2t. See the previous section. 261 static effects and observable structure of the firm's maximization problem. Here we attempt to assess the importance of the restrictions described in ' s e c t i o n s 2. a\u00abd 3 relative to the entire set of relations between comparative static effects and observable structure. We conclude that the most Important of the generally applicable relations may well have been specified In these sections. The discussion is highly speculative. First, consider the set of all special properties that can be imposed directly on Tr(x;a) and Incorporated Into our comparative static analysis, e.g. properties such as separability. Such properties can be useful in particular cases. However, these properties typically are observed either to hold or not to hold for a particular firm. Thus the specification of limits on the \"degree of deviation\" from these properties tends to be arbitrary rather than based on observation. In other words, any .particular special property is not generally appropriate for our quantitative comparative statics model. In addition, the set of properties that are always true for the Individual firm may be large. I.e. the set of parameters that can be correctly specified as varying numerically over all firms may be large. Nevertheless, the only properties of this type that have come to mind concern comparative static effects. In the remainder of this section, we note the following: (a) all types of changes In exogenous variables that are observable at the level of the Individual firm have been incorporated into our analysis, and (b) all types of comparative static effects that can be derived by the usual methods (primal, primal-dual, dual) for these changes In exogenous variables have been Incorpor-ated Into our analysis. Consider the following problems N I I I N I I I maximize R(x;ct\u00b0) - I c1 (x1 ;a') minimize I c (x1 ;a ) 1=1 1=1 _ subject to R(x) = R C*>3. This Is true even when an \"average\" of firms Is to be modelled. For example, It Is not clear how to average one firm where T T ( X ; C I ) Is separable and another firm where Tr(x;ct) is observed to be non-separable. N I 1 = 1 m a x i m i z e R ( x ; a \u00b0 ) - c ' ( x 1 ; a ' ) 2 6 2 S s u b j e c t t o I c 1 ( x 1 ;a') = C f = 1 w i t h e x o g e n o u s v a r i a b l e s (a 0,.., at*), (a 1,.., a^, R ) a n d (a 0,.., a^, C) r e s p e c -t i v e l y . An a r b i t r a r y s u b s e t o f i n p u t s c a n a l s o be c o n s i d e r e d f i x e d f o r t h e s e p r o b l e m s . T h e s e p r o b l e m s a p p e a r t o a c c o m m o d a t e a l l t y p e s o f c h a n g e s i n e x o g e n -o u s v a r i a b l e s t h a t a r e o b s e r v a b l e a t t h e l e v e l o f t h e I n d i v i d u a l f i r m , a n d r e l a t i o n s d e f i n i n g [ T T . J ( X * ) ] i n t e r m s o f c o m p a r a t i v e s t a t i c e f f e c t s o f i s o l a t e d c h a n g e s I n t h e s e v a r i a b l e s h a v e b e e n i n c o r p o r a t e d i n t o o u r m o d e l . ^ ' ^ ^ T h e f o r m o f t h e s e r e l a t i o n s b e t w e e n [ t i j . ( x * ) ] a n d c o m p a r a t i v e s t a t i c e f -f e c t s o f i s o l a t e d c h a n g e s i n e x o g e n o u s v a r i a b l e s f o r t h e a b o v e p r o b l e m s h a s b e e n d e r i v e d b y p r i m a l m e t h o d s r a t h e r t h a n b y p r i m a l - d u a l o r d u a l m e t h o d s . F u r t h e r r e l a t i o n s I n v o l v i n g c o m p a r a t i v e s t a t i c c h a n g e s i n e q u i l i b r i u m x a n d X c a n n o t b e d e r i v e d b y p r i m a l m e t h o d s ; b u t i t i s n o t i m m e d i a t e l y o b v i o u s t h a t r e l a t i o n s b e t w e e n [n ( x * ) ] a n d e q u i v a l e n t c o m p a r a t i v e s t a t i c e f f e c t s e x -p r e s s e d i n a d i f f e r e n t a n d m o r e o b s e r v a b l e f o r m c a n n o t b e d e r i v e d b y p r i m a l - d u a l o r d u a l m e t h o d s . N e v e r t h e l e s s , I t appears . t h a t p r i m a l - d u a l a n d d u a l m e t h o d s d o n o t l e a d t o a n y a d d i t i o n a l c o m p a r a t i v e s t a t i c p r o p e r t i e s t h a t c a n b e a s s o c i a t e d w i t h [ n | j ( x * > ] . 7 Summary o f M a j o r Q u a n t i t a t i v e R e s t r i c t i o n s In t h i s s e c t i o n , we s u m m a r i z e t h e m o s t I m p o r t a n t o f t h e p r e v i o u s l y e s t a b -l i s h e d r e l a t i o n s b e t w e e n ( a ) k n o w l e d g e o f v a r i o u s p a r a m e t e r s o f t h e p r o d u c e r p r o b l e m N I 1 = 1 m a x i m i z e i r ( x ; a ) = R ( x ; a \u00b0 ) - Z c ' ( x 1 , - a 1 ) fc>4. S e e s e c t i o n 3 . A s i m u l t a n e o u s c h a n g e i n t w o o r m o r e e x o g e n o u s v a r i a b l e s w o u l d b e r e a l i s t i c I f ( e . g . ) t h e f a c t o r s u p p l y o r p r o d u c t d e m a n d s c h e d u l e s ( o r p r o d u c -t i o n f u n c t i o n ) f a c e d b y o n e f a r m r e c e i v i n g c o m m u n i t y p a s t u r e a r e s i g n i f i c a n t l y a f f e c t e d b y t h e a c t i v i t i e s o f o t h e r f a r m s r e c e i v i n g c o m m u n i t y p a s t u r e . T h e e f f e c t s o f s i m u l t a n e o u s c h a n g e s I n e x o g e n o u s v a r i a b l e s c a n I n p r i n c i p l e b e I n c o r p o r a t e d e a s i l y I n t o o u r a p p r o a c h ; b u t s u c h m o d i f i c a t i o n s d o n o t a p p e a r a p p r o p r i a t e f o r t h e c o m m u n i t y p a s t u r e p r o g r a m s s t u d i e d a n d I n g e n e r a l w i l l n o t b e e a s y t o q u a n t i f y . fcfc. S e e t h e d i s c u s s i o n o f p r i m a l - d u a l a n d d u a l m e t h o d s t h a t I s i n c l u d e d i n A p p e n d i x 4. 263 (b) the maximization hypothesis, and (c> the comparative static effect 9x* for 3a7 this problem. These relations are presented In Figure 3. First, note that the (N + I)2 equations in the system [A] [K] = I can be reduced to (N + 2)(N + 1) equations without any loss of content, and the (S + I)2 2 equations in a system [A ] [L] = I can be reduced to (S + 2)(S + 1) equations 2. without any loss In content. For example, the restrictions implied by the (N + I)2 equations l n , j ] j c j l _-,4--!--- [K] = [M] = I _ c i I \u00b0_ in our model are expressed exactly by the (N + 2)(N + 1) right hand upper tri-2 angular equations of the system [M] = I: [o^T = [o^ -] or equivalently Ms s - I\u00bb . * o r a'' ('\u00bbj > such that i = 1,.., M I N + 1 and j > I. (3a) The argument for this statement can be sketched as follows. The matrix , \" [ T t i i J ! c ! ~ , c|T j o J (N+1)x(N+1) has full rank by the restriction that [n j j ] Is negative definite.^7 Thus the equations Mj j = Ij j (i = 1,.., N + 1) determi ne (K^ j ,.., KN + ^ j) for a I I j = 1,.., N + 1 and any feasible [TT|J3 - Therefore, the equations Mj N + ^ \u00bb ' i , N + 1 ( 1 = 1 \" \" N + ] ) d e T e r t n I n e ( K i , N + !\u00bb\u2022\u2022\u00bb *N + 1, N + I5 , n o u r model, and (given the value for N + i = KN + i ^ * h e eaua+'ons M i , N = Ij N (I = 1,.., N) determine (K^ N\u00bb*\u00ab\u00bb N }' e + c* T h u s + n e stations between any feasible matrix [A] and any symmetric matrix [K] that are Implied (el\\ See Theorem 3. ; (o%. The reader can verify that, tn contrast to the case of right hand upper triangular equations, necessary and sufficient conditions for determining [K] are not automatically satisfied In the cases of left hand upper triangular equations and of left or right hand lower triangular equations. 264 by t h e r e s t r i c t i o n [ A ] [ K ] = I c a n b e i n c o r p o r a t e d i n t o o u r m o d e l a s t h e ( N + 2 ) ( N + 1) e q u a t i o n s ( 3 2 ) . L i k e w i s e , t h e r e s t r i c t i o n s i m p l i e d b y t h e 2 ( S + 1 ) 2 e q u a t i o n s [ L ] = [ N ] = I \" [ * , \/ ] i c ! A \" i 1 1 ' .AT I L c i I \u00b0 i n o u r m o d e l a r e e x p r e s s e d e x a c t l y a s t h e ( S + 2 ) ( S + 1) r i g h t h a n d u p p e r t r i -2 a n g u l a r e q u a t i o n s o f t h e s y s t e m [N ] = I : N t : = I | : f o r a l l ( i , J ) s u c h t h a t i = 1 , . . , S + 1 a n d j > I . (33) ' \u00bbJ ' \u00bbJ \u2022\u2014 A s c a n b e s e e n f r o m F i g u r e 3 , t h e t o t a l n u m b e r o f e q u a t i o n s a n d v a r i a b l e s i n t h e s e t o f c o n s t r a i n t s i n c r e a s e s e x p o n e n t i a l l y w i t h t h e n u m b e r o f i n p u t s N a n d a l s o w i t h t h e n u m b e r o f d e c o m p o s i t i o n s . F o r N = 3 , 3 7 q u a d r a t i c e q u a t i o n s a n d 4 5 v a r i a b l e s a r e d e f i n e d w h e n e a c h o f t h e t h r e e p o s s i b l e d e c o m p o s i t i o n s , g i v e n o n e f i x e d i n p u t , i s I n c l u d e d i n t h e s y s t e m . F o r N = 4 , r e l a t i o n s A - C i n v o l v e 2 9 q u a d r a t i c e q u a t i o n s a n d 44 v a r i a b l e s , a n d t h e s e t o f a l l p o s s i b l e d e c o m p o s i t i o n s , g i v e n 1 t o 2 f i x e d I n p u t s , a d d s 7 6 q u a d r a t i c e q u a t i o n s a n d 76 v a r i a b l e s t o t h e s y s t e m . F o r N = 5 , r e l a t i o n s A - C d e f i n e 3 6 q u a d r a t i c e q u a -t i o n s a n d 6 2 v a r i a b l e s , a n d t h e s e t o f a l l p o s s i b l e d e c o m p o s i t i o n s w i t h 1 t o 3 f i x e d I n p u t s a d d s 2 3 5 q u a d r a t i c e q u a t i o n s a n d 2 3 5 v a r i a b l e s t o t h e s y s t e m . T h u s t h e s i z e o f t h e s y s t e m o f c o n s t r a i n t s I s p a r t i c u l a r l y s e n s i t i v e t o t h e n u m b e r o f d e c o m p o s i t i o n s t h a t a r e I n c l u d e d I n t h e s y s t e m . F o r N > 3 , I t i s q u i t e t e d i o u s t o I n c o r p o r a t e a l l s u c h d e c o m p o s i t i o n s I n t o t h e m o d e l . H o w e v e r , I n g e n e r a l k n o w l e d g e o f t h e s t r u c t u r e o f t h e f i r m ' s s t a t i c m a x i m i z a t i o n p r o b l e m w i l l r e l a t e o n l y ( o r p r i m a r i l y ) t o a s u b s e t o f d e c o m p o s i t i o n s . 265 (First page of Figure 3) (A) first order conditions for a maximum . , 3x IK,-.] 11 3a1 N quadratic equations (B) second order conditions for a maximum - fTtjj] = [H] [H] N(N + 1) quadratic equations H. . > 0 I.J (j = 1. ---.N) N bounds (C) long run decomposition (see Theorem 3) I iT I ' I [K] = I (N+1) x (N + 1) (N+2KN+1) . 2 independent quadratic equations iL , i . iU c. < c. i c. i i i 3x dx 3F 3F K. L < c'. | K. . < K. . i.J '.J K. L < R \u2022 K. . < K. U ' . J y LI '.J K. . L < -R 2 \u2022 K. . < K. . U U y \".J R L \u00a3 R \u00a3 R U y y y (i = 1, \u2022\u2022\u2022\u2022N) (i. i = 1, \u2022\u2022\u2022,N) (i = 1, \u2022\u2022\u2022,N and j = N+1) (i.j = N+1) 2(N + 1) bounds 2 bounds (D) decompositions, given fixed inputs (see Corollary 5); for each decomposition with N-S fixed inputs: I I T , A. 1 iA I W i J 1 I C i 3x iAT C i | i**S [L] = I (S+1) x (S + 1) (S + 2)(S+1) 2 independent quadratic equations 3CT i**S L. . L < c! : \u2022 L. . < L. ^ '.J | 0 ) i.J i,j 3x _ : L 7 < R \u2022 L . . < L . . 3F '-J y '.J I . J (i.j = 1,-\",S) (i = 1 , \u2022\u2022\u2022,N and j = S+1) 2(S+1) bounds F I G U R E 3 Summary of Major Constraints for the Quantitative Comparative Statics Model 7 0 The mark \" \u2014 \" is placed above any symbol that refers to a constant rather than an endogenous variable in the model. \" ^ F o r definitions of the symbols used here, see Theorem 3 and Corollary 5. R E 3 where y = F(x) and R(y) = R ( F ( x ) ) . V Y 266 (Second page of Figure 3) (D) (continued) 2(S + 1 ) 2 bounds (E) non -decompositions (output exogenous), given fixed inputs (see Corollary 6); for each non-decomposition with N-S fixed inputs: I i j j ^ l (P) - I | (S+2HS + D j n cj ependent quadratic equations (SxS) (SxS) i*S . : P. > < c! : \u2022 P. . < P. U (i.j = 1, ...,S) 3 A J '.) JOJ i.j i.j totals: (N+2MN+1) N(N+1) * N quadratic equations (N+2)(N+1) N(N + 1) + + 2N + 1 variables (S+2)(S+1) additional quadratic equations and variables for each decomposition 2 non-decomposition (C,D,E) with N-S fixed inputs F I G U R E 3 Summary of Major Constraints for the Quantitative Comparative Statics Model 1 G-(Footnotes fcfl andTO are the same as the previous page) 267 APPENDIX IV QUANTITATIVE COMPARATIVE STATICS AND DERIVED DEMAND: PROOFS Page 1. On Integrability in Comparative Statics 268 2. Lemma 4 272 3. Proposition 1 276 4. Constrained Maximization , 282 5. Theorem 3 285 6. Corollary 5 294 7. Corollary 6 302 8. Quantitative Comparative Statics of a Shift in a Firm's Product Demand Schedule 305 9. Primal-Dual and Dual Methods of Quantitative Comparative Statics 319 268 APPENDIX IV QUANTITATIVE COMPARATIVE STATICS AND DERIVED DEMAND: PROOFS 1. On Integrability in Comparative Statics The problem of determining the precise conditions corresponding to the comparative static implications of the maximization hypothesis has been called one of the major remaining challenges in the theory of compara-tive statics (Silberberg, 1971a), arid has been largely solved in the context of generalized duality theory (Epstein, 1978). Here we shall sketch a solution to this integrability problem in terms of primal methods of comparative statics and point out relations between the primal and dual approaches. Consider the general primal problem maximize Tr(x;a) . . . .(P) subject to g(x;a) = 0. with an interior solution x*(a) > 0. The only conditions that are placed on the Hessian [TT..(X*)] by the assumption of an interior solution and twice differentiability are negative semi-definiteness subject to constraint and symmetry. However, it is well known that the comparative static 9 x effect -r\u2014 is determined uniquely by [TT..(X*)] and g (x*) whenever 269 [TT..(X*)] is negative definite subject to constraint. Moreover, it can be shown that the condition [TT..(X*)] only negative semi-definite subject to constraint contradicts the notion of local comparative statics, i.e., the equations of the total differential are consistent with the implications of this condition only in the meaningless case where TT. (X*) - Xg. (x*) = 0 for all i . 1 Thus the conditions of symmetry and negative definiteness subject to constraint for [TT..(X*)] correspond exactly to the comparative static implications of the maximization hypothesis, i.e., are necessary and d X sufficient conditions for a solution \u2014 to the total differential of first d a order conditions for P to be consistent with the existence of a maximum at 2 3 x*(a) when Tr(x;a) is twice differentiate. ' Thus we have the following conditions for economic integrability: (a) given a primal Tr(x;a) that is twice differentiate, integrability occurs if and only if [TT..(X*)] is negative definite subject to constraint, and (b) given a dual TT(CO that is twice differentiate, integrability occurs if [TT ] is positive definite (when TT(X;a) = R(x) - Z ai* [ 7 Tij 1 9d = ' l a . . . .(e) has no solution 9x da We can show that given that [A] is an N x N matrix and C, X are N' x 1 vectors: [A]X = C has a unique solution x if and only if [A] has rank N,7 (f) [A] negative semi-definite only => [A]X =0 for at least one X $ 0. (g) By (e)-(g). if [TT.. ] is negative semi-definite only, the system ' 1 [ 7 rij ] da- = ' l a 9 x may have multiple solutions {-^ \u2014} which is statement B of the Proposition. By definition. 8 See Murdoch (1970), p. 112. [A] negative semi-definite only implies that there exists a scalar A* = 0 and a vector x* * 0 such that [A] x* = A*x* {see Madley, 1961, p. 256); so [A]x =0 for an x \u00a3 0 280 P P [TT.. ] negative semi-definite only => X'[TT.. ]x = 0 for some x t 0 X'[TT..P]X ^ 0 ij for all x. .(h) By (h) and Lemma 4, P 8 x*1 P 8 x* 9 [TT.. 1 negative semi-definite only => [TT.. ] ~ = 0. . . .(i) > 3 a 3 a Assuming that conditions 1-3 hold before and after da. N j* . \\ y x * ) P ! f - - c ] a ( x 1 * ; a ) = 0 \u2022 \u2022 \u2022 j = i * N j* 1 *;M*)P ITT\" = 0 \u2022 = 2, \u2022 \u2022 *,N. . . . .(k) j = l ' J d a By (i) and (k). N i* P * P 9 x [TTJ. ] negative semi-definite only => E T i j . ( x ) ^ ^ = 0 . . . .(I) 9 P 1 * i By (h), maximum Z'[TT.. ]z = 0 where z is N x 1, and {z } includes z1 = 1. P 1 1 1* Therefore, given TT(X) E TT(X) -c (X ;a), the {z*:z = 1} is the solution set for the problem \"maximize Z ' [ T T . . ^ ] Z subject to z' = 1\"; so maximum 0 1 1 * Z ' [ I T . J ]z =c11(x ;a) given z1 = 1. Thus, by Lemma 4-A, *i * *, * 1*^ lTr. Qj = cLfx1*;^; so \u00a3*\u2014 [TT. . P ] ^ - = 0 by Lemma 4-B 3xx IJ 3 X 1 1 1 9a ,J 9a and by TT(X)P = TT(X)^ - c ^ x ^ a ) . 281 1 I * which contradicts (j) for c l a(x ;a) t 0. By Theorem 1-A and TT(X) = TT(XP - c (x ;ct), j_1 \"J 3 a . =1 Ij dx1 . . . .(m) By Lemma 1, N * Q 8x j* _ 92TT(X*)Q \u00a3 7L.(x ) \u2014 = 1 \u2014 o - . . . .inj 1-^ ' 3 x 1 a? By Theorem 1-B, 82TT(X*)Q 3x \u2014 2 P,(x ) .(o) By (j) and (l)-(o), P 1 1 * if [TT.. ] is negative semi-definite only and c 1 a ( x ;a) \u00a3 0, * i * 1 * 8 x 1 1 * 8x 1 1 * 8 x then -r is undefined: p,(x c n i(x ; a ) 5 \u2014 \u2014 8a 1 8a 11 da 1 i * - c l a(x ;a) = 0 1 1 * 1 1 * by (j), whereas p^x ) - c^fx ;a) = 0 by the assumptions p tTTj. ] negative semi-definite only, ( k ) and Lemma 3 which is statement C of the Proposition. D 282 4. Comments on Constrained Maximization The problem P considered in our comparative static model is of the form \"maximize TT(X;OO\" rather than of the form maximize TT(X; a) j ... .(D subject to g(x;a) = 0 i=l,\u00ab\u00ab\u00ab,M which is the general classical problem. Here we shall point out that it seems difficult to extend our method of comparative statics to such a problem. How-ever, we shall also note that this does not appear to be a serious limitation of our approach. The main (or at least serious) difficulties in incorporating problem 1 into our approach stem from the following: the second order conditions for 9x* 9 A a solution to 1 do not require that a matrix relating (-5 , \u2014) to shift * j * parameters Tra(x ;a) and gQ(x ;a) be negative definite or semi-definite. Problem 1 can be expressed in Lagrange form as M . . maximize L(x,A;a) = Tr(x;a) - E A^g'(x;a) . . . . .(2) j=1 Total differentiating the first order conditions for 2 yields N i* * * 9 x * * .^ LxixJ ( x ' A ; a ) To\" = \" Lxia ( x ' X ; a ) M ?>Aj i * + Z g (x ;ct) i =1, ---.N j = 1 9 a 1 . . . .(3) N i* i * 9x' i * Z 9 i ( x ; a ) To -' = ~ g a ( x ; a ) i = 1, \u2014,N. j=l 1 283 Equations 3 can be expressed in matrix form as 0 1 C I x ( M x M ) | ( M x N) f G I L X | X X (N x M ) I (N x N) 3 X 3 a G a ( M x 1) ( M x i ) 3x 3a L . x a (N x i ) (N x i ) > ( M +N) x 1 ( M + N) .(4) using obvious notation. Denote the ( M + N) x ( M + N) matrix in 4 as [L]. This matrix cannot be negative definite (due to the M x M submatrix of O's). In addition, [L] has full rank by the usual second order conditions for a constrained maximum;1** so [L] cannot be negative semi-definite only. Thus we cannot specify [L] as negative definite, and instead must specify the more clumsy conditions that the determinant of [L] has the sign N of (-1) , the largest principal minor of [L] has a sign opposite to this, and successively smaller principal minors alternate in sign, down to the principal minor of order M + 1. However, apparently we can ignore problems P of the general classical form (1) without restricting our comparative static method in any * * serious way. The solution set x = x (a) for a problem 1 can also be obtained as the solution set for a suitably defined unconstrained problem maximize Tr(x;a)' . . . .(5) 10See Intrilligator (1971), pp. 496-7. 281 * * 11 where TT(X (a);a)' = TT(X (a); a) for all a. The form of Tr(x;aV implied by the particular problem 1 may not be obvious. However, we shall be interested only in specifying the ~ * restrictions on -r that are implied by a subset of possible o Oi forms Tr(x;a) and G(x;a) =0 for problem 1, and many of these restrictions can be incorporated into a set of equations * d X 12 G(-~ , p) = 0. In this case, defining quantitative com-o ot parative statics in terms of an unconstrained maximization problem does not lead to a serious loss in generality. Assuming that TT(X (a),a) > 0 for all a, we can \"simply\" construct * * 7T(x;a)' such that TT(X (a),a) 1 = TT(X (a);a) for all a and TT(X;O0' = 0 for all * * combinations (x,a) that do not satisfy the relation x = x (a). 12 In some respects, the comparative static effects of fixed factor proportions can be modelled more accurately in terms of (1) than in terms of an unconstrained maximization problem. We can incorporate some \u2014but not all \u2014 of the comparative static implications of fixed factor proportions into d X * a set of equations G ( - r \u2014 , p) = 0 (see Chapter 3). On the other hand, any a OL particular example \u2014but not the general case \u2014 o f fixed factor proportions 1 2 can easily be expressed as G(x) = 0 (e.g., x - 2x =0). 285 5. Theorem 3 Theorem 3. Suppose that conditions 1-2 are satisfied for a problem P N . . j maximize TT(X) E R(X) - E c (x ;a) . . . .(1) i=1 * and assume that this problem has a unique global solution x where the Hessian matrix for TT(X) is negative definite. Construct the related problem maximize TT(X) subject to R(x) = R(x ) \u2022 which can be expressed in Lagrange form as * maximize TT(X) - X(R(x) - R(x )). ... .(2) matrix [A] (N + 1) x (N x 1) where ij denotes the Hessian matrix for TT(X) at x , (N x N) 13 i _ i i This theorem is easily generalized to the case c = c (x;ct) (i = 1, \u2022\u2022\u2022,N); but the equations in the generalized theorem are somewhat more detailed than here, and the generalized theorem will not be employed in our research. Construct the symmetric TT.. 1 i c. \u2022j ; 1 ( N x N ) ; (1 x N ) i i C. i 1 ! 0 ( N x i ) ; (1 x i ) V J 286 9 A i 1 , 1 * 1, . N, N* N. c. r 9c (x ; a ) 9c (x ;a J \\ a n d ' n E 1 9x1 ' ' 3 xN > (N x 1 ) [A] necessarily has full rank, and denote its inverse as [K] [K] a ( N + 1 ) x ( N + 1 ) [A] ^ = always exists, Then (A) the comparative static effects for problem 2 are uniquely defined as follows: 9x'** _ 92c'(x'*;a!) 3aj 9xJ 9aJ ''' K. . i,j = 1 f - f N 9x ** K i = 1 \u2022 \u2022 \u2022 N i,N+1 9R 9X _ 32c'(x'*;a!) *~ \" 9x'9a' N + 1'' N+1,N+1 3aJ J 9a 9X = -K K i = 1 \u2022 \u2022 \u2022 N 9R where K. . = element (i,j) of matrix [K], and K. . = K. .(i,j = I, \u2022\u2022 \\N+1) ;and '\/J * i\u00bbJ J\/1 9 x (B) (a) the comparative static effects ( 9a ) t\"or problem 1 are unique, and \u2022 +u \u2022 v r is 9c'(x '* ;g') j. . 1 5 (b) given that Z E K. \u2022 ^ t - 1 , i=1 j = 1 1 9xJ !** 1 i i* i 2 i i* i i * * 1*Thus 3 x . = (3 C ( X ; a ) \/ 3 C, ( X- ; a ) ) 3 x. and 3 a' 9xJ9a' h'Sct' 9 a - * 2chx]*J)...jx!!l ( i,j = 1 , . . . , 9a1 9x' 9a' 9R N ) N N ar'fx'*^! 1 5A sufficient condition for E E K. \u2022 d 1 \u2022 ' J * - 1 (a) i=1 j=1 9x' is that K. \u201e, , S 0 (i = 1,\"\u00ab,N), which is equivalent to ruling out the possibility I , N +1 287 * 3 x -r- for problem 1 is uniquely defined in terms of 3 a 1 32cj(xj*;a!) * and the elements of [K] corresponding to 3 x J 3 a J dx-rr\u2014 and \u2014 \u2014 for problem 2, as follows: 3 a J 9 R 3 x 1 * = oW*-J) . K + K j R j x J , i J = 1 , . . . , N 3 a ' 3 x ' 9 a ' ' ' N + 1 3 a * * N i i* i i* 3R(x ) _ \" 3c'(x' ; a ' ) . 3 x ^ _ . . . . . N 9 a 1 i=1 3 x 3 a ' 7\\ Proof. Suppose that x is a unique interior global solution for the problem P N . . . maximize TT(X) = R(x) - Z c (x ; a ) . ... .(a) i = 1 Construct the related problem maximize TT(X) . . . .(b) 5 T subject to R(x) = R(x ) which can be expressed in Lagrange form as JT maximize Tr(x) - A(R(x) - R(x )) . . . . .(c) Footnote 15 (continued) of inferior inputs \u2014> o, i = N). Condition (a) would be violated only 3 R for a relatively few \"appropriate\" degrees of inferiority; so condition (a) is not a serious restriction. 288 By (a)-(b), x is the solution set to problem (c). ... .(d) By (c)-(d), TTj(x ) - ARj(x ) = 0 i = 1, \u2022\u2022 -,N \u2022 \u2022 . .(e) R(x*) = R(x*) which are the first order conditions for a solution to problem (c). Total differentiating (e). N * i * i * Z TT..(X ) dxJ - cj^i (x ;ot ) - Rj(x )dX N R , 'J - X Z ..(x*) dx' = 0 i = N . . . .(f) j=1 N * j _ Z R.(x ) dx - dR = 0 ... .(g) i=1 1 given (a). By (c), X d-rr(x*) 3R N i * * Z 7T.(x*) ~ =0 . . . .(h) i=l 1 3R by (d) and conditions 1-2. By (h), (f) reduces to By (a) and conditions 1-2, (g) can be rewritten as N i i* i i -Z c!(x' ; < x ) dx' - dR = 0 i=1 1 Construct the symmetric matrix i c. i ( N x N ) ! ( 1 x N ) c i : o i ( N x i ) ; ( i x i ) [A] ( N + 1 ) x ( N + 1 ) where TT.. ' J ( N x N ) = Hessian matrix for TT(X) at x* c. \/ . , M - r 1\/ 1 * K \u2122 i N \" ( 1 x N ) = (c^x ; a ),\u2022\u2022\u2022, c N(x ; a )J N , N * N , By definition, [A] has less than full rank if and only if N . . there exists a vector v t 0 such that Z v' IT.. - c. j=1 , J = 1 , N , c J Z vJ ! j=1 but this statement implies that 290 N N there exists a vector v ^ 0 such that Z I TT.. VV = 0 1=1 i=1 , J which contradicts the assumption that [TT...] is negative definite. Thus [TTJ.] negative definite => [A] has full rank. . . . .(***) By (i)-(l). [A] X = C ... .(m) ( N x i ) ( N x i ) where 1 N T X = (dx ,\"\u00ab,dx ,-dA) for problem (c) ( N x i ) T .(n) C = (c. i(x ;a )da ,\"-,c N(x ;a )da ,dRJ ( N x i ) l a N a By assumption, [A] 1 exists, and [A] 1 E [K] is symmetric . . . .(o) by the symmetry of [A]. By (o), X = [K] C . ... .(p) By (n)-(p), for problem (c) 291 \u2022 ** 9x \u2014, = c! j(x'*;a!) \u2022 K. . i,j = 1,---,N 9 A J JoJ U 9x'' 9R 9 X = K i = 1 \u2022 \u2022 \u2022 N i,N+1 ' . >n j = -c\\](j[j*j) - K N + 1 J J = l , \u2014 . N . . . . ( * * * ) 3a 1 A _ = - K 3R N+1, N+1 K. . = K. . i,j = 1, \u2022\u2022\u2022,N+1 where K.j = element (i,j) of [K], which is statement A of the Theorem. By Propositionl-A and the assumption that the Hessian matrix for * TT(X) ([ TT.J]) is negative definite at x , *. I 3 x. I for problem (a) contains only one 1 i 9a1 . . \u2022 .(q) vector j = 1,***,N. 9aJ By the implicit function theorem and [TT..] negative definite at x , * we can solve the first order conditions of problem (a) for x as a function of a: i* x (a) i = !,\u2022\u2022\u2022,N for problem (a). ... .(r) 292 _ * * Given R 5 R(x ), x is also the solution set for problem (c); * so we can also solve (e) of problem (c) for x as a function of (a,R): ; * ; * * _ _ * x = x (a,R) i=1,\"\u00ab,N given R E R(x ) . . . .(s) for problem (c). Substituting (r) into (s), i* i * * r N~* ^ x = x (a,R(x (a),\u00ab\u2014(a ) ) J i = !,\u2022\u2022\u2022,N (t) By (t) and the rule for the differential of a composite function, \u2022* :** \u2022** * 9x 9x ^ 9x 9R(x ) . Kl , , \u2014 j - = r \u2014 + \u2014\u2014\u2014 \u2022 \u2014p - I,J = 1,\"*,N , . . . .(u) 9aJ 9aJ 9 R 9 a1 where 9R(x ) 9 a I R.(x ) i * i=1 9 a' j = 1,'--,N .(v) By (u) and (***), i* * 9x _ j . , j*. L . K K . 9R(x ) j ; = 1 ... N r w i \u2014 y - - C j a , l X ,0fJ K i j + K i \/ N + 1 ''J ,N- \u2022 .IWj By (v) and conditions 1-2, 293 9R(x*) 3a' N I i . i i % c.(x ; a ) 3x 9 a) j = 1, ,N . .(x) By (w)-(x), I ( N x N ) Ki,N+1 ( N x l ) -f-c! | - i ( N x l ) | (1x1) (N+1) x (N+1) r 1* 1 9x' 9a1 \u2022 \u2022 N* 9x\u2122 3a' 9R(x*) _ 9a' (N+1) x 1 K i \u2022 K N,j| 0 j = 1, ,N \u2022 (y) where I E identity matrix and K. = N+1'st column of [K], ( N x N ) ( N ' x 1 ) Denote the (N+1) x (N+1) matrix in (y) as [L]. By (y) and the definition of a determinant. [L] N N . . . . \u2022 ^ = -1 - E Z K. ... \u2022 c.fx' ;a!) . ... .U) 1=1 |=1 '' N + 1 J Since (y) has a unique solution 9x 9R(x ) 1 { j a r b i t r a r y) i f a n d o n| y i f [ L] _ 1 exists <\u2022 9a1 ' 3 a' or equivalently |[L] | 0, (q) and (y)-(z) imply that g X (a) r is unique for problem (a) (j = 1,\"*,N) 9cxJ\u00b0 N N \u2022 J(xj*-a!) t -1 (b) given that Z Z K. N + 1 j 1 , o r j F * 9 x r- for problem (a) is uniquely defined in terms of 9aJ cjaj(x' a n C' t'16 e ' e m e n t s \u00b0^ corresponding to * * \u00a3^ r- and for problem (c) Cj = 1, ---,N), 9aJ 9R as follows: 3x'* _ i J * . J . . \u201e . \u201e . 9R(x ) j * * = c| ,(x' ;cr) \u2022 K, : +K : M a 1 . ^ 2 L \u00b1 j , j = 1 , . m*J- = Z c V * ; ^ ) j=1, 9aJ i=1 ' 9aJ which is statement B of the Theorem. D 6. Corollary 5 Corollary 5. Construct the problems 1 and 2, and the (N+1)x(N+1) matrices [A] and [K], as in Theorem 9 . Partition the Hessian matrix [TT..] and marginal factor cost vector c! of [A] as (Nxj\\|) ' (1xN) follows: (NxN) f A 1 TT.. 1 \u2022J j B 1 TT.. \u2022J (SxS) | (SxT) c 1 TT.. 1 \u2022'J | D u.. \"J (TxS) ! V 1 (TxT) (IxN) ( -A ! \u2022Bl i 1 \u2022 c. 1 c. > 1 4 (IxS) (IxT) where S+T = N. Construct the following symmetric matrix A ( S x S ) \u2022 .A : c ! ! (sxi) .A c. ! o ( I x S ) : ( i x i ) [ A n ] . (S+1)x(S+1) [A^] necessarily has full rank, and denote its inverse as [L] [A^] = [L] always exists. Construct the problem maximize TT(X) N . . . R(x) - Z c'(x ';a ) i=1 subject to x' = x' j = S+1, \"vN where x is the unique global solution to problem 1 Construct the related problem maximize TT(X) subject to R(x) = R(x ) j = S+1, \u2022 \u2022 \u00bb,N j j* x' = x' which can be expressed in Lagrange form as N maximize TT(X) - A(R(x)-R(x )) - Z yfx'-x' ) j=S+1 Then 296 (A) the comparative static effects for problem 4 are uniquely defined as follows: i**S 3x 32J(xj*;gJ) . L 3aj dxha) l , J U = 1 3x' a 3R Li,S+1 i = i , - . s 3 X S 9aJ 3 X 1 3R ' Zxho} S + 1 ' j j = i, \u2022\u2022\u2022,s LS+1,S+1 where L . . = element (i,j) of [ L ] , and L- . = L . .(i,j = 1, \u2022\u2022 *,S+1); ' 11 ' i J I\u00bb' and *S 3 x (B) (a) the comparative static effects for problem 3 are 3 a unique, and (b) given that E E L . _ \u2022 5 \u2014 r i=1 j=1 ''s 1 3xJ 3cJ(xJ ; a J ) , , 16 * -1, 16 . S S 3c'fx'*-ah Assuming that E I L. - . ^ . ^ ' ^ - i has implications analogous to those of assuming that E E K. N \u2022 i x . ; a ' i - 1 i=1 j=1 ''N+1 3xJ (see footnote to Theorem 3). 297 *S 9x * 9a1 3 2J(xJ*;oJ) for problem 3 is uniquely defined in terms of and the elements of [L] corresponding to **c **c 9 x 9 x \u2014\u2022\u2022 and \u2014 for problem 4, as follows: 9a1 9R :*c 2 i i * i * 9x' ^ = 9 c'(x' ;aJ) L + L 9R(x ) 3 9aj \" 9x' 9a' ' ''' ! ' S + 1 ' 9 a! * = x ' j = S+1, \u2022\u2022sN \u2022 (g) which are the first order conditions for a solution to (d). By argu-ments identical to (f)-(n) inthe proof of Theorem 3, [ A \u201e ] X S = C S (S+1)x(S+1) ( S x l ) ( S x l ) given (g) \u2022 (h) where (S+1)x(S+1) A TT.. 'J c i ( S x S ) ( S x l ) j A c. i 0 ( I x S ) (1x1) TT.. = submatrix for inputs i = 1, \u2022 \u2022 \u2022 , S of the Hessian ( S x S ) for TT(X) at x* * \" \" (0) continued on following page) 299 r _ , 1, 1* 1. S , S * S ' Cj = (c^x ;a ), \"',cs[x ;a )) (Sx1) S i S ^ x = (dx , \u2022\u2022\u2022,dx,-dX) . . . .(i) (S+1)xl _ 1 1 * 1 1 s \u00ab;*\u00ab;<;_ T c S = (cj a i(x ;a')da\\-.^c|^(x ;a )da ,dR) . (S+1)xi By definition, [A^] has less than full rank if and only if there s i i exists a vector v \u00a3 0 such that E v' TT.. - c. = 0 i = 1, \u2022 \u2022 \u2022 , S j = l , J ! v ' c i = 0 ; but this statement implies that S S there exists a vector v \u00a3 0 such that E E TT.. V'V' = 0 i = 1 j = 1 lj which contradicts the assumption that [TT..] (hence [TT.. ]) is negative definite. Thus [TT..] negative definite =*> [A n] has full rank (***) Then 1 (S+1)x(S+1) [A^] = [L] is symmetric . . . .(j) 300 by [A^] symmetric. By (h) and (j). S S X* = [L] cr By (i)-(k), for problem (d) the comparative static effects are uniquely defined as follows: **c c **c c 8x 3 9 X s 9x 3 9 A 3 d a ' d a ' 9R ' 9R 9x i**S 9a\/ c' f x ' -ah \u2022 L. . cjaT x , C C ) i,j 9x' = L. 9R ,S + 1 i = 1,-,S 9 A\" 9a! j = 1,-,S 9 A-9R \u2022S + 1,S+1 where L. . = element (i,j) of [L], and L. . = L. .(i,j=1, \u2022 \u2022 \u00ab,S+1) i, J ' i J J\u00bb1 which is statement A of the Corollary. By (b) and (e). * T T j ( x ) X * = x ' i * i = 1, \"^S j = S + 1, \u2022 \u2022 \u2022, N . .(I) 301 which are the first order conditions for an interior solution to problem (b). By (i) and (I), differentiating the first S first order conditions with respect to a} (j = 1 , \u2022\u2022\u2022,S) yields [TT ] 9x ij 9 a J ( S x S ) ( S x l ) \u2022 c | a i ( x J * ; a J ) a} = a , -\",0^ . . . .(m) where 9x 9a J 9x 1*S 9x S*S 9 a' 9a J [TT..] Since negative ( N x N ) r A l definite implies that [ 7 Tij is negative definite, ( S x S ) 9x 9a J for problem (b) contains one and only one vector 9x 9 a (j = 1 , \u2022\u2022\u2022,S). .(n) by (m), [TTJ.] negative definite and Proposition 1-A. By arguments ( N x N ) identical to (v)-(z) in the proof of Theorem 3 , 9x *S s s \u2022 \u2022 * \u2022 given that E E L . _ , \u2022 c!(xJ ;aJ) t - 1 , i=1 j = 1 ' ' S + 1 ' 9a\/ for problem (b) is uniquely defined in terms of cj^fx1 ;a}) and the elements of [L] corresponding to ... .(***) ( (***) continued on the following page) 302 \u2014 . and for problem (d) (j = 1,---,S), as 3a1 8R f0\"0WS: i ? \" = c I | o J ( x , * ; J ) ' L U + L i , s + 1 ' ^ X ^ \u2022 ' \u2022 -(***} i,j = 1, \"^S 3R( X*) S - I cW^a') .^L j=l,-,S. 3a1 i=1 ' 9aJ Statements (n) and (III) are equivalent to statement B of the Corollary, rj 7. Corollary 6 Corollary 6. Construct problems 1 and 3 as above, and partition the (negative definite) Hessian matrix [TTJ.] as above. Then the ( N x N ) comparative static effects for problem 3 are uniquely defined as follows: 8x'*S 92cj(xj*;aJ) D . . , _ 8a where P. . = element (i,j) of l T rij J (which always exists), ( S x S ) -J and P. . = P. ; (i,j = 1, -\".S). ' \u2022 J J \u20221 Proof. Construct the problems maximize TT(X) = R(x) maximize TT(X) = R(x) i ~T*~ subject to x' = x' where x* is a unique global solution to problem (a). Partition the * Hessian matrix of TT(X) at x as follows: A B TT.. \u2022 TT.. 'J ' 'J ( S x S ) ( S x T ) C D TT.. , TT.. 'J 1 ' J ( T x S ) i ( T x T ) . . . .(c) By the assumptions that (N' N^) is negative definite and symmetric, -1 A -1 [TT..] and [TT.. ] exist and are symmetric. . . . .(d) ( N x N ) 303 N . Z c (x1 ;a ) =1 N Z c (x1 ;a') = 1 (a) . . . .(b) j = S+1, \u2022 \u2022 \u00ab , N By (a)-(b), x is the solution set to (b) as well as (a). ... .(e) By conditions 1-2, the first order conditions for a solution to (b) are 304 TT.(X) = 0 i = 1, \u2014,S \u2022 (f) x' = x' j = S+1, \u2022 \u2022 \u00ab,N \u2022 (g) By (c)-(e) and (g), total differentiating (f) yields dx [TT.. ] \u2022 C IJ dxS -( S x l ) ( S x S ) ( S x l ) \u2022 (h) where C = (c|^(x1 ; a 1 I d a 1 , \u2022 \u2022 ^ c^^(x^ ; a S ) d a ^ ) . By ( d ) and (h), for (b) i*S 9 a i CiaJ ( X Pi,j i,j = 1, \"',S A -1 where P. . = element (i,j) of [TT.. ] , and P. . = P. . (i,j = 1, \u2022 \u2022 *,S) i \/J 'J '\/J J\/1 which is the Corollary. \u2022 305 8. A Theorem on the Quantitative Comparative Statics of a Shift in a Firm's Product Demand Schedule Relations between various potentially observable properties of a problem 0 N . . . maximize Tr(x;a) = R(x;a ) - \u00a3 c (x ;a) ... .(P) i=1 and * 8 x 0 \u2014jr- , when da implies a shift in the product demand schedule 8a faced by the firm, are presented in Theorem 4 and Corollary7.17 These relations differ from those specified in Theorem 3 and Corollaries 5-6 in one particularly important respect, which can be explained as follows. When a\u00b0 is a parameter in the product demand schedule, * \u2014Q\u2014 can be decomposed as 8 a : * \u2022 ** * 8x _ 8x m 8F(x ) 8a\u00b0 8F 8a\u00b0 i = 1, \u2022 \u2022 \u00ab,N ... .(1) ** 8 x where is the comparative static effect of dF for the problem 8F maximize TT(X) = P( F(x); a\u00b0) F(x) - Z c'(x';a) i subject to F(x) = F(x ) 17The proof of Corollary 7 is not presented here (Corollary 7 can be established in an abvious manner by the methods used in other proofs) 306 (y = F(x) denotes the firm's production function). In addition, m**L = ( 1 \/ R ) ? isiiA . 1 2 ^ . . . . ( 2 ) 9a y i=1 9x' 3a ** where R(F(x) ;a\u00b0) =R(x;a\u00b0) for all (x,a\u00b0). For a given v e c t o r \u2014 , 9F equations 1-2 constitute a homogeneous system of N+1 equations in N+1 9x* 9 F(x*) unknowns ( \u2014 \u201e , n ). Thus, equations 1-2 can determine the 8 a 9 a\" 9 x * unique \u2014 ^ only up to a scalar multiple, i.e., only ratios .9 a\" 1* i* N* i*-\\ 9X 1 , 9x' ... 9x\u2122 , 9x' ^ 0 0 ' * * ' 0 ' 0 9 a 9 a 9 a 9 a can be uniquely defined by 1-2. A similar statement holds for the de-*S 9 x composition of \u2014 g \u2014 (when a is a parameter in the product demand 9 a ** **c 9 x 9 x schedule). Therefore, knowledge o f \u2014 \u2014 o r {\u2014 } defined by all 9F 9F possible partitions of x into fixed and variable inputs (and knowledge of * ~ 1 1 * N N * 9x R , c^x ),\u00bb\u00bb\u00bb,C|Sj(x )) is insufficient to define the unique \u2014Q\u2014. The y 9 a additional restrictions due to the second order conditions for a maximum * 9 x only imply that the unique \u2014^- is determined up to a positive scalar ,*\u2022 . 1 8 ^ multiple. 18 The proof of this statement can be sketched as follows. The first order conditions imply that the \"correct\" Hessian [TT..*] and comparative static 9 x* effect \u2014 Q satisfy a system of equations of the form tTT,.*3 = [K] . ... .(a) IJ 9aU Nxi Exact knowledge of (\u00a3*\u2014 , R.c (x1 ),\u2022\u2022\u2022, cj(x\" )) implies only the following 9F y relation: 307 Thus restrictions must be placed on other parameters in order to 3 x * obtain both upper and lower bounds on \u2014g- by our methods. In particular, Sx* d C L knowledge of \u2014 r - (i \u00b1 0) and its various decompositions seems quite important 3 a in the quantitative comparative statics of changes in the firm's revenue 3 x * or benefits schedule (whereas, prior knowledge of \u2014g- and its decompositions 3 a is relatively unimportant in the quantitative comparative statics of changes in factor supply schedules). Theorem 4. Suppose that conditions 1-3 are satisfied for a problem P 0 N i i maximize TT(X) =. R(y;a).- I c (x ) . . . .(1) i=1 where y = F(x) is a scalar function, and assume that problem 1 [TT.. ] IJ ( N x N ) has a unique global solution x* where the Hessian JJ is 1 9 negative definite. Construct the related problem ll \u00bb [7T..*]][Y ' ^K-] = [K] -(b) (18 continued) - TT. Y * ^ 7 Y 'J 3a C where y is an arbitrary scalar. Given that E77j j * 3 's negative definite: ^ 0 t^jj*] 's negative definite if and only if y > 0. Thus, relation (b) plus the second order conditions has the solution set T dx* > . _-, 19The comparative static effect \u20223-XQ- is undefined when [TT..] is 3a ' only negative semi-definite (see Proposition 1). 308 0 N i i maximize TT(X) E R(y;a ) - E c (x ) i=1 .(2) subject to y = F(x*) L e t R = 3R(y*;\u00ab\u00b0) y sy = 32R(y*;a\u00b0) y a^ \u2022ao 8 x . 3 a o l y a \u00b0 y ' Construct matrix [A] as in Theorem 3, so that [A] 1 = [K] always exists. Then { ^ o r P r\u00b0b , e m 1 corresponds to the single solution to 3a the system [V 7^0 = - R i a 0 ( N x N ) d a ( N x i ) 2 0 l f R(y;a\u00b0) E P(y;a\u00b0)y, then Ry = P(y*;a\u00b0) + Py(y*;a\u00b0)y* and R y a \u00b0 = P a o ( y * ; + Py a\u00b0^ y* ; a\u00b0^ y** F o r t n e m o r e 9 e n e r a' case where the firm sells all y units at an identical price and also receives non-pecuniary benefits B(t) from the t'th unit of y, R(y;a\u00b0) = P(y;a\u00b0)y + \/ y B(t)dt. where Rjao = (R,aO.'\",RNaO) (Nxl) 309 T (B) the comparative static effects for problem 2 are defined in terms of [A] as follows: j ** 9F y ''N+1 j ** [ K ] = [A]\"1 => = 0 i = 1, \u2022 \u2022 \u2022, N 9 a \u2014 = - R y 2 'KN+1,N+1- i = 1,.\u00bb,N 9F [K] = [A]-' =>^ = R u. 0 ' \\ a c 9 a where K. . = element (i,j) of [A] 1, and K. . = K. . i \/ J 1 \u2022 J J ' 1 (i,j = 1, \u2022\u2022\u2022,N+1); and N N . (C) given that E E cj \u2022 K. N + 1 *-1 , * 3 X7j- is determined up to a scalar multiple by R , c! and 9 a -1 9 x** the elements of [A] corresponding to \u2014 \u2014 , i.e., the 9F 3x* 9F(x*) following system has as solution the {all y( Q~\u2022 0\u2014^ 9 a 9a (y an arbitrary scalar): 310 iiiL = R . K \u2022 iL 0 y i,N+1 \u201e 0 i = 1,\u00bb' fN 3 a 3 a 3F N Z i=1 3 x 3 a C i * Proof. Construct the problem N . . maximize TT(X) = R(y;a)y - Z c (x ) i=1 .(a) where y = F(x), or equivalently N j . maximize u(x) = R(x;a) - Z c (x ) i=1 .(b) where R(x;a) E R(F(x);a) .(c) Total differentiating the first order conditions for an interior solution to (b), N H X 1 * -Z TT..(X*)^\u2014 + R. (x ;a) = 0 i = 1,---,N. j = l I J 3 a a . .(d) Since a negative definite matrix has an inverse (see a-b in the proof of Proposition 1 ) , [TT..] negative definite => equations (d) has a unique 'J I***} *