NONPARAMETRIC TESTS OF REGULARITY CONDITIONS FOR PRODUCTION AND CONSUMPTION THEORY by CELIK PARKAN Dipl. Ing., Technical University of Istanbul, 1965 M.Sc., University of Pennsylvania, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Faculty of Commerce and Business Administration We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1975 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e fo r reference and study. I f u r t h e r agree tha t permiss ion fo r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed without my w r i t t e n p e r m i s s i o n . Department of Management Science, Faculty of Commerce and Business Admin The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date July ,-4, 1975 Chairman: P r o f e s s o r W.E. Diewert NONPARAMETRIC TESTS OF REGULARITY CONDITIONS FOR PRODUCTION AND CONSUMPTION THEORY By C e l i k Parkan ABSTRACT The t h e o r i e s of production and.consumption have proven to be valuable t o o l s i n p r e d i c t i n g v a r i a t i o n s i n demand f o r f a c t o r s of production (producer's demand) and f o r produced goods (consumer's demand) to changes i n p o l i c y instruments. An important d i r e c t i o n of i n q u i r y , i n the theory of production, has been on determining the f u n c t i o n a l form of production;.another one, i n the theory of consumption, has been on the observed behaviour of the consumer and the computation of index numbers. Econometricians, i n e s t i m a t i n g a production f u n c t i o n , have been using parametric e s t i m a t i o n techniques which always r e q u i r e the choice of a f u n c t i o n a l form f o r the unde r l y i n g production o p e r a t i o n . Also the c l a s s i c a l assumption about the consumer has been th a t any purchase i s such as to give a maximum of u t i l i t y f o r the money spent. I t i s w e l l known tha t economic agents' (producers' and consumers') o p t i m i z i n g behaviour i m p l i e s c e r t a i n r e s t r i c t i o n s on de r i v e d supply and demand p a t t e r n s ; e.g., see P.A. Samuelson or J.R. Hi c k s . T y p i c a l f u n c t i o n a l forms, f o r production, o f t e n used i n e m p i r i c a l a p p l i c a t i o n s are e i t h e r u n s u i t a b l e f o r modelling the demand f o r f a c t o r s of production or they do not always s a t i s f y the r e g u l a r i t y c o n d i t i o n s that c o s t minimizing or p r o f i t maximizing behaviour imposes. Neither have econometricians always checked whether these r e s t r i c t i o n s are c o n s i s t e n t w i t h the f u n c t i o n a l forms f o r demand and supply equations which they use i n t h e i r a p p l i e d econometric work. i i i This d i s s e r t a t i o n s t u d i e s the nonparametric methods f o r t e s t -i n g the consistency of observations on q u a n t i t i e s , p r i c e s , p r o f i t s , c o s t s and appropriate combinations of these with such r e g u l a r i t y c o n d i t i o n s as monotonicity, q u a s i - c o n c a v i t y , c o n c a v i t y , homothet-i c i t y , and s e p a r a b i l i t y . The approach has been th a t of u n i f y i n g the e x i s t i n g body of knowledge i n using l i n e a r programming as the t e s t instrument and making extensions e s p e c i a l l y to cover m u l t i p l e production, s e p a r a b i l i t y p r o p e r t i e s and nonparametric index numbers. The t h e o r e t i c a l work has been s u b s t a n t i a t e d by e m p i r i c a l t e s t s on a c t u a l data and the computer codes are presented to f a c i l i t a t e f u r t h e r research i n r e l a t e d areas. iv CONTENTS Page Chapter I I. INTRODUCTION 1 I I . SINGLE OUTPUT PRODUCTION FUNCTIONS 4 2.1 Regularity Conditions on Single Output Production 4 Functions 2.2 Notation and Ordering of the Observations 6 2.3 Test f o r a Non-decreasing Production Function 7 2.4 Tests f o r a Non-decreasing and Quasi-concave 7 Production Function 2.5 Tests f o r a Non-decreasing and Concave Production 11 Function 2.6 Tests f o r a Non-decreasing, Concave and Linear 13 Homogeneous Production Function 2.7 Test f o r a Non-decreasing, Quasi-concave and 15 Homothetic Production Function I I I . MULTIPLE OUTPUT PRODUCTION FUNCTIONS 19 3.1 Regularity Conditions on Mu l t i p l e Output 19 Production Function 3.2 Tests f o r a Non-decreasing and Quasi=TConcavect:.c^ 21 Production Function 3.3 Tests f o r a Non-decreasing, Concave Production 22 Function 3.4 Tests f o r a Non-decreasing, Concave and Linear 25 Homogeneous Production Function IV. EMPIRICAL EXAMPLES 26 4.1 An Example for Single Output Production 27 4.2 An Example f or M u l t i p l e Output Production " 30 4.3 Conclusion ' " 30 Chapter II I. INTRODUCTION I I . PRICE AND QUANTITY DATA 2.1 Test f o r a nondecreasing and Quasi-concave Production Function; P r i c e and Quantity Data 2.2 Test"for a Nondecreasing and Concave Production Function; P r i c e and Quantity Data 2.3 Test f o r a nondecreasing, Concave and Linear Homogeneous Production Function; P r i c e and Quantity Data I I I . PRICE AND PROFIT DATA 3.1 Test f o r a Nondecreasing and Concave Production Function P r i c e and P r o f i t Data 3.2 Test for a Nondecreasing, Concave and Linear Homogeneous Production Function; P r i c e and P r o f i t Data IV. PRICE DATA 4.1 Test f o r a Nondecreasing, Concave and Linear Homogeneous Production Function; P r i c e Data V. PROFIT (COST), PRICE AND OUTPUT QUANTITY DATA 5.1 Test f o r a Nondecreasing, Quasi-concave Production Function; P r o f i t (Cost), P r i c e and Output Quantity Data VI. AN EMPIRICAL EXAMPLE 6.1 A Multi-product Case 6.2 Conclusion Chapter I I I I. INTRODUCTION I I . TESTING THE CONSISTENCY OF CONSUMER DATA I I I . REVIEW OF INDEX NUMBER THEORY 3.1 Two Approaches to Index Numbers 3.2 The Test Approach and the Fisher Indexes 3.3 The Economic Approach 3.4 The Tornquist-Theil " D i v i s i a " Index v i Page IV. AGGREGATION AND HOMOGENEOUS WEAK SEPARABILITY 79 V. NONPARAMETRIC INDEX NUMBERS 80 5.1 Constructing Homothetic Nonparametric Index Numbers 81 5.2 Constructing Nonhomothetic Nonparametric Index Numbers 82 5.3 A Measure of V i o l a t i o n 83 VI. EMPIRICAL EXAMPLES 84 6.1 Microeconomic Data 85 6.2 Aggregate Data 86 6.3 Conclusion 105 I t e r a n c e s REFERENCES 107 APPENDIX I 111 APPENDIX II 126 COMPUTER PROGRAMS 120 APPENDIX I I I 150 COMPUTER PROGRAMS 154 APPENDIX IV 161 COMPUTER PROGRAMS 163 CHAPTER I NONPARAMETRIC TESTS OF REGULARITsY CONDITIONS FOR PRODUCTION FUNCTIONS: QUANTITY DATA lb I. INTRODUCTION Economists frequently attempt to predict how the demand for various factors of production (e.g., white c o l l a r labor, blue c o l l a r labor, machinery and equipment, structures) w i l l change i f various p o l i c y instruments (e.g., s o c i a l security or pension taxes, corporate p r o f i t taxes, property taxes, manufacturer's general sales tax, a g r i c u l t u r a l subsidies) are changed. Production function theory has proven to be a valuable t o o l i n t h i s endeavour. In order to s t a t i s t i c a l l y estimate a production function, i t i s necessary to choose a fun c t i o n a l form for the production function, that i s , a s p e c i f i c function which depends on a l i m i t e d number of unknown constants or parameters and i t gives the maximum amount of output which can be produced i n a given time period as a function of the quantities of the various inputs u t i l i z e d . T y p i c a l f u n c t i o n a l forms often used i n empirical applications are: (i) the fi x e d c o e f f i c i e n t s or Leontief [1941] production function, ( i i ) the constant factor shares or Cobb-Douglas [1928] production function, and ( i i i ) the constant e l a s t i c i t y of s u b s t i t u t i o n (CES) or Arrow, Chenery, Minhas, Solow [1961] production function. However, each of the above fun c t i o n a l forms i s unsuitable f o r modelling the demand for factors of production because they a p r i o r i impose severe r e s t r i c t i o n s on the pattern of s u b s t i t u t a b i l i t y between U) factors 1 . . Recently, new fun c t i o n a l forms have been proposed for production functions which do not impose any a p r i o r i r e s t r i c t i o n s on e l a s t i c i t i e s of su b s t i t u t i o n . Examples of new " f l e x i b l e " f u n c t i o n a l forms are: ( i ) the Generalized Leontief (Diewert [1971]), ( i i ) the Translog (Christensen, Jorgenson and Lau [1971]), and ( i i i ) the Generalized Cobb-Douglas (Diewert The CES fu n c t i o n a l form i s not p a r t i c u l a r l y r e s t r i c t i v e i n the case of only two inputs. 2 [1973c]). However, the a p p l i c a t i o n of these new f u n c t i o n a l forms to empirical data has r a i s e d a new problem - the estimated production functions do not always s a t i s f y the r e g u l a r i t y conditions that cost minimizing or p r o f i t maxi-mizing behaviour imposes on estimated production f u n c t i o n s . 2 I f t h i s problem arises i n an empirical a p p l i c a t i o n , then: (i) we may say that the assumption of cost minimizing or p r o f i t maximizing behaviour i s unwarranted, or ( i i ) the data that we are using may be inappropriate due to various aggregation errors, or ( i i i ) we may a t t r i b u t e our t h e o r e t i c a l l y unappealing r e s u l t s to the choice of an i n c o r r e c t f u n c t i o n a l form for the production function. It turns out that we can t e s t whether there e x i s t s any f u n c t i o n a l form f o r the production function which s a t i s f i e s various r e g u l a r i t y conditions and i s consistent with our observed data. A f r i a t [1967] l a i d the methodolog-i c a l foundations f o r these nonparametric t e s t s of r e g u l a r i t y conditions by extending the e a r l i e r work of F a r r e l [1957] i n the context of consumer theory. Hanoch and Rothschild [1972] and Diewert [1973b] advanced A f r i a t ' s work by developing constructive l i n e a r programming t e s t s . A f r i a t [1972] also developed nonparametric techniques to test the assumptions of production theory by pro-posing l i n e a r programming tests and e f f i c i e n c y measures, Hanoch and Rothschild [1972] extended A f r i a t ' s approach to include quasi-concave and homothetic production functions. zThe regularity conditions on production functions which are imposed by cost minimizing behaviour are generally of two types: (i) monotonicity \s^n^teic»BS9lMei^gafee^n#&tarttl5±iiged increase, output does not decrease and ( i i ) curvature conditions: i.e., level sets of the production function should be convex sets since any nonconvex portions w i l l never be revealed to us i f producers are minimizing costs subject to fixed factor prices. These conditions correspond to the production function being quasi-concave. For a review of the literature on this topic, see Diewert [1973d, pp. 21-36]. In addition, sometimes the property of constant returns to scale i s imposed on the production function (linear homogeneity) as is the somewhat weaker property of homo theticity. These terms w i l l be defined formally later in the paper. 3 The conclusion the above authors reach i s universal: before attempt-ing to estimate the parameters of a presupposed parametric production function which is to be consistent with the observations given, one should run a set of nonparametric tests to validate the assumptions inherent in the usual tech-nique of f i t t i n g a particular functional form. If the nonparametric tests f a i l , then this w i l l provide an indication that caution should be used in interpreting the results of f i t t i n g any functional form for the production function to that body of data. The objective of thete.hap.te'r i s twofold: f i r s t , we would l i k e to unify (and marginally extend) the approaches taken by Afriat [1972] and Hanoch and Rothschild [1972] to develop the theoretical foundations of nonparametric techniques for testing the properties of production functions for both the single and multiple output cases, and secondly, we would like to try out some of these tests on actual data to reach conclusions with respect to the merits of various tests. We consider a production operation which employs M factors (inputs) to produce P goods (outputs) and l e t : (.y-L*.'•->y-p'> x 1»---» x M) = (y;x) 3 represent this operation in terms of i t s outputs y^, i = 1,...,P, and inputs x., j = 1,...,M. The i th component y. of y and the j th component x. of x are the quantities of the i th good produced and the j. th factor used respec-tively. 3The nomenclature that i s used throughout fidre vect-orsfds \ as follows: (i) x 1 is a vector whose components are x ^ , . . . ,x^. x l t : is the row vector (x^ x^). Superscript i denotes dates or places. ( i i ) x 1 > >x : ] i f x^>xr^, k = 1,...,M; x 1 ^ , k = 1,. .. ,M; x\xi is x^xjj., k = 1,.. .,M, but xS^x 3 ; 0t = (0 0) ; l f c =(!,... ,1) . 4 The set of efficient input-output combinations (y;x) may satisfy the equation T(y;x) = 0 where T is an implicit function whose domain i s the P+M dimensional Euclidean space. From the function T one can obtain, at least in theory, the function t which determines the value of one of the outputs in terms of the other outputs and the inputs. For example: gives the maximum production of y^ when the vector y^ = (Ly^,...,y^) of other outputs and the vector x = (x^,...,x^) of inputs are known. The function t^ w i l l be called a production function. The plan of the jchapxte'rs isj aSj]fiojL-4pws :jin tthe rnext LS_e;epi-on we deal with the case of one output production functions; in Section III we generalize the tests for a single output to the case of multiple outputs. In Section IV we discuss the results of the given tests on several empirical examples which have recently appeared in the literature. The proofs of related theorems appear in Appendix I and Appendix II contains the relevant computer programs. II. SINGLE OUTPUT PRODUCTION FUNCTIONS 2.1 Regularity Conditions on Single Ouput Production Functions When there is only one output, y, produced by multiple inputs x^,...,x^we assume the existence of a relationship between y and x defined by y = f(x). The regularity conditions'we consider for the scalar function f are: (0) f is nondecreasing; (1) f is nondecreasing, continuous from above and quasi-concave; (2) f is nondecreasing, continuous and concave; (3) f is nondecreasing, continuous, concave and linear homogeneous; (4) f is nondecreasing, continuous from above, quasi-concave and homothetic. 5 From a p r a c t i c a l viewpoint, continuity never becomes a problem because i t cannot be contradicted with a f i n i t e set of observations. However, we have included i t i n our analysis f o r completeness. Of these conditions, sometimes i n the l i t e r a t u r e , nondecreasing i s refe r r e d to as monotonicity and concavity as diminishing returns and l i n e a r homogeneity as constant returns to scale. Below we give some d e f i n i t i o n s and theorems per t a i n i n g to production functions. D e f i n i t i o n 1. The function f(x) i s nondecreasing i f , f o r any 1 T . N J 2 „n 1 2 ^ _/ 1 N 2> x e R and x e R , x >_ x ==^> f (x ) >. f (x ). D e f i n i t i o n 2. The set X i s a closed set i f x 1 1 e X, n = 1, 2,... and l i m x = x —^> x ^ X. n-*» Theorem 1. The function f(x) i s continuous from above over R n i f and only i f , {x: f(x) >L y} i s closed f o r every yeR.1* Proof: SeeliRockafellarhi[il9iZ0s,.tpps\ 51]. Theorem 2. The function f(x) i s quasi-concave ovet R n i f and only i f , the set H^ = {x: f(x) >_ y} i s convex f o r every yeR. Proof: See Zangwill [1969, pp. 34]. D e f i n i t i o n 3. The function f(x) i s a proper concave 5 function over n 1 2 a convex set S C R i f ( i ) f o r every x e S and x er'S, and scalarAA, 0 < \ < 1, fOXx1 + (l-A)x 2) >, AfCx1) + (1-A) f ( x 2 ) , ( i i ) f(x) •<+» f o r every xeS, and ( i i i ) f(x) >-°° for at l e a s t one xeS. D e f i n i t i o n 4. The function f(x) i s homogeneous of degree k, k>0, i f f(t x ) = t f(x) for any t, t>0. ^By d e f n i t i o n the n u l l set § i s concave and closed. R i s the r e a l l i n e , R n i s the n dimensional Euledian space. Throug ^Thro.ughoufe the ^pap'er- «w,er<shal4^Trefer^ to CQneave production functions since, conditions'''Cli') tod., "(iii) of Definition 3 cannot be contradicted by any data.. 6 D e f i n i t i o n 5. The function f(x) i s l i n e a r homogeneous i f i t i s homogeneous of degree one. 2.2 Notation and Ordering of the Observations In t h i s section we s h a l l give d e f i n i t i o n s which order the observa-i c tions (y, x) , i = 1,...,N i n d i f f e r e n t ways to f a c i l i t a t e the ana l y s i s . k i D e f i n i t i o n 1. = {k: y =L y , k = 1 N}. D e f i n i t i o n 2.' M = {k: y k > y 1 , k = 1,...,N}.7 D e f i n i t i o n 3. The free disposal set for the i th observation i s BR. = {x: x > x 1}, x = R. = {x: x > x } x — As an example, l e t us consider the set of four observations where For t h i s x . 1) Furthermore, l e t y 4 = R4 R 3 4 X p "2 3 X 1 R l ex ( Figure 1 ample I = { l , 2, 3, 4), = (3, 4 >, L 3 = (3, 4}, L 2 = {2, 3, 4}, L± = (1, 2, 3, 4} , M 4 = <j>, M 2 = (3, 4}, M 2 = {2, 3, 4} and R ^ ^ l ^ . 1 , 4 , as d^ep^rcferi irr tn,e! f igu're'." 6Merely f o r n o t a t i o n a l convenience, from now on, we s h a l l assume that the observations have been put_i.n an ascendjLng order with respect to the "7Note that M = <h, the n u l l set. N y 7 Definition 4. The production function f (x) is called valid i f fCx 1 ) = y 1, i = 1,...,N. It is said to be valid for the i th observation i f f(x X) = y 1. 2.3 Test for a Non-decreasing Production Function This test does not really have much practical value; however, i t is heuristically useful as w i l l be clear later on. From Theorem 1 (Appendix I), given the set S = {(y, x ) 1 , i ~ 1,...,N} of observations i f y 1 = max_. {y : x 1 e Rj}» i = 1 N, then there exists a non-decreasing, valid production function for the observations. 2.4 Tests for a Non-decreasing and Quasi-concave Production Function The following definitions are useful: Definition 1.8 For any set S of vectors in Rn the convex hull of S consists of a l l the convex combinations of the vectors in S. k Definition 2. T± = {x: x = EX^x , E A ^ ^ 0 for a l l k such that k' i x CR., jcL./. Thus, T. is the set of a l l vectors which are convex combinations " j i i of the vectors in R^ with.jeL^. Alternatively, T^ = ^ (J where ^ y represents the convex h u l l . Fig. 2 shows T_^ , i = 1,...,4 for theirexample in Fig. 1. \ ^ T± =^ R1UR2UR3UR4 ^ = {^U^) x Figure 2 ^hi s definition is given as a theorem in Rockafellar [1970, pp. 12] By Theorem 2 (Appendix I) we have the following preliminary r e s u l t : Given the set S = {(y, x ) 1 , i = 1,...,N} of observations, i f y = max_. {y^ : x1eT^.}, i = 1,...,N, then there e x i s t s a non-decreasing quasi-concave, v a l i d production function f o r the observations. Corollary 3 (Appendix I) provides the following constructive al t e r n -a t i v e which makes up our f i r s t t e s t . Test l a Given the set S = {(y, x ) 1 , i = 1,...,N} of observations, i f ]i. % . . . . y 1 = min {u X: ZA.x^ y V l P , IA. = 1,A.>0, jeM., y ^ u ^ l i 7 1 - J J - i y ^ for i = 1,...;N-1, then there e x i s t s a non-decreasing, quasi-eoneaverovalidoproduction.. f unct-iono'f orrtheiobservations. The dual of the l i n e a r program i n Test l a leads as proved i n C orollary ^:(<appem:dix"t^Pie&ttdix I) to: Test l b 9 Given the set {(y, x ) 1 , i = 1,...,N} of observations, i f i * _ r 1 i t i i i t j i -n i _ . „ n , Y = max {y : w x <jL, w x J-y >P, w >0, ieM.}>l, i i Y ,w for i = 1,...N-l, then there e x i s t s a non-decreasing, quasi-concave, v a l i d production function for the observations. The . . The dual variables w.,...,w represent the input p r i c e s . 1 p 9Hanpch-Rothschild [1972, p. 262] derive t h i s t e s t using a separat-ing hyperplane argument. In t h e i r tes_t.the f i r s t constraint of the l i n e a r program i s a s t r i c t equality, i . e . w1 x 1 = l . This i s merely a normalization and does not present any problems. However, from computational viewpoint Test l b , without any equality constraints, i s more a t t r a c t i v e . 9 To i l l u s t r a t e how these tests work, we consider the following example where a production operation with two inputs i s observed four times. Let y 1 and x 1, i = 1, ...,4, denote the observed outputs and input vectors. Further-1 2 3 4 more, we assume that y y y y . Figure (3a) depicts the case when Test l a is successful at the f i r s t observation while in Figure (3b) i t f a i l s at the same observation. Figures (3c) and (3d) show how Test lb succeeds and f a i l s at the f i r s t observation. Figure 3 10 2 3 In Figure (3a) there does not e x i s t a convex combination of x , x 4 1 * and x which i s less than x . The best we can do i s x and yet, i t i s greater than x 1 which implies that y 1 > l . In Figure (3b) we can f i n d at le a s t one con-2 3 4 1 vex combination of x , x and x which i s less than x . One such vector i s * 1 x <x , hence, u<l. On the other hand, there are many hyperplanes separating x 1 from T^, Figure (3c), while there i s none i n Figure (3d). As a measure of v i o l a t i o n of the assumptions at the i th observation i * i * 1-u f o r Test l a and 1-y f o r Test lb can be used. The measure w i l l give, roughly, the percentage by which each component of the vector x 1 must be reduced i n order to make the i th observation consistent. Unfortunately, t h i s measure w i l l over-estimate some of the v i o l a t i o n s while working well f o r others. Figure 4 shows two possible cases with v i o l a t i o n . Figure 4 I f the f i r s t observation x"*" i s r e a l i z e d at A then |r* = ^ a n ^ t o make x 1 a U i i 1^ 1 "good" point each of i t s components should be m u l t i p l i e d by u . 1 0 I f x i s r e a l i z e d at B, although we can put i t back i n t o the "good" region by m u l t i -1* pplying each of i t s components by u t h i s can be achieved by reducing only i t s SB second component as l i t t l e as — . But such a precise measure of v i o l a t i o n cannot be very e a s i l y computed. 1 0 Or by a value £u 1* 11 2.5 Tests f o r a Non-decreasing and Concave Production Function From Theorem 5 (Appendix I) we have: Test 2 a 1 1 Given the set S = {(y, x ) 1 , i = 1,...,N} of observations, i f (5.1) i ± N . . N . . N y = F (x ) = max {£ A . y J : E A.x 3 < x 1, Z \. = 1, 3=1 3 3=1 3 Xi. ^ 0? j = 1,...,N} f o r i = 1,...,N, then there e x i s t s a non-decreasing, concave v a l i d production function f o r the observations. Theorem 6 (Appendix I) on the other hand, provides: Test 2 b 1 2 Given the set S = {(y, x ) 1 , i = 1,...,N} of observations, i f i * _ i Y = max ± ± y Y 1» P > w subject to / i i t v (P , w ) r i i r k i y -y > i k -x (5.2) k = 1,... ,N k ^ i i , , t i , p + 1 w = 1 w^O p Se-l l A f r i a t [1972, pp. 573-74] presents t h i s t e s t somewhat d i f f e r e n t l y . 1 2Hanoch-Rothschild [1972, p. 268] derive t h i s test throughaa d i f f e r -ent argument. 12 for i = 1, . . . ,N, and y >_0, then there exists a non-decreasing, ccncave, valid production function for the observations. The interpretation of variables p and w in the linear program (5.2) is obvious: they correspond to the dual variables of (5.1) and represent the prices of output and inputs. The value of the objective function of the linear program of Test 2a i s always at least as large as y 1, when testing for the i th observation. The difference A ^ = F(x 1) - y 1 represents the foregone production due to inefficiency or non-concavity of the production function. The ratio ^i is i y the increment in the production, as a percentage, which should be provided to put production back on the efficient surface. The-very same can be more directly obtained from Test 2b. Nor-malizing the constraints on p 1 and bmit'tlngdtheeMrnmMzingdcdnsfcrSint . p 1 + l^w1 = 1 in (5.2) we get i * i Y s max Y i , wi Y ; w (5.3) subject to i t , r i i r k i < y _ y i i -x -X \ - Y^O, k = 1,.. . ,N, k ^ i , w1>0, i * for i = 1,...,N. A_^ = Y > t n e optimal value of the objective function of i i (5.3). Note that Y 1 S n o w measured in the same units as y . The coefficients of the objective function of the linear program (5.1) are the observed outputs y^, j = 1,...,N. Once Test 2a succeeds at the i th observation then a post optimality analysis on the ranges of y 1 w i l l determine safety margins for adjusting the data on production quantities with 13 respect to this observation. As other observations are considered these margins can further be narrowed to some f i n a l l e v e l s . 2.6 Tests f o r a Non-decreasing, Concave and Linear Homogeneous Production Function Euler's Theorem states that i f the function f i s homogeneous of degree k then kkf(x) = x y f ( x ) , v f ( x ) = hJL >•••» l i . • T n e proof of t h i s theorem i s b r i e f l y as follows: i f f i s homogeneous of degree k then, k f(.ex) = e f ( x ) , f or e>o, which i s d i f f e r e n t i a t e d with respect to @ to get x t v f ( e x ) = k e k - 1 f ( x ) . The theorem follows as we substitute 6 = 1 . I f f i s l i n e a r homogeneous, f(x) = x t V f ( x ) . M u l t i p l y i n g t h i s by p, the output p r i c e , we get pf(x) = x t ( p V f ( x ) ) . Denoting p r o f i t s by TT and l e t t i n g the t o t a l cost be wfcx, a l i n e a r function of fac t o r prices w_., and quantities x j . = • 1,. .. ,-N, the following condition i s necessary for p r o f i t maximization: (6.1) cl (pf(x) - w.x) = 0, j = l,...,My dx. J This i n turn, implies that pVf(x) = w. On the other hand, i f f(x) i s concave pf(x) - wfcx w i l l be concave and (6.1) i s also s u f f i c i e n t f o r maximizing p r o f i t . Thus, i n order for the observations to be consistent with p r o f i t maximization and constant returns, we require that py = w*"x which i s a hyperplane through the o r i g i n . This condition i s added to Test 2b to obtain Test 3b13 Given the set S = {(y, x ) 1 , i = 1,...,N} of observations,:if (6.2) i * y = max yX>0 i l l Y J P > w subject to i i t , i , .t i 1 p + 1 w = 1, • i k I y y i k -x -x i i i t i _ i rt i _ p y -w x = 0 p >0 w >0, for i = 1,...,N, then there exists a non-decreasing, concave, linear homogeneous, valid production function for the observations. The dual of (6.2) leads, by Theorem 7 (Appendix I ) , to Test 3akk Given the set S = {(y, x ) 1 , i = 1,...,N> of observations, i f y 1 = F(x X) = max {ZA^y 3: ZA^x 3^x 1, )^>p, j = 1,...,N for i = 1,...,N, then there exists a non-decreasing, concave, linear homogeneous, valid production function for the observations. ,N, We can, as before, suggest the use of r±. a s a measure of violation y i of the hypotheses at the i th observation, where (S'%anoch-Rothschild [1972], pp. 268, 270]. : ' % f r i a t [1972, p. 573]. 15 (6.3) A. = F(x 1) - y 1 . Again, as before, i n (6.2) we normalize on p 1 and disregard constraint p 1 + l S ? 1 = 1 to obtain i * _ I. Y = max Y i i Y 5 w (6.4) subject to (1, w±1:) i i t i y - w x i " k " > < y -y > i k -x -x * = o, -y >0, k = 1, . .. ,N ~ Vr4 I i ^ f o r i = 1,...,N. A i i n (6.3) i s equal to y , the optimal value of the objective function of (6.4). Also, the post optimality analysis of Section 2.5 for Test 2a i s a p p l i c a b l e , f o r Test 3a. 2.7 Test for a Non-decreasing, Quasi-concave and Homothetic Production F u n c t i o n 1 5 In order to v i s u a l i z e how a.given set of data on output and inputs may not be consistent with homothetic production, we give the following example. 1 6 A production operation employing two inputs to produce one output i s considered. Suppose that we have recorded the values of y, x^ and x^ four times and found out that 1 2 3 4 y < y < y < y 15. The material i n t h i s section i s based on Section VI of Hanoch-Rothschild [1972] 16 This example i s due to Hanoch-Rothschild [1972, p. 271]. 16 where y 1 i s the output which i s produced by the input x 1 = (x^, x^), i = 1, 4 2 2 4 ....,4. Let us further assume that x = Xx , X>1, which places x and x on the same h a l f l i n e from the o r i g i n (Figure 5). Figure 5 There w i l l be a non-decreasing continuous from above, quasi-concave production function v a l i d f o r these observations because we are able t o . v e r i f y that: y 1 = max {y 3: x^T.} , i = 1,...,4. j 2 4 4 But, the slope of the y isoquant at x i s at least tan a, whereas the slope 2 2 of theyy isoquant at x i s at most tan 3. From the figure 3<a which implies 2 4 tan 3 < tan a. This contradicts homotheticity of x rand x . We now give D e f i n i t i o n 1. (Shephard [1970, p. 30]). The non-decreasing, con-tinuous from above, quasi-concave function f(x) i s homothetic i f there e x i s t s a transform cf> of f(x) which i s non-decreasing, concave and l i n e a r homogeneous with respect to x. 17 Let us define a function H.as a transform.of -f: H(x) = <j>[f(x)]. I f we assume that H i s l i n e a r homogeneous with.respect t o x and (f[.f]>0 f o r a l l f then :we can write W f T x 7 T H ( x ) = 1 o r ^ ^ [ f f x T T ) = x ' i i r('xiH x* L e t t i n g 1$ = Gj>[f(x )] we have HP—-- = 1 which implies that — i s .mapped onto the unit isoquant of H. l' In order f o r the observations {(fry, xxj^,-41== Xl.,-.-.- *NN}:oto-ebe'-Ocorisistent with a nondecreasing, continuous from above, quasi-concave v a l i d production . . i f unction f, the observation transforms {(1, 3=7), i = 1,...,N) must be consistent <j>i with a nondecreasing, concave and l i n e a r homogeneous production function. But the function H has these l a s t properties i f and only i f the following conditions are s a t i s f i e d . ( i ) <j)[f(x)] i s a nondecreasing function of f(x) or i},1^3 i f f (x 1) ^ f ( x j ) . ( i i ) There ex i s t s a set of normalized, non-negative p r i c e vectors {jwf :ww^ Q, ii==.llj. . .^N^usuchhthaththerprofittaththerprieesw^wp . ,v.^ w^) i s zero at the i th observation and non-positive elsewhere. This i s because of the l i n e a r homogeneity ( r e c a l l Euler's Theorem) and the consistency with p r o f i t maximization (or concavity). Thus, the value of the function , i . it>x g(w ) = w |^ , i t = w . X which represents a hyperplane i s equal to one at — r but not less than one at I 1 other points. Or equivalently: i , i t , x i .. w W *7f =1, 1 = 1,... ,N, i t x-^ w — r >_ 1, i = 1,...,N, j = 1,...,N, ±4 j • e -18 wX>p, i = 1,..1,N. $x>0, i = 1,...,N. Now, l e t us take ( i i i ) f i r s t . This can be achieved by normalizing <j> as follows: cf)1 = y"*" = min {y 3 : j = 1,...,N}. On the other hand, from ( i i ) we have w l tx 1 = cj) 1, i .= 1,...,N and w"Ltx-'>(f>-' = w^x 3, i = 1, ,N, j = 1, .. . ,N, i ^ j along with w 1^), i = 1,. . . ,N. Also, from ( i ) and ( i i ) we can write wltx"L > =w J tx J i f f ( x 1 ) ^ f(x~'), i 4 j , or i f i e L j , i 4 j . The foregoing establishes Theorem 7.1. There e x i s t s a non-decreasing, quasi-concave and homo-t h e t i c production function f v a l i d f o r observation {(y, x ) 1 , i = 1 N} i f and only i f , there e x i s t s w1, i = 1,...,N, such that I t 1 1 • r j • i AT! w x = y = mm {y : j = .1,. .. ,N} , (7.1) w l tx : 1 - w J tx : I ^ 0, i = 1,. .. ,N, j = 1,.. . ,N, i ± j , w x - w ^ x 3 > 0 i f i e L . , i ^ j , w 1^), i = 1, .. . ,N. Cle a r l y , a test f or the consistency of a set of observations {(y> x ) 1 , i = 1,...,N} with a production function havingt'the properties c i t e d i n Theorem 7.1 would be to attempt s o l v i n g (7.1). But, i f there does not exi s t any s o l u t i o n to (7.1) we cannot t e l l how bad the s i t u a t i o n may be. However, i t i s very easy to construct a l i n e a r program from (7.1) which w i l l make up f o r t h i s d e f i c i e n c y . Test 4 Given the set S = {(y, x*))11-, i •=!,...,N} of observations i f g*si 5umin =3 <= 0 3 , w 1 , i = l , . . . , N 19 subject to l t 1 1 • ' r j . T „ , ( I ) w x = y = mxnly : j = 1,...,N}, (7.2) (11) w x 3 - w3 x 3 + B>0, i = 1 N, j = 1,...,N, i * j , ~ i t i j t j>Q .j.f ieL., i 4 A, ( i i i ) w- x - vr x-VO i f i g L i \ l ? j', i 3 (iv) w >0, i = 1,...,N, (v) B>0, then there exists a non-decreasing, quasi-concave, homothetic and valid production function for the observations. Note that (7.2) has always at least one feasible solution. For example, we can always find a set {w1: i = 1,...,N, w1>0}wwhi,chssatisf ies ( i ) , ( i i i ) and (iv). Choosing a very large 3^ 0 ( i i ) and (v) are also satisfied. 2 The linear program (7.2) has Mf _-JILN + 2 constraints (excluding, ~2~ " ~-2~ of course, the non-negativity constraints) and MN + 1 variables which, even for a moderate number of observations and inputs, represents a very large programming problem. I I I . MULTIPLE OUTPUT PRODUCTION FUNCTIONS 3.1 Regularity Conditions on Multiple Output Production Functions Let the implicit function T determine the rule according to which a production operation works. 1 7 Thus, T(y 1 y p; x ± , • • • ,2^) = 0. 1 7T was introduced in the Introduction. 20 Let us further assume that we can express one of the outputs which we c a l l the reference output, as a function of the other outputs and the inputs as. follows yk t k ( y i " , , ' y k - i * yk+i'""' yp ; x i V or letting yk = ( yl'* ' * ' , y k - l ' yk+l""' yP^» k = 1»-"» p» yk = \ ( y k ; X )' where t^ i s a real valued function. Now, i f we replace y^ with -y^ . which i s the negative of a vector of outputs then along with x i t represents the extended vector of inputs (-y^ ; x). We introduce a new function f to determine the reference output as: ( l . l ) y k = f k ( ~ y k ; x ) = fck(yk; x )* We can write any output as a function of the appropriate extended vector of inputs. Similarly, letting x^ = (x^,.. ., xjc_^> xic+i»'"'' X M) » k = 1>«-'>M» we can write any input as a function of the extended vector of inputs (-y; x^)• 1 8 Since negative input can be regarded as positive output, the reference output i s determined as (1.2) -x k= g k(-y; x k), The regularity conditions we consider for multiple output production functions are: (1) f^ is non-decreasing and quasi-concave; •"•"Clearly, this extended vector of inputs i s slightly different from the previous one. 21 (2) f i s non-decreasing and concave, (3) i s non-decreasing, concave and l i n e a r homogeneous f o r k = 1,...,N. 3.2 Tests f o r a Non-decreasing and Quasi-concave Production Function Tests l a and lb of Section 2.4 are applicable to the multiple out-put productions providirigtwetfirjSt useiaeMnear^Erams^or^ a l l negative q u a n t i t i e s i n t o p o s i t i v e ones. L e t us consider a production operation with two outputs y^, y^ and one input x which are observed four 4 3 2 1 times. Let y^ = ^ ^ ~ ^ ± ' x ) a n < ^ also y ^ ^ i ^ y ^ y ^ ' Now i f the configuration of the extended.inputs ( - y ^ x ) 3 are as depicted i n Figure 6a and i f we apply Test l a to the 1 st observation the test w i l l a f a i l while i t should pass. (b) Figure 6 -T-he reason i s obvious from the f i g u r e ! However^ the following, transformation i t •_ i t = x 1 - ( d i - ' - V p - ^ + e l = ^ ~ ( d l » d 2 ) + where and d. = minix* : i = 1,...,N} = minted : i = 1,...,4> 2 i 2 i 2 x = t - y j ; x U x = i , . . . , 4 , with e small p o s i t i v e s c a l a r , by s h i f t i n g the o r i g i n from 0 to 0^ (Figure 6b), 22 enables us to.use Test l a and Test l b c o r r e c t l y since a l l q u a n t i t i e s are now s t r i c t l y p o s i t i v e . We need the p o s i t i v e quantity e to eliminate the p o s s i b i l -i t y of degeneracy which w i l l occur when t e s t i n g f o r the i th observation i f 1 k x. = min {x. : k = 1,...,N}, 2 k 2 for for j = 1,...,M+P-1. The measure of v i o l a t i o n i n th i s case w i l l not, as showniin Figure 6 b , produce the same value as f o r si n g l e output case and i t i s dependent on the value of e . 3 . 3 Tests f o r a Non-decreasing and Concave Production Function It turns out that the tests f o r mbnotonicity and concavity are independent of the output singled out (which i s not the case for the tests f o r quasi-concavity). To see t h i s , l e t us consider the following example where there i s an e f f i c i e n t production operation with two outputs y^, y^ and one input x (Figure 7 ) . Suppose the set {y^» y£> ^ °^ e f f i c i e n t input-output combinations s a t i s f i e s the equation: T ( y r .y 2;. x) = 0. Assume that we can express y^ i n terms of y^ and x as: y l = f l ( _ Y 2 ' X ) which we further assume, i s a non-decreasing, continuous and concave function. 23 Figure 7 -i Deinition 1. (Rockafellar.[1970, pp. 23]). If f i s a real valued function from R n.to R defined by y = f(x), xeS c Rn, yeR, i t s epigraph i s the set { (x, u): xeRn; ueR, u>f(x)}. Definition 2. (Rockafellar [1970, pp. 23]). A function f i s a convex function over S, i t s domain of definition, i f i t s epigraph i s convex. Clearly, since f i s convex over some set, -f is concave over the same set and the above definitions are equally applicable, with necessary changes in the inequalities of course, to concave functions. Hence, for our example, the set 2 E-L = {(-y2» x, px) : (-y2, x)eR , u^R, u ^ f ^ - y ^ x)} which i s the epigraph of - f ^ i s convex. On the other hand, i f y 2 = f 2 ( - y 1 ( x) 24 then the epigraph of - f r E2 = ^ ~ y l ' x' y2^ : ^ ~ y l ' X ^ e R ' y 2 e R ' y 2 = f 2 ^ ~ y l ' K ^ is nothing but the set E^. Also, the monotonicity and the continuity of are obvious. Thus, we conclude: i f there exists a production function having the properties of this section for some arbitrarily chosen k from 1,...,N then there exists one for any k from 1,...,N. Now, Test 2a and Test 2b are modi-fied for multiple production as follows: given the set ="^(v]c» _ yk' x^ 1» i = 1,...,N} of observations i f (3.1) N . N yk = F ( _ y k ' X ) ='inax { E V k : Z X i X j=l J j=l 3 X -"I X for i = 1, Z X . ~ 1, X .>p, j = 1,...,N}, j=l J J ,N, then there exists a non-decreasing, concave, valid production function 'for the observations, and given the set = {(yk> "Yj^» x) , i = 1, ...,N} of observations, i f 1* liq n Y = max Y ^ 0 subject to (3.2) Y , P » Pk» w (P k, P k, w ) f ' ' --'K yk i yk j . - Y^O, k = 1, - yk " yk ...,N, - x X J i , 1 t " i , , t i .. pk pk ' P k L °» P k =>°> w1 ^ 0, ig"i , i i i i i i . t , i " i t N p k = ( p r p2,..., p ^ , Pk+1,..», Pk+r---» P p) > (P k > P k ) = 25 for i = 1,...,N, then there exists a non-decreasing, concave, valid production function for the observations. 3.4 Tests for a Non-decreasing, Concave and Linear Homogeneous Production Function The addition of linear homogeneity to the conditions of Section 3.1 makes the surface y^ = ^(-y^* x) a cone. This is a special case and our con-clusion in the previous section remains intact for i t : the tests for monotonicity, concavity and linear homogeneity are independent of k. Figure 8 depicts the case when linear homogeneity i s introduced to the example in Figure 7. Figure 8 y i = f i ( ~ y 2 ' x ) y 2 = f 2 ( - y r x ) For the multiple output production we have the Tests 3a and 3b modi-fied as follows: given the set S^ = {(y^' ~y^> x) 1> 1 = 1,.-.,N} of observations i f i - N • T, = F(-y, , x) = max {I a .yrj :Ea. k •'k' a . j• 'k i 3 = 1 J J j A " y k < X X a_.>0, j = 1,. . . ,N} i = 1,...,N, then there exists a non-decreasing, concave, linear homogeneous, valid production function for the observations, and given the set S = {(y^; -y^, x ) 1 , i = 1,...,N} of observations i f i * - i Y = max Y >- 0 Y , P k, P k, w subject to ~t t , i V P k ' W > P k + 1 P k + 1 w =1, * i j ' " y k " ' yk" -yg. " y k X X > —Y iP, k = 1,...,N, k ^ i , i i , t t . i V k " ( p k ' W > -Yi, = 0, P k L 0, P k >, 0, w1 >, 0, i - 1,...,N, then there exists a non-decreasing, concave, linear homogeneous, valid production function for the observations. IV. EMPIRICAL EXAMPLES In this part, we discuss the results of the tests developed earlier applied to actual data on single and multiple output production operations. Here, we shall refer to Tests l a and lb of Section 2.4 as Test l,tTests 2a and 2b of Section 2.5 as Test 2,ean'dTests 3a and 3b of Section 2.6 as Test 3. 27 4.1 An Example for Single Output Production We have applied Test 1, Test 2 and Test 3 to the annual data for U.S. manufacturing from 1929 to 1971 of Berndt-Christensen [1973]. The four factors employed in the production of manufactured goods are equipment, structures, blue collar workers and white collar workers. Thus, the annual observations of these quantities for 43 years, from 1929 through 1971, make up the set S = { (y; x^, x^, x^, x^) 1, i = 1,...,43} where y; x^, x^, x^, and x^ represent the output and the inputsiintthatoorder. Table 2 gives the data used along with the summarized results of the tests. In the last three columns of this table we have listed the violations in terms of the measures suggested in the text (Section II ) . Table 1 tabulates the total C.P.U. time 2 0 (i.e., for a l l 43 observations) required for each test with double precision. Note that there are no violations in Test 1 (quasi-concavity), four extremely small violations in test 2 (monotonicity plus concavity) and nine extremely small violations in test 3 (monotonicity plus concavity plus linear homogeneity). Tests 1 2 3 a 8.067 18.211 17.754 b Not applied 25.648 27.452 C.P.U. time in seconds Table 1 A l l the computations have been carried out on an IBM 370/168. TABLE 2 Violation Indexes Observation • Number Year y X l x 2 x 3 x 4 Test 1 Test 2 Test 3 1 1929 0.777467 0. 614293 1. 036530 0.699089 1. 006579 - - -2 1930 0.700437 0. 627900 1. 101040 0. 582081 0. 952646 - - -3, 1931 0.593347 0. 616711 1. 105619 0. 471185 0. 791079 - - -4 1932 0.483868 0. 586922 1. 064210 0. 378V22 0. 607474 - - -5 1933 0.489019 0. 540050 1. 001619 0. 423.129 0. 550272 - - -6 1934 0.497059 0. 502044 0. 963399 0. 454383 0. 521588 - - -7 1935 0.550044 0. 476892 0. 927686 0. 514 L38 0. 603270 - - -8 1936 0.624121 0. 470097 0. 888512 0. 601L29 0. 713026 - - -9 1937 0.674741 0. 485566 0. 868074 0. 652106 0. 805027 - - 0.00008 10 1938 0.570284 0. 50972i 0. 874391 0. 507548 0. 733900 - - 0.01685 11 1939 0.637754 0. 502544 0. 848554 0. 602.'i64 0. 774717 - - 0.01122 12 1940 0 .692180 0. 507249 0. 831969 0. 671799 0. 833312 - - 0.00814 13 1941 0.843775 0. 533694 0. 838105 0. 892720 0. 875381 - - -14 1942 0.978586 0. 568752 0. 873377 1. 130390 0. 778314 - - -15 1943 1.110169 0. 567539 0. 845986 1. 369389 0 748648 - - -16 1944 1.147779 0 563229 0. 799029 1 344)70 1 042379 - - -17 1945 1.036280 0 577003 0 760319 1 145300 1 093080 - - -18 1946 0.959608 0 629963 0 766156 1 019 360 1 045190 - - -19 1947 1.027069 0 730955 0 888816 1 081300 1 074559 - - -20 1948 1.028959 0 881288 0 953040 1 109.L39 0 881595 - -21 1949 1.000000 1 000000 1 000000 1 000000 1 000000 - - -22 1950 1.059640 1 051720 1 017039 1 105 530 0 938076 - - -23 1951 1.148829 1 099859 1 016270 1 193 )10 1 079009 - - -24 1952 1.207120 1 186939 1 049359 1 208>69 1 265229 - - -25 1953 1.282379 1 262600 1 074710 1 262 509 1 424709 - -26 1954 1.227200 1 330660 1 096740 1 136L60 1 517139 . - - -27 1955 1.290250 1 398250 1 .114220 1 216229 1 539459 - - -28 1956- 1.330400 1 .450470 1 .137449 1 225220 1 .685430 - - -29 1957 1 .344110 1.550679 1 .178969 1.190709 1.817439 - - -30 1958 1.289929 1 .636889 1 .225849 1 .074,159 1 .880850 - - 0.00048 31 1959 1.366059 1 .637019 1 .249949 1 .173639 1 .932920 0 00090 0.00090 32 1960 1.376309 1 .632529 1 .245580 1 .163509 2 .027439 0 00055 0.00061 33 1961 1.359890 1 .657960 1 .253550 1 .130380 2 .039900 - - -34 1962 1.413289 1 .671269 1 .258280 1 .1964 20 2 .085799 0 00034 0.00036 35 1963 1.436769 1 .704309 1 .259450 1 .212549 2 .137469 - -36 1964 1 .472890 1 .751960 1 .268840 1 .250919 2 .170520 - - -37 1965 1.552759 1 .843699 1 .282740 1 .338590 2 .244840 - - -38 1966 1.660580 1 .995839 1 .321500 1 .440249 2 .380759 - - -39 1967 1.699590 2 .197590 1 .384580 1 .428060 2 .511729 - - -40 1968 1.759480 2 .352909 1 .439819 1 .465219 2 .609360 0 .00004 0.00004 41 1969 1.812840 2 .479659 1 .481990 1 .501249 2 .693450 - - -42 1970 1.762910 2 .626280 1 .528729 1 .41.1329 2 .671880 - - -43 1971 1.723470 2 .735410 1 .557819 1 .371590 2 .569409 - - — TABLE 3 Violation Indexes Observation Test 1 Test 2 Test 3 Number Year y l y 2 x l X 2 y = y l y=y2 y = x i y=x2 1 1929 136.300 55.800 173. 300 102. 700 - - - 0.0139 0.1759 2 1930 132.300 41. 300 165. 400 105. 000 0. 0004 - - 0.0339 0.1531 3 1931 128.800 31. 100 158. 200 104. 300 - - 0.0305 0.1045 4 1932 118.300 17. 600 141. 700 102. 100 - - - - -5 1933 113.800 18. 500 141. 600 96. 800 - ( *) - -6 1934 117.200 25. 100 148. 000 92. 100 - - - - -7 1935 124.300 30. 300 154. 400 90. 000 - - - - -8 1936 131.800 41. 100 163. 500 89. 500 - - - (*) -9 1937 139.800 44. 600 172. 000 90. 800 - - - 0.0410 10 1938 140.100 34. 300 161. 500 93. 200 - - - - — 11 1939 146.100 43. 800 168. 600 92. 200 - - - - 0.0011 12 1940 153.800 53. 300 176. 500 93. 000 - - - - 0.0239 13 1941 164.400 73. 100 192. 400 95. 600 - - - 0.0370 0.0986 14 1942 . 178.600 80i 800 205. 100 100. 000 0 0472 0.0472 0. 0558 0.0536 0.1183 15 1943 180.400 98 100 210. 100 99. 000 0.0326 0.0190 0. 0361 0.0334 0.0954 16 1944 188.800 103 200 208. 800 • 96. 700 - - - - -17 1945 192.300 92 600 202 100 94 900 - - - - -18 1946 195.800 77 300 213 400 94 800 - - • - - -19 1947 193.800 85 700 223 600 101 400 0 0056 0.0359 0. 0153 0.0002 0.0623 20 1948 203.900 93 500 228 200 108 700 - 0.0198 - - 0.0506 21 1949 206.100 91 300 221 300 116 100 - 0.0756 - - 0.1057 22 1950 214.900 113 900 228 800 121 200 - - - - 0.0953 23 1951 228.400 122 900 239 000 130 000 - - - - 0.0917 24 1952 237.300 123 000 241 700 137 300 - - 0. 0003 - 0.1070 25 1953 247.600 131 200 245 200 142 300 - • - - - 0.0900 26 1954 250;300 125.200 237 400 148 000 - - - - 0.0981 27 1955 262.900 143 .900 245 900 152 600 - - • - - 0.0865 28 1956 273.000 143 .300 251 600 160.800 - 0.0182 - - 0.0978 29 1957 281.100 141 .600 251 500 167 .700 - 0.0123 - - 0.0912 30 1958 288.000 130 .400 245 .100 173 .300 - - - - 0.0512 31 1959 300.700 145 .000 254 .900 176 .000 - - - - 0.0541 32 1960 310.000 147 .300 259 .600 181 .800 - - - - 0.0551 33 1961 320.400 145 .700 258 .100 187 .700 - - - - 0.0282 34 1962 335.000 160 .400 264 .600 191 .900 - - - - 0.0168 35 1963 346.300 169 .400 268 .500 198 .600 - - - - 0.0117 36 1964 363.300 181 .200 275 .400 206 . 200 - - - - 0.0031 37 1965 383.600 196 .300 285 .300 215 . 500 - - - - -38 1966 406.600 209 .900 297 .400 227 .200 (*) - - -39 1967 424.300 206 .900 305 .000 240 .700 CO - - - -0.4319 0.4822 0.4985 0.5645 0.5606 0.4749 0.3820 0.3182 0.2736 0.2731 0.2254 0.1864 0.1604 0.1236 0.1030 0.0733 0.0771 0.1066 0.1038 0.1057 0.1116 0.0976 0.1093 0.0894 0.0982 0.1010 0.0954 0.0713 0.0675 0.0597 0.0369 0.0335 0.0202 0.0078 (*) Test 1 is not applicable. 30 4.2 An Example for Multiple Output Production For this case we have used the data provided in Christensen-Jorgenson [1969, 1970] which consists of 39 annual observations from 1929 to 1967 on two outputs and two inputs. The outputs are consumption goods and investment goods while the inputs are private domestic labor and private domestic capital. We denote them by y^, y 2, x^ and x 2 respectively. Taking y^ as the output and -y 2 as the third input we have the transformed observa-tion set S = {(y^; ~y2> xi» x ^ ) 1 , i = l-: »~39}. We have applied Test 2 and Test 3 to this set. For Test 1, on the other hand, we have four transformed data sets. They are S^^ = { ( y ^ -y 2, x x, x ^ 1 , i = i,...,39}';' 52 = ^ y 2 ' ~ y l ' x l ' x2^ 1' 1 = 1>"-»39}; 5 3 = {(-x^ -y 1 5 -y 2, X2) 1, i = 1,...,39}; = {(-x 2; -y i s -y 2, x ^ 1 , i = 1,...,39}. Table 3 gives the data and the violation indexes at various observations for each test. 4.3 Conclusions A l l three of the tests have been successful at v i r t u a l l y a l l observations for the single output data. For the multiple output example while monotonicity-quasi-concavity conditions do not seem to be violated significantly the same conclusion certainly cannot be reached for the conditions of Test 2 and Test 3. Recalling that these tests are for monotonicity-concavity and monotonicity-concavity-linear homogeneity, one should exercise extra care in assuming the existence of these regularity conditions when trying to accommodate these figures with a production function having the underlying properties. The violations, especially for the multiple output example, seem to be consistent with the yarE.fia#i*pms in the data. As can seen from Table 1 •'he computation Mines • or- ' As can be seen from Table 1 the computation times for the Tests labelled a are significantly less than those for the tests labelled b. This difference w i l l probably become even greater when data with larger numbers of observations and input-outputs are attempted. It i s important to note that although the variables of thestest linear programs for Tests lb, 2b and 3b represent input-output prices, they cannot be suggested for practical use as such since they generally correspond to extreme cases and indicate only possibilities of theoretical value. 32a CHAPTER II NONPARAMETRIC TESTS OF REGULARITY CONDITIONS FOR PRODUCTION FUNCTIONS: PRICE AND QUANTITY DATA 3* I. INTRODUCTION Various nonparametric tests of such regularity conditions as monotonicity quasi-concavity, concavity, homotheticity for production functions have been suggested i n Chapter I for the cases when we have data available on input and output quantities only. In this Chapter we develop nonparametric tests of the same type of regularity conditions for production functions when there are data on prices, quantities, profits, costs and output quantities, or some combination of them. However, in doing so, we base our exposition completely on the material in Chapter I to the degree that both of these Chapters together establish a uni-fied methodology for nonparametric tests of regularity conditions for production functions when data on^quantities, prices, profits or costs, or some combina-tion of them are available. For a brief introduction to the development of nonparametric methods in production theory the reader i s also referred to Afriat [1972], Hanoch and Rothschild [1972]. Let us consider again a production operation which employs M factors (inputs) to produce P goods (outputs) and let the P + M dimensional row vector (y 1»"-.y ; Xi'---> X M) = (y; x) represent this operation in terms of i t s outputs y_^ , i = 1,. ..,P, and. inputs , j = 1,. . . ,M. The i th component of y_^ of y and the j th component x^ of x are the quantities of the i th good produced and the j th factor used respec-tively . The set of efficient input-output combinations (y;:.x) may satisfy the equation T(y; x) = 0 where T is an implicit function whose domain i s P+M R , the P+M dimensional Eucledian space. From the function T one can obtain, at least in theory, the function t which determines the value of one of the 33 outputs in terms of the other outputs and the inputs. For example: yk = Vvl'- •• ' y k - l ' y k + l , , - , ' y B ; x r * " » V gives the maximum production of when the vector y^ = (y^> • • • '^-1' yk+l' . ..,Vp) of other outputs and the vector x = (x^,...,x^) of inputs are known. In terms of a new function f we can determine the reference output y as y k = * k ( - y k ; x) = t f e(y k; x) . Similarly, letting xfc = (x.^ .. . . x ^ , \ + 1 ' " - » x M ) » k = 1,...,M, we can write, in theory, any input as a function of the extended vector of inputs (-y; Xj^) as -x^ = g k(-y; x^ )^ where the minus sign makes x k an output. The function t k w i l l be called a production function. We shall denote the prices of the goods produced by P = (p-^ • • • »Pp) » and the wages of the factors employed by w = (w1,...,wM). We also c a l l (p k; w) = (p , . .. .PJ^J Pk+l''"''pp' wI»"*»WM^ t h e e xtended vec-tor of input prices. For a production operation with multiple inputs and outputs we let the scalar valued function f = f k (or g^ )» for some k, represent the output and consider the following regularity conditions: (1) f i i s nondecreasing and quasi-concave; (2) f is nondecreasing and concave; (3) f i s nondecreasing, concave, and linear homogeneous. In the next Section, we develop tests of monotonicity plus quasi-concavity, concavity and concavity with linear homogeneity for the case when data on prices and quantities are available. We discuss the tests of monoton-i c i t y plus concavity and concavity with linear homogeneity for data on prices 34 and p r o f i t s i n Section 3. In Section 4, we give a t e s t of monotonicity, con-cavity and l i n e a r homogeneity f o r data on prices only. In Section 5, we give tests of monotonicity and quasi-concavity f o r data on p r i c e s , output q u a n t i t i e s and costs or p r o f i t s . An empirical example i s given i n Section 6. The compu-ter programs of most of the tests followiinithe.bAppe@dlixiIclI7-I. I I . PRICE AND QUANTITY DATA 2.1 Test f o r a nondecreasing and Quasi-concave Production Function; P r i c e and Quantity Data When we have observations on quantities only we can test i f there ex i s t s a nondecreasing, quasi-concave production function v a l i d f o r these observations using Test l a or lb i n Chapter I, Section 2. Here we consider tests when both p r i c e and quantity data are a v a i l a b l e . Let us now reconsider the test l i n e a r program of Test l a , Chapter I: i f : " (2.1) u 1* = min {u1 : i ^ j j i ^ i Q, S , l , ^ = ^ } > 1, u >. A J then the test i s successful f o r the i th observation. The dual variables w^ ,...,w^ of (2.1) are inter p r e t e d as the resource prices and they are the quantities to be determined i n Test l b , Chapter I which i s as follows: i f (2.2) Y 1 * = min {y1 : v/V" < 1, W ^ X^ - Y 1 i 0, jeM. , w1 > 0} > 1, i i Y , w then the tes t i s £s.uc-ees:sfull f or the i th observation. Let ( Y 1 » w w 1 ) be an optimal s o l u t i o n to (2.2) with y2' > 1. The con- . s t r a i n t s of (2.2) imply that i t i n w x <_ 1, i t i i . . w x =i Y > 1» J e M . 35 or (2.3) w l tx 1 < wl1:xj for jeM . Since cost minimization at observation i implies that x^ and x 3 must be such that the total cost wltx1" < wlt;x , for a l l j such that y 1 < f', at prices w^ , . .. ,wj^.1 If there are multiple outputs one of the outputs i s singled out, say i t i s the k th output, and x 3 is replaced by the extended vector (-y^ ; \x3) Inputs and;W^byf'thejextendedsvector of input prices (p 3; w3) for j = 1,...,N. Test 5 Given the set Z = {(y, x, p^, w),-i = 1,...,N} of observations i i , . . ,"i i N . r on outputs y , inputs x , and input prices (P^ » w ) i f (2.4) i t i " i t " i i t i " i t " j . ... , , j i w x -p^ y < w x -p^ y for a l l j such that y^ > y^, i t i " i t " i i t i " i t ~ i ,. ., . , i i w x -p, y < w x -p, y for a l l j such that yr; = y , for i = 1,...,N, then there exists a nondecreasing, quasi-w " concave valid production function for the observations and the observations are consistent with cost minimization. Obviously for a complete test, Test 5 should be applied to a l l differ ent goods singled out as the output, i.e., y^ for k = 1,...,P. Note, also, that (2.4) is equally applicable for normalized input prices. ^ h i s along with (2.3) could be obtained from the argument that when costs are being minimized i t should cost less i f the amount produced i s equal or less. 36 x r (a) x x 0 Figure 1 x. (b) In order to visualize how Test 5 works let us consider the examples in Figures l a and lb. A production operation with single output is assumed. 4 3 2 1 Let y > y > y > y . We are testing the 1st observation at which input prices are w\ The hyperplane w^x = w^ x"*" should be below.the hyperplanes w^x = w^Sc3 , j = 2, 3, 4. In Figure l a the hyperplanes w^Sc = w^x 3, j = 2, 3, 4, are above w X tx = w'^x"'' while in Figure lb there is at least one j-where y J > y 1, namely the 2nd observation.^, such that w^x 2 < w^ x"*". This last case represents a violation of the conditions. If the conditions (2.4) are not satisfied by some observation pair ( i , j) i t must be true that (2.5) i t , i j . - i t * i f ° r : *k > y k } A = w (x -x J) -p (y -y JM 2 1 > 0 for j £{q : y j = y£} which suggests the use of A „ as a measure of violation. A. . 12 <5 E max j i t i -"v£ w x -p k y£ Also represents the percentage by which the cost at the i th observation should be 37 reduced for consistency with the conditions and cost minimization. Note that A „ = 0 for some j , y 3 > y 1 implies that there is no violation and thus we can include this case as acceptable and slightly modify (2.4) to obtain i t i * i t " i i t j " i t " j . , j i W x _ p k y k = w p k y k J s y k = yk" Accordingly the violation at observation i is measured by A. . 6 i E m a. x>~r {i i t x i - i t - i : A i j > 0 }-J ) y kS y ik^k w x " pk y k 2.2 Test for a Nondecreasing. and Concave Production Function; Price and Quantity Data For a concavity test, we consider again the corresponding tests in Chapter I, Section 2?, for quantity data only. There, Tests 2a and. 2b were suggested to test the existence of a nondecreasing and concave production func-tion valid for a set of observations on inputs and outputs. The test linear program N max I X.y X 3=1 2 subject to N I X.x3 4 x \ j=l 3 N I X , = 1, j==l 1 3 )l >.0, of Test 2a has the following dual: i t i i (2.6) m n . V X + Y i x Y » v 38 subject to y J-v x J-y 4 0, j = 1,...,N, v 1 ^ 0, where v^,...,v^ are imputed input prices normalized on the output price for the i th observation. When the v 1 as well as y 1 and x 1, i = 1,...,N, are.known, i f the optimal solution to (2.6) is (y 1, v 1) then for a successful test i t i , i i v x + y = y , or equivalently i i t i i y -v x = Y » and from the constraints of (2.6) /o -,\ j i t j i i . i t i . - „ (2.7) y -v x J 4 Y = y -v x , j = 1,...,N, i 3 for i = 1,...,N. As we multiply both sides of (2.7) by p , the output prices, we obtain the following in terms of nonnormalized prices: n \ i i i t i i i i t i . .. „ (2.8) p y-w x J 4 p y -w x , j = 1,...,N. However, since for the multiple product case we use the extended price and input vectors, this in turn implies that we can write (2.8) as i t j i t j i t i i t i .. „ p y-w r < p y -w x , j = 1,...,N, where p 1 is the price vector whose components are p^,... ,pp", the output prices at observation i . Test 6 Given the set Z = { (y, x, ~e, w) 1, i = 1,...,N} of observat_ "?uts y 1, i'^Qutpu^pri'ceaup1?^ for a l l i = 1,...,N. Test 6 Given the set Z = {(y, x, p, w) , i = 1,...,N} of observations • i i . i . .., . • „ . i . con outputs y , inputs x , output prices p and input prices w i f (2.9) i t j i t i i t i i t i ., p yJ-w x J < p y -w x , j = 1,...,N9 ,N, then there exists a nondecreasing, concave ovi c r c a' nnn o^prDflClno nnnpaWD a-nrl •Kid ^ for i = 1,...,N, then there ex sts o decreasing, co cave d vs and valid production function for the observations, tion function for the observations. The interpretation of (2.9) i s that profit at the prices p 1, w1 should1, not be as large when the quantities are different than y 1, x 1. As an example let us consider the 3 observations in Figure 2. The hyperplanes p^-w^x = p^y^-w'Sc3 , j = 1, 3, which we denote by b and c in Figure 2a are below the hyperplane p^-wSc = p^^w^x 1, implying that Test 6 is successful -at the 2nd observation. On the other hand Figure 2b depicts the situation when the hyperplanes b and c are above a. Here the test f a i l s because, i n t u i -2 tively, y is not large enough to push point 2 to the efficient frontier. a \ 2 -x -x (a) (b) Figure 2 If (2.9) is not satisfied by some observation pair ( i , j ) , then i t i i t j i t i , i t i _ , A „ = p y -w x -p y +w x > 0, and 6. = max {A.. : A.. > 0} 1 j 1 3 1 J 40 can be used to indicate the amount by which the computed profit at the i th observation should be increased for consistency with the conditions. Clearly this in turn implies an increase in the output y 1 and/or a decrease in the inputs x^. 2.3 Test for a nondecreasing, Concave and Linear Homogeneous Production Function; Price and Quantity Data Tests 3a and 3b in Chapter I, Section 2^, were suggested for verifying the existence of a nondecreasing, concave and linear homogeneous production function valid for a given set of observations on input-output quantities. The test linear program of Test 3a i s , for the i th observation, as follows: N . N . i max il A . y 3 : £ X . x 3 < x 1, X ^0}. X 2=1 3 j=l 1 J Its dual is (2.10) min {v^x 1 : y^cV^x 3 < 0, j = 1 N, v 1 ^ 0}, i v where, again, v^,...,v^ are the normalized imputed input prices. For a success-f u l test, v l t x 1 = y 1 at the optimal solution in (2.10). This last condition becomes, i f v 1 in addition to (y, x ) 1 , i = 1,...,N, are known, v l t x 1 - y 1 =0, or along with the constraints of (2.10) (2.11) y^-V^x 1 = 0, y^-v^x 3 4 0, j = 1, .. .,N. Multiplying (2.11) by p 1, the output price, we obtain (2.12) p V W = 0, p^-w^x 3 4 0, j = 1,. ..,N. Again considering the multiple output case we replace p 1 with the 41 output price vector. Test 7 Given the set Z = {(y, x, p, w) 1, i = 1,...,N} of observations i i . i , . . i on outputs y , inputs x , output prices p and input prices w i f i t i i t i . p y -w x =0, i t j i t i n . i „ . , . P^VjI-w x J <0, j = 1 K, j ^ i , for i = 1,...,N, then there exists a nondecreasing, concave, linear homogeneous and valid production function for the observations. Here, we can use a measure of violation similar to the one suggested for Test 6 along with a check on the amount by which the computed profit p ^ y 1 - ^ ^ 1 deviates from zero. They are: .1 i t i i t i = p y -w x 6 2 = max {A.. = p 1 V-w1'^ : A.. > 0} -x. Figure 3 42 Let us consider an example where there are two inputs x^, x^ and one output y, Figure 3. As we apply Test 7 to observation 1, the hyperplane P t.. u • 1 It 1 1 l t 1 ' . which is represented by p y-w x = p y -w x = 0S p.as'sesisthr.ough the o r i -gin and point 1 while remaining above the hyperplane p^y-w^x = p^y^-w^Sc^. A measure of violation for this test would be A . = p i ty i-w i tx ± i f A . / 0 x J 1 and A . . = P^y^-w^x3 i f A . . > 0. As for the previous tests we could compute 6 . = max {A . . : A . . > 0 } which represents the largest violation. Note that the tests in this section did not involve linear programming problems. It was only necessary to check certain inequalities. In the follow-ing sections, we turn to tests involving price and/or profit.(or cost) data; i.e., we do not assume a complete knowledge of the quantities used and produced by the producer. The tests w i l l be for the multiple product case and they involve linear programming problems. III. PRICE AND PROFIT DATA 3.1 Test for a.Nondecreasing and Concave Production Function; Price and Profit Data Let us reconsider the dual (2.6) of the test linear program for mono-tonicity and concavity for quantity data only: r,i . i t i . i 2^ 1) Z mm v x +Y i i Y , v 43 subject to j i t j x± . .. .. y J-v x J-o < 0, , j = 1,...,N, i v > 0. For a successful test at the i th observation when the quantities (y, x ) 1 , i ='1,...,N, are known, Z 1 = y 1 or Z ^ y 1 = 0, i - 1,...,N. But y 1 - v l t x 1 is the profit in terms of normalized imputed prices v 1. Thus i f (y 1, v 1) is an optimal solution to (3.1), provided that the test is successful we have i t i , i i _ v x + Y~y = o> or I i t i l y -v x = y i which suggests that we interpret the dual variable yX as the normalized imputed profit. Assuming that normalized profits and input prices are known instead of the quantities then the linear program (3.1) can be recast as the following: „i _ . i t i i i Z = mm v x +Y -y (3.2) i i y ' x subject to y ^ V ^ - y 1 < 0, j = 1,...,N. Now we assume that we have the complete knowledge of .pricesC'of-inputs and outputs as well as profits, which we denote by TT, at N different times or places. Thus multiplying the constraints and the objective function of (3.2) i by p we get 4 4 r,i - . i t i i i i Z. = mm w x +i -p y ( 3 . 3 ) i i y > x subject to p1y-*-wltx:]--TT:i" <. 0 , j = N. I f a production operation i s p r o f i t maximizing the acceptable input output combinations are those that s a t i s f y ( 3 . 4 ) p V - w ^ x 1 i 7T>, j = 1.....N; I = 1,...,N, for given input output prices w3 , p 3. Incorporating ( 3 . 4 ) i n t o ( 3 . 3 ) we obtain the following l i n e a r programs: „i _ . / i t i , i i . Z - min (w x + ir - y ) (3.5) i I y , x subject to 1 j I t j i „ , p y -w x J-7T < 0 , j = 1 ,.. . ,N, p Jy 1-w : l tx : L-Tr : J < 0, j = 1 N, fo r i = 1,...,N. However some of the constraints i n (3.5) are redundant since (3.5) i s c a r r i e d out over y 1 and x 1 and i t i s always possible to f i n d f e a s i b l e values for y 3 and x 3 , j = 1,...,N, j 4 1 ,.. . ,N, j j ^ i i . OEOmitt-tnghfehemrf rom further consideration we obtain t h i s equivalent :ofc (t3%f5)i: r.h observation; i = „i _ . / i t i , i i N (2 6) Z = mm (w x +TT -y ) i i y , x subject to p-'y : L-w : l tx 1-iT : ] 4 0 , j = 1,...,N, 45 whose optimal objective function value must be equal to 0 f o r a successful t e s t . As f o r Test 2a, Chapter I, Section 2, we can use (3.6) for the mul-t i p l e output case by replacing w and x with the corresponding extended vectors and this would suggest the useVof the following a l t e r n a t i v e to the constraints of (3.6): i t i i t i j . . i „ p y -vr x - I T 4 0, j = 1,... ,N. Changing the objective function of (3.6) to that of a maximization we get i t , r i i t i i max p X -w x —rr (3.9) i i y , x subject if© j t i j t i j . , . . p y -w x 4 TT , j = 1,...,N. The sign constraints i n (3.9) w i l l presumably be y 1 ^ 0, x 1 ^ 0 and r e f l e c t the assumption that y^,...,y X represent outputs, x^,...,xj^ represent inputs. However, i t i s i n t e r e s t i n g to note that these sign constraints can be modified to provide some f l e x i b i l i t y i n f i x i n g any good as an input or output or even e i t h e r . For example, i f x^ represents the quantity of an input, the j th input, at the i th observation then x"!" > 0. However i f we want to consider i t as an J = output then -x"!" ^ 0 or x! < 0. S i m i l a r l y , i f y^ represents the quantity °f an output, the k th output, at the i th observation then y^ ^ 0, and y^ 4 0 i f we want to consider i t as an input. Also by leaving the sign of a var i a b l e unre-s t r i c t e d we can allow the l i n e a r program to determine whether the corresponding good should be an input or not i n order f o r the underlying r e g u l a r i t y conditions to be s a t i s f i e d . When prices and p r o f i t s are known we can consider the following equiv-alent f o r the objective function of (3.9): p l t y 1 - w l t x 1 whose la r g e s t value, 46 subject to the constraints thereby, is to be TT1 for a successful test at the i th observation. Although (3.9) constitutes the test we have been trying for, a more efficient alternative is i t s dual as given below. Test 8 Given the set P = {(TT, p, w) 1, i = 1,.. . ,N} of observations i . i , . . i . , -on profits TT , output prices p and input prices w i f N (3.10) TT1 = Z 1 = min £ A .TTJ A j=l 3 afibject to subject to N j=l 3 N I X vP 4 w1, j = l 3 X ^ o, for i = 1,...,N, then there exists a nondecreasing and concave production function consistent with the observations and profit maximizing behavior. The direction of inequalities can be reversed appropriately i f a good with price pj^ is to be taken as an input rather than an output or i f a good with price w3 is to be taken as an output rather than an input. The economic interpretation of (3.9) i s that the input output quanti-ties at each observation are such that at the observed prices they maximize the profit while at other prices they w i l l not bring any more than the profit the maximizing input output quantities at those prices do. If TT1 ^ Z 1 then = TT1 - Z 1 represents the amount by which the profit at the i the observation should be reduced to make the data consistent with the conditions. Note that A_^ ^ 0 always since Z 1 ^'TT 1. Thus as a measure of vio l a -tion of the conditions we can use 6 ^ = A^/TT 1 which is the percentage violation 47 of the p r o f i t at the i th observation. 3.2 Test f o r a Nondecreasing, Concave and Linear Homogeneous Production Function; P r i c e and P r o f i t Data When returns to scale are constant, p r o f i t maximization implies that I T 1 = 0, 3 i = 1,...,N. As we add these constraints to (3.10) the objective function disappears and = 0, j = 1,...,N, j ^ I; = 1 i s a f e a s i b l e s o l u -t i o n which always e x i s t s . Thus we have the necessary condition that TT = 0 f o r a l l observations. The s u f f i c i e n t condition i s obtained from (3.9) when a l l I T 1 = 0, i = 1,...,N; i . e . , s o l v i n g (3.9) with zero p r o f i t s . Test 9 Given the set P = {(IT, p, w) 1, i = 1,...,N} of observations i i , . i on p r o f i t s ir , on output p r i c e s p and input prices w ; (a) I T 1 = 0 f o r i = 1,...,N i s necessary, i t i i t iv (3.11) (b) Z 1 = max ( p y 1 V x ) l l y ,--x subject tosubject to p 3 t y X - w 3 t x 1 4 0, j = 1,...,N, y 1 =L 0, x 1 ^ 0, and Z 1 = 0 for i = 1,...,N i s s u f f i c i e n t f o r the existence of a production function f o r the observations, which i s non-decreasing, concave and l i n e a r homogeneous and the observations are consistent with p r o f i t maximizing behavior. Obviously i f T T 1 ^ 0, T T 1 i t s e l f represents the amount by which the p r o f i t i s too much (when T T 1 > 0) or too l i t t l e (when I T 1 < 0). But when I T 1 = 0 3Chapter I, pp.13 48 for a l l i = 1,...,N a l l we can do, perhaps, i s to suggest a somewhat a r b i t r a r y measure of v i o l a t i o n such as the following one: Assuming that ( y 1 , x 1) i s an optimal s o l u t i o n to (3.11) and Z 1 > 0, we can write i t i i t i " i t " i i i i t i _ i _ p y -w x = p k y k + p k y k - w x = Z > 0. Let us redefine the p r i c e of the k th good at t h i s period as i * , i t i " i t " i N , i p k = (w x -p k y k ) / y k , i ^ i i * i i i which makes Z , redefined i n terms of p k , P k , w , y and x , equal to zero. i * i Clearly p k < P k« Thus i i * P k " P k 5., = - ~ — - f o r k = 1 N lk i \ represents the percentage p r i c e reduction on the k th good necessary i n order f o r the data to be consistent with the r e g u l a r i t y conditions of the theorem. In the following section we introduce a test which i s very s i m i l a r to Test 9 for the case when there are observations on prices only. IV. PRICE DATA 4.1 Test for a Nondecreasing, Concave and Linear Homogeneous Production Function; P r i c e Data When we have observations on prices only we can use (3.11) to test monotonicity' and concavity by assuming l i n e a r homogeneity or zero p r o f i t s everywhere, i . e . , T T 1 = 0, i = 1,...,N. Test 10 Given the set W = {(p, w)"*", i = 1,...,N} of observations on ^ output prices p and input p r i c e s w , i f : i _ . i t i i t i . Z = max (p y -w x ) (4.1) i i V , x 49 subject to i t i j t i . ., „ p y -w x < 0, j = 1,...,N, i „ i ^ y >_ 0, x >_ 0, and Z 1 = 0, for i = 1,...,N, then there exists a non-decreasing, concave and linear homogeneous production function for the observations and the observations are consistent with profit maximization. The dual of (4.1) is the following f,easibiMt-yopr.ob)lemiwhichrcan also be obtained by substituting T T 1 = 0 in (3.10): N j=l J N I X w3 ^w1, j = l 3 X >_ 0, or i = 1,...,N. The violation measure developed for Test 9 could also be used here. V. PROFIT (COST), PRICE AND OUTPUT QUANTITY DATA 5.1 Test for a Nondecreasing, Quasi-concave Production Function; Profit (Cost), Price and Output Quantity Data Let us reconsider Test lb of Chapter I, Section 2,.as reproduced in (2.2): When there are observations on input output quantities i f (5.1) y 1* = max {y1 : w^x 1 <_ 1, w^x^y 1 _L 0, jeM., w1 >0} > 1, i i Y , w for i = 1,...,N, then there exists a nondecreasing, quasi-concave production function valid for the observations. 50 As we assume that we have observations on profits (or costs), input-output prices and output quantities instead of both input and output quantities we should consider profit maximizing or cost minimizing behavior as well in (5.1) . Also the maximization w i l l not be over the prices anymore. Now, l e t us f i r s t take the constraint w^x 1 4 1 in (5.1). This norm-alizing constraint can also be written as (5.2) w^x 1 = C1, where C1 is the total input cost. If there are more than one output (5.2) becomes ,_. Q v i t i " i t " i i (5.3) w x -p k y k = C , with the implicit assumption that y 1 i s the output that i s singled out. Also i i as was done in Section 2.1 assuming that (y » w ) is an optimal solution to (5.1) such that y1 > 1, we can obtain the following relationship directly from the constraints of (5.1): i t j i 1 i t i , .. . „ w x >_ y > 1 = w x for a l l jeM^, or in view of the new normalization, i t j i t i „i _ 1, . „ w x > w x . = C for a l l ieM.. This last inequality can be written, to take the observations into account correctly, as (5.4) w j tx 1 >: w^x 3 = CJ for all'IeM . This inequality represents a cost minimizing behavior. For the multiproduct case (5.4) w i l l be (5.5) w J t x l - p k . t y k > ^ f ° r 3 S U d l t h a t yk < yk' 51 Now, when we have observations on costs, input prices and output quantities, to test for monotonicity and quasi-concavity along with cost mini-mizing behavior, i f the quantity of the good singled out is dentoed by y^, we attempt to solve the following set of inequalities, which i s obtained by com-bining (5.3) and (5.5), for y^ and x 1 at the i th observation: i t i " i t ^ i _ i w x -p k y = C , w J tx 1-p k ty k > C2 for a l l j such that y 3 < y^, This problem i s equivalent to the linear programming problem 1* i Y = max Y (5.6) Y 1. y£> x 1 subject to i t i " i t ' i „i W X " Pk y k = C ' w;]t:x:L-pkty^ - y^C2 ^ 0 for a l l j such that y 3 < y^, y k >,°» x i°» i * and i f y > 1 the test is said to be successful at the i th observation. The sign constraints could be changed, as explained in Section 3.1 according to which goods are ^ consider edeasa.inpuits$s, which ones as outputs. The dual of linear program (5.6) i s our next test. Test 11a Given the set T = {(C, yfc, p^, w) 1, i = 1 N} of observations on costs C1, outputs y^ and input prices ,"1 i ^ i± . r n i _ . „i i 1 - '.U*. X n 1.. 1 ( 5 > y^ Z = min C y i , y , A subject to ^Recall that outputs other than the one singled out, namely yfc, a re._al s.o_inp.u.ts.. 52 subject to (i) ( i i ) ( i i i ) I.HC d = 1 IX w3- y^w1 i o j 3 over a l l j such that (iv) X ^ 0, and Z 1 > 1, for i = 1,...,N, then there exists a non-decreasing, quasi-concave production function valid for the observations and the observations are consistent with cost minimization. Suppose that Z 1 <^ 1, for some i , implying that the test f a i l s at the i th observation. Assuming that w >> 0, by (i) and (iv) A > 0 and by ( i i ) y 1 > or in words u 1 w i l l always be in the basis. Thus as C 1 i s increased C 1 ^ 1 w i l l increase although the value of y 1 w i l l remain unchanged. If ( y 1 , A) is an optimal solution to (5.7) and i f we use A as an increment operator, for a suc-cessful test we must have ( c 1 + A c ^ y 1 > 1, C ^ J 1 + AC^y 1 > 1, and since z1 = cV t „i .,1 AC 1 > l z Z _ = lz?_ c i f y 1 z1 which represents the least amount by which C . should be increased" 53 which represents the least amount by which G1. should be increased 5 for consistency with the hypothesized conditions and cost minimization. We now define the following measure of violation which is the percentage increase in the total cost C1: 6 . = - _ r - 1 for Z 1 < 1, i = 1,. ..,N. 1 z1" It should be clear that when Z 1 = 1, 6. =0, i.e., there i s no viola-l tion. When, on the other hand, we have observations on profits, input and output prices arid output quantities we are able to develop a test for the con-ditions of this section from the relationships above. For example, from (5.3) we can write i t I " i t " i „i i i i w x-p ky k =c = P k y k - * , o\ w l 1 i " i t " i i t i (5.8) ir - p ^ = p k y k- w x , and from (5.5) w ^ - p ^ y j > Cj = P^Y^ 1 for a l l j such that y j < y£ or (5.9) i r' I - pk yfc > ^ ^ k - ^ ^ 1 f ° r a 1 1 3 S U C h t h a t yk < yk' which also corresponds to a profit maximizing behavior^ 6 5Clearly an increase in C1 corresponds to a decrease in C3 for some or a l l j such that yJ < y 1. However from a practical viewpoint i t i s easier to consider an increase in C1 only. K i i bSince the output price p k as well as i t s quantity y^ i s assumed to be known profit maximization is equivalent to cost minimization. 54 Combining (5.8) and (5.9) we obtain " i t " i i t i i i i P k yk-w x = TT -p ky k, (5.10) p k t y k - w J ^ < 7r* 1~ pk yk f o r a 1 1 1 s u c n t n a t y k < y k ' y k >, °» x 1 .> 0, which should have a solution interms of (y^5 x 1) in order for the observations to be consistent with monotonicity, quasi-concavity and profit maximizing behavior. Finding a solution to (5.10) is equivalent to solving the following linear programming problem: Z = max Y (5.11) i ~ i i y * yk' x subject to " i t ' i j t i , j i i i i i i n e P^-W x -= r - m i i , • Ofor P k ty^-w J tx 1-( 7r ; 1-p ky k)Y : L i 0 for a l l j such that y 3 < y£, y k ^ 0, x ^ 0. The existence of a solution for (5.10) i s equivalent to Z 1 > 1 in (5.11). The difference between (5.6) , r-.the linear program when costs are known, and (5.11), that when profits are known, is the assumption that output price P k i s known f i n (5.11). The dual of (5.11), in fact, establishes our last test: Test l i b Given the set T = {(TT, yk» p, w) 1, i = 1,...,N} of obser-vations on profits IT 1, outputs y k and both input and output , i " i i s / i i s . r prices (p k; p , w ) = (p ; w ), i f 55 (5.12) min (P ky k _ T r l) y* y\,* .abject to subject to (i) Ix^pjy3-^1) = 1 (ii) lA.w^-^wJ _ 0 j 2 k (iii) I y k - A k = o over a l l j such that y j < y 1, k yk' (iv) X ^ o and Z 1 > 1, for i = 1,...,N, then there exists a non-• decreasing, quasi-concave production function valid for the observations and the observations are consistent with profit maximization. Let us suppose that Z 1 1 1 for some i . Assuming that w >> 0, by (i) and (iv) A > 0 and by ( i i ) y 1 > 0 implying that y 1 w i l l always be .in the basis. Thus as P ky k - 1 T* increases (P ky k _ 7 r^) V~ w i l l increase while the value of y 1 w i l l remain unchanged. Note that, though, P^y^-^1 I s increased either by i , , i i , . . i i , . decreasing TT , keeping P^ y^ s the revenue, constant or increasing Pkyk> keeping TT 1 constant. If (y 1, A) is anooptimal solution to (5.12) then for a success-fu l test • - i i i . , / i i i \ - i i -. C p k y k _ 7 T ( p k y k _ 7 T )-ly ' where A i s an increment operator. Since Z 1 E (P ky k - 7 T l) V1"1" w e can^write ., i i i , i , , i i 1. 1 A( pk yk - 7 r )y > l-(p ky K-T7 )y as 56 / r i o\ . / i i i \ l - Z 1 ,1 , . w i i i . (5.13) A ( p k y k " ^ ) > ~~T" = (~T' ~ ) ( p k y k - T r y z Hence (5.14) 6 = ( i ) - DCP^-^ 1) i s the l e a s t amont by which ( P k y k - T r l ) should be increased i n order f o r the i th observation to be consistent with the conditions. I t i s obvious that the same <5 can be looked at as representing the amount by which T T 1, the p r o f i t only, i s to be decreased or P^y^' t n e revenue only, i s to be increased. VI. AN EMPIRICAL EXAMPLE 6.1 A Multi-product Case We have applied Tests 5, 6, 7, 8, 10 and 11a to the data i n Christensen-Jorgenson [1969, 1970] which consist of 39 annual observations from 1929 to 1967 on two outputs and two inputs where the outputs are consump-t i o n goods and.ihves.tmentmgobdg90fiheainputs are p r i v a t e domestic labor and p r i v a t e domestic c a p i t a l . Table 1 summarizes the data: The quantities of the consumption goods and the investment goods are denoted by y^ and y^, t h e i r p r i c e s by p^ and p^; the quantities of the private domestic labor and c a p i t a l by x^ and x^, t h e i r p r i c e s by w^ and w^ . The r e s u l t s of Test 5 applied to t h i s data with y^, the consumption goods, singled out as the output are given i n Table 2 where the f i r s t column i s the reference observation i , the second column i s the observation k which hasj the greatest v i o l a t i o n , i . e . , <5. = A. v with 6. defined i n Section 2.1, the 1 XK. 1 t h i r d column i s 5^ and the fourth column i s the number of observations v i o l a t i n g . Table 3 summarizes the r e s u l t s of Test 6. The t h i r d column i s the 5^ of Section 2.2. 57 In Table 4 we have the r e s u l t s of Test 7. The t h i r d and fourth 1 2 columns are the v i o l a t i o n indexes 6. and 6. of Section 2.3. l I Although we do not have observations on p r o f i t s i n the data we have computed the p r o f i t at each observation as p ^ y 1 - wltx"L = TT1, i = 1,. ..,39, and used them along with the prices to apply Test 8. In Table 5 we have A_^ and 6^ as defined i n Section 3.1 as well.as the computed p r o f i t s ' " - ^ , i = 1, ...,39. Obviously as ir goes to zero the percentage v i o l a t i o n = A^/TT1 becomes les s meaningful. Test .10, f o r a nondecreasing, concave and l i n e a r homogeneous produc-tio n function when there are data on prices only, has been successful at a l l observations, therefore, the results, have not been separately reported. Since we do not have complete data for Test 11a, i . e . , we lack the observations on cost, we have, using the quantities and.prices, computed i i t i " i t " i i C = w x -p y for the output.singled out which i s y^, the consumption goods, i = 1,...,39. In Table 6 we have the v i o l a t i o n index 6^ of section 5.1 l i s t e d f o r each observation. 6.2 Conclusion Although Test 5 has been successful at most of the v i o l a t i o n s i f we a t t r i b u t e the very small v i o l a t i o n indexes at some of the observations to f a c -tors other than inconsistency of the data with monotonicity and.quasi-concavity, Tests 6 and 7 have f a i l e d at almost a l l . o f the observations with v i o l a t i o n s at moderate l e v e l s . Thus we conclude: there i s neither a nondecreasing, concave, and l i n e a r homogeneous nor a nondecreasing, and concave production function which i s consistent with the data of Christensen-Jorgenson and cost minimizing behavior. As explained e a r l i e r , Tests 8, 10 and 11a have been applied to the data derived from the o r i g i n a l one. However, i t should be noted that the r e s u l t s of Tests 6 and 8 are not i d e n t i c a l . While the number of v i o l a t i o n s 58 is s t i l l large the latter i s in general more successful. We should again remember that, when applying the tests involving linear programming, the solutions themselves are not of much practical value. For example, the dual variables of the test linear program for Test 8 represent input-output quantities but we cannot in fact suggest the implementation of any changes in quantities as indicated for consistency with the hypotheses of the test. TABLE 1 Outputs Inputs Output Prices Input Prices i s e r v a u x o n Number y l y2 X l X2 Pi p2 W l W2 1 136. 275 55. 781 173. 300 102. 700 . 0.547 0.509 0.324 0.456 2 132.291 41. 253 165. 400 105. 000 0. 525 0.489 0.311 0.365 3 128.840 31. 097 158. 200 104. 300 0. 488 0. 453 0.273 0.324 4 118. 260 17. 642 141. 700 102. 100 0. 429 0. 405 0.236 0.240 5 113. 791 18. 548 141. 600 96. 800 0. 422 0. 403 0.219 0.253 6 117. 234 25.064 148. 000 92. 100 0. 432 0.414 0.238 0.270 7 124. 285 30. 325 154. 500 90. 000 0. 454 0. 418 0.248 0.342 8 131. 804 41. 077 163. 500 89. 500 0. 450 0. 415 0.263 0.374 9 139. 840 44. 620 172. 000 .90. 800 0. 467 0. 442 0.285 0.395 10 140. 153 34. ,272 161. 500 93. 200 0. 443 0. 447 0.281 0.343 11 146. 147 43. 755 168. ,600 92. 200 0. 449 0. ,442 0.290 0.391 12 153. 778 53. ,265 176. ,500 93. 000 0. 452 0. ,447 0.300 - 0.435 13 164. 364 73. .076 192. ,400 95. 600 0. 479 0. ,506 0.337 0.532 14 178. ,567 80. ,802 205. ,100 100. ,000 . 0. ,535 0. .589 0.398 0.617 15 180. 380 98. ,066 210. .100 99. .000 0. ,600 0. .618 0.459 0.730 16 188. ,830 103. .207 208. ,800 96. ,700 0. ,614 0. .594 0.494 0.767 17 192. ,278 92, .600 202. .100 94. .900 0. ,637 0. .568 0.511 0.757 18 195. ,802 77. .297 213. .400 94. .800 0. ,718 0, .646 0.540 0.795 19 193. .836 85. .665 223, .600 101. .400 0. .794 0, .749 0.594 0.842 20 203. .862 93, .524 228. .200 108. ,700 0. .819 0, .777 0.639 0.862 21 206. .087. 91, .290 221. .300 116. .100 0. .796 0.791 0.647 0.797 22 214. .858 113, .906 228, .800 121.. . 200 0, .828 0.801 0.683 0.930 23 228. .406 122 .928 239, .000 130, .000 0, .880 0.864 0.742 0.999 24 237. .323- 122 .964 241, .700 137, ..100 0, .905 0 .880 0.782 0.997 25 247, .628 131 .165 245, .200 142, .100 0, .909 , o, .879 0.827 0.967 26 250, .337 125 .156 237 .400 148, .000 0, .927 0 .886 0.846 0.961 27 262 .884 143 .864 245 .900 152, .(.00 0, .936 0 .894 0.880 1.037 28 272, .994 143 .264 251 .600 160 .800 0 .956 0 .945 0.930 1.010 29 281, .133 141 .574 251 .500 167 . 700 0 .978 0 .989 0.978 1.009 30 287 .953 130 .421 245 .100 173 .300 1 .000 1 .000 1.000 1.000 31 300 .725 144 .979 254 .900 176 .000 1 .020 1 .013 1.042 1.067 32 310 .005 147 .263 259 .600 181 . 800 1 .044 1 .010 1.074 1.066 33 320 .353 145 .736 258 .100 187 . 700 1 .061 1 .012 1.103 1.079 34 334 .981 160 .431 264 .600 191 .900 1 .075 1 .019 1.144 1.151 35 346 .273 169 .410 268 .500 1.98 .(.00 1 .091 1 .022 1.180 1.179 36 363 .320 181 .165 275 .400 206 . 200 1 .106 1 .030 1.229 1.213 37 383 .562 196 .323 285 .300 215 .500 1 .137 1 .043 1.271 1.291 38 406 .587 209 .890 297 .400 227 .200 • 1 .174 1 .065 1.335 1.336 39 424 .326 206 .903 305 .000 240 . 700 1 .190 1 .097 1.387 1.283 60 TABLE 2 (Test 5) Reference Most Violating Violation Number of Violating Observation i Observation Index <5. Observations 1 16 0.0299 11 2 1 0.0394 21 3 8 0.0626 30 4 7 0.0496 36 5 16 0.0253 36 6 16 0.0223 30 7 16 0.0200 18 8 16 0.0171 7 9 16 0.0250 10 10 16 0.0376 18 11 16 0.0327 8 12 16 0.0317 5 13 16 0.0290 3 14 16 0.0479 3 15 16 0.0252 2 16 - - — 17 - -18 - - — 19 18 0.0174 2 20 22 0.0093 1 21 22 0.0274 2 22 - - -23 - - -244 27 0.0001 1 25 27 0.0009 1 26 27 0.0107 1 27 - - -28 - - -29 - - -30 34 0.0036 1 31 - - -32 - - -33 35 0.0047 2 34 - - -35 - - -36 - - -37 - - -38 - - -39 - - -TABLE 3 (Test 6) Reference Most Violating Violation Number of Observation i Observation Index 6 Violating Observations 1 38 129.321 30 2 39 1141.375 33 3 39 139.567 35 4 39 136.150 37 5 39 134.762 38 6 39 130.457 36 7 39 121.140 34 8 38 106.880 31 9 38 108.144 29 10 38 112.440 32 11 38 100.250 28 12 38 89.546 27 13 38 77/9.865 26 14 38 82.602 25 15 38 71.302 20 16 38 53.141 14 17 38 54.252 13 18 38 86.298 21 19 38 112.080 24 20 38 110.194 22 21 38 116.170 23 22 38 90.356 19 23 38 99.1.498 17 24 38 98.123 18 25 38 88.429 13 26 38 92.929 14 27 38 70.889 9 28 38 80.990 10 29 38 85.122 11 30 38 91.877 12 31 38 75.024 8 32 38 74.863 7 33 38 70.616 6 34 38 49.226 5 35 38 39.293 4 36 38 24.774 3 37 38 9.895 1 38 no — 39 38 10.096 1 1 TABLE 4 (Test 7) Reference Most Violating Violation Indexes Number of Observation i Observation Violating Observations 1 38 -0.0455 128.841 30 2 39 -0.1389 1141.236 33 3 39 -0.0210 139.546 35 4 39 -0.0666 136.084 37 5 39 -0.0061 134.755 38 6 39 0.9306 131.388 36 7 39 0.0300 121.170 34 8 38 -0.1147 106.879 31 9 38 0.1413 108.144 29 10 38 0.0583 112.440 32 11 38 0.0155 100.248 28 12 38 -0.0879 . 89.794 27 13 38 0.0088 79.865 26 14 38 -0.2041 82.602 25 15 38 0.1269 71.302 20 16 38 -0.0695 53.141 14 17 38 -0.0345 54.252 13 18 38 -0.0823 86.298 21 19 38 -0.1283 112.080 24 20 38 0.1119 110.194 22 21 38 0.5428 116.170 23 22 38 0.1548 90.356 19 23 38 -0.0007 91.498 17 24 38 -0.1658 98.123 18 25 38 0.0036 88.429 13 26 38 -0.1177 92.929 14 27 38 0.0357 70.889 9 28 38 -0.0295 80.990 10 29 38 -0.2112 85.122 11 30 38 -0.0259 91.877 12 31 38 0.2056 75.024 8 32 38 -0.2285 74.863 7 33 38 0.1665 70.616 6 34 38 0.0044 49.226 5 35 • 38 -0.0586 39.293 4 36 38 -0.1550 24.774 3 37 38 0.0483 9.895 1 38 - 0.2978 - £ 39 38 0.0674 10.096 1 TABLE 5 (Test 8) Reference Profit Violation Computed Observation i Deviation A.^ Index 6^ Profit TK 1 0.0992 2.20 -0.0456 2 0 -0.1389 3 0.0680] 3.25 SO.0210 4 0 -0.0666 5 0 -0.0061 6 0.9864 1.06 -0.9306 7 0.0926 3.09 0.0300 8 0 -0.1147 9 0.2713 1.93 0.1413 10 0.1778 3.04 0.0583 11 0.1528 9.83 0.0155 12 0.0565 0.64 -0.0879 13 20.1764 20.00 0.0088 14 0 -0.2041 15 0.3621 2.86 0.1>269 f 16 0.1838 2.63 -0.0695 17 0.2213 6.41 -0.0345 18 0.1872 2.27 -0.0823 19 0.1624 1.27 -0.1283 20 0.3618 3.24 0.1119 21 0.8384 5011.541 -0.5428 22 0.4824 3.13 0.1548 23 0.3532 500.00 -0.0007 24 0.1940 1.17 -0.1657 25 0.3645' 1.07 0.0035 26 0.2480 2.11 -0.1178 27 0.4297 12.00 0.0359 28 0.3549 12.00 -0.0295 29 0.1728 0.04 -0.2112 30 0.3547 13.7 -0.0258 31 0.6116 2.98 0.2056 32 0.1972 0.86 -0.2285 33 0.5771 3.47 0.1665 34 0.4424 , 100.00 0.0044 35 0.3902 6.67 -0.0586 36 0.3066 1.98 -0.1550 37 0.5397 11.13 0.0483 38 0.8063 2.7 0.2978 39 0.5557 8.2-5 0.0674 TABLE 6 (Test 11a) Reference Violation Index Observation <5. when y\ singled out 1 0.0008 2 0.0243 3 — 4 — 5 (*) 6 0.0309 7 — 8 0.0196 9 0.0076 10 0.0436 11 0.0184 12 — 13 — 14 — 15 0.0214 16 0.0243 17 — 18 — 19 — 20 — 21 — 22 0.0033 23 — 24 — 25 — 26 — 27 — 28 — 29 — 30 — 31 — 32 — 33 — 34 — 35 — 36 — 37 — 38 — 39 — Test i s not applicable. CHAPTER III NONPARAMETRIC INDEX NUMBERS AND TESTS FOR THE CONSISTENCY OF CONSUMER DATA 65b I. INTRODUCTION The commodities used in empirical applications of the theories of production and consumption are in reality aggregates of the underlying micro-economic commodities. While the decision variables remain at a microeconomic level they are determined at a macroeconomic level whereby only aggregates are considered. The reason for doing so is obvious. What is not so obvious is that i f one may aggregate any bundle of commodities. Among the methods of justifying aggregation the one which we w i l l consider i s that of Shephard's [1953, pp."*. 61-71] [1970, pp. 145-146]. He rr. . relates the problem of aggregation in the context of consumer to the property of homogeneous weak separability as follows: i f there exists a positive, con-cave and linear homogeneous u t i l i t y function of a set of commodities which i s consistent with the observed expenditures then that set of commoditites can be aggregatededihtoiahmacro commodity. But this aggregate commodity ("the inter-mediate input" or "real value added") is not directly observable. Neither i s the functional form for the aggregator function known. However, i t turns out that when there are observations on prices and quantities purchased i t i s possi-ble not only to test homogeneous weak separability, and thus justify aggregation, but also evaluate the aggregator function whose value represents the quantities of aggregate commodity or the nonparametric quantity indexes. Diewert [1973b] shows that in order to test the consistency of a con-sumer's expenditure configuration with the maximization of a u t i l i t y function which has certain properties a l l that i s needed i s to solve a linear programming problem, using data on prices and quantities purchased. As for the properties of the u t i l i t y function, he assumes i t i s continuous from above and subject to local nonsatiation. Along with this main result he proves that i f the test is successful then the u t i l i t y function can be taken to be increasing, continuous, and concave 'thusian alternativeymethodgof gesMmatifrgh^ u t i l i t y 66 function i s established. He bases his argument on the p a r t i t i o n i n g of the quantities into an exhaustive set of indifference groups. We have, i n the following Section, taken an approach based on a saddle point concept of nonlinear programming toward the same results as i n Diewert [1973b], while making s l i g h t l y different assumptions for the proper-t i e s of a u t i l i t y function. In Section 3 we review the main approaches to index number theory where Fisher's tests, economic index numbers and D i v i s i a index numbers are covered b r i e f l y . Section 4 i s on aggregation and homogeneous weak separa b i l i t y . In Section 5 the construction of nonparametric index num-bers i s discussed and as an application they are, i n Section 6, constructed along with the Fisher and D i v i s i a indexes using actual data. Anp'Appendix contains the relevant computer programming codes. Let us consider?a market of M commodities at prices P^,...,PM, and a consumer who has made purchases of these commodities i n quantities x^,...,x^. Let P*" = (P^,...,PM) and xfc = (x^,...,x^). The consumer i s constrained by the t o t a l amount of funds available to him for such an expenditure, which we denote by Y. Thus P tx 4 Y > 0. We assume the consumer's expenditure and income data which consist of a vector (P, x) and a scalar Y have been recorded on N occasions. Therefore, we have (P, x ) 1 = (P 1, x 1) and Y 1 for i = 1,...,N. The following d e f i n i t i o n s are relevant for the subsequent exposition: De f i n i t i o n 1. A set X C R n i s compact i f and only i f every sequence of points i n X has a l i m i t point i n X. D e f i n i t i o n 2. I l f f i s a r e a l valued function defined over XCR n then.f i i s said to be continuous from below i f for any XeR {x: xeX, f(x)<\} i s closed reiativeit'ooxs f rorr. above i r fo\ any Xe?.1 '{x: >.\,\. f(x)>X} ±r. * -67 II. TESTING THE CONSISTENCY OF CONSUMER DATA The classical assumption i s that the consumer determines his expendi-ture on occasion, i in such a way that a real valued function ^"""(x), the consum-er's u t i l i t y function, is maximized subject to a budget constraint. Formally, the consumer i s assumed to have solved the following programming problem: (2.1) max {(^(x) : P l t :x < Y 1}, i = 1,...,N. x eR If we assume that the consumer applies the same u t i l i t y function and has posi-tive income for expenditure on every occasion then (2.1) can be written as: (2.2) max {§(x) : p l t :x <_ 1}, i = 1,...,N, xeR where p 1 = P1/Y1, the normalized price vector for the i th occasion. If <f>(x) is a quasi-concave and continuous from above (upper semi-continuous) , so is the Lagrangean L(x, \) = cb(x) + X (1 - p l t :x), where xeXcR^, AeA = {X f X >_ 0}, because either of these properties are retained in a linear transformation. On the other hand, L(x, X) is a linear function of X. Therefore, i t is quasi-convex and.continuous from below (lower semi-continuous) with respect to X. We now consider a theorem which i s funde-mental for the analysis that follows: M M Theorem 1 (Sion [1957]) Let X C R and Y C R be two compact convex sets. If a real valued function f(x, y), x^X and ygY, defined over S = X*Y (the cartesian production of sets) is quasi-concave and continuous from above with respect to x, and quasi-convex and continuous from below with respect to y, then there exists a point (x°, y°) _S such that (2.3) f(x, y°) < f(x°, y°) < f(x°, y ) , for every xcX and yeY. Proof: See, Berge l i h^Sho^iMS-Houri [ 1962, p. 68.] D e f i n i t i o n 1. A function f ( x , y) i s s a i d to have a saddle point there e x i s t s a point (x°, y°) such that (2.3) i s s a t i s f i e d f o r every x^X an yeY. Let us define two functions that are r e l a t e d to a saddle point: (2.4) f*(y) = max f(x, y ) , xeX and (2.5) f^(x) = min f ( x , y) yeY I f should be ovserved that f*(x) < f ( x , y) < f * ( y ) , or (2.6) f^(x) < f * ( y ) . Theorem 2. (Zangwill [1969]) The following three statements are equivalent: (a) min max f(x, y) = max min f ( x , y ) , yeY xeX x ex yeY (b) f*(x°) = f*(y°), x°eX, y°eY, (c) f(x, y°) i f ( x ° , y°) < f(x°, y ) . Proof: See, Zangwill [1969, pp. 45-46]. In view of Theorem 1 the Lagrangean-'defined e a r l i e r possesses a saddle point, that i s , there e x i s t s a point (x°, X°) such that 69 (2.7) L(x, X°) < L(x°, X°) <; L(x°, X) fo r every xeX and XeA. However, by Theorem 2, max L(x, X°) = L(x°, X°) = min L(x°, X), xeX XeA Now, analogous to (2.5): fd>(x) i f l - p 1 x> 0 (2.8) L^(x) = min L(x, X) = min {<j>(x) + X ( l - p x)} = < XeA XeA [ -ss i f l - p 1 x < 0 because X >_ 0. Therefore, (2.9) max L A ( x ) = max min L(x, X) = max (<p(x) : p l t x <. 1} xeX xeX XeA xeX M as we disregard the p o s s i b i l i t y that L^(x) ->-*- -°° i n (2.8). But i f X = R , (2.9) i s the same as (2.1). By Theorem 2 and equation (2.9) x° solves the maximization problem (2.2). Thus, i f we rename x° as x 1, and X° as X 1 to correspond to the i th observation we have L(x, X 1) 4 L ( x 1 , X 1) which can be rewritten as: *(x) + x i ( i - P P l 1 : x ) < ^(x 1) + x ^ i - p 1 ^ 1 ) , M i i for every xeR . We have assumed that the values of p and x , i = 1,...,N, are known from observations, (2.10) can-equivalently be written as: (2.11) tp(x3') + X 1 ( l - p^x 3") < ^ ( x 1 ) + X 1 ( l - p ^ x 1 ) , for i = 1, .. . ,N, and j = 1,. . . ,N, where i ^ j . L e t t i n g ( p 1 = <p (x 1) , i = 1,... ,N, and r e c a l l i n g that the normalization of prices implies p ^ x 1 = 1, we obtain the 70 following equivalent of (2.11): (2.12) - <f,j + - 1) > 0, i , j = 1,...,N, I 4 j . Using the preceeding argument we can prove the following theorem: Theorem 3. If the data (p, x ) 1 , i = 1,...,N, were generated as a solution to the u t i l i t y maximization problem (2.2) where the u t i l i t y function $ is quasi-concave and continuous from above then the objective function of the following;-linear program attains i t s lower bound zero: N N (2.13) z = min J . J §f].. j=l i=l 1 J H ^ j j subject to •fr1 -4 1 +. X L(p l tx i - 1) - s*. + s~. = Ol I, j=l,...,N s?. > 0, s.. > 0, A 1 > 1 [ i ^ j Proof: If the data (p, x ) 1 , i = 1,...,N, were generated as a solu-tion to the maximization problem (2.2) then there would be values for cj)1 and A1, i = 1,...,N, which satisfy (2.12). On the other hand, in order to obtain a feasible solution to (2.12) Phase I of a linear program which we can formu-late by introducing appropriate surplus and a r t i f i c i a l variables to (2.12) i s considered. First, we transform the inequalities of (2.12) to equalities as: **" " <j>j + A i(p i tx j - 1) - S+ = 0 j I, j=l,...,N s +. > 0, A1 > 0 J l j = = Thenij, introducing the a r t i f i c a l variables _,0> i> j ^ l * • • • »N, i-4j , we obtain the following: N N 4) Z max j=l i=l 71 N N (2.14) Z = max £ . £ - g~ j = l 1=1 1 J subject to . (j,1 - ^ + ^ ( p ^ x 3 - 1) - + sT. = 0 1 i , j=l,...,N s*. > 0, > 0, A 1 > 0 J . i-fH 13 = 13 = = J C l e a r l y , the l i n e a r program (2.14) i s equivalent to (2.13). However, there exi s t s a t r i v i a l s o l u t i o n : ip1 = 0, A 1 = 0, s^. = 0, = 0; i , j = 1,...,N, •i¥j4^ j »' which does not have any s i g n i f i c a n c e , therefore we need a normalizing constraint on the A's. One normalization i s A1 ^ 1 , i = 1,...,N. Hence the theorem. Corollary 1. I f the data (p, x ) 1 , i = 1,...,N, were generated as a so l u t i o n to the u t i l i t y maximization problem (2.2) where the u t i l i t y function <p i s quasi-concave, continuous from above, and l i n e a r homogeneous then the objective function of the following l i n e a r program atta i n s i t s lower bound zero: N N (2.15) Z = min I • - J sT. j = l i = l 1 J subject to i t j , i _.+ „- • . i w -(p + p x J (p - » + .& = 0 1 l , 3=1,... ,N s + . > 0, s". > 0, # > 1 J i ^ j ij = 13 = * = i i - + Proof: Let (<j> , A , s.., s..)=b§ an optimal s o l u t i o n to (2.13). x> • o 1 j i j Consider the following N bhy^efipxhanes (2.16) L (x) E <jp + A 3 ( p J t x - 1), j = 1,...,N, *md l e t C l e a r l y , another normalization would be with respect to <pk, k being some index i n {1,...,N}, which we s h a l l use whenpsolving (2.15) to determine the nonparametric index numbers. 72 • and l e t (2.17) <j(x) = min {L. (x) : j = 1,...,N}. j ' 2 But by (2.12) (2.18) <ij + X2 ( p ^ x 1 - 1) > d,1. To o = To Since, by (2.17) and (2.18), cj^x1) = cb^, where x 1 i s a s o l u t i o n to the u t i l i t y maximization problem (2.2), (2.17) i s v a l i d . Therefore, cf>(x) i s l i n e a r homo-geneous only i f Iw (x) i s . L.(x) becomes l i n e a r homogeneous only i f the con-i 3 stant term i n the hyperplane equation (2.16) i s zero, i . e . , cfr = A q , j = 1,...,N. Geometrically, t h i s l a s t condition implies that the hyperplanes L_. (x) go through the o r i g i n of the coordinate system. By the addition of the constraints A 1 = cj)1 to (2.13) we obtain (2.15). Theorem 4. (Diewert [1973b]) I f the l i n e a r program (2.13) has a zero optimal s o l u t i o n , then the x 1, i = 1,...,N, can be generated as solutions to the maximization problem (2.2) where the u t i l i t y function i s quasi-concave and continuous from above. Moreover, the function <f> can be taken to be increas-ing, continuous, and concave i n x. Proof: See Diewert [1973b]. Co r o l l a r y 2. I f the l i n e a r programming problem (2.15) has a zero optimal s o l u t i o n , then the x 1 , i = 1,...,N, can be generated as solutions to the maximization problem (2.2) where the u t i l i t y function i s quasi-concave, continuous from above, and l i n e a r homogeneous. Moreover, the function can be taken to be increasing, continuous, concave, and l i n e a r homogeneous i n x. Proof: A straightforward extension of the proofs of Co r o l l a r y 1 and Theorem 4 establishes the proof. In the following sections we review the index number theory, and d i s -cuss the use of the material of th i s section i n constructing nonparametric index numbers for homothetic and nonhomothetic preferences as we l l as i n the j u s t i f i c a t i o n of aggregation. 73 I I I . REVIEW OF INDEX NUMBER THEORY 2 3.1 Two Approaches to Index Numbers Index numbers of p r i c e s and quantities are used with Increasing frequency i n macro-economic de s c r i p t i o n and sliding-scalerwage and other agree-ments. A p r i c e ((d_^anttifey)) index can be thought of as a function which expresses the l e v e l of prices (quaii-gigigs) o n occasion i r e l a t i v e to that on occasion 0, the base occasion. As occasions we s h a l l , i n t h i s study, consider d i f f e r e n t time periods. We now give the following general d e f i n i t i o n s f o r p r i c e and quanity indexes. De f i n i t i o n ~ 1 . A p r i c e index between two p r i c e s i t u a t i o n s 0 and i i s a function P(p^, p 1, x^, x^) of the prices p^, p 1 , and the quantities x^, x 1 i n these s i t u a t i o n s . D e f i n i t i o n 2. A quantity index between two p r i c e s i t u a t i o n s 0 and i i s a function Q(p^, pS x^, x 1) of the p r i c e s p^, p 1 , and the q u a n t i t i e s . There are two main approaches to index numbers: the test approach, and the economic approach. 3.2 The Test Approach and the Fisher Indexes In the test approach which was developed by Irr.vd!'ng Fisher [1911], '[1922] i t i s required that an index number s a t i s f y some or a l l of a number of tests corresponding to c e r t a i n a p r i o r i "reasonable" properties. I t i s through these tests that one can determine the f u n c t i o n a l form of the index number. The prop-e r t i e s associated with these tests are mostly algebraic r e l a t i o n s h i p s of prices and quantities at d i f f e r e n t periods, they do not have d i r e c t economic i n t e r -pretations . The t e s t s l e d Fisher to define the following index number 2. :> u u s s i l y reTherreviewairiFt [1973f ]. 74 The tests l e d Fisher to define the following index numbers which are commonly r e f e r r e d to as Fisher's i d e a l index numbers: „ . „ . . i t i i t 0 .';./_ (3-D P^P0, P1; x°, x 1) , ( f 0 t X i P o t X o ^ > ' p x p X and r , . n . ., i t i Ot i .1/2 / o o \ . / 0 l 0 i . ,rp x p x (3-2) QjCp , p ; x , x ) = <rit 0 P 0 t 0 J p x p X There has not yet been found.price and quantity index numbers which s a t i s f y a l l eight of Fisher's t e s t s . On the other hand there are many index numbers which s a t i s f y some of the tests (see F r i s c h [1930] and Samuelson [1967]). This, along with the l a c k o f economic i n t e r p r e t a t i o n mentioned e a r l i e r has prompted F r i s c h [1936], PP-olLfek- [1971] and others to lead another approach, the economic approach, which we now review b r i e f l y . 3.3 The Economic Approach Let us assume that a consumer's u t i l i t y , u, from consuming (purchas-ing) M goods i n quantities x^,...,x^, can be expressed as u = cp(x). I f the u t i l i t y function <p i s nondecreasing i n x and y the consumer i s bounded i n expend-ing by a budget constraint, p f cx < y, where p >> 0 and h i s income for expenditure y > 0, then (3.3) E(p; u) = min {pfcx : <p(x) ^ u} x>>0 i s h i s expenditure at prices p^,...,p for a given l e v e l of l i v i n g u. By d e f i n -i t i o n E i s . p o s i t i v e , l i n e a r homogeneous and concave i n p. Definit' D e f i n i t i o n 3. (Samuelson and Swamy [1974]) An economic p r i c e index, or a cost of l i v i n g index i s the r a t i o of the minimum costs of a given l e v e l of l i v i n g i n two p r i c e s i t u a t i o n s and i t i s mathematically defined as: 75 (3.4) P C ( P ° , P S i) = i l E l - L ^ i l ] , . E [ p U ; <p(x)] where x i s the reference quantity and the expenditure comparison i s made with respect to x. The economic p r i c e index defined as (3.4) s a t i s f i e s most of Fisher's tests and has well defined economic i n t e r p r e t a t i o n . However, there are two main problems with using P^ as p r i c e index:the determination of x and the true f u n c t i o n a l form of E (or e q u i v a l e n t l y i i t s dual, <p.) With respect to the f i r s t problem; of course the base period quantities x^,...,x^ , or the current period cond quantities x^,...,^ can be used as x ^ , — W i t h respect to the se problem, one can develop l i m i t s on e i t h e r side of P (see, f o r example, F r i s c h [1936] and Pollak [1971].) I f the u t i l i t y function i s homothetic i n x i t i s easy (see, f o r example Diewert [1974]) to show that (3.5) P p a = P c (p , p ; x ) = P£p(p , p ; x ) = P , V u T> , 0 i 0 i , . _ i t 0. , Ot 0 „ where P = P (p , p •; x , x ) = p x/, / p x , the Paasche p r i c e index, and ir 3. J. 3L D T> r ® i 0 i \ - i t i 1 Q t i , ' . . , _ ^ P^ a = P L a ( P » P > x > x ) = P x / P x » the Laspeyres p r i c e index. But, ! i t 0 i t i h p x p X The d e f i n i t i o n of an economic quantity index corresponding to (3.4), which we now review, has been proposed by Malmquist [1953] and Pollak [1971] i n the context of consumer theory. Eirstewe rieedtto define a distance function: (3.7) D(x; u) = max {X : c p O ^ f ) = u}, for x>>0, A or equivalently, X D(x; u) = max {X : — e L(u)}, A where L(u) = {y : cp(y) = u} i s a l e v e l set of the u t i l i t y function <p and i t i s 76 nonempty, closed, convex subset of the nonnegative orthant i n the M-dimensional space. In view of these assumptions D(x; u) i s p o s i t i v e , concave, and l i n e a r homogeneous i n x. I n t u i t i v e l y , D(x; u) represents the d e f l a t i o n proportion by which the given consumption x can be put on the u t i l i t y surface <j> (y) = u. D e f i n i t i o n 4. An economic quantity index for two given quantity s i t u a t i o n s x^ and x 1 i s the r a t i o of the d e f l a t i o n proportions needed to bring the respective quantities to the l e v e l corresponding to a reference well-being and i t i s mathematically defined as (3 . 8 ) Q F(x°, x 1; x) = D ^ \ \ D[<Kx); x U ] where x i s the reference quantity and the d e f l a t i o n proportions are compared with respect to x. Again, as f o r the economic p r i c e index, i f the preferences are homothetic i t can be shown that for any x>>0 ( 3 ' 9 ) Qpa^g E ( x°' X X'' X )^= QLa> v, n r v / 0 i 0 i X _ p^ibt'i A pVtic^O,. the Paasc'hp. q u a n c i t v where Q p a = Q p a ( p , p ; x , x 0 = p x / 'p x , the Paasche quantity index, a i l j ^La n , 0 i 0 • i \ _ l.'-^ b'A /, p^ oWo/ the L a s p o / r e s qu-_ bi r and Q L a = Q L a ( p , p ; x , x 0 = 'p x / *p x the Easpeyres quantity index. The geometric mean of Q and Q i s IT EL Lt 3. /o i i t i Ot i h B:io, «__•-_>*- = V p x p X the i d e a l quantity index. Given e i t h e r a p r i c e index or a quantity index, the other function can be defined i m p l i c i t l y by the following equation which i s Fisher's weak fact o r r e v e r s a l t e s t : /o i n -o, 0 i 0 i \ A / 0 1 0 i % i t i , Ot 0 U . l l ) P_(p , p ; x , x ) • Q(p , p ; x , x ) = p x / p x . The economic quantity index defined as (3.8) s a t i s f i e s most of Fisher's tests and has w e l l defined economic i n t e r p r e t a t i o n , but i t too suffers from the same problem:the economic indexes can be c a l c u l a t e d o n l y . i f the func-t i o n a l form of i s known. On the other hand, i t can be shown (see, Diewert |1973e] ) that the i d e a l quantity index , Q , w i l l c o r r e c t l y i n d i c a t e , i n an or d i n a l scale, welfare changes f o r any w e l l behaved u t i l i t y function which has i n d i f f e r e n c e surfaces of the form x tA(u)x = constant, A being a matrix. In t p a r t i c u l a r , i f the u t i l i t y function i s the homogeneous quadratic (x A x ) 2 , then by a theorem due to Byushgens [1925] and Konyus and Byushgens [1926], which i s formally stated below, the i d e a l quantity index i t s e l f becomes a cardinal i n d i c a t o r of o r d i n a l u t i l i t y . This i s also a fundamental property an economic quantity index i s required to have. Theorem 1* (Byushgens [1925], Rconyus and Byushgens [1926], F r i s c h [1936], Wald [1939], Pollak [1971], A f r i a t [1972]) Let p 1 » 0 f o r periods i=@,l,...,N and suppose that x 1 > 0 i s a s o l u t i o n to r , z s i t i t i - . max l<p(x) : p x <_ p x } x>0 t v N N ^ where <p(x) = (x Ax) 2 = [| lY x.a., x, ] 2, a„ = a, .. Then, j = l k = i 3 3 1 J /•o i o \ 4>(xl) n / 0 i 0 is . . (3.12) " ( J = Q i ^ p ' P ; x ' x )» 1 = 1 » - - - » N -cp(x ) Thus , given the base period normalization <p(x^) = 1, the i d e a l quantity index may be used to calculate ^(x 1) for i=l,...,N without having to estimate the unknown c o e f f i c i e n t s of A matrix. I f a quantity index Q(p^, p 1 ; x^, x 1) and a f u n c t i o n a l form for the u t i l i t y function <pp s a t i s f y equation f (3.12), then we say that Q i s exact f o r <p . There are other index numbers as we s h a l l see i n the next Section, which are exact for some fu n c t i o n a l form for <p(x). For more on exact index numbers 78 see Diewert [1974]. 3.£ The Torn q u i s t - T h e i l " D i v i s i a " Index Suppose that <b i s the homogeneous translog function (Christensen, Jorgenson, Lau [1971] [1973]): i M i '1 M M i i (3.13) l n cKx1) = a Q + , q _ l n x n + ijT I • Y j k In- x 1 In x£, i=l,...,N, M • M where £ a_ = 1, y j k =YYkj> a * d I Y j k = 0 f°r j=l,...,M. n=l k=l If we further assume that x1>>0 i s a s o l u t i o n to the u t i l i t y maximi-zation problem c ,c / \ i t i t 1. max icb(x) : p x <pp x } x>0 — where p1>>0, for i=l,...,N, and cb i s defined as (3.13), then, M . i k^+s0) i N M x n n n • n • ( } 77oC= \ * & = Q D ( P ' P ; X ' X } <p (x ) n=l x n where s^ = (pSc 1) / ( p ^ x 1 ) , the n th share of cost i n period i . The ri g h t hand side of (3.14) i s an exact quantity index f o r a homogeneous translog function. I t has been u t i l i z e d by Christensen and Jorgenson [1969] [1970] as a d i s c r e t e approximation to the D i v i s i a [1926] index and has been advocated by Tornqvist [1936] and T h e i l [1965] [1967] [1968]. On the other hand, i f the unit cost function for cb i s the translog unit cost function • * M i 1 M M (3.15) In c(p 1) = a + . _ a* l n p 1 + j . _ . _ y* l n p 1 l n p£, 1=1,...,N, n=l j = l k=l J J M * * M i where Y a„ = 1, y., = y, . and £ y = 0 f o r j=l,...,N, and i f x >>0 i s a n=l J J k=l J s o l u t i o n to the cost minimization problem 79 min {pSc : <p(x) > .1} x>0 then i hisKs0) / i \ M p n n n • r> • ( 3 . 1 6 ) —Z-Q' = II —Q = P D(p , p ; x , x ) c(p ) n=l p n where s 1 = ( p 1 x 1 ) / ( p ^ x 1 ) , the n th share of cost i n period i v The r i g h t hand side of ( 3 . 1 6 ) i s an exact p r i c e index for a translog unit cost function proposed by Tornqvist [ 1 9 3 6 ] , S h e l l [ 1 9 6 5 ] [ 1 9 6 7 ] [ 1 9 6 8 ] , Kloek [ 1 9 6 6 ] [ 1 9 6 7 ] . I t should be noted that Q D " P n ^ P*"^1 I p 0 t x ° i n general; i . e . , Pp and Q Q do not s a t i s f y the weak factor r e v e r s a l t e s t . However, given Q ^ , the the corresponding p r i c e index which s a t i s f i e s the weak factor r e v e r s a l test i s found by rv • r. i t i C. , 0 x 0 i . p x ( 3 . 1 7 ) P D(p , P ; x , x ) = = 0 J 0 , P x QD which we c a l l the derived D i v i s i a p r i c e index. S i m i l a r l y , the derived quan-t i t y index i s ( 3 . 1 8 ) Q D ( p ° , p 1 ; x°, x 1) = Q P 1 ^ 1 . P x P D In the following Section we look at the aggregation problem and the homogeneous weak s e p a r a b i l i t y of a u t i l i t y function. IV. AGGREGATION AND HOMOGENEOUS WEAK SEPARABILITY Suppose that the microeconomic commodities have been separated, with t t respect to some c r i t e r i o n , into two groups as y = (x^,...,^) , z = ( X j ^ , . . . ,x^ .) , Furthermore l e t (p(x) be the microeconomic u t i l i t y function of a consumer. When the problem i s considered from a producer's viewpoint <p(x) represents production 80 function. In order to j u s t i f y the aggregation of x^,...,x^ in t o i j i = ip(y) i t i s s u f f i c i e n t to show that y i s homogeneously weakly separable from z, or <t>(y; z) = <pl>(y); A, where if i s a macro u t i l i t y function and ip i s an aggregator function which i s assumed to be p o s i t i v e , concave, and l i n e a r homogeneous (see, Shephard [1953, pp. 61-71] [1970, pp. 145-146].) I f y i s homogeneously weakly separable from other inputs, z, aggre-gation of y into i p ( y ) i s achieved e a s i l y provided the f u n c t i o n a l form of ip i s known (Diewert [1974].) I f , on the other hand, the f u n c t i o n a l form of v p i s not known, also since i t s value cannot be d i r e c t l y observable the evaluation of i s not so obvious. In t h i s case Cor o l l a r y 1 of Section 2, which i s e f f e c -t i v e l y a l i n e a r programming t e s t , can be used to determine (a) i f the data on prices and quantities were generated as solutions to the u t i l i t y maximization problem (2.2) where the u t i l i t y function i s quasi-concave, continuous from above, and l i n e a r homogeneous, and (b) the values of the aggregator function when the answer to (a) i s a f f i r m a t i v e . C l e a r l y , i f the optimal z i n (2.15) i s equal to zero then by C o r o l l a r y 2 of Section 2 ip can be taken to be an increasing, continuous, concave, and l i n e a r homogeneous function of y, which i n turn implies the homogeneous weak s e p a r a b i l i t y of y from z. The optimal 1 N s o l u t i o n of the l i n e a r program, i p , . . . , i p , represents the values of the aggrega-tor function. Furthermore by means of a normalization, as w i l l be explained i n the next Section, the i p 1 which are, as a s o l u t i o n to the l i n e a r program, i n an o r d i n a l scale can be transformed i n t o a c a r d i n a l scale and thus represent the quantity indexes corresponding to y 1 , i=l,...,N. 5 , 1 V. NONPARAMETRIC INDEX NUMBERS Suppose again that there observations on p r i c e s , p , and quantities 81 purchased, x 1 , i = l , N , where p 1 and x 1 are M-dimensional vectors, i . e . , there are M commodities. 5.1 Constructing Homothetic Nonparametric Index Numbers We solve the following equivalent of the l i n e a r program (2.15) to 1 N f i n d the values of cb , . . . , c b , where we have incorporated the a d d i t i o n a l con-,m s t r a i n t cb - 1 to scale the u t i l i t i e s with respect to the u t i l i t y i n period m: N N 1 = 1 1=1 3 i 5 5 (5.1) z = min subject to -cb- 3 + p ^ x ^ c b 1 + s >_ 0 s T . > 0, c b 1 > 0 ,m . • r , cb =1, me {!,...,N}. i,j= l , . . . , N , i£j N^erlrJaattw.esh,av.et>pMttii't:edLthe ti&mta&i&i&gLconstraints X 1 ^ of (2.15) i n (5.1). Suppose an optimal s o l u t i o n to (5.1) i s (<^=1, <t>^ ,... ,cb^) such that z = 0. This implies, by the c o r o l l a r y , that the pricesquantity data used are consistent with the maximization of a u t i l i t y function,cb , which i s quasi-con-cave, continuous from above, and l i n e a r homogeneous, and by Cor o l l a r y 2 of Sec-t i o n 2, (j, can be taken to be increasing, continuous, concave, and l i n e a r homog-eneous . We use cb1 as quantity indexes with ^ = 1, the base period index. The corresponding p r i c e indexes can be calculated by (5.2) P. = ( p ^ x 1 ) / ( p 0 t x ° Q.), where = c b 1 . The homogeneous weak s e p a r a b i l i t y of a group of commodities can be tested by solving (5.1) f o r the price-quantity data corresponding to those com-modities. The r e s u l t i n g optimal s o l u t i o n i s then e f f e c t i v e l y used to represent 82 the aggregate q u a n t i t i e s . 5.2 Constructing Nonhomothetic Nonparametric Index Numbers For the nonhomothetic nonparametric index numbers we consider the following equivalent of the l i n e a r program (2.13) to determine the values of § L , . . . » < p ^ ,where we have incorporated the a d d i t i o n a l constraints cpm = 1 and ,n - , m n m n x <P = Q-J-CP » P 5 x , x ) : N N (5.3) z = min • £ • £ s subject to H 1=1 1 2 - s . . - D. . A 1 - cp 1 + cp3 < 0 s. . > 0 A 1 > 0 > i , j = l , . . . ,N,. i?tj , ,m , ,n , m n m n x cp = 1, cp = Q].(p , p ; x , x ) , i t i where D „ = p x J - 1. In (5.3) m can be 1, to correspond to the f i r s t period, and n can; be N, the l a s t period. Or, we may choose a s c a l i n g f o r cp 1 such that cp m = min {cp3/} = 1, j nt n mt n % , n _ • • . , m n m n. rp x p x . p x p X The; r a t i o n a l e foxapheydoubleinor^ -to have oneiriormalizafion fet.r(i5 s3>yiU±rioo.rder'(€o 3ivo^M -tke^MV'Ml^oser^og-oM&lion.- 'By norma-li'zinig. tehefcp.ciwiuth rSs'peretntsOTateHe2wt&lbi*ty\&±&M±M$v.e&t tpetffitf&bas* well±we obtain a- cWdinaluiyps^alddaopt^^ Pjri0.vi>desKa=gdodwb;asdis:afl.0ir acompahrisiori.swft'h". tto^Mistee*. indexesi?- FThfeen'ormalization ^^*oal©Be/6s.ea]^sis'ah"si>MM©B>o6.^#fcS^i,Jy and the cp1 e x h i b i t an e r r a t i c behavior. Suppose that <pm = cp^, cp n = cp^, and an optimal s o l u t i o n to (5.3) i s <p 1=l , cp 2 , | . . , ( p N _ 1 , <p N=Q I(p 1, p N; x 1 , x N) such that z = 0. Then, by the theorem, the price-quantity data used are consistent with the maximization of a u t i l i t y 83 function which i s quasi-concave and continuous from above. We use cp 1 as quan-t i t y indexes with period 1 being the base period. The corresponding p r i c e indexes are calculated by (5.2). 5.3 A Measure of V i o l a t i o n I f the optimal z>0, for the appropriate l i n e a r program, then we would l i k e to obtain some measure of v i o l a t i o n of the r e g u l a r i t y conditions hypothesized. In t h i s Section we suggest one such measure which develops as fo l l o w s : i n order f o r z to be zero, at aheoptimal s o l u t i o n , s „ = 0, i , j = l,...,N, i ^ j , f or both homothetic and nonhomothetic cases. The nonhomothetic case: Suppose that an optimal s o l u t i o n to (5.3) i s ( c p 1 , A 1 , s „ ; i , j = l , . . . ,N, i ^ j ) with at le a s t one s _ > 0. Let Ax = (Ax^J,... ,Ax^) represent a pertur-bation vector whose components are determined as the so l u t i o n to the following system of l i n e a r equations which can be obtained from (5.3) by making s „ = 0: (5.4)- - [ p l t ( x 3 + Ax 3) - 1]XX - (p 1 + cp3 = 0 f o r a l l i such that s7.>0, (5.4) . p 1 Ax 3 = 0 f o r a l l i such that s. .=0, or equivalently, A ^ ^ A x 3 = <p3 - (p 1 - [ p X t x 3 - l j A 1 for a l l i such that sT.>0, (5.5) 1 J i t i — p Ax = 0 for a l l i such that s. .=0. Cle a r l y , (5i:3)"-whd!ehshasaN-li\equ^ f o r a l l i for which there ex i s t s at l e a s t one s.. > 0, i=l,...,N. 13 The homothetic case: For the homothetic case a l l that i s needed i s to replace A 1 , i n (5.5), by cp 1 and divide through by (p 1 > 0 to obtain 84 p l t : x 3 = ((p3/'?1) - p^x 3 for a l l i such that s..>0, (5-6) . t • "J p x = 0 for a l l i such that s..=0. ij It should be noted, however, that if-.sMv. • >:.Ncther.e, are M - N degrees bfrfreedom for determining the Ax3 while i f M < N Ax3 are overdetermined by large number of observations. More importantly, i t is in general necessary to solve the testing linear program after correcting each x 3, by solving (5.4) or (5.5). Therefore when there are many violations at significant levels the measure suggested here may not be easily calculated. In the following Section we discuss the actual calculations of the nonparametric indexnnumbers, both homothetic and nonhomothetic, as well as the Fisher indexes and the Divisia indexes for the price-quantity data of Cummings and Meduna [1973]. VI. EMPIRICAL EXAMPLES Cummings and Meduna [1973] provide data on Canadian consumer expen-ditures, i.e., prices of commodities and quantities purchased, for the years 1947-1971. Using the Divisia indexes they have aggregated the microeconomic data to obtain 13 commodities which they further divide into four groups as follows:services (medical, other); nondurables (food, alcohol, tobacco, energy, other); semi-durables (clothing, other); durables (motor vehicles, housing, land, other). We hav.elused their data, as explained i n the following Sections, as i f they represent the microeconomic observations on prices and quantities of 13 commodities. The data are given in Tables l a , lb, 2a, and 2b. Because of the size restrictions of the linear programming code used we have divided the 25 observations into two sets of 13 observations each. The f i r s t set covers the years 1947-1959, the second set 1959-1971. Thus 1959 i s common to both sets . 85 6.1 Microeconomic Data In t h i s Section we have used the "microeconomic" data of Cummings and Meduna [1973] on 13 commodities to calculate various index numbers covered e a r l i e r i n the text. Homothetic nonparametric index numbers: We have solved the l i n e a r program (5.1) f o r the two sets of data on prices and quantities such that ^ = 1, i . e . , with a normalization with respect to the firstse<ji i n each case. The optimal value of z for the f i r s t set of 13 observations has been computed to be 0.023474 while the second set has had an optimal z equal to zero. For the f i r s t set the optimal s „ values which are i n the basis at a p o s i t i v e l e v e l are a l l le s s than 0.01. In view of t h i s and the very small z we have assumed that the v i o l a t i o n of homotheticity i s not s i g n i f i c a n t , therefore we have not calculated x as suggested i n (5.6). The r e s u l t i n g index numbers turn out not to be a f f e c t e d noticeably as can be ob-served upon examining the subsequent tables. In order to obtain the combined quantity indexes f o r the e n t i r e set, 1959 which appear i n Table 3, we have m u l t i p l i e d the quantity indexes Q =1, ,. .. ,0^^^^ jwhich are i n an optimal s o l u t i o n to the l i n e a r program for the 1959 second set of observations by Q = 1.355202, the quantity index for 1959 obtained from the optimal s o l u t i o n to the l i n e a r program for the f i r s t set of observations. We have calculated the corresponding p r i c e indexes by using (5.2). Nonhomothetic nonparametric index numbers: 1 13 We have solved the l i n e a r program (5.3) with <j> = 1, cb = 1 13 1 13 Q (p , 'p ; x , x ) = 1.352015 f o r the f i r s t set of 13 observations, and 1 13 1 13 1 13 cb = 1,. <p = Q T(p , p ; x , x ) = 1.269831 f o r the second set of 13 obser-vations. The optimal z has been zero for both of the cases. The combined quantity indexes which we have calculated as explained above appear i n Table 4. The corresponding p r i c e indexes have been calculated by using (5.2). Fisher indexenumbers: The straightforward c a l c u l a t i o n s , for the e n t i r e set of observations, using the Fisher's i d e a l p r i c e and quantity index number formulae (3.1) and ((3.2) have y i e l d e d the figures of Table 5. D i v i s i a index numbers: F i r s t , by (3.16) we have calculated the D i v i s i a p r i c e indexes for the e n t i r e set of observations from which, by using (3.18), we have derived the quantity indexes and tabulated them i n the f i r s t two columns of Table 6. Then, by (3.14) we have calculated the D i v i s i a quantity indexes from which, by using (3.17), we have derived the p r i c e indexes. They are tabulated i n the l a s t two columns of Table 6. 6.2 Aggregate Data We have considered the aggregation of the same microeconomic com-modities as Cummings and Meduna [1973] do. The four aggregates are:services, nondurables, semi-durables, and durables. These aggregate commodities appear i n the appropriate tables always i n the above order. To j u s t i f y the aggre-gations we have tested the existence of an aggregator function i n each case, which i s p o s i t i v e , concave, and l i n e a r homogeneous, by so l v i n g the l i n e a r prog-ram (5.1) for each set of data. Aggregation with homothetic nonparametric index numbers and nonpa-rametric index numbers for the aggregate data: We have solved the l i n e a r program (5.1) with each of the four sets . of data corresponding to those i n s e r v i c e s , nondurables, semi-durables, and durables. The optimal z i n each case has been les s than 0.0001 providing the required j u s t i f i c a t i o n of aggregation. To c a l c u l a t e the aggregate quantity i n period i , X 1, for the set A of microeconomic commodities we have used the f o l -lowing: 87 (6.1) X 1 = ( P ^ ^ ) ' * 1 * i-l,-...,N,. where again ( cb^ "=l, c|?,.. . , <jF) is an optimal solution to (5.1) with price-quan-t i t y data for the aggregate commodity A, p^ and x^, i=l,...,N, and p^ and x^ are the base period prices and quantities. The price indexes have been found, as before, by which i s of course nothing but (5.2). The resulting prices (price indexes) and quantities of four aggregate commodities are tabulated i n Table 7. Using 'Che data i n Table 7, exactly as i n the case when we have cal-culated the homothetic nonparametric indexes from the microeconomic data, we have solved the linear program (5.1) for the two sets of observations. The optimal z has been 0.175064 for the f i r s t set and 0.077671 for the second set of observations. Assuming that these violations are not significant, we have combined the optimal solutions to the linear programs to construct the homo-thetic nonparametric quantity index numbers for the entire set of aggregate data. The corresponding price indexes have been calculated, as usual, by (5.2) . They are given i n Table 8. Also using the data in Table 7, again similar to the calculation of nonhomothetic nonparametric indexes for the microeconomic data, we have 1 13 solved (5.3) with cb - 1, _.. •= 1.350435 for the f i r s t 13 observations and 1 13 cb = 1, cb = 1.270435. The optimal z has been 0.074116 for the f i r s t set, 0.014165 for the second set. Assuming that these violations are not signi-ficant, we have combined the optimal solutions i n the usual manner to obtain the quantity indexes tabulated in.Table 9 along with the derived price index numbers.. Aggregation.with Fisher index numbers and Fisher index numbers for the aggregate data: Table l a Price Indexes Year Medical Other Food Alcohol Tobacco Energy Other Services Services Non-Durab: 1947 0.4981 0.5582 0.6517 0.8090 0.8467 0.7886 . 0.6642 1948 0.5519 0.6098 0.7976 0.8301 0.8907 0.8698 0.7409 1949 0.5817 0.6397 0.8120 0.8262 0.9085 0.9010 0.7829 1950 0.5920 0.6631 ' 0.8289 0.8469 0.9318 0.9377 0.7830 1951 0.6433 0.7196 0.9475 0.9161 1.0234 0.9750 0.8639 1952 0.7122 0.7520 0.9488 0.9178 1.0508 0.9953 0.8889 1953 0.7429 0.7731 0.9084 0.9172 0.9284 1.0065 0.8989 1954 0.7866 0.7980 0.9078 0.9205 0.9135 1.0066 0.9047 1955 0.8105 0.8194 0.9077 0.9222 0.9161 0.9992 0.9123 1956 0.8459 0.8520 0.9183 0.9245 0.9180 1.0102 0.9195 1957 0.8979 0.8895 0.9580 0.9517 0.9172 - 1.0354 0.9421 1958 0.9378 0.9276 0.9966 0.9644 0.9176 1.0200 0.9607 1959 0.9627 0.9549 0.9877 0.9822 0.9798 1.0107 0.9976 1960 0.9819 0.9779 0.9909 0.9924 0.9973 0.9989 1.0144 1961 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1962 1.0299 1.0241 1.0218 1.0201 1.0036 0.9958 1.0034 1963 1.0604 1.0499 1.0531 1.0237 1.0035 0.9859 1.0037 1964 1.0885 1.0881 1.0633 1.0520 1.0149 0.9926 1.0092 1965 1.1286 1.1385 1.0933 1.0610 1.0458 0.9951 1.0298 1966 1.1672 1.1987 1.1601 1.0753 1.0901 1.0159 1.0564 1967 1.2476 1.2739 1.1640 1.0921 1.1357 1.0474 1.0817 1968 1.3142 1.3503 1.2052 1.1733 1.2713 1.0895 1.0945 1969 1.3803 1.4377 1.2373 1.2122 1.3333 1.1170 1.1289 1970 1.4747 1.5167 1.2552 1.2212 1.3514 1.1576 1.1661 1971 1.6031 1.6061 1.2664 1.2360 1.3805 1.2115 1.1807 Table lb Price Indexes Year Clothing Other Motor Housing Land Other Semi-Durables Vehicles Durables 1947 0.7004 0.5768 0.6298 0.5363 0.1907 0.7209 1948 0.8437 0.6591 . 0.6980 0.6212 0.2107 0.7209 1949 0.8785 0.7081 0.7428 0.6481 0.2193 0.8304 1950 0.8787 0.7181 0.7702 0.6818 0.2467 0.8577 1951 0.9707 0.8232 0.8727 0.8068 0.2929 0.9558 1952 0.9807 0.8517 0.8946 0.8321 0.3447 0.9796 1953 0.9729 0.8615 0.9123 0.8419 0.3522 0.9996 1954 0.9572 0.8581 0.9072 0.8446 0.5151 0.9742 1955 0.9484 0.8670 0.8521 0.8576 0.5467 0.9859 1956 0.9630 0.9047 0.8762 0.8847 0.6389 1.0033 1957 0.9696 0.9431 0.9526 0.9281 0.7574 1.0120 1958 0.9704 0.9707 0.9635 .0.9269 0.8319 1.0098 1959 0.9819 0.9895 1.0009 0.9451 0.8989 1.0068 1960 0.9888 0.9943 1.0131 0.9851 0.9351 0.9961 1961 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1962 1.0119 1.0323 0.9902 0.9993 1.0722 1.0051 1963 1.0398 1.0573 0.9959 1.0125 1.1171 1.0095 1964 1.0671 1.0698 0.9633 1.0503 1.1487 1.0142 1965 1.0871 1.0875 0.9643 1.1034 1.1354 1.0135 1966 1.1347 1.1316 0.9556 1.1829 1.3060 1.0411 1967 1.1900 1.1852 0.9882 1.2708 1.4032 1.0873 1968 1.2357 1.2522 1.0241 1.3359 1.6325 1.1080 1969 1.2798 1.3075 1.0771 1.4023 1.8549 1.1426 1970 1.2966 1.3572 1.1385 1.4612 1.9321 1.1565 1971 1.2961 1.3803 1.1271 1.5209 2.1056 1.1472 Table 2a Quantities Year Medical Other Food Alcohol Tobacco Energy Other Services Services SN'on-Durables 1947 62.8178 336.9223 363.2 61.6 47.0 81.0430 59.1363 1948 62.8101 336.7741 339.9 62.4 48.0 80.4620 57.7096 1949 66.5480 344.2469 325.6 64.2 48.3 81.9374 53.1292 1950 67.4519 343.9550 34331 65.0 48.6 83.1782 57.6213 1951 68.8142 346.4647 343.5 64.8 43.9 87.6535 55.7497 1952 71.9850 356.5573 345.5 71.8 45.3 91.8742 56.8884 1953 75.2292 368.4157 359.1 72.1 49.2 86.5642 58T2537 1954 78.8151 363.0288 370.5 69.8 50.9 103.1243 58.3744 1955 81.3733 373.7091 378.9 72.3 53.7 112.3883 60.9813 1956 88.9720 386.9655 394.8 75.6 56.2 120.9167 65.4202 1957 90.4676 390.4000 397.0 76.1 59.5 126.0799 68.2362 1958 97.3232 384.3018 392.1 76.4 62.8 132.0692 69.0601 1959 103.5591 399.5136 399.6 77.5 63.7 140.3441 69.1534 1960 111.9933 408.0034 407.1 78.1 62.9 146.3603 69.4393 1961 123.7714 415.8919 399.4 79.7 65.6 148.6751 72.0547 1962 128.7156 423.1016 399.4 81.2 67.9 155.5423 75 74363 1963 131.4352 432.1978 398.9 84.2 68.1 164.0638 79.5420 1964 138.3965 442.7872 409.1 84.2 68.1 167.6769 84.2798 1965 147.2502 458.7387 411.3 90.2 70.1 171.8423 88.3681 1966 148.9702 472.7667 402.9 94.0 71.1 173.2833 93.7923 1967 152.6879 496.4628 413.4 98.3 71.0 174.6835 97.1894 1968 159.3377 497.4608 405.3 95.6 67.8 175.9511 100.5649 1969 163.6039 502.7356 413.8 98.8 66.2 178.6521 103.5517 1970 165.0784 501.9054 423.3 105.2 69.3 183.3012 103.8650 1971 160.9649 500.9330 440.2 111.9 69.4 184.3525 109.0244 Table 2b Quantities Year Clothing Other Motor Housing Land Other Semi-Durables Vehicles Durables 1947 185.8296 82.5197 50.3615 536.6964 69.4505 83.0256 1948 180.7889 93.8176 58.5842 549.1060 70.5847 90.5746 1949 174.1012 96.5588 70.5784 552.2546 70.2375 96.0744 1950 173.1203 102.5152 90.5534 572.6273 73.5161 103.7143 1951 170.4048 101.0387 102.6978 582.2215 74.9383 106.7824 1952 171.8655 100.2412 111.5661 587.2420 75.4238 111.5040 1953 174.8763 101.9153 124.5732 604.4740 75.3009 117.4303 1954 175.0879 102.3951 128.9915 624.7326 75.1537 123.3796 1955 179.5909 105.4837 144.3373 656.8284687 75.6564 132.3288 1956 186.9391 109.1385 159.4698 687.0941 76.0241 143.2088 1957 190.6533 108.7907 163.4028 701.1263 75.5380 150.3823 1958 194.2590 108.7939 166.9769 729.7188 75.8828 156.8640 1959 198.5905 110.0535 172.6037 754.6835 76.0486 164.1914 1060 201.3829 109.2511 176.5273 768.2365 75.8652 170.1423 1961 203.2368 109.3496 180.6938 779.8234 75.9197 176.9517 1962 206.7944 109.0166 190.3609 790.6373 75.8280 184.6533 1963 208.7504 108.9630 203.5821 801.5321 75.6753 193.0380 1964 212.4342 110.8029 219.5218 815.7592 75.3187 203.0678 1965 216.9063 112.8239 238.7562 830.5192 74.8504 214.2421 1966 220.1602 115.3934 253.5834 837.5820 74.0877 226.6255 1967 223.5426 117.2578 263.1430 841.7429 73.1599 237.5630 1968 227.4157 118.4010 274.0165 854.0965 72.4541 248.9752 1969 232.3.450 119.9471 282.4893 873.8266 71.8687 261.4810 1970 233.4541 119.2140 276.1131 885.1791 71.2887 270.5428 1971 241.1156 121.5385 282.5391 910.2052 71.5034 285.5865 Table 3. Homothetic Nonparametric Indexes f o r Microeconomic Data Year P r i c e Index Quant.^Index 1947 1 .000000 1.000000 1948 1.142581 1.005699 1949 1 .187932 1.009727 1950 1.225230 1.050173 1951 1,382341 1.061393 1952 1.419715 1.086252 1953 1.421169 1. 121087 1954 1.434796 1.151496 1955 1.444596 ' 1. 203498 1956 1.484022 1 .265540 1957 1 .547710 1.289757 1958 1. 575312 1 .313967 1959 1.603050 1.355202 1960 1.632741 1.382907 1961 1.651885 1.405627 1962 1.671670 1 .433995 1963 1.700466 1.463979 1964 1.735755 1.503798 1965 1 .786894 1.549721 1966 1.876215 1.577137 1967 1 .969655 1.613915 1968 2.068784 1.631276 1969 2.167222 1.665504 1970 2.247797 1.683644 1971 2.319879 1.720877 Table 4. Nonhomothetic Nonparametric Indexes f or Microeconomic Data year Price*; Index Ouant. Index 1947 1.000000 1.000000 1948 1.145120 1.003469 1949 1. 189747 1.008187 1950 1.225643 1.049819 1951 1.382887 1.060974 1952 1.42054 7 1.085616 1953 1.422033 1.120406 1954 1.436506 1.150126 1955 1.446933 1.201554 1956 1.487152 1.262877 1957 1.55144 2 1.286654 1958 1.578990 1.310905 1959 1.606829 1.352015 1960 1.637244 1.379103 1961 1.656939 1.401340 1962 1.677331 1.429155 1963 1.706640 1.458682 1964 1.742396 1 .498066 1965 1.793909 1.543660 1966 1.883390 1.571129 1967 1.976693 1.608169 1968 2.076008 1.625600 1969 2. 174079 1.660251 1:970 2.254612 1.678555 1971 2.325347 1.716830 Table 5. Fisher Indexes for Microeconomic Data Year P r i c e Index Quant. Index 1947 1.000000 1.000000 1948 1.144914 1.003650 1949 1.190146 1.007848 1950 1.227468 1.048256 1951 1.385117 1.059262 1952 1.422978 1.083754 1953 1.424354 1.118572 1954 1.438248 1.148722 1955 1.447826 1.200799 1956 1.487453 1.262606 1957 1.551634 1.286478 1958 1.579053 1.310834 1959 1.606800 1.352015 1960 1.636508 1.379700 1961 1.655674 1.402385 1962 1.675392 1.430781 1963 1.704101 1.460826 1964 1.739245 1.500746 1965 1.790347 1.546695 1966 1.879580 1.574276 1967 1.973051 1.611096 1968 2.072244 1.628509 1969 2.170704 1.662785 1970 2.251181 1.681063 1971 2.322852 1.718630 Table 6. Divisia Indexes for Microeconomic Data YEAR PRICE INDX QUANT INDX PRICE INDX QUANT INDX DIVISIA DERIVED DERIVED DIVISIA 1947 1.000000 1.000000 1.000000 1 .000000 1948 1 .144888 1.003673 1.144896 1.003666 1949 1.190113 1.007876 1.190105 1.007883 1950 1.227427 1.048294 1.227397 1 .048319 1951 1.385065 1.059306 1.385046 1.059320 1952 1.422924 1.083802 1.422917 1.083807 1953 1 .424299 1.118624 1.424304 1. 118620 1954 1.438264 1.148720 1.438191 1 . 148778 1955 1.447839 1.200802 1.447783 1. 200849 1956 1.487456 1.262618 1.487414 1.262653 1957 1.551635 1.286494 1.551602 1.286522 1958 1 .579042 1.310863 1.579027 1.310875 1959 1.606780 1.352057 1.606786 1.352052 1960 1.636477 1.379750 1.636502 1 .379728 1961 1.655638 1.402441 1.655673 1.402411 1962 1 .675345 1.430849 1.675404 1 .430799 1963 1.704043 1.460905 1.704123 1.460837 1964 1.739180 1.500836 1.739282 1.500749 1965 1.790264 1.546803 1.790397 1.546689 1966 1.879483 1.574395 1.879644 1.574260 1967 1.972941 1.611227 1.973129 1.611073 1968 2.072121 1.628649 2.072331 1.628484 1969 2. 170565 1 .662939 2.170807 1.662753 1970 2.251023 1.681231 2.251300 1 .681025 1971 2.322675 1.718804 2.322984 1.718576 Table 7. Aggregate Data:Homothetic Nonparametric P r i c e Indexes and Quantities Year P r i c e Indexes of Four Aggregate Commodities 1947 1.000000 1.000000 1.000000 1 . 000000 1948 1.156845 1.094768 1.185920 1. 141869 1949 1;182091 1. 149282 1. 246 140 1. 195523 1950 1.209075 1.188046 1.250892 1. 254867 1951 1.344559 1.289540 1.398700 1. 463767 1952 1.356342 1.359840 1.424415 1. 513679 1953 1.314086 1.4014 17 1.422402 1. 535789 1954 1.313338 1.453357 1.405362 1. 556389 1955 1.313198 1.493238 1.401905 1. 565750 1956 1.325672 1.553782 1.437153 1. 621508 1957 1.369967 1. 627262 1. 464520 1. 711667 1958 1.400489 1.697872 1.480190 1. 722177 1959 1.406219 1.746837 1.501702 1. 761441 1960 1.411289 1.787337 1.511121 1. 813812 1961 1.418028 1. 826026 1.525311 1. 836520 1962 1.436862 1.872475 1.554188 1. 842680 1963 1.457369 1. 9 2158 2 1.595217 1. 865965 1964 1.473020 1.987079 1.629125 1. 9 04 8 53 1965 1.503036 2.074679 1.658391 1. 964201 1966 1.567122 2.175087 1.729161 2. 076236 1967 1.591326 2.314806 1.812615 2. 208255 1968 1.662995 2.449899 1.893501 2. 320503 1969 1.712063 2.599976 1.966619 2. 443149 1970 1.746075 2.751156 2 .009334 2. 540252 1971 1.777747 2. 93 1643 2.020737 2. 613748 Aggregate Quantities 1947 429.515137 219.359421 177.752365 392. 645020 1948 413.536133 219.251480 180.759781 410. 083252 1949 403.293213 225.292877 177.605179 422. 847168 1950 420.025391 225.586823 180.460739 452. 042236 1951 419.007080 227.665390 177.727 112 466. 857910 1952 430.067627 234.879166 178.265320 478.092041 1953 439.471924 243. 1 17981 18 1.338852 499. 064209 1954 459.670410 241 .986435 181.774521 516. 308838 1955 478.013916 249.236588 186.730194 548.049805 1956 502.635986 260.625244 193.966568 579. 6 16211 1957 512.531250 263.319824 196. 281 189 593. 439209 1958 516.339600 263.710693 198.700531 614. 795166 1959 529.114014 275.464111 202.366028 635. 659424 1960 538.705078 284.753906 203 .660416 648. 3850 10 1961 539.784180 295.539551 204.932602 660. 698975 1962 549.575195 302.199707 207.048935 675. 904297 1963 560.050781 308.671143 208.288147 693. 317383 1964 573.093750 318.276367 211.908264 714. 346924 1965 585.938721 331.838623 216. 169922 737. 574707 1966 587.770508 340.484131 219.987839 754. 151611 1967 601.560059 355.509766 223.427933 * 765. 619629 1968 594.465332 359.656250 226.711487 782. 481201 1969 605.396729 364.851074 230.947449 802. 944092 1970 622.397461 365.184326 231 .167847 810. 3120 12 1971 643.314209 362.455322 237.670288 834. 471924 Table 8. Homothetic Nonparametric Indexes for the Aggregate Data i n Table 7 Year P r i c e Index Quant. Index 1947 1.000000 1.000000 1948 1. 144712 1.003824 1949 1.189978 1.007939 1950 1.227494 1.048234 1951 1.385571 1.058916 1952 1.423349 1.083475 1953 1.424718 1 .118293 1954 1.438970 1.148154 1955 1.448778 1.200021 1956 1.488573 1.261668 1957 1.552965 1.285390 1958 1.580751 1.309443 1959 1.608704 1.350436 1960 1.638790 1.377799 1961 1.658081 1.400372 1962 1.677919 1.428651 1963 1.706778 1 .453563 1964 1.742241 1.498197 1965 1.793677 1.543 857 1966 1.883430 1.571094 1967 1.977254 1.607711 1968 2.076739 1.625025 1969 2.175514 1.659153 1970 2.256164 1.677400 1971 2.326957 1.715641 Table 9. Nonhomothetic Nonparametric Indexes for the Aggregate Data in Table 7 ' — Year Price Index Quanta Index 1947 1.000000 1.000000 1948 1.144510 1.004001 1949 1.189254 1.008602 1950 1.223751 1.051439 1951 1.380411 1.062874 1952 1.418030 1.087540 1953 1.419806 1.122162 1954 1.434472 1. 151753 1955 1.445449 1.202785 1956 1.486504 1.263424 1957 1.551403 1.286683 1958 1.579782 1.310245 1959 1.608705 1.350435 1960 1.613906 1.399043 1961 1.63024 3 1.424284 1962 1.648226 1.454389 1963 1.671038 1.489758 1964 1.711246 1.525332 1965 1.766253 1.567827 1966 1.857710 1.592845 1967 1.948498 1.631437 1968 2.057553 1.640177 1969 2.159804 1.671222 1970 2.250061 1.681948 1971 2.326960 1.715639 .fable 10. Aggregate DatarFisher P r i c e Indexes and Quantities P r i c e Indexes of Four Aggregate Commodities 1947 1 .000000 1 .000000 1 .000000 1.000000 1948 1. 155738 1.094661 1 .187039 1.142160 1949 1 . 180838 1 .149130 1 .247603 1.195605 1950 1.207903 1. 187901 1 .253364 1.254716 1951 1 .344186 1 .289375 1 .401625 1.463232 1952 1.355997 1.359580 1 .427406 1.513231 1953 1.313515 1 .401028 1 .425391 1.535294 1954 1.312663 1.452426 1 .408310 1.555398 1955 1 .312398 1 .492289 1 .404853 1.564197 1956 1.324859 1.55271 1 1 .440238 1.619582 1957 1.368871 1.626107 1 .468129 1.709338 1958 1.398993 1.696270 1 .484152 1.719663 1959 1 .404848 1 .745185 1 .505731 1.758734 1960 1.409834 1.785686 1 .515144 1.810906 1961 1.416476 1.824382 1 .529322 1.833474 1962 1.435137 1.870795 1 .558362 1.839498 1963 1 .455343 1.919861 1 .599506 1.86 26 27 1964 1.470927 1.985208 1 .633497 1 .900898 1965 1.500959 2 .072632 1 .662879 1.959715 1966 1.565020 2. 172947 1 .733845 2.070640 1967 1 .589107 2.312372 1 .817528 2.202009 1968 1 .660556 2.447467 1 .898603 2,313519 1969 1 .709603 2.597436 1 .971904 2.435495 1970 1.743503 2.748507 2 .014672 2.532101 1971 1 .774935 2 .929080 2 .026056 2.605076 Aggregate Quantities 1947 429 .515137 219 .359436 177.752365 392.644775 1948 413 .932373 219 .272903 180.589401 409. 978271 1949 403.720215 225.322601 177.396729 422. 618359 1950 420 .432373 225 .613754 180.104263 452. 094971 1951 419 .122070 227 .693726 177.355515 467. 026855 1952 430 .175293 234 .922714 177.891006 478.230713 1 953 439.660156 243 . 183746 180.957932 499. 221680 1954 459 .903320 242 .139175 181.392975 516. 633057 1955 478 .301025 249 . 392654 186.337021 548. 589111 1956 502 .940186 260 .802002 193.549347 580. 299072 1957 512 .935791 263 .503906 195.796280 594. 240723 1958 516 .885254 263 .956299 198. 167511 615.686035 1959 529 .623291 275 .720215 201.821579 €36.628174 1960 539 .253174 285 .012207 203. 1161 19 649. 414795 1961 540 .366455 295 .800049 204. 39 1479 661. 785156 1962 550 .226318 302.464844 206.490219 677. 061768 1963 560 .820068 308 .941406 207.725327 694. 546387 1964 573 .898438 318 .569092 211.336334 715. 819092 1965 586 .738281 332 .158691 215.581284 739. 247559 1966 588.546875 340 .810303 219.388260 756. 172363 1967 602 .386475 355 .874268 222.818268 767. 772705 1968 595 .325439 360 .003418 226.096024 784. 824707 1969 606 .252930 365 . 196533 230.321991 805. 446289 1970 623 .300049 365 .524414 230.548660 812. 898682 1971 644 .316406 362 .760498 237.038940 837.226563 Table 11. Fisher Indexes for the Aggregate Data in Table 10 Year Price Index Ouant. Index 1947 1.000000 1 .000000 1948 1.144902 1 .003658 1949 1.190133 1.007855 1950 1.227453 1.048263 1951 1.385099 1.059268 1952 1.422959 1 .083759 1953 1.424336 1. 118575 1954 1.438231 1 . 148723 1955 1.447812 1.200798 1956 1.487437 1 .262604 1957 1.551616 1.286476 1958 1.579033 1.310832 1959 1.606778 1 . 352014 1960 1.636484 1 .379695 1961 1.655649 1. 402378 1962 1.675366 1 .430775 1963 1.704074 1.460819 1964 1.739219 1.500739 1965 1.790317 1.546688 1966 1.879543 1.574269 1967 1.973011 1.611089 1968 2.072202 1.628501 1969 2.170658 1.662775 1970 2.251 133 1.681052 1971 2.322801 1.718610 Table. ;12. Aggregate Data:Divisia P r i c e Indexes and Derived Quantities P r i c e Indexes of Four Aggregate Commodities 1947 1 .000000 1. 000000 1.000000 1 . 000000 1948 1 . 155746 1 . 094661 1.187048 1. 142161 1949 1. 180842 1 . 149130 1.247614 1 . 195605 1950 1.207903 1. 187901 1.253374 1. 254715 1951 1 .344193 1 . 289373 1.401637 1 . 463231 1952 1.356001 1. 359583 1.427419 1. 513231 1953 1.313522 1 . 401030 1.425402 1. 535293 1954 1.312671 1. 452432 1. 408320 1. 555580 1955 1 .312407 1 . 492293 1.404863 1 . 564382 1956 1.324864 1. 552717 1.440248 1. 619767 1957 1.368874 1 . 6261 10 1.468137 1 . 709538 1958 1.398994 1 . 696272 1.484159 1. 719861 1959 1.404848 1 . 745187 1.505738 1 . 758936 1960 1 .409831 1 . 785687 1.515152 1. 811110 1961 1.416471 1 . 824382 1.529329 1. 833676 1962 1.435130 1 . 870795 1.558368 1. 839701 1963 1 .455336 1 . 919860 1.599509 1. 862826 1964 1.470915 1. 985205 1.633500 1. 901105 1965 1.500942 2. 072625 1 .662881 1. 959925 1966 1.565001 2. 172938 1.733846 2. 070865 1967 1.589085 2. 312360 1.817527 2. 202245 1968 1.660527 2. 447455 1.898603 2. 313766 1969 1.709569 2. 597423 1.971901 2. 435750 1970 1.743464 2. 748494 2.014667 2. 532360 1971 1.774891 2. 929061 2.026052 2. 605339 T t " " ' - . ''V .-'nr • Aggregate Quantities 1947 429.515137 219. 359436 177.752365 392. 644775 1948 413.929932 219. 273026 180.588120 409. 978027 1949 403.719238 225. 322922 177.395416 422. 818848 1950 420.432861 225. 614319 180.103378 452. 097168 1951 419.121094 227. 694901 177.354599 467. 029053 1952 430.175781 234. 923676 177.890137 478.233398 1953 439.660400 243. 185211 180.957291 499. 226318 1954 459.904785 242. 140549 181.392776 516. 577393 1955 478.302490 249. 394440 186.337204 548. 529297 1956 502.943359 260. 804443 193.549881 580. 238525 1957 512.940918 263. 506836 195.797531 594. 178223 1958 516.891113 263. 959717 198.169266 615. 622559 1959 529.630371 275. 724854 201.823639 636. 565186 1960 539.262207 285. 017090 203.118500 649. 352295 1961 540.377197 295. 805420 204.394287 661. 723877 1962 550.238525 302. 471191 206.493469 676. 999268 1963 560.833008 308. 948242 207.729141 694. 485107 1964 573.913818 318.576904 211.340698 715.754883 1965 586.756348 332. 167725 215.586197 739. 183838 1966 588.566406 340. .820557 219.393539 756. 107422 1967 602.408447 355. 885742 222.824234 767. 708496 1968 595.349121 360. ,015625 226.102386 784. 760498 1969 606.279541 365. 209717 230.328796 805. 382813 1970 623.329346 365. ,53 80 86 230.555862 812. 836914 1971 644.349609 362. 774658 237.046799 837. 164551 102 Table 13. D i v i s i a Indexes f or the Aggregate Data i n Table 12 YEAR PRICE INDX QUANT INDX PRICE INDX QUANT INDX DIVISIA DERIVED DERIVED DIVISIA 1947 1.000000 1.000000 1.000000 1.000000 1948 1.144904 1.003656 1.144906 1.003655 19 49 1.190132 1.007858 1.190137 1.007854 1950 1.227448 1.048273 1.227459 1 .048265 1951 1.385098 1.059278 1.385110 1.059269 1952 1.422955 1.083776 1.422976 1 .083759 1953 1.424330 1.118598 1.424359 1.118575 1954 1.438288 1. 148699 1.438324 1.148670 1955 1.447865 1.200778 1 .447910 1.200741 1956 1.487486 1. 262590 1.487538 1.262547 1957 1.551663 1.286469 1.551727 1.286416 1958 1.579076 1.310831 1.579151 1.310769 1959 1.606820 1.352020 1.606905 1.351950 1960 1.636521 1.379709 1.636613 1 .379632 1961 1.655682 1.402401 1.655779 1.402318 1962 1.675395 1.430805 1.675502 1 .430714 1963 1.704096 1.460858 1.704212 1.460758 1964 1.739234 1.500787 1.739362 1.500677 1965 1.790327 1.546746 1.790463 1.546628 1966 1.879555 1.574332 1.879705 1.574206 1967 1.973016 1.611164 1.973186 1.611026 1968 2.07220 1 1.628585 2.072389 1 .628437 1969 2. 170650 1.662871 2.170860 1.66271 1 1970 2.251121 1.681156 2.251348 1.680987 1971 2.322780 1.718725 2.323023 1.713546 Table 14. Aggregate Data:Divisia Quantities and Derived P r i c e Indexes ' Year, P r i c e luaexes ~ofx Four Aggregate Commodities 1947 1.0G0000 1 .ccoooo 1.000000 1.000000 1948 1.155748 1.094661 1. 187040 1. 142163 1949 1.180847 1 .149130 1.247601 1. 195566 1950 1.207916 1. 187904 1.253358 1.254X24 1951 1.344225 1 .289381 1.401620 1.463128 1952 1.356037 1.359588 1.427403 1.513128 1953 1 .313563 1.401035 1.425386 1.535190 1954 1.312681 1.452432 1.408310 1.555287 1955 1.312416 1 .492296 1.404854 1.564094 1956 1.324880 1.552720 1.440241 1.619473 1957 1 .368897 1 .626121 1.468135 1.709229 1958 1.399024 1.696278 1.484159 1.719555 1959 1 .404881 1.745200 1.505743 1.758630 1960 1.409873 1.785704 1.515158 1.810802 1961 1.416519 1 .824397 1.529337 1.833373 1962 1.435184 1.870811 1.558380 1.839401 1963 1.455394 1.919881 1.599524 1.862534 1964 1.470979 1.985231 1.63 3517 1.900812 1965 1.501017 2.072659 1.662904 1.959637 1966 1.565084 2. 172980 1.733873 2.070567 1967 1.589173 2 .312413 1.817560 2.201938 1968 1.660625 2.447515 1.898640 2.313447 1969 1 .709679 2 .597494 1.971945 2.435420 1970 1.743586 2.748572 2.014716 2.532028 1971 1.775027 2.929150 2.026105 2.605011 Aggregate^ Quantities 1947 429.515137 1948 413.928955 1949 403.717773 1950 -420.427979 1951 419.110840 1952 430.164307 1953 439.646973 1954 459.901123 1955 478.299316 1956 502.937256 1957 512.931885 1958 516.880127 1959 529.617432 1960 539.246094 1961 540.359131 1962 550.218262 1963 56 0.810791 1964 573.888916 1965 586.727295 1966 588.535645 1967 602.374512 1968 595.314453 1969 606.240479 1970 623.285889 1971 644.300293 219.359436 177.752365 392.644775 219.272903 180.589249 409.977051 225.322800 177.397446 422.832520 225.613739 180.105499 452.129639 227.693726 177.356689 467.062012 234.922928 177.892029 478.265869 243.183929 180.959122 499.259766 242.140381 181.394196 516.674072 249.393875 186.338257 548.629883 260.803711 193.550629 580.343994 263.504883 195.797760 594.285400 263.958496 198.169174 615.731934 275.722656 201.823090 636.675781 285.014648 203.117645 649.462402 295.803467 204.393021 661. 83.3C08 302.468506 206.491959 677.109375 308.944824 207.727249 694.594482 318.572754 211.338501 715.865479 332.162354 215.583298 739.292969 340.813965 219.390091 756.215820 355.877686 222.820145 767.816162 360.006592 226.097931 784.868408 365.199707 230.323715 805.492188 365.527588 230.550171 812.943359 362.763916 237.040497 837.269775 Table 15. D i v i s i a Indexes f or the Aggregate Data i n Table 14 YEAR PRICE INDX QUANT INDX PRICE INDX QUANT INDX DIVISIA DERIVED DERIVED DIVISIA 1947 1.000000 1.000000 1.000000 1.000000 1948 1. 144905 1.003655 1.144907 1.003654 1949 1.190120 1.007870 1.190125 1 .007865 1950 1.227422 1.048295 1.227432 1.048286 1951 1.385074 1.059297 1.385086 1.059287 1952 1.422933 1.083793 1 .422954 1.083776 1953 1.424313 1. 118611 1.424339 1 . 118590 1954 1.438190 1.148776 1.438227 1. 148747 1955 1.447772 1.200855 1.447815 1 .200820 1956 1.487393 1 .262669 1.487442 1.262628 1957 1.551574 1.286543 1.551632 1.286494 1958 1.578988 1.310904 1.579056 1.310847 1959 1.606736 1.352091 1.606812 1 .352028 1960 1.636441 1.379777 1.636526 1.379705 1961 1.655605 1.402466 1.655698 1 .402388 1962 1.675323 1.430867 1.675422 1. 430782 1963 1.704026 1.460918 1.704136 1.460824 1964 1.739170 1.500843 1.739293 1.500737 1965 1.790270 1.546796 1.790404 1.546680 1966 1.879499 1.574378 1.879648 1.574254 1967 1.972965 1.611206 1.973133 1.611069 1968 2.072153 1.628622 2.072336 1.628478 1969 2. 170607 1.662905 2. 170812 1.662748 1970 2.251083 1.681184 2.251304 1.681020 1971 2.322747 1.718750 2.322990 1 .718570 V 105 In order to aggregate.with Fisher index numbers we have f i r s t calcu-lated the Fisher p r i c e and quantity indexes by (3.1) and (3.2). for each of the four sets of commodities. Then, by (6.1), we have determined the aggregate q u a n t i t i e s . These are tabulated i n Table 10. Using the aggregate data i n Table 10 we have calculated the Fisher p r i c e and quantity index numbers of Table 11 as i n the case of microeconomic data. Aggregation with D i v i s i a index numbers and D i v i s i a index numbers for the aggregate data: The aggregation with D i v i s i a index numbers i s s i m i l a r to the Fisher aggregation. The exception of course i s that we have used (3.18) and (3.17) to obtaih^qu'antity indexes from p r i c e indexes and p r i c e indexes from quantity indexes re s p e c t i v e l y . In Table 12 we have tabulated the p r i c e indexes P 1 for the aggregate commodity A calculated by (3.16) and the quantities calculated by „i / i t ' i , ., D i X A - ( P A V 1 P A • Using the aggregate data i n Table 12 we have calculated the D i v i s i a p r i c e indexes and the derived quantity indexes as w e l l as the D i v i s i a quantity indexes and the derived p r i c e indexes a l l of which appear i n Table 13. In Table 14 we have tabulated the p r i c e indexes derived by (3.17) aadmtheeaggregate quanti-t i e s calculated by (6.1), both from the quantity indexes calculated by (3.14). Then, using the aggregate data i n Table 14, we have calculated the D i v i s i a p r i c e indexes and dea?ivedithe quantity indexes, and calculated the guantity indexes and derived the p r i c e indexes of Table 15. 6.3 Conclusion One of the most important advantages of the n6rip~a"ramet~r-i-c''tests suggested i n t h i s study should become apparent when i t i s noted that the s i z e of the l i n e a r programs (5.1) and (5.3) i s independent of the number of com-106 madities. The l a r g e r of the two, (5.3), has (N, - 1)N constraints and 2 N + N - 2 v a r i a b l e s , where N i s the number of observations. Even.for moder-ately large number of observations the l i n e a r program remains manageable. There-fore the suggested nonparametric tests are easy to apply. They can be used not only as a test instrument of homogeneous weak s e p a r a b i l i t y but also i n evalu-ating u t i l i t i e s or quantity indexes. I t i s i n t e r e s t i n g to note that, f or the Cummings-Meduna data, when using the microeconomic prices and quantities the nonhomothetic nonparametric indexes are extremely close to both Fisher and D i v i s i a indexes. The homothetic nonparametric indexes are also very close to the others probably with the exception of the f i r s t few observations. The same degree of closeness between the nonparametric and the other index numbers has not been achieved with the laggrdg-ate (datat,owh^ehv:islpirbb;ablyoidue3.jt03ithelyiolations however small. 107 References A f r i a t , S.N. [1967], "The Construction of U t i l i t y Functions from Expenditure Data," International Economic Review, VIII (January, 1967), 67-77. A f r i a t , S.N. [1972], " E f f i c i e n c y Estimation of Production Functions," In t e r n a t i o n a l Economic Review, XIII (October, 1972), 568-598. Arrow, K.J., H6Bs@henery, B.S.Minnas, and R.M.Solow [1961], "Capital-Labor S u b s t i t u i t i o n and Economic E f f i c i e n c y , " Review of Economics and S t a t i s t i c s , 63 (1961), 225-250. Berge, C. and A.Ghouila-Houri [1962] , Programming, Games and Transportation NetWork, London: Methuen and Co. Ltd., New York: John Wiley and Sons Inc. Buscheguennce, S.S. [1925], "Sur une classe des hypersufaces: a propos de ' 1' index id£al'de M. Irv. Fischer," Recueil Mathematique (Moscow) 32, 625-631. Berndt, E.R. and L.R.Christensen [1973], "The Translog Function and the S u b s t i t u i t i o n of Equipment, Structures, and Labor i n U.S. Manu-factu r i n g 1920-68," Journal of Econometrics, VI (March 1973). Cobb, C. and P.H.Douglas [1928]] "A Theory of Production," American Economic Review 18, 139-165. Christensen, L.R. , D.W.Jorgenson and L.J.Lau [1971], "Conjugate Duality and the Transcendental Logarithmic Production Function," Econometrica, 39 (1971), 255-256. Christensen, L.R. , and D.W.Jorgenson [1969], "The Measurement of U..S. Real Ca p i t a l Input, 1920-67," Review of Income and Wealth, Series 15, December, 293-320. Christensen, L.R. and D.W.Jorgenson 1970 , "U.S.RReal Product and Real Factor Input, 1920-67," Review of Income and Wealth, Series 16, March, 19-50. 108 Cummings, E.D. and L.Meduna [1973], "The Canadian Consumer.Accounts," Research Projects Group, Department of Manpower and Immigration, Ottawa. Diewert, W.E. [1971] , "An A p p l i c a t i o n of the Shephard Duality Theorem: A Gene-r a l i z e d Leontief Production Function," Journal of P o l i t i c a l Economy, 79 (1971), 481-507. Diewert, W.E. [1973a], "Fu c t i o n a l Forms f o r P r o f i t and Transformation Functions," Journal of Economic Theory, VI (June, 1973), 284-316. Diewert, W.E. [l973b], " A f r i a t and Revealed Preference Theory," The Review of •Economic 'Studies-,- XL (July, 1973), 419-425. Diewert, W.E. [l973c], " S e p a r a b i l i t y and a Generalization of the Cobb-Douglas Cost, Production and I n d i r e c t U t i l i t y Function," Research Projects Group, Department of Manpower and Immigration, Ottawa. Diewert, W.E. . [l973d], "Applications of Duality Theory," Research Projects Group, Department of Manpower and Immigration, Ottawa. Diewert, W.E. [l973e], "Harberger's Welfare Indicator and Revealed Preference Theory," Tech. Report 104, IMSSS, Fourth Floor, Encina H a l l , Stanford Un i v e r s i t y , August, 1973. Diewert, W.E. [l973f], Mimeographed Lecture Notes on Aggregation and Index Number Theory, October, 1973. Diewert, W.E. [l974l, "Homogeneous Weak Se p a r a b i l i t y and Exact Index Numbers," Tech. Report 122, IMSSS, Fourth Floor, Encina H a l l , Stanford Univer-s i t y , January, 1974. F a r r e l , M.J. [1957], "The Measurement of Economic E f f i c i e n c y , " Journal of The Royal S t a t i s t i c a l Society, CXX, (Part 3, 1957), 253-281. Fisher, I. [1911], The Purchasing Power of Money, London: Macmillan. Fisher, I. [19221, The Making of Index Numbers, Boston: Houghton M i f f l i n . F-r-i^eh^ R>.. [1930] , 109 F r i s c h , R. [1936], "Annual Survey of General Economic Theory: The Problem of Index Numbers," Ecoriometrica 4-, 1-39. Kloek, T... tl'967], "On Quadratic Approximation of Cost of L i v i n g and Real Income Index Numbers," Report 6710, Econometric I n s t i t u t e , Netherlands School of Economics (mimeo). Konyus, A.A. and SySsByushgens [1926], "On the Problem of the Purchasing Power of Money," The Problems of Economic Conditions (supplement to the Economic B u l l e t i n of the Conjuncture I n s t i t u t e ) 2_, 151-171. Leontief, W.W. [1941], The Structure of the American Economy 1919-1929, Cambridge, Mass.:;;: Harvard U n i v e r s i t y Press, 1941. Malmqvist, S. [1953], "Index Numbers and Inference Surfaces," Trabajos de E s t a d i s t i c a , 4, 209-242. Nerlove, M. [1961], "Returns to Scale i n E l e c t r i c i t y Supply," Tech. Report 96, IMSSS, Serra House, Stanford U n i v e r s i t y , May, 1961. Pollak, R.A. [1971], "The Theory of the Cost of L i v i n g Index," Research Discus-sion Papersoion Pape5?fN6?.el<l>f O f f i c e of-dPr^GesgafldnEivingsCondiiitionsva-U. S. . .. Bureau of Labor S t a t i s t i c s , Washington, D.C. Rockafellar, R.T. [1970], Convex An a l y s i s , Princeton Un i v e r s i t y Press, 1970. Samuelson, P.A. and S.Swamy [1973], "Invariant Economic Index Numbers and Canonical Duality: Survey and Synthesis," H.I.E.R. Discussion Paper No. 313, Harvard U n i v e r s i t y , August, 1973. Shephard, R.W. [1953], Cost and Production Functions, Princeton Un i v e r s i t y Press. Shephard, R.W. [1970], Theory of Cost and Production Functions, Princeton Uni-v e r s i t y Press. Sion, M. 1957 , "Sur une ge n e r a l i s a t i o n du theoreme minimax," C.R.Acad., S c i . , P a r i s , 244, 2120, 1957. T h e i l , H. [1965], "The Information Approach to Demand Analy s i s , " Econometrica 33, 67-87. T h e i l , H. [1967], Economics and Information Theory, Amsterdam: North-Holland Publishing Co. T h e i l , H. 11968], "On the Geometry and the Numerical Approximation of Cost of L i v i n g and Real Income Indices," De Economist 116, 677-689. Tornqvist, L. [1936], "The Bank of Finland's Consumption P r i c e Index," Bank of Finland Monthly B u l l e t i n , No. 10, 1-8. U.B.C. LIP [1972,[1974], A Linear Programming Package, Uni v e r s i t y of B r i t i s h Columbia Computing Centre, Vancouver, B.C. Zangwill, W.I. .. [1969] , Nonlinear Programming, P r e n t i c e - H a l l Inc., Englewood C l i f f s , N.J., 1969. I l l APPENDIX I Theorem 1. There ex i s t s a non-decreasing production function f v a l i d f o r observations {(y, x ) 1 , i = 1,...,N} i f and only i f i J T / - is r j 1 „ •, y = f ( x ) = max {y : x eR.} j 2 f o r i = 1,...,N. Proof: Let f(x) be a non-decreasing production function v a l i d f o r observations {(y, x ) 1 , i = 1,...,N}. Thus y 1 = f C x 1 ) , i = 1,...,N and i f x 1 ^ x2 then f ( x X ) >_ fix2) or y 1 >_ y2 . Also by d e f i n i t i o n x 1 >_ x2 i f and only i f x 1eR.. Since J and then y 1e{y J : x XeR.}, i k , i i „ , y > y e{y : x eR..}, y = f ( x ) = max {y : x eR.}, j 2 for i = 1,...,N. On the other hand l e t i jzr i \ - r j i TS i y = f ( x ) = max {y : x eR.}, 3 J k 1 for i = 1,...,N. I f x < x then {y2 : x keRj} S {y2 : xieR^}, by d e f i n t i o n of R.. This i n turn implies that 3 i k i 1 max {y J : x eR.} 4 max {y J : x eR.} 3 J 3 2 QED. Theorem 2. A non-decreasing, quasi-concave production function f v a l i d f o r observations {(y, x ) 1 , i = 1,...,N} e x i s t s - i f and only i f 112 1 r / Is - r j l _ i y = f ( x ) = max {y J : x eT.}, j 3 f o r i = 1,...,N. f(x) so defined i s continuous from above. Proof: Let f(x) be a non-decreasing, quasi-concave production function v a l i d f o r observations {(y, x ) 1 , i = 1,...,N}. Thus y 1 = fCx 1 ) for i = 1,...,N. Now, suppose x 1eT^. By the d e f i n i t i o n of T., x 1 can be written as a convex combination of vectors r belonging to the sets R^, keL , so x l = I a k r k > \ll ak = 1 3 1 1 ( 1 k e L - i keL " keLt Since, by hypothesis, f i s non-decreasing r keS = {x : f(x) >_ f (x j)}, f o r keL_.. Also since f i s quasi-concave, again by hypothesis, S^ i s convex and x 1 = J a, r " e S keL. k 3 k J Thus ieL and fCx 1 ) >_ f(x*) or f f x 1 ) >_ f ( x ) e { f ( x J ) : - ^ T }, y 1 = fCx 1 ) = max {y 3 : x V . } , j J f o r i = 1,.. . ,N. Now, l e t y 1 = f (x 1) = max {{y 3 : x^T.} for i = 1, .. . ,N. k 1 k 3 i 1 Consider observations x and x . I f x <_Jx then {y j : x k E T } £ { y j : x ^ T }, which implies that i k i 1 max {y : x ET.} < max {y J : x eT.}. k 1 In order to prove quasi-concavity of f, assuming that x eV(y) and x"eV(y) k 1 where V(y) = ^ {x : f(x) >>_y} , the convex combination of x and x should be shown to be i n V(y). Now, xkeV(y) then f ( x k ) = max {y 3 : xkeT.} >_'yt 3 3 and x 1 E V ( Y ) then fCx 1) = max {y 3 : x1eT.} 4 >.y. j J k k 1 1 k 1 Let f ( x ) = y and f ( x ) = y with corresponding Tfc and 1 . I f y = y = y k 1 then T = T = T and A X + (1 - X)x E T , 0 4 X 4 1, and by monotonicity f [ X x k + (1 - X)x 1] = y > y. k 1 k 1 If y > y then T f c c and Xx + (1 - X)x eT 1 which implies f [ X x k + (1 - X)x 1] > y 1 > y. By Theorem 1 of Section 2.1 U(y)={x : f(x) 4 y} must be closed for every y e R i n order f o r f(x) to be continuous from above. I t can be assumed, N i i 1 N without loss of generality, that y 4 y , y J 4 y , j = 2 N - l . If y > y 1 N then U(y) = cp,. the n u l l set, which i s closed by d e f i n i t i o n . I f y < y < y there i s some k, k = 1,...,N, such that U(y) = T^ the closedness of which cannot be contradicted by the f i n i t e set of observations {x : xeT^}. QED. Lemma '1 y 1 = max {y 3 : x 1eT.} i f and only i f x 1eT. implies that 3 2 2 1 3 • 1 vr y 4 y , 1 = l,...,N. Proof: Let y 1 = max {y 3 : x 1eT.}. i 3 k i i i Cle a r l y i f x eT then y e{y : x eT.}. Since y i s greater than or equal to k J k i i i any such y then x eT_. implies that y _. y . On the other hand, consider the set A. = {y 3 : x 1eT.}. i l 3 114 Since x ^ X » y 1 ^ . I f x^T.. implies that y 1 _ y J then y 1 _ i yeA.j^ and 1 r j i -i y = max {y J : x eT.}. 3 3 ^ Corollary 1 x eT^ implies y >_ y J i f and only i f y 1 < y3 implies that x1eT?^';');i = 1.....N. 3 Proof: Elementary, by contradiction. Theorem 3. Let T = / U R.N, J c {1, ,N} . There e x i s t s at \ j e j / l e a s t one vector x' i n the set x Sj = {x : x = JX kx k, _ X f c = 1, X k > 0, keJ}), such that x' < x" i f and only i f x"eT. Proof: By the d e f i n i t i o n of convex h u l l , DJe-fani'tiion 2 of Section 2.4, any vector i n T can be written as a convex combination of the vectors x^eR., 3 j e J . 1 2 1 2 Now, l e t x" = Xx + (1 - X)x . I f x eR., x eR. then since R. i s 3 J 3 i 1 2 a convex set x"eR. and thus x J < x". I f , on the other hand, x eR., x eR, , 3 3 & 1 i 2 k keJ, then one can write x = x + u, and x = x + v where u >_ 0 and v >_ 0. Also x" = X(x j + u) + (1 - X)(x k + v) = Xxj + (1 - X)x k + Xu + (1 - X)v. Thus x" >_ Xx3 + (1 - X)x k = x'eSj. Let x' = £ x x k , J x k = 1, X k >_ 0, keJ. Suppose x' 4 x" f o r some x". I f xxx + u = x" then u ^ 0 and x' + u = Y x i x k + u = >.J >\, ( x k + u) = x". , T k k keJ keJ I t i s clear however that (x"1 + u ) e E J f ° r any j e { l , . . . ,N} as long as u > 0. i c Tj i s the set complementary to T^ Hence x" i s a convex combination of vectors i n R , j e J , and i t i s i n T, again by the d e f i n i t i o n of convex h u l l . QED. Corollary 2 Let T =( U R. ) , Jc {1,••.,N}. \ j e J V There does not e x i s t a vector x'eS*. = {x : x = £x,x , \\, =1, X, 4 0, keJ} J K K . K. such that x' 4 x" f o r some x" i f and only i f x"eT C. Proof: Elementary, using Theorem 3, by contradiction. Theorem 4. I f x > 0, k = 1 N, min {y1 : Jx .x3 - ]x±x~ 4 0, Jx. = 1, X. 4 0, jeM., y 1 4 0} > 1, y,. X i f and only i f y 1 < y 3 implies that x l e T ^ f o r i = l y ...,N-1. 2 i i i c Proof: SSq^p.oseayy.,,<i^ia$]i^d:es'efehat jkyeT^sttBy • the d e f i n i t i o n of i k k i c M., y < ye{y : keM.}. Let y = min {y : keM.}. C l e a r l y x eT which by the 1 1 P . k 1 P i c d e f i n i t i o n of T. implies that x eT, f o r any jeM.. By Corollary 2 there does 3 P 1 not e x i s t a vector x' i n S. = {x : x = Tx.x3, Yx. .= 1, X. > 0, jeM.}, i L 3 L 3 ' j = ' J 1 ' such that x'4 x 1. I f y'Sc1 4 x' even the smallest y 1 > 1 regardless of the values of X. > 0, ieM.. 3 = 1 Now, i f min {y1 : Ix^x3 4 y ^ 1 , = 1, X.. 4 0, j e M ^ y 1 4 0} > 1, Pv X then there does not e x i s t a convex combination x' of x 3, jeM 1, such that x' 4 x 1 because otherwise there would be a f e a s i b l e y < y 1. Thus, by i c Corollary 2, x eT.. 3 QED. ? N N-l 1 Rec a l l that i t i s assumed that y 4 y 4 ... 4 y . 116 k C o r o l l a r y 3 I f x >=0, k = 1,...,N, there e x i s t s a non-decreasing, continuous from above, quasi-concave production function f v a l i d f o r the obser-vations {(y, x);, i = 1,...,N} i f and only i f (1) min {u 1 : I V - * 3 " V±^L < 0, J A ^ = 1, A . > 0, jeM_^, y 1 4 0} > 1, y^AA for i = 1 , . . . j N j l l s necessary and Proof: By Theorem 2 y 1 = fCx 1 ) = max {y 3 : x 1eT.}, 3 2 ' f o r i = 1,...,N, i s necessary and s u f f i c i e n t f o r the existence of a non-decreasing quasi-concave production function f v a l i d for the observations {(y, x ) 1 , i = 1,...,N} and f(x) i s i t s e l f a continuous from above function. By Lemma 1 y 1 = max {y 3 : x l e T j } i f a n ( i only i f x l e T j implies that y 1 4 y 3 . By Corollary 1, however, x l e T j implies that y 1 4 y 3 i f and only i f y 1 < y 3 i c implies that x ET^. Thus Theorem 4 establishes the proof. Q E D . k C o r o l l a r y 4 I f x 4 0, k = 1,...,N, there e x i s t s a non-decreasing, continuous from above, quasi-concave production function f v a l i d f or the obser-vations {(y, x ) 1 , i = 1,...,N} i f and only i f (2) max {y 1 : w^x 1 4 1, w 1 1 :x 3 - y 1 > 0, jeM., w 1 > 0} > 1, i i Y,w for i = 1,...,N-1. Proof: The dual of the Linear Program (1) i s that of minimizing the Lagrangean £ = - y 1 + w l t ( y i x 1 - I A x 3 ) + X i(. I A - 1) + J w 6 A + By 1 jeM. jeM. ieM1. 1 J 1 J 1 117 over w1, y 1, <S , j e M ^ and 3 subject to i t i , „ w x +3 = 1, w l t :x J - Y 1 - cS = 0 , jeM ±, w1 _L 0, 6 ^ 0, jeM i, 3 _L 0. Since 3"_L0, 6^ 2L0, jeM^, t h i s can e a s i l y be put i n t o the form of (2). QED. Theorem 5. There e x i s t s a non-decreasing, continuous and concave production function f v a l i d f o r the observations {(y, x ) 1 , i = 1,...,N} i f and only i f (3) y 1 = fCx 1) = max {^.y j : ^A.x 3 < x 1, [ l . = 1, i . > 0, j = 1,. ..,N> A 3 3 3 3 f o r i = 1,...,N. Proof: Suppose there e x i s t s a non-decreasing concave production function f v a l i d f o r the observations {(y, x ) 1 , i = 1,...,N}. By the d e f i n i -t i o n of a non-decreasing function N . N . . N I X . x i 4 x 1 implies that f(£ A.x 3) 4 fCx 1) f o r J A. = 1. j = l 3 j = l 3 j = l J Also by the d e f i n i t i o n of concavity N . N . I A f ( x J ) < f ( _ A x J) < f C x 1 ) . j = l J j = l J From the v a l i d i t y of f at (y, x ) 1 , f ( x X ) = y 1 , i = 1,...,N. Thus N N N y 1 >, max { J A .yJ : £ A .xJ 4 x 1, £ A . = 1, A. ^ 0, j = 1,... ,N}. A j = l 3 j = l 3 j = l 3 3 But since ie{l,...,N} the equality holds i n the^'above. On the other hand l e t fCx 1) be defined as i n (3). f C x 1 ) , so defined, i s a l i n e a r function of x 1 because i f the optimal b a s i s f o r the l i n e a r program i n (3) i s B then 118 fCx 1) = ( B ^ x V y . Now consider two r e a l i z a t i o n s x 1 and x 1 of x 1 such that f(x^) and f(x^) e x i s t . Let x 1 4 x 1, thus x 1 = x 1 + r, r 4 0. The defining linear.programs are N N . . . N f (x..) ~ max A.y J .: £ A ,xJ + Ax 1 < x|, £ A. = 1, A. 4 0, j = 1,... A j = l J j - ' l J 1 1 j = l J J N . N . N f ( x 1 ) = max {£ A.y J : £ A.xJ + Ax 1 + Ar < x 1 + r, ]> A. = 1, A. >. 0, A j = l 2 3=1 2 1 j = l J 2 mi 3=1,...,N}. (4) (5) Since 0 < A . < 1, A . r < r thus any f e a s i b l e s o l u t i o n of (4) i s a f e a s i b l e = J = 3 = s o l u t i o n of (5). Consequently fCx 1) > f ( x | ) . The non-decreasing, concave function f whose value at x 1 i s y 1 , i = 1,...,N, i s also continuous from above because the closedness of C(y) = {x : f(x) >>y} for any yeR cannot be contradicted by empirical data. C(y) i s eitherxempty, whenayy>'ymax {y~* : j = 1 N}, or there e x i s t s an j observation, say the k th observation, such that C(y) = C(y k) = {x : f(x) > y k} = {x k + 1,...,x N}, which i s closed. On the other hand since a concave function i s always con-tinuous from below over the i n t e r i o r of the domain of d e f i n i t i o n (Rockafellar [1970, pp. 51, 84]) f defined as i n (3) i s continuous. QED. Theorem 6. There e x i s t s a non-decreasing, continuous, concave pro-duction function f v a l i d f o r the observations {(y, x ) 1 , i = 1,...,N} i f and only i f there e x i s t s an (1 + f.) dimensional vector ( p 1, w 1) and a s c a l a r y 1 such that i f i * i y = max Y i i i Y , P , w 119 subject to i i t . * i ' r k i y y i - y 1 > 0, k = 1, -i k k 4 i -x -x (6) , i i t . , £3)' . (p , w ) 13 = 1, p 1 ^ 0, w1 >_ 0, i * Y >_ 0, f o r i = 1,...,N. Proof: The addition of the constraint i i t , r i - i r k . y y -i k -x -x I 4 > - Y > 0, to the constraints of (6) has the e f f e c t of fo r c i n g Y 1 to have a non-positive value. This, i n turn, implies that y** 4 0 or that the condition y 1 * <_ 0 i s i * reduced to y = 0. (6) can be rewritten as i * _ i Y = max Y i i i Y > P > w i * subject to i i t , r 1 r k _ y y - ) i k -x -x j ) - Y > 0, k = 1,...,N, (7) (p 1, wlt:) 1 = 1 p 1 >_ 0, w1 >^ 0. The dual of (7) i s that of minimizing the Lagrangean 1 = (1,...,!)- with the appropriate dimension. 120 L = Y 1 + y X [ l - ( p \ wlt:) 1] + I X v[;(p\ w X t) k=l - Y 1] + (P, w l t : ) 6 over 6 , y , and X, subject to r 1 i r k -i * y y • i k -x J (8) ( i ) 1 - I A, = 0 k=l ( i i ) 6 - y \ l + 1 X k=l fc r i i r k i y y i k . -x . - ~x - o, ( i i i ) « > 0, U 0 . M u l t i p l y i n g ( i ) by y 1 , ( i i ) by ( p 1 , w1) from the r i g h t , disregarding the non-negative 6 , and s u b s t i t u t i n g them i n the Lagrangean the dual problem i s transformed into the following: subject to (9) min y y 1 , XX I K(*x1T y k ) k=i k N I Xk(y.'± -0) < y 1!, k=l N 0. I k=l X > 0. v , / k i . i , X k(x - x ) < y 1, However (9) can be rewritten as: mm y i , y , X subject to N k=l k = 1, 121 N , ... . N (10) I \ y > -y1 + y 1 I A K , k=l k=l N . N I \ x < y1! + x 1 J A,, k=l K k=l k A >. 0. Remembering that yX < 0 (10) can be put i n t o the following form: I min y i , y , A N subject to I A, = 1, 1 V k l y - z. \y i n i k=l N ' - I k=l N . V , k l 2, A, x < x , k=l k A 4 0, which i s equivalent to: N . i v , k • min y - I A, y= l s A k=l k N 5 N k , l . subject to I AK = AMX*k* X<'x\ k=l " k , : 5 : Lk=l x >p, xi°» whose objective function i s to be zero, by the dual theorem of l i n e a r program-ming, at an optimal s o l u t i o n . Disregarding the constant i n the objective function and converting the problem to that of maxi'Mvza'tfron the proof i s complete by Theorem 5. QED. 122 Corollary 7 There e x i s t s a non-decreasing, concave and l i n e a r homogeneous production function f v a l i d f o r the observations {(y, x ) 1 , i = 1, ...,N} i f and only i f there e x i s t and (r+1) dimensional vector (p 1, w 1) and a scala r 6 1 such that: i * i y = max Y i i i Y , p , w r i - l y r v i -i y subject to . x xt (p , w ) < i -x -k -x i , k + i (11) ( p 1 , w X t ) l = 1, i i i t i _ p y - w x =0, 1 n 1 n p >_ 0, w >_ 0, i * Y ^ 0, f o r i = 1,...,N. Proof: Obvious by the d e f i n i t i o n of l i n e a r homogeneity, D e f i n i t i o n 5, Section 2*1, Euler's Theorem, Section 2.6, and Theorem 6. Theorem 7. There e x i s t s a non-decreasing, concave and l i n e a r homo-geneous production function f v a l i d f or the observations {(y, x ) 1 , i = 1,...,N} i f and only i f N . N y 1 = f (x 1) = max £ A .y"3 : £ A .xJ <•> x 1, A ^ 0}, j = l 3 3=1 3 f o r i = 1,...,N. Proof: The dual of (11) i s that of minimizing the Lagrangean N = Y 1 + ( p \ w 1 1 : ) 6 + y X [ l - (p 1, w l t : ) l ] + la [ ( p 1 , .w1') k = l K k^Ji-i i y y y - i , . A / i i t . • - T ] + 6 (P , w ) i k i -x _-x -x 123 over 6 , y , a and subject to: N 1 -I \ = o, k=l fc k * i i ' y " r k - i y • r i n y N; N 6 - u lk+ • £ ak '< k=l- k=l k=i k-tl i -x — k -x 4 >+ e i -x = o, 6 > 0, a ^ 0, or equivalently min- y 1 J", xi e subject to (12) N j I a i y k = i k 9y i -y + y , N k Yd, x k = l k k ^ i Ox <. x + y 1, L e t t i n g a > 0. \ ~ 62' °1 = °' 92 = °' a n d B = ^ a i ' - ••»ai_i''',ei» a 1 + 1 » " ' » a N ) (12) can be written i n terms of y 1, 3 and 6^ as mm subject to N , (1 + e j y 1 - J 3 ,y* < y 1 2 k=l k 124 (13) J / ] / X * 1 + d + K.— J. 3 > o, e 2 4 0. By the dual theorem of linear programming (13) has an optimal solution ( y , 3 , 9,, ) such that y 4 0 i f and only i f y 4 0 in (11) . When the constraint y 1 4 0 i s appended to (13) the preceding statment i s mod-i * i * if i e d as y = 0 i f and only i f y > 0, and (13) is rewritten as follows: N , max I 3 vy - (8 + Dy 1 3, 6 9k=l K subject to N I 3, xK - (e + Dx1 < 0, (14) k=l k 2 340, e 2 > 0, Since + 1 > 0 (14) can be written as: N k i . -ax (e + i ) d | g | - f = # ) 3, e 2 ^ k=i©2+i (15) subject to N #T3X r x , . 3k ~ - 4 0, k = 1.....N, e 2 4 0. This last problem has an optimal solution at which i t s objective function is i * equal to zero i f and only i f y 4 0 in (11). On the other hand since 02+l > 0 N 3, • (e 2 + D (I e+T y - y1) = 0 k=i 2 125 v Bk k i 3. i f and only i f ) _ y = y . Thus with the transformation X . = k = l G 2 + 1 k e 2 + 1 k = 1,...,N, (15) can be put into the following form: i N k f(x ) = max J A.y X k=l k subject to N V , k l I X.x <x , k=l k A ^ 0, and f(x 1) = y 1 i f and only i f y1 =0 in (11). QED. 126 APPENDIX II In t h i s appendix we have s i x computer programs a l l coded i n FORTRAN IV. The f i r s t f i v e arrange the data for the te s t s Test l a , Test 2a, Test 2b, Test 3a, Test 3b and the l a s t one i s a l i n e a r programming code which c a r r i e s out the te s t s . 1. Preparation of the Data The data are prepared exactly the same way for a l l the tests except for an a d d i t i o n a l parameter i n Test l a and stored i n a f i l e . Let us c a l l t h is f i l e DATA where the numbers representing the input and output quantities are stored. I f the number of f a c t o r s i s N l and the number of observations i s N2 then they are stored i n DATA as follows: J = 1 Store observations of Jth input from 1 to N2 into f i l e DATA with /FORMAT (4E20.6) F i r s t transformed inputs then^ r e a l inputs. J = J+l Store output observa/; tons vations 1 to N2 into, f i l e DATA with FORMAT (4E20.6) (1) In case of multiple output t h i s number i s the sum of the number of i n -puts arid the number of outputs minus one. As mentioned elsewhere those outputs transformed into inputs should a l l have negative signs. 1 2 7 Once the observations are recorded as s p e c i f i e d above then the follow-ing steps are ca r r i e d out. (a) Choose Program l a for Test l a Program 2a for Test 2a Program 2b f o r Test 2b Program 3a f o r Test 3a Program 3b for Test 3b (b) Run the program chosen i n (a) with a parameter card containing N l , the number inputs (including the outputs that are considered as inputs i n case of multiple output), N2 , the number of observations, N , the index of the observation which i s the f i r s t one (2) of the data segment considered , N10, the index of the observation which i s the f i r s t one of (3) a ser i e s of observations >tblbe tested » N12, the index of the observation which i s the l a s t one of (3) a ser i e s of observations to be tested , N8 , i s applicable only f o r Test l a and i t i s the number of Out-puts fottmulSi^l-g^prddil'Gt, 1 f or si n g l e product case, N5 , i s 1 i f objective function c o e f f i c i e n t ranging i s required from the l i n e a r program, 0 otherwise* N6 , i s 1 i f r i g h t hand side ranging i s required from the l i n e a r program, 0 otherwise.} (2) For example, the data may have 40 observations. I f one wants to use only the l a s t 24 of them the data segment of i n t e r e s t consists of observations 17,18,...,40 with N=17. (3) For example i f one wants to test only observations 25 and 26 then N10=25 and N12=26. 128 TOL, i s the tolerance l e v e l which i s t y p i c a l l y set equal to IO" 5. These quantities must be printed with FORMAT] (8I4.F20.12) f or Program l a and FORMAT (7I4,F20.12) f o r the r e s t . There are, however, some r e s t r i c t i o n s to the values of the above parameters that should be observed. (i) 1 < N < N2 - 1 < 100 f o r Programs 2a, 2b, 3a and 3b, 1 | N.< N2 - 2 < 100 for Program l a ( i i ) Nl< 50 for a l l t e s t s . Naturally N< N10 < N12. Also Program l a assumes that the output, actual or transformed, i s not equal at two d i f f e r e n t observations. I f t h i s assumption i s ever v i o l a t e d i t can be corrected by making the equal outputs _g d i f f e r e n t as l i t t l e as 10 . When running the required program the input unit 1 i s assigned to f i l e DATA and the output unit 7 i s assigned to another f i l e which we c a l l (4) LPDATA . The l i n e a r programming code reads i t s data from LPDATA. (c) Run Program the LP code, assigning the input unit 5 to LPDATA. The output i s printed out. 2. An Example When using Program 3a to test the consistency of Christensen-Jorgenson (1969, 1970) data with monotonicity, continuity and concavity we have followed the steps of the flow diagram below. (4) Depending on the system i t might be more s u i t a b l e to have LPDATA as a scratch f i l e . (5) Program 0 has been adapted from UBC LIP [14 ] . Create f i l e s DATA and LPDATA 1 Store data i n f i l e DATA i n the s p e c i f i e d format. * There are 1 output, 3 inputs, 39 observations. The t h i r d i n -put has minus sign at a l l obser-vations . 1 Run PROGRAM 2A. JL=DATA 7=LPDATA Nl=3, N2=39, N=l, N10=l, N12=39 N5=0, N6=0. , 1 Run Program 0. 5 = LPDATA. 130. $LIST PROG 1 A 1 C 2 C 3 c 4 c 5 6 7 13 8 9 1 0 11 12 13 14 . 1 15 10 16 17 18 19 20 21 22 23 24 5 25 26 27 28 6 29 8 30 31 32 33 34 35 36 7 37 38 39 40 41 42 43 44 45 3 46 47 48 49 50 51 52 53 54 55 PROGRAM 1A. TEST FOR MONOTONICITY AND QUASI-CONCAVITY. DIMENSION E (100,50) #IHD (100) ,KIN (100) ,ABIB (50) READ (5,13) N1fN2,NfN10/N12,N8,N5,N6,TOL FORMAT (8I4,F20.12) N3=1 N4 = 0 N1 1 = NI +1 N15=N2-1 DO 1 J=1,N11 READ (1,10) (E (I,J) ,1=1 ,N2) CONTINUE FORMAT (4E20.6) EPS=0. 1 IF (N8 .EQ. 1) GO TO 8 DO 5 J = 1 , N 1 1 AM IN (J) =E (N , J) NP=N+1 DO 5 I=NP,N2 IF (E(I,J) . GE. AMIN(J)) GO TO 5 AM IN (J) =E (1,3) CONTINUE DO 6 J = 1, N11 DO 6 I=N,N2 E ( I , J) = E ( I , J) -A MIN (J) +EPS CONTINUE A=1. &M=-1. AZ = 0. MM=0 DO 7 ID=N,N2 IND(ID) = ID KIN (ID) =ID CONTINUE DO 4 J=N,N15 C=E(J,N11) K=J IJ=J+1 DO 3 I = IJ,N2 IF (E (I,N11) . GE. C) GO TO 3 C=E(I,N11) K=I CONTINUE IF (K . EQ. J) GO TO 4 II=KIN(K) IK=KIN(J) IND(.II)-J IND(IK) = K KT=KIN (K) KIN (K) = KIN (J) KIN (J) =KT DO 2 M=1 , N1 1 T=E (J,M) 56 E (J,M) = E (K, M) 131 57 E(K,M)=T 58 2 CONTINUE 59 4 CONTINUE 60 DO 24 K=N10,N12 61 N7=N2- IND (K) +1 62 IF (N7 . EQ. 1) GO TO 25 6 3 WRITE (7,14) K 64 14 FORMAT (» TEST 1 A. REFERENCE OBSERVATION .',12) 65 N 9=N 7-1 66 N13=N7 + 1 67 WRITE (7,16) N11,N7,N3,N4,N5,E6,TCI 68 16 FORMAT (614, F20. 12) 69 J = 1 70 WRITE (7,11) J,N7,A 71 11 FORMAT (2I4,F20.12) 72 DO 21 J=1,N1 73 JJ=J+1 74 DO 20 1=1 ,N9 75 I1=IND (K) +1 76 WRITE (7,11) J J , I , E ( I 1 , J ) 77 20 CONTINUE 78 RT= -E (IND (K) , J) 79 WRITE (7,11)JJ,N7,RT 80 WRITE (7,11) J J , N 13, AZ 81 21 CONTINUE 82 JJ=JJ+1 83 DO 22 I=1,N9 84 WRITE (7,11) J J , I , A 85 22 CONTINUE 86 WRITE (7,11) JJ,N13,A 87 WRITE (7,12) MM 88 12 FORMAT (14) 89 GO TO 24 90 25 WRITE (6,26) K 9 1 26 FORMAT (' THIS TEST IS NOT APPLICABLE FOB OBSERVATION ',12) 92 24 CONTINUE 93 WRITE (6,23) N1,N2,N 94 23 FORMAT (« NUMEER OF FACTORS=' , 14/ 95 1* NUMBER OF OBSERVATIONS= ' ,14/ 96 2« INITIAL OBS ERV ART ION IWDEX= 1, 14 ) 97 STOP 98 END END OF FILE $ L I S T P R 0 G 2 A 1 C P R O G R A M 2 A . T E S T F O R M O N O T O N I C I T Y A N D C O N C A V I T Y . 2 C 3 C 4 C 5 D I M E N S I O N E ( 1 0 0 , 5 0 ) 6 R E A D ( 5 , 1 3 ) N 1 , N 2 , N , N 1 0 , N 1 2 , N 5 , N 6 , T O L 7 1 3 F O R M A T ( 7 I 4 , F 2 0 . 1 2 ) 8 N 3 = 1 9 N 4 = 1 1 0 N 1 1 = N 1 + 1 1 1 D O 1 J = 1 , N 1 1 1 2 R E A D ( 1 , 1 0 ) ( E ( I , J ) , 1 = 1 , N 2 ) 1 3 1 C O N T I N U E 1 f t 1 0 F O R M A T ( 4 E 2 0 . 6 ) 1 5 D O 2 4 K = N 1 0 , N 1 2 1 6 W R I T E ( 7 , 1 4 ) K , E ( K , N 1 1 ) 1 7 1 4 F O R M A T ( • T E S T 2 A . O D T P D T A T R E F . O B S . ' . 1 2 , * I S 1 8 N 7 = N 2 - N + 1 1 9 W R I T E ( 7 , 3 ) N 1 1 , N 7 , N 3 , N 4 , N 5 , N 6 , T O L 2 0 3 F O R M A T ( 6 I 4 , F 2 0 . 1 2 ) 2 1 J = 1 2 2 D O 1 9 I = N , N 2 2 3 I M = I - N + 1 2 4 H R I T E ( 7 , 1 1 ) J , I M , E ( I , N 1 1 ) 2 5 1 9 C O N T I N U E 2 6 D O 2 1 J = 1 , N 1 2 7 J J = J + 1 2 8 D O 2 0 I = N , N 2 2 9 I M = I - N + 1 3 0 W R I T E ( 7 , 1 1 ) J J , I M , E ( I r J ) 3 1 1 1 F O R M A T ( 2 I 4 , F 2 0 . 1 2 ) 3 2 2 0 C O N T I N U E 3 3 I M = I M + 1 3 4 W R I T E ( 7 , 1 1 ) J J , I M , E ( K , J ) 3 5 2 1 C O N T I N U E 3 6 A = 1 . 3 7 J J = J J + 1 3 8 D O 2 3 I = N , N 2 3 9 I M = I - N + 1 4 0 W R I T E ( 7 , 1 1 ) J J , I M , A 4 1 2 3 C O N T I N U E 4 2 I M = I M + 1 4 3 W R I T E ( 7 , 1 1 ) J J , I M , A 4 4 H M = 0 4 5 W R I T E ( 7 , 1 2 ) M M 4 6 1 2 F O R M A T ( 1 4 ) 4 7 2 4 C O N T I N U E 48 WRITE (6,2) N1,N2,N 49 2 FORMAT (• NUMBER OF FACTORS=«,14/ 50 1* NUMBER OF OBSERVATIONS=•,14/ 51 2» INITIAL OBSERVARTION INDEX=',I4) 52 STOP 53 END END OF FILE $LIST PROG2B 1 C PROGRAM 2B. TEST FOR MONOTONICITY AND CONCAVITY. 2 C 3 C 4 C 5 DIMENSION E (100,50) ,D (50) 6 READ (5,13) N1,N2,N,N10,N12,N5,N6,TOL 7 13 FORMAT (7I4,F20.12) 8 N3=1 9 N4=1 10 N7=N2-N+1 11 N8=N1+2 12 N9 = N1 + 3 13 N11=N1+1 14 N13=N1 +4 15 A=1. 16 AM=-1. 17 AZ=0. 18 DO 1 J=1,N11 19 READ (1,10) (E(I,J), 1=1,N2) 20 1 CONTINUE 21 10 FORMAT (4E20.6) 22 DO 24 K=N10,N12 23 WRITE(7, 14) K 24 14 FORMAT (' TEST 2B. REFERENCE OBSERVATION :',I2) 25 WRITE (7,3) N7,N9,N3,N4,N5,N6,TOL 26 3 FORMAT (614,F20.12) 27 J=1 28 WRITE (7,11) J,N8,A 29 11 FORMAT (2I4,F20.12) 30 WRITE (7,11) J,N9,AM 31 DO 21 JK=»,N2 32 IF (JK .EQ. K) GO TO 21 33 J=J + 1 34 DO 20 1=1,N1 35 D(I) =E (K,I) -E (JK,I) 36 WRITE (7,11) J,I,D(I) 37 20 CONTINUE 38 D (Nl 1) =E (JK,N11) -E (K,N11) 39 WRITE (7,11) J,N11,D(N11) 40 WRITE (7,11) J,N8,A 41 WRITE (7,11) J,N9,AM 42 WRITE (7,11) J,N13,AZ 43 21 CONTINUE 44 J=J+1 45 DO 26 1=1,811 46 WRITE (7,11) J, I, A 47 26 CONTINUE 48 WRITE (7,11) J,N13,A 49 MM=0 134 50 WRITE (7,12) MM 51 12 FORMAT (14) 52 24 CONTINUE 53 WRITE (6,2) N1,N2,N 54 2 FORMAT (» NUHBER OF FACTORS=•,14/ 55 1» NUHBER OF OBSERVATIONS31' ,14/ 56 2 » INITIAL OBSERVARTION INDEX=',I^) 57 STOP 58 END END OF FILE JLIST PROG3A 1 C PROGRAM 3A. 2 C TEST FOR MONOTONICITY, CONCAVITY AND LINEAR HOMOGENEITY. 3 C 4 c 5 c 6 DIMENSION E (100,50) 7 READ (5,13) N1,N2,N,N10,N12,N5,N6,TOL 8 13 FORMAT (7I4,F20.12) 9 N3=0 10 N4 = 1 11 H11=111+1 12 N7=N2-N+1 13 DO 1 J=1,N11 14 READ (1,10) (E(I,J),1=1,N2) 15 1 CONTINUE 16 10 FORMAT (4E20.6) 17 DO 24 K=N10,N12 18 WRITE (7,14) K,E(K,N11) 19 14 FORMAT (' TEST 3A. OUTPUT AT REF. OBS. * ,12, 1 IS ',F12. 20 WRITE (7,3) N1,N7,N3,N4,N5,N6,TOL 21 3 FORMAT (6I4,F20.12) 22 J=1 23 DO 19 I=N,N2 24 IM=I-N+1 25 WRITE (7,11) J,IM,E(I,N11) 26 19 CONTINUE 27 DO 21 J=1,N1 28 JJ=J+1 29 DO 20 I=N,N2 30 IM=I-N+1 31 WRITE (7,11) JJ,IM,E(I ,a) 32 11 FORMAT (2I4,F20.12) 33 20 CONTINUE 34 IM=IM+1 35 WRITE (7,11) JJ,IM,E(K,J) 36 21 CONTINUE 37 MM=0 38 WRITE (7,12) MM 39 12 FORMAT (14) 40 24 CONTINUE 41 WRITE (6,2) N1,N2,N 42 2 FORMAT (' NUMBER OF FACTORS=•,14/ 4 3 1» NUMBER OF OBSERVATIONS=»,14/ 44 2« INITIAL OBS ERVARTION INDEX=«,I4) 4 5 STOP 46 END END OF FILE 1 3 5 $LIST PROG3B 1 C 2 C 3 c 4 c 5 c 6 7 8 13 9 10 11 12 13 14 15 16 17 18 19 20 21 1 22 10 23 24 25 14 26 27 3 28 29 30 11 31 32 33 34 35 36 37 38 20 39 40 41 42 43 44 21 45 46 47 48 49 25 50 51 52 53 54 55 26 PROGRAM 3B. TEST FOR MONOTONICITY, CONCAVITY AND LINEAR HOMOGENEITY. DIMENSION E (100,50) ,D(50) READ (5,13) N1,N2,N,N10,N12,N5,N6,TOL FORMAT (714,F20.12) N3=2 N4 = 1 N7=N2-N+2 N8=N1*2 N9=Nl+3 N11 = N1+1 N13=N1+4 A-1. AM=-1. AZ=0. DO 1 J=1,N11 READ (1,10) (E (I, J) ,1=1, N2) CONTINUE FORMAT (4E20.6) DO 24 K=N10,N12 WRITE(7,14) K FORMAT (* TEST 3B. REFERENCE OBSERVATION : *,12) WRITE (7,3) N7,N9,N3,N4,N5,N6,TOL FORMAT (6I4,F20.12) J=1 WRITE (7,11), J,N8,A FORMAT (2I4,F20. 12) WRITE (7,11) J,H9,AM DO 21 JK=N,N2 IF (JK .EQ. K) GO TO 21 J=J+1 DO 20 1=1,N1 D(I)=E (K,I)-E (JK,I) WRITE (7,11) J,I,D(I) CONTINUE D (N 11) =E (JK, N11) -E (K, N11) WRITE (7,11) J,N11,D(N11) WRITE (7,11) J,N8,A WRITE (7,11) J,N9,AM WRITE (7,11) J,N13,AZ CONTINUE J=J + 1 DO 25 1=1,N1 B=-E(K,I) WRITE (7,11) J,I,B CONTINUE WRITE (7,11) J,N11,E(K,N11) WRITE (7,11) J,N13,AZ J-J+1 DO 26 1=1,N11 WHITE (7,11) J,I,A CONTINUE 56 WRITE (7,11) J,N13,A 57 MM=0 58 WRITE (7,12) MM 59 12 FORMAT (14) 60 24 CONTINUE 61 WRITE (6,2) N1,N2,N 62 2 FORMAT (• NUMBER OF FACTORS= *,14/ 63 1« NUMBER OF OBSERVATIONS31 * ,14/ 64 2 » INITIAL OBSERVARTION INDEX=«,I4) 65 STOP 66 END END OF FILE SSIGHOFF 137 PROGRAM 0 C A LINEAR PROGRAMMING CODE. LOGICAL ERR,NOPT EXTERNAL LIPIN,LIP,IIPOOT COMMON /COM 1/1,J,K,L,NCHK,NCHK1,M,N,MP1, NP, C TD,GMIN,PHIMAX,MMP,NP1,KMPP2,MM,NOEJ, C NRHS /COM2/SAVRHS (301),SAVOBJ(300) INTEGER*2 NCHK (301) ,NCHK1 (301) REAL*4 TITLE (20) DIMENSION E (300,300) LOGICAL*1 1ST (244) EQUIVALENCE (TITLE (1), 1ST (1) ) 5 FORMAT (214,F20.12) 10 FORMAT (20A4) 15 FORMAT (614, F20. 12,12) 20 FORMAT(///•PARAMETER CARD ECHO: NUMBER CF INEQUALITIES= »,16/ C24X,« NUMBER OF EQUALITIES=•,I7/25X,1NUMBER OF ACTIVITIES^ *,17/ C30X'TOLERANCE LEVEL= «,F20.12) 2 5 FORMAT ( M •, 20 A4) 30 FORMAT(35X,•MAXIMIZING'///) 35 FORMAT(35X,•MINIMIZING1///) C** READ PARAMETER LIST ILOG=0 ITAB=0 C** READ TITLE CARD 100 READ (5,10,END=200) TITLE WRITE (6,25) TITLE C** READ PARAMETER CARD READ (5,15,END=200)M,N,NP,MM,NOBJ,NRHS,TD,IVICL IF(TD.EQ.O.O)TD=1.E-6 MMP=M-NP WRITE (6,20)MMP,NP,N,TD / V IF (MM. NE. 1) GOTO 110 WRITE (6,30) GOTO 120 110 WRITE(6,35) 120 IF(M.LE.0.OR.N.LE.0.OR.NP.LT.0)GOTO 170 IF(M.GT.300.OR.N.GT.300)GOTO 170 IF(MMP.LT.O.OR.NP.GT.N)GOTO 170 NP1=N+1 MP1 = M + 1 C** CALL THE LIP PACKAGE MAX=300 CALL LIPIN (E, MAX, ERR , IT A E) IF(.NOT.EBR)GOTO 140 WRITE (6,130) 130 FORMAT(»0*** DATA ERRORS... PROCEEDING WITH NEXT PROBLEM') GOTO 150 140 CALL LIP (E,MAX,NOPT,ILOG) IF(NOPT)GOTO 150 CALL LIPOUT (E,MAX) C** RELEASE LP TABLEAU SPACE 150 CONTINUE GOTO 100 170 WRITE(6,180) 180 FORMAT(•0*** INVALID VARIABLE ON PARAMETER CARD*) 190 CONTINUE 1 3 8 READ (5,5,EN D=200) I , J ,X IF(I.GT.O) GOTO 190 GOTO 100 200 WRITE(6,210) 210 FORMAT(•0*** END-OF-DATA REACHED'/) 220 STOP END SUBROUTINE LIP (E,MAX,OPT,ILOG) C THE TABLEAU IS STORED IN ARRAY E. C MAX PROVIDES FOR OBJECT TIME DIMENSIONING OF E C NUMBER OF CONSTRAINTS = H C LAST NP CONSTRAINTS ARE EQUALITIES C NUMBER OF ACTIVITIES = H C TD IS THE TOLERANCE C OPT IS SET .TRUE. IF OPTIMIZATION FAILED TO OCCUR C ILOG=1 IF ITERATION LOG IS TO BE PART OF OUTPUT C LOGICAL OPT DIMENSION E (300,300) INTEGER*2 NCHK (301),NCBK1(301) COMMON/COM1/1,J,K,L,NCHK,NCHK1,M,N,MP1,NP,TC,GMIN,PHIMAX,MMP,NP1, XMMPP2,MM,NOEJ,NRHS/COM2/SAVRHS(301) ,SAVOBJ (300) C0MM0N/C0M3/IFEAS,I0PT 5 FORMAT('PHASE I I BEGINS *) 7 FORMAT(/•ITERATION LOG:'/ C'ITERATION',5X,'OBJ. FUN.»,8X,•VAR. IN VAR. OUT'/) 8 FORMAT(* FEASIBLE') C C MAXIMIZATION OR MINIMIZATION ? C SAVE OBJ. COEFF.'S 6 RHS'S FOR RANGING PROCEDURES. C IF(MM-1)180,150,180 150 IF(NOBJ-1)170,160,170 160 DO 165 J=1,N SAVOBJ (J) =E (1 , J) 165 E (1 , J)=-E (1, J) GO TO 3 170 DO 175 J=1,N 175 E(1,J)=-E(1,J) GO TO 3 180 IF(NOBJ-1)3,185,3 185 IF(MMP.NE.M)GO TO 3 DO 190 J=1,N 190 SAVOBJ (J)=-E (1, J) 3 IF(NP.GE.M.OR.NRHS.NE.1)GO TO 99 J=MMP+1 EO 4 1=2, J 4 SAVRHS (I) = E (I,NP1) C 99 OPT=.FALSE. C C SET ITERATION LOG 'ON' OR 'OFF' C ASSIGN 83 TO LOG IF(ILOG.NE.1)GO TO 11 ASSIGN 82 TO LOG WRITE(6,7) C C SETTING UP ARRAY NCHK1. IT RECORDS VARIABLES NOT IN SOL. C 11 CO 12 1=1,N 12 NCHK1 (I)=I C C SETTING UP ARRAY *'NCHK'' . IT RECORDS WHICH VARIABLES ARE IN SOL. C IF (MMP.EQ.0)GO TO 14 L=MMP+1 0=NP1 C C PUT SLACK VARIABLES INTO THE INITIAL SOLUTION C DO 13 1=2,L NCHK (I) =J 13 J=J+1 C C ARE THERE ANY EQUALITIES TO BE PROCESSED? C IF(NE.EQ.0)GO TO 9 14 CO 16 I=MMPP2,MP1 16 NCHK(I)=0 CALL LIP1 (E,MAX,ILOG,S10) C C MINIT PHASE2 BEGINS. C 9 WRITE(6,5) IFEAS=1 C IFEAS IS FEASIBLITY FLAG I0PT=1 C IOPT IS OPTIMALITY FLAG 11=0 C I I IS A SUBOPTIMIZATICN FLAG N1=NP C N1 WILL COUNT ITERATIONS NOW. C C TEST FOR OPTIMALITY,FEASIBIIITY ... IN THAT CRCER. C 15 GO TO (17,25) ,IOPT 17 DO 20 J=1,N IF(E (1 ,3) .GE.-TD) GO TO 20 GO TO 25 20 CONTINUE IOPT=2 C 25 GO TO (26,50) ,IFEAS 26 DO 30 1=2,MP1 IF(E (I,NP1) .GE.-TD) GO TO 30 GO TO 55 3 0 CONTINUE C C FEASIBLE ? C IFEAS=2 WRITE (6,8) C C OPTIMUM SOLUTION OCCURS WHEN IFEAS=IOPT=2 C SEARCH FOR COL. & ROW PIVOTS WHEN IFEAS=IOPT=1 C «• COL. PIVOT ONLY WHEN IOPT=1 S IFIAS=2 C ROW PIVOT •» ••• IOPT=2 8 IFEAS=1 C 140 50 IF(IOPT.EQ.2)RETURN GO TO 70 55 IF(IOPT.EQ.2)GO TO 40 60 CALL LIPC(IM,JMIN,E,MAX,810) CALL LIPR (IMAX,JM,E,MAX,810) IF (GMIN . NE. 1. E6 0)GO TO 61 GO TO 64 61 IF(PHIMAX.NE.-1.E60)GO TO 63 GO TO 72 63 11=1 PHIMAX=PHIMAX*E(IMAX,NP1) GMIN=GMIN*E (1 , JMIN) IF(ABS(PHIMAX) .LE.ABS(GMIN))GO TO 72 64 CALL LIPT(IMAX,JM,E,MAX,810) L=JM K=IMAX GO TO 80 40 CALL LIPR (IMAX,JM,E,MAX,810) GO TO 64 70 CALL LIPC (IM,JMIN,E,MAX,810) 72 CALL LIPT (IM,JMIN,E,MAX,810) L=JMIN K=IM C 80 GO TO LOG, (82,83) 82 N1=N1+1 IF(MM.NE.I) E (1,NP1)=-E (1,NP1) WRITE (6,6) N1,E (1,NP1) ,NCHK (K) ,NCHK1 (L) C C SUBOPTIMIZATION ATTEMPT. C IF(MM.NE.I) E(1 ,NP1)=-E (1 ,NP1) 83 IF(II.NE.1)GO TO 15 11=0 C CAN'T BRING SAME VARIABLE IN OR OOT TWICE IN A ROW IF(JM.EQ.JMIN .OR. IMAX.EQ.IM)GO TO 15 IF(ABS (PHIMAX).LE.ABS(GMIN))GO TO 84 C CHECK TO SEE IF POTENTIAL PIVOT IS CK IF(E (1,JMIN).GE.-TD.OR.E (IM,NP1) .LE.TD.OR.E(IM,JMIN) .LE.TD)GOT015 GO TO 72 C CHECK TO SEE IF POTENTIAL PIVOT IS OK 84 IF(E (IMAX,NP1) .GE.-TD.OR.E (1,JM).LE.TD.OR.E (IMAX,JM) .GE.-TC)GOTO 15 GO TO 64 C 10 GPT=.TRUE. RETURN 6 FORMAT (2X,I4,7X,G15.6,4X,I4,4X,I4) END SUBROUTINE LIPIN(E,MAX,ERR,LPRINT) C LIPIN READS DATA CHECKING FOR INPUT ERRORS ANE SITS UP TAELEAU C ERR IS SET .TRUE. WHEN INPUT ERROR(S) ARE ENCOUNTERED C MM=1 IMPLIES MAXIMIZATION, MM=0 IMPLIES MINIMIZATION C LPRINT=1 CAUSES INPUT TABLEAU TO BE PRINTED C NOBJ=1 CAUSES 'LIPOBJ' TO BE CALLED POST-OPTIMALLY C NRHS=1 CAUSES 'LIPRHS' TO BE CALLED POST-OPTIMALLY LOGICAL ERR COMMON/COM1/1,J,K,L,NCHK,NCHK1,M,N,MP1,NP,TD,GMIN,PHIMAX,MMP,NP1, XMMPP2,MM,NOBJ,NRHS/COM2/SAVRHS(301) ,SAVOE0 (300) INTEGER*2 NCHK (30 1) , NCHK 1 (30 1) DIMENSION E (300,300) 1 4 1 DATA COLUMN/'COL.•/ ERR=.FALSE. 4 FORMAT (214,F20.12) MMPP2=MMP+2 C C INITIALIZE MATRIX TO ZEROES C DO 20 J=1,NP1 DO 20 1=1,MP1 20 E(I,J)=0.0 C C READ MATRIX ELEMENTS C 25 READ(5,4,END=99,ERR=83)I,J,X IF(I.LE.0)GO TO 30 IF(I.GT.MP1.0R.J.LT.1.0R.J.GT.NP1)GO TO 94 IF(J.NE.NP1)G0 TO 27 IF(I.LT.MMPP2 .OR. X.GE.0.0) GO TO 27 ERR=.TRUE. WRITE(6,87)I,NP1,X 2 7 E(I,J)=X GO TO 2 5 C 30 WRITE(6,31) E (1 ,NP1)=0.0 C C PRINT INPUT TAELEAU ? C IF(LPRINT.NE.1)RETURN WRITE (6,50) IF(MMP.NE.0)WRITE (6,56)MMP IF(NP.NE.O)HRITE(6,57)NP DO 54 J=1,NP1,10 , L=J + 9 IF(L.GT.NP1)L=NP1 WRITE(6,51) (COLUMN,I,I=J,L) DO 54 1=1,MP1 54 WRITE (6,59) I , (E (I,K) ,K=J,L) C RETURN C C 31 FORMAT ('ODATA LOADED •) 50 FORMAT('OINPUT TABLEAU:*/ X« FIRST ROW CONTAINS OBJECTIVE FUNCTION *) 56 FORMAT(' NEXT «,I3,» ROWS CONTAIN INEQUALITY CONSTRAINTS') 57 FORMAT (• NEXT ',13,' ROWS CONTAIN EQUALITY CONSTRAINTS') 51 FORMAT (////10X, 10 (A4 ,13 ,5X) ) 59 FORMAT('ROW',13,2X,10G12.5) 87 FORMAT (• *** RHS OF EQUALITY CONSTRAINT •,215,F20.12,* IS NEGATIVE X») C 93 FORMAT(' *** MATRIX ELEMENT •,215,F20.12, • HAS IMPROPER SUBSCRIPT' X) 94 ERR=.TRUE. WRITE (6,93)I,J,X GO TO 25 C C FATAL INPUT ERROR ENCOUNTERED C ATTEMPT TO EXHAUST DATA S PROCEED WITH NEXT PBOELEH C 95 CONTINUE READ (5,4,END=99,ERR=95)1,0,X IF(I.GT.0)GO TO 95 RETURN C 82 FORMAT {* 0 *** ERROR DETECTED IN DATA TRANSFER') 83 ERR=.TRUE. WRITE (6,82) GO TO 95 C 98 FORMAT('0 *** END-OF-DATA REACHED'/) 99 WRITE (6,98) STOP END SUBROUTINE LIP1 (E , MAX, HOG , *) C LIP1 IS APPLIED TO EQUALITY CONSTRAINTS IF ANY C0MM0N/C0M1/I,J,K, 1,NCHK,NCHK1,M,N,MP1,NP,TD,GMIN,PHIMAX,MMP,NP1, XMMPP2,MM,N0BJ,NRHS/C0M2/SAVRHS(301) ,SAVOBJ (300) INTEGER + 2 NCHK (30 1) ,NCHK1 (301) DIMENSION E (300,300) 5 FORMAT ('PHASE I BEGINS') WRITE (6,5) C ASSIGN 100 TO LOG IF(ILOG.EQ.1)ASSIGN 95 TO LOG C C FOR EACH ROW TO WHICH A VARIABLE HAS NOT BEEN INTRODUCED A SEARCH IS C MADE FOR THE BEST PIVOT. C DO 90 NR=1,NP C C PRIMAL PRICING SECTION. C GMIN=-1.E60 IM=0 LAST=0 9 GNEW=1.E60 DO 10 J=1,N X=E (1, J) IF(X.NE.GMIN .OR. J.LE.LAST)GO TO 11 JMIN=J GO TO 12 11 IF(X.GE.GNEW.OR.X.LE.GMIN)GO TO 10 GNEW=X JMIN=J 10 CONTINUE IF(GNEW.EQ.1.E60)GO TO 600 GMIN=GNEW 12 LAST=JMIN C C SEARCH DOWN KEY-COLUMN FOR PIVOT. C THMIN=1.E60 C MMPP2=ROW NO. OF 1ST EQUALITY IF ANY, MP1=R0W NO. OF LAST EQUALITY DO 40 I=MMPP2,MP1 C IF NCHK(I)=/=0 THE ROW ALREADY HAS A VARIABLE C CHANGE IN TD SIGN TO COMPENSATE FOR DIV. EY 0 IF(NCHK (I).NE.0 .OR. E (I,JMIN) .LE.TD) GO TO 40 THETA=E(I,NP1)/E(I,JMIN) IF(THETA.GE.THMIN)GO TO 40 THMIN=THETA IMIN=I 40 CONTINUE IF(THMIN.EQ.1.E60)GO TO 9 IM=IMIN C C TRANSFORM THE LP TABLEAU IN THE DSUAL WAI. C E(IM,JMIN) IS THE PIVOTAL ELEMENT. C PIVOT=E (IM, JMIN) DO 20 J=1,NP1 20 E (IM,J) = E (IM,J)/PIVOT DO 42 1=1,MP1 IF(I.EQ.IM .OR. E ( I , JMIN) .EQ.O.0) GO TO 42 PIVOT=E(I,JHIN) DO 41 J=1,NP1 41 E(I,J)=E(I,J)-E(IM,J)*PIVOT 42 CONTINUE C C RECORD INFO. ABOUT VARIABLE BROUGHT INTO SOL. C NCHK (IM) = NCHK1 (JMIN) C GO TO LOG, (95, 100) 95 TEMP=E (1,NP1) IF (MM.NE.1)TEMP=-TEMP WRITE (6,6) NR,TEMP,NCHK (IM) C C REMOVE UNIT VECTOR FROM CONSIDERATION. C 100 IF(JMIN.EQ.N)GO TO 300 DO 200 I=1,MP1 200 E(I,JMIN)=E (I,N) NCHK1 (JMIN)=NCHK1 (N) 300 DO 400 1=1,MP1 400 E(I,N)=E (I,NP1) NP1=N N=N-1 C 90 CONTINUE C RETURN 600 WRITE(6,7) RETURN 1 6 FORMAT (2X,I4,7X,G15.6,4X,I4) 7 FORMAT(•0*** NO SOLUTION•) END SUBROUTINE LIPT (IM,JMIN,E,MAX,*) C PERFORMS TRANSFORMATIONS ON THE REDUCED TABLEAU. C E(IM,JMIN) IS THE PIVOTAL ELEMENT. COMMON/COM1/I,J,K,L,NCHK,NCHK1,M,N,KP1,NP,TD,GMIN,PHIMAX,MMP,NP1, XMMPP2,MM,NOBJ,NRHS/COM2/SAVRHS(301) ,SAVOBJ (300) INTEGER*2 NCHK (301),NCHK1 (301) DIMENSION E (300,300) 5 FORMAT ('0 *** NO SOLUTION•) IF(IM.NE.O . AND. JMIN.NE.O) GO TO 10 WRITE(6,5) RETURN 1 10 PIVOT=1./E(IM,JMIN) 1 4 4 DO 20 J=1,NP1 E (IM , J) = E (IM, J) *PIVOT 20 CONTINUE E (IM , JMIN) =PIVOT DO 30 1=1,MP1 IF(I.EQ.IM .OR. E (I,JMIN).EQ.0.0) 60 TC 30 DUMMY=E ( I , JMIN) DO 31 J=1,NP1 E ( I , J) = E ( I , J) -E (IM, J) * DUMMY 31 CONTINUE E (I ,JMIN)=-DUMMY*PIVOT 30 CONTINUE C C SWAP INFORMATION ABOUT VARIABLES IN AND CUT CF SOLUTION. C I=NCHK (IM) NCHK (IM)=NCHK1 (JMIN) NCHK 1 (JMIN)=1 RETURN END SUBROUTINE LIPC (IM,JMIN,E,MAX,*) C PERFORMS CALCULATIONS OVER COLUMNS TC DETERMINE THE PIVOT ELEMENT. COMMON/COM1/1,J,K,L,NCHK,NCHK1,M,N,MP1,NP,TD,GMIN,PHIMA X,MMP,NP1, XMMPP2,MM,NOBJ,NRHS/COM2/SAVRHS (301) ,SAVCEJ (300) COM MON/COM3/IFEAS,IOPT INTEGER*2 NCHK (301) ,NCHK1 (301) DIMENSION E (300,300) C C PRIMAL PRICING SECTION. C THMIN=-1.E60 IM=0 LAST=0 90 GNEW=1.E60 DO 10 J=1,N X=E (1 , J) IF (X.NE.THMIN .OR. J.LE.LAST)GO TO 11 JMIN=J GO TO 12 11 IF(X.GE.GNEW .OR. X.LE.THMIN)GC TC 10 GNEW=X JMIN=J 10 CONTINUE IF(GNEW.GE.-TD)RETURN THMIN=GNEW 12 LAST=JMIN C C SEARCH DOWN KEY-COLUMN FOR PIVOT. C GMIN=1.E60 DO 50 1=2,MP1 IF(E (I,JMIN).LE.TD .OR. E (I,NP1).LT.-TE)GO TO 50 C CHANGE IN TOLERANCE SIGN IS TO COMPENSATE FOR DIVISION BY ZERO THETA=E(I,NP1)/E(I,JMIN) IF(THETA.GE.GMIN)GO TO 50 GMIN=THETA IMIN=I 50 CONTINUE IF(GMIN.NE.1.E60)GO TO 100 IF(IFEAS.NE.2) GOTO 90 C C TEST FOR DNBOUNDEDNESS. C DO 53 1=2,MP1 IF(E ( I , JMIN) .GT.TD) GO TO 90 53 CONTINUE WRITE(6,55) 55 FORMAT(•0*** PRIMAL OBJECTIVE FUNCTION IS UNECU NEED'/ X» DUAL PROBLEM HAS NO FEASIBLE SOLUTIONS') RETURN 1 100 IM=IMIN RETURN END SUBROUTINE LIPR (IMAX , JM , E, M AX ,*) C PERFORMS CALCULATIONS OVER ROWS TO DETERMINE PIVOT ELEMENT. COMMON/COM 1/I,J,K,L,NCHK,NCHK1, M , N , MP1 , NP ,T B,GMIN,PHIMAX,MMP,NP 1, XMMPP2,MM,NOBJ,NRHS/COM2/SAVRHS(301) ,SAVOBJ (300) COMMON/COM3/IFEAS,IOPT INTEGER*2 NCHK (30 1) , NCHK 1 (30 1) DIMENSION E (300 ,300) C C DUAL PRICING SECTION. C DELMAX=-1.E60 JM=0 LAST=0 90 GNEW=1.E60 DO 10 1=2,MP1 X=E (I,NP1) IF(X.NE.DELMAX .OR. I.IE.LAST) GO TO 11 IMAX=I GO TO 12 11 IF(X.GE.GNEW .OR. X. LE.DELMAX)GO TO 10 GNEW=X IMAX=I 10 CONTINUE IF(GNEW.GE.-TD)RETURN BELMAX=GNEW 12 LAST=IMAX C C SEARCH ACROSS KEY-ROW FOR PIVOT. C PHIMAX=-1.E60 DO 50 J=1,N IF(E(IMAX,J).GE.-TD .OR. E (1,J) .LT.-TD)GO TO 50 BELTA=E (1,J)/E (IMAX,J) IF(DELTA.LE.PHIMAX)GO TO 50 PHIMAX=DELTA JMAX=J 50 CONTINUE IF(PHIMAX.NE.-1.E60)GO TO 100 IF(IOPT.NE.2)GOTO 90 C C TEST FOR INCONSISTENCY. C DO 3 6 J=1,N IF(E (IMAX,J).LT.-TD)GO TO 90 36 CONTINUE WRITE (6,37)IMAX 37 FORMAT(* 0*** PRIMAL PROBLEM HAS NO FEASIBLE SOLUTIONS'/ C DUAL OBJECTIVE FUNCTION IS UNBOUNDED•/ C INCONSISTENT CONSTRAINT IS NUMEER ',13) RETURN 1 100 JM=JMAX RETURN END SUBROUTINE LIPOUT (E,MAX) DIMENSION E (300,300) COMMON/COM1/1,J,K,L,NCHK,NCHK1,M,N,MP 1,NP,TD,GMIN,PHIMAX,MMP,NP1, XMMPP2,MM,NOBJ,NRHS/COM2/SAVRHS (301) ,SAVOEJ (300) INTEGER*2 NCHK (301),NCHK 1 (30 1) 3 FORMAT(//'DUAL SOLUTION VECTOR IS INCOMPLETE SINCE NE > 0') 4 FORMAT (//* DUAL SOLUTION VECTOR:•/• VARIABLE',9X,'VALUE'/) 5 FORMAT(//'OPTIMAL VALUE CF THE OBJECTIVE FUNCTICN= ',G20.7) 6 FORMAT (//'PRIMAL SOLUTION VECTOR:'/' VARIABLE',9X,'VALUE•/) 7 FORMAT (2X,I4,4X,G20.7) 8 FORMAT(//'THE REDUCED COSTS:•/• VARIABIE',9 X,'VALU E'/) 9 FORMAT (2X, 14 , 4X ,G20 . 7, • SLACK',13) C C PRINT OPTIMAL PRIMAL FUNCTION VALUE C IF (MM. NE. 1) E (1 , NP 1) =-E (1,NP1) WRITE (6,5) E (1 ,NP1) C C SORT AND PRINT PRIMAL SOLUTION VECTOR C WRITE (6,6) N1 = 0 NOLD=N+NP NP10LD=N0LD+1 DO 13 1=2,MP 1 K=9999 DO 10 J=2,MP1 L=NCHK(J) IF(L.LE.N1 .OR. L.GE.K) GO TO 10 K=L IND=J 10 CONTINUE N 1=K IF(K.LT.NP10LD)GO TO 12 J=K-NOLD WRITE (6,9) K,E (IND,NP1) , J GO TO 13 12 WRITE(6,7)K,E(IND,NP1) r-x-..<: mi . D r c „ . 0* r o ^ v • i T . w O K i l . . ; v j j . ; c C SORT AND PRINT REDUCED COSTS S DUAL SOL. VECTOR. C JJ=0 11=0 IF(N.LE.0)GO TO 20 N1 = 0 DO 15 1=1, N K=9999 DO 14 J= 1 ,N L=NCHK1 (J) IF(L.LE.N1 .OR. L.GE.K)GO TO 14 IND=J 14 CONTINUE N1=K IF(K.GE.NP10LD)GO TO 18 IF(JJ.NE.O)GO TO 19 WRITE (6,8) JJ=1 GO TO 19 18 K=K-NOLD IF ( I I . NE. 0) GO TO 19 11=1 WRITE (6,4) 19 WRITE (6,7) K,E (1 ,IND) 15 CONTINUE 20 IF(MMP.NE.M)WRITE (6,3) C C POST-OPTIMAL PROCEDURES CALLED IN NOW IF REQUESTED C IF(NOBJ.EQ.1) CALL LIPOBJ (E, MAX) IF(NRHS.EQ.1)CALL LIPRHS (E,MAX) RETURN END SUBROUTINE LIPOBJ (E,MAX) COM MON/COM1/1, J , K , L ,NCHK,NCHK1,M,N,MP1,NP,TD,GMIN,PHIMAX,MMP,NP1, CMMPP2,MM,NOBJ,NRHS/COM2/SAVRHS (301) ,SAVOEJ (300) DIMENSION E (300,300) INTEGER*2 NCHK (301) ,NCHK1 (301) IF (MMP. EQ. M) GO TO 10 WRITE (6,34) RETURN C C COMPUTE AND PRINT RANGE OF OBJECTIVE FUNCTION COEFFICIENTS. C 10 WRITE(6,35) NOLD=N+NP BO 15 J=1,N C NECESSARY TO PROTECT AGAINST VALUES WHICH MIGHT EE SLIGHTLY NEGATIVE 15 IF(E (1, J) .IT. 0.0) E (1, J)=0.0 DO 40 1=1,NOLD TD=SAVOBJ (I) DO 20 J=2,MP1 IF(NCHK (J).EQ.I)GO TO 25 20 CONTINUE C VARIABLE NOT IN BASIS,THEREFORE LL=NONE S UL=REDUCED COST DO 21 K=1,N IF(NCHK1 (K) .EQ.I) GO TO 23 21 CONTINUE 23 X=TD + E(1,K) IF(MM.NE.1)GOTO 400 WRITE (6,36)I,TD,X GO TO 40 4 00 X=-X TD=-TD WRITE (6,38) I,X,TD GOTO 40 25 GMIN=1.E60 PHIMAX=-1.E60 DO 26 K=1,N Y=E(J,K) 148' IF (Y) 29,26,30 29 X=-E(1,K)/Y IF (X.LT. GMIN) GMIN=X GO TO 26 30 X=-E(1,K)/Y IF(X.GT.PHIMAX)PHIMAX=X 26 CONTINUE IF(GMIN.NE.1.E60)GO TO 31 X=TD+PHIMAX C NO UPPEH BOUND FOUND IF(MM.NE.1)GOTO 100 WRITE (6,38)I,X,TD GO TO 40 100 X=-X TD=-TD WRITE(6,36)I,TD,X GOTO 40 31 IF(PHIMAX.NE.-1.E60)GO TO 32 C NO LOWER BOUND FOUND. X=GMIN+TD IF(MM.NE.1)GOTO 200 WRITE (6,36)I,TD,X GO TO 40 200 X=-X TD=-TD WRITE(6,38)I,X,TD GOTO 40 32 X=TD+GMIN Y=TD+PHIMAX IF(MM.EQ.1)GOTO 300 TD=-TD TEMP=-X X=-Y Y=TEMP 300 WRITE (6,39)I,Y,TD,X 40 CONTINUE RETURN 34 FORMAT(•0 *** OBJECTIVE RANGING NOT POSSIBLE WBFN NE > 0») 35 FORMAT{'OOBJECTIVE FUNCTION COEFFICIENT RANGING:'/ C» COEFF.»,9X,'LOWER BOUND',9X,'COST/PROFIT',9X,'UPPER EOUN D•/ C) 36 F0RMAT(2X,I4,11X,•-INFINITY',4X,2G20.7) 3 8 FORMAT(2X,14,4X,2G20.7,7X,'+INFINITY *) 39 FORMAT(2X,I4,4X,3G20.7) END SUBROUTINE LIPRHS (E,MAX) COMMON/COM1/1,J,K,L,NCHK,NCHK1,M,N,MP1,NP,TD,GMIN,PHIMAX,MMP,NP1, CMMPP2,MM,NOBJ,NRHS/COM2/SAVRHS (301) ,SAVOEJ (300) DIMENSION E (300,300) INTEGER*2 NCHK (301) ,NCHK1 (301) IF(MMP.NE.0)GO TO 10 WRITE(6,34) RETURN C C COMPUTE AND PRINT RANGE OF BBS'S C 10 L=1 WRITE(6,81) NOLDP1=N+NP+1 K=MMP+N+NP DO 15 J=2,MP1 i 4 y C NECESSARY TO PROTECT AGAINST VALUES WHICH MIGHT EE SLIGHTLY NEGATIVE. 15 IF(E (J,NP1) .LT.0.0) E (J,NP1)=0.0 DO 85 J=NOLDP1,K TD=SAVRHS (L + 1) GMIN=1.E60 EHIMAX=-1.E60 DO 20 JJ=1,N IF(NCHK1 (JJ).EQ.J)GO TO 25 20 CONTINUE C INVERSE COLUMN IS UNIT VECTOR, & THEREFORE NOT PRESENT X=-E (L + 1 ,NP1) +TD WRITE (6,38)L,X,TD GO TO 85 25 DO 65 1=2,MP1 Y=E(I,JJ) IF(Y)67,65,68 67 X=- E (I,NP1) / Y IF (X.LT.GMIN) GMIN=X GO TO 65 68 X=-E (I,NP1)/Y IF(X.GT.PHIMAX)PHIMAX=X 65 CONTINUE IF(GMIN.NE.1.E60)GO TO 69 C NO UEPER BOUND X=PHIMAX+TD WRITE (6,38) L,X,TD GO TO 85 69 IF(PHIMAX.NE.-1.E60)GO TO 70 C NO LOWER BOUND X=TD+GMIN WRITE (6,36)L,TD,X GO TO 85 70 X=GMIN+TD Y=PHIMAX+TD WRITE(6,39)L,Y,TD,X 85 L=L+1 IF(MMP.NE.M)WRITE(6,82) RETURN 34 FORMAT('0 *** RHS RANGING NOT POSSIBLE WHEN NE = M') 36 FORMAT(2X,I4,11X,'-INFINITY',4X,2G20.7) 38 FORMAT(2X,I4,4X,2G20.7,7X,'+INFINITY') 39 FORMAT (2X,14,4X,3G20.7) 81 FORMAT('ORIGHT HAND SIDE RANGING:'/ C NUMBER',9X,'LOWER BOUND',13X,'RHS',13X,»UPPER BOUND'/) 82 FORMAT(/'RIGHT HAND SIDE RANGING IS INCOMPLETE SINCE NE > 0') END $SIG 150. APPENDIX I I I In t h i s Appendix we have s i x - computer programs a l l coded i n FORTRAN IV. The f i r s t three, namely, Programs 5, 6 and 7 do the computations fo r Tests 5, 6 and 7. The second three, namely, Programs 8, 10 and 11A arrange the data i n the format the l i n e a r programming code Program 0 (Appendix II) requires to carry out the te s t s . We s h a l l , i n what follows, explain the input-output organization of these programs i n d i v i d u a l l y . Let us assume that there are NI inputs, NO outputs on which N2 obser-vations recorded. The data f o r Programs 5, 6 and 7 are prepared and stored as explained below. Two f i l e s are created and the quantity data are stored i n one, the pr i c e data i n the other. F i l e 1 - Store the quantity observations on the j th input f o r a l l i = 1,...N2 with FORMAT (4E20.6), j = 1,...,NI; the negative of the quantity observations on the k th output f o r a l l i = 1,...,N2 with FORMAT (4E20.6), k = 1, . .., (NO^ -1) ; the quantity observations on the output 1 f o r a l l i = 1,...,N2 with FORMAT (4E20.6). F i l e 2 - Store the p r i c e observations on the j th input, i n the same order as for the input q u a n t i t i e s , f o r a l l i = 1,...,N2 with FORMAT (4E20.6), j = 1,...,NI; the p r i c e observations on the j th output, i n the same order as f o r the output q u a n t i t i e s , f o r a l l i = 1,...,N2 with FORMAT (4E20.6), j = 1,...,N0. Input: The input i s almost.the same f o r a l l three of Programs 5, 6 and 7. There i s only one parameter card which contains the values of the The output i s , i n case of multiple output, the one which has been singled out while the others, as explained i n the text, become input by m u l t i -p l y i n g them with minus one. 151 following parameters i n the order given: NI the number of inputs. NO the number of outputs. N2 the number of observations. N the index of the observation which i s the f i r s t one of the data seg-ment considered, 2 and i s applicable only to Program 5. I t i s omitted fo r the other programs. N10 the index of theoobservation which i s the f i r s t one of a se r i e s of observations to be t e s t e d . 3 N12 the index of the observation which i s the l a s t one of a series of observations to be t e s t e d . 3 These parameters must be input with FORMAT (614) f o r Program 5 and FORMAT (514) for Programs 6 and 7. Output: Program 5: Ai.1 4 i f A^. > 0, f o r a l l j such that y, > i t i " i t " i 1 J w x -p k y k and i = N10,...,N12. Program 6: A ± _ . 5 > 0, j = 1,...,N2, i = N10,...,N12. 2 F o r example, the data may have 40 observations, we may want to use only the l a s t 24 of them. Then the data segment of i n t e r e s t consists of obser-vations 17, 18,...,40 and N = 17. 3For example, i f we want to test only observations 25 and 26 then N10 = 25, N12 = 26. 4 R e f e r to Section 2.1 f o r the d e f i n i t i o n of A... ij 5Refer to Section 2.2 f o r the d e f i n i t i o n of A . . . 152 Program 7: A > 0, j = 1,...,N2, i = N10,...,N12. On the other hand, the data are prepared d i f f e r e n t l y f o r Programs 8, 10 and 11a. Program 8: Create two f i l e s : F i l e 1 - Store the p r o f i t observations f o r a l l i = 1,...,N2 with FORMAT (4E20.6), F i l e 2 - Store the p r i c e observations on the j th input f o r a l l i = 1,...,N2 with FORMAT (4E20.6), jt = 1,...,NI; the p r i c e observations on the j th output for a l l i = 1,...,N2 with FORMAT (4E20.6), j = 1,...,N0. Input: There i s one parameter card, i t contains the values of the following parameters i n that order: NI, NO, N,M2, N10, N12, which are the same parameters as for Program 5, and TOL which approximates zero i n the l i n e a r pro-gramming code PROGRAM 0 whencchecking the optimality of a b a s i c f e a s i b l e s o l u t i o n . They are read.in with FORMAT (614, F20.12). The value of TOL i s t y p i c a l l y 10~ 5 -6 - 10 Output:- T T 1 and Z 1, i = 1,...,N2, Program 10: Create a f i l e . F i l e 1 - Store the p r i c e observations on the j th input f o r a l l i = 1,...,N2 with FORMAT (4E20.6), j = 1,...,NI; the p r i c e observations on the j th output for a l l i = 1,...,N2 with FORMAT (4E20.6). Input: Same as for Program 8. Output: Z 1, i = 1,...,N2. Program 11a: Create a f i l e . F i l e 1 - Store the cost observations for a l l i = 1,...,N2 with FORMAT (4E20.6); the output observations (output si n g l e d out from multiple output) f o r Refer to Section 2.3 for the d e f i n i t i o n of A . . . a l l i = 1,...,N2 with FORMAT (4E20.6); the p r i c e observations on the j th input f o r a l l i = 1,...,N2 with FORMAT (4E20.6), j = 1,...,NI; the p r i c e observations on the j th.output (those outputs which are regarded as inputs) f o r a l l i = 1,...,N2 with FORMAT (4E20.6), j = 1,...,(N0-1). Input: Same as for Program 8. Output: Z 1, i = 1,...,N2. As an example we have the following flowchart f o r running Program 11a for the Christensen-Jorgenson (1969-70) data which have 39 observations on 3 outputs and 2 inputs. Creat F i l e s DATA and LPDATA \ f Store Cost observations.in DATA with FORMAT (4E20.6) \ f Store output observations i n DATA with FORMAT (4E20.6) \ t Store Pric e observations.in DATA with FORMAT (4E20.6) \ f Run PROGRAM 11a with input unit 1 assigned to f i l e DATA, output unit 7 assigned to f i l e DBDATA<:.ul?DATA 7=LPDATA, NI=2, N0=2, N2=39, N=l, N10=l, N12=39, TOL=0.000001 \ f Run PROGRAM 0 with input unit 5 assigned to f i l e LPDATA: 5=LPDATA 154 $LIST FR0G5 1 C 2 C 3 C 4 c 5 c 6 7 8 9 10 11 13 12 28 13 14 14 15 16 17 18 19 20 1 21 10 22 23 24 25 7 26 27 28 29 30 31 32 33 34 3 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 2 51 4 52 53 54 55 PROGRAM 5. TEST FOR MONOTONICITY AND COASI-CONCAVITY WITH COST MINIMIZATION. (PRICE AND CUA NT ITY DATA.) DIMENSION E(50,10) ,IND (50) ,KIN (50 ) , P (50 ,10 ) ,S (50 ,50) DIMENSION SUM (50) READ (5,13) NI,N0,N2 ,N,N10,N12 WRITE (6,14) c WRITE (6,28) FORMAT (614) FORMAT (1X,'REF. OBS.»,4X,»OBS. CCM PRE. *,6 X, ' EELT A'/) FORMAT (• 1 TEST 5.'//) N11=NI+N0 N1 = N11-1 N15=N2-1 DO 1 J=1,N11 READ (1,10) (E (I , J) ,1=1 , N2) READ (2,10) (P (I, J) ,I=1,N2) CONTINUE FORMAT (4E20.6) DO 7 ID=N,N2 IND (ID) = ID KIN (ID) =ID CONTINUE DO 4 J =N , N 1 5 C=E(J,N11) K=J IJ=J+1 DO 3 I = IJ,N2 IF (E (1,1111) . GE. C) GO TO 3 C=E (I,N 11) K=I CONTINUE IF (K . EQ. J) GO TO 4 II=KIN (K) IK=KIN (J) IND (II) =J IND(IK) = K KT=K IN (K) KIN (K) =KIN (J) KIN (0) =KT DO 2 M=1,N11 T=E(J,M) E (J,M) = E(K,M) E (K ,M) =T T=P (J,M) P (J , M) =P (K,M) P(K,M)=T CONTINUE CONTINUE DO 26 K=N10,N12 IS=IND (K) SUM (IS) =0. DO 26 K1=IS,N2 56 S(K1,K)=0. 57 DO 25 J=1,N1 58 SUM (IS) =SUM (IS)+P (IS, J) *E (IS,J) 59 S (K1 ,K) = P (IS, J ) * (I (IS,J)-E(K1,J) ) +S (K 1,K) 60 25 CONTINUE 61 IF (S(K1,K) .IE. 0.) GO TO 26 62 S (K 1,K) =S (K1 ,K)/SUM (IS) 63 WRITE (6,27) K, KIN (K 1) , S (K 1, K) 64 27 FORMAT (4X,13,12X,12,7X ,F10.6) 65 26 CONTINUE 66 STOP 67 END END OF FILE $LIST PSOG6 1 C PROGRAM 6. TEST FOR MCNOTGNICIT Y AND CCKCAVITY 2 C WITH COST MINIMIZATION. (PRICE AND QUANTITY DATA.) 3 C 4 C 5 C 6 DIMENSION E (50,50) ,P (50,50) , S (50 ) ,SUKJ50) 7 READ (5,13) NI, NO, N2,N10,N 1 2 8 WRITE (6,14) 9 WRITE (6,28) 10 13 FORMAT (514) 11 28 FORMAT (1X,'REF. OBS. • , 4X, '0BS . COMPRD. ', 6X, 'DELTA ' 12 14 FORMAT ('1 TEST 6.'//) 13 N11=NI + N0 14 N1=N11-1 15 N15=N2-1 16 DO 1 J=1,N11 17 READ (1,10) (E ( I , J) , 1= 1, N2) 18 READ (2,10) (P (I,J) ,1 = 1 ,N2) 19 1 CONTINUE 20 10 FORMAT (4E20.6) 21. DO 20 K=N10 ,N12 22 SUM (K) =0. 2 3 DO 20 M=1 ,N2 24 S(M)=0. 25 DO 21 0=1,N1 26 SUM (K) =SUM (K) +P (K,J) *E (K,J) 27 S (M)=S (M)+P (K, J)*E (M, J) 28 21 CONTINUE 29 SUM (K) = E (K, N1 1) *P (K, N 11) -SUM (K) 30 S (M) =E (M ,N 1 1) *P(K,N11)-S (M) 3 1 S (M)=S (M) -SUM (K) 32 IF (S(M) .LE. 0.) GO TO 20 33 WRITE (6,27) K,M,S(M) 34 27 FORMAT (4X,13 ,12X ,12,6X ,F11.6) 35 20 CONTINUE 36 STOP 37 END END OF FILE $LIST PROG7 1 C PROGRAM 7.. TEST FOR MONOTONICIT Y, CONCAVITY 2 C AND LINEAR HOMOGENEITY WITH COST MINIMIZATION. 3 C 3.25 C (PRICE AND QUANTITY DATA.) 4 C 5 C 6 DIMENSION E (50,50) , P (50, 50) ,S (50, 50) 7 READ (5,13) NI,NO,N2,N10,N12 8 WRITE (6,14) 9 WRITE (6,28) 10 13 FORMAT (514) 11 28 FORMAT (1X,'REF. OBS.* ,4X,'OBS. CCMPRD. • ,6X,•EELTA• 12 -14 FORMAT ( »T TEST 7. »//) 13 N11=NI+NO 13.25 N1 = N11 -1 14 N15=N2-1 15 DO 1 J=1,N11 16 READ (1,10) (E (I , J) ,1 = 1 , N2 ) 17 READ (2,10) (P (I,J) ,1=1,N2) 18 1 CONTINUE 1 9 10 FORMAT (4E20.6) 20 DO 20 K=N10,N12 21 DO 20 M=1,N2 22 S (M ,K) =0. 23 DO 21 J=1,N1 24 S(M,K) =S (M,K) +P (K,J) *E (E,J) 25 21 CONTINUE 26 S (M , K) =E(M,N11) *P (K,N11)-S (M,K) 27 IF (M . EQ. K) GO TO 2 28 IF (S(M,K) .LE. 0.) GC TO 20 29 4 WRITE (6,27) K,M,S (M,K) 30 27 FORMAT (4X,I3,12X,I2,6X,F11.6) 31 GO TO 20 32 2 IF (S(M,K) .NE. 0.) GO TC 4 33 20 CONTINUE 34 STOP 35 END ENE OF FILE $LIST PROG8 1 C PROGRAM 8. TEST FOR MONOTONICITY AND CONCAVITY 2 C WITH PROFIT MAXIMIZATION. (PRICE AND PROFIT DATA.) 3 C 4 C 5 C 6 DIMENSION P (50 , 1 0) ,PR (50) 7 READ (5,18) N I, NO, N2, N, N 10, N 1 2, TOL 8 18 FORMAT (6I4,F20.12) 9 MM=0 10 N3=0 11 N4 = 0 12 N5=0 13 N6 = 0 14 AM=-1 15 ' N11=NI+N0 16 N1=N 11-1 17 N14=N2 + 1 18 READ ( 1, 10) (PR (I) ,1=1 ,N2) 19 DO 1 J=1,N11 20 READ ( 2, 10) (P ( I , J) ,1 = 1 ,N2) 21 1 CONTINUE 22 10 FORMAT (4E20.6) 1 23 DO 5 K=N10,N12 24 WRITE (7,6)K,PR(K) 25 6 FORMAT (• TEST 8. PROFIT AT REFERENCE OBSERVATION*, 26 113, » IS : • ,F20.6) 27 WRITE (7,7) N 11, N2, N3, N4, N5, N6, TOL 28 7 FORMAT (6I4,F20.12) 2 9 JJ=1 30 DO 8 1=1,N2 31 WRITE (7,9) JJ,I,PR(I) 32 9 FORMAT (2I4,F20.6) 3 3 8 CONTINUE 34 DO 11 J=1,NI 35 JJ=JJ+1 36 DO 12 1=1,N2 37 WRITE (7,9) J J , I , P ( I , J) 3 8 12 CONTINUE 39 WRITE (7,9) JJ,N14,P (K,J) 40 11 CONTINUE 41 IF (NO .LT. 1) GO TO 17 4 2 DO 16 J = 1 ,NO 43 JJ=JJ+1 44- DO 19 1=1,N2 45 PT=-P (I,NI + J) 46 WRITE (7,9) JJ,I,PT 47 19 CONTINUE 48 PT=-P (K,NI + J) 49 WRITE (7,9) JJ,N14,PT 50 16 CONTINUE 51 17 CONTINUE 52 JJ=JJ+1 53 WRITE (7,15) HH 54 5 CONTINUE 55 15 FORMAT (14) 56 STOP 57 END END OF FILE $LIST PROG10 1 C PROGRAM 10. TEST FOR MONOTONICITY, CONCAVITY 2 C AND LINEAR HOMOGENEITY WITH PROFIT MAXIMIZATION. 3 C (PRICE DATA. ) 4 C -5 C 6 C 7 DIMENSION P (50, 10) 8 READ (5,18) NI,NO,N2,N,N10,N12 ,TCI 9 FORMAT (6I4,F20.12) 10 MM = 0 11 N4 = 1 12 N5 = 0 13 N6 = 0 14 A = 1. 15 AM=-1. 16 N1=NI+NO-1 17 N11 = N1 + 1 158 18 N14=N2+1 19 N18=N2 + N0 20 N20=N18+1 21 DO 1 J=1 , H1 1 22 READ (2,10) (P (I , J) ,1=1 ,N2) 23 1 CONTINUE 24 10 FORMAT (4E20.6) 25 DO 5 K=N10,N12 26 WRITE (7,6) K 27 6 FORMAT (' TEST 10. REFERENCE 0 BS ER VA TION: * , 12) 28 WRITE (7,7) N11 ,N18,NO,N4,N5,E6 ,TCI 29 7 FORMAT (6I4,F20.12) 30 JJ=1 31 DO 8 I=N14,N18 32 WRITE (7,9) JO,I,AM 33 9 FORMAT (2I4,F20.6) 34 8 CONTINUE 3 5 DO 11 J= 1 , NI 36 JJ=JJ+1 3 7 DO 12 1=1, N2 38 WRITE (7,9) J J , I , P (I,J) 3 9 12 CONTINUE 40 WRITE (7,9) JJ,N20,P (K,J) 41 11 CONTINUE 42 IF (NO .LT. 1) GO TO 17 4 3 DO 16 J=1,NO 44 JJ=JJ+1 4 5 DO 19 1=1, N2 46 WRITE (7,9) J J , I , P (I , NI + 0) 47 19 CONTINUE 48 N19=N2+J 49 WRITE (7,9) JJ,N19,A 50 WRITE (7,9) JJ,N20,P (K,NI + J) 51 16 CONTINUE 52 17 CONTINUE 53 JJ=JJ+1 54 WRITE (7,15)MM 55 5 CONTINUE 56 15 FORMAT (14) 57 STOP 58 END END OF FILE $LIST PRO G 11A 1 C PROGRAM 11 A. TEST FOR MONOTONICITY AND QUASI-CONCAVITY 2 C WITH COST MINIMIZATION. . (INPUT PRICE , OUTPUT QUANTITY 3 C AND COST DATA.) 4 C 5 C 6 C 7 DIMENSION P (100,50) ,1ND (1 00 ) , KI N (100 ) ,CT (100 ) ,Y (100) 8 READ (5,13) NI,NO,N2,N,N10,N12,TOL 9 13 FORMAT (6I4,F20.12) 1 0 N3 = 1 1 1 N4 = 0 12 N5 = 0 13 N6=0 14 N15=N2-1 159 15 N1 1=NI + 1 16 N18=NI + N0 17 N14=N18-1 1 8 READ (1,10) (CT (I) , T= 1, N 2) 19 READ (1,10) (Y(I) ,1 = 1 ,N2) 20 DO 1 J=1,N14 21 READ ( 1,10) (P (I,J) ,1 = 1 ,N2) 22 1 CONTINUE 23 10 FORMAT (4E20.6) 24 8 A=1. 25 AM=- 1. 26 AZ=0. 27 MM = 0 28 DO 7 ID=N,N2 29 IND (ID) =ID 30 KIN(ID) = ID 31 7 CONTINUE 32 DO 4 J= N,N15 33 C=Y (KIN (J) ) 34 K= 3 35 IJ=J + 1 36 DO 3 I=IJ,N2 37 IF (Y(KIN(I)) .GE. C) GO TO 3 38 C=Y (KIN (I) ) 39 K = I 40 3 CONTINUE 41 IF (K .EQ. 3) GO TO 4 42 II=KIN (K) 4 3 IK=KIN (J) 44 IND (II) = J 45 IND(IK)=K 4 6 KT= KIN (K) 47 KIN (K) =KIN (3) 48 KIN (J)=KT 49 DO 2 M=1,N14 50 T=P (3, M) 51 P (J,M) =P (K , M) 52 P(K,M)=T 53 2 CONTINUE 54 T= CT (J ) 55 CT(J)=CT(K) 56 CT (K) = T 57 4 CONTINUE 58 DO 24 K=N10,N12 59 N7=IND (K) 60 IF (N7 . EQ. 1) GO TO 25 61 WRITE (7,14) K 62 14 FORMAT {' TEST 11. REFERENCE OBS 63 N9=N7-1 64 N13=N7 + 1 65 WRITE (7,16) N18,N7,N3,N4 ,N5 ,N6 ,TCI 66 16 FORMAT (6I4,F20.12) 67 J=1 68 WRITE (7,11) J, N7, CT (IND (K) ) 69 11 FORMAT (2I4,F20.12) 70 DO 21 J=1,NI 71 JJ=J + 1 72 DO 20 1=1 ,N9 73 11 = IND (K) - I 74 WRITE (7,11) OJ,I,P(I1,J) ,12) 160 75 20 CONTINUE 76 RT= -P (IND (K) , J) 77 WRITE (7, 11}JJ,N7,RT 78 WRITE (7,11) JJ,N13,AZ 79 21 CONTINUE 80 DO 27 J=N11,N14 81 JJ=JJ+1 82 DO 28 1=1,N9 83 I1 = IND (K) - I 84 RT=-P(I1,J) 85 WRITE (7,11) 0J,I,RT 86 28 CONTINUE 87 WRITE (7,11) JJ,N7,P (IND (K) ,0) 88 WRITE (7,11) JJ,N13,AZ 89 27 CO NT IN UE 90 0J=JJ+1 91 DO 2 2 I=1,N9 92 I1 = IND (K) - I 93 WRITE (7,11) J J,I,CT (11) 94 22 CONTINUE 95 WRITE (7, 11)JJ,N13,A 96 WRITE (7,12)MM 97 12 FORMAT (14) 98 GO TO 24 99 25 WRITE (6,26) K 100 26 FORMAT (• THIS TEST IS NOT APPLICABLE FOR OBSERVATION «,I2) 101 24 CONTINUE 102 STOP 103 END END OF FILE 161 APPENDIX IV In t h i s Appendix we have two computer programs coded i n FORTRAN IV. The f i r s t one i s IDLCON which arranges the data on prices and quantities i n the format necessary for. the l i n e a r programming code Program 0 to do the nonhomotheticity t e s t and calculatedthe nonhomothetic nonparametric quantity indexes. The second program, HOMCON, arranges the same data, again f o r the l i n e a r programming code, to do the homotheticity t e s t and cal c u l a t e the homo-t h e t i c nonparametric quantity indexes. IDLCON: Create F i l e s 1, 2, and 3. F i l e 1 - Store firstfehre p r i c e observations on the j t h commodity i n a l l periods i=l,...,N f o r j=l,...,M, with FORMAT (4F20.8). Then store the quan-t i t y observations i n the same order and with the same 'format. Input - The parameter card contains: . M the number of commodities N the number of observations KRF the base period index NRF the second normalization period index PHI Fisher quantity index of period NRF with respect to period KRF TOL zero approximation i n the l i n e a r programming code, t y p i c a l l y IO" 5 - i o - 6 . Output - Quantities and normalized p r i c e s are stored i n F i l e 2, data arranged for the l i n e a r programming code are stored i n F i l e 3. HOMCON: Create F i l e s 1,2, and 3. F i l e 1 - Same as i n IDLCON. Input - The parameter card contains: 162 M the number of commodities N the number of observations TOIL, the tolerance l e v e l t y p i c a l l y 10 - 10 Output - Same as i n IDLCON. Example: The following flowchart can be used as a guide when running IDLCON: Create F i l e s 1 , 2, and 3 \ Store p r i c e and quantity observations, i n that order i n F i l e 1 with FORMAT(4F20.8) ( Run INLCON with input unit 2 assigned to F i l e 1, output u n i t 3 to F i l e 2, and out-put u n i t 7 to F i l e 3 \ / Run Program 0 with input u n i t 5 assigned to F i l e 3 unit 163 $LIST IDLCON 1 C TEST FOB THE EXISTENCE OF A NON-HOMOTHETIC 2 c PREFERENCE FUNCTION USING HOMOTHETIC INDEXES 3 c 4 c 5 DIMENSION P (50,50) ,X (50,50) ,D(50>50) ,C(50) 6 READ (5,18) M, N, KRF,NRF/PHI, TOL 7 18 FORMAT (414 ,2F16.8) 8 READ (2, 16) ( (P ( I , J) , 1= 1,N) ,J=1,M) 9 READ (2,16) ( (X(I,J) ,1=1 ,N) ,J=1,M) 10 16 FORMAT (4F20.8) 11 DO 13 1=1 ,N 12 C(I)=0. 13 DO 12 J=1,M 14 C(I)=C (I) +P(I,J) *X(I,J) 15 12 CONTINUE 16 DO 13 J=1,M 17 P(I,J) =P (I , J ) / C ( I ) 18 13 CONTINUE 19 WRITE (3,16) ( (P ( I , J) ,1=1 ,N) , J=1 ,M) 20 WRITE (3,16) ( (X ( I , J) ,1=1, N) ,J=1 ,M) 21 DO 17 1=1,N 22 DO 17 K=1,N 23 D(I,K)=-1. 24 DO 17 J= 1,M 25 D (I,K) =D (I,K)+P ( I , J)*X (K, J) 26 17 CONTINUE 27 WRITE (6,40) M,N,KRF,NRF,PHI 28 40 FORMAT (» NO. OF GOODS=',I2,* NO. OF OBS. =',12/ 29 1» PHI (*,I2,,)=1 . • PHI(*,I2, ,) = ,,F12.6) 30 A=1. 31 AM=-1. 32 MM=0 33 NM=N-1 34 N1=NM*N 35 N2=N1+N 36 N7=N1+2*N~2 37 N6=N7+1 38 K=1 39 KM=0 40 WRITE (7,6) 41 6 FORMAT ('• TEST FOR NON-HOMOTHETIC PREFERENCE FUNCTION1) 42 WRITE (7,7) N 1,N7,MH,MM,MM,MM,TOL 43 7 FORMAT (6I4,E20.12) 44 DO 5 I=1,N1 45 WRITE (7,1) K,I,A 46 5 CONTINUE 47 II=N2 48 DO 3 1=1,N 49 N3=N1+I 50 IF (I .NE. KRF) GO TO 8 51 JJ=N2 52 DO 20 J=1,N 53 IF (I .EQ. J) GO TO 20 54 K=K + 1 55 KM=KM+1 56 . IF (J . EQ, NRF) GO TO 21 57 JJ=JJ+1 58 WRITE (7,1) K,KM,AM 59 DM=-D(I,J) 60 WRITE (7, 1) K,N 3, DM 61 WRITE (7,1) K,JJ,A 62 DM= D(I,J)+1. 63 WRITE (7,1) K,N6,DM 64 GO TO 20 65-" 21 WRITE (7,1) K>KM,AM 66 DM=-D(I,J) 67 WRITE (7,1) K,N3,DM 68 DM= D(I,J)+1.-PHI 69 WRITE (7,1) K,N6 ,DM 70 20 CONTINUE 71 GO TO 3 72 8 IF (I . NE, NRF) GO TO 9 73 JJ=N2 74 DO 22 J=1,N 75 IF (I .EQ. J) GO TO 22 76 K=K + 1 77 KM=KM+1 78 IF (J .EQ. KRF) GO TO 24 79 JJ=JJ+1 80 WRITE (7, 1) K,KM, AM 81 DM=-D(I,J) 82 WRITE (7, 1) K,N3,DM 83 WRITE (7,1) K,JJ,A 84 DM= D ( I , J) +PHI 85 WRITE (7,1) K,N6 ,DM 86 GO TO 22 87 24 WRITE (7,1) K,KM,AM 88 DM=-D(I,J) 89 WRITE (7,1) K,N3,DH 90 DM= D ( I , J) +PHI-1. 91 WRITE (7,1) K,N6 ,DM 92 22 CONTINUE 93 GO TO 3 94 9 11=11+1 95 JJ=N2 96 DO 10 J=1,N 97 IF (I .EQ. J) GO TO 23 98 K=K + 1 99 KM=KM+1 100 IF (J . EQ. KRF) GO TO 11 101 IF (J .EQ. NRF) GO TO 14 102 JJ=JJ*1 103 WRITE (7,1) K,KM, AM 104 DM=-D(I,J) 105 WRITE (7,1) K,N3 ,DM 106 IF (I .GT. J) GO TO 19 107 WRITE (7,1) K,II,AM 108 WRITE (7,1) K,JJ,A 109 GO TO 15 110 19 WRITE (7,1) K,JJ,A 111 WRITE (7,1) K,II,AM 112 15 WRITE (7,1) K,N6,D(I,J) 113 GO TO 10 114 11 WRITE (7, 1) K,KM, AM 115 DH=-D(I,J) 116 WRITE (7, 1) K,N3,DM 117 WRITE (7,1) K,IT, AM 118 DM= D(I,J ) - 1 . 119 WRITE (7,1) K,N6,DM 120 GO TO 10 121 14 WRITE (7,1) K,KM,AN 122 DM=-D ( I , J) 123 WRITE (7,1) K,N3 ,DM 124 WRITE (7, 1) K,II,AM 125 DM= D(I,J)-PHI 126 WRITE (7, 1) R,N6,DM 127 GO TO 10 128 23 JJ=JJ+1 129 10 CONTINUE 130 3 CONTINUE 131 1 FORMAT (2I4,F14.6) 132 WRITE (7,1) MM 133 STOP 134 END 165 END OF FILE 166 $LIST HOMCON 1 C TEST FOR THE EXISTENCE OF A HOMOTHETIC PREFERENCE 2 c FUNCTION 3 c f 4 c 5 DIMENSION P (20,20) ,X (20,20) rD (20 ,20) ,C (20) 6 READ (5,18) M,N,TOL 7 18 FORMAT (2I4,F20.12) 8 READ (2,16) ( (P ( I , J) ,1=1 ,N) , J=1, M) 9 READ (2,16) ( (X ( I , J) ,1=1 ,N) , J=1,M) 10 16 FORMAT (4F20.8) 1 1 DO 13 1= 1 , N 12 C(I)=0 . 13 DO 12 J=1 , M 14 C(I)=C (I)+P(I,J)*X(I,J) 15 12 CONTINUE 16 DO 13 J=1,M 17 P (1,3) =P (I,J) /C (I) 18 13 CONTINUE 19 WRITE (3,16) { (P (I , J) , 1= 1 , N) , J= 1 , M) 20 WRITE (3,16) ((X(I,J),I=1,N), J=1 , M) 21 DO 17 1=1,N 22 DO 17 K=1,N 23 D ( I , K) =-1 . 24 DO 17 J=1,H 25 D ( I , K) = D ( I , K) +P ( I , J) *X (K,J) 26 17 CONTINUE 27 MM = 0 28 AM=-1. 29 AZ = 0 . 30 A= 1 . 31 NM=N-1 32 N1=NM*N 33 N7=N1+NM 34 N6 = N7+ 1 35 K=1 36 KM=0 37 WRITE (7,6) 38 6 FORMAT (• TEST FOR HOMOTHETIC PREFERENCE FUNCTION') 39 WRITE (7,7) N1,N7,MM,MM,MM,MM,TOL 40 7 FORMAT (6I4/E20.12) 4 1 DO 5 1=1,N1 42 WRITE (7,1) K,I,A 43 5 CONTINUE 44 DO 10 J=2,N 45 K=K+1 46 KM=KM+ 1 47 WRITE (7,1) K , K M , A M 48 N5=N1+J-1 49 WRITE (7,1) K,N5,A 50 DM=D(1,J)+1. 51 WRITE (7,1) K , N 6 , D M 52 10 CONTINUE 53 DO 2 1=2,N 54 N4=N1+I-1 55 K=K+1 167 56 KM=Kfl+ 1 57 WRITE (7,1) K,KM,AM 58 DM = -D(I, 1)-1 . 59 WRITE (7,1) K,N4,DM 60 WRITE (7,1) K,N6,AM 61 DO 2 J = 2,N 62 N5=N1+J-1 63 IF (I .EQ. J) GO TO 64 K=K+1 -65 KM=KM+1 66 DM = -D(I,J) -1. 67 WRITE (7,1) K,KM,AM 68 IF (I .GT. J) GO TO 69 WRITE (7,1) K , N 4 , D M 70 WRITE (7,1) , K,N5,A 71 GO TO 8 72 9 WRITE (7,1) K , N 5, A 73 WRITE (7,1) K , N 4 , D M 74 8 WRITE (7,1) K,N6,AZ 75 2 CONTINUE 76 1 FORMAT (2I4,F10.6) 77 WRITE (7,1) MM 78 STOP 79 END END OF FILE I
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Nonparametric tests of regularity conditions for production...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Nonparametric tests of regularity conditions for production and consumption theory Parkan, Celik 1975
pdf
Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.
Page Metadata
Item Metadata
Title | Nonparametric tests of regularity conditions for production and consumption theory |
Creator |
Parkan, Celik |
Publisher | University of British Columbia |
Date Issued | 1975 |
Description | The theories of production and consumption have proven to be valuable tools in predicting variations in demand for factors of production (producer's demand) and for produced goods (consumer's demand) to changes in policy instruments. An important direction of inquiry, in the theory of production, has been on determining the functional form of production; another one, in the theory of consumption, has been on the observed behaviour of the consumer and the computation of index numbers. Econometricians, in estimating a production function, have been using parametric estimation techniques which always require the choice of a functional form for the underlying production operation. Also the classical assumption about the consumer has been that any purchase is such as to give a maximum of utility for the money spent. It is well known that economic agents' (producers' and consumers') optimizing behaviour implies certain restrictions on derived supply and demand patterns; e.g., see P.A. Samuelson or J.R. Hicks. Typical functional forms, for production, often used in empirical applications are either unsuitable for modelling the demand for factors of production or they do not always satisfy the regularity conditions that cost minimizing or profit maximizing behaviour imposes. Neither have econometricians always checked whether these restrictions are consistent with the functional forms for demand and supply equations which they use in their applied econometric work. This dissertation studies the nonparametric methods for testing the consistency of observations on quantities, prices, profits, costs and appropriate combinations of these with such regularity conditions as monotonicity, quasi-concavity, concavity, homotheticity, and separability. The approach has been that of unifying the existing body of knowledge in using linear programming as the test instrument and making extensions especially to cover multiple production, separability properties and nonparametric index numbers. The theoretical work has been substantiated by empirical tests on actual data and the computer codes are presented to facilitate further research in related areas. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-08 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0100108 |
URI | http://hdl.handle.net/2429/19764 |
Degree |
Doctor of Philosophy - PhD |
Program |
Business Administration - Management Science |
Affiliation |
Business, Sauder School of Operations and Logistics (OPLOG), Division of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
Download
- Media
- 831-UBC_1975_A1 P37_5.pdf [ 10.25MB ]
- Metadata
- JSON: 831-1.0100108.json
- JSON-LD: 831-1.0100108-ld.json
- RDF/XML (Pretty): 831-1.0100108-rdf.xml
- RDF/JSON: 831-1.0100108-rdf.json
- Turtle: 831-1.0100108-turtle.txt
- N-Triples: 831-1.0100108-rdf-ntriples.txt
- Original Record: 831-1.0100108-source.json
- Full Text
- 831-1.0100108-fulltext.txt
- Citation
- 831-1.0100108.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
data-media="{[{embed.selectedMedia}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0100108/manifest