ESSAYS IN THE ECONOMICS OF UNCERTAINTY by LARRY GEDALEAH EPSTEIN B . S c , U n i v e r s i t y of Manitoba, 1968 M.A. , Hebrew U n i v e r s i t y , Jerusalem, 1970 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Economics) We accept th is thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1977 Larry Gedaleah E p s t e i n , 1977 In present ing th is thes is in p a r t i a l fu l f i lment o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho lar ly purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l ica t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of C^&Wiamtf The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date Am£ *7. 19m i i ABSTRACT There have been several recent advances in the theory of choice under uncerta inty that have extended the r e s t r i c t i v e mean-variance framework. Working within the context of a model of expected u t i l i t y maximizat ion, Rothschi ld and S t i g l i t z (1970) and Diamond and S t i g l i t z (1974) present i n t u i t i v e l y appeal ing and t h e o r e t i c a l l y sound d e f i n i t i o n s of "greater r i s k or uncer ta in ty" . Moreover, they show that t h e i r d e f i n i t i o n s are useful as well as c o n s i s t e n t , in that they may be used to der ive comparative s t a t i c s r e s u l t s in which economists are i n t e r e s t e d . In the f i r s t part of the thes is we argue that the above analyses and most re la ted ones are r e s t r i c t e d to models where both the dec is ion var iab le and the exogenous random var iab le that def ines the s t o c h a s t i c environment, are s c a l a r s . Then we extend many of the d e f i n i t i o n s and r e s u l t s to the context o f a general mu l t i va r ia te dec is ion problem. In p a r t i c u l a r , a genera l ized not ion of r i s k independence i s shown to be re levant to behaviour under uncer ta in ty . This general ana lys is i s then appl ied to two s p e c i f i c dec is ion problems: f i r s t , the standard two-period consumer choice problem where current consumption must be decided upon subject to uncerta inty about future income and p r i c e s ; and second, the corresponding problem in the theory of the f i r m , where a competi t ive f i rm must make some production dec is ions subject to uncerta inty about the pr ices that w i l l p reva i l f o r some products and fac tors of product ion . We extend e a r l i e r s tudies of these problems by consider ing disaggregated models, by adopting t h e o r e t i c a l l y cons is tent d e f i n i t i o n s of i i i increased uncer ta inty and by i n v e s t i g a t i n g the ro le of production f l e x i b i l i t y in determining f i rm behaviour under uncer ta in ty . In both the consumer and producer models the c r u c i a l proper t ies of preferences and technology are pointed out and f l e x i b l e funct iona l forms are hypothesized that are amenable to empir ica l es t imat ion . The theory of d u a l i t y plays an important part throughout the formulat ion and ana lys is of both models. F i n a l l y the bas ic theory of producer behaviour analysed above i s appl ied to aggregate U.S. manufacturing data f o r the 1947-71 p e r i o d . We assume that the c a p i t a l stock dec is ion must be made one per iod before the cap i ta l comes into opera t ion , subject to expectat ions about uncertain future p r i c e s , while a l l other fac tors and outputs may be adjusted f u l l y to current p r i c e s . An added important ingredient of the model is the d i s t i n c t i o n between the c a p i t a l stock and u t i l i z a t i o n (deprec ia t ion) d e c i s i o n s , the l a t t e r being made in each per iod a f t e r that p e r i o d ' s pr ices are known. The consistency of the model with the data is inves t iga ted and the empir ica l s i g n i f i c a n c e of our formulat ion of the c a p i t a l u t i l i z a t i o n dec is ion i s t e s t e d . iv TABLE OF CONTENTS Chapter Page I. INTRODUCTION 1 II. SOME BASIC CONCEPTS AND RESULTS IN THE THEORY OF CHOICE UNDER UNCERTAINTY 4 II I . MULTIVARIATE RISKS AND CHOICE UNDER UNCERTAINTY 9 1. Some D e f i n i t i o n s 9 2. The Comparative S t a t i c s 13 3. Risk Independence 18 IV. SOME BASIC CONCEPTS IN DUALITY THEORY 23 V. CONSUMER CHOICE UNDER UNCERTAINTY 32 1. Some Introductory Remarks 32 2. The Model 33 3. The E f f e c t s o f Increased Uncerta inty 37 4. Risk Independence 38 5. Some Empir ica l Impl icat ions 43 VI . PRODUCTION FLEXIBILITY AND THE BEHAVIOUR OF THE COMPETITIVE FIRM UNDER PRICE UNCERTAINTY 50 1. Introduct ion 50 2. The Model 53 3. The Overa l l Impact o f Uncerta inty 55 4. S h i f t s in Fixed Costs and Pr ice Expectat ions 58 5. The Marginal Impact of Uncerta inty 61 6. Technologica l Risk Independence 67 7. An Example 71 8. Some Empir ica l Impl icat ions 73 VI I . AN EMPIRICAL ANALYSIS OF THE EFFECTS OF PRICE UNCERTAINTY ON THE DEMAND FOR CAPITAL AND ITS RATE OF UTILIZATION 76 1. The Model 82 2. Expectat ions 85 3. Funct ional Forms 88 4. S tochas t ic S p e c i f i c a t i o n and Est imat ion Procedure . . 91 5. Data Construct ion and Sources 96 6. Empir ica l Results 100 7. Summary and Conclusions 117 VI I I . CONCLUDING REMARKS 119 V Page FOOTNOTES . . . 122 (BIBLIOGRAPHY 134 APPENDIX A: PROOFS OF THEOREMS IN CHAPTER III 142 APPENDIX B: PROOFS OF THEOREMS IN CHAPTER V 145 APPENDIX C: PROOFS OF THEOREMS IN CHAPTER VI 147 APPENDIX D: DATA FOR U.S. MANUFACTURING 1947-71 152 APPENDIX E: SOME ESTIMATION RESULTS . 155 vi LIST OF TABLES Table Page VI1.1 Summary of "Long Run" Models Estimated 105 VII.2 Summary of "Short Run" Models Estimated I l l VI I .3 Short Run Demand and Supply E l a s t i c i t i e s , For Selected Years and Models 113 D.l P r ice and Quantity^ Data 152 D.2 Tax Var iab les and " E f f e c t i v e " Pr ices of "New" and "Used" Capi ta l 153 D. 3 Capi ta l Stock Ser ies Estimated From Publ ished Data 154 E. l Parameter Estimates From Selected Models (With Labour the Only Var iab le Factor in the Short Run) 155 E.2 Parameter Estimates From Selected Models (With Labour and Energy Var iab le Factors in the Short Run) 156 E.3 Capi ta l Stocks and Depreciat ion Rates Implied by Selected Models 157 v i i ACKNOWLEDGEMENTS I am p a r t i c u l a r l y indebted to Erwin Diewert fo r convincing me to enter the Ph.D. programme and for the encouragement and advice he has provided me during the l a s t several y e a r s . Several people have a s s i s t e d me during the w r i t i n g o f th is t h e s i s . I am grate fu l to Ern ie Berndt , Erwin Diewert, Keizo Nagatani and Alan Woodland f o r valuable suggestions and e x c e l l e n t s u p e r v i s i o n . Discussions with David Rose and Terry Wales on the ana lys is in Chapter VII were b e n e f i c i a l . Keith Wales d id~a l l the computer programming and made p o s s i b l e the est imat ion of the complicated model in Chapter VII . Many thanks are due May McKee fo r typing a lengthy and d i f f i c u l t manuscr ipt . F i n a l l y , I would l i k e to acknowledge g r a t e f u l l y three years of support from the Canada Counci l in the form of a Doctoral Fe l lowship . I. INTRODUCTION There have been several recent advances in the theory o f choice under uncer ta inty that have served to extend the r e s t r i c t i v e mean-variance ana lys is that had l a r g e l y dominated the scene in e a r l i e r y e a r s . Working within the context of a model o f expected u t i l i t y maximizat ion, Rothschi ld and S t i g l i t z (1970) present a d e f i n i t i o n of "greater r i s k or uncerta inty" that i s i n t u i t i v e l y more appeal ing then the equating of r i s k with variance as in the mean-variance approach. Moreover, and th is i s c r i t i c a l , they show (1971) that t h e i r d e f i n i t i o n i s useful as well as c o n s i s t e n t , in that i t may be used to derive comparative s t a t i c s resu l ts in which economists are i n t e r e s t e d . Another "natura l" d e f i n i t i o n of greater r i s k i s developed by Diamond and S t i g l i t z (1974). It i s re la ted to a corresponding "natura l" d e f i n i t i o n of "one ind iv idua l being more r i s k averse than another" and aga in , many q u a l i t a t i v e comparative s t a t i c s r e s u l t s are der ived . The c r u c i a l p roper t ies of the u t i l i t y index in s ign ing these r e s u l t s are those summarised in measures o f r i s k aversion that are s t ra ight forward extensions o f the Arrow (1965) and Pra t t (1964) measures, d iscussed in the context o f u t i l i t y funct ions def ined over a s c a l a r var iab le such as t o t a l wealth. We argue below that the above analyses and most re la ted ones, are l a rge ly r e s t r i c t e d to models where both the dec is ion var iab le and the exogenous random var iab le that def ines the s t o c h a s t i c environment, are s c a l a r s . The f i r s t main ob ject ive of the t h e s i s , t h e r e f o r e , i s to extend many of the d e f i n i t i o n s and r e s u l t s to the context of a general mul t ivar ia te dec is ion problem. Subsequently we apply th is general ana lys is to two s p e c i f i c dec is ion problems: f i r s t , the standard two-period consumer choice problem where 2. current consumption must be decided upon subject to uncer ta inty about future income and p r i c e s ; and second, the corresponding problem in the theory of the f i r m , where a competit ive f i rm must make some production dec is ions subject to uncer ta inty about the pr ices that w i l l p reva i l f o r some products and fac tors o f p roduct ion . E x i s t i n g studies o f these problems are e i t h e r h igh ly aggregated, o r e l se i m p l i c i t l y assume that preferences or technology s a t i s f y some r e s t r i c t i v e s e p a r a b i l i t y c o n d i t i o n s . Most do not adopt the above mentioned d e f i n i t i o n s of increased uncer ta in ty , employing instead others that are much more d i f f i c u l t to j u s t i f y . F i n a l l y , the p o s s i b i l i t y of ex post adjustment to the p r i ces that are eventua l ly r e a l i z e d i s l a r g e l y ignored in s tudies of the f i r m . We propose to inves t iga te the ro le o f production f l e x i b i l i t y in determining f i rm behaviour under uncer ta in ty . The formulat ion and comparative s t a t i c s analyses of both the consumer and producer problems w i l l be f a c i l i t a t e d by the use o f dua l i t y theory. Many a p p l i c a t i o n s of d u a l i t y theory have been surveyed by Diewert (1974). However, with only few exceptions that we mention below, they have been conf ined to d e t e r m i n i s t i c choice models. Thus the research we undertake w i l l demonstrate the broader usefulness of dua l i t y theory. In p a r t i c u l a r , the c r u c i a l proper t ies of preferences and technology in models o f opt imizat ion subject to uncerta inty w i l l be simply expressed in terms o f appropriate dual f u n c t i o n s , and f l e x i b l e funct iona l forms, that allow the econometric ana lys is of the models, w i l l be hypothesized. One of these funct iona l forms is used in Chapter VII to estimate technology given that c a p i t a l stock dec is ions are made subject to uncerta inty about future p r i c e s . As f a r as we know, very few e x i s t i n g econometric 3. f a c t o r demand studies have taken into account the fac t tha t , t y p i c a l l y , many dec is ions have imp l ica t ions fo r the future about which expectat ions are not p e r f e c t l y c e r t a i n . The procedure we ou t l ine should enable us to determine e m p i r i c a l l y the e f f e c t of p r i c e uncer ta inty on the demand f o r c a p i t a l and i t s expected rate of u t i l i z a t i o n . The remainder of t h i s thes is may be o u t l i n e d as fo l lows: In II we. descr ibe in more de ta i l some of the recent work in choice under uncerta inty mentioned above, and in III some mul t i va r ia te extensions are developed. Some bas ic concepts in d u a l i t y theory are presented in IV. Chapters V and VI apply these concepts and the ana lys is in III to the consumer and producer prob lemsrrespect ive ly . (Proofs of many o f the techn ica l propos i t ions are located in the Appendices. ) F i n a l l y , VII contains an empir ica l i n v e s t i g a t i o n of the theory o f producer behavior discussed in VI . A b r i e f summary concludes the t h e s i s . 4. II. SOME BASIC CONCEPTS AND RESULTS IN THE THEORY OF CHOICE UNDER UNCERTAINTY Rothschi ld and S t i g l i t z (1970) have provided an answer to the question "when i s a (sca la r ) random var iab le $ 'more v a r i a b l e ' than another (sca lar ) random var iab le 0?" They consider the fo l lowing three poss ib le d e f i n i t i o n s : D e f i n i t i o n I1.1: $ i s more var iab le than 0 i f : D.l $ = 0 + Z , where "=" means "has the same d i s t r i b u t i o n d d as" and Z i s a random var iab le such that E[Z/e] = 0 f o r a l l 0. D.2' EU(0) > EU($) f o r a l l concave u t i l i t y funct ions U. "r t D.3 T( t ) = (G(x) - F(x))dx > 0 fo r a l l t E [ 0 , 1 ] , and •'0 T ( l ) = 0, where G and F are the cumulative d i s t r i b u -t ion funct ions corresponding to $ and 0 r e s p e c t i v e l y and where, f o r s i m p l i c i t y , the range of each random var iab le i s r e s t r i c t e d to the in te rva l [ 0 , 1 ] . D.l and D.2 are e a s i l y i n t e r p r e t e d . The f i r s t says that <f i s equal to 0 plus a disturbance term ( n o i s e ) , and the second that a l l r i s k averters p re fe r 0 to $. D.3 i s the second order s t o c h a s t i c dominance condi t ion (see Hadar and Russel l (1969), Hanoch and Levy (1969)) app l ied to random var iab les with the same expected va lue . It i s equivalent to the condi t ion that the d i s t r i b u t i o n G may be obtained from F by a sequence o f s t e p s , each o f which involves s h i f t i n g some weight from the centre of the d i s t r i b u -t ion to i t s t a i l s in such a way as to leave the mean unchanged. Thus a l l three are i n t u i t i v e l y p l a u s i b l e d e f i n i t i o n s and the major r e s u l t of the Rothschi ld and S t i g l i t z paper i s to show that they are a l l equ iva len t , and hence extremely appea l ing , d e f i n i t i o n s of increased v a r i a b i l i t y . 5. I f $ and 0 s a t i s f y D e f i n i t i o n 1, then (the d i s t r i b u t i o n of) $ is sa id to cons t i tu te a mean preserv ing spread of (the d i s t r i b u t i o n of) e. It i s not obvious that a mean preserv ing spread of e i s always the appropriate notion of increased v a r i a b i l i t y . For example, when e represents the leve l of future p r i c e s , by increased uncerta inty about future p r i ces we could mean, depending on the way in which expectat ions are formed, increased uncerta inty about the rate o f i n f l a t i o n , which in in tegrated form may be represented by a spread in expectat ions that keeps E( log e) constant . In genera l , a mean preserv ing spread of any funct ion f (e) may be appropr ia te . Consider the dec is ion problem (1) max EV[x; 0], x>0 where x i s a (sca la r ) dec is ion var iab le that must be chosen subject to uncerta inty about the (sca lar ) exogenous va r iab le e. The i n d i v i d u a l enter ta ins expectat ions concerning the poss ib le future values of e, descr ibed by the random var iab le 0, and chooses x to maximize expected u t i l i t y . 1 Denote by x* the so lu t ion to (1) , assuming i t e x i s t s . Diamond and S t i g l i t z (1974) have provided another natural notion of increased v a r i a b i l i t y by taking the funct ion f above to be V[x* ; 0]. More f o r m a l l y , we have: D e f i n i t i o n I I .2: Suppose that (the d i s t r i b u t i o n of) $ may be obtained from (the d i s t r i b u t i o n of) 0 by an i n f i n i t e s i m a l v a r i a t i o n of a continuous parameter and l e t x* solve (1) given 0. Then $ const i tu tes a marginal mean u t i l i t y preserv ing increase in r i s k of 0 i f V [x* ; $] i s a mean preserv ing spread of V [x * ; 0]. Thus a mean u t i l i t y preserv ing spread corresponds to a s h i f t to the t a i l s of the d i s t r i b u t i o n keeping expected u t i l i t y constant . Note that D e f i n i t i o n 2 , unl ike D e f i n i t i o n 1, i s r e s t r i c t e d to marginal changes in d i s t r i b u t i o n s , "because" of the dependence o f the u t i l i t y funct ion on the . 2 x chosen. The importance of these d e f i n i t i o n s i s that the economic consequences of increased v a r i a b i l i t y , def ined in e i t h e r way, are determined by s t ra ight forward proper t ies o f the u t i l i t y index. Consider ing the choice problem (1 ) , we have: Theorem I1.1; (Rothschi ld and S t i g l i t z (1971)): A mean preserv ing spread of e reduces ( increases) x* i f V v ( x ; e ) i s concave (convex) in e . Theorem 11.2 (Diamond and S t i g l i t z ) : Suppose that V Q > 0. A marginal mean u t i l i t y preserv ing spread of o reduces ( increases) x* i f The l a t t e r i n e q u a l i t i e s are reversed i f V. < 0. Theorem 1 i s appl ied by Rothschi ld and S t i g l i t z to p o r t f o l i o , consumption savings and production problems and the condi t ions f o r the re levant funct ion to be concave or convex are genera l ly phrased in terms o f the Arrow-Prat t concepts of r e l a t i v e and absolute r i s k a v e r s i o n . The re levant condi t ion in Theorem 2 i s a l ready phrased in terms of the proper t ies o f the "extended" Arrow-Prat t measure o f absolute r i s k aversion with respect to uncer ta inty in e , ( - v e e ( x * ; e ) / V 0 ( x * ; e ) ) . The charac te r i za t ion of u t i l i t y functions f o r which — d X hence f o r which the e f f e c t of a mean u t i l i t y preserving spread vanishes , suggests i t s e l f as worthy o f a t t e n t i o n . Keeney (1973) has considered such a p r o p e r t y , though with a d i f f e r e n t mot iva t ion , and has made the fo l lowing d e f i n i t i o n s : - U x * ; e ) V 0 ( x * ; e ) > < = 0 and D e f i n i t i o n I I .3: Consider a u t i l i t y funct ion W ( e ^ , e n ) . e . . i s sa id to be r i s k independent of e . (denoted e . RI e . ) i f J ' J -W 0 0 ~ = 0. If i s a vec to r , then by 0. RI we mean that 0. i s Y V 1 1 r i s k independent of each component of 0 . D e f i n i t i o n II .4: Consider a u t i l i t y funct ion W ( e ' , 0"), where ( e 1 , 0") cons t i tu tes a p a r t i t i o n of ( e - j , e n ) . Then 0 1 i s u t i l i t y independent of 0" (denoted 0' UI 0") i f W can be expressed in the form W(0', 0") = a(0") + b(0") .w . (e ' ) ; that i s , i f preferences over l o t t e r i e s on 0 1 are independent of 0". Keeney proves that these two proper t ies are equ iva len t . Theorem II . 3 : Let 0^ be a vector with components 0., j E J ^ { l , 2 , . . . , n } . 3 Then e. RI 0 J i f and only i f e. UI 0 a. I t fo l lows that fo r a l l funct ions V(x;0) of the form (2) V(x;e) = a(x) + b(x)v(e) , mean u t i l i t y preserv ing spreads have no e f f e c t . Keeney (1972) and Pol lak (1973) a lso prove the fo l low ing : Theorem II.4: Let 0 be the two dimensional vector (0-|,0 2). Then 0^ RI 0 2 aridc02 RI 0-j i f and only i f W has the form W ( e l f e 2 ) = c Q + c ] w 1 ^ ) + c 2 w 2 ( © 2 ) + c 3 w ^ e ^ w 2 ( © 2 ) , 1 2 3 for" some funct ions w and w . 8. The above discussion surrounding the decision problem (1) is restricted to the case where both x and e are scalars. This assumption is required in the definitions of increased variability,- in deriving the economic consequences in Theorems 1 and 2, in the definition of risk independence and consequently in Theorems 3 and 4. Clearly, in many economic problems the relevant x and e variables are vectors, for example when a consumer or producer chooses a vector of consumption or factor demands subject to uncertainty about a vector of future prices. Therefore, the first main objective of the thesis is to extend the above analysis to the case where (1) is a general multivariate decision problem. 9. II I . MULTIVARIATE RISKS AND CHOICE UNDER UNCERTAINTY 1. Some D e f i n i t i o n s We now consider the problem (1) max EV[x;e] , x>0 which i s to be in terpre ted p r e c i s e l y as 11.(1) except that x = ( x - j , . . . , x n ) and e = (0-],...,0 ). We assume that V(x ; e ) i s cont inuously d i f fe rent ! 'ab le to the t h i r d order , s t r i c t l y inc reas ing or decreasing in each e . , and s t r i c t l y concave in x , f o r (x,e) in the nonnegative orthant of R n + m . Since we s h a l l be in te res ted in the e f f e c t s on the optimal demand x* of an increase in the v a r i a b i l i t y of expecta t ions , we require mul t ivar ia te extensions o f the R o t h s c h i I d - S t i g l i t z and D iamond-St ig l i t z d e f i n i t i o n s . The d e f i n i t i o n of a mean u t i l i t y preserv ing spread app l ies equa l ly wel l to vector random v a r i a b l e s , s ince i t i s phrased in terms of a mean preserv ing spread of the s c a l a r random var iab le V[x*; © ] . The d e f i n i t i o n of a mean preserv ing spread presents somewhat more of a problem, s ince the equivalence of the analogues of D.1-D.3 i s not c l e a r when 0 and $ are v e c t o r s . M u l t i v a r i a t e s t o c h a s t i c dominance has received considerable a t t e n t i o n , (see Levy and Paroush (1974), L e v h a r i , Paroush and PeTeg (1975)) but at present seems too complex to be of any help in d e r i v i n g q u a l i t a t i v e comparative s t a t i c s r e s u l t s . There fore , we propose to use the vector vers ion of D.l as the d e f i n i t i o n of a mean preserv ing spread. It has the advantage of being o p e r a t i o n a l , as descr ibed below, as wecll as being p l a u s i b l e in as much as i t impl ies D.2. Moreover, we conjecture that D . l , D.2 and an analogue of D.3 are in fac t equ iva len t , though the equivalence w i l l probably be d i f f i c u l t to e s t a b l i s h . We note that Hartman (1972) has 10. a lso used D.l in h is ana lys is of the e f f e c t of p r i c e uncerta inty on investment. The fo l lowing d e f i n i t i o n s are adopted: D e f i n i t i o n 111.1: Let 0., $., Z., i = l , 2 , . . . , m , be random var iab les such that ( * ! , . . . , $ „ ) has the same d i s t r i b u t i o n as (e^ + Zls...,0m + Z m), and v 1 m 1 1 m m ' E[Z.j/an 0., j = l , 2 , . . . , m ] = 0. Then the j o i n t d i s t r i b u t i o n of the var iab les $ is a mean preserv ing spread o f the j o i n t d i s t r i b u t i o n of the var iab les o- . I f only Z. , ..., Z. are nonzero, we s h a l l speak of a (simultaneous) mean preserv ing spread in the d i r e c t i o n s i^ , i^ . D e f i n i t i o n 111.2: " Let 0. and , i = l , 2 , . . . , m , be random var iab les and l e t x* solve (1) when expectat ions are descr ibed by the 0^'s. I f the $..'s may be obtained from the 0.'s by an i n f i n i t e s i m a l v a r i a t i o n of a continuous parameter, we s h a l l speak o f a (marginal) mean u t i l i t y preserv ing increase in r i s k i f V[x*;$] i s a mean preserv ing spread o f V[x*;0]. While recent s tudies have concentrated on developing the not ion of increased v a r i a b i l i t y in expected u t i l i t y models, increased c o r r e l a t i o n has to date been considered only in the context o f mean-variance a n a l y s i s . D e f i n i t i o n 1 suggests the fo l lowing d e f i n i t i o n of increased c o r r e l a t i o n : D e f i n i t i o n III .3 : Let 0., $ . , i = 1 , 2 , . , , , m , and Z-, Z'., j = 1,2, be random var iab les such that : (a) Z. and Z'- are i d e n t i c a l l y d i s t r i b u t e d with zero means, j = 1,2 ; J J (b) Z = (Z-j , Z 2) and Z' = (Z-j.Zp are d i s t r i b u t e d independently o f 0 = (e-|,...,e ) and $ = ( « 1 , . . . , $ m ) r e s p e c t i v e l y ; (c) Z-j and Z£ are independently d i s t r i b u t e d , whi le E[Z-j Z^] > 0 (<0); (d) ( t ^ Z j , $2+Z^, $ 3, * m) and ( e ^ Z ^ 02+Z2, e 3, e m) have the same j o i n t d i s t r i b u t i o n . 11. Then the j o i n t d i s t r i b u t i o n of the var iab les $. represents a mean preserv ing increase (decrease) in c o r r e l a t i o n , with respect to the f i r s t two c o - o r d i n a t e s , of the j o i n t d i s t r i b u t i o n of the e. 's . E s s e n t i a l l y , an increase in c o r r e l a t i o n i s def ined as that change in a d i s t r i b u t i o n due to the add i t ion of a noise vec to r , a f t e r subt rac t ing out the "equivalent" independent noise in each d i r e c t i o n . Note that in D e f i n i t i o n 3 we have required that the added noise be s t o c h a s t i c a l l y independent of G, while e a r l i e r we required a weaker form of zero cond i t iona l means. For s i m p l i c i t y we s h a l l henceforth assume that the stronger condi t ion h o l d s , though the weaker condi t ion i s s u f f i c i e n t f o r many o f the resu l ts below. With th is fu r ther s t i p u l a t i o n , a simulataneous mean preserv ing spread in any two d i r e c t i o n s may be thought o f as c o n s i s t i n g of the sum of mean preserv ing spreads in each d i r e c t i o n separate ly and an increase (or decrease) in c o r r e l a t i o n . The e f f e c t s o f increased v a r i a b i l i t y ( c o r r e l a t i o n ) w i l l be shown to depend c r i t i c a l l y on the proper t ies o f r i s k (cor re la t ion ) aversion measures that are mu l t i va r ia te genera l i za t ions o f the Arrow (1965) and Pra t t (1964) measure of absolute r i s k a v e r s i o n . In the mult idimensional case , we do not have a unique r i s k aversion measure and a family of d i r e c t i o n a l measures must be cons idered . There fore , l e t e = ( e T - - " e m ) represent i n i t i a l expec ta t ions , Z = (Z-| Z m ) pure n o i s e , and K = ( c i - j , . . . , ? ) a vector o f real numbers s a t i s f y i n g (2) (a) ? k > 0 (<0), where V k = 9V/39k > (<) 0 , (b) c k = 0 i f Z k = 0, and (c) l f 0 . 12. Define the function -n-(b) implicitly by (3) EV[x; e ] + bZ r..., Gm + b Z j = EV[x; 8 ] - i r ^ , .... e m - TTSJ . Thus ir represents the risk premium associated with the added variability bZ, calculated relative to the direction £. A two-fold differentiation of (3), with evaluation at b = 0, yields - J _ £(1.15) EV...(x;e) (4) TT"(0) = ^ ;^.(x;e) 5 92V(x;e)/3e.9e, . I 5 k EVk(x;e) 1 J 1 J Letting A be any variance-covariance matrix, we define a family of risk measures RA'^(x;e) by - I L EV i j (x;e) (5) R A' ?(x;e) = . I C k EVk(x;e) k It w i l l be convenient to have a separate notation for these measures when i n i t i a l expectations 0 are nonstochastic. So we define -J A . J V.j(x;e) (6) r A' 5(x;&) = : . I h V k ( x ; e ) Each RA'? and r A ' ? has the familiar interpretation as a risk premium for small risks. When A'. ^ = 1 and A . . =0 for ( i , j ) t ( i Q , i 0 ) , and 0 0 p when K = (0, 0, 1, 0 0), r f l'^ = -V. . (x;e)/V. (x;e), a risk ' , * V o no no aversion measure with respect to uncertainty in the i direction. But in a multivariate model more than these "unidirectional" measures are of interest. Indeed, the whole family of measures defined by (5) and (6) w i l l be shown to be important in describing choice under uncertainty. 13. A s i m i l a r argument may be used to a r r i v e at the fo l lowing measures o f th th aversion with respect to increased c o r r e l a t i o n between the i and j co-ord inate expecta t ions: 1 i E - E V ^ - U i O ) (7) C ( x ; 0 ) = u , and I 5 k EV k ( x ;e ) k 1 i F - V . ^ X ^ ) (8) c 1 , J » 5 ( x ; e ) = — L I l S k ^V k (x ;e ) k where E s a t i s f i e s (2) (a ) , (c) and ^ = 0 i f k / i , j . These measures represent per uni t c o r r e l a t i o n premiums f o r small increases in c o r r e l a t i o n . There i s an aversion to c o r r e l a t i o n i f c 1 ' ' - ' ' ^ > 0 (V . . < 0) and an ' J a f f i n i t y i f c 1 " 3 ' 5 < 0 ( V . , > 0 ) . 2. The Comparative S t a t i c s The f i r s t order condi t ions corresponding to (1) are: (9) EV (x*;e) = 0 , i = 1 ,2 , . . . . n x i We assume that (9) uniquely def ines the optimal demand x* . Let D be the determinant |EV V ( x * ;e ) | and denote by D. . the ( i , j ) t h c o f a c t o r . C a l l x i x j 1 J x. and x. s t o c h a s t i c subst i tu tes (complements) i f D. . /D > (<) 0. The e f f e c t of a marginal s h i f t in expectat ions in the d i r e c t i o n E, i . e . , 0 e + bE f o r small b, i s e a s i l y shown to be given by 14. The impact of a marginal mean preserv ing spread may be found by s u b s t i t u t i n g 0 + bZ f o r 0 in (9 ) , where Z i s pure n o i s e , t o t a l l y 9X* d i f f e r e n t i a t i n g and eva luat ing at b = 0. We f i n d that T o t a l l y d i f f e r e n t i a t i n g again we have b=0 (11) I EV l ! 3 X i X j 8b 2 = — (- I b=0 8 x i k,z i = 1 , 2 , . . . , n , with so lu t ions 9x? D.. ( 1 2 ) 3Risk(A) I IxT ( " A k £ E V k J i ) ' i = l , 2 , . . . , n , where ^ E E (Z f Z j ) and 9X* 1 2 9 X* 9Risk(A) 9b' b=0 There are two "natura l" decompositions o f (12). F i r s t , making use o f (5) and (10), we can write (13) 3 x T A 3xt 9 A c D i k 9Risk(A) = _ R ' 3Shi f t (? ) + £ { \ C k V I 9 ^ R ' "TT ' i = 1 , 2 , . . . ,n The impact o f increased uncerta inty i s decomposed in to a s h i f t e f f e c t and a s u b s t i t u t i o n e f f e c t as fo l lows: l e t TT be that s h i f t in expectat ions in d i -r e c t i o n ^ which i s equ iva len t , in terms of maximum expected u t i l i t y , to the added uncer ta in ty . Then the f i r s t term on the r i g h t hand s ide o f (13) represents the e f f e c t of the s h i f t in expecta t ions , and the second represents the change due to movement along a contour of constant expected u t i l i t y . 15. The sign of the s h i f t e f f e c t depends on the sign of 3ShVft(g) anc* o n whether there is r i s k aversion ( R A ' ? > 0 ) , or r i s k a f f i n i t y (R A '^ < 0 ) . The sign of the s u b s t i t u t i o n term depends on the s t o c h a s t i c s u b s t i t u t i o n proper t ies of the x^ 's and on the way in which r i s k aversion i s a f fec ted by varying l e v e l s of each x^. For example, i f x. and x^ are s t o c h a s t i c subst i tu tes and i f R A '^ increases with x^, a p o s i t i v e term i s contr ibuted to the overa l l impact. I f there i s zero s u b s t i t u t a b i l i t y between the x ^ ' s , (V add i t i ve in x ) , the s u b s t i t u t i o n term i s p o s i t i v e (negative) i f r i s k aversion decreases (increases) with x i . A d i f f i c u l t y with using (13) to der ive q u a l i t a t i v e r e s u l t s i s that the proper t ies o f R A '^ are genera l ly d is t r ibu t ion -dependent ; that i s , apart from some r e s t r i c t i v e spec ia l c a s e s , assumptions about V are not s u f f i c i e n t to imply proper t ies o f R A ' ^ , independent of the underly ing p r o b a b i l i t y d i s t r i b u t i o n . This d i f f i c u l t y i s not shared by the fo l lowing decomposition o f (12) which i s obtained using (6): 3x* 3xv . _ . . Du> D.. i = 1 , 2 , . . . , n . This equation may be in te rpre ted as fo l lows: i f , f o r each poss ib le value e b 2 of e , we d iv ide the change e ->• 8i + bZ i n t o e ->• e + r ( x * ; e ) - £ -+0 + bZ, the e f f e c t on x* of the f i r s t "nonuniform s h i f t " in expectat ions i s r e f l e c t e d by the f i r s t expression in (14). In the l i m i t as b 0, the second change in d i s t r i b u t i o n s const i tu tes a marginal mean u t i l i t y preserv ing spread q whose e f f e c t on x* i s given by the second term in (14). 16. Diamond and S t i g l i t z (p. 342) suggest that a mean preserv ing increase in r i s k can be decomposed in to a mean u t i l i t y preserv ing spread plus some "other" change. Equation (14) e s t a b l i s h e s the decomposition p r e c i s e l y and shows tha t , in the mu l t i va r ia te model, there are several such decompositions and mean u t i l i t y preserv ing spreads corresponding to the many poss ib le choices o f A and E. Moreover, our ana lys is points out the d i f fe rence between the decompositions in (13) and (14). The former involves a s h i f t t o , and movement a long , a contour of constant maximum expected u t i l i t y , while the l a t t e r involves corresponding movements at each value assumed by V as G var ies over i t s range. The two co inc ide given • in i t i a l nonstochast ic expec ta t ions , but in general i t i s only the l a t t e r that inc ludes the e f f e c t of a mean u t i l i t y preserv ing spread as a component and in which the d i s t r i b u t i o n - f r e e r i s k measures r A '^ play a 4 v i t a l r o l e . Equation (14) can help in determining the q u a l i t a t i v e impact on x* of increased v a r i a b i l i t y . Since I E^ > 0 un i formly , the s u b s t i t u t i o n 8 A E proper t ies o f the x u ' s and the signs of —— r determine the e f f e c t of ax k the mean u t i l i t y preserv ing spread , genera l i z ing the Diamond and S t i g l i t z theorem stated in II. For example, i f x. i s a s t o c h a s t i c subst i tu te with a l l other goods and i f r i s k aversion increases (decreases) with each x k , k f i , and decreases ( increases) with x^, the e f f e c t of the mean u t i l i t y preserv ing spread i s to increase (reduce) x * . On the other hand, ~ - (I E V ) can often be uniformly s igned , (as in the model of the f i rm below, f o r example), and so fu r ther assumptions concerning a t t i tudes towards r i s k , i . e . , the signs of r A ' ^ , can be found to s ign the " s h i f t e f f e c t " . Note that f o r any given var iance-covar iance s t ructure of the added n o i s e , there are many poss ib le sets o f condi t ions s u f f i c i e n t to s ign the overa l l impact , depending on the choice o f E. 17. Comparing these r e s u l t s with the corresponding ana lys is in II, we see the compl icat ions introduced by the mul t i va r ia te framework. When x i s a v e c t o r , the e f f e c t s of nonmarginal spreads (such as in Theorem I1.1) are in general impossible to der ive and the e f f e c t s of marginal spreads depend both on the curvature in e of the terms V v (x;e) and on the s u b s t i t u t i o n x i p roper t ies o f the x ^ ' s . When e i s a v e c t o r , the e f f e c t of a l l mean u t i l i t y preserv ing spreads cannot be descr ibed by proper t ies o f a s i n g l e measure of r i s k avers ion . Rather, there are many mean u t i l i t y preserv ing spreads each o f which is re la ted to a d i f f e r e n t r i s k aversion measure. We have analysed only those mean u t i l i t y preserv ing spreads that can be der ived from mean preserv ing spreads as a component of a S lu tsky - type decomposition of the l a t t e r , but we show l a t e r that in a sense they const i tu te a "very large" subset . F i n a l l y , the e f f e c t s o f increased c o r r e l a t i o n , between expectat ions f o r and f o r example, may be determined in an analogous f a s h i o n . We f i n d that 9x* D. . c <15> »Corr(k.i) ° - | 5 T E , > . f • 1 - 1 - 2 The fo l lowing decompositions are obta ined: 9x* . 9 x t a r k ' ^ D i i ( 1 6 ) 9Corr(k,0 = - C 3Shi f t (? ) + E{1 ? j V I ~5x" J J J 9 X* D D •: • J J 1 D. . D 18. These equations may be in te rpre ted exact ly as above. Note that the second expression in (17) represents the e f f e c t of a marginal mean u t i l i t y preserv ing increase in r i s k " induced" by an increase in c o r r e l a t i o n of expectat ions 0^ and e . When x i s 1-d imensional , t h i s e f f e c t has the sign of — c ' , that i s , a "mean u t i l i t y preserv ing increase in c o r r e l a t i o n " oX increases (reduces) x* i f the c o r r e l a t i o n aversion measure decreases ( increases) with x. We have thus demonstrated the importance of the signs of terms of the form r — r and - — c 5 in determining the q u a l i t a t i v e impact of o X • o X • J J increases in v a r i a b i l i t y and c o r r e l a t i o n . Roughly speak ing, mean u t i l i t y preserv ing increases in r i s k and c o r r e l a t i o n induce s u b s t i t u t i o n away from goods x- that increase aversion ( T | — , r A ' ^ > 0 - 3 — c ^ ' ^ ' ^ > 0 ) , and towards 3 x j T 8'Xj goods x . that reduce aversion ( ~ - r A '^ < 0, ~— c ^ * * " ^ <; 0 ) . The border-J 9Xj 9 X j l i n e cases - r r — r ' - 0, c ' = 0, immediately suggest themselves as OA' OA• J J worthy o f s p e c i a l a t ten t ion and represent m u l t i v a r i a t e extensions o f Keeney's no.tion of r i s k independence to which .we now t u r n . 3. Risk Independence Recal l the d e f i n i t i o n s o f r i s k independence and u t i l i t y independence in II. Note that whereas the l a t t e r i s p e r f e c t l y genera l , the former re fers only to aversion and r i s k in s c a l a r a t t r i b u t e s . Keeney (1973; p. 32) claims that s ince the r i s k aversion funct ion i s only def ined f o r s c a l a r a t t r i b u t e s , mult idimensional analogues of the above theorems cannot be proven, and hence, in cons ider ing choice among uncertain vector a t t r i b u t e s , he speaks of u t i l i t y independence o n l y . The d iscuss ion above, however, suggests the fo l lowing genera l i za t ion of r i s k independence: 19. D e f i n i t i o n I I I .4: Let 9' be a vector with components e . , i . e N 1 <= { l , 2 , . . . , n } , and def ine - I A W..(0) I 5 k w . ( e ) ke'N' K K where A i s a var iance-covar iance matrix and E s a t i s f i e s the appropriate version of (2) . The vector 0' i s sa id to be r i s k independent of e . , (again denoted 0 1 RI e . ) , i f - | - r A , ? ( 0 ) = 0 f o r a l l such A and 0. 0' RI 0 J, J 00 • J J where 0 i s a vec to r , i f e 1 i s r i s k independent of each component of 0 . When 0' i s a s c a l a r , t h i s d e f i n i t i o n co inc ides with d e f i n i t i o n I I .3. The former, however, can be used to provide a genera l i za t ion of Theorems II.3 and I I .4. Theorem III . 1 : The vector 0' i s r i s k independent of the vector 0^ i f and only i f i t i s u t i l i t y independent of 0 . Theorem III .2: Let 0 = (<j>,i|0, where dj and i|» are vec to rs . Then <|> RI ip and ip RI <j) i f and only i f W can be wr i t ten in the form W(<M) = c Q + c ^ U ) + c 2 w 2 U ) + c3w1(<|>) w 2 U ) . 7 Y^.e'ssTheselresults may now be app l ied to the funct ion V(x;0) and the opt imizat ion problem (1). By the f i r s t theorem 0 i s r i s k independent of x- i f and only i f J (18) V(x;e) = a(x) + b ( x ) v ( x . ; e ) , where x. = ( x ^ x ^ ^ , x^, x p ) 20. I f there i s zero s u b s t i t u t a b i l i t y between x. and the other goods, or i f x i s 1 -d imensional , x* i s unaffected by marginal mean u t i l i t y preserv ing increases in r i s k and c o r r e l a t i o n when V has th is form, e i s r i s k independent of x i f and only i f (19) v(x;e) = a(x) + b(x)v(e) . In that case x* i s t o t a l l y unaffected by mean u t i l i t y preserv ing spreads. A form of the converse is a lso t r u e . Theorem III .3: I f e Rl x , mean u t i l i t y preserv ing changes in d i s t r i b u t i o n have no e f f e c t on x * . Conversely , suppose that f o r a l l "su i tab le" A and 5 and f o r a l l i , ~ - f A ' ^ ( x ; e ) i s uniformd.yysigned with respect to e , f o r 3 x i each x. I f x* i s unaffected by a l l mean u t i l i t y preserv ing changes in d i s t r i b u t i o n then e Rl x . By mean u t i l i t y preserv ing spreads (and increases in c o r r e l a t i o n ) we mean throughout only those that are der ivable from mean preserv ing spreads (and increases in c o r r e l a t i o n , r e s p e c t i v e l y ) as in sec t ion 111.2. But note that f o r preferences s a t i s f y i n g (19) the e f f e c t s of a l l mean u t i l i t y preserv ing spreads van ish . Thus the subset of spreads we consider is large in the sense that i f behaviour i s unaffected by a l l members of the subset , i t i s unaffected b y s a l l mean u t i l i t y preserv ing spreads. Risk independence a lso has strong impl ica t ions f o r the r i s k aversion measures R A '^ (and C A ' ^ ) , and the decomposition in (13) (and (16)) . For example, e Rl x. impl ies that T ^ — R A ' ? = 0 f o r a l l A and % and f o r a l l 1 oX.j p r o b a b i l i t y d i s t r i b u t i o n s f o r 0. The reverse imp l ica t ion i s a lso true 21. s ince r A '^ and R A '^ co inc ide given cer ta in expecta t ions . S i m i l a r l y e RI x i f and only i f R A '^ = 0 f o r a l l i , A , E and p r o b a b i l i t y d i s t r i b u t i o n s . 9 x i In that case , 3Risk ( A ) " K a S h i f t U ) and the marginal impact of increased uncerta inty i s the same as the e f f e c t o f a s h i f t in expectat ions in the d i r e c t i o n E, and o f s i z e - R A ' ^ . Moreover, as is impl ied by these and e a r l i e r r e s u l t s , and as can be v e r i f i e d d i r e c t l y from (13) and (14), when e RI x the two decompositions coinc ide the s u b s t i t u t i o n term in (13) corresponds to the e f f e c t of a mean u t i l i t y preserv ing change and van ishes , leav ing only the s h i f t e f f e c t . Theorem 4 summarizes and extends these r e s u l t s . Theorem 111.4: Let x*(e) denote the s o l u t i o n to (1) given the random expectat ions e, and denote by x(e) the s o l u t i o n when expectat ions equal e with c e r t a i n t y . Define the r i s k premium ir^ by EV(x;e) = V(x; 0 - •TT'-E) where 0 i s the expected value of 0, and E s a t i s f i e s the appropriate version of (2) . Then: (a) e RI x i f and only i f T ^ - R A ' ? = 0 f o r a l l i , A , E arid p r o b a b i l i t y 3 x i d i s t r i b u t i o n s ; (b) i f e RI x , Ihe TT ? i s independent of x and X * ( G ) = x(0 - 1^(0)5). Thus when 6 Rl x , knowledge o f c e r t a i n t y equivalents (0 - ^E) and the "cer ta in ty" demand funct ion x, i s s u f f i c i e n t to determine the " s t o c h a s t i c " demand funct ion x* . With some add i t iona l assumptions things can be made even s imp!er r . 22. Theorem III .5 : Define i r . by EV(x; e ^ , . . . , e. _^ , , e.j+i> •••> 9 m ) = V(x; 0 i , . . . , 8 . _ - | , 0 . —n-^, e . + p . . . , e ) , i = l , 2 , . . . , m . Suppose that the e..'s are mutually r i s k independent and that e. Rl x f o r each i . Then f o r each i , i r . depends only on Suppose fu r ther that the 0 . . 1 s are _ _ Q independently d i s t r i b u t e d . Then x*(e) = x(e-j - ir-j 0 - i r m ) . This concludes our ana lys is of the general choice problem (1). In Chapters V and VI the ana lys is i s app l ied to the s p e c i f i c dec is ion problems in two-period models of consumer and producer behaviour. F i r s t , however, we descr ibe b r i e f l y those r e s u l t s from the theory o f d u a l i t y that w i l l be exp lo i ted below. 23. IV. SOME BASIC CONCEPTS IN DUALITY THEORY 1 Beginning with consumer theory , l e t u(x,y) be a d i r e c t u t i l i t y funct ion o f the vectors x = ( x ^ , . . . , x n ) and y = ( y - j , . . . , y n ) , which may be thought of as f i r s t and second per iod consumption vectors r e s p e c t i v e l y . Assume that u s a t i s f i e s the fo l lowing set of c o n d i t i o n s : Condit ion A: ( i ) u(x,y) i s r e a l - v a l u e d and def ined fo r a l l x ,y > 0; ( i i ) u(x,y) i s j o i n t l y continuous with respect to x > 0, y > 0; ( i i i ) u(x,y) i s nondecreasing f o r x,y > 0; and ( iv ) u(x,y) i s quasiconcave f o r x ,y > 0 . Denote byqq the vector of second per iodoor future p r i ces and by s the to ta l income a v a i l a b l e f o r future consumption, i . e . , sav ings . We def ine the var iab le i n d i r e c t u t i l i t y funct ion g ( s ; q ; x) as fo l lows: (3) g(x; q ; x) E max {u(x ,y ) /q -y < s:}, s > 0, q » 0 and x > 0. y>0 Consider the fo l lowing set of condi t ions on the funct ion g: Condit ion B: ( i ) g (s ;q ;x ) i s rea l -va lued and def ined f o r s > 0, q>>0, x > 0; ( i i ) g (s ;q ;x ) is j o i n t l y continuous with respect to s > 0, q » 0 and x > 0; ( i i i ) f o r f i xed s , x > 0, g (s ;q ;x ) is nonincreasing and q u a s i -convex in q fo r q > >0; ( iv) f o r f i xed x > 0, g (s ;q ;x ) i s homogeneous of degree zero in ( s ; q ) ; and (v) f o r f i x e d q>>0, g (s ;q ;x ) i s nondecreasing and q u a s i -concave in (s;x) f o r s > 0, x > 0. 24. The funct ion g gives the maximum u t i l i t y the consumer can a t ta in given f i r s t per iod consumption x , future p r i ces q and savings s . I f g s a t i s f i e s B, then f o r x f i x e d at x Q , g ( s ; q ; x Q ) i s an i n d i r e c t u t i l i t y funct ion , while i f q is f i x e d at q Q , g ( s ; q Q ; x ) i s a d i r e c t u t i l i t y funct ion of savings and f i r s t per iod consumption. Diewert (1975) and Epste in (1975b) have shown that there i s a one-to-one correspondence between the set of d i r e c t u t i l i t y funct ions u(x,y) s a t i s f y i n g A and the set o f va r iab le i n d i r e c t u t i l i t y funct ions g (s ;q ;x ) s a t i s f y i n g B. That i s , any u s a t i s f y i n g A def ines a unique funct ion g , by (3) , that s a t i s f i e s B, and converse ly , given any g s a t i s f y i n g B, there e x i s t s a unique u s a t i s f y i n g A that "generates" i t through (3) . Moreover u(x,y) i s l i n e a r homogeneous i f and only i f g i s l i n e a r homogeneous in ( s , x ) . Thus u and g are equiva lent representat ions of preferences but use o f the l a t t e r i s advantageous in the ana lys is o f the two per iod consumer problem, as we show below. Since the curvature o f g with respect to s and q r e f l e c t s the consumer's a t t i tudes towards uncerta inty in savings and future pr ices r e s p e c t i v e l y , i t s i n v e s t i g a t i o n i s o f i n t e r e s t . Diewert (1975) has shown that the consumer i s averse to uncer ta inty in savings (g concave in s) i f and only i f he i s r i s k averse with regard to the future consumption bundle (u(x,y) concave in y ) . Convexity o f g in s i s not i n c o n s i s t e n t with condi t ions A.ofe'Band.as the case i f |u | ( - l ) n < 0 . (See Hanoch (1974), y i y j Theorem 2 ) . Diewert has a lso shown that g i s convex in q i f and only i f u ( x , y A ) i s convex in A > 0. Hanoch (Theorem 3) has proved that there is r i s k - l o v i n g with respect to p r ices i f and only i f income r i s k aversion i s not too l a r g e , in the sense that the r e l a t i v e r i s k aversion measure - s g s s / g s i s less than 2. I f n > 1, g cannot be s t r i c t l y concave in q , 25. but i t can be s t r i c t l y concave in any s i n g l e p r i c e or in a l l p r ices separa te ly . (As an example, consider funct ions of the form g(s ;q ;x ) n a . = n (q.-/s) + v ( x ) , a. > 1.) A necessary condi t ion f o r concavi ty in i=l 1 1 any s i n g l e p r i c e i s given by the fo l lowing theorem: Theorem I V . l : I f g (s ;q ;x ) i s concave in q. , then u(x, y - | , . . . , y^_-|» o y^ A > y. +1, y n ) i s concave in A > 0, or e q u i v a l e n t l y , u(x,y) is o o concave in 1 /y. . n o Proof: The " inverse" of (3) i s the transformation def ined by u(x,y) = min {g (s ;q;x ) /qy < s} = m in ' {g ( l ;q ;x ) /qy < 1} q,s>0 q>0 Solv ing f o r q . ^ from the budget cons t ra in t and s u b s t i t u t i n g in to g , we obtain u(x,y) = min j g( l ; q 1 , . . . , q . _ ] (1 - J. q . y . ) q-|, • . • »q1- _-| »q i +•]»•• - q n o o q 1 q n ; x ) / I q ^ - < l j There fore , u equals the minimum of a fami ly o f funct ions each of which i s concave in 1 /y. , and hence i s i t s e l f concave, in 1 /y. . Q . E . D . n o o Again we may i n t e r p r e t th is as s t a t i n g that a necessary condi t ion f o r aversion to r i s k s in q . i s that r i s k aversion with respect to the n o corresponding commodity be s u f f i c i e n t l y great . (Note that u i s concave in u 1/y,- i f and only i f -, y i _ y i \ Unfor tunate ly , we have not yet ' • " I T — 2 - >- 2-> o u y , o been able to prove or disprove the s u f f i c i e n c y o f the c o n d i t i o n . 26. Suppose that : ( i ) preferences e x h i b i t l oca l nonsat ia t ion in y , i . e . , u(x,y) i s increas ing in at l e a s t one y . f o r every x ,y and hence g J i s increas ing in s ; and ( i i ) g i s decreasing in each component of q . Then two other equivalent representat ives o f preferences that w i l l prove to be useful below can be obtained by appropriate invers ions o f the equation (2) u = g (s ;q ;x ) . So lv ing the equation f o r s , we obtain the funct ion e(u;q;x) which i s given by (3) e(u;q;x)- = min {q-y /u(x ,y ) > u} , q>>0, x > 0. y>0 e i s the var iab le expenditure funct ion , g i v i n g fo r each x and q the minimum expenditure on y necessary to a t ta in u t i l i t y l e v e l u. Roughly, e i s charac ter i zed by concavi ty and l i n e a r homogeneity in q , convexity in x, and is i n c r e a s i n g in u and q and nonincreasing in x . So lv ing f o r q- in (2) we obtain the funct ion f J ( u , s , q . , x ) , (q . = ( q 1 , . . . , q . _ 1 , q j + 1 » . . . , q n ) ) which i s given by (4) f J ( u , s , q . , x ) = max {q . /g (s ;q ;x ) > u} qj>0 J The in te rp re ta t ion o f the funct ion f J i s c l e a r . Each f J i s character i zed by convexity and l i n e a r homogeneity in ( s , q ) , concavi ty in (s ,x) and is decreasing in u and nondecreasing in ( s , x ) . 27. Turning to the production s i d e , l e t F[y^, z ; x] be a transformation f u n c t i o n , where y = ( y 1 , y 1 ) , ^ = ( y 2 , . . . , y ), z = .(z 1,... ,z n^) and x = ( x - , , . . . , x ). We think of y and z as var iab le outputs and fac tors i n 3 r e s p e c t i v e l y and o f x as a vector o f f i x e d f a c t o r s . Thus F [y^ ,z ;x ] gives the maximum production of the f i r s t output given the vector o f other outputs y-| to produce, var iab le fac tors z and f i x e d fac tors x. Assume that F s a t i s f i e s the fo l lowing set of c o n d i t i o n s : Condit ion C: ( i ) F [y^ , z ;x ] i s an extended r e a l - v a l u e d funct ion ( i . e . , i t can take on the value -°° f o r f i n i t e (y- | ,z;x)) def ined f o r each (y-| ,z;x) > 0, non-negative whenever i t i s f i n i t e , and F(0) = 0; ( i i ) F i s cont inuous; 2 ( i i i ) F i s a (proper) concave f u n c t i o n ; ( iv ) F i s non-decreasing in (z;x) and non- increas ing in y-|; (v) f o r every x , the set { ( y 1 , y 1 ; z ) / y 1 < F[y. | ,z;x]} i s bounded from above. When F takes on the value -°° we think of y as being so l a r g e , or z and x so s m a l l , that i t i s impossible to produce any non-negative amount of y-| . Property ( i i i ) imposes that the technology be convex, ( iv ) i s a f ree d isposal assumption while (v) asser ts that with x g iven , the set of p o s s i b l e production a c t i v i t i e s i s bounded. Let us now suppose that the producer 's f i x e d inputs are f i xed at x and that he can buy and s e l l va r iab le inputs and outputs at the f i x e d p o s i t i v e p r i ces w = ( w - j , w ) >> Gl-and p = (p-j,p-|) = (p-j ,p 2> • • • >Pn )>>0. Then the producer 's var iab le p r o f i t funct ion may be def ined as fo l lows: 28. (5) TT(P,W;X) = max {p-y - w-z /y , < F [ y , , z ; x ] } . y»z Consider the fo l lowing condi t ions on the funct ion T T : Condit ion D: ( i ) TT(P,W;X) i s non-negative and bounded above by p-b - w-c f o r f i x e d vectors b and c i f p , w » 0, x > 0 and x i s bounded from above; ( i i ) f o r every x > 0 TT i s cont inuous, convex and l i n e a r homogeneous in w; ( i i i ) f o r every (p,w)>>0, TT i s cont inuous, non-decreasing and concave in x . Diewert (1973) has proved the f o l l o w i n g : I f F s a t i s f i e s C then TT def ined by (5) s a t i s f i e s D. Moreover, given any funct ion TT s a t i s f y i n g D there e x i s t s a unique F s a t i s f y i n g A that "generates" TT though (5) . Thus F and TT are equiva lent representat ions of technology. Moreover, F exh ib i t s 3 constant returns to s c a l e i f and only i f TT i s l i n e a r homogeneous in x. For any f i x e d x, say x Q , TT(P,W;X 0) i s a p r o f i t funct ion i f i t s a t i s f i e s the appropriate condi t ions in D, and there i s a dua l i t y between the "shor t - run" technologies F [ y ^ , z ; x Q ] and the p r o f i t funct ions Tr(p,w;x Q ) . Note that unl ike the case in the consumer model, the producer 's a t t i tude towards uncerta inty in v a r i a b l e p r ices i s unambiguous. Convexity o f TT in (p,w) impl ies p r i c e r i s k a f f i n i t y . Much of the usefulness of the dual representat ion of the technology stems from the fo l lowing lemma: H o t e l l i n g ' s Lemma: I f Tr(p,w;x) s a t i s f i e s D and is in add i t ion d i f f e r e n t i a t e with respect to (p,w) at (p*,w*)>>0, x* > 0 , then we have 29. (6) y | ( p * , w * ; x * ) = ~ TT(P*,W*;X*) , i = l , 2 , . . . , n 1 , z j . (p*,w*;x*) = ^ TT(P*,W*;X*) , j = 1 , 2 , . . . , n 2 , 3 where y * (p * ,w* ;x * ) and z * (p* ,w* ;x* ) solve the p r o f i t maximization problem in (5) given p* , W* and x* . The extreme usefulness o f t h i s lemma in standard analyses of behaviour under c e r t a i n t y i s that i t enables us to obtain funct ional forms f o r demand and supply funct ions cons is ten t with p r o f i t maximization simply by choosing a funct iona l form f o r ir and d i f f e r e n t i a t i n g i t with respect to input and output p r i c e s . By hypothesiz ing a funct ional form that is l i n e a r in parameters, we obtain a l i n e a r system of demand and supply equations in terms o f p r i c e s and f i xed f a c t o r s , which are often considered exogenous, at l e a s t at the micro l e v e l . Thus standard l i n e a r regression techniques may be app l ied to estimate the technology. I f the funct ional form i s f l e x i b l e , in that i t contains the number o f parameters needed to provide a second order approximation to an a r b i t r a r y p r o f i t funct ion s a t i s f y i n g the appropriate r e g u l a r i t y c o n d i t i o n s , then a l l the c r u c i a l e l a s t i c i t i e s o f s u b s t i t u t i o n are estimated e m p i r i c a l l y in t h i s way. Apart from i t s advantages in empir ica l work, H o t e l l i n g ' s Lemma tremendously s i m p l i f i e s comparative s t a t i c s ana lyses . (See f o r example Diewert (1972a) and (1974), and Epstein (1974).) F i n a l l y , note that the counterpart of H o t e l l i n g ' s Lemma in consumer theory , Roy's Ident i ty (see Diewert (1974), p. 125), . 4 generates s i m i l a r advantages f o r the dual formulat ion o f consumer theory. As we descr ibe in more d e t a i l below, not a l l o f these advantages extend to models with uncer ta in ty . In p a r t i c u l a r , d u a l i t y theory does not always al low us to derive e x p l i c i t demand and supply funct ions as e a s i l y 30. as above. On the other hand, the theory proves to be v i r t u a l l y i n d i s p e n s i b l e f o r the empir ica l implementation of the models we cons ider ; that i s , while the est imat ion of these models i s not as s t ra ight forward as above, i t i s much more d i f f i c u l t i f the models are formulated in terms of u t i l i t y or transformation f u n c t i o n s . A lso the s i m p l i f i c a t i o n of comparative s t a t i c s analyses due to the use o f dua l i t y theory i s l a r g e l y re ta ined . H o t e l l i n g ' s Lemma can be used to der ive pr ice e l a s t i c i t i e s that w i l l serve as a d e s c r i p t i o n of the technology in Chapter VI I . In analogy with the Allen-Uzawa p a r t i a l e l a s t i c i t y of s u b s t i t u t i o n between two inputs fo r a given leve l o f output , we may def ine e l a s t i c i t i e s of t ransformation e . . and <j>.. between var iab le products and fac tors r e s p e c t i v e l y , as foltlows: (7) 0- - = TT TT / i r TT U PiP j Pi Pj ' i j = "w.W^W.jW.j (ci> - •) i s a normal izat ion of 3y-/3p- (3z./3w.) so that e. . (<{>•.:) i s invar ian t to the uni ts of measurement and e . . = e . . (ct. . = tf)^). The p a r t i a l (short run) p r i c e e l a s t i c i t i e s e.. and of the var iab le outputs and fac tors r e s p e c t i v e l y can be def ined as ( 8 ) e i j = yn- 3 P J ' 6 i j = ^ " ^ 7 Then the fo l lowing r e l a t i o n s ho ld : .31. (9) - Sj e , j and • ^ * u th where S . = p . y . / i r i s the j var iab le output 's share o f short run p r o f i t s vJ J J and V. = W.Z./TT i s the j f i x e d i n p u t ' s share of short run p r o f i t s . The J J J corresponding d e f i n i t i o n s and r e l a t i o n s f o r cross e l a s t i c i t i e s between var iab le products and fac tors are c l e a r . 32. V. CONSUMER CHOICE UNDER UNCERTAINTY ' 1. Some Introductory Remarks Recently in the l i t e r a t u r e there have appeared several s tud ies o f consumer behaviour under uncer ta in ty . For example, Lei and (1968), Sandmo (1970) and Dreze and Modigl iani (1972) examine the consumer's optimal consumption-savings dec is ion when there i s uncerta inty about future income and the rate o f return to sav ings . The Sandmo paper has been extended by Block and Heineke (1972) in order to al low f o r a simultaneous consumption and labour supply d e c i s i o n . In another paper (1973), the same authors concentrate on the e f f e c t of uncerta inty about wages and non-labour income on the supply of labour . F i n a l l y , Betancourt (1973) invest iga tes the e f f e c t of i n t e r e s t rate uncerta inty on labour supply and consumption. In each o f these papers, the consumer i s assumed to maximize the expected value of a u t i l i t y funct ion of a ce r ta in form, but there is l i t t l e d iscuss ion of the j u s t i f i c a t i o n f o r hypothesiz ing such a form. For example, in the f i r s t three papers, u t i l i t y is a funct ion of aggregate current and future consumption, in the Block and Heineke papers i t depends on current consumption, l e i s u r e and s a v i n g s , and on income and l e i s u r e r e s p e c t i v e l y , while Betancourt postu lates a u t i l i t y funct ion of current and future ind ices of the consumption of goods and le i 'sure . Of course , the conclusions reached in each of the papers depend upon the hypothesized form of the u t i l i t y funct ion and hence upon the i m p l i c i t assumptions about consumer behaviour which may be necessary to j u s t i f y the hypothesized representat ion of pre ferences . The need f o r caution in th is regard was expressed by Mossin (1969) who demonstrated that under temporal uncerta inty the representat ion of preferences in terms of a 33. u t i l i t y funct ion of wealth may be inappropr ia te . Because the current consumption and labour supply dec is ions must be made before i t i s known which future pr ices and income w i l l p r e v a i l , H icks ' aggregation cannot be invoked to j u s t i f y aggregation over current consumption bundles, and in genera l , the consumer's ob jec t ive reduces to the maximization of the expected value o f aggregate consumption and savings only i f the i n t e r -temporal u t i l i t y funct ion s a t i s f i e s some r e s t r i c t i v e s e p a r a b i l i t y c o n d i t i o n s . Thus a general study of even aggregate consumption and savings behaviour requires a disaggregated a n a l y s i s . In th is chapter we present such a general ana lys is o f consumer behaviour under uncer ta in ty . The ana lys is i l l u s t r a t e s c l e a r l y the l i m i t a t i o n s o f the above aggregative s t u d i e s . It a lso j u s t i f i e s the form of the u t i l i t y funct ion used by Block and Heineke (1972). In a d d i t i o n , the model we develop enables us to examine the e f f e c t s of uncer ta inty about future pr ices on current behaviour , a l a r g e l y ignored problem in the l i t e r a t u r e . We proceed as f o l l o w s : in sect ion 2 the model i s formulated and in sec t ion 3 the e f f e c t s of increased uncer ta inty about future income and pr ices are d iscussed b r i e f l y . Next the notion of r i s k independence is inves t iga ted in the context of the consumer's dec is ion problem. F i n a l l y , in sect ion 5 we discuss the impl ica t ions of the ana lys is f o r the empir ical est imat ion of consumer preferences under uncer ta in ty . 2. The Model A consumer faces a two per iod time hor i zon . There are n goods in the economy with x = (x - | , . . . , x n ) , x > 0 and p = ( p - | , . . . , p n ) , p >> 0, represent ing f i r s t per iod consumption and pr ices r e s p e c t i v e l y . Expected 34. second per iod discounted p r i ces Q = ( Q i » . . . , Q ) and the expected two per iod discounted sum of incomes I are random v a r i a b l e s ; that i s , there i s a measure space (fi, F, y ) , where F i s a a -a lgebra of subsets of fi c o n s i s t i n g of a l l poss ib le future states of the wor ld , and where y i s a p r o b a b i l i t y measure def ined on F which gives the p r o b a b i l i t y assigned to each of these states by the consumer, such that ," i = 1,2 n , and I are measurable real valued funct ions def ined on fi. We assume that that i s , expected pr ices and income are e s s e n t i a l l y bounded away from zero and e s s e n t i a l l y bounded from above. As a r e s u l t of uncerta inty in future income and p r i c e s , the consumer i s uncertain of h is second per iod consumption f o r any chosen f i r s t per iod consumption bundle. Thus in making his dec is ions the consumer must choose among vectors of the form ( x - | , . . . , x n , Y ^ , . . . , Y n ) , where Y ^ , . . . , Y are random v a r i a b l e s . The bas ic assumption of the model is that the consumer maximizes the expected value of a von Neumann-Morgenstern u t i l i t y f u n c t i o n , subject to the const ra in t that h is to ta l discounted expenditure cannot exceed h is to ta l discounted income no matter which state o f the world i s eventua l ly r e a l i z e d ; i . e . , the consumer solves the fo l lowing maximization problem: (1) 0 < 5 < Q. < M , i = 1 , 2 , . . . ,n , ( a .e . [ y ] ) , f o r some s c a l a r s 6 and M; and 0 < 6 < I < M (2) max u(x,Y(wj; dy(w x>0 J Y>0 subject to p x + Q (w)Y(w) < I(w) fo r every w in fi, where u(x,y) s a t i s f i e s condi t ions A of IV. 3 35. Since second per iod consumption is chosen a f t e r the consumer knows which state of the world has been r e a l i z e d , i f we break the maximization in (2) in to two stages - maximization with respect to Y f o r given x, and then maximization with respect to x - the f i r s t maximization may be taken ins ide the in tegra l and we obtain (3) max x>0 J g(I(:w) - p t x ; Q(w); x) dy(w) , p^x < I(w) f o r every w in n 4 where g i s the var iab le i n d i r e c t u t i l i t y funct ion corresponding to u. Formulating the consumer choice problem in t h i s way in terms of the var iab le i n d i r e c t u t i l i t y funct ion makes i t mathematically t r a c t a b l e . In p a r t i c u l a r we may der ive the fo l lowing f i r s t order condi t ions fo r an i n t e r i o r maximum x*:^ (4) E [ g x ( I - p V ; Q; x* ) ] = P^Hg^I - p t x * ; Q; x * ) ] , i = 1,2 , . . . , n . The optimal f i r s t per iod consumption bundle is def ined by the condi t ion that the expected value o f the marginal u t i l i t y o f each good i s equal to the product o f the p r i c e of that good with the expected value of the marginal u t i l i t y o f sav ings . The d iscuss ion to fo l low centers around a comparative s t a t i c s ana lys is o f equations (4). Before proceeding with that analysis,'".however, we show with the help o f the above model, that the H i c k s ' Aggregation Theorem i s i n v a l i d f o r choice under temporal uncer ta in ty . It i s c l e a r that the 36. consumer's preferences between pa i rs of the form ( c , S ) , where c i s aggregate current expenditure and S is s t o c h a s t i c savings i s represented by the funct ion V ( c , S ) , where (5) V(c,S) = max x>0 p x<c g(S(w); Q(w); x) dy(w) E s s e n t i a l l y because the optimal consumption bundle x* must be chosen before i t i s known which future p r ices and savings w i l l p r e v a i l , (and hence the maximization cannot be taken ins ide the i n t e g r a l ) , V(c ,S) cannot, in genera l , be expressed as the expected value o f a u t i l i t y funct ion of aggregate consumption and s a v i n g s . ^ This impl ies that the analyses c i t e d above, with the poss ib le exception o f Block and Heineke (1972), are v a l i d only f o r s p e c i a l cases where such aggregation i s p o s s i b l e , such as when there i s zero s u b s t i t u t i o n between any two goods in the f i r s t p e r i o d , or when the underly ing u t i l i t y funct ion u(x,y) i s f u n c t i o n a l l y separable and so can be expressed as v ( X ( x ) , y ) , where X i s a nonnegative s c a l a r - v a l u e d funct ion 2 and v i s a u t i l i t y funct ion def ined on the p o s i t i v e orthant of R . In g e n e r a l , the var iab le i n d i r e c t u t i l i t y funct ion and the model descr ibed above are necessary even fo r a study o f aggregate consumption behaviour . ' 7 F i n a l l y we remark that our model j u s t i f i e s the hypothes is , made in Block and Heineke (1972), that consumers maximize the expected value of a u t i l i t y funct ion of disaggregated f i r s t per iod consumption and sav ings . As a bonus, our ana lys is provides a way to examine the e f f e c t s of uncerta inty about future p r ices on current consumption. 37. 3. The E f f e c t s o f Increased Uncertainty We now apply the d iscuss ion of III.2 to the dec is ion problem (3) i d e n t i f y i n g e with (I,Q) and V with g(I - px; q ; x ) . It i s a s t ra ight forward matter to show that ax* D. H J E [ g > c where D and D. . are the determinant and cofac tors of the matrix ,2 1 J ^dx~dx~ Eg(I-px*;Q;x*.)) . Thus our e a r l i e r terminology regarding "1 3 s t o c h a s t i c s u b s t i t u t a b i l i t y and complementarity i s cons is ten t with the standard terminology in consumer theory. For s i m p l i c i t y we consider only one-dimensional r i s k s so that the re levant r i s k aversion measures are s - g s s ( s ' ^ x ) qi ~ V i ( s s q ; x ) ( 7 ) r = ^ T i T q l T r a n d r ' J = g (x;q;x) • J = l - 2 - - " -q j Suppose that i n i t i a l expectat ions are c e r t a i n and l e t income expectat ions undergo a marginal mean preserv ing spread. Adopting the obvious n o t a t i o n , 111.(13) impl ies that sxt 8x* . D.. ( 8 ) 3 R l s k ( l ) = " r 3Sh1f t ( l ) + g s l dx7 r S ( I - p x ' q ; x ) T T ' i = 1 » 2 , . . . j n A s i m i l a r argument with respect to expectat ions of j future pr ice y i e l d s (9) 9X* 1 8Risk(Q.) J q i -r J 9 X * 3Shi f t (Q , ) '"j k d x k 38. D.. px;q;x) - J - , i , j = l , 2 , . . . , n . The i n t e r p r e t a t i o n of these equations in terms o f a s h i f t component and a component due to a mean u t i l i t y preserv ing spread i s obvious from the d iscuss ion in III. Epstein (1975b) has discussed s u f f i c i e n t condi t ions to sign these impacts and has demonstrated the r e s t r i c t e d v a l i d i t y of the comparative s t a t i c s analyses c a r r i e d out i n the aggregative s t u d i e s . The analogues of (8) and (9) f o r the producer model are looked at c l o s e l y in VI , where the e f f e c t s o f increased c o r r e l a t i o n are a lso i n v e s t i g a t e d . Thus we do not go in to more d e t a i l here and turn instead to the problem of c h a r a c t e r i z i n g preferences cons is ten t with various forms o f r i s k independence. 4. Risk Independence The behavioural imp l ica t ions of r i s k independence were d iscussed in II I . Here we attempt to charac ter i ze the consumer preferences cons is tent with such hypotheses. q^ Rl x k requires that ° 0 ) P X ; q i X ) " f e - p k * ) v i ( I - PXiq;X> " ° • However the to ta l de r iva t i ve vanishes f o r a l l p i f and only i f each of the p a r t i a l de r iva t ives van ishes . Thus q^ Rl x k in g(I - px;q;x) f o r a l l p i s equ iva lent to q^ Rl ( s ; x k ) in g ( s ; q ; x ) . S i m i l a r l y s Rl x k in g(I - px;q;x) 39. fo r a l l p i s equiva lent to s RI (s.x^) in q ( s ; q ; x ) . Henceforth by r i s k independence we sha l l mean independence with reference to g ( s ; q ; x ) . Theorem V. I : (a) It i s impossible that s and q each be r i s k independent of s and hence that (s ,q) RI ( s , x k ) f o r any k. - r s ( a l s (b) s RI (s,x) i f and only i f g (s ;q ;x ) = a(x,q) + e (x,q)e, V H ; , where a and 6 are each o-homogeneous in q and r (q) i s homogeneous o f degree -1 in q . (c) q RI (s ,x) i f and only i f g (s ;q ;x ) = a(x) + g(x)h(s;q) where h(s;q) corresponds to homothetic preferences and has constant r e l a t i v e r i s k aversion R E -s n s s / h s . Moreover, n l - R + c , R f 1 h(s;q) = [ifiTJ ' R = 1 where a(q) i s p o s i t i v e and l i n e a r homogeneous. Parts (b) and (c) charac te r i ze the sets of preferences fo r which x* i s unaffected by mean u t i l i t y preserv ing spreads in income and p r i c e expectat ions r e s p e c t i v e l y . (We observe that Hanoch (1974) has noted that the funct ions h in (c) charac te r i ze the c lass of homothetic preferences with a constant measure of r e l a t i v e income r i s k ave rs ion . ) That these two sets o f preferences have a nu l l i n t e r s e c t i o n i s the content o f (a) . Turning to mutual p r i c e and income r i s k independence we can prove: Theorem V .2 : Let q = (q^,q 2 ) be two dimensional and consider g ( s ; q ) , the var iab le i n d i r e c t u t i l i t y funct ion with current consumption x suppressed. Then: (a) q-| and q 2 are mutually r i s k independent i f and only i f 40. (11) g ( s ; q r q 2 ) = c Q + c ] h ^ q ^ s ) + c 2 h 2 ( q ' 2 / s ) + cj^ ( q ^ s j h ^ q ^ s ) (b) q-| Rl s and q 2 Rl s i f and only i f in addi t ion to (11) each h 1 has constant r e l a t i v e r i s k a v e r s i o n , h ^ v ) A; v ^ V O - a ^ , a. f 1 A. log v a i = 1 (c) s,q-j and q 2 are each mutually r i s k independent i f and only i f in add i t ion e i t h e r ( i ) c-j = c 2 = 0 , a^ f 1, a 2 1, i . e . Cobb-Douglas pre fe rences , or ( i i ) c 3 = 0 and a.| = a 2 -The proof fol lows from Theorem II.4 a f t e r some manipulations and i s omit ted, as i s the n-dimensional genera l i za t ion of the theorem. Combinations of (11) and (c) of Theorem 1 above charac te r i ze the preferences for which c e r t a i n t y equivalents f o r random p r i c e expectat ions are s u f f i c i e n t to descr ibe optimal consumption x * . Risk independence in the consumer model i s seen to correspond to a very r e s t r i c t i v e family o f pre ferences . Thus we invest iga te b r i e f l y a weaker property which has correspondingly weaker, but s t i l l s i g n i f i c a n t , behavioural implicatdions and which i s re la ted to the work of S t i g l i t z (1969), Deschamps (1973) and Hanoch (1974). They consider an (ordinary) i n d i r e c t u t i l i t y funct ion of income and p r i c e s , h ( s ; q ) , and charac te r i ze those funct ions cons is ten t with hypotheses concerning the measures of r i s k aversion A = -h / h and R = -sh / h . However, they do not discuss the behavioural imp l ica t ions of these hypotheses and indeed, i t i s c l e a r from our e a r l i e r observat ions regarding choice under temporal uncerta inty tha t , at most, the proper t ies they consider are re la ted to t imeless p o r t f o l i o problems. Some of the corresponding proper t ies in terms of the var iab le i n d i r e c t u t i l i t y f u n c t i o n , however, are re la ted to r i s k independence and hence are o f i n t e r e s t f o r the consumer problem. 41. Consider the fo l lowing hypotheses: (12) (S) r S ( s ; q ; x ) = F s [ g ( s ; q ; x ) ,q] q j ^ r (s ;q;x) = F J [ g ( s ; q ; x ) , q ] Two comments are in order . F i r s t note that -for r i s k "aversion q,-measures r J say , of th is form, a l l mean u t i l i t y preserv ing spreads in q . , s t a r t i n g with cer ta in expecta t ions , have no e f f e c t on q . . This i s so J j because s t a r t i n g with cer ta in expectat ions and the tangency of the i n d i f f e r e n c e surface and the budget l i n e at x * , the to ta l d e r i v a t i v e in (10) vanishes q . at any x* i f and only i f r J is unaffected by changes in x and s that leave the ind iv idua l on the same i n d i f f e r e n c e s u r f a c e . Second*the above hypotheses, p a r t i c u l a r l y ( S ) , are c l o s e l y re la ted to those inves t iga ted by S t i g l i t z , Deschamps and Hanoch in the context of the i n d i r e c t u t i l i t y funct ion; , namely that A and R are constant , independent of p r i c e s , or constant on each i n d i f f e r e n c e s u r f a c e . Theorems 1 and 2 above demonstrate the s i g n i f i c a n c e o f the constancy of R f o r the two per iod consumer problem. Constancy along i n d i f f e r e n c e surfaces i s shown to be impossible f o r A but is charac ter i zed fo r R by Hanoch. However, (S ) , with absolute income r i s k aversion constant when s and x ( rather than s and q) vary along an i n d i f f e r e n c e s u r f a c e , i s both p o s s i b l e and meaningful f o r consumer react ions to increased uncer ta in ty . We may adopt Hanoch's approach to der ive the fo l lowing charac te r i za t ion o f preferences cons is ten t with (S) . The charac te r i za t ion is in terms of the var iab le expenditure funct ion def ined in IV. 42. Theorem V . 3 : The r i s k aversion measure r s s a t i s f i e s (S) i f and only i f the corresponding va r iab le expenditure funct ion i s of the form f (v ,q ) I H(x,q) d V + G ( x ' q ) ' rU (13) e(u;q;x) = -where f u ( u , q ) = r (u,q) i s homogeneous of degree -1 in q and H and G are homogeneous of degrees -1 and 1 r e s p e c t i v e l y in q . When r i s independent of the leve l of u t i l i t y , we have s Rl (s ,x ) and g(s ;q;x ) is given by Theorem 1(b). In genera l , however, the var iab le i n d i r e c t u t i l i t y funct ion i s given i m p l i c i t l y by (13) and the inverse r e l a t i o n between the funct ions e and g. Turning to (Q-)> the corresponding charac te r i za t ions are more complex and we have succeeded in obta in ing them only in spec ia l cases . When n = 1, g (s ;q ;x ) = u (x ,s /q ) and = ^- [2 - s r s ] . Thus (Q) i s true i f and only i f the measure o f r e l a t i v e savings r i s k aversion s r s i s a funct ion of u t i l i t y and p r ices , and the proof o f Hanoch's Theorem 4 may be adapted to der ive the corresponding u t i l i t y f u n c t i o n s . In the n-dimensional case , consider the stronger hypothesis q • q • (14) (Qj) r J = F J [ g ( s ; q ; x ) , q..] whentecqa = ( q - , , . . . , q j . _ 1 , q j + 1 An analogue of Theorem 3 may now be es tab l i shed using the funct ion f J of IV.(4) to def ine pre fe rences . 43. " i Theorem V.4: The r i s k aversion measure r J s a t i s f i e s (Q!) i f and only i f 3 the corresponding funct ion f J ( d e f i n e d by IV.(4)) is of the fo rm (15) f J ( u , s , q \ , x ) = fU f ( v , q \ ) I H ( s , q J , x ) d v + G ( s , q . , x ) , Hi where f u (u>qj ) = r J i s homogenouss-of degree -1 in q and H and G are homogenous o f degrees -1 and 1 r e s p e c t i v e l y in ( s , q . ) . 5. Some Empir ica l Impl icat ions Equation (4) can be used to estimate pre ferences . One need only hypothesize a funct iona l form f o r g l i n e a r in parameters. Then, given current p r i c e and quant i ty data and knowledge of future p r i c e and income expecta t ions , " r e l a t i v e l y " s t ra ight forward econometric techniques may be app l ied to estimate g. Note, however, that things are not as s t ra ight forward as in the corresponding c e r t a i n t y model in that we do not have endogenous, dependent var iab les on the l e f t hand s i d e , and the supposedly exogenous pr ices and expectat ions alone on the r igh t hand s i d e . This would require a "genera l ized" theory o f d u a l i t y and a "genera l ized" Roy's Ident i ty that would give us e x p l i c i t funct iona l forms f o r demand x * ( I ; p ; Q ) , as a funct ion of current p r i ces and random income and p r i c e expecta t ions . Lau (1974) c l a i m s , without p roof , that such an extension o f d u a l i t y theory to choice under uncerta inty is s t ra igh t fo rward ; however Epstein (1975a) has looked at the problem c l o s e l y . He f inds that though a "general ized" i n d i r e c t u t i l i t y funct ion may be def ined and charac te r i zed and the dual sets of p r o p e r t i e s , corresponding to A and B of IV, wr i t ten down, they are so complex so as to be of l i t t l e 44. or no value f o r empir ica l work. The r e l a t i v e l y simple homogeneity, monoton-i c i t y and curvature proper t ies o f B are replaced by exceed-ing ly complex curvature and s e p a r a b i l i t y proper t ies that we cannot expect to impose or t e s t r e a d i l y . As an a l t e r n a t i v e to having e x p l i c i t a lgebra ic expressions f o r demand funct ions and ad jo in ing an ;add i t i ve disturbance term, consider the fo l lowing s t o c h a s t i c s p e c i f i c a t i o n f o r (4): n (16) E[g (I - I p . (x* + v . ) ; Q ; x * + v)] •j j=l <J J J n = P,-E[g (I - I p . (x* + v . ) ; Q;x* + v ) ] , i = 1,2,...,n , where v = ( v - | , . . . , v n ) , and each Vj i s a vector (VJ-J , . . . , v . y ) , T being the number of o b s e r v a t i o n s , and v i s assumed to be mul t i va r ia te normal with zero mean. Assumptions about the var iance-cOvar iance matrix corresponding to v do not concern us here , but in general we propose the fo l lowing est imat ion procedure: fo r each set of parameter va lues , solve (16) numerical ly f o r the observed res idua ls and search over parameter values to maximize the l i k e l i h o o d o f the observed data . Since (16) i s j u s t an i m p l i c i t representat ion f o r (17) x. = f ^ U p i Q ) + v., . t h i s procedure amounts to maximum l i k e l i h o o d est imat ion of (17) and hence provides maximum l i k e l i h o o d estimates of g , with a l l the standard hypothesis t e s t i n g techniques being a v a i l a b l e . Such a numerical invers ion o f (4) or (16) should be computat ional ly f e a s i b l e i f the number o f consumption goods i s s u f f i c i e n t l y s m a l l . 45. In any case , note that est imat ion o f the model i s much more d i f f i c u l t i f preferences are expressed in terms o f the d i r e c t u t i l i t y f u n c t i o n . This i s because the f i r s t order condi t ions corresponding to (2) inc lude the unobservable future contingent consumption plan Y as an argument. While preferences may s t i l l be i d e n t i f i e d given an appropriate funct iona l form s p e c i f i c a t i o n , the computations would be complex and expensive. E s s e n t i a l l y what would be involved i s the numerical so lu t ion o f the f i r s t order condi t ions f o r Y to determine the unobservable contingent plans Y as a funct ion o f x , p, I and p r i c e expectat ions Q, and s u b s t i t u t i o n in to those f i r s t order condi t ions that def ine the optimal x. Moreover, t h i s procedure must be c a r r i e d out f o r each poss ib le s ta te of the wor ld . The var iab le i n d i r e c t u t i l i t y f u n c t i o n , on the other hand already contains the so lu t ion o f t h i s second stage opt imizat ion problem. Thus unobservable Y i s replaced by income and p r i c e expectat ions I and Q, about which reasonable hypotheses may genera l ly be made from observed incomes and p r i c e s , and est imat ion i s r e l a t i v e l y s t ra igh t fo rward . Note that i f p r i c e and income expectat ions are a lso considered unobservable, we w i l l genera l ly be unable to i d e n t i f y preferences nonmatter how they are formulated. The a p p l i c a b i l i t y o f the model is r e s t r i c t e d by i t s two-period frame-work, a d i r e c t a p p l i c a t i o n of which requires that consumers go through l i f e planning only one per iod ahead. More g e n e r a l l y , suppose that : ( i ) they have a planning horizon o f (T+1) periods but the intertemporal u t i l i t y 1 T funct ion u(x,y , . . . , y ) has the form I T I T I T u(x,y , . . . , y ) = v ( x , f 1 ( y 1 , . . . , y 1 ) , . . . , f n ( y n , . . . , y n ) ) , where yz = ( y ^ , . . . ) i s the ( t + l ) s t per iod consumption vec to r , y^ i s 46. the consumption of the i good in per iod (t+1), and each f . i s homothetic; and ( i i ) , expectat ions are such that at any t ime, uncerta inty about a l l T periods that fo l low i s expected to be removed in the next p e r i o d . Then the two-period framework i s a lso v a l i d with the future consumption vector def ined by the aggregator funct ions f. , and Q represent ing expectat ions 8 i concerning the corresponding pr ice i n d i c e s . - ' However, we can generate data f o r the l a t t e r from observed pr ices and then estimate preferences only by hypothesiz ing s p e c i f i c funct iona l forms fo r the f . . This problem! has been d iscussed in the context of c e r t a i n t y models by Darrough (1975) and Donovan (1976). If we are w i l l i n g to adopt s i m i l a r " s o l u t i o n s " , our model may be appl ied to aggregate time s e r i e s da ta , f o r example, to estimate preferences taking into account the uncerta inty that t y p i c a l l y surrounds 9 future expecta t ions . j The e f f e c t s of p r i c e and income uncer ta inty on current consumption (and labour supply) may thus be determined e m p i r i c a l l y . Note a l s o , that whereas we have u n t i l now assumed that the uncerta inty in income was exogenous, the model i s r e a d i l y extended to permit a simul-:'; ; • taneous p o r t f o l i o and consumption d e c i s i o n . In such an extended model, assets are held fo r the future purchasing power they y i e l d . Thus the hedging e f f e c t on the demand f o r money and bonds, ( i . e . , the e f f e c t of the c o r r e l a t i o n between the expected rate of i n f l a t i o n and the expected nominal return to bonds) , recent ly d iscussed by Boonekamp (1975), can be est imated. A mul t iper iod extension of the model proposed in t h i s sec t ion is c l e a r l y d e s i r a b l e , however i t would be t r a c t a b l e and useful only in spec ia l cases . Samuelson (1969), Leyhari and Sr in ivasan (1969) and Hahn (1970) have considered mul t iper iod models with s t rongly add i t i ve u t i l i t y funct ions co of the form I 3 t u ( c t ) , g . > 0 and c t denoting aggregate consumption in t=0 1 47. the t pe r iod . G e n e r a l l y , q u a l i t a t i v e r e s u l t s are forthcoming only i f f u r t h e r assumptions, such as independence of expectat ions over time or a p a r t i c u l a r funct ional form f o r u, are made. A disaggregated a n a l y s i s , with future pr ices uncer ta in , would, of course , be much more compl icated. Cox. (1975) considers such a model where a consumer solves a s t o c h a s t i c contro l problem to determine consumption, l e i s u r e and p o r t f o l i o composit ion. Moreover, the intertemporal u t i l i t y funct ion i s p e r f e c t l y genera l . This complicated T - p e r i o d problem i s reduced to a two-period problem by apply ing the dynamic programming method o f backward o p t i m i z a t i o n , i . e . by s o l v i n g , f o r given per iod d e c i s i o n s , the maximum value of the ob jec t ive funct ion with respect to a l l future dec is ion v a r i a b l e s . Th is new f u n c t i o n , c a l l i t J , then serves as the basis of a two-period problem which determines optimal current d e c i s i o n s . Cox i s able to show that the funct ion J i n h e r i t s some of the proper t ies of u, and these are used to der ive e a s i l y some comparative s t a t i c s r e s u l t s . Fama (1970) has s i m i l a r l y argued that the consumer's l i f e t i m e planning problem may be reduced to a two-period problem. However, t h i s approach has the fo l lowing l i m i t a t i o n s : f i r s t , only a few second order proper t ies may be shown to be inher i t ed by J . In p a r t i c u l a r the e f f e c t s of s h i f t s in pr ice expectat ions (a second order property) and increased v a r i a b i l i t y of expectat ions (a t h i r d order property) cannot be determined in t h i s way. Expectat ions are i n e x t r i c a b l y embedded in J , and i t i s impossible to completely charac te r i ze the proper t ies of J f o r expectat ions in a given parametric c l a s s , say. Thus the funct ion J i s v i r t u a l l y useless f o r empir ical work; we can nei ther impose nor tes t f o r the condi t ions that i t correspond to the so lu t ion of a control problem. Only in a two-period model are these l i m i t a t i o n s absent , and in that case the Cox approach i s s i m i l a r to ours . 48. To conclude, consider the fo l lowing funct ional form f o r g (s ;q ;x ) that w i l l enable us to estimate preferences given the assumptions discussed above s u f f i c i e n t to j u s t i f y a two-period framework: (18) g ( S i „ x ) - S U,t + I b i d X) • s " I C . . K q - \ : \ + s k J d i j + ^ J e i ^ + I V i + 9o • where a . . = a . • , b. . = b . . and c . . , = c . . , . It i s r e a d i l y v e r i f i e d that a s u f f i c i e n t cond i t ion f o r t h i s funct ion to s a t i s f y B i s that a l l the parameters are non-negat ive. Even i f some of the parameters are negat ive , g i s f requent ly a l o c a l representat ion of a va r iab le i n d i r e c t u t i l i t y f u n c t i o n . 1 ^ g corresponds to l i n e a r homogeneous preferences i f and only i f the d ^ ' s , e ^ ' s , and f . - ' s van ish . Thus (18) i s more general than the func t iona l_ fo rm- in Epstein (1975b)^which imposed l i n e a r homogeneity. F i n a l l y , g contains s u f f i c i e n t l y many parameters to assume any set of f i r s t and second order de r iva t i ves at a p o i n t , plus a l l t h i r d order de r iva t i ves of the form g , g . Comparative s t a t i c s s q i x k q i q j x k ana lys is of (4) shows that these t h i r d order proper t ies are s u f f i c i e n t to determine consumer react ions to increased v a r i a b i l i t y and c o r r e l a t i o n , and so we may c a l l such funct iona l forms " s u f f i c i e n t l y f l e x i b l e " . We note that the funct iona l form in Epstein (1975b) i s t h i r d order f l e x i b l e and''so~ contains "too many" parameters, while that given in Diewert (1975) (equation 2.17) i s only f l e x i b l e to the second order and so places a p r i o r i r e s t r i c t i o n s on behaviour under uncer ta in ty . The matter of f l e x i b i l i t y requires fu r ther comment. Although f l e x i b l e 49. funct iona l forms are capable of approximating an a r b i t r a r y funct ion well at a p o i n t , they do not n e c e s s a r i l y provide a good approximation over a range of observat ions as measured by t h e i r a b i l i t y to s a t i s f y the appropr iate r e g u l a r i t y c o n d i t i o n s . The ana lys is in Wales (1.975) suggests that t h e i r global performance i s genera l ly bet ter the smal ler the v a r i a t i o n in the determining v a r i a b l e s . The issue of loca l versus global performance i s even more ser ious in models with uncer ta in ty , fo r then we want the funct ional form to be well-behaved not only over the range of actual observa-t i o n s , but a lso over the range determined by the consumer's random expecta-t ions. ' Moreover, genera l ly speaking, the l a t t e r i s small only i f the uncerta inty in expectat ions can reasonably be ignored in the est imat ion in the f i r s t p lace . It does not seem that much can be done to improve matters , with the exception of ensuring that the postulated funct iona l form i s capable of being well -behaved over a large range fo r some parameter va lues , as is the case with (18). As argued by Wales, however, even i f the r e g u l a r i t y condi t ions are not s a t i s f i e d , the use o f f l e x i b l e funct iona l forms might be r a t i o n a l i z e d as a convenient method f o r summarizing market behaviour, white at the same time maintaining the homogeneity and adding up proper t ies of the demand equat ions. The usefulness of (18) or any other funct iona l form depends on the way in which i t meshes with the p r o b a b i l i t y d i s t r i b u t i o n descr ib ing the random expecta t ions . This is d iscussed below in the context of an empir ica l ana lys is of producer behaviour. We do not propose to estimate a vers ion of the consumer model in t h i s t h e s i s , though i t may be the subject of future research . The matter has been d iscussed at t h i s point only to c l a r i f y the potent ia l usefulness of the t h e o r e t i c a l ana lys is that we have presented. 50. VI . PRODUCTION FLEXIBILITY AND THE BEHAVIOUR OF THE COMPETITIVE FIRM UNDER PRICE UNCERTAINTY 1. Introduct ion The recent i n t e r e s t in the theory of the f i rm under demand or p r i c e uncerta inty o r ig ina tes with the ana lys is of the f i r m ' s optimal inventory-output p o l i c y in M i l l s (1962) and Arrow, Kar l in and Scar f (1958) and (1962). 1 Later work along s i m i l a r l i n e s includes Orr (1967) and Zabel (1967) and (1969). These studies t y p i c a l l y assume s p e c i f i c funct iona l forms, e . g . , quadrat ic production costs and/or l i n e a r inventory storage c o s t s , and r i s k n e u t r a l i t y on the part of the f i r m . Two-period models that maintain the l a t t e r assumption are found in Nelson (1961), T i s d e l l (1968) and Smith (1969) and (1970). Hartman (1972) and (1973) analyses the e f f e c t of future product and f a c t o r p r i c e uncer ta inty on the rate of investment in a s i n g l e cap i ta l s tock , given that the f i rm is a p r ice taker in a l l markets but encounters convex adjustment costs in changing the leve l of i t s s tock . Some q u a l i t a t i v e e f f e c t s of uncerta inty are der ived by assuming, in (1972), that the technology exh ib i ts constant returns to s c a l e , and in (1973), that expecta-t ions are temporally independent. In both cases the f i rm maximizes the expected value of the sum of discounted cash f lows. P r o f i t r i s k aversion is assumed by Dhrymes (1964), who adopts a mean-variance approach and by McCall (1967) and Zabel (1971) who assume that the f i r m ' s von Neumann-Morgenstern u t i l i t y index exh ib i t s constant absolute r i s k a v e r s i o n , the former in the context of a s t a t i c model and the l a t t e r in the framework of his two e a r l i e r s t u d i e s ; 51. The most complete s tudies of the e f f e c t s o f p r i c e uncerta inty on the o competi t ive f i rm are those by Sandmo (1971) and Batra and Ul1 ah (1974). They assume that the dec is ion on the volume of output to be produced must be taken p r i o r to the sa les date, at which time the market p r i c e becomes known, and i s determined as the s o l u t i o n to (1) max E U[Py - C ( y ) ] , y>o where P i s a random var iab le descr ib ing the f i r m ' s sub jec t ive p r ice expecta-t i o n s , y i s the leve l o f production of the s i n g l e output , C(y) i s the minimum cost of producing y and U i s a von Neumann-Morgenstern u t i l i t y f u n c t i o n . A complete comparative s t a t i c s ana lys is i s c a r r i e d out of the f i r s t order condi t ions corresponding to (1) and i t i s shown that some of the standard r e s u l t s of the theory of the f i rm under c e r t a i n t y are no longer v a l i d . In p a r t i c u l a r , f i xed costs may a f f e c t the leve l of production and output may increase or decrease in response to an upward s h i f t in p r i c e expecta t ions , depending on the way in which the measure of absolute r i s k aversion changes with the leve l o f p r o f i t s . When U" < 0, output i s shown to be smal ler when expectat ions are random than when they are ce r ta in with the same expected va lue . This is re fe r red to as the overa l l e f f e c t of uncer ta in ty . A marginal increase in the v a r i a b i l i t y of expectat ions i s shown by Batra and U l lah to reduce y i f absolute r i s k aversion i s decreas ing . We modify and genera l ize these studies in three p r i n c i p a l ways. F i r s t , we adopt the d e f i n i t i o n s of increased v a r i a b i l i t y d iscussed in II and III. Sandmo and Batra and Ul lah represent increased uncerta inty by a m u l t i -p l i c a t i v e s h i f t of the random var iab le coupled with a t r a n s l a t i o n so as to 52. preserve the mean. Since the r e s u l t s are s e n s i t i v e to the d e f i n i t i o n s adopted, i t i s c l e a r l y des i rab le to adopt t h e o r e t i c a l l y cons is tent d e f i n i t i o n s such as those of Rothschi ld and S t i g l i t z , and Diamond and S t i g l i t z . With the exception of the paper by Dhrymes, other s tudies consider s i n g l e product f irms e x c l u s i v e l y . G e n e r a l l y , a s i n g l e dec is ion v a r i a b l e , t y p i c a l l y output , i s determined subject to uncer ta inty about a s i n g l e future v a r i a b l e , t y p i c a l l y the corresponding p r i c e . We consider the case of a genera l , mu l t ip roduc t -mu l t i f ac to r technology where several dec is ion var iab les must be chosen subject to uncer ta inty about several product and f a c t o r p r i c e s , f o r example. Such a mu l t i va r ia te ana lys is allows one to inves t iga te the e f f e c t s of increased c o r r e l a t i o n in expecta t ions . F i n a l l y and most impor tant ly , we consider producer behaviour given that the technology permits some ex post f l e x i b i l i t y in ad jus t ing to the pr ices that are eventua l ly r e a l i z e d . In the Sandmo, Batra and Ul lah model, a l l dec is ions must be made ex ante . Turnovsky (1973) has extended the model to allow the f i rm to modify i t s i n i t i a l d e c i s i o n s , at an add i t iona l c o s t , a f t e r i t learns the true s e l l i n g p r i c e of i t s product , captur ing the notion of production f l e x i b i l i t y in the sense introduced by Hart (1942). However, he assumes that only the future demand f o r the s i n g l e output is uncertain and does not address h imsel f to any of the comparative s t a t i c s quest ions mentioned above. Ex post adjustment a lso plays a ro le in the dynamic analyses re fe r red to above, but they s u f f e r , perhaps n e c e s s a r i l y because of the complexity of mul t iper iod models, from r e s t r i c t i v e funct iona l forms and s c a l a r dec is ion v a r i a b l e s . In the Smith models, cap i ta l stock i s determined ex ante subject to uncerta inty about future demand, while i t s 53. rate of u t i l i z a t i o n and labour are adjustable ex post . However, he assumes r i s k n e u t r a l i t y and most of his q u a l i t a t i v e r e s u l t s are der ived assuming a Cobb-Douglas or CES technology. The ana lys is of behaviour in the general s i t u a t i o n , with an a r b i t r a r y technology and where some dec is ions are made ex ante and some ex p o s t , w i l l be shown to be s i m p l i f i e d by using the var iab le p r o f i t funct ion to descr ibe the technology. This chapter proceeds as fo l lows: in sec t ion 2 the model i s formulated. Ignoring many of the bounded problems which complicate Turnovsky's a n a l y s i s , in sec t ion 3 we s i m p l i f y the presentat ion and proof of some of his r e s u l t s . In the next two s e c t i o n s , some of the comparative s t a t i c s questions not addressed by Turnovsky are examined and some of the r e s u l t s of Sandmo and Batra and Ul1 ah are genera l i zed . Sect ion 6 deals with r i s k independence in the context of the producer 's dec is ion problem and an example is given in sect ion 7 to i l l u s t r a t e many of our f i n d i n g s . F i n a l l y , a f l e x i b l e funct iona l form f o r the va r iab le p r o f i t funct ion i s hypothesized in sec t ion 8 and the est imat ion of technology given that producers operate in a s t o c h a s t i c environment, i s b r i e f l y d i s c u s s e d . 2. The Model Consider a technology descr ibed by an intertemporal concave transforma-t ion funct ion F ( x , y , z ) , where we genera l ly i n t e r p r e t x = ( x - , , . . . , x ) as i n r the vector of f i r s t per iod (ex ante) i n p u t s , z = (z q z ) as the 2 vector of future (second per iod or ex post) i n p u t s , and y = ( y - | » . . . , y ) as the vector o f outputs forthcoming in the second p e r i o d . With x , z and y are associa ted pr ices q , W and P,where W and P are vector random var iab les r e f l e c t i n g uncer ta inty about (discounted) future f a c t o r and product p r i c e s . The f i rm i s assumed to possess a von Neumann-Morgenstern u t i l i t y funct ion V, def ined over p r o f i t s , such that i t solves 54. (2) max EV[P-y - W-z - q .x] x,y,z>0 subject to F ( x , y , z ) = 0 , where x i s chosen ex ante and y and z ex post . Breaking the maximization in to two s tages , we der ive the fo l lowing equivalent problem: (3) max EV[g(P,W;x) - q-x] , where x>0 y g (P»w;x) = max {p-y - w - z / F ( x , y , z ) = 0} , z,y>0 4 is the var iab le p r o f i t funct ion dual to F. Since y and z are chosen ex post and opt imal ly subject to the p rev ious ly chosen x, random p r o f i t s as a funct ion of x and expectat ions P and W are simply TT = g(P,W;x) - q -x , and (3) f o l l o w s . Suppose that n 3 (4) g (p»w;x ) = I p . - f ' f x ) , where each f i s inc reas ing and concave. i = l 1 This i s the case of zero f l e x i b i l i t y - * analysed in the s i n g l e product case by Sandmo and Batra and U l l a h . Outputs are completely determined ex ante by s e l e c t i n g x. There i s no p o s s i b i l i t y of a l t e r i n g t h e i r l e v e l s by apply ing the f a c t o r of production z , nor i s i t poss ib le to change the mix of outputs ex post . The f i r s t order condi t ions corresponding to (3) are: (5) E [ V ' . ( g x . - q,-)] = 0 , i = 1,2 n r 55. The second order condi t ions require concavi ty in x of E[V(g(P,W;x) - q - x ) ] , f o r which i t is s u f f i c i e n t , though not necessary , to assume V" < 0. For s i m p l i c i t y , th is i s a maintained hypothesis throughout our d i s c u s s i o n . We a lso assume that a unique s o l u t i o n to (5) e x i s t s , which i s denoted x * . In the next three sect ions we analyse the f i r s t order condi t ions (5) . In sect ions 4 and 5 a comparative s t a t i c s ana lys is i s performed. F i r s t , however, some global r e s u l t s are der ived f o r a s p e c i a l case of the above model. The d iscuss ion below w i l l be s i m p l i f i e d by adopting the fo l lowing nota t ion : (6) x (p ,w ,q ) , y(p,w,q) and z(p,w,q) denote the so lu t ions to max {p-y - w-z - q - x / F ( x , y , z ) = 0} , x,y,z>0 and y(p,w:;x) and z(p,w;x) denote the short run supply and demand f u n c t i o n s , i . e . , the so lu t ions to max {p-y - w - z / F ( x , y , z ) = 0} . y,z>0 3. The Overa l l Impact o f Uncerta inty Assume that there i s a s i n g l e ex ante f a c t o r and a s i n g l e product produced ex post ; i .e. , n^ = n^ = 1, and that W = w is nonstochas t ic , and consider the e f f e c t s on current f a c t o r demand of future product p r i c e 5 6 uncer ta in ty . ' Denote by x the so lu t ion to (3) when p r i c e expectat ions are cer ta in and equal to P = E [ P ] . In the l i t e r a t u r e , x* - x has been c a l l e d the overa l l impact o f uncer ta in ty . The fo l lowing theorem descr ibes condi t ions under which i t s s ign i s determinate. Theorem V I . 1 : (a) I f g > 0 and i f g is concave in p, then x* < x . xp — X — (b) I f g < 0 and i f g i s convex in p, then x* > x . 56. In the case of zero f l e x i b i l i t y , g(p,w;x) = p f ( x ) , g = p f ' ( x ) which A i s concave in p, and g = f ' ( x ) > 0, so that r i s k aversion (V" < 0) alone px impl ies that x* < x, a r e s u l t which has been noted in several s t u d i e s , i n c l u d i n g Sandmo (1971) and Batra and Ul1 ah (1974). Given production f l e x i b i l i t y , more than jiust r i s k aversion is required to enable us to say unambiguously that the overa l l impact of uncerta inty i s to reduce f a c t o r demand. Indeed, the opposite impact i s a p r i o r i equa l ly l i k e l y , depending on the sign of g . The s i g n i f i c a n c e of the l a t t e r i s made c l e a r by not ing xp that q 3 p ( p . W . q ) - ^ . 9X — and so — and g„„ have the same s i g n . The r e l a t i v e s i z e s of x* and x , 3 p 3 px 3 the re fo re , depend in part on whether x is a "normal" f a c t o r in the sense t h a t , given c e r t a i n t y , a higher product p r i c e , and hence a higher output l e v e l , induces a greater u t i l i z a t i o n of x . The overa l l impact of uncerta inty on the expected leve l of production i s a lso determinate under c e r t a i n c o n d i t i o n s . Theorem VI .2: Let y be optimal given c e r t a i n expectat ions P = E [ P ] , and l e t E [y* ] be the expected value of output fo l lowing from opt imizat ion subject to the s t o c h a s t i c expectat ions P. Then': ( a ) I f g n 1 S concave in p and i f e i t h e r , x* < x and g > 0, o r , p px x* > x and g n v < 0, then E[y* ] < y . px -(b) I f g i s convex in p and i f e i t h e r , x* < x and g > 0, o r , p px x* > x and g < 0, then E[y* ] < y . 57. Given zero f l e x i b i l i t y , (a) appl ies and expected output i s reduced unambiguously by uncer ta in ty . Though most of our d iscuss ion i s concerned, as above, with the e f f e c t on a given f i rm of increased uncer ta in ty , we can a lso compare the behaviour of d i f f e r e n t f irms fac ing the same uncer ta in ty . Let x ° be the s o l u t i o n to (3) given V" = 0. Then we can prove the fo l low ing : Theorem VI .3 : (a) I f g v n > 0 , then x* < x ° . xp -(b) I f g v n < 0, then x* > x ° . xp The e f f e c t of r i s k aversion on f a c t o r demand depends p r e c i s e l y on whether x i s a "normal" or " i n f e r i o r " f a c t o r in the sense descr ibed above. C l e a r l y x ° > x* unambiguously given zero f l e x i b i l i t y . Combining Theorems 1 and 3 we see that both x* > x ° > x , ( i . e . , a l l f irms increase f a c t o r demands from t h e i r c e r t a i n t y l e v e l s and the r i s k averse f i rm more so then the r i s k neutral f i r m ) , and x* < x ° < x , ( i . e . , a l l f irms reduce f a c t o r demands from t h e i r c e r t a i n t y l e v e l s and the r i s k averse f i rm more so than the r i s k neutral f i r m ) , are p o s s i b l e . Theorem VI .4: Adopting the obvious n o t a t i o n , we have that (a) I f g p x > 0, then E [y* ] < E [ y ° ] . (b) I f g _ x < 0, then E[y* ] > E [ y ° ] . Combining Theorems 3 and 4 we see that i f g is uniformly signed with respect px to p, then E[y* ] = E [ y ° ] according as x* = x ° . I f there are several ex ante fac tors o f product ion , extension of Theorems 1 and 3 may be der ived i f the technology i s separable in the sense that g = 0, i f i f j . I f g < 0 f o r a l l i and j , we can f i n d condi t ions x i x j x i x j " that imply x* = x^ f o r at l eas t one i . In genera l , however, with many fac tors 58. we can i n f e r l i t t l e about the overa l l impact o f uncerta inty and we must be s a t i s f i e d with determining what has been c a l l e d in the l i t e r a t u r e , the marginal impact o f uncer ta in ty . That i s the subject of sect ion 5. 4. S h i f t s in Fixed Costs and Pr ice Expectat ions^ Both Sandmo and Batra and U l lah point out that two of the standard r e s u l t s of the theory of the f i rm under c e r t a i n t y , that f i x e d costs do not a f f e c t production and that supply curves are upward s l o p i n g , are not v a l i d fo r the f i rm operat ing in a s t o c h a s t i c environment. Appropr iate g e n e r a l i z a -t ions of t h e i r r e s u l t s to the case of production f l e x i b i l i t y ex post are r e a d i l y der ived in the context of the simple model considered in the l a s t s e c t i o n . To determine the e f f e c t of higher f i x e d c o s t s , subt ract B from the argument of the u t i l i t y funct ion in (5) , t o t a l l y d i f f e r e n t i a t e (5) and evaluate at B = 0. We f i n d that . . . a y * E [ V " . ( g x - q ) ] where D is the second order de r i va t i ve with respect to x of the ob jec t ive funct ion and is negative in s i g n . Adopting an argument o f Sandmo, i t fo l lows that < « t : < : > ° 1 f d 7 - V - ( > - ' ° • o where A( i r) = - V " (TT ) / V ' (TT) i s the absolute measure o f r i s k avers ion . Decreasing absolute r i s k aversion impl ies 8x*/8B < 0 when there are no ex post adjustment p o s s i b i l i t i e s , as found by Sandmo and Batra and U l l a h , g but not in genera l , depending on the "normali ty" o f the ex ante f a c t o r . 5 9 . Denote by 9 x * / 3 S h i f t (P) the impact on x* o f a marginal s h i f t in the expected pr ice d i s t r i b u t i o n . S u b s t i t u t i n g P + b f o r P in ( 5 ) , t o t a l l y d i f f e r e n t i a t i n g and eva luat ing at b = 0 , we der ive , q , _ ^ x * - E C V g x p ^ g p v"(g x -q)] i y ; 3Shif t (P) D Assuming that x* responds p o s i t i v e l y to product p r i ces given c e r t a i n t y (g > 0 ) , x* responds p o s i t i v e l y to a s h i f t in va r iab le expectat ions i f px E [ V " g p ( g x - q)] = -E[A g p V ' ( g x - q)] > 0 . Since g x p > 0 , a s u f f i c i e n t condi t ion i s that — (A*g p ) < 0 , or do) 1 ^ < "fee. u u ; A d-rr - 2 9 P With no f l e x i b i l i t y , g = 0 and so decreasing absolute r i s k aversion i s r r s u f f i c i e n t to ensure an upward s lop ing supply f u n c t i o n . When there is some production f l e x i b i l i t y , an element of p r i c e r i s k a f f i n i t y i s introduced 3PP (g > 0 ) , and only i f ^ i s s u f f i c i e n t l y negative to o f f s e t th is in the 3 X* sense that ( 1 0 ) i s s a t i s f i e d , can we say that 3 $ h i f t ( P ) > ^ unambiguously. An a l t e r n a t i v e s u f f i c i e n t condi t ion can be der ived as fo l lows: a f t e r some manipulations we f i n d that (11) V ' 9 x p + ( g x - q ) g p V " - V ' g x p ( l - R ) • V " g x p ( ^ - g) • V"q g x p ( x - ? E _ , where R = —n-V"(ir)/V (TT) i s the measure of r e l a t i v e r i s k avers ion . Making use of the homogeneity proper t ies of g and of H o t e l l i n g ' s Lemma, we can show that i f 60. (12) (a) z(p,w;x) < y(p,w;x) • --— z(p,w;x) / -^- y (p ,w ;x ) , and (b) x < y(p,w;x) / -^- y(p,w;x) , then the l a s t two brackets on the r i g h t hand s ide of (11) are n e g a t i v e . ^ Therefore , the fo l lowing condi t ions ensure that 9 x * / ° S h i f t ( P ) > (<) 0: (13) ( i ) R < 1 ( i i ) 9 x p >_ (<) 0 ( i i i ) (12) J 1 g If g (p»w;x ) = p f ( x ) , z = 0 and x - -P— > 0 i f and only i f f (x ) i s concave. 9 px Thus with no production f l e x i b i l i t y , R < 1 and a concave technology ensure an upward s lop ing supply f u n c t i o n . q F i n a l l y , note that 9 x * / ° q i s more e a s i l y s igned . Since 1 E [ V + x * ( g x - q ) V M ] , i t i s negative i f E[V"-(g x-q)] > 0. Apply ing the Sandmo argument, (see footnote 8 ) , the f a c t o r demand curve i s downward dA s lop ing i f ^ • g p x < 0. A l t e r n a t i v e l y , s ince V + x * ( g x - q ) V " = V ' ( l - R ) + (x* g v - g ) V " , 3 x * / 9 q i s negative i f R < 1. A — In the general mul t ip roduct , m u l t i f a c t o r model, these q u a l i t a t i v e r e s u l t s can be extended only in spec ia l cases , such a s , f o r example, when only one 1 o future p r ice i s uncertain or there is p r o f i t r i s k n e u t r a l i t y (V" = 0 ) . Such spec ia l cases are considered in the ana lys is of the marginal impact o f uncer ta in ty , to which we now t u r n . 61. 5. The Marginal Impact of Uncertainty We now apply the d i s c u s s i o n of Chapter III to the dec is ion problem (3) , making the obvious i d e n t i f i c a t i o n of e with (p,w) and V with the composite funct ion V[g(p,w;x) - q - x ] . From (5) i t fo l lows that (14) dx* dq. TT= constant - D J I D EV 1 where D and D. . are the determinant and cofactors the matrix ' J 2 (9 EV(g(P,W;x*) - q - x * ) / 9 x . 9 x / l " s » a n d where the t o t a l de r i va t i ve denotes the e f f e c t on x* of a change in q . that is compensated by a lump sum addi t ion to p r o f i t s that keeps TT = g(P,W;x*) - q - x * constant in every state of the wor ld. When V" = 0, the d e r i v a t i v e equals 9 x * / 9 q j so that our e a r l i e r terminology regarding s t o c h a s t i c s u b s t i t u t a b i l i t y and complementarity is cons is ten t with customary terminology in the theory of the f i rm i f we i n t e r p r e t these proper t ies as r e f e r r i n g to gross demand f u n c t i o n s . For s i m p l i c i t y we consider only one-dimensional r i s k s so that the re levant r i s k aversion measures are (15) Pit P J P J V j 1 5 2 , . . . , n g , wn-r J V v q '.vi j = 1 , 2 , . . . , n . These measures r e f l e c t the a t t i tude towards p r i c e uncerta inty der ived from aversion towards r i s k in p r o f i t s , v" and J 62. and the p r i c e r i s k a f f i n i t y due to the p o s s i b i l i t y of ex post adjustment to the eventual market p r i c e , ' g p - P i V w . —>LJ_ < o , J J < 0 ° P J " ° W J 13 Pi w i I f there i s no ex post f l e x i b i l i t y , r J , r 0 > 0 and the f i rm i s p r ice r i s k averse. I f there is p r o f i t r i s k n e u t r a l i t y , r^ J , rW j ' < 0 and the f i rm is r i s k a f f i n e with respect to p r i c e s . In g e n e r a l , the r e l a t i v e magnitudes of the cont r ibut ions o f p r o f i t r i s k aversion and production f l e x i b i l i t y determine the a t t i tude towards p r ice uncer ta in ty . The technolog ica l component of these measures, or the measures them-selves when V" = 0, can be fur ther i n t e r p r e t e d . By H o t e l l i n g ' s Lemma, short run supply and demand funct ions are given by y.-(p,w;x) = g (p,w;x) and z-(p,w;x)=-g- ; (p ,w;x ) . Define the short run e l a s t i c i t i e s Wj p i 3 ~ (16) a i : j ( p , w ; x ) = — y. (p,w;x) , i , j = 1 , 2 , . . . , n 3 , and "I J w. ii . j . j (p»w;x ) = z. (p,w;x) , i ,j = 1 , 2 , . . . , n 2 . Then, i f V" = 0, the r i s k aversion measures can be expressed as (17) r J = ^ - a . , ( p , w , x ) , r J = f y..v(p;w;x) I Pj J J W j J J demonstrating the r e l a t i o n s h i p between technologica l f l e x i b i l i t y and p r i c e r i s k a f f i n i t y . Other things e q u a l , l a rger own short run e l a s t i c i t i e s ( in absolute value) f o r the ex post products and factors,, imp!ies greater a f f i n i t y fo r uncer ta inty in any s i n g l e future p r i c e . 63. One would a lso expect r i s k a f f i n i t y to be la rger the fewer the number of dec is ions that must be made ex ante , o r , adopting the terminology of Dreze and Modig l ian i (1972; p. 314), we would expect the expected value of information to be p o s i t i v e . That th is is indeed the case i s confirmed by not ing that in l i g h t of the above, th is asser t ion is equiva lent to the Le C h a t e l i e r P r i n c i p l e . Note that th is i s true even i f V" f O . 1 ^ Aversion towards c o r r e l a t i o n o f the i and j product p r ices say , i s r e f l e c t e d by the measures (18) c P i ' P J ' ^ = z l 0 l j ( p , w ; x ) + g J " - V " 3 r P i , P j , P j _ -1 p . a . ^ p . w i x ) + g p _ P r o f i t r i s k aversion impl ies an aversion to increased c o r r e l a t i o n whi le the f l e x i b i l i t y in production impl ies an aversion ( a f f i n i t y ) i f the i and th j products are gross complements (subst i tu tes ) in the short run. Since the Le C h a t e l i e r P r i n c i p l e does not apply to cross e l a s t i c i t i e s , i t i s p o s s i b l e f o r the producer to be more averse to c o r r e l a t i o n when a l l dec is ions may be made ex post ' than when some must be made ex ante. For example, compare the a t t i tude towards increased c o r r e l a t i o n o f the and product pr ices o f ^a^producer who mustcmake2a2sealaK=i.nputndecision .ex ante , with that of a producer who can make a l l dec is ions ex pos t . Adapting P o l l a k ' s (1969) proof of the Le C h a t e l i e r P r i n c i p l e , i t can be shown that the l a t t e r i s less (more) averse to c o r r e l a t i o n i f (p,w,q) • -^ p— (p,w,q) > 0 (< In p a r t i c u l a r , the expected value of information with respect to greater c o r r e l a t i o n i s negative i f the c e r t a i n t y demand funct ion x(p,w,q) responds q u a l i t a t i v e l y d i f f e r e n t l y to p. and p . . 64. Turning to the e f f e c t s on x* of a marginal increase in the v a r i a b i l i t y th of the j product p r i c e expecta t ion , f o r example, we may apply 111.(14) to derive that 9 X * P,-_ D L <19> 8Rii * T P T T = I ^ ' \ P . + v" g p ( g X k - q k » r j ] + PCV' 9 P j 4 r P d ] ^ 1 , 1 = l t 2 — ^ * The decomposition of r J in (15) induces a corresponding decomposition o f (19) whichaaliliows us roughly to d i s t i n g u i s h the cont r ibut ions o f the u t i l i t y index and technolog ica l f l e x i b i l i t y to the overa l l impact. We now consider some s p e c i a l c a s e s . I f i n i t i a l expectat ions are c e r t a i n , g = q ! / at the i n i t i a l optimum x k K and we obtain (20) 9 X * 9Risk(P. ) J 9 p . p . 9 X * 9ETPTT " V ' 9 P , l - 2 Pj k 9 x k -V" 9 X * 9 P j wurj where 9X* 9P: x^(p,w,q) , (p,w) being the i n i t i a l c e r t a i n expectat ions, I f there is p r o f i t r i s k n e u t r a l i t y 9 x * / 9 R i s k ( P . ) i s equal to the large bracketed expression and so depends on the "cer ta in ty r e l a t i o n " between x^ and p . , the subst i tut ion-complementar i ty r e l a t i o n s among the X i / s , and the J r q 1 1.-P--. The l a t t e r i n d i c a t e the way in which the short run . - -'TIS —9 terms 9 X r -P J P J .th e l a s t i c i t y o f supply of the j product i s a f fec ted by the l eve ls of var ious ex ante f a c t o r s . From (20) , ~ - a . . . ( p , w ; x ) < 0 (>0) induces a reduct ion o X ^ J J ( increase) in x k and in a l l f ac tors gross ly complementary (subst i tu tab le ) 65. with i t , cons is tent with the expectat ion that increased uncer ta inty should c a l l f o r the adoption of more f l e x i b l e production techniques. P r o f i t r i s k avers ion leads to a greater reduct ion (smal ler increase) in x* only i f 9x- ( p , w , q)/9p . > 0. If there i s zero f l e x i b i l i t y , the large bracket vanishes and V" < 0 alone impl ies that x* f a l l s fo r a l l i . p i I f i n i t i a l expectat ions are s t o c h a s t i c but V" = 0, r J i s given by (17) and from (19) i t fo l lows that a mean u t i l i t y ( i . e . p r o f i t ) increase in r i s k •f*h with respect to the j product p r ice increases x* i f 9a--/9x,, < (>) 0 fo r i J J K -a l l f ac to rs x^ that are gross subst i tu tes (complements) with x^. To sign the e f f e c t o f a mean preserv ing spread in P . , we have to make some assumptions J about the signs of g _ ( = ay . (p ,w;x)/9x.,) , k = l , 2 , . . . , n n . With no ex post x k p j j 1 f l e x i b i l i t y at a l l , g i s l i n e a r in future p r ices and we get the well-known r e s u l t that x* is independent of the degree of uncer ta in ty . F i n a l l y , suppose that n-| = n 2 = n 3 = 1 and that W = w i s c e r t a i n . Then <21> M T = ^ { E [ ( v ' g x p + v " g p ( g x - q ) ) r p ] + E [ V g p A _ r P ] } , where r p = + g p [^jr-] . The fo l lowing theorem descr ibes condi t ions under which the impact on x* o f increased uncerta inty i s determinate in s i g n . Theorem VI .5 : Assume that dA/dir < 0, dR/dir > 0 and that (12) i s s a t i s f i e d by the technology. Then the e f f e c t of a mean u t i l i t y preserv ing spread in p r i c e is to reduce ( increase) x* i f g > (<) 0 and ~ (J^-) < (>) 0. 8 A mean px - - ax g p preserv ing spread in expectat ions reduces x* i f g x p > 0, g x p p < 0, g p p p < Q and R < 1. 66. Comparison with the much stronger Theorem 1 demonstrates the added complexity in determining the e f f e c t o f a mean preserv ing spread due to i n i t i a l s t o c h a s t i c expecta t ions . (However, Theorems 2 through 4 genera l i ze d i r e c t l y . For example, i t fo l lows from H o t e l l i n g ' s Lemma that i f g p i s concave (convex) in p and g_ v - 9 x * / 3 R i s k [ P . ] < (>) 0, then 9E [y* ] /9Risk[P] < (>) 0. px - - - -I f we assume that the degree of r i s k aversion may be represented by a continuous parameter as in the Diamond a n d j S t i g l i t z paper, we may apply t h e i r Theorem 4 to show that i f g > (<) 0, then x* decreases ( increases) with px -increas ing r i s k a v e r s i o n , genera l i z ing Theorem 3. F i n a l l y , exact ly as in the proof of Theorem 4, H o t e l l i n g ' s Lemma impl ies that E[y*.] i s reduced ( increased) by increas ing r i s k aversion i f g > (<) 0.) px When there i s no f l e x i b i l i t y , Theorem 5 becomes: Theorem VI .6 : Suppose that there i s zero f l e x i b i l i t y in product ion . Then the e f f e c t o f a mean u t i l i t y preserv ing change i s negative i f d A / d i r < 0, and dR/dTr > 0. A mean preserv ing change in expectat ions reduces x* i f dA/dir < 0, dR/dTr > 0 and R < I.15 Note that in c o n t r a d i c t i o n with the Batra and U l lah f i n d i n g , the proof of the theorem shows that decreasing absolute r i s k aversion i s cons is ten t with increased uncerta inty inducing a higher output l e v e l ; e . g . , i f 1 -a V(TT) = TT / 1 - a , where a > 1 i s the constant measure of r e l a t i v e r i s k a v e r s i o n , and i f the technology i s l i n e a r homogeneous. The analogue of Theorem 5 f o r the case of f a c t o r p r i c e uncer ta in ty , given c e r t a i n product p r i c e expecta t ions , i s the fo l low ing : 67. Theorem VI .7: Suppose that dA/oV < 0, d R/diT > 0 and that the technology s a t i s f i e s (12) (a 1 ) and ( b 1 ) . (See footnote 11). Then the e f f e c t o f a mean u t i l i t y preserv ing spread in f a c t o r p r ices i s to reduce ( increase) x* i f g 9 W X - 0 and — {-—-) > (<) 0. A mean preserv ing spread increases x* i f 9,,„ > 0, 9 „ , , w > 0 , g „ „ , > 0 and R < 1. 3wx - 3xww - ' 3www -F i n a l l y , consider the e f f e c t on x* of an increase in c o r r e l a t i o n between product and f a c t o r expectat ions in th is s i m p l i f i e d model. From 111.(17) we get W = rr *W\X + \ ( + E[vgw fx m y , where c p ' w ' w ( p , w ; x ) = 1 a p w ( p , w ; x ) + g w ^ - and a p w = j jrr y(p,w;x) . I f V" = 0, the impact has the same sign as g p w x which vanishes given zero f l e x i b i l i t y . On the other hand, the e f f e c t of a mean u t i l i t y preserv ing increase in c o r r e l a t i o n , (wiith u t i l i t y compensated by v a r i a t i o n s in the future factor . , p r i c e ) , has the s ign of 8 a ( p , w ; x ) / 3 x . Again t h i s may be in terpre ted as the adoption of a more f l e x i b l e technique, where f l e x i b i l i t y in the face of increased c o r r e l a t i o n is measured by the cross e l a s t i c i t y a p w . 6. Technological Risk Independence Consider now the problem of r i s k independence in the context of our model of the f i r m , and adopt the new notat ion whereby the technology is represented by the transformation funct ion F(x ,y) with the dual v a r i a b l e p r o f i t funct ion g ( p ; x ) , where components o f y and p can r e f e r e i t h e r to outputs or inpu ts . Though most of our d iscuss ion concerns r i s k independence 68. r e l a t i v e to the p r o f i t funct ion g, and i s thus p a r t i c u l a r l y re levant to the d e c i s i o n problem when V" = 0, the fo l low ing theorem deals with r i s k independence r e l a t i v e to the composite funct ion V[g(p;x) - q - x ] : Theorem VI .8: Suppose that V" < 0. Then there does not e x i s t a technology such that p Rl x , with respect to the funct ion V[g(p;x) - q-x - B ] , f o r a l l l eve ls of f i xed costs B. From From Theorem II I .3 i t fo l lows that i f the f i rm i s not p r o f i t r i s k n e u t r a l , there e x i s t s a leve l of f i xed costs B and a mean u t i l i t y preserv ing spread of p r i c e expectat ions that a f fec ts i t s demand f o r current i n p u t s . ^ Henceforth, we assume that V" = 0 and consider r i s k independence with respect to g . The fo l lowing theorem charac ter i zes the technologies cons is ten t with the r i s k independence of pr ices from some of the ex ante inputs . Theorem VI .9 : Let x = (x ,x ) where each x i s a v e c t o r . The fo l lowing are equiva lent : (a) p Rl x 1 ; (b) g(p;x) = a(x)h(p;x ) fo r some funct ions a and h; (c) the transformation f r o n t i e r is descr ibed by F ' [ x 2 , ^ „ \ ] = 0, f o r some F 1 . a^x; I t i s r e a d i l y shown that the technologies of the theorem are p r e c i s e l y those fo r which a l l r a t ios y . ( p ; x ) / y • (p ;x ) , of short run demand and supply f u n c t i o n s , are independent o f x , and thus they may be sa id to be "homothetic in x^ in the short r u n " . From (c) the fo l lowing equivalent representat ion may be der ived: (23) G [ x 2 , y ] = H(x) , with G homothetic in y . 69. Examples include the Cobb-Douglas technology and, when x = x^ i s a s c a l a r , a l l l i n e a r homogeneous t e c h n o l o g i e s . ^ When x = x^, the technologies are simply "homothetic in x". The r e s u l t s of Chapter III show that i t i s p r e c i s e l y f o r these technologies that the e f f e c t s o f a l l mean p r o f i t preserv ing spreads vanish and that c e r t a i n t y equivalents play the ro le descr ibed in Theorem I I I .4. The c l a s s of technologies cons is ten t with mutual r i s k independence of p r i ces i s much more r e s t r i c t e d : Theorem VI .10: Suppose p = (p-| ,p 2 ) i s two-dimensional and consider the var iab le p r o f i t funct ion g(p) with the dependence on f i x e d fac to rs suppressed. I f both pr ices are product p r ices mutual r i s k independence i s imposs ib le , while i f at l e a s t one va r iab le good i s a f a c t o r , only the Cobb-Douglas (short run) technology i s cons is tent with mutual r i s k independence. It fol lows that in the case of only two goods va r iab le ex pos t , f a c t o r demand funct ions can be constructed from c e r t a i n t y equivalents as in Theorem I I I .5 i f and only i f there i s at l e a s t one f a c t o r va r iab le ex pos t , and the var iab le p r o f i t funct ion has the form g(p;x) = a ( x ) - h ( p ) , with the short run technology dual to h being Cobb-Douglas. In higher dimensions, the c h a r a c t e r i z a t i o n of technologies cons is ten t with to ta l mutual p r i c e r i s k independence i s not so neat . I t fo l lows from Theorem 11.4, however, and from the proof o f Theorem 10 above, that in P 2 P 3 1 three dimensions we must have g(p) = p^ h ( — , — ) , h (p 2 ,Pg) = c Q + c-jh (p 2 ) 2 1 2 ' ^ ^ + c 2 h (p^) + c^h (p 2 )h ^ 3). where each h has constant r e l a t i v e r i s k avers ion . Various combinations of c o e f f i c i e n t values and choices of h 1 may be cons is ten t with independence. An example of such a funct ion in n-dimensions, with only fac tors va r iab le ex pos t , i s the negative of the General ized Leont ie f cost 70. f u n c t i o n , i . e . , - £ b . . p? p? with b . . = b . - , when b . . = 0 f o r a l l i . Some fur ther r e s u l t s f o r the n-dimensional case are contained in the fo l lowing theorem: Theorem VI .11: Let p = .(p-j P n ) and l e t g(p) be the var iab le p r o f i t funct ion with x suppressed, (a) Suppose that only products are va r iab le ex pos t , and that a l l second order de r iva t i ves g . . are uniformly signed (outside ofLa^bounded r e g i o n ) . Then i t i s impossible f o r a l l the pr ices to be g l o b a l l y mutually r i s k independent, (b) Consider the case of both products and fac tors v a r i a b l e ex post . I f i and j are products that are gross (short run) s u b s t i t u t e s , i . e . , g^ . < 0, then ( g - ) and ^ — ( g ^ ) both assume p o s i t i v e values " f r e q u e n t l y " , i . e . , outs ide of any bounded s e t . I f i i s a f a c t o r and j a product such that g . . > 0 ( - — y . < 0 ) , then g • • J 1~- (g4i) i s " f requent ly" n e g a t i v e . 1 8 1 3 Part (a) i s a g e n e r a l i z a t i o n , though in a somewhat weakened form, of the f i r s t part of Theorem 10. The monotonic i ty , curvature and homogeneity proper t ies of the v a r i a b l e p r o f i t funct ion are to a great extent i n c o n s i s t e n t with t o t a l p r i c e r i s k independence when only products are va r iab le ex post . The r e s u l t s in (b) are somewhat s u r p r i s i n g . When products i and j are gross s u b s t i t u t e s , a higher p r i c e fo r i reduces the potent ia l blow from an unusual ly 16whps,but a lso reduces the net advantage to be expected from an unusual ly high p . . A p r i o r i , i t i s not c l e a r whether the potent ia l loss or gain i s J reduced to a greater extent and hence what the e f f e c t on r i s k aversion might be. S i m i l a r l y , when i i s a f a c t o r and j a product such that ay.: (p;x ) / a p . < 0, a higher p^, making the use of i more expensive, increases the losses and gains to be expected from unusually low and hight'prices fo r j , r e s p e c t i v e l y . 71. Part (b) s ta tes that the net e f f e c t on r i s k aversion in these two cases i s determinate, though i t i s r e a d i l y demonstrated that in s i m i l a r s i t u a t i o n s with the many other poss ib le product -product , p roduc t - f ac to r and f a c t o r - f a c t o r s u b s t i t u t i o n r e l a t i o n s , the net e f f e c t i s indeterminate. An add i t iona l determinate r e s u l t dea l ing with the r i s k independence of pr ices from ex ante factors i s the fo l low ing : Theorem VI.12: Consider g(p;x) and l e t i be a product . I f g < 0, then _.a_ i P-jP-j (— ) i s "frequently" p o s i t i v e . It fo l lows that i f y.. i s an " i n f e r i o r " product with respect to the f a c t o r x ^ ( 3 y . ( p ; x ) / 9 x k < 0 ) , then p. cannot be r i s k independent of x^. 7. An Example Suppose that a s i n g l e product i s produced ex post with two. fac tors o f production - c a p i t a l , chosen ex ante , and labour , determined ex pos t . Consider the family of technologies whose var iab le p r o f i t funct ion has the form (24) g(p,w;x) = <j>(x)h(p,w) , where <j>(x) i s increas ing and s t r i c t l y concave, h(p,w) is a p r o f i t funct ion and w is the wage ra te . Some of the impl ica t ions of th is funct iona l form f o r behaviour under uncerta inty were pointed out above. Here we use the r e s u l t s of sect ions 3, 4 and 5 of th is chapter to der ive some fur ther i m p l i c a t i o n s . We make use of the fo l lowing proper t ies o f these p r o f i t f u n c t i o n s : 72, J _ ( ^ ) = A ( ? w w . ) = 0 a n d h e n c e g x p p > 0 , g x w w > 0 ; p w ( i i i ) ^ y ( p , w ; x ) = Iff (p,w;x) < 1 ; ( i v ) h p w < 0 a n d h e n c e 9 p w x < 0 ; (v) g„„„ < 0 and g < 0 ( v a l i d f o r the Cobb-Douglas 3ppp - 3www - •. x 3 technology o n l y ) . It i s now a s t ra ight forward matter to v e r i f y the fo l lowing a s s e r t i o n s : Wage rate uncer ta inty leads to a greater demand f o r c a p i t a l than in the case of c e r t a i n expectat ions with the same expected va lue . The corresponding impact of p r i c e uncer ta inty may be p o s i t i v e or negative depending on the extent o f p r o f i t r i s k a v e r s i o n . I f V" = 0 , p r i c e uncerta inty increases the demand f o r c a p i t a l . The expected value of the demand f o r labour is smal ler f o r s t o c h a s t i c wage expectat ions than fo r c e r t a i n expectat ions with the same expected 19 va lue . The corresponding impact o f p r i c e uncerta inty may be p o s i t i v e or negat ive . The more r i s k averse the producer, the less (more) c a p i t a l employed given p r i c e (wage) uncer ta in ty . The expected value of production ( labour employed) decreases with the degree of p r o f i t r i s k aversion of a producer fac ing p r i c e (wage) uncer ta in ty . I f dA/d7r < (>) 0 , an increase in f i xed costs lowers ( increases) the demand f o r c a p i t a l given e i t h e r p r i c e or wage uncer ta in ty . I f R < 1, an increase in expected pr ices (wages) increases (reduces) the demand f o r cap i ta l given s t o c h a s t i c p r i c e (wage) expecta t ions . 73. (g) The demand curve f o r c a p i t a l i s downward s l o p i n g i f dA/dir < 0, given e i t h e r p r i c e or wage uncer ta in ty . (h) I f V" = 0, mean u t i l i t y ( i . e . p r o f i t ) preserv ing spreads of p r i c e or wage expectat ions have no e f f e c t on the demand f o r c a p i t a l . I f V" < 0, dA/diT < 0 and dR/dir > 0, both mean u t i l i t y preserv ing spreads have negative impacts. ( i ) I f V" = 0, a mean preserv ing spread in p r i c e or wage expectat ions 20 increases the demand f o r c a p i t a l , ( j ) I f V" = 0, an increase in the c o r r e l a t i o n between p r i c e and wage expectat ions reduces the demand f o r c a p i t a l . 8. Some Empir ica l Impl icat ions Our d iscuss ion of the implementation of the consumer model with regard to the ro le played by d u a l i t y theory and the problems of s t o c h a s t i c s p e c i f i c a t i o n app l ies equa l ly here. The l im i ted a p p l i c a b i l i t y of a two-per iod model a lso a p p l i e s , though, in a producer context once-and- for -a l1 capaci ty dec is ions are frequent and i n t e r e s t i n g phenomena that can r e a l i s t i c a l l y be modelled by (3) . (The p r o f i t s argument in the u t i l i t y funct ion might be N be replaced by { J l / ( l+ r ) T r ( P t , W t ; x ) - qx}, P t and W t denoting expectat ions f o r the t planning p e r i o d , N being the number of periods in the planning horizon and r a ce r ta in rate of i n t e r e s t used by the f i rm to discount c e r t a i n re tu rns ; ) Other a p p l i c a t i o n s are d iscussed in the next s e c t i o n . The main d i f f e r e n c e between the consumer and producer models i s that the composition o f funct ions in (3) makes the problem of funct iona l form s p e c i f i c a t i o n , and subsequently e s t i m a t i o n , extremely d i f f i c u l t . Thus in the remainder of the t h e s i s , and in p a r t i c u l a r in the empir ica l ana lys is in Chapter VI I , the assumption o f p r o f i t r i s k n e u t r a l i t y i s made. We fee l th is 74. i s j u s t i f i e d as a f i r s t step s ince most e x i s t i n g econometric studies have t o t a l l y ignored uncer ta in ty , while the very few that haven' t have made the same assumption. With p r o f i t r i s k n e u t r a l i t y and given data on r e a l i s e d pr ices and quant i t i es and random expecta t ions , the technology may be estimated from the f i r s t order condi t ions corresponding to (3) i f we hypothesize a funct ional form fo r TT. Consider , t h e r e f o r e , the fo l lowing funct iona l form: (26) h (p r w;x) = (I x k K I a . j s " 1 s ^ ' + £ a .p \ - £ a! w ^ + I c i k x i x k + I d k x k TT(P,W;X) E p 1 h ( p 1 / p 1 , w / p p x ) , where P = ( P r P - | ) » S = (p-,,w), a. = +1 , i = 1 , 2 , . . . ^ ^ - l -1 , i = n^ n2+n2~l a i j = a j i ' b i j k = b j i k ' c i k = c k i It i s easy to v e r i f y that TT s a t i s f i e s a l l of the homogeneity, mono-tondci ty and curvature proper t ies of D, though not n e c e s s a r i l y D ( i ) , i f a l l the c o e f f i c i e n t s are nonnegative. The arguments of Diewert (1973) may be appl ied to show that TT w i l l f requent ly be a loca l representat ion o f a var iab le p r o f i t f u n c t i o n . Note that in that case , h w i l l be the loca l representat ion o f a normalized p r o f i t f u n c t i o n . (See Lau (1969) or (1974).) 75. This funct iona l form has several des i rab le p r o p e r t i e s . F i r s t , i t i s " s u f f i c i e n t l y f l e x i b l e " , in the sense that i t may assume any set o f f i r s t and second order de r iva t i ves at a p o i n t , plus a l l t h i r d order der iva t ives of the form Tr IT ,, V , TT,, 1 ( v ... .Our comparative s t a t i c s ana lys is of '(5) p i p j x k ' Pi w j xk' w i w j x k - r r ' • -shows that these d e r i v a t i v e s are s u f f i c i e n t to determine react ions to increased uncer ta in ty . Second, each TT i s a simple funct ion of p r i c e s , x k being a sum of terms each of which i s a product of funct ions of i n d i v i d u a l p r i c e s . Thus the expected value of TT which appears in the f i r s t order x k condi t ions (5) , may be r e a d i l y computed numerical ly given the p r o b a b i l i t y d i s t r i b u t i o n descr ib ing expecta t ions . In f a c t , the in tegra t ion may be performed a l g e b r a i c a l l y f o r the d i s t r i b u t i o n s p e c i f i e d below. Note that ne i ther of the funct iona l forms hypothesized by Diewert (1973), nor the t rans log va r iab le p r o f i t funct ion (see Chr is tensen , Jorgenson and Lau (1971) and Diewert (1974)), have these p r o p e r t i e s . They are a l l f l e x i b l e to the second order and would make the necessary i n t e g r a t i o n very d i f f i c u l t . Two s p e c i a l cases of i n t e r e s t fo l low from parametric r e s t r i c t i o n s in (26). The technology exh ib i t s no ex post f l e x i b i l i t y , (TT i s l i n e a r in p r i ces and hence the short run supply and demand funct ions -rr (p,w;x) and -TT , , (p,w;x) Pi w are p r ice i n e l a s t i c ) , if and only if a . . = b . . . ••= 0 f o r . a l l i , j , k : This is 1 J 1 J K s i m i l a r to the "put ty -c lay" technology d iscussed by Fuss (1976), except that he considers a technology with a s i n g l e output that is exogenous to the f i r m . Our model represents an a l te rnat ive :approach to t e s t i n g f o r ' a . " p u t t y - c l a y " 21 technology under the maintained hypothesis o f "put ty-semiputty" . The technology dual to TT exh ib i ts constant returns to sca le i f and only i f b i j k = b i k = b i k = d k = 0 f o r a 1 1 i ' J > k - B o t n 0 + ~ these hypotheses w i l l be tested in the empir ica l i n v e s t i g a t i o n that f o l l o w s . 76. VI I . AN EMPIRICAL ANALYSIS OF THE EFFECTS OF PRICE UNCERTAINTY ON THE DEMAND FOR CAPITAL AND ITS RATE OF UTILIZATION E x i s t i n g empir ica l s tud ies of systems of f a c t o r demands generated by opt imiz ing behaviour subject to technolog ica l const ra in ts have v i r t u a l l y ignored , in d e r i v i n g t h e i r est imates , the s t o c h a s t i c environment in which producers t y p i c a l l y f u n c t i o n ; and subsequently have had nothing to say about the way in which producers ' dec is ions are l i k e l y to be a f fec ted by increas ing or reducing the degree of uncer ta in ty . Most of the recent s tudies that have estimated " f l e x i b l e " funct iona l , forms in attempting to determine the nature of s u b s t i t u t i o n among i n p u t s , (see Berndt and Christensen (1973, 4 ) , Chr is tensen , Jorgenson and Lau (1973), Berndt and Wood (1975) and Woodland (1975), f o r example) assume that a l l inputs and outputs are p e r f e c t l y var iab le wi th in the observat ion per iod to changes in current p r i c e s . Such technologies are c a l l e d decomposable by Jorgenson (1973). Under such a technology the ob jec t ive of the f i rm is simply to maximize p r o f i t at each point of time—random p r i c e expectat ions play no r o l e . Woodland (1974) relaxes the assumption of per fec t v a r i a b i l i t y by supposing that q u a n t i t i e s adjust f u l l y to "planning p r i c e s " which d i f f e r from current p r i c e s . E a r l i e r s tudies by Nadir i and Rosen (1969), Coen and Hickman (1970) and Schramm (1970) and the recent work reported by Brech l ing (1975) argue that because of costs assoc ia ted with changes in the l e v e l s of f a c t o r s , fac tors adjust to current p r ices only g radua l ly . In a l l c a s e s , optimal dec is ions must be made subject to expectat ions about the f u t u r e , but such expectat ions are taken to be held with c e r t a i n t y . The incorporat ion of the v a r i a b i l i t y of expectat ions i s a prime ob jec t ive of the empir ica l i n v e s t i g a t i o n of a cost of adjustment model of investment undertaken by 77. Craine (1975). However, he hypothesizes a Cobb-Douglas cons tan t - re tu rns -t o - s c a l e technology which, by Hartman (1972) and the comparative s t a t i c s ana lys is c a r r i e d out in Chapter VI, - • , has extremely strong a p r i o r i imp l ica t ions f o r the e f f e c t s of uncer ta inty on the rate of investment. Moreover, Craine qu i te i n e x p l i c a b l y , in view of his stated o b j e c t i v e , supposes that f irms use cond i t iona l expectat ions of random var iab les rather than the random expectat ions themselves, in s o l v i n g f o r optimal investment, thus making the choice problem d e t e r m i n i s t i c J One ob jec t ive of the empir ica l i n v e s t i g a t i o n we propose i s to estimate technology recogniz ing that some inputs are only imper fect ly var iab le and so must be determined subject to random expectat ions about the f u t u r e . B a s i c a l l y , we assume that the c a p i t a l stock dec is ion must be made one per iod before the c a p i t a l comes into opera t ion , subject to expectat ions about future p r i c e s , while a l l other fac tors and outputs may be adjusted f u l l y to current p r i c e s . These assumptions c l e a r l y require some comment. F i r s t , a v a i l a b l e evidence suggests that phys ica l c a p i t a l i s the most imper fect ly adjustable f a c t o r in the short run and our model adopts an extreme i n t e r p r e t a t i o n of th is ev idence. The q u a s i - f i x i t y of other fac tors should obviously be taken into account in future research . Second, we have a r b i t r a r i l y imposed a one per iod lag in the c a p i t a l stock d e c i s i o n . It would c l e a r l y be des i rab le to der ive such a lag from intertemporal opt imiz ing behaviour s u b j e c t , f o r example, to costs of c a p i t a l stock adjustment. But even i f such costs of adjustment are assumed separable from the production f u n c t i o n , such a model y i e l d s e m p i r i c a l l y implementable equations only i f there i s a s i n g l e c a p i t a l good and there are 78. constant returns to sca le in product ion . (See Craine (1975).) As suggested e a r l i e r , the l a t t e r assumption is too strong to adopt as a maintained hypothesis in an i n v e s t i g a t i o n of the e f f e c t s of p r i c e uncer ta in ty , and so we have opted f o r the above ad hoc lag s p e c i f i c a t i o n . Note that the lag s t ruc ture s p e c i f i c a t i o n s in many investment demand s tud ies (see Jorgenson (1965), f o r example) are no less ad hoc, being at best appended to a t o t a l l y s t a t i c theory. On the other hand, Treadway (1974) has shown that the f l e x i b l e a c c e l e r a t o r adjustment mechanism with constant adjustment c o e f f i c i e n t s i s r a t i o n a l i z e d by costs of adjustment only f o r a very r e s t r i c t i v e fami ly of t echno log ies . (In genera l , the f l e x i b l e acce le ra to r i s optimal only l o c a l l y in the neighbourhood o f the "desired s t o c k " , assuming s t a t i c expecta t ions , while the adjustment matrix depends in a complicated way on exogenous v a r i a b l e s . ) Th is r e f l e c t s d i r e c t l y on the genera l i t y of the empir ica l s tudies re fe r red to above that employ the f l e x i b l e a c c e l e r a t o r . F i n a l l y , Woodland's (1974) "planning p r i c e " model a l s o imposes an a r b i t r a r y lag s t r u c t u r e , at l e a s t . i m p l i c i t l y . On a more p o s i t i v e note , there i s substant ia l empir ica l agreement that the c a p i t a l stock d e c i s i o n , i . e . , investment, depends mostly on var iab les lagged one year or more, (see Evans (1969; p. 98-101)) , mainly because of the appropr ia t ions and de l i ve ry lags between the approval of appropr ia t ions and the placement of an o rder , and the actual expenditures when the c a p i t a l good i s d e l i v e r e d . There i s i n e v i t a b l y an add i t iona l lag between expenditures and the time the new capaci ty is in tegrated in to the production p rocess , though i t has apparently been assumed in the l i t e r a t u r e to be i n s i g n i f i c a n t . In any case , a lag in the c a p i t a l stock dec is ion would seem to have more empir ica l support than the assumption of per fec t v a r i a b i l i t y made in other s t u d i e s . More general d i s t r i b u t e d lag s t r u c t u r e s , such as employed by 79. Jorgenson (1965), could be hypothesized in future work, but i t i s not f e l t that th is more complicated ad hocery would be o f any use at th is s tage . Future research should concentrate on endogenizing the lag s t ruc ture and in p a r t i c u l a r , i n v e s t i g a t i n g the way in which i t i s a f fec ted by varying degrees of uncer ta in ty . The reader w i l l probably have not iced our continued reference to c ap i ta l stocks as opposed to se rv ice f lows , and the relevance to our arguments o f the d i s t i n c t i o n between the two. While stocks may reasonably be taken to be f i x e d in the short run , t h e i r rates of u t i l i z a t i o n would be expected to be v a r i a b l e . Indeed in several o f the f l e x i b l e funct iona l form studies c a p i t a l stock data are adjusted by estimated rates of u t i l i z a t i o n , so that i t i s c a p i t a l se rv ices rather than stocks that are taken to be p e r f e c t l y v a r i a b l e . • A second.ob jec t ive of our proposed study i s to d i s t i n g u i s h between the "long run" stock d e c i s i o n and the "short run" u t i l i z a t i o n d e c i s i o n . Jorgenson and Christensen (1969) c o n s t r u c t • i n d i c e s of r e l a t i v e u t i l i z a t i o n of c a p i t a l stocks which are subsequently adopted in the paper by C h r i s t e n s e n , Jorgenson and Lau . The i n d i c e s , fo l lowing the work or ig ina ted by Foss (1963), are based on the consumption of e l e c t r i c i t y r e l a t i v e to the capac i ty to consume e l e c t r i c i t y as measured by i n s t a l l e d horsepower of e l e c t r i c motors, and are constructed by l i n e a r i n t e r p o l a t i o n between benchmark years when capaci ty estimates are a v a i l a b l e . In s p i t e of using these va r iab le u t i l i z a t i o n r a t e s , the rates of phys ica l deprec ia t ion o f the c a p i t a l stocks are i n v a r i a b l y taken to be constant through t ime. Depreciat ion in use is ignored , though Keynes (1935, 69-70) argued that th is "user cos t" " . . . cons t i tu tes one of the l i n k s between the present and the f u t u r e . For in dec id ing h is sca le of production an entrepreneur has to exerc ise a choice between using up his c a p i t a l now and preserv ing i t to be used l a t e r on . . . " 80. The s tudies by Nadi r i and Rosen, Coen and Hickman and Brech l ing d i s t i n g u i s h between stock and u t i l i z a t i o n dec is ions and inc lude a l l the re levant costs at l e a s t in the s p e c i f i c a t i o n of t h e i r model. However, in the est imat ion some of the cost va r iab les are dropped because of lack of da ta , causing biases of (unknown s i z e and d i r e c t i o n i n v t h e i r est imates . Moreover, they use the r e s t r i c t i v e Cobb-Douglas production funct ion and ignore uncertainty."^ We propose to adopt the framework developed by Hicks (1946, Ch. XV), Malinvaud (1953) and recent ly d iscussed from an econometric point o f view by Diewert (1972, 49-53). The f i rm is assumed to take a stock of "unused" c a p i t a l at the beginning of the p e r i o d , combine i t with va r iab le fac tors during the per iod and produce outputs which inc lude the depreciated c a p i ta l stock l e f t over at the end of the p e r i o d . Since we assume in the Jorgenson t r a d i t i o n that a per fec t market e x i s t s f o r used c a p i t a l goods, the rate of u t i l i z a t i o n whether in the form of the length of time that c a p i ta l i s operated or the i n t e n s i t y of i t s operat ion per uni t o f t ime, i s determined by short run p r o f i t maximization and is i m p l i c i t in the d e s c r i p t i o n of the technology. Models where the rate of u t i l i z a t i o n i s an e x p l i c i t f a c t o r in the, production funct ion and determines phys ica l d e p r e c i a t i o n , such as the above empir ica l models and the t h e o r e t i c a l papers of Smith (1969) and (1970) and Taubman and Wilkinson (1970), can be regarded as spec ia l cases of t h i s 4 more general model. The proposed framework has several advantages: f i r s t , i t i s s u f f i c i e n t l y general to incorporate i m p l i c i t l y any maintenance the f i rm may decide to undertake in the form of maintenance labour or replacement p a r t s . The model makes the rate of deprec ia t ion endogenous and i s capable of r e f l e c t i n g 81. i t s secu la r and c y c l i c a l v a r i a b i l i t y . The deprec ia t ion rate estimates that are genera l ly used are based on sparse benchmark f igures and are at best estimates of "average" r a t e s . S t a t i s t i c s Canada (13-522, p. 87) has c a l l e d these estimates the weakest l i nk in t h e i r set o f c a p i t a l stock estimates f o r Canadian manufacturing i n d u s t r i e s . Ihe fo l lowing ana lys is w i l l al low us to construct a l t e r n a t i v e estimates of c a p i t a l stock cons is ten t with endogenous rates of u t i l i z a t i o n and p r o f i t maximization subject to a technology represented by a f l e x i b l e funct iona l form. Our s t rategy and object ives may be b r i e f l y summarized as fo l lows: technology w i l l be represented by a var iab le p r o f i t f u n c t i o n . Thus the consistency o f the model with the data may be tested by v e r i f y i n g that the p r o f i t funct ion s a t i s f i e s the r e g u l a r i t y condi t ions s p e c i f i e d in IV over the range of p r ices and quant i t i es def ined by actual observations and random expecta t ions . Estimates generated fo r c a p i t a l stock and deprec ia t ion rates serve as fur ther tes ts of the model. Reasonable r e s u l t s should c o n s i s t of a monotonical ly increas ing c a p i t a l stock se r ies and deprec ia t ion rates between 0 and 1. Imp l i c i t in the model is a notion of the rate of u t i l i z a -t ion of c a p i t a l d i r e c t l y re la ted to the rate of deprec ia t ion i f we assume r e l a t i v e l y constant maintainence expenditures through time. Thus we may roughly compare our r e s u l t s with other measures of u t i l i z a t i o n , such as that employed by Christensen and Jorgenson and those surveyed by P h i l l i p s (1963). Note that the above assumption is i m p l i c i t in most of the fo l lowing d iscuss ion as we often use "deprec ia t ion" and " u t i l i z a t i o n " interchangeably . Frequent ly , as d iscussed in IV, a technology i s descr ibed in terms of e l a s t i c i t i e s of s u b s t i t u t i o n . Of p a r t i c u l a r i n t e r e s t to us w i l l be the p r i c e f l e x i b i l i t y of the rate of c a p i t a l u t i l i z a t i o n . 8 2 . F i n a l l y , the model provides a framework wi th in which the e f f e c t s o f p r i c e uncerta inty can be determined. Using the comparative s t a t i c s r e s u l t s o f VI , the e f f e c t on the demand f o r c a p i t a l of mean and mean u t i l i t y preserv ing changes in expectat ions can be de r i ved . Given our s p e c i f i c a t i o n of expecta t ions , there i s a natural measure of uncerta inty and appropr iate e l a s t i c i t i e s with respect to changes in uncerta inty may be de f ined . We turn now to a d e t a i l e d d e s c r i p t i o n o f the model. 1. The Model Technology is descr ibed by a concave transformation funct ion y = F [ z , K , K ] where <in any p e r i o d , K i s the vector of "beginning of per iod" c a p i t a l stock and z i s the vector of inputs ( z ^ , . . . , z n ) app l ied during the p e r i o d , which produce output y and "end o f per iod" stocks R. At any point in t ime, current product and f a c t o r p r ices are known but future pr ices are uncer ta in . The f i rm is assumed to have sub jec t ive p r o b a b i l i t y d i s t r i b u t i o n s concerning these pr ices and to s e l e c t a s t ra tegy that maximizes the expected value of the discounted sum of an t ic ipa ted p r o f i t s over an i n f i n i t e future planning hor i zon ; that i s , i t solves co (1) max t=0 (1+r) ¥ [ P t F [ z t , K t , K t ] - W t - z t - Q t . I t ] subject to K t = Kj._i + I K > 0 g i v e n , 1 = 0 o 3 o 83. where - P^, and Q t are the product , va r i ab le f a c t o r and cap i ta l asset p r i c e s . At t = 0, a l l p r ices fo r t > 0 are random var iab les but they are nonstochast ic f o r the current p e r i o d . The f i rm i s a p r i c e taker in a l l markets i n c l u d i n g that f o r c a p i t a l goods. - r i s the rate of discount assumed to be cer ta in over the planning h o r i z o n , and fo r s i m p l i c i t y , a lso taken to be constant . - z t , i<t are chosen given knowledge of pr ices up to and i n c l u d i n g the t per iod but subject to uncer ta inty about p r ices in subsequent periods'. - gross investment in per iod t , 1^, i s chosen in (t-1) and producers, aware of the lag inherent in the process , determine 1^ . according to the des i red future l e v e l s of c a p i t a l s tock . The dec is ion i s thus made subject to uncerta inty about future p r i c e s . - K t = i<t_.| + 1^ i s the c a p i t a l accumulation equation that replaces the customary K t = (l-6)K t_-| + fo r some constant geometric rate of deprec ia t ion s. S u b s t i t u t i n g the cap i ta l accumulation equation into the ob jec t ive funct ion we obtain ( 2 ) 7 T „ E X T ^ t C P t F [ z t > K t > V - w t ' z t - V K t + V t - i i ' Z^,N^,N|. t-u u + n where z t and are chosen given pr ices f o r per iod t and K t i s determined in per iod (t-1) subject to uncerta inty about t per iod p r i c e s . C l e a r l y , t h e r e f o r e , z^ i s chosen opt imal ly to maximize short run p r o f i t s in each periiod and t per iod va r iab le p r o f i t s are equal to , ( P t , W t ; K t , K t ) - Q t K t + 84. where TT i s the var iab le p r o f i t funct ion dual to F with K and K f i x e d . The f i r s t order condi t ion with respect to K t i s (3) T T f E t ( Q t + 1 ) = - ^ ( P t » w t ; K t , K t ) , where E t denotes the expected value subject to information a v a i l a b l e at time t and P t » w t are actual p r ices in that p e r i o d . Thus i<t i s a lso chosen to maximize short run p r o f i t s with respect to the " e f f e c t i v e " p r ice F i n a l l y , we can rewrite (2) as (5) max E j —]—p [TT(P. ,W. ,q ;K. ) - Q . K . ] , K t t=0 ( H r ) * t t t its t t where TT now denotes the va r iab le p r o f i t funct ion with the s i n g l e f i xed f a c t o r K, and K t i s chosen in ( t - 1 ) . Note that q"t i s a random var iab le when viewed from any per iod before t . Optimal cap i ta l stock K£ i s determined by ( 6 ) E t - l Q t = E t -1 V ^ V ^ ' t = 1 . 2 , . . . Thus the opt imal i ty condi t ions are st ra ight forward extensions of Jorgenson's (1963) myopic c r i t e r i a : va r iab le fac tors and outputs are chosen to maximize short run p r o f i t s while the stock of cap i ta l i s chosen to maximize the expected value of next p e r i o d ' s p r o f i t s , E^_^[Tr ( P ^ > W ^. ,q^;K^) - Q^K^l. 85. Note that K£ i s indeterminate i f there are constant returns to sca le in product ion . Hence the model requires that F be s t r i c t l y concave in i t s arguments, or equ iva len t ly that r r (p,w;q;K) be s t r i c t l y concave in K. Note a lso that c a p i t a l stock i s determined wi th in a two-period framework and so the comparative s t a t i c s ana lys is of . i u ; >.VI may be appl ied to t h i s model. Apply ing H o t e l l i n g ' s Lemma, we der ive the fo l lowing set o f e x p l i c i t and i m p l i c i t demand and supply equat ions: z t = - V P f V v M R t - • ' q f o t ' V V M q t = E t - i V p t ' w t ' V K t ) > where q t = E t _ - | Q t , p^, w^ denote r e a l i z e d pr ices and P^, W^ and q"t are as be fore . To implement the model we must s e l e c t a funct iona l form f o r TT and s p e c i f y expectat ion formation and s t o c h a s t i c d is turbances . 2. Expectat ions Since most econometric theory i s based upon the normal d i s t r i b u t i o n , the est imat ion of random expectat ions by regress ion ana lys is w i l l be s i m p l i f i e d i f thepprobab i l i t y d i s t r i b u t i o n descr ib ing expectations i s c l o s e l y re la ted to the normal d i s t r i b u t i o n . We intend to assume lognormally d i s t r i b u t e d expectat ions which have the fo l lowing des i rab le p r o p e r t i e s : 86. the d i s t r i b u t i o n s are s ing le -peaked , "reasonably" shaped and ru le out the p o s s i b i l i t y of nonposi t ive p r i c e s ; s ince d log x = dx /x , expectat ions in d i s c r e t e time in terms of the log of pr ices may be in terpre ted as the in tegra l of normally d i s t r i b u t e d expectat ions about the rate of i n f l a t i o n through continuous t ime; i f log x is normally d i s t r i b u t e d with mean y and 2 o 9 variance a , then f o r any a, E ( x a ) = exp{ay + ha a } so that the i n t e g r a l s in the f i r s t order condi t ion (6) are r e a d i l y evaluated given the funct ional form VI The techniques descr ibed in Box and Jenkins (1970) w i l l be used to f i t an ARIMA (autoregressive in tegra ted , moving average) process to the log of each p i r c e , and we hypothesize that the f i r m ' s expectat ions are p r e c i s e l y those that are def ined by the process . To s i m p l i f y matters , p r i c e expecta-t ions are assumed to be independently d i s t r i b u t e d . For i l l u s t r a t i v e purposes, consider the fo l lowing simple process: (8) log p t + 1 = p Q + P-J log p t + a t + 1 a t + - | being normally d i s t r i b u t e d white n o i s e . Then y ^ , the expected value of log P t + i given information at t , w i l l be taken to be (9) y t + 1 = P Q + $} log p t , where p Q and p-| are parameter estimates of p Q and p-| r e s p e c t i v e l y . Est imation of (8) a lso y i e l d s an estimate of the variance o f the noise which w i l l be 2 2 taken to represent , the variance of log P t + i • Thus y t + - j and w i l l be derived from est imat ion of (8) and subsequently subst i tu ted in to the equations de f in ing optimal c a p i t a l s t o c k s , where they w i l l be viewed as exact representat ions of expecta t ions . 87. Rose (1972) has discussed the optimal e r r o r - l e a r n i n g proper t ies of ARIMA-based expectat ions and the advantages of generat ing an expectat ions s e r i e s independently of the regress ion f o r which the se r ies is requ i red . Never the less , the above procedure i m p l i e s , perhaps u n r e a l i s t i c a l l y , that the var iance of expectat ions i s constant through t ime, and other s p e c i f i c a t i o n s could be inves t iga ted in future r e s e a r c h . One a l t e r n a t i v e to the above might be to maintain (9) but to argue that the degree of uncer ta inty in sub ject ive expecta t ions , ra ther than being determined by the noise under ly ing the complete p rocess , i s l a r g e l y 7 in f luenced by recent p r e d i c t i o n e r r o r s . Thus, fo r example, we might hypothesize that the e r ro r in the p r e d i c t i o n f o r per iod k, and 0 < A < 1. When X = 1, 2 CTt+1 1 S the maximum l i k e l i h o o d estimate of the var iance of e^ in periods t -T to t , assuming e t i s normal ly , i d e n t i c a l l y and independently d i s t r i b u t e d with zero mean. For ^smal ler values o f X, more weight i s placed on more recent 2 2 p e r i o d s ; f o r A = 0, c r t + 1 = e t . Both (9) and (10) would be subs t i tu ted into the cap i ta l stock equation from which A could be est imated. F i n a l l y , instead of (10), we could have (10) 2 i 2 i /N a t+ l = J x e t - i / J A » w h e r e e k = 1 o g p k " ^o " 1^ l o g p k-1 (•11) that i s , the v a r i a b i l i t y of expectat ions i s equal to a weighted average Q of future p r e d i c t i o n er rors - a sor t o f " r a t i o n a l i t y " hypothesis . Combinations of (10) and (11) represent fur ther p o s s i b i l i t i e s . 88. Note that in a l l cases an exact representat ion o f preferences would be adopted. It would appear to be extremely d i f f i c u l t to estimate and i d e n t i f y the complete s t r u c t u r a l model c o n s i s t i n g o f s t o c h a s t i c equations d e f i n i n g expectat ions and the demand and supply equations given below. 3. Funct ional Forms Suppose there e x i s t s a s i n g l e c a p i t a l good; modifying VI.(26) s l i g h t l y we have (12) TT(p,w;q;K) = K [ a p " V / 2 + q 2 I a ^ 1 + p 3 I a.. w^wT 1 + a 'q - I a-w i + cp] + log K 2 [ b p - % q 3 / 2 + q 2 £ b ^ T 1 + p 3 I b^-w^w" 1 + b'q - I b1!wi + dp] + t - K [ Y L P - I Y ^ W J + y n + 2 q ] i a 1 d = a j l b i j = b j i • This funct iona l form has a l l the des i rab le proper t ies o f VI.(26) with respect to f l e x i b i l i t y in model l ing react ions to increased uncer ta in ty , y i e l d i n g i n t e r e s t i n g s p e c i a l cases by simple parameter r e s t r i c t i o n s , and "meshing" well with lognormally d i s t r i b u t e d expecta t ions . In a d d i t i o n , i t i s more f l e x i b l e in model l ing strong p r i c e f l e x i b i l i t y and because log K has replaced K 2 , we avoid the need to compute the square root of a negative number, a problem which could a r i s e in the est imat ion o f models with endogenous deprec ia t ion 89. In the absence o f a s u i t a b l e index of technology we represent the e f f e c t of techn ica l change on the technology by the l i n e a r time t rend. (S imi la r s p e c i f i c a t i o n s are adopted by Parks (1971) and Woodland (1975).) Thus techn ica l change i s taken to be exogenous and disembodied, and we assume that there i s per fec t f o r e s i g h t with respect to future changes. By H o t e l l i n g ' s Lemma, the parameters y., i = l , . . . , n + 2 , descr ibe the impact o f technica l change on the optimal ra t ios y ^ / K ^ z^/K^, j = 1 n , and K t / K t . A few words are in order regarding other approaches in the l i t e r a t u r e to model l ing techn ica l change. Chr is tensen , Jorgenson and Lau (1973) include an index of techn ica l change as a f ree f i xed f a c t o r . Kohli (1975) hypothesizes f a c t o r augmenting change at rates that are estimated parametr ical In both cases several add i t iona l parameters must be est imated, while the l a t t e r approach would have the fur ther disadvantage in our model of s i g n i f i c a n t l y inc reas ing the degree o f n o n l i n e a r i t y of the est imat ing equation Berndt and Christensen (1973,4), Berndt and Wood (1975) and Woodland (1974) hypothesize Hicks neutral techn ica l change which adds only one new parameter. But with our mult iproduct nonl inear homogeneous technology, Hicks n e u t r a l i t y i s more d i f f i c u l t to def ine and to j u s t i f y a p r i o r i . Several parameter r e s t r i c t i o n s in (12) are o f i n t e r e s t . The technology exh ib i ts no short run ( i . e . , with f ixed cap i ta l stock) f l e x i b i l i t y i f and only i f a = a . = b = = a.^ = b^. = 0 f o r a l l i , j . Short run demand and supply funct ions are then p r i c e i n e l a s t i c . There are constant returns to sca le in production i f and only i f b = b^ . = b ^ = b' = b'. = d = 0 f o r a l l i , j . A change in w- has no e f f e c t on the demand f o r the i** 1 f a c t o r z . , i f j , a f t e r a l lowing f o r output e f f e c t s but with f i xed cap i ta l s tock , i f and only i f a . - = b . . = 0. Of spec ia l i n t e r e s t are the proper t ies o f the optimal 1 J "I J rate of d e p r e c i a t i o n . It i s r e a d i l y seen .that K and y are produced non jo in t l y 90. exac t ly when TT - = 0 = a = b, while the rate of deprec ia t ion i s rH independent of the i va r iab le f a c t o r p r i c e in the short run i f and only i f ff,, n = 0 = a-; = D i • Depreciat ion i s qeometric i f b = b' = b. = 0 fo r W j Cj 1 1 ' 1 . a l l i . In order to s i m p l i f y the e s t i m a t i o n , t h i s hypothesis i s maintained below. F i n a l l y , deprec ia t ion proceeds at a constant geometric rate (= 1 - a 1 ) , the usual assumption, i f and only i f a = a- = b = b.. = b' = Y n + 2 = 0 f o r a l l i . 1 0 For reasons to be given below we w i l l a lso consider the fo l lowing s l i g h t modi f ica t ion of (12) where the log term i s replaced by a quadrat ic in c a p i t a l and the parameter d enters in a d i f f e r e n t way: (12') Tr(p,w;q;K) = K [ a p " % q 3 / 2 + q 2 £ a ^ T 1 + p 3 £ a^.w^v*: 1 + a'q - I aiwi + cp] , 2 r k -h-3/2 . -2 r , ,-1 , 3 v k ...-1...-1 + (K - d.54)^[bp^q J / d + q* I ^-w" 1 + p J J b ^ w^Wj + b'q + I b\(p-vi)l + tK[ Y - ,P - I'Yi-nWi +-Y N + 2 q] » a i j = a j i ' b i j = b j i The two funct iona l forms, which we c a l l below funct iona l forms 1(12) and 2(12') r e s p e c t i v e l y , share many of the proper t ies d iscussed above. A property which they do not share and which has not been mentioned i s that funct iona l form 2 (1) i s ( i s not) invar ian t to the choice o f the uni ts in which quant i t i es are measured. However, we do not consider th is to be a ser ious defect of (12). The choice of uni ts o f measurement i s no more a r b i t r a r y than i s the choice of one member o f the fami ly of f l e x i b l e funct iona l forms to represent the technology. 91. 4. S tochast ic S p e c i f i c a t i o n and Est imat ion Procedure Representing the technology by the v a r i a b l e p r o f i t funct ion (12), (a s i m i l a r procedure i s fol lowed when (12') i s used) , we assume that the representat ion i s exact and that any devia t ions of production and f a c t o r demands from p r o f i t maximizing l e v e l s are due to random errors in op t im iza t ion . Equation (6) can be solved e x p l i c i t l y f o r the optimal c a p i t a l s tock . Assuming lognormal expectat ions and appending a d d i t i v e disturbance terms, we have the fo l lowing system of equat ions: , . . . , n, K t = K t [ 3 / 2 a p ^ q 3 2 + 2q £ a ^ T 1 + a' + Y n + 2 t ] + log K 2 [3 /2 bp~hqh + 2q £ b ^ T 1 + b ' l + u t , K. 2B + e t+1 q-A t K t = K t-1 + h t = 1,2 , . . . , KQ g iven , /•* r where: - B = b exp(-1/2 y + 3/2 y- + i / 8 CJD + 9/8 al + I b l exp ( 2 y--y w_ + 2a\ + 1/2 a2_) + 1 b i j e x p < % - \ - %+ 9-/2 4+y2 \ + 1 / 2 v } 92. + b' exp(y- + 1/2 a|) - I b! exp(y w_ + 1/2 a 2 ) + d exp(y p + 1/2 a 2) , - A = a exp( -1/2 y p + 3/2 y- + 1/8 a 2 + 9/8 a 2) + I a i exp(2y- - yw_ + 2a| + 1/2 <,£. ) + I exp(3y p - y w - - yw_ + 9,/2 a 2 + 1/2 al + 1/2 a 2 ) + (a 1 + ( t + l ) Y n + 2 ) - e x p(y- + 1/2 a | ) + (c+(t+l) Y l.)-exp(v + 1/2 a 2) - I(aj + ( t + l ) Y i + 1 ) • exp(y w_ + 1/2 a 2_) , - u ^ , u t , and e t are normally d i s t r i b u t e d disturbance terms with zero means, uncorre lated with pr ices and with the y ' s and a ' s , - time subscr ip ts are deleted where there i s no ambiguity, 2 2 2 - vn> y-> y w , a , a- and a denote the means and variances of the random expectat ions formed in t for (the log of ) p r ices in t+1. Viewing pr ices and expectat ions as exogenous var iab les that determine the endogenous var iab les y t , z t , K t , K t and I t , we could estimate the system (13) with s t ra ight forward nonl inear techniques i f we had data fo r K and K . 1 1 However, e x i s t i n g c a p i t a l stock se r ies are constructed on the basis of constant exogenous deprec ia t ion rate estimates and so are incons is ten t with our model, l eav ing the system u n d e r i d e n t i f i e d in genera l . The problem of s t r u c t u r a l equation model est imat ion has received some at tent ion in the l i t e r a t u r e , (see Goldberger (1971) and G r i l i c h e s (1973) f o r an overview of some of the problems and proposed s o l u t i o n s ) , and two main methods of i d e n t i f i c a t i o n are cons idered . 93. The f i r s t , in terms of our model, r e l i e s on the mutual independence o f the errors u y and uA fo r a l l i . C l e a r l y such independence i s u n l i k e l y in (13), i f not imposs ib le , because o f the under ly ing production funct ion c o n s t r a i n t . I f we had pooled c r o s s - s e c t i o n and t ime-ser ies da ta , the c r o s s -sec t iona l independence of disturbances might be s u f f i c i e n t to i d e n t i f y the model. Working with time s e r i e s data a lone , however, as we do in t h i s t h e s i s , we must re ly on the second general procedure, which i s to hypothesize an exact r e l a t i o n s h i p between the unobservable var iab les and other exogenous or endogenous var iab les of the system. Thus we take the K equation to be exact (u = 0). Further j u s t i f i c a t i o n f o r t h i s s p e c i f i c a t i o n fol lows by not ing that in standard production models where deprec ia t ion rates are taken to be exogenous, they are a lso taken to be exact ; that i s , other s tud ies fo l low an analogous procedure except that our K equation i s replaced by = (1-6)K^, where 6 i s an exogenous point estimate o f the constant rate of d e p r e c i a t i o n , which estimate i s subject to s i g n i f i c a n t measurement e r ro r as d iscussed above. Thus the est imat ion technique we propose i s cons is ten t with that adopted in other production funct ion s t u d i e s , s u f f e r i n g from s i m i l a r and apparently unavoidable biases due to er rors in v a r i a b l e s . To complete the s t o c h a s t i c s p e c i f i c a t i o n we note that the vector of disturbances u^ . = (Uy^j u ^ u n t , e t ) i s l i k e l y to be both contemporaneously and temporal ly c o r r e l a t e d . To accomodate both p o s s i b i l i t i e s we hypothesize that u^ . i s generated according to the f i r s t - o r d e r autoregressive process (14) u t = u t - 1 R + e t , t = 2 T , where T i s the number of observat ions a v a i l a b l e and R is an (n+2)x(n+2) matrix o f au tocor re la t ion c o e f f i c i e n t s , assumed f o r s i m p l i c i t y to be d i a g o n a l , i . e . , 94. 0 (15) R = 0 pn+2 The vectors e^ . = (e It 9 " • • 9 e ( n + 2 ) t ) ' * = 2 9 • • • 9 T are assumed to be normally and independently d i s t r i b u t e d with zero mean and constant nonsingular covariance matrix ft. The s t o c h a s t i c s p e c i f i c a t i o n so def ined w i l l be re fe r red to below as the homoscedastic s p e c i f i c a t i o n . Since there is no reason to be l ieve that opt imizat ion errors become r e l a t i v e l y l ess important as the l e v e l s of output and fac tors i n c r e a s e , most of the est imat ion i s c a r r i e d out maintaining what we c a l l the heteroscedast ic s p e c i f i c a t i o n . This involves rep lac ing Uy t, u i t and e t in (13) by K t u y t ' K t u i t a n d K t e t r e s P e c t 1 v e l y -Adopting the heteroscedast ic s p e c i f i c a t i o n , f o r example, we proceed as fo l lows: Given K Q and gross investment data 1^, we use the' c a p i t a l accumulation equation to compute a c a p i t a l stock s e r i e s K t fo r any set o f parameters de f in ing the technology. Using the computed {K^}^_^, we compute the observed res idua ls corresponding to u y , u^ and e . Rewrite the cap i ta l equation in terms of the observable gross investment as f o l l o w s : Because the system i s r e c u r s i v e , maximizing the l i k e l i h o o d of the observed res idua ls i s equivalent to maximizing the l i k e l i h o o d of the observed y , z and I v a r i a b l e s . Thus a search procedure over parameter values y i e l d s maximum l i k e l i h o o d es t imates , which solve (16) I 2B " K t + K f £ t t+1 q-A 95. (17) max [ ~ ™ ( log 2TT + 1) - T/.2 log | n | - N/2 £ log K 2 ] , a ,p t where: - a i s the vector of parameters de f in ing the technology , p = (p-| ,- . . . »p + 2 ) i s the vector o f f i r s t order autoregression c o e f f i c i e n t s , N = n+2, the number of equations in the system, i s the sample covariance matrix (corresponding to n), and the ob jec t ive funct ion i s the concentrated (with respect to n) l o g - l i k e l i h o o d f u n c t i o n . The use of maximum l i k e l i h o o d est imat ion requires some comment, s ince at the leve l of aggregation at which the model i s a p p l i e d , i t may be inappropr ia te to assume that p r i ces are exogenous and uncorre la ted with the d is turbances . This poss ib le s imul tane i ty may r e s u l t in incons is ten t parameter est imates . On the other hand, Berndt and Christensen (1973) have found in t h e i r ana lys is of aggregate U.S. manufacturing data that estimates obtained from maximum l i k e l i h o o d and instrumental va r iab le est imat ion d id not d i f f e r g r e a t l y . Though t h i s f i n d i n g i s model s p e c i f i c , i t i s not unreasonable to expect the same to be true in our c a s e . Moreover, maximum l i k e l i h o o d est imat ion i s employed in a large number of f a c t o r demand studies (and in aggregate consumption s t u d i e s , where a s i m i l a r problem e x i s t s ) in the l i t e r a t u r e . Refinements in the est imat ion procedure are l e f t f o r future work. Hypothesis t e s t i n g i s c a r r i e d out by the l i k e l i h o o d r a t i o t e s t . The l i k e l i h o o d r a t i o i s the r a t i o of the l i k e l i h o o d maximized under the nu l l hypothesis to the l i k e l i h o o d maximized under the a l t e r n a t i v e hypothes is . 2 Minus twice the logari thm of th is r a t i o i s asymptot ica l ly x d i s t r i b u t e d , the number of degrees of freedom being equal to the number of add i t iona l 96. cons t ra in ts required by the nul l hypothesis . Thus the j u s t i f i c a t i o n of the 12 hypothesis t es ts (and est imates) used in th is paper are asymptot ic . 5. Data Construct ion and Sources The model is app l ied to the aggregate U.S. manufacturing sec tor f o r the 1947-71 p e r i o d . The da ta , reported in Appendix D, are based l a r g e l y on Berndt and Wood (1975), where they are ex tens ive ly d i s c u s s e d . We descr ibe only the modi f ica t ions necessary to make them s u i t a b l e fo r our purposes. The fo l lowing var iab les are re levant : ( i ) y , p : quant i ty and p r i c e ind ices of value added ( i i ) z - | > w i : quant i ty and p r i c e ind ices of labour se rv ices ( i i i ) z^viy} quant i ty and p r i c e ind ices of energy consumption ( iv ) y g . p ^ quantity,/and p r i c e ind ices of gross production (v) M, p^: quant i ty and p r i c e ind ices of nonenergy intermediate mater ia ls (v i ) I ,q: quant i ty and p r i c e ind ices of gross investment ( v i i ) K Q : 1947 stock o f "new" cap i ta l ( v i i i ) r: nominal a f t e r - t a x rate of d iscount . Data on var iab les ( i i ) - (v) are taken from Berndt and Wood. We 6 estimate models where labour i s the only short run f a c t o r (apart from c a p i t a l u t i l i z a t i o n ) and a lso where both labour and energy are short run f a c t o r s . In the f i r s t case , p i s constructed as a D i v i s i a index o f P Q , p^ and w 2 and y i s then def ined so as to maintain the equa l i t y py = nominal value added by labour and c a p i t a l . In the second case , p i s constructed as a D i v i s i a index of p G and p M arid y i s def ined so that py = nominal value added by 13 l abour , c a p i t a l and energy. 97. The use of real value-added as an output measure in a p r o f i t maximizing model i s v a l i d i f : ( i ) the quant i t i es of gross output and the "excluded" fac to rs move in f i xed proport ions (Leont ie f aggregat ion) , ( i i ) the p r ices o f gross output and the "excluded" fac tors move in f ixed proport ions (Hicks' aggregat ion) , or ( i i i ) the " inc luded" va r iab les (K, R and L o r , K, K, L and E) are weakly separable in the gross production f u n c t i o n . The Berndt and Wood d iscuss ion of the matter suggests that ( i ) and ( i i ) cannot be invoked to j u s t i f y the s p e c i f i c a t i o n of c a p i t a l - l a b o u r value added. They tes t fo r weak s e p a r a b i l i t y and r e j e c t i t but t h e i r t e s t i s in 15 fac t a j o i n t t es t of weak s e p a r a b i l i t y and other unwanted r e s t r i c t i o n s . Since t h e i r r e s u l t s a lso depend on the maintained assumptions in t h e i r a n a l y s i s , which d i f f e r from ours , and s ince the c a p i t a l - l a b o u r value added s p e c i f i c a t i o n i s common in the l i t e r a t u r e , we a lso adopt i t . J u s t i f i c a t i o n of the l a b o u r - c a p i t a l energy value added s p e c i f i c a t i o n is not as d i f f i c u l t . We have computed the simple c o r r e l a t i o n of the p r i ces of output and nonenergy intermediate inputs and the c o r r e l a t i o n o f the corresponding q u a n t i t i e s . In each c a s e , the c o r r e l a t i o n c o e f f i c i e n t was l a r g e r than .99, so that e i t h e r ( i ) or ( i i ) can be used to argue that the value added s p e c i f i c a t i o n should provide a good approximation. Both value added s p e c i f i c a t i o n s are inves t iga ted below, though not with the in ten t ion o f choosing between them. Note that they c o n s t i t u t e d i s t i n c t and t o t a l l y independent hypotheses. Returning to the da ta , we obtain data f o r gross investment in each of equipment and s t r u c t u r e s , in both real and constant d o l l a r s , from Berndt and Christensen (1973). Adding the constant d o l l a r f igures we obtain I and the p r ice d e f l a t o r i s der ived from the va lue -preserv ing e q u a l i t y . S i m i l a r l y we use t h e i r c a p i t a l stock estimates to obtain a s e r i e s (K t )_fo,r aggregate c a p i t a l s tock . Using t h i s s e r i e s and the perpetual inventory formula 98. R t = (1 -<5 ) - j + 1^ we compute a s e r i e s of deprec ia t ion rates 6 t, with - 1 fi average value 5 = .107. This exogenous estimate of the rate of c a p i t a l deprec ia t ion can be imposed on the model by s e t t i n g a 1 = 1-6 = .893, and a = a.. = b = b. = b' = y n + 2 = 0 f o r a ^ i - I n f a c t , the models that we estimate based on publ ished c a p i t a l stock data are based on R .^ However, because of the small v a r i a b i l i t y of the deprec ia t ion rates 5^ over t ime, the est imat ion r e s u l t s are not l i k e l y to d i f f e r s i g n i f i c a n t l y , (see a lso TABLE 3 of Appendix D) , and so we often i d e n t i f y the two models below. For example, in t e s t i n g the hypothesis that the rate of deprec ia t ion i s independent o f p r i c e s , i s unaffected by technica l change and is equal to the estimate based on publ ished data , (to which we r e f e r as the model with exogenous d e p r e c i a t i o n ) , we compare the l i k e l i h o o d funct ions from the u n r e s t r i c t e d model and from the model using K t app l ied to (13) , with the R and c a p i t a l accumulation equations deleted and a' = .893, a = a-| = b = b. = b' = Y n + 2 = 0 f o r a l l i . The nul l hypothesis i s tested by supposing that minus twice the logari thm of the r a t i o of the l i k e l i h o o d funct ions has a c h i r s q u a r e d i s t r i b u t i o n with 2n+5 degrees o f freedom. It i s h igh ly doubtful that t h i s approximation has s i g n i f i c a n t l y a f fec ted the tes t r e s u l t s , the l a t t e r genera l ly being qu i te d e c i s i v e . Of course when deprec ia t ion i s endogenous ( i . e . , estimated j o i n t l y with the other parameters o f the technology) we requi re only a benchmark 1947 c a p i t a l stock K. This i s taken to be ^g^y I m p l i c i t i s the assumption that the er rors involved in using exogenously estimated deprec ia t ion rates over the 1929-47 p e r i o d , (which i s the b a s i s , along with 1929 benchmarks, of the computed K t ) , average out to zero . Only a few var iab les remain. Note that an e f f e c t i v e rate o f return to c a p i t a l such as computed by Christensen and Jorgenson (1969), i s 99. i n c o n s i s t e n t with our model because of the endogeneity of the rate of d e p r e c i a t i o n . However, we take the approximate average of the Christensen and Jorgenson estimates for the postwar p e r i o d , 10%, to represent the constant , nominal , a f t e r - t a x rate of d iscount . The essen t ia l nature of the r e s u l t s reported below are u n l i k e l y to have been a l t e red by th is s i m p l i f i e d hypothesis . F i n a l l y , tax var iab les were excluded from the formulat ion of the t h e o r e t i c a l model fo r s i m p l i c i t y , but are included in the es t imat ion . The Christensen-Jorgenson treatment of taxes may be modif ied in the context of our model and the fo l lowing formulae der ived f o r the " e f f e c t i v e " p r ices of " n e w a n d "used" c a p i t a l r e s p e c t i v e l y : (18) q t = E t q t + 1 n E t q t+1 q t ~ ( l+r ) ( l - T r j r - - n - T C t + i " ^ + i n-DUM t + rTC t + 1)D t + 1]+TP t + 1 V , t+1. J ^ 1 ; 1 " T C t + l - v t + 1 ( l - D U M t + 1 . T C t + 1 ) D t + 1 | , where: v .^ = s ta tu tory corporat ion p r o f i t s tax r a t e , T C t = e f f e c t i v e rate of investment tax c r e d i t , TP^ = e f f e c t i v e corporat ion property tax r a t e , DUMj. = rate at which the deprec ia t ion base must be wr i t ten o f f because of the investment tax c r e d i t , Dj. = present value of deprec ia t ion deductions on a d o l l a r ' s investment over the l i f e t i m e of the asset (assuming a rate of discount of 10% and that the corporat ion p r o f i t s tax rate i s expected to remain constant at i t s current va lue ) . 100. Because tax p rov is ions with respect to equipment and s t ructures d i f f e r regarding deprec ia t ion allowances and the investment tax c r e d i t , we need data on tax var iab les f o r each of these subaggregates. This i s obtained from Christensen and Jorgenson (1969) and from the data const ruct ion in Berndt and Wood. The tax var iab les TC^. and D^ f o r aggregate c a p i t a l are computed as weighted averages of the corresponding subaggregate v a r i a b l e s , the value share of investment in each subaggregate forming the D E t weights . DUMt i s equal to D U M E t — ^ , where DUMEt i s 1 in 1962,3 and 0 otherwise and D E t i s the analogue of D t f o r equipment. F i n a l l y v^ and TP^. are taken d i r e c t l y from the above sources . Note that no p rov is ion i s made f o r deducting the i n t e r e s t changes of debt f inances or c a p i t a l losses from tax l i a b i l i t i e s . A l s o , we use the s ta tu tory p r o f i t s tax r a t e . The e f f e c t i v e rate computed by Christensen and Jorgenson i s incons is ten t with our model. F i n a l l y , per fec t f o r e s i g h t with respect to future tax va r iab les i s assumed. 6. Empir ica l Results The d iscuss ion d iv ides n a t u r a l l y in to three s e c t i o n s , dea l ing with the est imat ion of expec ta t ions , the ana lys is of the demand f o r c a p i t a l under uncer ta in ty , and the e f f e c t s o f the short run v a r i a b i l i t y of the rate of u t i l i z a t i o n of c a p i t a l . Expectat ions Using the computer program TIME (the current vers ion of which i s due to David Rose) and annual data fo r the 1947-71 p e r i o d , the fo l lowing models were i d e n t i f i e d and est imated: 101. (19) c a p i t a l stock p r i c e s : (1 - . 3615B) ( l -B ) log q t = .0193 + a t (.1963) (.0079) a 2 = .000545 R 2 = .9830 a Q = 2.35 R ( l - B ) = - 0 9 8 0 wage r a t e s : ( l - B ) l o g w n = .0423 + a t (.0057) a 2 = .000784 R 2 = .9874 Q = 2.49 R 2 = 0 energy p r i c e s : ^ (1-B) log w 2 t = .0094 + a t (.0035) a 2 = .000254 R 2 = .8912 a Q = 3.47 RZ}_B) = 0 value added d e f l a t o r s (subtract ing out energy and a l l other intermediate m a t e r i a l s ) : ( l - B ) l o g p t = .0167 + (1 - .4686B + .5480 B 2 ) a t (.0078) (.1531) (.1556) a2 = .00145 R 2 = .8269 a Q = 1.20 R p _ B ^ = .1415 102. value added de f l ec to rs (subtract ing out a l l nonenergy intermediate m a t e r i a l s ) : ( l - B ) l o g p t = .0171 + a (.0083) a2 = .00164 a Q = 5.38 R 2 = .7906 R(l-B) = ° B denotes the backsh i f t opera tor , ( e . g . B log p t = log Pt_-|)> and {a t > t a sequence of independently and i d e n t i c a l l y d i s t r i b u t e d normal var ia tes with zero mean. The standard er rors of the c o e f f i c i e n t estimates are given in parentheses below the c o e f f i c i e n t s . A lso ind ica ted f o r each equation i s _2 the estimate of the var iance of the d is turbance , R fo r the undi f ferenced _2 and d i f fe renced ser ies (RQ _:B.y)and the Box and Pierce Q s t a t i s t i c fo r a lag o f 4 p e r i o d s . Note that three of the ser ies are represented by simple c o n s t a n t - r a t e -o f - i n f l a t i o n models. A l l models were o v e r f i t t e d in an attempt to uncover add i t iona l s t ruc ture that was suggested at the i d e n t i f i c a t i o n stage or by the. au tocor re la t ion o f the res idua ls from the above models. No add i t iona l s i g n i f i c a n t s t ruc ture was found, however, which i s not t e r r i b l y s u r p r i s i n g in l i g h t of the h igh ly aggregated nature o f the data . The autoregressive parameter in the model f o r c a p i t a l stock pr ices i s s i g n i f i c a n t l y d i f f e r e n t from zero only at a 10% confidence l e v e l . At 5% we cannot r e j e c t the hypothesis of a c o n s t a n t - r a t e - o f - i n f l a t i o n model but i t i s u n l i k e l y that th is would s i g n i f i c a n t l y a f f e c t any of the empir ica l r e s u l t s reported below. The above equations are now taken to be an exact representat ion o f 18 ~ 9 producers' expecta t ions , with the a from each equation represent ing the a uniform degree of uncer ta inty through the sample p e r i o d . As w i l l become 103. evident s h o r t l y , the e s s e n t i a l features of the est imat ion r e s u l t s are u n l i k e l y to be a f fec ted by adopting the a l t e r n a t i v e measures of uncer ta inty (10) or (11). Note that the next per iod forecasts o f q t may be obtained from (18) and (19). Uncerta inty and the Demand f o r Cap i ta l A l l estimates were obtained using the Nonl inear Monitor from the U . B . C . S t a t i s t i c a l Centre , using a convergence to lerance f o r the percentage change in each parameter from one i t e r a t i o n to another less than .01. Though in a nonl inear model some doubt always remains about the l o c a l versus global nature of the maximum to which the a lgor i thm converges, a l l the estimates we report were obtained beginning from at l eas t two d i f f e r e n t s t a r t i n g va lues . Moreover, a l l the short run models d iscussed below were very wel l behaved. Convergence was f a i r l y rap id and no l o c a l , nonglobal maxima were found. Thus we are reasonably conf ident that we do indeed have maximum l i k e l i h o o d est imates . Several models were estimated as descr ibed in TABLE 1. (Some of the parameter estimates are presented in Appendix E.) In view of the s t a r t l i n g uni formity of the r e s u l t s , i t was not f e l t that anything would be gained by going through the other poss ib le permutations and combinations. The bas ic message of the f ind ings i s c l e a r : the model provides l i t t l e explanat ion of the demand f o r c a p i t a l . Though R 's are high when lagged endogenous var iab les are included as explanatory v a r i a b l e s , they are otherwise extremely low. Autocor re la t ion in the c a p i ta l equation hovers in a small in te rva l about 1 and there are even two unstable models. In a l l c a s e s , the theory adds l i t t l e to the explanatory power of a f i r s t order autoregressive model, to which, as v e r i f i e d by an examination of the 104. r e s i d u a l s , the estimated c a p i t a l equations have v i r t u a l l y been reduced. (Note that the autoregressive parameter in a f i r s t order autoregressive model appl ied to c a p i t a l stock data (K t ) i s estimated to be 1.03.) There i s some v a r i a t i o n in the r e s u l t s between models. With funct iona l 2 form 1, the R 's in the c a p i t a l equation are very large negat ive . Est imat ing thesee,,models without a l lowing f o r a u t o c o r r e l a t i o n , we found that res idua ls , were large and uniformly s igned. This suggested that perhaps the problem was the lack of a constant term in the c a p i t a l equat ion. Funct ional form 2 was s p e c i f i e d p r e c i s e l y in order to introduce a constant and i t resu l ted in the res idua ls and t h e i r average being considerably reduced in s i z e , again before a l lowing f o r a u t o c o r r e l a t i o n . However, the e r rors were s t i l l h ighly autocorre la ted and a f t e r a l lowing f o r au tocor re la t ion in the es t ima t ion , we were no be t te r o f f than with funct iona l form 1. The f i t in the c a p i t a l equation improved but that was more than o f f s e t by the de te r io ra t ion of the f i t s in the other equat ions , r e s u l t i n g in a lower l i k e l i h o o d va lue . Homoscedastic disturbances were hypothesized so as to s i m p l i f y the model in an attempt to uncover the reasons under ly ing our r e s u l t s . This change in s p e c i f i c a t i o n had l i t t l e e f f e c t as d id the s p e c i f i c a t i o n of endogenous d e p r e c i a t i o n . Energy was included as a separate f a c t o r to al low f o r the p o s s i b i l i t y that an i n v a l i d c a p i t a l - l a b o u r value added s p e c i f i c a t i o n was the problem, but the r e s u l t s were e s s e n t i a l l y una l te red . F i n a l l y , in order to determine whether the problem was i n v a l i d cross equation parameter c o n s t r a i n t s , we estimated the c a p i t a l equation a lone. Though a l l parameters could not be estimated in the s i n g l e equat ion , several s impler equations were estimated and the r e s u l t s (not reported here) were i d e n t i c a l to those reported in the tab le - the equations reduced to autoregressive models. TABLE VI1.1: SUMMARY OF "LONG RUN" MODELS ESTIMATED Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 variable factors labour labour labour labour labour labour, energy stochast ic spec i f i ca t ion heteroscedastic heteroscedastic heteroscedastic homoscedastic homoscedastic heteroscedastic functional form 1 1 2 2 2 1 depreciat ion exogenous 19 endogenous exogenous exogenous endogenous exogenous log L* -165.35 -161.89 -181.10 -181.87 -165.09 -167 .85 2 0 OUTPUT .9578 .9782 .9080 .9624 .9694 R 2 LABOUR (transformed equat ions)** ENERGY .9465 .9766 .8299 .9671 .9648 .9957 CAPITAL .7902 .9799 .9633 .5765 .5513 OUTPUT .9276 ; -.1068 .6096 .6599 .8687 .9627 R 2 LABOUR (or ig inal equat ions)** ENERGY .8110 .9573 .1774 -.6452 .5598 .9150 .9920 CAPITAL -19.56 -136.8 .0814 -3.690 -1.485 -11.0 OUTPUT .5928 .9773 .5856 .4304 .4113 .6665 LABOUR p s .4711 .5272 .4092 .4437 .3142 \ , .2469 ENERGY - • .5309 CAPITAL .9947 .9886 1.029 .9498 .9692 .9941 * Concentrated log - l i ke l ihood values, including constant term. **By "transformed equations" we mean those equations obtained from the "or ig ina l equations", I .e . , the appropriate version of (13), by applying the standard transformation to r id the disturbances of f i r s t order autocorre la t ion. o R 2 ' s for the transformed equations of Model 3 were not ca lcula ted. 106. In t r y i n g to expla in the poor performance of our models, we might invoke the usual "explanat ions" . The r e s u l t s may be due, at l e a s t in p a r t , to poor data ( for c a p i t a l , in p a r t i c u l a r ) , our assumptions of p r i c e tak ing behaviour and the t h e o r e t i c a l v a l i d i t y of aggregating over producers and goods, and our treatment o fppr ices as exogenous var iab les in the e s t i m a t i o n , none of which may be v a l i d in r e a l i t y . Wales (1975) has shown that the s p e c i f i c a t i o n of the wrong funct iona l form, even i f i t i s f l e x i b l e to the second order and even i f the data are p e r f e c t l y cons is tent with the theory , may lead to poor s t a t i s t i c a l r e s u l t s . However, the uniformly bad r e s u l t s we obtained with two d i f f e r e n t funct iona l forms lead us to be l ieve that the problem l i e s elsewhere. Our assumption of zero costs of adjustment may be quest ioned. There i s no doubt that a f l e x i b l e acce le ra to r s p e c i f i c a t i o n fo r the adjustment of actual towards desi red cap i ta l s tock , by i n c l u d i n g lagged cap i ta l stock as an explanatory var iab le ,would provide a good f i t . But as argued e a r l i e r , such a s p e c i f i c a -t ion i s e s s e n t i a l l y ad hoc, and i t i s not c l e a r that such a model would not a lso reduce e s s e n t i a l l y to an autoregressive equat ion. It i s poss ib le that the au tocor re la t ion matrix has nonzero o f f diagonal elements, with the large au tocor re la t ion c o e f f i c i e n t s we found being induced by our m i s s p e c i f i c a t i o n . The a l t e r n a t i v e of a f u l l au tocor re la t ion matrix i s not a t t r a c t i v e , however, because i t seems u n l i k e l y that the theory w i l l gain a la rger ro le in exp la in ing the data r e l a t i v e to the a u t o c o r r e l a t i o n . A l s o , i t i s qui te p o s s i b l e that an (almost) unstable au tocor re la t ion s t ruc ture would remain. None of the above represent s a t i s f a c t o r y explanat ions of our . f indings. To varying degrees the problems we have ra ised are found in a l l f a c t o r demand studies and yet none of those c i t e d e a r l i e r produced s t a t i s t i c a l r e s u l t s near ly as poor as ours : This led us to enquire whether there was a bas ic 107. d i f f e r e n c e between our study and others in the l i t e r a t u r e that employ, the theory of d u a l i t y and f l e x i b l e funct iona l forms, and one point stands out which we be l ieve may l a r g e l y expla in t h e i r success and our f a i l u r e in conf i rming very s i m i l a r t h e o r i e s : none of the other s tudies contain an equation l i k e our c a p i t a l equation where the leve l of a f a c t o r demand i s to be explained s o l e l y by p r i c e s . They a l l take at l e a s t one quant i ty to be exogenous (e .g . output in cost minimizat ion models as ii(n Woodland (1975), or cap i ta l stock in models of va r iab le p r o f i t maximization as in Woodland (1974)), or expla in f a c t o r cost shares , rather than l e v e l s , as in Berndt 21 and Christensen (1973,4) and Berndt and Wood (1975). Cor re la t ion between quant i t i es would seem to enhance the p r o b a b i l i t y of reasonable s t a t i s t i c a l r e s u l t s . We have managed to f i n d in the l i t e r a t u r e one empir ica l ana lys is of n e o c l a s s i c a l f a c t o r demand theory based on mul t iper iod p r o f i t maximization without adjustment costs where f a c t o r demands are to be explained by pr ices a lone. Brech l ing (1975, Chapter 3) app l ies the Cobb-Douglas funct iona l form to quar te r ly U.S. data f o r a l l manufacturing i n d u s t r i e s , durable goods i n d u s t r i e s and nondurable goods i n d u s t r i e s r e s p e c t i v e l y . He does not allow fo r au tocor re la t ion in his est imat ion but f inds that in a l l three cases the Durbin-Watson s t a t i s t i c s are c lose to z e r o , ( for the c a p i t a l equation they range from .06 to .12) , implying au tocor re la t ion parameters comparable to ours . Though one might be tempted to argue that his r e s u l t s are caused by the s p e c i f i c a t i o n of the r e s t r i c t i v e Cobb-Douglas funct iona l form, our a n a l y s i s , a l b e i t app l ied to a s l i g h t l y d i f f e r e n t set of da ta , suggests that th is i s not the case. Our data do not lend any empir ica l support to the n e o c l a s s i c a l theory of p r o f i t maximization subject to e i t h e r of two d i f f e r e n t f l e x i b l e funct iona l forms. 108. A necessary condi t ion f o r the p r o f i t maximization model we have formulated is that there be decreasing returns to s c a l e , i . e . , that TT be s t r i c t l y concave in K. To inves t iga te the v a l i d i t y o f t h i s condi t ion we estimated the short run vers ions ( i . e . , with the c a p i t a l equation deleted) o f Models 1, 2 and 4. The p r o f i t funct ion was s t r i c t l y concave in K over the en t i re sample in the f i r s t case , over h a l f the sample in the second and in both cases the hypothesis of constant returns to sca le was re jec ted at .05. The hypothesis could not be re jected in Model 4 even at a .10 leve l of conf idence . Jorgenson (1972) has surveyed the evidence in the l i t e r a t u r e on returns to s c a l e in U.S. manufacturing and argues that constant returns to sca le cannot be r e j e c t e d . But the evidence he c i t e s and our f ind ings as w e l l , are not unanimous and i t i s not c l e a r that th is i s the source of our poor r e s u l t s . A p r i n c i p a l ob jec t ive of our ana lys is was to determine the e f f e c t s of the v a r i a b i l i t y of expectat ions on the demand f o r c a p i t a l and investment. In l i g h t of the poor performance o f the model in exp la in ing the. demand fo r c a p i t a l there i s no point in present ing the e l a s t i c i t i e s of the l a t t e r with respect to expected rates of i n f l a t i o n of a s s e t , product and f a c t o r p r i ces and the var iances surrounding these expecta t ions , which dould be ca lcu la ted from the parameter estimates in any of the models. S i m i l a r l y , the consistency of the models with the monotonicity and curvature proper t ies impl ied by the theory i s of l i t t l e i n t e r e s t , though they were l a r g e l y s a t i s f i e d , at l e a s t in models where an exogenous rate of deprec ia t ion was imposed. These matters could be inves t iga ted in two a l t e r n a t i v e s which could reason-ably be considered in l i g h t of the poor performance of our model: a model o f p r o f i t maximization with costs of adjustment and a mul t iper iod cost minimizat ion problem with no costs of adjustment. Both are cons is ten t with constant or inc reas ing returns to s c a l e . Thefformer requires numerous 109. s i m p l i f y i n g assumptions to be put in to implementable form, while the l a t t e r can be developed in a t o t a l l y analogous fashion to o u r s , with the va r iab le cost funct ion rep lac ing the var iab le p r o f i t funct ion as the representat ion of the under ly ing technology. Both of these suggestions are c l e a r l y the subjects of future research and are beyond the scope of t h i s t h e s i s . Given our model's f a i l u r e to expla in the demand f o r c a p i t a l , we ignore the l a t t e r in the remainder of t h i s chapter and concentrate on the short run d e c i s i o n s : output , labour and energy demands and the u t i l i z a t i o n o f a given c a p i t a l s tock . That i s , we drop the c a p i t a l equation from the system to be est imated. Of course the appropriate p r i c e q f o r "used" c a p i t a l depends to some extent on the way in which the c a p i t a l dec is ion i s made. But formula (18) i s conceivably a lso v a l i d in a l t e r n a t i v e s p e c i f i c a t i o n s of the demand f o r c a p i t a l ; f o r example, in a model with costs of adjustment that depend on net ( rather than gross) investment. Thus we continue to use i t . The short run equations c l e a r l y have l i t t l e to say about the e f f e c t s of v a r i a b l e expectat ions and the focus of the ana lys is s h i f t s to the endogeneity of the rate of deprec ia t ion o f c a p i t a l . The Endogenous Depreciat ion of Capi ta l We estimate (13) without the c a p i t a l equat ion , adopting the hetero-s c e d a s t i c e r ro r s p e c i f i c a t i o n and funct iona l form 1, because, unl ike (12 ' ) , i t renders the short run equations l i n e a r in parameters when deprec ia t ion i s f i x e d . Because we are no longer concerned with the e f f e c t s of uncer ta in ty , second order f l e x i b i l i t y is s u f f i c i e n t and we set b = b^ = b..j> =, 0; f o r a l l i , j . Again we assume that deprec ia t ion is geometric (b' = 0 ) , a maintained hypothesis in most of the l i t e r a t u r e . The rate of geometric d e p r e c i a t i o n , however, depends on pr ices in genera l . n o . The models estimated are descr ibed in TABLE 2 and the corresponding parameter estimates and t h e i r standard er rors are l i s t e d in Appendix E. Both labour alone and labour and energy are taken in turn to be va r iab le f a c t o r s , while in each case we f i r s t impose a constant rate of deprec ia t ion and then allow the rate of deprec ia t ion to be determined endogenously. In t h i s way the p a r t i c u l a r impl ica t ions of the l a t t e r hypothesis are h igh-l i g h t e d . We not ice immediately that the f i t s are much bet ter than those discussed above. The R 's are h i g h , even without the benef i t of lagged endogenous v a r i a b l e s , and the au tocor re la t ion parameters are well i n s i d e the s tab le reg ion . A lso ind ica ted in the table are the regions where the r e g u l a r i t y condi t ions impl ied by the theory are s a t i s f i e d by the est imates. Monotonici ty in p r i ces corresponds to p o s i t i v e f i t t e d values f o r the supply and demand f u n c t i o n s , while convexity in p r ices obtains when a l l eigenvalues of the p r i c e Hessian of the p r o f i t funct ion are nonnegative. The remaining r e g u l a r i t y condi t ions are e a s i l y determined, but note that decreasing returns to sca le i s no longer a necessary condi t ion in t h i s model of short run p r o f i t maximiza-t i o n . In a l l cases we say that the r e g u l a r i t y condi t ions are s a t i s f i e d at a given p r i c e - c a p i t a l conf igura t ion i f they are s a t i s f i e d given the, point estimates of the parameters of the technology. In genera l , no j o i n t s t a t i s t i c a l t es t appears to be a v a i l a b l e to t e s t f o r the convexity o f the p r o f i t funct ion in p r i c e s , though in some cases (Models 7 and 10) convexity corresponds to a p o s i t i v e value f o r a s i n g l e parameter and so can be tested 22 fo rma l ly . TABLE VII.2: SUMMARY OF "SHORT RUN" MODELS ESTIMATED Model 7 Model 8 Model 9 Model 10 Model 11 Model 12 Model 13 Model 14 variable factors depreciation other specifications log L* R2 OUTPUT (original , A B 0 U R equations)* L A B U U R ENERGY p' s OUTPUT LABOUR ENERGY Regions where regularity conditions are sat is f ied: * * (1) monotonicity in prices (2) convexity in prices (3) monotonicity in capital (4) concavity in capital (5) positive profits labour exogenous -126.71 .9588 .9359 .4796 .3164 • X**** / / / labour exogenous CRS*** and short run price in f lex ib i l i ty -128.33 .9503 .9322 .4530 .3205 labour endogenous -116.50 .9168 .9666 .5780 .1585 X / 1947-63 • labour endogenous price inflexible rate of depreciation -124.55 .9522 .9209 .4581 .3952 labour,energy labour,energy labour,energy exogenous exogenous / X**** V • • -131.68 .9647 .9345 .9841 .5114 .3139 .6567 X • / / CRS*** -134.39 .9589 .9196 .9846 .5875 .3582 .5132 X / • endogenous CRS*** -124.02 .9568 .9052 .9914 .3198 .3962 -.0004 X / / / labour,energy endogenous CRS*** and constant price inflexible rate of depreciation ,t -127.38 .9535 .9073 .9874 .' ' .4048 .3613 .2718 X / • • 107 .342 .097 31 125 Average depreciation rate .107 Capital stock in 1971 126 126 *See the notes at the bottom of Table VI1.1. * V indicates everwhere in the sample and X indicates nowhere in the sample. ***Constant returns to scale. ****Convexity cannot be rejected at .05 confidence level. .107 TToT .055 126 126 209 .075 162 112. F i n a l l y , average deprec ia t ion rates and the 1971 c a p i t a l stocks impl ied by the models are l i s t e d . When deprec ia t ion i s exogenous, the f igures correspond to publ ished data as d iscussed above. Otherwise they ind ica te what depreciate rates and c a p i t a l stocks are cons is tent with rates of u t i l i z a t i o n determined by p r o f i t maximizat ion. Depreciat ion rates are equal to 1 - K t / K t , and K t and K t are computed r e c u r s i v e l y . TABLE 3 o f Appendix E contains the corresponding complete time s e r i e s . Consider f i r s t the s i n g l e - v a r i a b l e - f a c t o r , m o d e l s . When deprec ia t ion i s exogenous (Models 7 and 8) we have the ra ther s u r p r i s i n g r e s u l t that a constant returns to sca le Leont ie f ( f ixed proport ions) technology cannot be re jected even at a leve l o f confidence of .10. (The c r i t i c a l x- value is 2 X 3 = 6.25 and the computed s t a t i s t i c i s 3.24.) The degree of s u b s t i t u t a b i l i t y between labour and c a p i t a l in; aggregate two f a c t o r models has received considerable a t tent ion in the l i t e r a t u r e , most of i t i n v o l v i n g the use of the CES production f u n c t i o n . While the evidence on whether a Cobb-Douglas technology i s cons is ten t with the fac ts i s not unanimous, there i s much greater concensus that the e l a s t i c i t y of s u b s t i t u t i o n is p o s i t i v e so that a Leont ie f technology i s re jec ted . Two major points d i f f e r e n t i a t e our s p e c i f i c a t i o n from most o thers . F i r s t , we assume c a p i t a l stock i s given in any per iod and that output and labour demand are explained by pr ices and c a p i t a l v i a short run p r o f i t maximization. That a s i m i l a r s p e c i f i c a t i o n does not appear to have been adopted before i s undoubtedly due to the f a c t that with the CES funct iona l form the short run supply and demand funct ions are nonl inear in parameters and hence d i f f i c u l t to est imate. Second, we have not adopted the common s p e c i f i c a t i o n of Hicks neutral techn ica l change. It . might be i n t e r e s t i n g to determine whether the tes t r e s u l t s are a f fec ted by a change in s p e c i f i c a t i o n , though we are not aware of any evidence of the TABLE VII.3: SHORT RUN DEMAND AND SUPPLY ELASTICITIES, FOR SELECTED YEARS AND MODELS MODEL 9 MODEL 12 MODEL 14 1948 1958 1968 1948 1958 1968 1948 1958 1968 -.768 -.240 -.185 .427 -.211 -.073 -.098 -.057 .003 £ y w r -.064 -.008 -.003 .443 .274 .166 .064 .097 .099 e y q .831 .248 .188 V .062 .013 .005 -.791 -.472 .268 -.113 -.167 -.161 -.312 .122 .069 .684 .396 .218 .045 .040 .069 .250 .109 .063 .775 1.26 1.59 — : -.239 -.355 -.349 : -.536 -.903 -1.24 . e yw 2 -.017 -.063 -.093 .034 -.040 -.102 .107 .076 ' .050 .157 .127 .091 — — .144 .637 1.03 .312 .411 1.13 E z2 wl .524 .443 .340 .814 .753 .623 E z 2 w 2 — - -.668 -1.08 -1.37 -.502 -1.16 -1.75 e_ -2 2 q % 2 114. s u p e r i o r i t y of the Hicks neutral s p e c i f i c a t i o n on t h e o r e t i c a l or empir ica l grounds. In any case , given our s p e c i f i c a t i o n , we cannot r e j e c t the hypothesis that the data are cons is tent with short run p r o f i t maximization subject to a g l o b a l l y well -behaved technology. Technical change causes the ou tput -cap i ta l r a t i o to increase and the l a b o u r - c a p i t a l r a t i o to decrease over t ime. When deprec ia t ion i s made endogenous (Model 9 ) , convexity in p r ices i s v i o l a t e d g l o b a l l y and i r - - and are negative throughout the sample. The hypothesis o f constant returns to sca le and a p r i c e i n f l e x i b l e rate of deprec ia t ion (and hence a lso o f to ta l short run p r i c e f l e x i b i l i t y ) are each 2 r e j e c t e d , with^ x values 9.84 and 16.10 r e s p e c t i v e l y , compared with the .01 2 c r i t i c a l value x2 = 9 .21. The impl ied deprec ia t ion rates are u n r e a l i s t i c a l l y large and imply a f a l l i n g c a p i t a l stock through t ime. However, i f we const ra in deprec ia t ion rates so that they are independent of f a c t o r and product p r ices (a = a-j = 0) as in Model 10, we obtain a c a p i t a l stock se r ies remarkably s i m i l a r to R .^ In fac t we cannot re jec t the hypothesis 2 that the rate of deprec ia t ion is constant and equal to .107. (The x 2 s t a t i s t i c i s 4.32 and the .10 c r i t i c a l value i s x^ = 4 .32) . Nor can we re jec t the Leont ie f technology (Model 8) at .10 leve l of s i g n i f i c a n c e . 2 2 (The x< s t a t i s t i c i s 7.56 while the c r i t i c a l value .is x g = 9.24.) Short run demand and supply e l a s t i c i t i e s impl ied by Model 9 are l i s t e d in TABLE 3, where e y p E | ^ ( p , q , w ; K ) , e^- = g ^- (p ,q ,w;K) • | - and s i m i l a r l y f o r the remaining terms. Consistent with the nonconvexity of the p r o f i t f u n c t i o n , the own e l a s t i c i t i e s e , e z , e ^ - have signs which make l i t t l e yp 1 -j q economic sense. The small magnitudes o f e , and e i n d i c a t e that the yw-, z-,p p r i c e of labour has l i t t l e e f f e c t on output and the demand f o r labour i s unaffected by changes in the product p r i c e , fo r a given c a p i t a l s tock . 115. Higher output p r i ces appear to reduce the r a t e . o f c a p i ta l u t i l i z a t i o n apprec iab ly , contrary to what we might have expected, while higher wages tend to increase i t , though to a smal ler degree. C l e a r l y , however, in l i g h t o f the inconsis tency of the model with the theory , these e l a s t i c i t i e s cannot be taken very s e r i o u s l y . Consider now the models where.energy i s included as a d i s t i n c t f a c t o r . Test ing f o r short run p r i c e i n f l e x i b i l i t y and constant returns to sca le (CRS) when deprec ia t ion is exogenous, we d e c i s i v e l y re jec t the former, (the p computed s t a t i s t i c i s %^ = 17 .8) , but cannot re jec t the l a t t e r at a .05 2 l eve l of confidence (x 3 = 5.42). In l i g h t o f t h i s r e s u l t , the f a c t that CRS i s a standard assumption in the l i t e r a t u r e and in order to reduce the number of parameters in the models with endogenous deprec ia t ion to more manageable p ropor t ions , we maintain the CRS hypothesis in the remainder o f the a n a l y s i s . Maintaining CRS and exogenous d e p r e c i a t i o n , short run p r i c e i n f l e x i b i l i t y 23 i s r e j e c t e d , but the e l a s t i c i t i e s l i s t e d in TABLE 3 are i n c o n s i s t e n t with p r o f i t maximizing theory , as again convexity is g l o b a l l y v i o l a t e d , e Z 2 W 2 i s the s i n g l e o w n - e l a s t i c i t y that i s c o r r e c t l y signed throughout the sample. There would appear to be some gross s u b s t i t u t a b i l i t y between labour and energy. (Reca l l ing IV.(9) we can expla in the small s i z e of E z l w 2 r e l a t i v e to e z w by the much smal ler share o f energy in t o t a l va r iab le p r o f i t s r e l a t i v e to the share o f l abour . ) But again these f igures must be in te rpre ted with extreme cau t ion . For example, the upward "bias" in the estimate of e,„,, (-IT,, ,, ) may be co inc ident withaa downward "b ias" in the z^w-i W-JW-J estimate of e (-TT ), so that l i n e a r homogeneity in p r i ces is maintained. 116. Al lowing the rate of deprec ia t ion to be determined endogenously, we are unable to r e j e c t at .10 leve l o f confidence a constant geometric rate 2 2 of d e p r e c i a t i o n . (From Models 13 and 14, the x s t a t i s t i c i s x^ = 6.72 and the c r i t i c a l value i s 7.78.) Moreover, the l a t t e r model impl ies a "reasonable" deprec ia t ion rate of 7.5%, which i s s i g n i f i c a n t l y d i f f e r e n t from the 10.7% f igure estimated from data . (From Models 12 and 14, the 2 2 x s t a t i s t i c i s x-| = 14.02.) The asymptotic standard e r r o r of a 1 impl ies an asymptotic 95% confidence in te rva l f o r the rate of deprec ia t ion of (.064, .087) and f o r the 1971 stock of c a p i t a l of (148, 178). Test ing f o r an endogenous and constant rate of deprec ia t ion and short run p r i c e i n f l e x i b i l i t y we Reject the hypothesis d e c i s i v e l y (Xy = 26.34) . However, convexity i s again g l o b a l l y v i o l a t e d . To the, extent that cross e l a s t i c i t y estimates are s t i l l meaningful , they ind ica te that labour and 24 energy are gross subs t i tu tes in the short run. This has important imp l ica t ions f o r i t i n d i c a t e s , f o r example, that higher energy p r i ces increase ( s l i g h t l y ) the demand f o r labour in the short run with c a p i t a l stock f i x e d , a f t e r accounting f o r the change in output. Converse ly , higher wages induce a l a rger demand f o r energy in the short run and thus can counteract energy conservat ion measures. Long run e f f e c t s can be determined only by s p e c i f y i n g the investment d e c i s i o n . In Models 12 and 14 technica l change causes an upward trend in the output-cap i ta l r a t i o and downward trends in the l a b o u r - c a p i t a l and energy-cap i ta l r a t i o s . The i n s i g n i f i c a n c e of t h i s f a c t o r as a determinant of the r a t i o s was tes ted maintaining the s p e c i f i c a t i o n o f Model 14 and was re jected at a l l reasonable l e v e l s of s i g n i f i c a n c e . 117. 7. Summary and Conclusions Our i n i t i a l ob jec t ives in undertaking the research reported in t h i s chapter were to : ( i ) tes t the n e o c l a s s i c a l theory o f p r o f i t maximization recogniz ing that some producer dec is ions are made subject to uncertain expectat ions and d i s t i n g u i s h i n g between cap i ta l stocks and se rv ice f lows; ( i i ) determine the e f f e c t o f p r i c e uncer ta inty on the demand f o r c a p i t a l ; ( i i i ) t es t the s t a t i s t i c a l s i g n i f i c a n c e of our formulat ion of the c a p i t a l u t i l i z a t i o n d e c i s i o n ; ( iv ) provide maximum l i k e l i h o o d estimates of the rate of c a p i t a l deprec ia t ion cons is tent with the theory of p r o f i t maximizat ion. A model was formulated in the context ofv>which these ob jec t ives could be pursued, but they were only p a r t i a l l y achieved. We were unable to say anything about the e f f e c t s o f p r i c e uncerta inty because we found that the var ious s p e c i f i c a t i o n s of the n e o c l a s s i c a l model o f p r o f i t maximization that were considered were incons is ten t with the data f o r the U.S. manufacturing sec tor during the 1947-70 p e r i o d . Some p o s s i b l e causes of the incons is tency were d iscussed and suggestions f o r fu r ther research were made. Next we considered a model of short run p r o f i t maximization f o r a given c a p i t a l s tock , with labour being the only var iab le f a c t o r of product ion . The hypothesis that the rate of cap i ta l u t i l i z a t i o n i s independent o f product and f a c t o r p r i c e s was r e j e c t e d . However, the rates o f deprec ia t ion impl ied by the model were u n r e a l i s t i c a l l y l a r g e . When energy was added as a f a c t o r in the production p rocess , a constant geometric rate of deprec ia t ion could not be r e j e c t e d . Moreover the estimated rate of 7.5% was not g ross ly unreasonable and was s i g n i f i c a n t l y d i f f e r e n t from the rate of 10.7% der ived from publ ished data . It i s not s u r p r i s i n g that the r e s u l t s in the two cases d i f f e r e d in t h i s r e s p e c t , as they involve two genera l ly d i s t i n c t and independent maintained hypotheses, nor is i t abso lu te ly c l e a r from our 118. f ind ings or other evidence that e i t h e r is the c l e a r l y super ior s p e c i f i c a t i o n . But we might be tempted to i n t e r p r e t the r e s u l t s so that the s i g n i f i c a n c e o f va r iab le u t i l i z a t i o n and the unreasonable deprec ia t ion rate estimates in the case o f " labour only" are due to the e r r o r in the s p e c i f i c a t i o n of c a p i t a l - l a b o u r real value added as an output measure. On the other hand, when energy i s proper ly taken into account, deprec ia t ion rates are reasonable and var iab le cap i ta l u t i l i z a t i o n ceases to be s t a t i s t i c a l l y s i g n i f i c a n t ; the l a t t e r being cons is ten t w i t h , though not equivalent t o , the reasoning under ly ing the const ruc t ion of cap i ta l u t i l i z a t i o n rates based on energy consumption. The most important reservat ion in adopting th is i n t e r p r e t a t i o n is the f a c t that most of the models est imated v i o l a t e d convexity in p r i ces and thus were i n c o n s i s t e n t with p r o f i t maximizing behaviour. Of course convexity was not re jec ted s t a t i s t i c a l l y , but s ince a point examination of convexity i s the best we can do, the r e s u l t s are s t i l l u n s a t i s f a c t o r y . Several of the points mentioned in our d iscuss ion of the s t a t i s t i c a l estimates of the demand fo r c a p i t a l may be the cause of the incons is tency and provide leg i t imate subjects f o r future work. Th is chapter has developed and demonstrated a workable framework wi th in which such work can proceed. 119. VI I I . CONCLUDING REMARKS In the f i r s t part of the thesis we provided multidimensional general iza-t ions of notions that have hi therto been considered almost exc lus ive ly in re la t ion to unidimensional r i s k s . We then establ ished corresponding general izat ion of known resu l t s , both with respect to the marginal impact of uncertainty and the character izat ion of r i sk independence. Some in teres t ing Slutsky-type decompositions were obtained, which c l a r i f i e d the re la t ionsh ip between the ef fects of mean preserving spreads, mean u t i l i t y preserving spreads and sh i f t s of expectat ions. A "natura l " de f in i t i on of increased cor re la t ion was presented and a corresponding analysis of the ef fects of increased cor re la t ion was carr ied out. This general analysis was then applied to two spec i f i c decision problems. F i r s t , we considered a two-period consumer choice problem where a vector of current consumption a c t i v i t i e s must be decided upon subject to uncertainty about future income and p r i ces . The need for such a disaggregate analysis was establ ished and the l im i ta t ions of e a r l i e r aggregate studies in the l i t e ra tu re demonstrated. Preferences exh ib i t ing r i sk independence were characterized and the character izat ions were related to other work in the l i t e r a t u r e . It was argued that our character izat ions alone were relevant to the behaviour of a consumer in the above two-period framework. Second, a deta i led analysis was carr ied out of the behaviour of a competitive f i rm which faces pr ice uncertainty and which has ex post adjustment p o s s i b i l i t i e s . The l a t t e r were ignored in most e a r l i e r studies whose comparative s ta t i cs resul ts were shown to be special cases of our r esu l t s . 120. In both. the consumer and producer models those proper t ies of preferences and technology that are p a r t i c u l a r l y re levant to behaviour under uncer ta inty were pointed out . In p a r t i c u l a r , t h i r d order proper t ies were shown to be important in determining the impacts of increased v a r i a b i l i t y and c o r r e l a t i o n . Consequently, the funct iona l forms discussed by Diewert (1974), when appl ied in the context of p r i c e or income uncer ta in ty , are not as f l e x i b l e as has prev ious ly been thought, s ince they are capable o f prov id ing only second order approximations to a r b i t r a r y preferences and techno log ies . (Our ana lys is can a lso be used to point out . the assumptions i m p l i c i t in other " r e s t r i c t i v e " funct iona l forms that have u n t i l now been evaluated on the basis o f second order propert ies '^, i . e . , e l a s t i c i t i e s of s u b s t i t u t i o n . Thus the Cobb-Douglas funct iona l form f o r a technology has been c r i t i c i z e d because i t imposes a p r i o r i the value of the e l a s t i c i t y of f a c t o r s u b s t i t u t i o n . We showed that i t a lso imposes severe r i s k independence condi t ions and so has strong imp l ica t ions f o r the react ions to increased uncer ta inty and c o r r e l a t i o n . ) In l i g h t of the above, " s u f f i c i e n t l y " f l e x i b l e funct iona l forms were hypothesized and t h e i r po tent ia l a p p l i c a t i o n in empir ica l work was d i s c u s s e d . Another cont r ibu t ion of the thes is l i e s in the demonstration o f the usefulness and l i m i t a t i o n s of d u a l i t y theory in models of dec is ion making under uncer ta in ty . The formulat ion and ana lys is o f the consumer and producer problems were great ly f a c i l i t a t e d by the use of d u a l i t y , but un l ike the s i t u a t i o n in standard c e r t a i n t y models, d u a l i t y theory d id not provide us with e x p l i c i t funct iona l forms f o r demand and supply f u n c t i o n s . However, i t was argued that d u a l i t y theory was v i r t u a l l y indispensable f o r the empir ica l ana lys is o f consumer and producer behaviour in the framework o f temporal uncer ta in ty . 121. F i n a l l y the bas ic theory of producer behaviour which we analysed was appl ied to aggregate U.S. manufacturing data from 1947-71. We assumed that the cap i ta l stock dec is ion must be made one per iod before the c a p i t a l comes in to opera t ion , subject to random expectat ions about future p r i c e s , while a l l other fac tors and outputs may be adjusted f u l l y to current p r i c e s . An added important ingredient of the model was the d i s t i n c t i o n between c a p i t a l stock and u t i l i z a t i o n (deprec ia t ion) d e c i s i o n s , the l a t t e r being made in each per iod a f t e r that p e r i o d 1 s s p r i c e s are known. Various s p e c i f i c a t i o n s of the n e o c l a s s i c a l model of expected p r o f i t maximization were estimated and a l l were found to be incons is ten t with the data . An attempt was made to expla in our resu l ts in l i g h t of the much more favourable f ind ings of other re la ted s t u d i e s , and some suggestions f o r fu r ther research were made. When the c a p i t a l d e c i s i o n was l e f t unexplained and producers were assumed to maximize short run p r o f i t s , the s i g n i f i c a n c e of the endogenous rate of u t i l i z a t i o n was found to depend on the measure of output adopted and on the fac to rs inc luded in the production process . When energy was included as a f a c t o r o f product ion and an index o f the rea l value added by c a p i t a l , labour and energy was used as a measure of output* we could not r e j e c t the hypothesis that cap i ta l u t i l i z a t i o n and deprec ia t ion were p r i c e independent. However, even though the s t a t i s t i c a l r e s u l t s obtained were much bet ter than f o r the f u l l model, the model of short run p r o f i t maximization was s t i l l i n c o n s i s t e n t with the da ta , as the convexity condi t ions impl ied by the theory were l a r g e l y v i o l a t e d . FOOTNOTES 122. Chapter II 1. Note that lower case l e t t e r s e, tt), i|>, z . . . , w i l l genera l ly r e f e r to de te rmin is t i c var iab les and upper case 9 , $, Z . . . , to random v a r i a b l e s . Note a lso that given a funct ion h( ), 9h/3a. and 9rhf/9a,.9a. w i l l be denoted by h (or h.) and h (or h . . ) , r e s p e c t i v e l y . l i I J 2. For nonmarginal changes in d i s t r i b u t i o n s , requ i r ing that V[x*($) ;$] be a mean preserv ing spread o f V [ x * ( e ) ; e ] does not lead to any q u a l i t a t i v e r e s u l t s f o r the e f f e c t s of increased v a r i a b i l i t y . A more prec ise formulat ion of D e f i n i t i o n 2 , in terms of d i s t r i b u t i o n f u n c t i o n s , may be found in the Diamond and S t i g l i t z paper. 3. Keeney proves the corresponding r e s u l t in 3 dimensions and Pol lak genera l izes i t to n dimensions. Chapter III 1. The key in obta in ing (10) and (12) below and subsequently the decompositions (13) and (14) is the repeated interchanging of the order of d i f f e r e n t i a t i o n of EV. 2. That i s , TT i s def ined by max EV[x; 0 + bZ] = max EV[x; 0 - TTE;]. x>0 x>0 The d e t a i l e d v e r i f i c a t i o n of the asser t ions beTow fo r unidimensional increases in v a r i a b i l i t y in the context of a consumer model may be found in Epstein (1975b). The proof here i s s i m i l a r and i s omit ted. 3. See Appendix A f o r a proof of these a s s e r t i o n s . 4. This d i s t i n c t i o n is not made c l e a r in the one footnote (p. 342) which Diamond and S t i g l i t z devote to the q u e s t i o n . 123. 5. Let Z k and be random var iab les with zero means, £(1^1^) = 1, and s t o c h a s t i c a l l y independent from e. Consider the new expectat ions where e k and e & are replaced by e k + b Z k and e £ + bZ^ r e s p e c t i v e l y . Then 3Corr(k",£) = 1/2 b=0 9Risk(e k ) " sRiskCe^) adopting the obvious n o t a t i o n . 6. In a s i m i l a r fashion we can prove the fo l lowing propos i t ion which genera l izes the well-known r e s u l t of Prat t (1964) f o r u t i l i t y funct ions of s c a l a r a t t r i b u t e s : Let W(e) and V(e) be u t i l i t y funct ions of the vector e such that the corresponding r i s k aversion measures r^'^ and r^'^ are equal f o r a l l A and £ . Then there e x i s t constants a and b such that W(e) = aV(e) + b. This i s e a s i l y proved as in Kihlstrom and Mirman (1974) i f we r e s t r i c t a t tent ion to o r d i n a l l y equiva lent u t i l i t y f u n c t i o n s , and indeed the key in our proof is to show that i f r^'^ = r^'^ f o r a l l , or at l e a s t " s u f f i c i e n t l y many" A and £ , then W = F (V) . Our argument, however, shows that ord ina l equivalence is an imp l ica t ion of t o t a l l y natural d e f i n i t i o n s and hypotheses in the mult idimensional case . 7. Working through the d e t a i l s of the proof one can show that i f <j> = [$•.,...,4n ) , ^ = ( i j ; 1 , . . . , ^ n ) , <f> Rl and ip. Rl <f> f o r some i Q , 1 2 o then Rl ij). 8. Keeney (1973) demonstrates the use o f c e r t a i n t y equ iva len ts , given r i s k independence and s t o c h a s t i c independence, in the assessment of mu l t i va r ia te u t i l i t y f u n c t i o n s . We are merely s p e l l i n g out the assoc ia ted impl ica t ions fo r the demand f u n c t i o n s . 124. Chapter IV 1. A more rigourous and de ta i l ed exposi t ion may be found in Diewert (1975), Epstein (1975b), Diewert (1973) and Jorgenson and Lau (1974). 2. By proper we mean that F never assumes the value +°° and is greater than f o r at l e a s t one (y^,z ,x) > 0. 3. Diewert 's (1973) proof of the d u a l i t y i s f o r technologies with constant returns to sca le but i t i s e a s i l y modif ied to e s t a b l i s h th is more general d u a l i t y . 4. Roy's Ident i ty in the context of the problem in (1) states tha t , given the required d i f f e r e n t i a b i l i t y , the u t i l i t y maximizing bundle y * ( s ; q ; x ) i s given by y * = - g q _ ( s ; q ; x ) / g s ( s ;q;x) f o r each i . Chapter V 1. This chapter i s based l a r g e l y on Epstein (1975b). But his ana lys is is extended, p a r t i c u l a r l y in sect ion 4, in l i g h t of the resu l ts of Chapter II I . 2. The q u a l i f i c a t i o n "a.e.[y]" or "almost everywhere with respect to p" i s i m p l i c i t in the remainder of the thes is where c a l l e d f o r . 3. It i s not obvious at th is stage that the maximum e x i s t s . However, s ince (2) i s equiva lent to the problem (3) below, where the maximum e x i s t s because o f the cont inu i ty of the va r iab le i n d i r e c t u t i l i t y funct ion and assumption (1) , the maximum is well d e f i n e d . 4. That the consumer w i l l ensure that p*x < I(w) fo r every w in n i s reasonable i f he w i l l s tarve i f he has no savings in the second p e r i o d . The model may be e a s i l y extended so that a form of s o c i a l insurance w i l l prevent him from s t a r v i n g . Under these circumstances aggregate f i r s t per iod expenditure w i l l be bounded by f i r s t per iod income and c r e d i t a v a i l a b i l i t y , and the consumer may v i o l a t e the cons t ra in t p*x < l(w) f o r 125. s u f f i c i e n t l y u n l i k e l y w and thus r i s k going bankrupt and having to s u b s i s t on welfare payments. The model may a lso be extended in other d i r e c t i o n s . For example, uncerta inty in income may be endogenized to allow fo r a simultaneous consumption and p o r t f o l i o d e c i s i o n . A lso the model is s u f f i c i e n t l y general to allow f o r a simultaneous consumption and labour supply dec is ion as i.ncBlpck and.Heineke i(rl972): -tin;, that case , some of the goods are l e i s u r e goods corresponding to the d i f f e r e n t types of labour in the economy, the corresponding pr ices are the appropriate wage r a t e s , and income is made up of nonlabour income plus income der ived from the consumer's stock of leri'sure goods. 5. We consider only i n t e r i o r so lu t ions throughout and we assume that the second order condi t ions corresponding to (4) are s a t i s f i e d . (This w i l l be the case , f o r example, i f g (s ;q ;x ) i s concave in (s;x) f o r f i x e d q ) . The funct ion g is assumed to be s u f f i c i e n t l y smooth so that the d i f f e r e n t i a t i o n ind ica ted in the res t of the chapter may be c a r r i e d out . 6. The von Neumann-Morgenstern independence condi t ion or the "compounding of p r o b a b i l i t i e s " condi t ion is v i o l a t e d by preferences descr ibed by V ( c , S ) . For a more d e t a i l e d and r igorous argument the reader i s asked to r e f e r to Mossin (1969). 7. A study by Betancourt (1973) of the e f f e c t s of uncerta inty about the rate of i n t e r e s t on labour supply and consumption has as an e x p l i c i t ob jec t ive the ana lys is of the problem f o r general intertemporal u t i l i t y f u n c t i o n s , i . e . , f o r intertemporal u t i l i t y funct ions which are not separab le . Unfor tunate ly , the author invokes the H icks ' Aggregation Theorem i n order to j u s t i f y the hypothesis that consumers maximize the expected value of a u t i l i t y funct ion of aggregate ind ices of present and 126. future consumption. Thus his ana lys is is a c t u a l l y v a l i d f o r a fami ly o f u t i l i t y funct ions not very much la rger than the fami ly of separable u t i l i t y f u n c t i o n s . 8. For d e t a i l s on the ro le of homothetic s e p a r a b i l i t y , the existence of quant i ty and p r i c e ind ices and budget d e c e n t r a l i z a t i o n see Blackorby, Primont and Russel l (1975). 9. For a d iscuss ion of the condi t ions under which aggregation over consumers is v a l i d see Diewert (1976). The d iscuss ion i s r e a d i l y extended to incorporate uncer ta inty i f we assume that i n d i v i d u a l s have i d e n t i c a l expecta t ions . 10. More d e t a i l s may be found in Epste in (1975b) and Diewert (1973), the l a t t e r in the context of funct ional forms fo r var iab le p r o f i t f u n c t i o n s . Chapter VI 1. McCall (1971) surveys many of the subsequent c o n t r i b u t i o n s . 2. A corresponding ana lys is of the monopol is t ic f i rm is c a r r i e d out by Leland (1972). 3. Most of Turnovsky's d iscuss ion re la tes to a s i t u a t i o n where i n i t i a l dec is ions are merely production plans which are not completely put in to e f f e c t before the p r i c e is known. However, he states that the same approach may be used to y i e l d s i m i l a r q u a l i t a t i v e resu l ts when i n i t i a l quant i t i es are a c t u a l l y produced before the true s e l l i n g p r i c e is known. It i s to t h i s i n t e r p r e t a t i o n of his ana lys is that we r e f e r g e n e r a l l y . A f t e r the t h e s i s was completed, the author became aware of a paper by Hartman (1976) that uses a formulat ion s i m i l a r to ours in model l ing ex post f l e x i b i l i t y . He considers only a s i n g l e output , two f a c t o r model 127. and h is comparative s t a t i c s r e s u l t s are r e s t r i c t e d to the analogue of our Theorem V . l and to the determination of a m.p .s . given p r o f i t r i s k n e u t r a l i t y . 4. Note that we w i l l often require the s t r i c t concavi ty of g , and hence of F, in x. 5. Instead of assuming that n-j = 1, suppose that x is weakly separable in the technology and hence that the var iab le p r o f i t funct ion has the form g(p,w;<)>(x)). Suppose fu r ther that <j> i s concave and increas ing and def ine the dual cost funct ion C(<J>;q) = min{q-x/c|>(x) > <f>}. Then, from (3) , the x>0 producer solves max EV[g(P,W;cf>) -~C(<j>;q)] and i t i s easy to show that a l l x>0 of the resu l ts to" fo l low in the rest of the paper f o r the case of 1 ex ante f a c t o r apply equa l ly well here to the level of <j> chosen. The s i g n i f i c a n c e of th is i s tha t , f o r example, when the technology i s of the form y = f(z,<j>(x)) + <)>(x), <j>(x) can be thought of as the current production of the s i n g l e output , p r e c i s e l y the model analysed by Turnovsky. On the other hand, when there is no f l e x i b i l i t y in product ion , g(p,w;<|)) = pf(<j>) and the opt imizat ion problem becomes max EV[py - Q ( y ; q ) ] » where Q(y;q) = C( f (y);q) i s the cost funct ion dual to f(<j>(x)). This i s the s i n g l e product , m u l t i f a c t o r , zero f l e x i b i l i t y model considered by Sandmo and Batra and U l l a h . 6. Analogues of the resu l ts to f o l l o w , fo r the case o f f a c t o r p r i c e uncer ta in ty , may be r e a d i l y de r i ved . In e i t h e r case, note the r e l a t i v e s i m p l i c i t y of our proofs compared with those used by Turnovsky f o r his corresponding r e s u l t s . This i s due to our assumption of i n t e r i o r s o l u t i o n s , the -formulation in terms of the v a r i a b l e p r o f i t funct ion and the a p p l i c a t i o n of H o t e l l i n g ' s Lemma. .128. 7. Turnovsky d i d not address himsel f to any of the matters that we consider in sect ions 4 and 5. 8. For example, suppose that dA/du < 0 and g > 0. Then V"- (g -q) — p X — X = - A V ' - ( g v - q ) > - A * V ' - ( g v - q ) , where A * = A(g(p* ,w;x*) - qx*) and p* i s X — A def ined by g x (p * ,w ;x * ) = q , and so E [ V " - ( g x - q ) ] > - A * E [ V • ( g x - q ) ] = 0. 9. I f P is ce r ta in but W s t o c h a s t i c and n^ = 1, we get 3 x * / 3 B < (>) 0 i f (dA/dTr)g > (<) 0. Thus the greater r i s k aversion induced by xyv -l a rger f i x e d c o s t s , given decreasing absolute r i s k a v e r s i o n , increases (decreases) the demand fo r the ex ante f a c t o r i f ex ante and ex post fac tors are gross subst i tu tes (complements). 10. Simply note that g = p g p + w g w . , s g ^ = - : p g « x ; ' + % w g w x and that y(p,w;x) =.g , z(p,w;x) = - g w -11. S i m i l a r l y 3 x * / 3 S h i f t (W) < U) 0, given W s t o c h a s t i c i f : ( i ) R < 1 ( i i ) g x w < (>) o ( i i i ) (12) (a 1 ) y(p,w;x) > z ( p , w ; x ) y ( p , w ; x ) / ^ - z ( p , w ; x ) (b 1 ) x < z(»p,w ; x ) / ^ - z ( p , w ; x ) . Note that in th is case and in the case of condi t ions (13) the asserted i n e q u a l i t y holds even i f both f a c t o r and product pr ices are uncer ta in . 12. The indeterminacy in the general model is analogous to that in a mul t iasset p o r t f o l i o model demonstrated by Cass and S t i g l i t z (1972). 13. The mathematical proof of the convexity o f g - in pritces;7and'the consequent p r i c e r i s k a f f i n i t y , i s s t ra ight forward and i s given in Diewert (1974). I n t u i t i v e l y , i t might be "explained" as fo l lows: i f production were held constant , p r o f i t s would be l i n e a r in p r i c e s . However, roughtly speaking, higher (lower) product pr ices and lower (higher) f a c t o r p r ices induce higher (lower) l e v e l s o f p roduct ion , and so the 1 129. increase in p r o f i t s consequent upon a favourable r e a l i z a t i o n of pr ices i s l a rger than the f a l l in p r o f i t s consequent upon an equa l ly unfavourable r e a l i z a t i o n . Convexity and p r i c e r i s k a f f i n i t y r e s u l t . 14. This i s because we are comparing e l a s t i c i t i e s and r i s k measures assuming a p o s i t i o n of long run e q u i l i b r i u m . A proof of the Le C h a t e l i e r P r i n c i p l e using the var iab le p r o f i t funct ion may be found in Diewert (1974). Indeed Diewert proves a mult idimensional vers ion of the P r i n c i p l e which shows that the expected value of information is a lso p o s i t i v e f o r mult idimensional r i s k s . 15. In f a c t , from Theorem II.1 these condi t ions imply that output dec l ines given any (nonmarginal) mean preserv ing spread. However, the corresponding asser t ion in Theorem 5 is v a l i d only f o r marginal changes. 16. Unfor tunate ly , we have not been able to extend th is r e s u l t , or provide a counterexample when f i x e d costs are i d e n t i c a l l y zero . Note that i f the technology s a t i s f i e s "Vx, p 3q such that x = x ( p , q ) " , then Theorem II I .3 may be strengthened by de le t ing the "uniformly s igned" hypothes is . It i s th is stronger vers ion that we invoke here and below. 17. A f r o n t i e r descr ibed by F ' [ x 2 , -TTT"] = 0 can be "solved" fo r a(x) and J L ct(x) 2 2 2 expressed as a(x) = G[x , y ] . It fo l lows that G[x , Xy] = Xa(x) = xG[x ,y ] and so G i s l i n e a r homogeneous, and hence homothetic, in y . Conversely , i f h i s a monotonic funct ion such that h(G['x ,y]) i s l i n e a r homogeneous 2 2 v i n y , then G[x ,y ] = H(x) i f and only i f G[x , -7*T T ] = 1. where a(x) = h ( H ( x ) ) / h ( l ) . Turning to the Cobb-Douglas, note that a . i f and only i f 130: where n a. ei a(x) = n X ] , a. = j ^ — ^ V - k ) F i n a l l y , F [y ,x ] = 0 i f and only i f F [ y / x , 1] = 0 i f there are constant returns to sca le and x i s a s c a l a r . 18. For the proofs of Theorems 11 and 12 below i t i s s u f f i c i e n t to assume that the hypothesized i n e q u a l i t i e s are v a l i d outside of some bounded r e g i o n , rather than g l o b a l l y . 19. This appl ies to the Cobb-Douglas technology o n l y . 20. This fol lows from Theorem II.1 by the convexity of g v in (p,w). A 21. Fuss a lso permits the technique to be va r iab le ex ante , whereas our approach does so only i f there is a f ixed one-one correspondence between the technique and the ex ante fac tors chosen. Fuss 1 approach i s d iscussed fu r ther below. Chapter VII 1. We have been able to f i n d only one other empir ica l study in the l i t e r a t u r e that takes uncer ta inty in to account in est imat ing technology, namely Harkema and van der Loef f (1976), which fol lows up the e a r l i e r d iscuss ion o f Z e l l n e r , Kmenta and Dreze (1966). In t h e i r model, i n d i v i d u a l producers view the technology as s t o c h a s t i c because o f unpredictable v a r i a t i o n s in the weather and in machine or labour performance, f o r example. Expected p r o f i t maximization i s assumed to be the ob jec t ive of f irms and a CES production funct ion i s est imated. 2. Harkema and van der Loef f (1976) go even f u r t h e r . They d isregard cap i ta l stock data completely and use energy consumption as a proxy fo r c a p i t a l s e r v i c e s . 131. 3. They f requent ly do, however, d i s t i n g u i s h between the stock and flow dec is ions with respect to the labour input as w e l l , which we do not . 4. Lucas (1970) and Winston and McCoy (1974) have emphasized the importance of rhythmic f a c t o r pr ices ( e . g . overtime wage premiums) as the major determinant of the degree of c a p i t a l u t i l i z a t i o n , to the exc lus ion of deprec ia t ion c o s t s . 5. Maccini (1973) has inves t iga ted the e f f e c t s of d e l i v e r y lags on the demand fo r investment in the context of a d e t e r m i n i s t i c , cost of adjustment model. Note that i f we i n t e r p r e t the lag in our model to be a d e l i v e r y l a g , then we are assuming that cap i ta l goods ordered in ( t -1) are pa id fo r at the market p r i c e that i s eventua l ly r e a l i z e d in t . 6. The formula may be found in Kendall and Stuar t (1963), 168-9. 7. Some favourab le , though f a r from c o n c l u s i v e , evidence was found f o r t h i s hypothesis in the recent experimental study of the formation of random expectat ions by Schmalensee (1976). 8. This use of r a t i o n a l i t y in the context of random expectat ions i s borrowed from Boonekamp and Davidson (1976). 9. The part of the funct iona l form without the time trend cannot be wr i t ten as compactly as VI.(26) when K and q are v e c t o r s , and so the l a t t e r was def ined in VI . 10. Note that our model provides a natural d e f i n i t i o n of the existence of an index of c a p i t a l s e r v i c e s , namely the weak s e p a r a b i l i t y of K and K in F; i . e . , F [ z , K , K ] = f ( z , J ( K , K ) . If^J i s ;linear' t'homdgehedusr'the r e n t a P p r i c e •< < of the c a p i t a l se rv ice may be def ined as a(q,q) = min {qK-qK/J(K,K) > 1} , K,K though ne i ther the rental p r i c e nor the index J is re levant to a problem where K i s chosen ex ante and K ex p o s t . I f z = (L ,E) where L denotes 132.. labour and E energy, E may be used as a proxy f o r c a p i t a l se rv ices i f and only i f there is zero s u b s t i t u t a b i l i t y between E and J in f . However there are two problems in t e s t i n g these hypotheses: f i r s t , they are not r e a d i l y expressable in terms of the va r iab le p r o f i t funct ion i T (p ,w ;q ;K ) and we would have to reformulate the model in terms of T T ( P , W ; K , K ) , which complicates the es t imat ion . Secondly and more important ly , B lackorby , Primont and Russel l (1975) have recent ly demonstrated the l i m i t e d f l e x i b i l i t y of funct iona l forms such as VI . (26) and VII . (12) in model l ing s e p a r a b i l i t y . It would appear to be extremely d i f f i c u l t to test the hypothesis of weak s e p a r a b i l i t y i s o l a t e d from other r e s t r i c t i v e condi t ions and in view of i t s l i m i t e d relevance to a temporal dec is ion problem, we w i l l not consider the matter f u r t h e r . 11. By assuming that technique as well as c a p i t a l stock is var iab le ex ante, Fuss and McFadden (1971) der ive funct iona l forms f o r demand and supply funct ions that are l i n e a r in the parameters o f the technology. T h e i r approach requires est imat ing s i g n i f i c a n t l y more parameters than does ours . 12. See Go ld fe ld and Quandt (1972), fo r example. 13. Diewert (1976b) d iscusses the j u s t i f i c a t i o n of the use of D i v i s i a i n d i c e s . 14. For a more d e t a i l e d d iscuss ion see Parks (1971), Berndt and Wood (1975) and e s p e c i a l l y Diewert (1975). 15. See footnote 10. 16. 6^ . ddld not vary g r e a t l y , going from .102 to .111. 17. These estimates are based on pr ices from 1949 to 1970. It was f e l t that the s t o c h a s t i c process would bet ter approximate expectat ions during the sample per iod i f we excluded the h igh ly e r r a t i c p r i c e behaviour in the ear ly postwar,years and in 1971, at the s t a r t of the recent boom in energy p r i c e s . 133. 18. The s i n g l e exception was fo r the expectat ion formed in 1947 of the 1948 c a p i t a l stock p r i c e . The process in (19) would requi re the 1946 p r i c e . Instead we assumed that th is i n i t i a l expectat ion was based on an estimate of a c o n s t a n t - r a t e - o f - i n f l a t i o n model of asset pr ices during 1947-71. 19. A geometric rate of deprec ia t ion was imposed. Note a lso that when deprec ia t ion is endogenous (exogenous), the demand f o r investment (cap i ta l stock) is the dependent va r iab le in the " c a p i t a l " equat ion. 20. We were not qui te able to a t ta in convergence in th is model, but our numerous attempts d id convince us that we were not f a r from the minimum. These r e s u l t s , however, should not be viewed as f i n a l . 21. In empir ica l consumption s t u d i e s , the quant i ty income or t o t a l expenditure i s usua l ly one of the var iab les exp la in ing the consumption of each good. 22. Lau (1974) has discussed procedures f o r t e s t i n g f o r quasi convexity and convexity and f o r est imat ing with such const ra in ts imposed. However, they are computat ional ly d i f f i c u l t and are beyond the scope of th is t h e s i s . 23. This fol lows from the f a c t that when Model 11 is modif ied by imposing a . . = 0 f o r i , j = 1,2, the l i k e l i h o o d funct ion is only -140.58. 24. 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For each p o s s i b l e value e of 0 d iv ide the change L 2 AC in to e rjy> 6 + jr r (x* ;e) -£ "[Try*9 + b z - T h e n i n t n e l i m i t as b -> 0 we have: (a) The e f f e c t on x* of ( i ) i s equal to E ( r A > 5 \ - ± - (£ E% V £ ) ^ - ) . k k J2» (b) ( i i ) def ines a marginal mean u t i l i t y preserv ing spread in expecta-- D.. t ions whose e f f e c t on x* equals j[ E , ^ ( £ E^ V^) rA ' ?J-j!p . Proof : (a) i s e s t a b l i s h e d by s u b s t i t u t i o n of ( i ) in to the f i r s t order condi t ions (9) and d i f f e r e n t i a t i o n . It fo l lows that the impact of ( i i ) i s the r e s i d u a l , 9x * / 3 R i s k ( A ) - the impact o f ( i ) , which by (14) is the des i red r e s u l t . It remains to prove, t h e r e f o r e , that ( i i ) const i tu tes a mean u t i l i t y preserv ing spread. Let x(b) solve max EV(x; 0 - l£ r A' S(0 ; x*k) . x>0 The necessary condi t ion concerning the cond i t iona l expected values w i l l be s a t i s f i e d as b 0 i f 2 2 E[V[x(b) ; 0 + bZ ] /V [x (b ) ; 0 - ^ r A ' C ? ] ] = V[x(b) ; 0 - | j - rA'h] , .2 f o r a l l poss ib le values of V[x(b) ; 0 - jr r £]> where e q u a l i t y , both here and below, i s understood to hold to the second order in b. This equation is impl ied by 143. E[V[x(b) ; e + bZ] /e ] = V[x(b) ; 0 - ^ r A ' ^ ] f o r a l l e, and so we need only prove the l a t t e r . This is r e a d i l y done — b _ Q = 0 imply that the f i r s t order terms in b vanish on both s i d e s . There fore , the l e f t hand s ide is equal to an. s ince E[Z/e] = 0 and T T - 1 V[x* ;e ] + 2! (_ i x i 3b^ + I E(Z b=0 i , j iz j ) w i o5 and the r igh t hand s ide i s equal to V[x* ;e ] + • I i x i 9b^ b=0 k j The two s ides are thus equal by the d e f i n i t i o n of r A ' ? Theorem III .1 : That e' UI e J impl ies e' Rl e J i s e a s i l y v e r i f i e d . To prove the converse, suppose f o r s i m p l i c i t y that e' = (e-|, 0 2) i s two-dimensional , e J = e 3 i s a s c a l a r , and 0 = (e-|,e 2,e 3), though the proof is e a s i l y extended to the general case . Now 80. 2 2 5 k W k k=l K K = 0 i f and only i f I , J , K. •144. This equa l i t y is v a l i d f o r a l l admissible A and K i f and only i f W.. Wk W i j 3 - W..^ = 0, i . e . , ^ - = 0 , f o r a l l i , j , k = 1,2. F ix ing 3 k • i and j and apply ing th is e q u a l i t y in turn f o r k = 1,2, we see that 3 W l TQ~ W^~~^ = ®' ^ n u s e 3 1 S w e a ' < l y separable from (e-j,62) and hence (Goldman and Uzawa (1964, Theorem 2) W can be expressed in the form W(e) = F [ f ( e i , e 2 ) > 3 W i i 3 F l l Now T T — (n-^) = 0 impl ies that T T — i f — = 0» and hence, by Theorem 1, " 3 w k 3 e 3 F l FC^ .e^ ] = a (e 3 ) + b(e3)A(<f>), which proves that (e - j^) i s u t i l i t y independent of 6 3 . Theorem 111.2: Apply Theorem 3 and the c o r o l l a r y on page 33 of Keeney (1973) Theorem III .3 : We need only prove the l a t t e r a s s e r t i o n . By hypothesis f o r a l l i , which, by the s t r i c t concavi ty of V in x impl ies that f o r a l l i , A and £ . Since I s V is s t r i c t l y p o s i t i v e and 3r A ' ^ ( x * ; e)/3x. i s uniformly signed with respect to e, i t fo l lows that 3r ' ^ (x* ;eyax . = 0 "almost everywhere" with respect to e. Suppose now that V is such that f o r each x and e there e x i s t s a random v a r i a b l e 0 such that x i s optimal given 0 and e is a poss ib le r e a l i z a t i o n of 0 with nonzero p r o b a b i l i t y . For such funct ions V i t fo l lows that e Rl x. (Note, the re fo re , that the second asser t ion of the theorem i s v a l i d only f o r such u t i l i t y f u n c t i o n s . This 145: does not appear to be a s i g n i f i c a n t r e s t r i c t i o n , however.) Theorem II I .4: (a) has been proved above. (19) impl ies that ^ is independent o f x and hence that x* solves max V(x; 0 - T T^(0)C ) , proving (b) . x>0 Theorem III .5: For s i m p l i c i t y suppose that e = (<j>,^ ) i s two-dimensional . A s l i g h t extension of Theorem 2 shows that the above r i s k independence assumptions are v a l i d i f and only i f V has the form V(x;<M) = c Q ( x ) + c ^ x ^ U ) + c 2 ( x ) v 2 U ) + c 3 (x)v-, (<f>)v2(*). From th is i t fo l lows that n.. = T T ^ (Q. ) . Now EV(x;0,w) = c (x) + c-j (x)v-^ ($-TT-| ) + C 2 ( X ) V 2 ( T - T T 2 ) + c 3 ( x )v 1 ( 5 - T r ^ - V g f y - T T 2 ) = V(x; * - T ^ , $ - T T 2 ) , by the assumed s t o c h a s t i c independence of $ and The desi red r e s u l t f o l l o w s . APPENDIX B: PROOFS OF THEOREMS IN CHAPTER V Theorem V . 1 : (a) It fo l lows from Theorem 1 of Hanoch (1974) that q RI s impl ies — ( - s g c c / g _ ) = 0. But thn'is i s i n c o n s i s t e n t with s RI s , i . e . , OA O O O W Ks/'s) = °- I (b) S u f f i c i e n c y of the funct iona l form i s c l e a r . On the other hand, s RI (s ,x) impl ies that - g s s / g s = r (q) . g (s ;q;x ) homogeneous o f degree zero in (s ,q) impl ies that r (q) i s homogeneous o f degree -1 in q . A two- fo ld in tegra t ion of the d i f f e r e n t i a l equation produces the desi red funct iona l form with the homogeneity proper t ies o f a and 6 i n h e r i t e d from the" homogeneity o f g . 146. (c) Again s u f f i c i e n c y i s c l e a r . From Theorem I I I . l and the zero homogeneity of g in (s ,q) i t fol lows that q RI (s ,x) i f and only i f g(s ;q ;x ) = a(x) + g ( x ) h ( s , q ) , where q RI s in h ( s , q ) . By Theorem 1 and C o r o l l a r y 3 of Hanoch (1974), q RI s impl ies that R E - s h s s / h s = R(q) , and; i f R(q) f 1 , alql -ll-R(q) b(q) Where R(q) and b(q) are zero-homogeneous in q and a(q) i s p o s i t i v e and l i n e a r homogeneous in q . But t h i s funct iona l form i s i n c o n s i s t e n t with q RI s and the funct iona l form of Theorem I I I . l unless R(q) i s a constant f u n c t i o n , -,1-R and b(q) i s a constant or i s a constant mul t ip le of 1 1-R iTql The . latter i s i n c o n s i s t e n t with the homogeneity proper t ies of a(q) and b(q) . Applying a s i m i l a r argument f o r R=l, we a r r i v e at 1 1-R n l - R iT[qT + c , R i 1 h(s ,q ) c log ifqT Theorem V . 3 : The proof i s i d e n t i c a l to that of Hanoch's Theorem 4. Simply note that >the measure of r i s k aversion r s can be expressed as uu (u;q;x) e u (u;q;x) . ~ and'so<ivej_the d i f f e r e n t i a l equation e11M(u;q;x) uu e u (u;q;x) = rs(u,q) Theorem V..4: -The proof i s i d e n t i c a l to Theorem V.3 a f t e r observing that A - f m i (u's'q-:>x) given (S • ) , r -1 - - - 1 ' •• - ^ - u u J = r J ( u , q . ) = —pr ^ J ( u , s , q . , x ) u 147. APPENDIX C: PROOFS OF THEOREMS IN CHAPTER VI Theorem VI .1 : (a) V" < 0, g > 0 and (5) imply that q = E l " V ' • 9 x ( p » w i x * ) J xp - E V , < E[g (P ,w;x* ) ] . I f g v i s concave in p, we get that q < g v ( P , w ; x * ) . ~ A A — X But q = g v (P ,w;x) and so g v (P ,w;x) < g v ( P , w ; x * ) . There fore , g v v < 0 impl ies A A — X XX ™ that x* < x. (b) i s proved s i m i l a r l y . Theorem VI .2: (a) By H o t e l l i n g ' s Lemma, E[y* ] = E[g (P,w;x*)] < g (P,w;x*) < 9 p ( P » w ; x ) = y . S i m i l a r l y fo r (b) . E [ V ' . g x ( P , w ; x * ) ] Theorem VI .3 : (a) E [ g x ( P , w ; x 0 ) ] = q = ^ < E [ g x ( P , w ; x * ) ] , given that V" < 0 and g > 0. il iherefore, g < 0 impl ies that x° > x * . — px — XX — ~ S i m i l a r l y fo r (b) . Theorem VI .4: (a) E [y* ] = E [g p (P ,w;x * ) ] < E [ g p ( P , w ; x 0 ) ] = E [ y 0 ] . S i m i l a r l y f o r (b) . Theorem VI .5: The e f f e c t o f a mean u t i l i t y preserv ing spread has the opposite s ign of 3X 3X -g 9 p v The l a t t e r is equal to _3_ 3X f^Ss) + g A + g (g -q) ^ = ± - ( ~ ^ \ \ % J V V y x H ; dir 3x V, g / + a — 9 p x drr + ^ [g (g -q) - -n-g ] d-rr L y p v y x H ' y p x J 148. The r e s u l t fo l lows noting that 9 p ( V q ) " *9px = [ gp gx " "px ] + q ( x gpx " V and r e c a l l i n g the argument fo l lowing (11). From I I I . (11) i t fo l lows that the e f f e c t of a mean preserv ing spread has the s ign of £ J V[g(Psw;x) - q-x] = V ' g p p x + g p p ( g x - q ) V " + 2 V ' « g p g p x + gp (g x -q )V '"= V ' g p p x - A g p p V ' ( g x - q ) + g p g x p [ 2 V " + .V" ] • + V " ' g p [ g p ( g x - q ) - * g x p ] = V ' g p p x - A g p p V (g x-q) + VxP[-v' d V " v"(R-i)] + V " g p [ g p ( g x - q ) - , g x p ] . Noting that dA/dir > 0 impl ies V " > 0, each term, with the exception of the second, i s negative in s i g n . That the s ign of the l a t t e r is a lso negative can be shown by noting that — ( A - g p p ) < 0 and apply ing the Sandmo argument o f footnote 8. Theorem VI .6: Apply Theorem VI.5 to the var iab le p r o f i t funct ion g (p»w;x ) = p f ( x ) , not ing that such a technology possesses property (12). Theorem VI .7 : The proof i s s i m i l a r to that of Theorem V I .5 . 149. Theorem VI .8: Suppose fo r s i m p l i c i t y that x i s a s c a l a r . Then p^ RI x f o r any component p^ of p impl ies that _8_ 8X P-P' = A ( T r ) g + g d 4 ^ - (g -q) v ; yp..x y p ^ fa v y x H ' I f t h i s equation is v a l i d f o r a l l B, the to ta l de r iva t i ve _d_ dB = J L _ 1 _ L B-qx=constant 3 8 x 3 q may be appl ied to each s ide of the equat ion , y i e l d i n g dA/dir = 0. As in the proof of Theorem 3, r i s k independence impl ies that x i s weakly separable from p in g . There fore , g(p;x) = F [ f ( p ) , x ] and p RI x impl ies fu r ther that _9_ 3X f f A F, So lv ing th is d i f f e r e n t i a l equation we get g(p;x) = F [ f ( p ) j x ] = - ^ log [a(x) - Ab(x)c(f(p)>)] But th is funct iona l form cannot possess the homogeneity proper t ies of a var iab le p r o f i t f u n c t i o n , e s t a b l i s h i n g the i m p o s s i b i l i t y . Theorem VI .9: (a )=^(b) : By Theorem 3, g(p;x) = a(x)h(p;x ) + B(x). But g l i n e a r homogeneous in p impl ies that 3(x) = 0 i d e n t i c a l l y . (Applying 2 2 E u l e r ' s equation to g , we obtain h(p;x ) - £ p ^ (p;x ) = -e(x)/a(x) = . c , 2 1 a constant , fo r f i x e d x . Therefore h i s homogeneous of degree c+1 in p and so Aa(x)h(p;x 2 ) + xe(x) = A C + 1 a ( x ) h ( p ; x 2 ) + B(x), which impl ies c = 3(x) = 0.) 150: ^ ^ 2 2 (b)<S=^(c): Simply note that a (x)h(p;x ) = h (a (x )p ;x ). Since mu l t ip l i ca t ion of pr ices i s "dual" to the d iv i s ion of quant i t ies , the equivalence fo l lows. (b) =^ (a ) : Apply Theorem I I I . l . P 2 Theorem VI.10: By the l i near homogeneity of g(p), g(p-|,p 2) = p^h(—). I t fol lows that p 2 Rl p-j i f and only i f h has constant re la t i ve r i sk avers ion, i . e . , r a y 1 _ a + b, a f 1 h(y) = j l a l n y + b, c* = 1 Some manipulation shows that p-| Rl p 2 i f and only i f b = 0. But the resu l t ing funct ional form cannot sa t i s f y the monotonicity and convexity propert ies of a p ro f i t function i f both prices are product p r i ces . I f both are factor p r i ces , then necessar i ly g = -ap^1 P 2 ~ a , a > 0, 0<Ca><l, while i f there i s one factor and one product g = a p j + o t p 2 ° l , a , a > 0. Theorem VI.11: (a) by homogeneity and convexity, Vi 3 j such that g. • < 0. But g. > 0 and g . . < 0 imply that g . . . > 0 " f requent ly" , (because a decreasing function that i s bounded below cannot be concave in any "neighbourhood of °°"). Therefore, g,-g-ji - g- • g-n- i s " frequently" pos i t i ve J ' J J J J ' J and hence p. Rl p. i s impossible. (b) Suppose i and j are products and that b._- < 0. Then g . . . , q . . q . . 1 J 1 J J g.j. and hence 7 ^ — ( j ^ ) a n c l (g~T~) a r e " " f r e c l u e n t l y " pos i t i ve . I f i i s a "1 3 3 factor and j a product, g. < 0 and g . . > 0 imply that g . . . < 0 " f requent ly" , I I J I J J (because an increasing function that i s bounded above cannot be convex in any •' 9 g i i "neighbourhood of °°"), and hence 9j9jj-j ~ 9jj9-jj a n c * g p - ( g ~ ^ a r e "f r e 9 u e n t l y " 3 negative. Theorem VI .12: The proof is i d e n t i c a l to that of Theorem VI . APPENDIX D: DATA FOR U.S. MANUFACTURING, 1947-71 TABLE D . l : PRICE AND QUANTITY DATA* y G p G -1 w l z 2 w2 M y * * p** p*** I q 1947 256 .73 .76423 75 .486 .59741 10.808 .71768 159.51 .75359 86.632 .78771 97.433 .78000 11.015 .62920 1948 239 23 .82433 73 600 .68975 10.045 .93484 141 28 .79522 87.659 .86090 97.697 .86855 10.310 .67033 1949 249 96 .81322 69 996 .69051 11.024 .85880 150.09 .80050 88.701 .83045 99.764 .83325 8.0400 .69050 1950 284 64 .83173 74 485 .73801 11.718 .87156 171 68 .84726 101.12 .80173 112.84 .80898 7.6490 .72272 1951 307 88 .90771 81 619 .79924 12.770 .89838 181 39 .91707 113.76 .89345 126.52 .89409 9.9710 .79794 1952 319 39 .89845 85 603 .82412 12.858 .91805 187 29 .90401 119.32 .88706 132.15 .89025 9.8220 .81601 1953 352 13 .89451 91 156 .85703 13.902 .91508 207 95 .89710 130.35 .88773 144.22 .89057 9.8530 .83140 1954 323 61 .90965 85 861 .86841 14.043 .93554 188 46 .90891 121.15 .90751 135.19 .91041 10.019 .83211 1955 363 49 .92321 90 555 .90281 14.480 .96368 211 06 .93320 138.07 .90290 152.48 .90906 10.043 .85984 1956 368 19 .96373 93 379 .94502 15.260 .98433 215 36 .97466 137.60 .94410 152.84 .94827 12.236 .92998 1957 369 20 .98778 93.591 .98358 16.480 .99047 216 47 .98496 136.25 .99188 152.73 .99173 12.467 .98759 1958 343 82 1 .00000 88 209 1.00000 15.635 1.00000 200 02 1.00000 127.72 1.00000 143.36 1.00000 9.724 1 .00000 1959 381 32 1 .0057 94 456 1.0361 16.664 .98147 225 31 .98549 139.41 1.0408 156.08 1.0344 8.800 1 .0148 1960 388 16 1 .0097 95 383 1.0651 16.951 .99058 224 56 1.0094 146.58 1.0128 163.56 1.0103 10.009 1 .0221 1961 388 63 1 .0064 93 765 1.0872 17.202 .98774 222 88 1.0122 148.48 1.0003 165.72 .99876 9.6770 1 .0239 1962 419 88 1 .0058 98 089 1.1263 17.909 .98817 237 05 1.0154 164.92 .99403 182.87 .99321 10.209 1 .0368 1963 422 08 1 .0091 99 785 1.1553 19.053 .96698 254 55 1.0034 168.48 1.0227 187.56 1.0169 11.010 1 .0423 1964 469 88 1.0061 102 39 1.2008 19.101 .99735 263.17 1.0188 187.62 .98920 206.79 .98961 12.635 1 .0587 1965 509 32 1 .0188 108 29 1.2279 19.639 .99496 286.05 1.0365 203.61 .99635 223.33 .99589 15.519 1 .0802 1966 554 89 1 .0266 115 96 1.2751 20.809 1.0055 303 08 1.0692 231.59 .97015 252.46 .97283 18.362 1 .1123 1967 557.03 1 .0541 117 44 1.3180 22.036 .99899 311 31 1.0733 223.79 1,0323 245.89 1.0290 17.704 1 .1512 1968 590 41 1 .0748 121 03 1.3972 22.589 1.0291 332.39 1.0813 235.45 1.0699 258.12 1.0660 17.212 1 .1796 1969 610 85 1 .1060 124.33 1.4721 23.766 1.0513 336.03 1.1557 251.54 1.0426 275.40 1.0430 18.309 1 .2330 1970 584 52 1 .1350 119 19 1.5565 25.886 1.0471 324. 18 1.1662 234.76 1.1001 260.67 1.0948 16.966 1.2977 1971 599 75 1 .1711 115. 38 1.6490 24.946 1.1819 349.10 1.1679 225.56 1.1757 250.52 1.1762 14.876 1 .3571 •Quant i t ies are in b i l l i o n s of 1958 d o l l a r s . **Refers to value added by capi ta l and labour. * * *Refers to value added by c a p i t a l , labour and energy. . . - ; jjj 153. TABLE D.2: TAX VARIABLES AND "EFFECTIVE" PRICES OF "NEW" AND "USED" CAPITAL V TC TP DUM D q q 1947 .380 0 .01620 0 .36784 .90968 .81824 1948 .380 0 .01480 0 .37182 .97787 .87956 1949 .380 0 .01480 0 .37454 1.0428 .87730 1950 .420 0 .01570 0 .37938 1.2444 .95094 1951 .508 0 .01510 0 .37452 1.4265 1.2543 1952 .520 0 .01460 0 .37524 1.4189 1.2782 1953 .520 0 .01530 0 .37594 1.3062 1.1753 1954 .520 0 .01560 0 .52497 1.2879 1.1581 1955 .520 0 .01640 0 .53766 1.3298 1.1953 1956 .520 0 .01690 0 .55524 1.4470 1.3009 1957 .520 0 .01640 0 .56916 1.5123 1.3596 1958 .520 0 .01620 0 .58170 1.4782 1.3286 1959 .520 0 .01640 0 .60615 1 .5013 1.3491 1960 .520 0 .01660 0 .60632 1.5088 1.3554 1961 .520 0 .01710 0 .60603 1.4669 1.3167 1962 .520 .02708 .01780 1. 0576 .60876 1.4832 1.3311 1963 .520 .03334 .01790 1. 0592 .60780 1.3804 1.2888 1964 .500 .03519 .01830 0 .65162 1.3736 1.2798 1965 .480 .03942 .01840 0 .65054 1.4118 1.2658 1966 .480 .03840 .01760 0 .64396 1.4588 1.3083 1967 .480 .03800 .01720 0 .64416 1.5702 1.2782 1968 .528 .04190 .01780 0 .64956 1.6367 1.4680 1969 .528 .03003 .01810 0 .64763 1.7012 1.6423 1970 .492 .01402 .01780 0 .64762 1.7056 1.5646 1971 .480 .02356 .01800 0 .68297 154. TABLE D.3: CAPITAL STOCK SERIES ESTIMATED FROM PUBLISHED DATA* Capi ta l stock assuming K uniform deprec ia t ion rate o f 10.7% 1947; 54.0 54.0 1948 58.8 58.5 1949 60.8 60.3 1950 62.2 6.1 &5 1951 65.7 64.9 1952 68.6 67.8 1953 71.2 70.4 1954 73.7 72.9 1955 75.9 75.1 1956 80.0 79.3 1957 83.9 83.3 1958 84.6 84.1 1959 84.4 83.9 1960 85.3 84.9 1961 85.9 85.5 1962 86.9 86.6 1963 88.5 88.3 1964 91.6 91.5 1965 97.2 97.2 1966 105 105 1967 111 112 1968 116 117 1969 121 123 1970 125 127 1971 126 128 * In b i l l i o n s of 1958 d o l l a r s . APPENDIX E: SOME ESTIMATION RESULTS TABLE E . l : PARAMETER ESTIMATES FROM SELECTED MODELS (WITH LABOUR THE ONLY VARIABLE FACTOR IN THE SHORT RUN)* Model 1 Model 2 Model 3 Model 4 Model 5 Model 7 Model 8 Model 9 Model 10 a 0 -.7438 (.1870) 0 0 -.0083 (.1043) 0 0 -.9472 (.2263) 0 a' .8930 1.949 (.2852) .8930 .8930 .8514 (.1449) .8930 .8930 2.309 (.3691 ) . .9326 (.0232) c 1.158 (.2558) -1.897 (.7991) .4482 (.3527) .4058 (.2629) 1.233 (.3709) .0385 (1.670) 1.491 (.0935) -2.246 (.7804) -.4935 (.6057) d 2.062 (1.014) 12.98 (2.277) 1.372 (.0489) 2.108 (.0822) 1.158 (.1044) 12.08 (12.21) 0 16.06 (3.697) 14.16 (4.445) a l 0 .0876 (.0468) 0 0 .0561 (.0536) 0 0 .0869 (.0437) 0 a i 1.267 (.1604) -.1186 (.3020) .8584 (.1818) .2236 (.2258) 1.154 (.2208) 1.160 (.5221) 1.252 (.0289) -1.071 (.5328) .6112 (.4250) b i -.0723 (.1322) 8.842 (1.563) .0110 (.0038) .0191 (.0029) .0013 (.0248) 1.866 (3.795) 0 .13.95 (2.160) 3.918 (2.608) a l l .2584 ' (.1335) -.3703 (.1793) .0917 (.0525) .4237 (.0798) -.0360 (.0743) -.0511 (.0363) 0 .0060 (.0622) -.0031 (.0378) bn -2.143 (.9781 ) 2.294 (1.0992) -.0292 (.0049) -.0113 (.0019) -.0893 (.0256) 0 0 0 0 Y i .0240 (.0072) .1223 (.0382) .0833 (.0189) .0325 (.0091) .0958 (.0093) .0471 (.0303) .0232 (.0061) .2098 (.0547) .0576 (.0179) Y2 -.0136 (.0055) .0478 (.0152) .0115 (.0082) .0166 (.0076) .0242 (.0066) -.0106 (.0100) -.0095 (.0019) .0662 (.0183) .0029 (.0137) Y 3 0 -.0055 (.0035) 0 0 -.0063 (.0010) 0 0 -.0113 (.0038) -.0024 (.0026) p l .5928 (.1109) .9773 (.0397) .5856 (.0780) .4304 (.0689) .4113 (.0763) .4796 (.1457) .4530 (.1292) .5780 (.1659) .4581 (.1345) p 2 .4711 (.0800) .5272 (.1499) .4092 (.0561) .4437 (.0453) .3142 (.1015) .3164 (.0985) .3265 (.1172) .1585 (.1850) .3952 (.1095) p 3 .9947 (.0039) .9886 (.0097) 1.029 (.0447) .9498 (.0188) .9692 (.0717) * The asymptotic standard errors of the estimates are in parentheses. Those parameter values without standard errors are Imposed rather than estimated. Imposed throughout is b = b 1 = b 1 = 0. 156. TABLE E.2: PARAMETER ESTIMATES FROM SELECTED MODELS (WITH • LABOUR AND ENERGY VARIABLE FACTORS IN THE SHORT RUN) Model 11 Model 12 Model 13 Model 14 a c d a n b 2 a l l a 12 a 22 Y l Y 2 Y 3 p l P 2 p 3 p 4 0 .8930 -.2353 15.81 0 0 1.220 -.0513 1.896 1.621 -.0419 -.0332 .0523 .0533 -.0121 .0027 0 .5114 .3139 .6567 (.5374) (3.792) (.3939) (.0827) (2.651) (.5941) (.0392) (.0198) (.0169) (.0118) (.0082) (.0019) (.1359) (.0960) (.1400) 0 .8930 1 .996 0 0 0 1.520 1 .722 0 0 -.0460 -.0427 .0632 .0125 -.0178 -.0017 0 .5875 .3582 .5132 (.2166) (.1386) (.0198) (.0421) (.0206) (.0170) (.0092) (.0050) (.0010) (.1184) (.0938) (.1447) -.0513 .9648 1 .783 0 .0402 .0007 1.179 .2084 0 0 .0314 -.0614 .0655 -.0174 -.0236 -.0056 .0014 .3198 .3962 -.0004 .0643) ,1048) .1873) .0203) .0022) .1235) .0177) .0431) .0169) .0191) .0126) .0062) .0010) .0012) .1130) .1022) .2135) 0 .9251 1.679 0 0 0 1.220 .1879 0 0 .0343 -.0610 .0721 .0018 -.0169 -.0040 0 .4048 .3613 .2718 (.0063) (.1756) (.1278) (.0152) (.0399) (.0155) (.0134) (.0065) (.0041) (.0008) (.0978) (.0840) (.1213) * Asymptotic standard e r rors are in parentheses. Those parameter values without standard e r rors are imposed rather than est imated. Imposed throughout i s b = b' = b, = b . . = 0 fo r i ,j = 1,2. TABLE E .3 : CAPITAL STOCKS AND DEPRECIATION RATES IMPLIED BY SELECTED MODELS* Model 9 Model 10 Model 13 Model 14 Cap i ta l Depreciat ion. Stock Rate Capi ta l Depreciat ion Stock Rate Cap i ta l Deprec ia t ion Stock Rate Cap i ta l Deprec ia t ion Stock Rate 1947 54.0 .079 54.0 .070 54.0 .079 54.0 .075 1948 60.0 .073 60.5 .072 60.0 .074 60.2 .075 1949 63.6 .109 64.2 .074 63.6 .074 63.8 .075 1950 64.3 .201 67.1 .077 66.5 .074 66.6 .075 1951 61.4 .322 71.9 .079 71.6 .068 71.6 .075 1952 51.4 .353 76.0 .082 76.5 .067 76.1 .075 1953 43.1 .303 79.7 .084 81.2 .066 80.2 .075 1954 40.1 .286 83.0 .086 85.9 .064 84.2 .075 1955 38.7 .324 85.9 .089 90.4 .062 88.0 .075 1956 38.4 .359 90.5 .091 97.0 .059 93.6 .075 1957 37.1 .361 94.8 .093 104 .055 99.1 .075 1958 33.4 .349 95.7 .096 108 .053 101 .075 1959 30.6 .338 95.3 .098 111 .050 103 .075 1960 30.2 .375 96.0 .100 115 .050 105 .075 1961 28.6 .376 96.0 .103 119 .050 107 .075 1962 28.0 .400 96.4 .105 124 .048 109 .075 1963 27.8 .366 97.3 .107 129 .046 112 .075 1964 30.3 .399 99.5 .110 135 .046 116 .075 1965 33.7 .397 104 .112 145 .045 123 .075 1966 38.7 .453 111 .114 156 .044 132 .075 1967 38.8 .398 116 .117 167 .040 140 .075 1968 40.6 .476 120 .119 178 .036 147 .075 1969 39.6 .592 124 .121 190 .034 154 .075 1970 33.1 .521 126 .124 200. .031 159 .075 1971 30.8 125 . • 209 162 .075 * C a p i t a l stocks are in b i l l i o n s o f 1958 d o l l a r s . CXI
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Essays in the economics of uncertainty Epstein, Larry Gedaleah 1977
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Title | Essays in the economics of uncertainty |
Creator |
Epstein, Larry Gedaleah |
Date Issued | 1977 |
Description | There have been several recent advances in the theory of choice under uncertainty that have extended the restrictive mean-variance framework. Working within the context of a model of expected utility maximization, Rothschild and Stiglitz (1970) and Diamond and Stiglitz (1974) present intuitively appealing and theoretically sound definitions of "greater risk or uncertainty". Moreover, they show that their definitions are useful as well as consistent, in that they may be used to derive comparative statics results in which economists are interested. In the first part of the thesis we argue that the above analyses and most related ones are restricted to models where both the decision variable and the exogenous random variable that defines the stochastic environment, are scalars. Then we extend many of the definitions and results to the context of a general multivariate decision problem. In particular, a generalized notion of risk independence is shown to be relevant to behaviour under uncertainty. This general analysis is then applied to two specific decision problems: first, the standard two-period consumer choice problem where current consumption must be decided upon subject to uncertainty about future income and prices; and second, the corresponding problem in the theory of the firm, where a competitive firm must make some production decisions subject to uncertainty about the prices that will prevail for some products and factors of production. We extend earlier studies of these problems by considering disaggregated models, by adopting theoretically consistent definitions of increased uncertainty and by investigating the role of production flexibility in determining firm behaviour under uncertainty. In both the consumer and producer models the crucial properties of preferences and technology are pointed out and flexible functional forms are hypothesized that are amenable to empirical estimation. The theory of duality plays an important part throughout the formulation and analysis of both models. Finally the basic theory of producer behaviour analysed above is applied to aggregate U.S. manufacturing data for the 1947-71 period. We assume that the capital stock decision must be made one period before the capital comes into operation, subject to expectations about uncertain future prices, while all other factors and outputs may be adjusted fully to current prices. An added important ingredient of the model is the distinction between the capital stock and utilization (depreciation) decisions, the latter being made in each period after that period's prices are known. The consistency of the model with the data is investigated and the empirical significance of our formulation of the capital utilization decision is tested. |
Subject |
Economics - Mathematical models |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-21 |
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. |
IsShownAt | 10.14288/1.0094198 |
URI | http://hdl.handle.net/2429/20669 |
Degree |
Doctor of Philosophy - PhD |
Program |
Economics |
Affiliation |
Arts, Faculty of Vancouver School of Economics |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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