1.109476 1963 2.347976 1.058602 0.154408 1.443560 0.660243 1.207313 1964 2.516343 1.097354 0.167456 1.510000 0.702184 1.307317 1965 2.686661 1.154747 0.190346 1.619491 0.763447 1.380739 1966 2.815390 1.241190 0.219303 1.338196 0.839970 1.232211 1967 2.828762 1.340864 0.232885 1.328888 0.913000 1.071073 1968 2.809657 1.456240 0.242194 1.570020 0.954506 1.127606 1969 2.890357 1.557259 0.258157 1.817067 0.991791 1.289792 1970 2.799147 1.686883 0.267666 1.008121 1.056741 1.044481 1971 2.764914 1.814393 0.219643 1.245930 1.121957 1.111256 1972 2.906985 1.953254 0.218953 1.575181 1.164555 1.238862 1973 3.072781 2.140171 0.240360 2.229533 1.214828 1.588370 1974 3.126569 2.484959 0.269613 2.151007 1.280471 1.866524 1975 2.927767 2.840254 0.280483 1.621759 1.340268 1.569852 1976 3.007986 3.310478 0.252980 1.712065 1.382314 1.602402 1977 3.027801 3.643374 0.268715 1.921505 1.416345 1.894035 1978 3.172290 3.878281 0.289846 2.705790 1.446666 2.368748 1979 3.266791 4.213563 0.321056 3.601053 1.480658 2.986773 1980 3.259662 4.738964 0.356681 2.351243 1.547863 2.812982 cap-S year quantity price 1961 0.414444 1.000000 1962 0.420784 1.107176 1963 0.430535 1.190106 1964 0.448799 1.248288 1965 0.478177 1.308641 1966 0.517172 1.181744 1967 0.553215 1.079364 1968 0.574461 1.152858 1969 0.594162 1.334347 1970 0.622544 1.071653 1971 0.645474 1.147545 1972 0.658373 1.296258 1973 0.675827 1.706880 1974 0.701327 2.043999 1975 0.727206 1.550815 1976 0.744526 1.611738 1977 0.761717 1.864861 1978 0.779613 2.275004 1979 0.800990 2.884777 1980 0.834024 2.530481 Appendix B. The Canadian Input-Output Data, 1961-1980 225 Table B.20: The data for sector III, the manufacturing sector (domestic market-oriented) Sector III: Manufacturing, Domestic Market-Oriented net output resources exp-manuf year quantity price quantity price quantity price 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 12.037386 12.987668 13.605353 14.735647 15.823795 16.931363 17.241228 18.071040 19.055620 19.055881 20.057231 21.429801 22.864381 23.424968 22.408561 23.881598 24.314540 25.561322 26.803966 26.585478 1.000000 1.013297 1.030103 1.040056 1.053652 1.085651 1.107805 1.121121 1.159219 1.187352 1.216044 1.271673 1.402400 1.658326 1.862847 1.935665 2.048225 2.236829 2.528423 2.823797 1.828422 1.863090 1.946189 2.098514 2.166830 2.257699 2.354801 2.395088 2.439193 2.512004 2.691736 2.766351 2.621073 2.449706 2.654289 2.796468 2.987332 3.123728 3.172526 3.133351 1.000000 1.062329 1.057284 1.052316 1.080163 1.144946 1.158415 1.163200 1.197145 1.218478 1.191584 1.325867 1.758798 2.228840 2.335712 2.317120 2.410752 2.795465 3.299165 3.620208 1.169241 1.259862 1.351255 1.447883 1.584562 1.737049 1.728370 1.781939 1.832271 1.865613 1.934796 2.094266 2.232044 2.285692 2.098421 2.177060 2.128802 2.219740 2.236875 2.132916 1.000000 1.018057 1.022763 1.041157 1.069105 1.101336 1.122732 1.141395 1.180501 1.243472 1.244704 1.274907 1.418403 1.766253 2.002042 2.132037 2.299330 2.477568 2.966844 3.394111 dom-manuf services imports year quantity price quantity price quantity price 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.751402 1.881884 1.993663 2.130058 2.283580 2.436646 2.480078 2.539887 2.674857 2.682302 2.754310 2.916878 3.086557 3.245139 3.245954 3.307732 3.361782 3.578964 3.861903 3.952335 1.000000 1.004487 1.009156 1.020611 1.041980 1.070147 1.110855 1.151481 1.201688 1.258386 1.305960 1.359511 1.436884 1.598958 1.788455 1.977327 2.123620 2.280534 2.428958 2.717233 1.612199 1.600771 1.663543 1.822574 1.976544 2.180249 2.205295 2.313007 2.560571 2.459276 2.814057 3.130511 3.467886 3.841500 3.381120 3.637338 3.547960 3.843679 4.244077 4.146493 1.000000 1.079881 1.141548 1.140279 1.114379 1.126500 1.125105 1.124609 1.165566 1.203446 1.120774 1.159296 1.314222 1.666344 1.800503 1.818480 2.030523 2.238825 2.634669 3.210584 Appendix B. The Canadian Input-Output Data, 1961-1980 226 Table B.20 (continued) Sector III: Manufacturing, Domestic Market-Oriented labc inventories cap-M&E year quantity price quantity price quantity price 1961 3.866455 1.000000 0.390087 1.000000 0.790091 1.000000 1962 4.007279 1.028578 0.402435 1.215633 0.816207 1.159242 1963 4.094212 1.062301 0.424859 1.252342 0.842033 1.208093 1964 4.265746 1.099703 0.437419 1.384666 0.875287 1.375407 1965 4.471031 1.148536 0.466149 1.384827 0.928024 1.442232 1966 4.658253 1.235270 0.505314 1.318669 1.001514 1.400918 1967 4.656392 1.323675 0.528980 1.190127 1.075553 1.274247 1968 4.634462 1.409570 0.542721 1.342566 1.133056 1.336079 1969 4.683603 1.514101 0.565547 1.410297 1.184919 1.401385 1970 4.556537 1.626044 0.586728 1.196164 1.238159 1.281890 1971 4.532065 1.743428 0.500339 1.503670 1.287277 1.469134 1972 4.611433 1.882876 0.520267 1.685027 1.332858 1.570679 1973 4.764969 2.058962 0.565381 1.946266 1.395390 1.737622 1974 4.833621 2.380140 0.628511 2.131463 1.475864 1.856973 1975 4.752283 2.722542 0.644599 2.040733 1.555075 1.920806 1976 4.723326 3.150614 0.623038 2.102718 1.626614 2.005039 1977 4.565358 3.472681 0.619148 2.108469 1.688481 2.118139 1978 4.685567 3.684217 0.645320 2.455621 1.734585 2.483261 1979 4.815412 4.019026 0.687416 2.948763 1.782644 2.890741 1980 4.775804 4.572550 0.707758 2.674895 1.813132 2.874828 cap-S year quantity price 1961 0.606574 1.000000 1962 0.623151 1.119752 1963 0.642046 1.152039 1964 0.665170 1.245956 1965 0.699341 1.294947 1966 0.746907 1.287254 1967 0.791364 1.253040 1968 0.828119 1.322063 1969 0.866487 1.395417 1970 0.906805 1.283205 1971 0.940722 1.517903 1972 0.963127 1.670885 1973 0.988821 1.921457 1974 1.025331 2.057440 1975 1.062938 2.019534 1976 1.092270 2.220401 1977 1.124662 2.255490 1978 1.161963 2.505986 1979 1.199084 2.862710 1980 1.228090 2.727067 Appendix B. The Canadian Input-Output Data, 1961-1980 227 Table B.21: The data for sector IV, the services sector Sector IV: Services net output resources exp-manuf year quantity price quantity price quantity price 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 25.447810 26.676252 27.940061 30.025948 32.214360 34.453836 35.558786 36.930822 38.535515 39.557438 42.250994 44.775514 47.911077 50.275018 51.862215 54.699236 56.292097 58.398141 61.554562 63.767157 1.000000 1.004829 1.020665 1.038074 1.067979 1.114736 1.161715 1.201863 1.261842 1.325342 1.381803 1.445799 1.550809 1.749947 1.968710 2.165522 2.326202 2.498532 2.653385 2.967861 0.875907 0.852661 0.894182 0.955285 1.007009 1.156012 1.104906 1.086179 1.136589 1.163318 1.196356 1.329988 1.423878 1.364078 1.365785 1.432804 1.600500 1.746025 1.879707 1.937400 1.000000 1.034028 1.033111 1.039502 1.047827 1.072106 1.092991 1.104841 1.115011 1.139757 1.174812 1.234390 1.476695 2.142910 2.468426 2.757956 3.123400 3.571692 4.067483 4.809709 1.280840 1.264061 1.352863 1.470477 1.492395 1.569492 1.635840 1.653321 1.671295 1.707698 1.870303 1.898392 1.815247 1.838916 1.794571 1.974918 1.985405 1.953228 1.997412 1.980460 1.000000 1.032044 1.053248 1.088668 1.122017 1.156154 1.189471 1.249679 1.306078 1.314224 1.348444 1.466019 1.670257 1.890350 2.053709 2.221193 2.423828 2.693428 3.118340 3.317524 dom-manuf services imports year quantity price quantity price quantity price 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 2.850217 3.080031 3.233912 3.475665 3.763027 3.986007 4.027994 4.169141 4.249828 4.355433 4.839141 4.978960 5.249498 5.233725 5.225492 5.499952 5.665069 5.783386 5.950791 5.940092 1.000000 1.014433 1.027262 1.055144 1.080331 1.111411 1.140687 1.156044 1.195202 1.235274 1.267082 1.317792 1.425952 1.687207 1.882823 1.989473 2.107197 2.275023 2.557222 2.861909 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.032118 0.717764 0.705851 0.793994 0.863286 0.978423 1.031074 1.082051 1.219990 1.214786 1.970859 2.170307 2.380077 2.505475 2.399229 2.511302 2.689136 2.819863 3.057310 3.139288 1.000000 1.517208 1.577892 1.688200 1.723218 1.743120 1.761693 1.723657 1.752065 1.851175 1.306229 1.355255 1.458526 1.710694 1.913355 2.011226 2.232756 2.482589 2.776580 3.070752 Appendix B. The Canadian Input-Output Data, 1961-1980 2 2 8 Table B.21 (continued) Sector IV: Services labor inventories cap-M&E year quantity price quantity price quantity price 1961 11.234121 1.000000 0.499994 1.000000 1.460116 1.000000 1962 11.619290 1.023880 0.520299 0.956480 1.518303 1.040389 1963 11.895067 1.063034 0.548213 1.006311 1.574607 1.073694 1964 12.366934 1.102978 0.561931 1.110753 1.647954 1.127365 1965 13.090155 1.160666 0.572043 1.094808 1.751854 1.145944 1966 13.535069 1.253518 0.597625 1.199915 1.895576 1.154594 1967 13.665839 1.351107 0.581371 1.217706 2.070180 1.137151 1968 13.666817 1.451743 0.533125 1.311124 2.246863 1.162876 1969 14.123674 1.569194 0.505476 1.391735 2.436544 1.166433 1970 14.110210 1.688741 0.539809 1.495045 2.631226 1.205674 1971 14.462436 1.837723 0.442178 1.630938 2.814189 1.244395 1972 15.067784 1.968243 0.361624 1.937491 3.046717 1.294961 1973 15.971162 2.166338 0.372176 2.175214 3.373524 1.347917 1974 16.929037 2.450203 0.348171 2.383500 3.764153 1.428437 1975 17.256626 2.816291 0.447577 2.540867 4.158640 1.503969 1976 17.533465 3.171177 0.488874 2.730098 4.533461 1.561888 1977 17.944677 3.394450 0.486190 2.087035 4.868069 1.644222 1978 18.519227 3.573324 0.401134 2.866397 5.194545 1.879136 1979 19.221697 3.748176 0.345426 3.959155 5.583736 2.077162 1980 19.717578 4.296999 0.359586 4.338340 6.015682 2.224588 cap-S year quantity price 1961 5.825608 1.000000 1962 6.049437 1.007619 1963 6.285030 1.026021 1964 6.558808 1.064720 1965 6.882067 1.112967 1966 7.235363 1.180646 1967 7.569433 1.226004 1968 7.890240 1.304001 1969 8.234933 1.348724 1970 8.581558 1.422943 1971 8.969297 1.513358 1972 9.443831 1.585623 1973 9.988914 1.652258 1974 10.582015 1.747954 1975 11.195258 1.933909 1976 11.826278 2.243971 1977 12.535674 2.385420 1978 13.231093 2.540867 1979 13.870268 2.628307 1980 14.377347 2.957135 Appendix B. The Canadian Input-Output Data, 1961-1980 229 Sector I: Resources inventories cap-M&E cap-S year quantity price quantity price quantity price 1961 0.227087 1.000000 0.848760 1.000000 0.734308 1.000000 1962 0.203089 1.047303 0.839489 1.036271 0.781798 1.009704 1963 0.215036 1.030674 0.848161 1.061460 0.826633 1.042815 1964 0.232245 1.048629 0.872371 1.094989 0.872262 1.056381 1965 0.222952 1.100948 0.904941 1.125648 0.922479 1.108032 1966 0.215263 1.320082 0.949056 1.182138 0.982136 1.235355 1967 0.221975 1.455095 1.008377 1.208545 1.047189 1.348057 1968 0.217291 1.626530 1.061347 1.266151 1.108862 1.454962 1969 0.227130 1.826663 1.096548 1.348249 1.170762 1.624284 1970 0.252924 1.972613 1.115754 1.425845 1.236791 1.736279 1971 0.251413 1.777211 1.135326 1.420743 1.309196 1.833680 1972 0.231957 1.941780 1.179486 1.464467 1.375157 1.951953 1973 0.169166 2.569619 1.254671 1.518346 1.432420 2.161825 1974 0.151474 3.849088 1.350557 1.819539 1.495302 2.694249 1975 0.142889 4.440838 1.462862 2.133749 1.565270 3.124458 1976 0.149687 4.590336 1.591711 2.271819 1.635938 3.494099 1977 0.156511 4.553407 1.729738 2.358169 1.702191 3.698859 1978 0.142418 5.391131 1.834384 2.665340 1.785242 4.015207 1979 0.120502 6.779880 1.931452 3.125832 1.874581 4.624146 1980 0.113386 9.064909 1.997021 3.862775 2.026919 5.466408 land year quantity price 1961 0.086268 1.000000 1962 0.091847 1.030529 1963 0.098377 1.075770 1964 0.106453 1.122775 1965 0.115995 1.229504 1966 0.126522 1.443417 1967 0.137918 1.654847 1968 0.148022 1.882855 1969 0.155267 2.087160 1970 0.159798 2.211249 1971 0.170117 2.246751 1972 0.184393 2.462951 1973 0.194197 2.783838 1974 0.207830 3.715461 1975 0.231935 4.717755 1976 0.252724 5.459365 1977 0.276813 5.479471 1978 0.310122 5.968442 1979 0.354174 6.871396 1980 0.413645 9.166418 Table B.22: Capital data series based on an exogenous bond rate, sector I Appendix B. The Canadian Input-Output Data, 1961-1980 230 Sector II: Manufacturing, Export Market-Oriented inventories cap-M&E cap-S year quantity price quantity price quantity price 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 0.076313 0.076596 0.083965 0.091586 0.102387 0.115252 0.123113 0.128381 0.134569 0.138769 0.121653 0.122501 0.129907 0.145378 0.152757 0.149906 0.149504 0.157234 0.172016 0.182643 1.000000 1.004770 1.005430 1.041521 1.088788 1.265452 1.406697 1.612187 1.818726 1.973151 1.852294 1.914694 2.111803 2.909187 3.511676 3.781411 3.792476 4.239261 5.207380 6.869955 0.472856 0.486746 0.505983 0.537510 0.583267 0.641311 0.697485 0.727795 0.752919 0.800110 0.852064 0.888178 0.920078 0.965346 1.013320 1.046277 1.070925 1.088803 1.107604 1.153381 1.000000 1.026605 1.064638 1.139774 1.217316 1.313045 1.329949 1.388745 1.513458 1.616918 1.603145 1.656954 1.775663 2.209229 2.636296 2.814122 3.003033 3.415912 3.978079 4.950691 0.310777 0.315613 0.322750 0.335343 0.356255 0.384495 0.410196 0.425820 0.440558 0.461652 0.480921 0.493828 0.505614 0.522270 0.541596 0.554417 0.567320 0.579464 0.591812 0.614269 000000 021858 047083 083789 140548 248588 339978 426479 571085 698275 703257 789545 927482 381916 741109 019335 ,114732 376527 .828683 ,641175 Table B.23: Capital data series based on an exogenous bond rate, sector II Appendix B. The Canadian Input-Output Data, 1961-1980 231 Sector III: Manufacturing, Domestic Market-Oriented inventories cap-M&E cap-S year quantity price quantity price quantity price 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 0.164678 0.169764 0.178758 0.184200 0.195846 0.211290 0.222683 0.228521 0.238340 0.248481 0.207857 0.215227 0.232139 0.256425 0.263251 0.254863 0.253324 0.264671 0.280701 0.288731 1.000000 0.995672 0.995883 1.024868 1.065092 1.243829 1.380210 1.546356 1.738964 1.886586 1.770582 1.811295 2.001246 2.846886 3.321801 3.429188 3.368375 3.731241 4.586677 6.133201 0.516827 0.532623 0.548827 0.570569 0.605757 0.654497 0.703041 0.740182 0.772203 0.804503 0.833562 0.860768 0.899922 0.953438 1.009977 1.065924 1.116926 1.156359 1.194850 1.219294 1.000000 1.042179 1.074446 1.164916 1.249673 1.355496 1.364515 1.424216 1.543129 1.647465 1.620201 1.644342 1.731891 2.132474 2.533978 2.652419 2.829180 3.237520 3.750944 4.650824 0.383919 0.393531 0.404951 0.419491 0.441394 0.471838 0.500381 0.524520 0.549957 0.576534 0.598815 0.613483 0.629924 0.653947 0.681459 0.707355 0.736231 0.768397 0.796496 0.815957 000000 021812 037866 069006 139481 248308 343386 421186 553902 679964 701685 786588 917496 353066 705036 987202 091905 359088 801051 618980 Table B.24: Capital data series based on an exogenous bond rate, sector III Appendix B. The Canadian Input-Output Data, 1961-1980 2 3 2 Sector IV: Services inventories cap-M&E cap-S year quantity price quantity price quantity price 1961 0.233352 1.000000 1.315189 1.000000 5.552619 1.000000 1962 0.241952 0.991364 1.352735 1.041860 5.762240 1.009823 1963 0.253990 0.986217 1.386427 1.064014 5.975821 1.024876 1964 0.260214 1.019418 1.433827 1.099282 6.225290 1.071754 1965 0.264663 1.063537 1.510808 1.129523 6.515181 1.142834 1966 0.276367 1.240609 1.622925 1.201201 6.818234 1.304128 1967 0.270340 1.392451 1.760261 1.245401 7.112892 1.450180 1968 0.249144 1.592179 1.900871 1.307811 7.413251 1.594196 1969 0.237284 1.804711 2.047023 1.395441 7.737688 1.786155 1970 0.251856 1.989671 2.192816 1.492537 8.064885 1.923506 1971 0.211834 1.888489 2.328460 1.478687 8.427107 1.921061 1972 0.176606 1.948844 2.487258 1.509143 8.866944 2.043214 1973 0.179386 2.108799 2.706953 1.568481 9.371714 2.286863 1974 0.168302 2.795965 2.975054 1.885795 9.907648 2.955057 1975 0.209924 3.354079 3.245710 2.193212 10.440952 3.495112 1976 0.230993 3.610360 3.506838 2.273766 10.999748 3.866969 1977 0.226609 3.540756 3.741300 2.405548 11.656002 3.980235 1978 0.190276 3.957381 3.969147 2.757107 12.316026 4.319935 1979 0.163974 4.672649 4.234024 3.122160 12.922131 4.878309 1980 0.172828 6.299277 4.522550 3.705302 13.400917 5.987700 Table B.25: Capital data series based on an exogenous bond rate, sector IV Appendix C Computer Programs for Implementing the Nonparametric Efficiency Tests C.l Introduction to the computer programs Implementation of the nonparametric efficiency tests given in part I is illustrated by the follow-ing sample FORTRAN programs. The technical efficiency tests (tests 1-3) and the allocative efficiency tests (tests 4-6) require the solution of linear programming problems. The user-callable subprogram LIPSUB originally written by Dennis O'Reilly of the University Computing Services of the University of British Columbia is used. There are three subroutines: 1. TESTQ — performs the technical efficiency tests 1, 2 and 3 which require quantity data; 2. TESTPQC — performs the allocative efficiency tests 4, 5 and 6 assuming partial profit maximization and which require price and quantity data; and 3. TESTPQ — performs the allocative efficiency tests 7, 8 and 9 assuming complete profit maximization and which require price and quantity data. The variant of the efficiency test incorporating the no technological regress assumption (listed in appendix A) can be performed by specifying the integer variable JTECH appropriately in the main calling program; set JTECH=1 if the no technological regress assumption is desired to be incorporated in the efficiency test, or set JTECH=0, otherwise. For the allocative efficiency tests 4 and 7 which assume a convex technology, if it is desired to restrict the optimal profits to be nonnegative (as described in subsection 7.2.6), then either set JP0SP=1 or JPMU=1 in the main calling program. Altogether, the above three subroutines are capable of performing twenty-two kinds of efficiency tests. The subroutines need not be modified by the user and all data inputting can be made in the main calling programs. 233 Appendix C. Computer Programs for Implementing the Nonparametric 234 Sample main programs calling the subroutines are also given. Since the main calling pro-grams will be specific to the data being processed and the efficiency tests desired, for illustrative purposes, we use the data on sector I, the resources sector.1 Note that for the resources sec-tor (I), there are 20 observations (1961-1980) and 10 goods (see table B.17). The sample main programs are in the files: 1. MAINQl — calls the subroutine TESTQ, 2. MAINPQC1 — calls the subroutine TESTPQC, and 3. MAINPQ1 — calls the subroutine TESTPQ. Variables whose values have to be specified by the user are noted in the listings. For allocating space, the dimensioning is done through the PARAMETER option; the user has to specify the value of the following variables: JOBS = number of observations (J), JGDS = number of goods (iV), JGOB = number of observations + number of goods (J + N), JOBS2 = number of observations + 2 (J + 2), JGDS2 = number of goods + 2 (N + 2), and JGOB2 = number of observations + number of goods + 2 (J + N + 2). For the tests requiring the solution of linear programming problems, a detailed version of the results which includes the linear programming tableaus and solutions can be printed out by specifying JDET=1. 2 The quantity variable is stored in the variable Z(n,j) where the first dimension refers to the goods index and the second dimension refers to the observation index. The price variable is stored in the variable P(n, j) and follows the same dimensioning convention as the quantity variable. If only the technical efficiency tests (1-3) are to be performed, the user need not enter 'See appendix B for a detailed description and listing of the data. 2This can be helpful in debugging when a problem is encountered in the solution of the linear programming problem. For most purposes, JDET=0 will suffice. Appendix C. Computer Programs for Implementing the Nonparametric . 235 the price data. The efficiency direction vector 7 is stored in the variables JDIR(-) of dimension JGDS. The user has to specify the values of the individual components JDIR(n) as +1, -1 or 0. For tests 4-6 which assume partial profit maximization, the set S containing the indices of goods with respect to which the producer can optimize and the set E containing the indices of the reference goods are specified through the variables JSET(-) and JNUM(-) 3, respectively, both of dimension JGDS. Let JSET(n)=l if good n is in S; set JNUM(rc)=l if good n is in E. For tests 3, 6 and 9 which assume a quasiconcave technology, a good with respect to whose technology the test is to be performed has to be singled out; this good is specified by JFNC=n where n is the index of the good chosen to play an asymmetric role. For test 8 which assumes a convex cone technology and complete profit maximization, the normalizing good is specified by JNORM=n where n is the index of the good chosen. The desired efficiency test is performed by specifying JTEST=£ where t is the number (listed in table C.26) corresponding to the efficiency test and calling the appropriate subroutine. A summary of the different efficiency tests, their informational requirements and options are listed in table C.26. Listings of the sample main calling programs and subroutines follow. Sample results using the data on the resources sector (I) are also given. C.2 Listings of computer programs C.2.1 Subroutines TESTQ, TESTPQC and TESTPQ Listing of subroutine TESTQ SUBROUTINE TESTQ(Z,JGDS,JGDS2,JOBS,J0BS2,JG0B2,JDIR, 1 TABL0,JVIN,JV0UT,0PTIM,T0L, 2 X, JPLAM, ZP, ZDIF, ZPDIF, JVIO, VVIO, JPASS, JGEZ, ZW, 3 JFRD.JPASSW.JPASSS) C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c C TEST 1. TECHNICAL EFFICIENCY TEST C QUANTITY DATA C CONVEX TECHNOLOGY C C TEST 2. TECHNICAL EFFICIENCY TEST C QUANTITY DATA C CONSTANT RETURNS TO SCALE TECHNOLOGY C 3In the sample computer,programs below, the reference goods are called "numeraire" goods. Appendix C. Computer Programs for Implementing the Nonparametric . 236 test and subroutine called informational requirements options 1. Test 1: TESTQ, JTEST=1 JOBS, J0BS2, JGDS, JGDS2, JG0B2, Z(n,j), JDIR(n) JDET=0/1 JTECH=0/1 2. Test 2: TESTQ, JTEST=2 JOBS, JOBS2, JGDS, JGDS2, JGOB2, Z(n,j), JDIR(n) JDET=0/1 JTECH=0/1 3. Test 3: TESTQ, JTEST=3 JOBS, J0BS2, JGDS, JGDS2, JG0B2, Z(n,i), JDIR(n), JFNC=n JDET=0/1 JTECH=0/1 4. Test 4: TESTPQC, JTEST=4 JOBS, J0BS2, JGDS, JGDS2, JGOB2, Z(n,j), P(n,j), JSET(n), JNUM(n) JDET=0/1 JTECH=0/1 JPOSP=0/1 or JPMU=0/1 5. Test 5: TESTPQC, JTEST=5 JOBS, JOBS2, JGDS, JGDS2, JGOB2, Z(n, j), P(n,j), JSET(n), JNUM(n) JDET=0/1 JTECH=0/1 6. Test 6: TESTPQC, JTEST=6 JOBS, JOBS2, JGDS, JGDS2, JG0B2, Z(n,j), P(n,j), JSET(n), JNUM(n), JFNC=n JDET=0/1 JTECH=0/1 7. Test 7: TESTPQ, JTEST=7 JOBS, JOBS2, JGDS, JGDS2, JGOB, Z(n, j), P(n,j), JDIR(n) JTECH=0/1 JPOSP=0/1 or JPMU=0/1 8. Test 8: TESTPQ, JTEST=8 JOBS, JOBS2, JGDS, JGDS2, JGOB, Z(n, j), P(n,j), JDIR(n), JNORM=n JTECH=0/1 9. Test 9: TESTPQ, JTEST=9 JOBS, JOBS2, JGDS, JGDS2, JGOB, Z(n, j), P(n,j), JDIR(n), JFNC=n JTECH=0/1 Table C.26: Summary of the different efficiency tests, informational requirements and options Appendix C. Computer Programs for Implementing the Nonparametric 237 C TEST 3. TECHNICAL EFFICIENCY TEST C QUANTITY DATA C qUASICONCAVE TECHNOLOGY C C JTECH = 0 ASSUMES NO TECHNICAL CHANGE C =1 ASSUMES NO TECHNOLOGICAL REGRESS C IMPLICIT REAL*8(B-I,K-Z),CHARACTER*12(A) DIMENSION Z(JGDS,JOBS),JDIR(JGDS),TABLO(JGDS2,J0BS2), 1 JVIN(JGDS2),JV0UT(J0BS2).OPTIM(JOBS), 2 X(JG0B2),JPLAM(JOBS),ZP(JGDS),ZDIF(JGDS).ZPDIF(JGDS), 3 JVIO(JOBS).VVIO(JOBS),JPASS(JOBS),JGEZ(JOBS), 4 ZW(JGDS),JFRD(JOBS),JPASSW(JOBS),JPASSS(JOBS) CHARACTER A0FL0W*8 DATA AOFLOW/'******•*'/ COMMON /CONST/ CZERO,JDET,JTEST,JFNC,JNORM,JTECH C C SWITCH JTEST INDICES C IF (JTEST .EQ. 3) THEN JTEST = 1 C (QUASICONCAVE CASE) GO TO 700 END IF IF (JTEST .EQ. 1) THEN JTEST = 2 C (CONVEX CASE) GO TO 700 END IF IF (JTEST .EQ. 2) THEN JTEST = 3 C (CONSTANT RETURNS TO SCALE CASE) GO TO 700 END IF 700 CONTINUE C C PRINT TEST HEADING C IF (JTEST .EQ. 1) THEN WRITE(6,420) JFNC ELSE IF (JTEST .EQ. 2) THEN WRITE(6,400) ELSE WRITE(6,410) END IF IF (JTECH .EQ. 0) THEN WRITE(6,422) ELSE WRITE(6,424) END IF 400 FORMAT('1','TEST 1. TECHNICAL EFFICIENCY TEST'/ 1 ' ',' QUANTITY DATA'/ 2 ' ',' CONVEX TECHNOLOGY'//) 410 FORMAT('1','TEST 2. TECHNICAL EFFICIENCY TEST'/ 1 ' ',' QUANTITY DATA'/ 2 ' ',' CONVEX CONE TECHNOLOGY'/ 3 ' ',' (CONSTANT RETURNS TO SCALE)'//) 420 FORMAT('1','TEST 3. TECHNICAL EFFICIENCY TEST'/ 1 ' ',' QUANTITY DATA'/ 2 ' ',' QUASICONCAVE TECHNOLOGY'/ 3 '0',' NOTE: TEST PERFORMED WITH RESPECT TO'/ 4 ' ',' TECHNOLOGY OF GOOD N=',14//) 422 FORMAT(' ',9X,'( TEST ASSUMES NO TECHNICAL CHANGE )'//) 424 FORMAT(' ',9X,'( TEST ASSUMES NO TECHNOLOGICAL REGRESS )'//) Appendix C. Computer Programs for Implementing the Nonparametric C C SOLVE LP FOR JI=l,JOBS C DO 5 JI = l.JOBS C C INITIALIZE VALUES FOR LP SUBROUTINE C JCONST = JGDS + 1 JEQUAL = 1 JMAXIM = 1 JFOBJ = 0 JFRHS = 0 IF (JTEST . E q . 3) THEN JCONST = JGDS JEQUAL = 0 END IF C C ZERO TABLEAU C DO 10 J1=1,JGDS2 DO 15 J2=1,J0BS2 TABL0(J1,J2) = O.DO 15 CONTINUE 10 CONTINUE C C SET UP TABLEAU C , JU = JOBS ^ IF (JTECH . E q . 1) JU=JI C C SET UP CONSTRAINTS AND RHS (TEST 1) C IF (JTEST . E q . 1) THEN JCONST = JGDS J = 0 DO 90 J J = l . J U IF (Z(JFNC.JJ) .GE. Z(JFNC.JI)) THEN J = J+l JGEZ(J) = J J END IF 90 CONTINUE JCGEZ = J DO 91 JJ = 1,JCGEZ JR = 1 DO 92 JI = l.JGDS IF (JI .EQ. JFNC) GO TO 92 JR = JR+1 TABLO(JR.JJ) = -Z(J1,JGEZ(JJ)> 92 CONTINUE TABL0(JGDS+1,JJ) = l.DO 91 CONTINUE JNVARS = JCGEZ+1 JV1 = JNVARS+1 JR = 1 DO 93 JI = l.JGDS IF (JI .EQ. JFNC) GO TO 93 JR = JR+1 TABLO(JR,JNVARS) = JDIR(J1)*Z(J1,JI) TABLO(JR.JVl) = - Z ( J l . J I ) 93 CONTINUE TABL0(JGDS+1,JV1) = l.DO ELSE C SET UP CONSTRAINTS AND RHS (TESTS 2,3) C DO 20 J2=1,JU DO 22 J1=1,JGDS TABL0(J1+1,J2) = -Z(J1,J2) 22 CONTINUE Appendix C. Computer Programs for Implementing the Nonparametric 239 TABL0(JGDS2,J2) = l.DO 20 CONTINUE JNVARS = JOBS+1 IF (JTECH .EQ. 1) JNVARS=JI+1 JV1 = JNVARS+1 DO 30 J1=1,JGDS JR = Jl+1 • , v , TABLO(JR,JNVARS) = JDIR(J1)*Z(J1,JI) TABL0(JR,JV1) = - Z ( J l . J I ) 30 CONTINUE TABLO(JGDS2.JV1) = l.DO END IF C C SET UP OBJECTIVE FUNCTION C TABLO(1,JNVARS) = l.DO C C SUM OF LP VARIABLES C JLPV = JNVARS + JCONST C C PRINT OUT INPUT TO LP SUBROUTINE C IF (JDET . E q . 0) GO TO 1234 WRITE(6,100)JI,JCONST,JNVARS,JEqUAL,TOL 100 FORMAT('1','OBSERVATION 1=',14/ 1 ' ',5X,'NUMBER OF CONSTRAINTS 3',14/ 2 ' ',5X,'NUMBER OF VARIABLES 3 ' ,14/ 3 ' ',5X,'NUMBER OF EqUALITY CONSTRAINTS 3',14/ 4 ' ' , 5 X , ' T O L E R A N C E 3 ' , F 1 0 . 6 / / ) WRITE(6,105) 105 FORMAT(' ' , 'TABLEAU IS: ' ) IF (JTEST . E q . 1) THEN CALL DPRMAT(TABLO,JGDS2,J0BS2,JGDS+1,JV1,1,1, 1 JGDS2,1) ELSE IF (JTEST . E q . 2) THEN CALL DPRMAT(TABLO,JGDS2,J0BS2,JGDS2,JV1,1,1,JGDS2,1) ELSE CALL DPRMAT(TABLO,JGDS2,J0BS2,JGDS+1,JV1,1,1, 1 JGDS2.1) END IF 1234 CONTINUE C C CALL LIPSUB SUBROUTINE C CALL LIPSUB(TABLO,JGDS2,JCONST,JNVARS,JEQUAL,JMAXIM, 1 JFOBJ,JFRHS,TOL,JVIN,JVOUT,BBOBJ,UBOBJ, 2 BBRHS,UBRHS,*999) JNP1 3 JNVARS+1 OPTIM(JI) = TABLO(l.JNPl) IF (JMAXIM .NE. 1) OPTIM(JI) = -OPTIM(JI) C IF (JDET . E q . 0) GO TO 2345 WRITE(6,200)OPTIM(JI) WRITE(6,300) (JVIN(J),TABLO(J.JNPl),J=2,JCONST+1) 2345 CONTINUE C C CHECK FOR NEGATIVE VALUES IN THE OPTIMAL BASIS C JWARN = 0 DO 60 J 3 2,JCONST+1 IF (TABLO(J.JNPl) . L T . -CZERO) JWARN=1 60 CONTINUE IF (JWARN . E q . 1) WRITE(6,888)JI C Appendix C. Computer Programs for Implementing the Nonparametric 240 C PRINT SUMMARY FOR OBSERVATION I C C CALCULATE WEAKLY EFFICIENT VECTOR ZW C ( SOME GOODS, BUT NOT ALL, MAY HAVE C ZERO SHADOW PRICES ) C DO 65 JN = l.JGDS ZW(JN) = (l.DO + (OPTIM(JI)*JDIR(JN))) * Z(JN.JI) 65 CONTINUE DO 70 J = l .JLPV X(J) = O.DO 70 CONTINUE DO 72 J = 2.JC0NST+1 X(JVIN(J)) = TABLO(J.JNPl) 72 CONTINUE C C COUNT POSITIVE LAMBDA'S C JPL - 0 IF (JTEST . E q . 1) JU = JCGEZ DO 74 J = l . J U IF (X(J) .GT. CZERO) THEN JPL = JPL+1 JPLAM(JPL) = J END IF 74 CONTINUE C C CALCULATE EFFICIENT VECTOR ZP C ( ZP IS (IDEALLY) PARETO EFFICIENT, ALL GOODS C HAVE POSITIVE SHADOW PRICES BUT PROCEDURE DOES C NOT GUARANTEE THAT ZP CALCULATED IS SUCH ) C AND DETERMINE IF EFFICIENT VECTOR ZW C IS IN FREE DISPOSAL SECTION OF THE C TECHNOLOGY SET JFRD(JI) = 0 IF ( JTEST . E q . 1) THEN DO 94 JG = l.JGDS IF (JG . E q . JFNC) THEN ZP(JG) = Z(JG.JI) ELSE ZP(JG) = O.DO DO 95 J J = 1,JCGEZ ZP(JG) = ZP(JG) + (X(JJ)*Z(JG,JGEZ(JJ))) 95 CONTINUE END IF ZDIF(JG) = ZP(JG) - Z(JG,JI) IF (ABS(Z(JG,JI)) .GT. CZERO) THEN ZPDIF(JG) = ZDIF(JG)/Z(JG,JI) ELSE ZPDIF(JG) = 9999.999D0 END IF IF ((ABS(ZPDIF(JG)) .GT. CZERO) .AND. (JG .NE. JFNC)) 1 JFRD(JI) = 1 94 CONTINUE ELSE DO 76 JG = l.JGDS ZP(JG) = O.DO DO 78 J J = l . J U ZP(JG) = ZP(JG) + (X(JJ)*Z(JG.JJ)) 78 CONTINUE ZDIF(JG) = ZP(JG) - Z(JG.JI) IF (ABS(Z(JG,JI)) .GT. CZERO) THEN ZPDIF(JG) = ZDIF(JG)/Z(JG,JI) ELSE ZPDIF(JG) = 9999.999D0 Appendix C. Computer Programs for Implementing the Nonparametric 241 END IF IF (ABS(ZPDIF(JG)) .GT. CZERO) JFRD(JI) = 1 76 CONTINUE END IF IF (JTEST .EQ. 1) THEN WRITE(6,500) JI,0PTIM(JI),(JGEZ(JPLAM(J)),J=1,JPL) ELSE IF (JPL .GT. 0) THEN WRITE(6,500) JI .OPTIM(JI) , (JPLAM(J),J=l ,JPL) ELSE WRITE(6,501) JI.OPTIM(JI) END IF END IF WRITE(6,510) IF (JTEST .EQ. 1) THEN DO 96 J = l.JGDS IF (J .EQ. JFNC) GO TO 96 IF (ZPDIF(J) .GT. 9000.DO) THEN WRITE(6,521) J , J D I R ( J ) , Z W ( J ) , Z P ( J ) , Z ( J , J I ) . 1 ZDIF(J).AOFLOW ELSE WRITE(6,520) J , J D I R ( J ) , Z W ( J ) , Z P ( J ) , Z ( J , J I ) , 1 ZDIF(J),ZPDIF(J) END IF 96 CONTINUE WRITE(6,525) JFNC,Z(JFNC,JI) ELSE DO 97 J = l.JGDS IF (ZPDIF(J) .GT. 9000.DO) THEN WRITE(6,521) J , J D I R ( J ) , Z W ( J ) , Z P ( J ) . Z ( J , J I ) , 1 ZDIF(J),AOFLOW ELSE WRITE(6.520) J . J D I R ( J ) , Z W ( J ) , Z P ( J ) , Z ( J , J I ) , 1 ZDIF(J),ZPDIF(J) END IF 97 CONTINUE END IF IF (OPTIM(JI) .GT. CZERO) THEN WRITE(6,550) JI ELSE IF (JFRD(JI) .EQ. 1) THEN WRITE(6,555) JI ELSE WRITE(6,560) JI END IF 550 FORMAT('-' . 'HENCE, OBSERVATION I=',I4, 1 ' IS TECHNICALLY E-INEFFICIENT.') 555 F0RMATO-' , 'HENCE, OBSERVATION 1=',14, 1 ' IS WEAKLY TECHNICALLY E-EFFICIENT.') 560 FORMATC-' , 'HENCE, OBSERVATION I=',I4, 1 ' IS TECHNICALLY E-EFFICIENT.') GO TO 5 999 WRITE(6,900) JI 5 CONTINUE C C PRINT SUMMARY FOR DATAFILE C WRITE(6,600) JNF = 0 JNP = 0 JMAX = 0 VMAX = O.DO JNPW = 0 JNPS = 0 DO 80 JJ = l.JOBS IF (OPTIMQJ) .GT. CZERO) THEN Appendix C. Computer Programs for Implementing the Nonparametric 242 JNF = JNF+1 JVIO(JNF) = J J VVIO(JNF) = OPTIM(JJ) IF (OPTIMOJ) .GT. VMAX) THEN JMAX = JJ VMAX = OPTIM(JJ) END IF ELSE JNP = JNP+1 JPASS(JNP) = JJ IF (JFRD(JJ) . E q . 1) THEN JNPW = JNPW+1 JPASSW(JNPW) = J J ELSE JNPS = JNPS+1 JPASSS(JNPS) = J J END IF END IF 80 CONTINUE IF (JNF .GT. 0) THEN WRITE(6,610) (JVIO(J),VVIO(J),J=l ,JNF) WRITE(6,615) JNF,VMAX,JMAX END IF WRITE(6,620) JOBS,JNP,JNF IF (JNP .GT. 0) THEN IF (JNPS .EQ. JOBS) THEN WRITE(6,625) ELSE WRITE(6,630) IF (JNPW .GT. 0) 1 WRITE(6,632) JNPW,(JPASSW(J),J=1,JNPW) IF (JNPS .GT. 0) 1 WRITE(6,634) JNPS,(JPASSS(J),J=1,JNPS) END IF END IF IF (JNF .GT. 0) THEN WRITE(6,640) ELSE WRITE(6,650) END IF 600 FORMAT('1','SUMMARY FOR DATAFILE: ' / / ) 610 FORMAT(' ',5X,'VIOLATIONS ARE AT OBSERVATIONS:'/ 1 (' ' 15X ' 1 = ' 1 4 ' DELTA*=' F10.6)) 615 FORMAT('0',10X,'TOTAL NUMBER OF VIOLATIONS FOR TEST IS ' , 1 4 / 1 ' ',10X,'MAXIMUM DELTA*=',F10.6, ' AT OBSERVATION 1 = ' , 2 1 4 , ' . ' / / ) 620 FORMAT(' \ 5 X , ' 0 U T OF ' , 1 4 , ' OBSERVATIONS, ' , 1 4 , 1 ' PASS AND ' , 1 4 , ' FAIL THE TEST. ' ) 625 FORMAT(' ' , 1 0 X , ' A L L OBSERVATIONS ARE ' , 1 'TECHNICALLY E-EFFICIENT.') 630 FORMAT(' ' . l O X . ' T H E OBSERVATIONS CONSISTENT WITH T H E ' , 1 ' HYPOTHESIS ARE:') 632 FORMAT(' ' , 1 5 X , ' ( ' , 1 4 , ' ) WEAKLY TECHNICALLY E-EFFICIENT', 1 ' OBSERVATIONS:'/(' ' , 2 0 X , ' 1 = ' , 1 4 ) ) 634 FORMAT(' ' , 1 5 X , ' ( ' , 1 4 , ' ) TECHNICALLY E-EFFICIENT', 1 ' OBSERVATIONS:'/(' ' , 2 0 X , ' 1 = ' , 1 4 ) ) 640 FORMATCO',5X,'CONCLUSION: OVERALL, THE DATA IS NOT CONSISTENT'/ 1 ' ' , 5 X , ' WITH THE HYPOTHESIS.') 650 FORMATCO' ,5X, 'CONCLUSION: OVERALL, THE DATA IS CONSISTENT'/ 1 ' ' , 5 X , ' WITH THE HYPOTHESIS.') RETURN 200 FORMAT(/,' ',5X,'OPTIMUM=',G20.12/) 300 FORMATCO' ,5X, 'PRIMAL SOLUTION:'/ Appendix C. Computer Programs for Implementing the Nonparametric 1 ' ',71,'VARIABLE',8X,'VALUE'/ 2 (' ' ,8X,I5,7X,G20.12)) 500 FORMAT (' 2' SUMMARY FOR OBSERVATION I = ' , I 4 , ' : ' / / 1 ' ' ,5X, 'DELTA*=' ,F10 .6 / 2 ' ' ,5X,'POSITIVE LAMBDA''S: ' , I4 ,9 (1X, I4 ) / 3 (' ' 5X ' ' 1 4 9(1X 14))) 501 FORMAT{'2','SUMMARY FOR OBSERVATION f='',14,' : ' / / 1 ' ' ,5X, 'DELTA*=' ,F10.6 / 2 ' ' ,5X,'POSITIVE LAMBDA''S: 0') 510 FORMATC'0','GOODS',2X,'EFFICIENCY\4X,'WEAKLY E F F . ' , 5 X , . 1 'STRONGLY E F F . ' , 5 X , 2 'VECTOR Z(I) ' ,7X, 'DIFFERENCE',4X, 3 'PROPORTIONAL'/ 4 ' ' , I X , ' N O . \ 2 X , ' D I R . VECTOR',5X,'VECTOR Z * ' , 5 7X,'VECTOR Z * * ' , 6 26X, 'Z**-Z(I ) ' , 6X, 'DIFFERENCE' / 7 ' ' , 8 6 X , ' ( Z * * - Z ( I ) ) / Z ( I ) ' / / ) 520 FORMATC ' ,I4 ,7X,I2,4X,4G17.5,4X,F8.5) 521 FORMATC ' ,I4,7X,I2,4X,4G17.5,4X,A8) 525 FORMATCO',14,49X.G15.5) 888 FORMATCO',5X,'WARNING: THERE IS AT LEAST ONE NEGATIVE'/ 1 ' \ 5 X , ' VALUE IN THE OPTIMAL BASIS FOR T H E ' / 2 ' \ 5 X , ' LP PROBLEM AT OBSERVATION 1=',14, ' . ' / 3 ' \ 5 X , ' DATA SCALING PROBLEMS MAY E X I S T . ' / ) 900 FORMATCO',5X,'PROBLEM ENCOUNTERED IN LP SUBROUTINE'/ 1 ' ' , 5 X , ' A T OBSERVATION 1=' ,14 , ' . ' / ) END Listing of subroutine TESTPQC SUBROUTINE TSTPqC(Z,P,PI,JGDS,JGDS2,JOBS,J0BS2,JG0B2, 1 TABLO,JVIN.JVOUT.OPTIM,TOL, 2 EPS.ZP.ZDIF.ZPDIF.X.JPLAM, 3 JVIO,VVIO,JPASS,JGEZ,JSET,JNUM,JINS,JNOTS) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C C TEST 4. CONSTRAINED ALLOCATIVE EFFICIENCY TEST C PRICE AND QUANTITY DATA C CONVEX TECHNOLOGY C C TEST 5. CONSTRAINED ALLOCATIVE EFFICIENCY TEST C PRICE AND QUANTITY DATA C CONSTANT RETURNS TO SCALE TECHNOLOGY C C TEST 6. CONSTRAINED ALLOCATIVE EFFICIENCY TEST C PRICE AND QUANTITY DATA C QUASICONCAVE TECHNOLOGY C C JTECH = 0 ASSUMES NO TECHNICAL CHANGE C = 1 ASSUMES NO TECHNOLOGICAL REGRESS C C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * IMPLICIT REAL*8(B-I,K-Z),CHARACTER*12(A) DIMENSION Z(JGDS,0:JOBS),P(JGDS,0:JOBS),PI(0:JOBS,0:JOBS), 1 TABLO(JGDS2,J0BS2),JVIN(JGDS2),JV0UT(J0BS2), 2 OPTIM(JOBS).EPS(JOBS).ZP(JGDS).ZDIF(JGDS).ZPDIF(JGDS), 3 X(0:JGOB2).JPLAM(JOBS), 4 JVIO(JOBS).VVIO(JOBS),JPASS(JOBS),JGEZ(JOBS), 5 JSET(JGDS).JNUM(JGDS),JINS(JGDS),JNOTS(JGDS) COMMON /CONST/ CZERO,JDET,JTEST,JFNC,JNORM,JTECH,JPOSP.JPMU Appendix C. Computer Programs for Implementing the Nonparametric . 244 C C C DATA AVMAX/' UNBOUNDED'/ PRINT TEST HEADING DO 1 JN = l.JGDS Z(JN,0) = O.DO P(JN,0) = O.DO CONTINUE IF (JTEST .EQ. 6) THEN WRITE(6,420) JFNC ELSE IF (JTEST . E q . 4) THEN WRITE(6,400) ELSE WRITE(6,410) END IF IF (JTEST .EQ. 4) THEN IF ((JPOSP . E q . 0) .AND. (JPMU . E q . 0)) THEN WRITE(6,430) ELSE WRITE(6,435) END IF END IF IF (JTECH . E q . 0) THEN WRITE(6,422) ELSE WRITE(6,424) END IF 400 FORMAT('1 1 2 410 FORMAT('1 1 ' 2 3 ' 420 FORMAT('1 1 2 3 '0 4 422 FORMAT(' 424 FORMAT(' 430 FORMAT(' 435 FORMAT(' 1 C C C C** c** c c c c c c TEST 4. CONSTRAINED ALLOCATIVE EFFICIENCY TEST' / PRICE AND qUANTITY DATA'/ CONVEX TECHNOLOGY'//) TEST 5. CONSTRAINED ALLOCATIVE EFFICIENCY T E S T ' / PRICE AND qUANTITY DATA'/ CONVEX CONE TECHNOLOGY'/ (CONSTANT RETURNS TO S C A L E ) ' / / ) TEST 6. CONSTRAINED ALLOCATIVE EFFICIENCY TEST' / PRICE AND QUANTITY DATA'/ qUASICONCAVE TECHNOLOGY'/ NOTE: TEST PERFORMED WITH RESPECT T O ' / TECHNOLOGY OF GOOD N=',14//) , 9 X , ' ( TEST ASSUMES NO TECHNICAL CHANGE ) ' / / ) , 9 X , ' ( TEST ASSUMES NO TECHNOLOGICAL REGRESS ) ' / / ) , 9 X , ' ( OPTIMAL PROFITS UNRESTRICTED IN SIGN ) ' / / ) , 9 X , ' ( OPTIMAL PROFITS RESTRICTED ' , 'TO BE NONNEGATIVE ) ' / / ) SOLVE LP FOR JI=1,J0BS DO 5 JI = l.JOBS INITIALIZE VALUES FOR LP SUBROUTINE JMAXIM = 1 JFOBJ = 0 JFRHS = 0 ZERO TABLEAU DO 15 10 10 J1=1,JGDS2 DO 15 J2=1,J0BS2 TABL0(J1,J2) = CONTINUE CONTINUE O.DO Appendix C. Computer Programs for Implementing the Nonparametric 245 C C SET UP TABLEAU C JU = JOBS IF (JTECH . E q . 1) JU=JI JL = 1 JADJ = 0 C C COUNT GOODS IN S AND GOODS NOT IN S C DO 18 J = 1,JGDS IF ((JTEST . E q . 6) .AND. (J . E q . JFNC)) GO TO 18 IF (JSET(J) . E q . 1) THEN JI = Jl+1 JINS(Jl) = J ELSE J2 = J2+1 JN0TSQ2) = J END IF 18 CONTINUE JCINS = JI JCNOTS = J2 IF (JCINS . E q . 0) THEN WRITE(6,810) RETURN END IF 810 FORMAT(' ','CONSTRAINED OPTIMIZATION TEST CANNOT'/ 1 ' ' , ' B E PERFORMED; THE SET S CONTAINING'/ 2 ' ' , ' T H E GOODS WITH RESPECT TO WHICH T H E ' / 3 ' ','PRODUCER CAN OPTIMIZE HAS NO ELEMENTS.') C C TO CHECK IF THERE IS AT LEAST ONE GOOD C NOT IN S (TEST 5 (CRS) ONLY) IF ((JTEST . E q . 5) .AND. (JCNOTS . E q . 0)) THEN WRITE(6,800) RETURN END IF 800 FORMATCl'.'CONSTRAINED OPTIMIZATION TEST FOR A CRS'/ 1 ' ','TECHNOLOGY CANNOT BE PERFORMED; THERE ' / 2 ' ' , ' A R E NO CONSTRAINTS IN THE LP PROBLEM.') C C SET UP CONSTRAINTS AND RHS (TEST 6) IF (JTEST .EQ. 6) THEN JEqUAL = 1 J = 0 DO 30 JJ = 1,JU IF (Z(JFNC.JJ) .GE. Z(JFNC.JI)) THEN J = J+l JGEZ(J) = JJ END IF 30 CONTINUE JCGEZ = J JNVARS = JCGEZ JV1 = JNVARS+1 JR = 1 DO 32 JI = 1,JCNOTS JR = JR+1 DO 34 J2 = 1,JCGEZ TABL0(JR,J2) = -Z(JN0TS(J1),JGEZ(J2)) 34 CONTINUE TABLO(JR,JV1) = -Z(JN0TS(J1),JI) 32 CONTINUE Appendix C. Computer Programs for Implementing the Nonparametric 246 JR = JR+1 DO 36 J2 = l . J V l TABLO(JR.J2) = l.DO 36 CONTINUE JCONST = JR-1 C C SET UP OBJECTIVE FUNCTION (TEST 6) C DO 38 J2 = 1,JCGEZ DO 39 JI = l .JCINS TABLO(1.J2) = TABLO(1.J2) + 1 (P(JINS(J1),JI)*Z(JINS(J1),JGEZ(J2))) 39 CONTINUE 38 CONTINUE ELSE C C SET UP CONSTRAINTS AND RHS (TEST 4,5) C CC* CC* JNVARS = JU JV1 = JNVARS+1 JCONST = JCNOTS +1 JEQUAL = 1 IF ((JTEST . E q . 4) .AND. (JPMU . E q . 1)) JEQUAL = 0 IF (JTEST . E q . 5) THEN JCONST = JCNOTS JEQUAL = 0 END IF C IF ((JTEST . E q . 4) .AND. (JPOSP . E q . 1)) THEN JNVARS = JU+1 JV1 = JNVARS+1 JL = 0 JADJ = 1 END IF C JR = 1 DO 20 JI = 1,JCNOTS JR = JR+1 DO 22 J2 = J L . J U J2P = J2 IF (JADJ . E q . 1) J2P = J2+1 TABL0(JR,J2P) = -Z(JN0TS(J1),J2) 22 CONTINUE TABLO(JR.JVl) = -Z(JN0TS(J1),JI) 20 CONTINUE IF (JTEST .EQ. 4) THEN JR = JR+1 DO 24 J2 = l . J V l TABLO(JR.J2) = l.DO 24 CONTINUE END IF C C SET UP OBJECTIVE FUNCTION (TESTS 4,5) C DO 26 J2 = J L . J U J2P = J2 IF (JADJ . E q . 1) J2P = J2+1 DO 29 JI = l .JCINS TABL0(1,J2P) = TABL0(1,J2P) + (P(JINS(J1),JI)*Z(JINS(J1),J2)) 29 CONTINUE 26 CONTINUE END IF C C SUM OF LP VARIABLES C JLPV = JNVARS + JCONST Appendix C. Computer Programs for Implementing the Nonparametric 247 C C PRINT OUT INPUT TO LP SUBROUTINE C IF (JDET .EQ. 0) GO TO 1234 WRITE(6,100)JI,JCONST,JNVARS,JEQUAL.TOL 100 FORMAT('1','OBSERVATION 1=',14/ 1 ' ',5X,'NUMBER OF CONSTRAINTS=',14/ 2 ' ',5X,'NUMBER OF VARIABLES 3 ' ,14/ 3 ' ',5X,'NUMBER OF EQUALITY CONSTRAINTS 3',14/ 4 ' ' , 5 X , ' T O L E R A N C E 3 ' , F 1 0 . 6 / / ) WRITE(6,105) 105 FORMAT(' ' , 'TABLEAU IS: ' ) CALL DPRMAT(TABLO,JGDS2,J0BS2,JR,JV1,1,1,JGDS2,1) 1234 CONTINUE C C CHECK IF JEQUAL < JNVARS C JNOLP = 0 IF (JNVARS .EQ. JEQUAL) THEN JNOLP = 1 X( l ) = l.DO GO TO 82 END IF C C CALL LIPSUB SUBROUTINE C CALL LIPSUB(TABLO,JGDS2,JCONST,JNVARS,JEQUAL,JMAXIM, 1 JFOBJ,JFRHS,TOL,JVIN,JVOUT,BBOBJ,UBOBJ, 2 BBRHS,UBRHS,*999) JNP1 = JNVARS+1 OPTIM(JI) = TABL0(1,JNP1) IF (JMAXIM .NE. 1) OPTIM(JI) = -OPTIM(JI) C IF (JDET . E q . 0) GO TO 2345 WRITE(6,200)OPTIM(JI) WRITE(6,300) (JVIN(J),TABLO(J,JNP1),J=2,JCONST+1) 2345 CONTINUE C C CHECK FOR NEGATIVE VALUES IN THE OPTIMAL BASIS C JWARN = 0 DO 60 J = 2,JCONST+1 IF (TABLO(J,JNP1) . L T . -CZERO) JWARN=1 60 CONTINUE IF (JWARN . E q . 1) WRITE(6,888)JI C C CALCULATE ACTUAL PARTIAL PROFIT, VALUE OF NUMERAIRE C GOODS, LOSS MEASURE C 82 PI(JI.JI) = O.DO VNUM = O.DO DO 40 Jl=l ,JCINS P I ( J I . J I ) = PI (JI .J I ) + (P(JINS(J1),JI)*Z(JINS(J1),JI)) 40 CONTINUE DO 42 JI = l.JGDS IF (JNUM(Jl) . E q . 1) 1 VNUM = VNUM + (P(J1,JI)*ABS(Z(J1,JI))) 42 CONTINUE C C CHECK FOR ZERO VALUES OF VNUM C IF (ABS(VNUM) . L E . CZERO) THEN EPS(JI) = 999.999999D0 WRITE(6,700) JI GO TO 5 Appendix C. Computer Programs for Implementing the Nonparametric END IF 700 FORMAT('2'WARNING: FOR OBSERVATION I = ' , I 4 , \ THE VALUE'/ 1 ' ' , ' OF NUMERAIRE GOODS IS ZERO; HENCE, ' / 2 ' ' , ' EPSILON(S) IS UNDEFINED.') C CALCULATE LOSS MEASURE EPS(JI) C IF (JNOLP . E q . 1) THEN OPTIM(JI) = PI (JI .J I ) EPS(JI) = O.DO GO TO 84 END IF EPS(JI) = (OPTIM(JI) - P I ( J I . J I ) ) / VNUM C C PRINT SUMMARY FOR OBSERVATION I C C COUNT POSITIVE LAMBDA'S C DO 70 J = l . JLPV X(J) = O.DO 70 CONTINUE DO 72 J = 2,JCONST+1 X(JVIN(J)) = TABLO(J.JNPl) 72 CONTINUE C IF (JADJ . E q . 1) THEN DO 71 J = l .JLPV X ( J - l ) = X(J) 71 CONTINUE END IF C 84 JPL = 0 JPLAM(l) = 0 IF (JTEST . E q . 6) JU = JCGEZ DO 74 J = J L . J U IF (X(J) .GT. CZERO) THEN JPL = JPL+1 JPLAM(JPL) = J END IF 74 CONTINUE C C CALCULATE EFFICIENT VECTOR ZP C IF ( JTEST . E q . 6) THEN DO 94 JG = l.JGDS IF (JG .EQ. JFNC) THEN ZP(JG) = Z(JG.JI) ELSE ZP(JG) = O.DO DO 95 JJ = 1,JCGEZ ZP(JG) = ZP(JG) + (X(JJ)*Z(JG,JGEZ(JJ))) 95 CONTINUE END IF ZDIF(JG) = ZP(JG) - Z(JG.JI) ZPDIF(JG) = ZDIF(JG)/Z(JG.JI) 94 CONTINUE ELSE DO 76 JG = l.JGDS ZP(JG) = O.DO DO 78 JJ = J L . J U ZP(JG) = ZP(JG) + (X(JJ)*Z(JG,JJ)) 78 CONTINUE ZDIF(JG) = ZP(JG) - Z(JG.JI) ZPDIF(JG) = ZDIF(JG)/Z(JG,JI) 76 CONTINUE END IF Appendix C. Computer Programs for Implementing the Nonparametric 249 C C PRINT RESULTS FOR OBSERVATION I C I F (JTEST .Eq . 6) THEN WRITE(6,500) J I , E P S ( J I ) , ( J G E Z ( J P L A M ( J ) ) , J = 1 , J P L ) ELSE WRITE(6,500) J I . E P S ( J I ) , ( J P L A M ( J ) , J = 1 , J P L ) END I F WRITE(6,510) WRITE(6,505) JCINS W R I T E ( 6 , 5 2 0 ) ( J I N S ( J ) , J N U M ( J I N S ( J ) ) , P ( J I N S ( J ) , J I ) , Z P ( J I N S ( J ) ) , 1 Z ( J I N S ( J ) , J I ) , Z D I F ( J I N S ( J ) ) , 2 Z P D I F ( J I N S ( J ) ) , J = 1 , J C I N S ) I F (JCNOTS .GT. 0) THEN WRITE(6,515) JCNOTS WRITE(6,525) ( J N O T S ( J ) , Z P ( J N O T S ( J ) ) , Z ( J N O T S ( J ) , J I ) , 1 ZDIF(JNOTS(J)) ,ZPDIF(JNOTS(J)) ,J=1,JCNOTS) END I F I F (JTEST .Eq . 6) WRITE(6,517) JFNC,Z(JFNC,JI ) WRITE(6,530) V N U M , P I ( J I , J I ) , O P T I M ( J I ) I F (EPS(J I ) .GT. CZERO) THEN WRITE(6,550) J I ELSE WRITE(6,555) J I END I F GO TO 5 999 WRITE(6,900) J I EPS(J I ) = 999.999999D0 5 CONTINUE C C PRINT SUMMARY FOR DATAFILE C WRITE(6,600) JNF = 0 JNP = 0 JMAX = 0 VMAX = O.DO DO 80 JJ = l.JOBS I F (EPS(JJ) .GT. CZERO) THEN JNF = JNF+1 JVIO(JNF) = JJ VVIO(JNF) = EPS(JJ) I F (EPS(JJ) .GT. VMAX) THEN JMAX = JJ VMAX = EPS(JJ) END I F ELSE JNP = JNP+1 JPASS(JNP) = JJ END I F 80 CONTINUE I F (JNF .GT. 0) THEN WRITE(6,610) JBOU = 0 DO 88 J=1,JNF I F (VVIO(J) .GT. 999.DO) THEN JBOU = 1 WRITE(6,612) JVIO(J).AVMAX ELSE WRITE(6,614) J V I O ( J ) , V V I O ( J ) END I F 88 CONTINUE I F (JBOU .Eq . 1) THEN WRITE(6,617) JNF,JMAX ELSE WRITE(6,615) JNF,VMAX,JMAX Appendix C. Computer Programs for Implementing the Nonparametric 250 END IF END IF WRITE(6,620) JOBS,JNP,JNF IF ((JNP .GT. 0) .AND. (JNP .LT. JOBS)) 1 WRITE(6,630) (JPASS(J),J=l,JNP) IF (JNF .GT. 0) THEN WRITE(6,640) ELSE WRITE(6,650) END IF RETURN 200 FORMAT(/, 300 FORMATCO 1 2 0 500 FORMAT('2 1 2 3 (' 505 FORMAT(' 515 FORMATCO 517 FORMATCO 510 FORMATCO 1 2 3 4 5 6 7 ' 8 520 FORMAT(' 525 FORMAT(' ' ',5X,'0PTIMUM='.G20.12/) ',5X,'PRIMAL SOLUTION:'/ ',7X,'VARIABLE',8X,'VALUE'/ ',8X,I5,7X,G20.12)) ','SUMMARY FOR OBSERVATION I=',I4,':'// ',5X,'EPSIL0N(S)=',F10.6/ ',5X,'POSITIVE LAMBDA''S: ',I4,9(1X,I4)/ ',5X,' ',I4,9(1X,I4))) ','(',14,') GOODS IN S:') ','(',14,') GOODS NOT IN S:') ',I4,47X,G17.5) ','GOODS',3X,'NUMERAIRE',4X,'PRICE VECTOR',5X, 'ALLOC. EFF.',6X, 'VECTOR Z(I)',7X,'DIFFERENCE',4X, 'PROPORTIONAL'/ ',1X,'N0.',1X,'(=>EFFICIENCY',7X,'P(I)', 10X,'VECTOR Z*', 27X,'Z*-Z(I)',6X,'DIFFERENCE'/ ',5X,' DIR. VECTOR)',68X, '(Z*-Z(I))/Z(I)>//) ',I4,7X,I2,4X,4G17.5,4X,F8.5) ',I4,30X,3G17.5,4X,F8.5) 530 FORMATCO' ,5X, 'VALUE OF NUMERAIRE G00DS='.G17.7/ 1 ' ',5X,'PARTIAL PROFIT AT CURRENT PRODUCTION PLAN=',G17.7/ 2 ' ',5X,'PARTIAL PROFIT AT ALLOCATIVELY EFFICIENT ', 3 'PRODUCTION PLAN=',G17.7) 550 FORMATC-','HENCE, OBSERVATION I=',I4, 1 ' IS ALLOCATIVELY INEFFICIENT.') 555 FORMATC-','HENCE, OBSERVATION I=',I4, 1 ' IS (CONSTRAINED) ALLOCATIVELY EFFICIENT.') 600 FORMATCl' , 'SUMMARY FOR DATAFILE:'//) 610 FORMAT(' ',5X, 612 FORMAT(' ',15X 614 FORMAT(' ',15X 615 FORMATCO',10X 1 ' ',10X 2 617 FORMATCO' ,10X 1 ' ',10X 2 620 FORMAT(' ',5X, 1 630 FORMAT (' \10X 1 2 (' ' 15X '1=' 14)) 640 FORMAT('0',5X,'CONCLUSION: 1 ' ' 5X 650 FORMATCO' *5x) 'CONCLUSION: 1 ' ',5X, VIOLATIONS ARE AT OBSERVATIONS:') 'I=',I4,\ EPSILON(S)=',A10) '1=' 14 ' EPSILON(S)=' F10.6) 'TOTAL NUMBER OF VIOLATIONS FOR TEST IS ',14/ 'MAXIMUM EPSILON(S)=',F10.6,' AT OBSERVATION 1=' 14,'.'//) 'TOTAL NUMBER OF VIOLATIONS FOR TEST IS ',14/ 'MAXIMUM EPSILON(S)= UNBOUNDED AT OBSERVATION 1= 14,'.'//) OUT OF ',14,' OBSERVATIONS, ',14, PASS AND ',14,' FAIL THE TEST.') 'THE OBSERVATIONS CONSISTENT WITH THE', ' HYPOTHESIS ARE:'/ OVERALL, THE DATA IS NOT CONSISTENT'/ WITH THE HYPOTHESIS.') OVERALL, THE DATA IS CONSISTENT'/ WITH THE HYPOTHESIS.') Appendix C. Computer Programs for Implementing the Nonparametric 888 FORMAT( 1 2 3 900 FORMAT( 1 2 3 4 END 0',5X,'WARNING: THERE IS AT LEAST ONE NEGATIVE'/ \ 5 X , ' VALUE IN THE OPTIMAL BASIS FOR T H E ' / \ 5 X , ' LP PROBLEM AT OBSERVATION 1=',14, ' . ' / ' , 5 X , ' DATA SCALING PROBLEMS MAY E X I S T . ' / ) 2','SUMMARY FOR OBSERVATION I = ' , I 4 , ' : ' / / ',5X,'PROBLEM ENCOUNTERED IN LP SUBROUTINE.'/ ' ,5X, 'THERE IS NO FINITE SOLUTION TO THE L P ' / ',5X,'PROBLEM; HENCE, THE OBJECTIVE FUNCTION I S ' / ',5X,'UNBOUNDED. EPSILON(S) IS UNBOUNDED.') Listing of subroutine TESTPQ SUBROUTINE TESTPQ(Z,P,PI,JDIR,JGDS,JOBS,EPS,ZDIF,ZPDIF, 1 JVIO.VVIO.JPASS,ZN,JEFF,JBEST) *********************************************** C C TEST 7. UNCONSTRAINED ALLOCATIVE EFFICIENCY TEST C PRICE AND QUANTITY DATA C CONVEX TECHNOLOGY C C TEST 8. UNCONSTRAINED ALLOCATIVE EFFICIENCY TEST C PRICE AND QUANTITY DATA C CONSTANT RETURNS TO SCALE TECHNOLOGY C C TEST 9. UNCONSTRAINED ALLOCATIVE EFFICIENCY TEST C PRICE AND QUANTITY DATA C QUASICONCAVE TECHNOLOGY C C JTECH = 0 ASSUMES NO TECHNICAL CHANGE C = 1 ASSUMES NO TECHNOLOGICAL REGRESS C C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * IMPLICIT REAL*8(B-I,K-Z),CHARACTER*12(A) DIMENSION Z(JGDS,0:JOBS),P(JGDS,0:JOBS).JDIR(JGDS), 1 PI(0:JOBS,0:JOBS), 2 EPS(JOBS).ZDIF(JGDS).ZPDIF(JGDS), 3 JVIO(JOBS).VV